file_path
stringlengths 11
79
| full_name
stringlengths 2
100
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list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.not_mem_Iio
|
[] |
[
1056,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1055,
1
] |
Mathlib/RingTheory/Multiplicity.lean
|
multiplicity.mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nhp : Prime p\nh : Finite p a ∧ Finite p b\n⊢ multiplicity p (a * b) = multiplicity p a + multiplicity p b",
"tactic": "rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ←\n PartENat.natCast_get (finite_iff_dom.1 h.2), ←\n PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add,\n PartENat.natCast_inj, multiplicity.mul' hp]"
},
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nhp : Prime p\nh : ¬(Finite p a ∧ Finite p b)\n⊢ ⊤ = multiplicity p a + multiplicity p b",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nhp : Prime p\nh : ¬(Finite p a ∧ Finite p b)\n⊢ multiplicity p (a * b) = multiplicity p a + multiplicity p b",
"tactic": "rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nhp : Prime p\nh : ¬(Finite p a ∧ Finite p b)\n⊢ ⊤ = multiplicity p a + multiplicity p b",
"tactic": "cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h]"
}
] |
[
585,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
576,
11
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.IsRoot.eq_zero
|
[] |
[
494,
4
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
493,
1
] |
Mathlib/Algebra/Group/WithOne/Basic.lean
|
MulEquiv.withOneCongr_refl
|
[] |
[
152,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/Topology/MetricSpace/Contracting.lean
|
ContractingWith.dist_le_of_fixedPoint
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : MetricSpace α\nK : ℝ≥0\nf : α → α\nhf : ContractingWith K f\nx y : α\nhy : IsFixedPt f y\n⊢ dist x y ≤ dist x (f x) / (1 - ↑K)",
"tactic": "simpa only [hy.eq, dist_self, add_zero] using hf.dist_inequality x y"
}
] |
[
285,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
1
] |
Mathlib/Data/List/Basic.lean
|
List.surjective_tail
|
[] |
[
872,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
870,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.mul_re
|
[] |
[
128,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Algebra/Hom/Centroid.lean
|
CentroidHom.comp_assoc
|
[] |
[
218,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Analysis/Complex/UnitDisc/Basic.lean
|
Complex.UnitDisc.re_neg
|
[] |
[
201,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/Algebra/Order/Hom/Ring.lean
|
OrderRingHom.toOrderMonoidWithZeroHom_eq_coe
|
[] |
[
216,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean
|
MeasureTheory.AeMeasurable.ae_eq_of_forall_set_lintegral_eq
|
[
{
"state_after": "α : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ (∫ (x : α) in s, ENNReal.toReal (f x) ∂μ) = ∫ (x : α) in s, ENNReal.toReal (g x) ∂μ",
"state_before": "α : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\n⊢ f =ᵐ[μ] g",
"tactic": "refine'\n ENNReal.eventuallyEq_of_toReal_eventuallyEq (ae_lt_top' hf hfi).ne_of_lt\n (ae_lt_top' hg hgi).ne_of_lt\n (Integrable.ae_eq_of_forall_set_integral_eq _ _\n (integrable_toReal_of_lintegral_ne_top hf hfi)\n (integrable_toReal_of_lintegral_ne_top hg hgi) fun s hs hs' => _)"
},
{
"state_after": "α : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ ENNReal.toReal (∫⁻ (a : α) in s, ENNReal.ofReal (ENNReal.toReal (f a)) ∂μ) =\n ENNReal.toReal (∫⁻ (a : α) in s, ENNReal.ofReal (ENNReal.toReal (g a)) ∂μ)\n\ncase hf\nα : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ 0 ≤ᵐ[Measure.restrict μ s] fun x => ENNReal.toReal (g x)\n\ncase hfm\nα : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ AEStronglyMeasurable (fun x => ENNReal.toReal (g x)) (Measure.restrict μ s)\n\ncase hf\nα : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ 0 ≤ᵐ[Measure.restrict μ s] fun x => ENNReal.toReal (f x)\n\ncase hfm\nα : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ AEStronglyMeasurable (fun x => ENNReal.toReal (f x)) (Measure.restrict μ s)",
"state_before": "α : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ (∫ (x : α) in s, ENNReal.toReal (f x) ∂μ) = ∫ (x : α) in s, ENNReal.toReal (g x) ∂μ",
"tactic": "rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae]"
},
{
"state_after": "no goals",
"state_before": "case hf\nα : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ 0 ≤ᵐ[Measure.restrict μ s] fun x => ENNReal.toReal (g x)\n\ncase hfm\nα : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ AEStronglyMeasurable (fun x => ENNReal.toReal (g x)) (Measure.restrict μ s)\n\ncase hf\nα : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ 0 ≤ᵐ[Measure.restrict μ s] fun x => ENNReal.toReal (f x)\n\ncase hfm\nα : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ AEStronglyMeasurable (fun x => ENNReal.toReal (f x)) (Measure.restrict μ s)",
"tactic": "exacts [ae_of_all _ fun x => ENNReal.toReal_nonneg,\n hg.ennreal_toReal.restrict.aestronglyMeasurable, ae_of_all _ fun x => ENNReal.toReal_nonneg,\n hf.ennreal_toReal.restrict.aestronglyMeasurable]"
},
{
"state_after": "case e_a\nα : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ (∫⁻ (a : α) in s, ENNReal.ofReal (ENNReal.toReal (f a)) ∂μ) =\n ∫⁻ (a : α) in s, ENNReal.ofReal (ENNReal.toReal (g a)) ∂μ",
"state_before": "α : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ ENNReal.toReal (∫⁻ (a : α) in s, ENNReal.ofReal (ENNReal.toReal (f a)) ∂μ) =\n ENNReal.toReal (∫⁻ (a : α) in s, ENNReal.ofReal (ENNReal.toReal (g a)) ∂μ)",
"tactic": "congr 1"
},
{
"state_after": "case e_a\nα : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ (∫⁻ (a : α) in s, f a ∂μ) = ∫⁻ (a : α) in s, g a ∂μ\n\nα : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, g x < ⊤\n\nα : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, f x < ⊤",
"state_before": "case e_a\nα : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ (∫⁻ (a : α) in s, ENNReal.ofReal (ENNReal.toReal (f a)) ∂μ) =\n ∫⁻ (a : α) in s, ENNReal.ofReal (ENNReal.toReal (g a)) ∂μ",
"tactic": "rw [lintegral_congr_ae (ofReal_toReal_ae_eq _), lintegral_congr_ae (ofReal_toReal_ae_eq _)]"
},
{
"state_after": "no goals",
"state_before": "case e_a\nα : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ (∫⁻ (a : α) in s, f a ∂μ) = ∫⁻ (a : α) in s, g a ∂μ",
"tactic": "exact hfg hs hs'"
},
{
"state_after": "α : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ (∫⁻ (x : α) in s, g x ∂μ) ≤ ∫⁻ (x : α), g x ∂μ",
"state_before": "α : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, g x < ⊤",
"tactic": "refine' ae_lt_top' hg.restrict (ne_of_lt (lt_of_le_of_lt _ hgi.lt_top))"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ (∫⁻ (x : α) in s, g x ∂μ) ≤ ∫⁻ (x : α), g x ∂μ",
"tactic": "exact @set_lintegral_univ α _ μ g ▸ lintegral_mono_set (Set.subset_univ _)"
},
{
"state_after": "α : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ",
"state_before": "α : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, f x < ⊤",
"tactic": "refine' ae_lt_top' hf.restrict (ne_of_lt (lt_of_le_of_lt _ hfi.lt_top))"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.137396\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhgi : (∫⁻ (x : α), g x ∂μ) ≠ ⊤\nhfg : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) = ∫⁻ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\nhs' : ↑↑μ s < ⊤\n⊢ (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ",
"tactic": "exact @set_lintegral_univ α _ μ f ▸ lintegral_mono_set (Set.subset_univ _)"
}
] |
[
572,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
551,
1
] |
Mathlib/Data/Finsupp/Interval.lean
|
Finsupp.mem_rangeIcc_apply_iff
|
[] |
[
86,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.map_leftInverse
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf✝ f : R →+* S₁\ng : S₁ →+* R\nhf : LeftInverse ↑f ↑g\nX : MvPolynomial σ S₁\n⊢ ↑(map f) (↑(map g) X) = X",
"tactic": "rw [map_map, (RingHom.ext hf : f.comp g = RingHom.id _), map_id]"
}
] |
[
1310,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1308,
1
] |
Mathlib/Analysis/NormedSpace/Star/Multiplier.lean
|
DoubleCentralizer.norm_def'
|
[] |
[
564,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
563,
1
] |
Mathlib/Order/SuccPred/Basic.lean
|
Order.le_pred_iff_eq_bot
|
[] |
[
875,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
874,
1
] |
Mathlib/Data/Set/List.lean
|
Set.range_list_get?
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.1563\nl : List α\n⊢ range (get? l) = insert none (range (some ∘ fun k => nthLe l ↑k (_ : ↑k < length l)))",
"state_before": "α : Type u_1\nβ : Type ?u.1563\nl : List α\n⊢ range (get? l) = insert none (some '' {x | x ∈ l})",
"tactic": "rw [← range_list_nthLe, ← range_comp]"
},
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type ?u.1563\nl : List α\nn : ℕ\n⊢ get? l n ∈ insert none (range (some ∘ fun k => nthLe l ↑k (_ : ↑k < length l)))\n\ncase refine'_2\nα : Type u_1\nβ : Type ?u.1563\nl : List α\n⊢ none ∈ range (get? l)\n\ncase refine'_3\nα : Type u_1\nβ : Type ?u.1563\nl : List α\n⊢ range (some ∘ fun k => nthLe l ↑k (_ : ↑k < length l)) ⊆ range (get? l)",
"state_before": "α : Type u_1\nβ : Type ?u.1563\nl : List α\n⊢ range (get? l) = insert none (range (some ∘ fun k => nthLe l ↑k (_ : ↑k < length l)))",
"tactic": "refine' (range_subset_iff.2 fun n => _).antisymm (insert_subset.2 ⟨_, _⟩)"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.1563\nl : List α\nn : ℕ\n⊢ get? l n ∈ insert none (range (some ∘ fun k => nthLe l ↑k (_ : ↑k < length l)))\n\ncase refine'_2\nα : Type u_1\nβ : Type ?u.1563\nl : List α\n⊢ none ∈ range (get? l)\n\ncase refine'_3\nα : Type u_1\nβ : Type ?u.1563\nl : List α\n⊢ range (some ∘ fun k => nthLe l ↑k (_ : ↑k < length l)) ⊆ range (get? l)",
"tactic": "exacts [(le_or_lt l.length n).imp get?_eq_none.2 (fun hlt => ⟨⟨_, hlt⟩, (get?_eq_get hlt).symm⟩),\n ⟨_, get?_eq_none.2 le_rfl⟩, range_subset_iff.2 <| fun k => ⟨_, get?_eq_get _⟩]"
}
] |
[
53,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/Topology/Bases.lean
|
TopologicalSpace.exists_dense_seq
|
[
{
"state_after": "case intro.intro\nα : Type u\nt : TopologicalSpace α\ninst✝¹ : SeparableSpace α\ninst✝ : Nonempty α\ns : Set α\nhs : Set.Countable s\ns_dense : Dense s\n⊢ ∃ u, DenseRange u",
"state_before": "α : Type u\nt : TopologicalSpace α\ninst✝¹ : SeparableSpace α\ninst✝ : Nonempty α\n⊢ ∃ u, DenseRange u",
"tactic": "obtain ⟨s : Set α, hs, s_dense⟩ := exists_countable_dense α"
},
{
"state_after": "case intro.intro.intro\nα : Type u\nt : TopologicalSpace α\ninst✝¹ : SeparableSpace α\ninst✝ : Nonempty α\ns : Set α\nhs : Set.Countable s\ns_dense : Dense s\nu : ℕ → α\nhu : s ⊆ range u\n⊢ ∃ u, DenseRange u",
"state_before": "case intro.intro\nα : Type u\nt : TopologicalSpace α\ninst✝¹ : SeparableSpace α\ninst✝ : Nonempty α\ns : Set α\nhs : Set.Countable s\ns_dense : Dense s\n⊢ ∃ u, DenseRange u",
"tactic": "cases' Set.countable_iff_exists_subset_range.mp hs with u hu"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u\nt : TopologicalSpace α\ninst✝¹ : SeparableSpace α\ninst✝ : Nonempty α\ns : Set α\nhs : Set.Countable s\ns_dense : Dense s\nu : ℕ → α\nhu : s ⊆ range u\n⊢ ∃ u, DenseRange u",
"tactic": "exact ⟨u, s_dense.mono hu⟩"
}
] |
[
325,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
322,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean
|
LinearPMap.smul_graph
|
[
{
"state_after": "case h\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.518983\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type u_4\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny : M\nf : E →ₗ.[R] F\nz : M\nx : E × F\n⊢ x ∈ graph (z • f) ↔ x ∈ Submodule.map (LinearMap.prodMap LinearMap.id (z • LinearMap.id)) (graph f)",
"state_before": "R : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.518983\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type u_4\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny : M\nf : E →ₗ.[R] F\nz : M\n⊢ graph (z • f) = Submodule.map (LinearMap.prodMap LinearMap.id (z • LinearMap.id)) (graph f)",
"tactic": "ext x"
},
{
"state_after": "case h.mk\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.518983\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type u_4\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny : M\nf : E →ₗ.[R] F\nz : M\nx_fst : E\nx_snd : F\n⊢ (x_fst, x_snd) ∈ graph (z • f) ↔\n (x_fst, x_snd) ∈ Submodule.map (LinearMap.prodMap LinearMap.id (z • LinearMap.id)) (graph f)",
"state_before": "case h\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.518983\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type u_4\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny : M\nf : E →ₗ.[R] F\nz : M\nx : E × F\n⊢ x ∈ graph (z • f) ↔ x ∈ Submodule.map (LinearMap.prodMap LinearMap.id (z • LinearMap.id)) (graph f)",
"tactic": "cases' x with x_fst x_snd"
},
{
"state_after": "case h.mk.mp\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.518983\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type u_4\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny : M\nf : E →ₗ.[R] F\nz : M\nx_fst : E\nx_snd : F\nh : (x_fst, x_snd) ∈ graph (z • f)\n⊢ (x_fst, x_snd) ∈ Submodule.map (LinearMap.prodMap LinearMap.id (z • LinearMap.id)) (graph f)\n\ncase h.mk.mpr\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.518983\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type u_4\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny : M\nf : E →ₗ.[R] F\nz : M\nx_fst : E\nx_snd : F\nh : (x_fst, x_snd) ∈ Submodule.map (LinearMap.prodMap LinearMap.id (z • LinearMap.id)) (graph f)\n⊢ (x_fst, x_snd) ∈ graph (z • f)",
"state_before": "case h.mk\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.518983\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type u_4\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny : M\nf : E →ₗ.[R] F\nz : M\nx_fst : E\nx_snd : F\n⊢ (x_fst, x_snd) ∈ graph (z • f) ↔\n (x_fst, x_snd) ∈ Submodule.map (LinearMap.prodMap LinearMap.id (z • LinearMap.id)) (graph f)",
"tactic": "constructor <;> intro h"
},
{
"state_after": "case h.mk.mpr\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.518983\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type u_4\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny : M\nf : E →ₗ.[R] F\nz : M\nx_fst : E\nx_snd : F\nh : ∃ y, y ∈ graph f ∧ ↑(LinearMap.prodMap LinearMap.id (z • LinearMap.id)) y = (x_fst, x_snd)\n⊢ (x_fst, x_snd) ∈ graph (z • f)",
"state_before": "case h.mk.mpr\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.518983\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type u_4\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny : M\nf : E →ₗ.[R] F\nz : M\nx_fst : E\nx_snd : F\nh : (x_fst, x_snd) ∈ Submodule.map (LinearMap.prodMap LinearMap.id (z • LinearMap.id)) (graph f)\n⊢ (x_fst, x_snd) ∈ graph (z • f)",
"tactic": "rw [Submodule.mem_map] at h"
},
{
"state_after": "case h.mk.mpr.intro.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.518983\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type u_4\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny : M\nf : E →ₗ.[R] F\nz : M\nx_fst : E\nx_snd : F\nx' : E × F\nhx' : x' ∈ graph f\nh : ↑(LinearMap.prodMap LinearMap.id (z • LinearMap.id)) x' = (x_fst, x_snd)\n⊢ (x_fst, x_snd) ∈ graph (z • f)",
"state_before": "case h.mk.mpr\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.518983\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type u_4\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny : M\nf : E →ₗ.[R] F\nz : M\nx_fst : E\nx_snd : F\nh : ∃ y, y ∈ graph f ∧ ↑(LinearMap.prodMap LinearMap.id (z • LinearMap.id)) y = (x_fst, x_snd)\n⊢ (x_fst, x_snd) ∈ graph (z • f)",
"tactic": "rcases h with ⟨x', hx', h⟩"
},
{
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{
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},
{
"state_after": "case h.mk.mp.intro.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.518983\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type u_4\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\nz : M\nx_fst : E\nx_snd : F\ny : { x // x ∈ (z • f).domain }\nhy : ↑y = (x_fst, x_snd).fst\nh : z • ↑f y = (x_fst, x_snd).snd\n⊢ ↑y = x_fst ∧ x_fst = x_fst ∧ z • ↑f y = x_snd",
"state_before": "case h.mk.mp.intro.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.518983\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type u_4\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\nz : M\nx_fst : E\nx_snd : F\ny : { x // x ∈ (z • f).domain }\nhy : ↑y = (x_fst, x_snd).fst\nh : z • ↑f y = (x_fst, x_snd).snd\n⊢ ∃ a a_1, ↑a_1 = a ∧ a = x_fst ∧ z • ↑f a_1 = x_snd",
"tactic": "use x_fst, y"
},
{
"state_after": "no goals",
"state_before": "case h.mk.mp.intro.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.518983\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type u_4\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\nz : M\nx_fst : E\nx_snd : F\ny : { x // x ∈ (z • f).domain }\nhy : ↑y = (x_fst, x_snd).fst\nh : z • ↑f y = (x_fst, x_snd).snd\n⊢ ↑y = x_fst ∧ x_fst = x_fst ∧ z • ↑f y = x_snd",
"tactic": "simp [hy, h]"
}
] |
[
779,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
757,
1
] |
Mathlib/LinearAlgebra/BilinearForm.lean
|
BilinForm.compLeft_apply
|
[] |
[
598,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
597,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.add_mul_limit_aux
|
[
{
"state_after": "a b c : Ordinal\nba : b + a = a\nl : IsLimit c\nIH : ∀ (c' : Ordinal), c' < c → (a + b) * succ c' = a * succ c' + b\nc' : Ordinal\nh : c' < c\n⊢ (a + b) * succ c' ≤ a * c",
"state_before": "a b c : Ordinal\nba : b + a = a\nl : IsLimit c\nIH : ∀ (c' : Ordinal), c' < c → (a + b) * succ c' = a * succ c' + b\nc' : Ordinal\nh : c' < c\n⊢ (a + b) * c' ≤ a * c",
"tactic": "apply (mul_le_mul_left' (le_succ c') _).trans"
},
{
"state_after": "a b c : Ordinal\nba : b + a = a\nl : IsLimit c\nIH : ∀ (c' : Ordinal), c' < c → (a + b) * succ c' = a * succ c' + b\nc' : Ordinal\nh : c' < c\n⊢ a * succ c' + b ≤ a * c",
"state_before": "a b c : Ordinal\nba : b + a = a\nl : IsLimit c\nIH : ∀ (c' : Ordinal), c' < c → (a + b) * succ c' = a * succ c' + b\nc' : Ordinal\nh : c' < c\n⊢ (a + b) * succ c' ≤ a * c",
"tactic": "rw [IH _ h]"
},
{
"state_after": "a b c : Ordinal\nba : b + a = a\nl : IsLimit c\nIH : ∀ (c' : Ordinal), c' < c → (a + b) * succ c' = a * succ c' + b\nc' : Ordinal\nh : c' < c\n⊢ a * succ c' + ?m.427842 ≤ a * c\n\na b c : Ordinal\nba : b + a = a\nl : IsLimit c\nIH : ∀ (c' : Ordinal), c' < c → (a + b) * succ c' = a * succ c' + b\nc' : Ordinal\nh : c' < c\n⊢ Ordinal\n\na b c : Ordinal\nba : b + a = a\nl : IsLimit c\nIH : ∀ (c' : Ordinal), c' < c → (a + b) * succ c' = a * succ c' + b\nc' : Ordinal\nh : c' < c\n⊢ b ≤ ?m.427842",
"state_before": "a b c : Ordinal\nba : b + a = a\nl : IsLimit c\nIH : ∀ (c' : Ordinal), c' < c → (a + b) * succ c' = a * succ c' + b\nc' : Ordinal\nh : c' < c\n⊢ a * succ c' + b ≤ a * c",
"tactic": "apply (add_le_add_left _ _).trans"
},
{
"state_after": "a b c : Ordinal\nba : b + a = a\nl : IsLimit c\nIH : ∀ (c' : Ordinal), c' < c → (a + b) * succ c' = a * succ c' + b\nc' : Ordinal\nh : c' < c\n⊢ a * succ (succ c') ≤ a * c",
"state_before": "a b c : Ordinal\nba : b + a = a\nl : IsLimit c\nIH : ∀ (c' : Ordinal), c' < c → (a + b) * succ c' = a * succ c' + b\nc' : Ordinal\nh : c' < c\n⊢ a * succ c' + ?m.427842 ≤ a * c",
"tactic": "rw [← mul_succ]"
},
{
"state_after": "no goals",
"state_before": "a b c : Ordinal\nba : b + a = a\nl : IsLimit c\nIH : ∀ (c' : Ordinal), c' < c → (a + b) * succ c' = a * succ c' + b\nc' : Ordinal\nh : c' < c\n⊢ a * succ (succ c') ≤ a * c",
"tactic": "exact mul_le_mul_left' (succ_le_of_lt <| l.2 _ h) _"
},
{
"state_after": "a b c : Ordinal\nba : b + a = a\nl : IsLimit c\nIH : ∀ (c' : Ordinal), c' < c → (a + b) * succ c' = a * succ c' + b\nc' : Ordinal\nh : c' < c\n⊢ b ≤ b + a",
"state_before": "a b c : Ordinal\nba : b + a = a\nl : IsLimit c\nIH : ∀ (c' : Ordinal), c' < c → (a + b) * succ c' = a * succ c' + b\nc' : Ordinal\nh : c' < c\n⊢ b ≤ a",
"tactic": "rw [← ba]"
},
{
"state_after": "no goals",
"state_before": "a b c : Ordinal\nba : b + a = a\nl : IsLimit c\nIH : ∀ (c' : Ordinal), c' < c → (a + b) * succ c' = a * succ c' + b\nc' : Ordinal\nh : c' < c\n⊢ b ≤ b + a",
"tactic": "exact le_add_right _ _"
}
] |
[
2477,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2466,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.tr_reaches
|
[
{
"state_after": "case inl\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₂ : σ₂\nb₁ : σ₁\naa : tr b₁ a₂\nab : Reaches f₁ b₁ b₁\n⊢ ∃ b₂, tr b₁ b₂ ∧ Reaches f₂ a₂ b₂\n\ncase inr\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₁ : σ₁\nab✝ : Reaches f₁ a₁ b₁\nab : TransGen (fun a b => b ∈ f₁ a) a₁ b₁\n⊢ ∃ b₂, tr b₁ b₂ ∧ Reaches f₂ a₂ b₂",
"state_before": "σ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₁ : σ₁\nab : Reaches f₁ a₁ b₁\n⊢ ∃ b₂, tr b₁ b₂ ∧ Reaches f₂ a₂ b₂",
"tactic": "rcases reflTransGen_iff_eq_or_transGen.1 ab with (rfl | ab)"
},
{
"state_after": "no goals",
"state_before": "case inl\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₂ : σ₂\nb₁ : σ₁\naa : tr b₁ a₂\nab : Reaches f₁ b₁ b₁\n⊢ ∃ b₂, tr b₁ b₂ ∧ Reaches f₂ a₂ b₂",
"tactic": "exact ⟨_, aa, ReflTransGen.refl⟩"
},
{
"state_after": "case inr\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₁ : σ₁\nab✝ : Reaches f₁ a₁ b₁\nab : TransGen (fun a b => b ∈ f₁ a) a₁ b₁\nb₂ : σ₂\nbb : tr b₁ b₂\nh : Reaches₁ f₂ a₂ b₂\n⊢ ∃ b₂, tr b₁ b₂ ∧ Reaches f₂ a₂ b₂",
"state_before": "case inr\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₁ : σ₁\nab✝ : Reaches f₁ a₁ b₁\nab : TransGen (fun a b => b ∈ f₁ a) a₁ b₁\n⊢ ∃ b₂, tr b₁ b₂ ∧ Reaches f₂ a₂ b₂",
"tactic": "have ⟨b₂, bb, h⟩ := tr_reaches₁ H aa ab"
},
{
"state_after": "no goals",
"state_before": "case inr\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₁ : σ₁\nab✝ : Reaches f₁ a₁ b₁\nab : TransGen (fun a b => b ∈ f₁ a) a₁ b₁\nb₂ : σ₂\nbb : tr b₁ b₂\nh : Reaches₁ f₂ a₂ b₂\n⊢ ∃ b₂, tr b₁ b₂ ∧ Reaches f₂ a₂ b₂",
"tactic": "exact ⟨b₂, bb, h.to_reflTransGen⟩"
}
] |
[
903,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
898,
1
] |
Mathlib/RingTheory/Polynomial/Content.lean
|
Polynomial.natDegree_primPart
|
[
{
"state_after": "case pos\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nh : ↑C (content p) = 0\n⊢ natDegree (primPart p) = natDegree p\n\ncase neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nh : ¬↑C (content p) = 0\n⊢ natDegree (primPart p) = natDegree p",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\n⊢ natDegree (primPart p) = natDegree p",
"tactic": "by_cases h : C p.content = 0"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nh : ¬↑C (content p) = 0\n⊢ natDegree (primPart p) = natDegree p",
"tactic": "conv_rhs =>\n rw [p.eq_C_content_mul_primPart, natDegree_mul h p.primPart_ne_zero, natDegree_C, zero_add]"
},
{
"state_after": "case pos\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nh : p = 0\n⊢ natDegree (primPart p) = natDegree p",
"state_before": "case pos\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nh : ↑C (content p) = 0\n⊢ natDegree (primPart p) = natDegree p",
"tactic": "rw [C_eq_zero, content_eq_zero_iff] at h"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nh : p = 0\n⊢ natDegree (primPart p) = natDegree p",
"tactic": "simp [h]"
}
] |
[
288,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/CategoryTheory/Adjunction/Mates.lean
|
CategoryTheory.transferNatTransSelf_symm_of_iso
|
[
{
"state_after": "C : Type u₁\nD : Type u₂\ninst✝² : Category C\ninst✝¹ : Category D\nL₁ L₂ L₃ : C ⥤ D\nR₁ R₂ R₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ : L₃ ⊣ R₃\nf : R₁ ⟶ R₂\ninst✝ : IsIso (↑(transferNatTransSelf adj₁ adj₂).symm f)\n⊢ IsIso (↑(transferNatTransSelf adj₁ adj₂) (↑(transferNatTransSelf adj₁ adj₂).symm f))",
"state_before": "C : Type u₁\nD : Type u₂\ninst✝² : Category C\ninst✝¹ : Category D\nL₁ L₂ L₃ : C ⥤ D\nR₁ R₂ R₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ : L₃ ⊣ R₃\nf : R₁ ⟶ R₂\ninst✝ : IsIso (↑(transferNatTransSelf adj₁ adj₂).symm f)\n⊢ IsIso f",
"tactic": "suffices IsIso ((transferNatTransSelf adj₁ adj₂) ((transferNatTransSelf adj₁ adj₂).symm f))\n by simpa using this"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\nD : Type u₂\ninst✝² : Category C\ninst✝¹ : Category D\nL₁ L₂ L₃ : C ⥤ D\nR₁ R₂ R₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ : L₃ ⊣ R₃\nf : R₁ ⟶ R₂\ninst✝ : IsIso (↑(transferNatTransSelf adj₁ adj₂).symm f)\n⊢ IsIso (↑(transferNatTransSelf adj₁ adj₂) (↑(transferNatTransSelf adj₁ adj₂).symm f))",
"tactic": "infer_instance"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\nD : Type u₂\ninst✝² : Category C\ninst✝¹ : Category D\nL₁ L₂ L₃ : C ⥤ D\nR₁ R₂ R₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ : L₃ ⊣ R₃\nf : R₁ ⟶ R₂\ninst✝ : IsIso (↑(transferNatTransSelf adj₁ adj₂).symm f)\nthis : IsIso (↑(transferNatTransSelf adj₁ adj₂) (↑(transferNatTransSelf adj₁ adj₂).symm f))\n⊢ IsIso f",
"tactic": "simpa using this"
}
] |
[
275,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
271,
1
] |
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
|
intervalIntegral.integral_comp_smul_deriv'
|
[] |
[
1421,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1416,
1
] |
Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean
|
GeneralizedContinuedFraction.squashSeq_nth_of_not_terminated
|
[
{
"state_after": "no goals",
"state_before": "K : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_n gp_succ_n : Pair K\ns_nth_eq : Stream'.Seq.get? s n = some gp_n\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\n⊢ Stream'.Seq.get? (squashSeq s n) n = some { a := gp_n.a, b := gp_n.b + gp_succ_n.a / gp_succ_n.b }",
"tactic": "simp [*, squashSeq]"
}
] |
[
120,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/Order/Basic.lean
|
Pi.compl_def
|
[] |
[
772,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
770,
1
] |
Mathlib/Topology/Instances/Rat.lean
|
Rat.totallyBounded_Icc
|
[
{
"state_after": "no goals",
"state_before": "a b : ℚ\n⊢ TotallyBounded (Icc a b)",
"tactic": "simpa only [preimage_cast_Icc] using\n totallyBounded_preimage Rat.uniformEmbedding_coe_real (totallyBounded_Icc (a : ℝ) b)"
}
] |
[
124,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
8
] |
Mathlib/Topology/SubsetProperties.lean
|
IsClopen.diff
|
[] |
[
1577,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1576,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.setOf_app_iff
|
[] |
[
285,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.factorization_prod
|
[
{
"state_after": "case h\nα : Type u_1\nS : Finset α\ng : α → ℕ\nhS : ∀ (x : α), x ∈ S → g x ≠ 0\np : ℕ\n⊢ ↑(factorization (Finset.prod S g)) p = ↑(∑ x in S, factorization (g x)) p",
"state_before": "α : Type u_1\nS : Finset α\ng : α → ℕ\nhS : ∀ (x : α), x ∈ S → g x ≠ 0\n⊢ factorization (Finset.prod S g) = ∑ x in S, factorization (g x)",
"tactic": "ext p"
},
{
"state_after": "case h.refine'_1\nα : Type u_1\nS : Finset α\ng : α → ℕ\nhS : ∀ (x : α), x ∈ S → g x ≠ 0\np : ℕ\n⊢ ↑(factorization (Finset.prod ∅ g)) p = ↑(∑ x in ∅, factorization (g x)) p\n\ncase h.refine'_2\nα : Type u_1\nS : Finset α\ng : α → ℕ\nhS : ∀ (x : α), x ∈ S → g x ≠ 0\np : ℕ\n⊢ ∀ {a : α} {s : Finset α},\n a ∈ S →\n s ⊆ S →\n ¬a ∈ s →\n ↑(factorization (Finset.prod s g)) p = ↑(∑ x in s, factorization (g x)) p →\n ↑(factorization (Finset.prod (insert a s) g)) p = ↑(∑ x in insert a s, factorization (g x)) p",
"state_before": "case h\nα : Type u_1\nS : Finset α\ng : α → ℕ\nhS : ∀ (x : α), x ∈ S → g x ≠ 0\np : ℕ\n⊢ ↑(factorization (Finset.prod S g)) p = ↑(∑ x in S, factorization (g x)) p",
"tactic": "refine' Finset.induction_on' S ?_ ?_"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_1\nα : Type u_1\nS : Finset α\ng : α → ℕ\nhS : ∀ (x : α), x ∈ S → g x ≠ 0\np : ℕ\n⊢ ↑(factorization (Finset.prod ∅ g)) p = ↑(∑ x in ∅, factorization (g x)) p",
"tactic": "simp"
},
{
"state_after": "case h.refine'_2\nα : Type u_1\nS : Finset α\ng : α → ℕ\nhS : ∀ (x : α), x ∈ S → g x ≠ 0\np : ℕ\nx : α\nT : Finset α\nhxS : x ∈ S\nhTS : T ⊆ S\nhxT : ¬x ∈ T\nIH : ↑(factorization (Finset.prod T g)) p = ↑(∑ x in T, factorization (g x)) p\n⊢ ↑(factorization (Finset.prod (insert x T) g)) p = ↑(∑ x in insert x T, factorization (g x)) p",
"state_before": "case h.refine'_2\nα : Type u_1\nS : Finset α\ng : α → ℕ\nhS : ∀ (x : α), x ∈ S → g x ≠ 0\np : ℕ\n⊢ ∀ {a : α} {s : Finset α},\n a ∈ S →\n s ⊆ S →\n ¬a ∈ s →\n ↑(factorization (Finset.prod s g)) p = ↑(∑ x in s, factorization (g x)) p →\n ↑(factorization (Finset.prod (insert a s) g)) p = ↑(∑ x in insert a s, factorization (g x)) p",
"tactic": "intro x T hxS hTS hxT IH"
},
{
"state_after": "case h.refine'_2\nα : Type u_1\nS : Finset α\ng : α → ℕ\nhS : ∀ (x : α), x ∈ S → g x ≠ 0\np : ℕ\nx : α\nT : Finset α\nhxS : x ∈ S\nhTS : T ⊆ S\nhxT : ¬x ∈ T\nIH : ↑(factorization (Finset.prod T g)) p = ↑(∑ x in T, factorization (g x)) p\nhT : Finset.prod T g ≠ 0\n⊢ ↑(factorization (Finset.prod (insert x T) g)) p = ↑(∑ x in insert x T, factorization (g x)) p",
"state_before": "case h.refine'_2\nα : Type u_1\nS : Finset α\ng : α → ℕ\nhS : ∀ (x : α), x ∈ S → g x ≠ 0\np : ℕ\nx : α\nT : Finset α\nhxS : x ∈ S\nhTS : T ⊆ S\nhxT : ¬x ∈ T\nIH : ↑(factorization (Finset.prod T g)) p = ↑(∑ x in T, factorization (g x)) p\n⊢ ↑(factorization (Finset.prod (insert x T) g)) p = ↑(∑ x in insert x T, factorization (g x)) p",
"tactic": "have hT : T.prod g ≠ 0 := prod_ne_zero_iff.mpr fun x hx => hS x (hTS hx)"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_2\nα : Type u_1\nS : Finset α\ng : α → ℕ\nhS : ∀ (x : α), x ∈ S → g x ≠ 0\np : ℕ\nx : α\nT : Finset α\nhxS : x ∈ S\nhTS : T ⊆ S\nhxT : ¬x ∈ T\nIH : ↑(factorization (Finset.prod T g)) p = ↑(∑ x in T, factorization (g x)) p\nhT : Finset.prod T g ≠ 0\n⊢ ↑(factorization (Finset.prod (insert x T) g)) p = ↑(∑ x in insert x T, factorization (g x)) p",
"tactic": "simp [prod_insert hxT, sum_insert hxT, ← IH, factorization_mul (hS x hxS) hT]"
}
] |
[
255,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Equiv.Perm.prodExtendRight_apply_ne
|
[] |
[
836,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
834,
1
] |
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.aecover_Iio_of_Iic
|
[] |
[
196,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Data/List/Pairwise.lean
|
List.Pairwise.imp_mem
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.4602\nR S T : α → α → Prop\na : α\nl✝ l : List α\n⊢ ∀ {a b : α}, a ∈ l → b ∈ l → (R a b ↔ a ∈ l → b ∈ l → R a b)",
"tactic": "simp (config := { contextual := true }) only [forall_prop_of_true, iff_self_iff,\nforall₂_true_iff]"
}
] |
[
120,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Logic/Basic.lean
|
xor_iff_not_iff
|
[] |
[
479,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
479,
1
] |
Std/Classes/LawfulMonad.lean
|
SatisfiesM.seqLeft
|
[] |
[
160,
54
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
157,
11
] |
Std/Data/Rat/Lemmas.lean
|
Rat.normalize_mul_normalize
|
[
{
"state_after": "case mk'\nn₁ n₂ : Int\nd₁ d₂ : Nat\nz₁ : d₁ ≠ 0\nz₂ : d₂ ≠ 0\nnum✝ : Int\nden✝ : Nat\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs num✝) den✝\ne₁ : normalize n₁ d₁ = mk' num✝ den✝\n⊢ mk' num✝ den✝ * normalize n₂ d₂ = normalize (n₁ * n₂) (d₁ * d₂)",
"state_before": "n₁ n₂ : Int\nd₁ d₂ : Nat\nz₁ : d₁ ≠ 0\nz₂ : d₂ ≠ 0\n⊢ normalize n₁ d₁ * normalize n₂ d₂ = normalize (n₁ * n₂) (d₁ * d₂)",
"tactic": "cases e₁ : normalize n₁ d₁ z₁"
},
{
"state_after": "case mk'.intro.intro.intro\nn₂ : Int\nd₂ : Nat\nz₂ : d₂ ≠ 0\nnum✝ : Int\nden✝ : Nat\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs num✝) den✝\ng₁ : Nat\nzg₁ : g₁ ≠ 0\nz₁ : den✝ * g₁ ≠ 0\ne₁ : normalize (num✝ * ↑g₁) (den✝ * g₁) = mk' num✝ den✝\n⊢ mk' num✝ den✝ * normalize n₂ d₂ = normalize (num✝ * ↑g₁ * n₂) (den✝ * g₁ * d₂)",
"state_before": "case mk'\nn₁ n₂ : Int\nd₁ d₂ : Nat\nz₁ : d₁ ≠ 0\nz₂ : d₂ ≠ 0\nnum✝ : Int\nden✝ : Nat\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs num✝) den✝\ne₁ : normalize n₁ d₁ = mk' num✝ den✝\n⊢ mk' num✝ den✝ * normalize n₂ d₂ = normalize (n₁ * n₂) (d₁ * d₂)",
"tactic": "rcases normalize_num_den e₁ with ⟨g₁, zg₁, rfl, rfl⟩"
},
{
"state_after": "case mk'.intro.intro.intro.mk'\nn₂ : Int\nd₂ : Nat\nz₂ : d₂ ≠ 0\nnum✝¹ : Int\nden✝¹ : Nat\nden_nz✝¹ : den✝¹ ≠ 0\nreduced✝¹ : Nat.coprime (Int.natAbs num✝¹) den✝¹\ng₁ : Nat\nzg₁ : g₁ ≠ 0\nz₁ : den✝¹ * g₁ ≠ 0\ne₁ : normalize (num✝¹ * ↑g₁) (den✝¹ * g₁) = mk' num✝¹ den✝¹\nnum✝ : Int\nden✝ : Nat\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs num✝) den✝\ne₂ : normalize n₂ d₂ = mk' num✝ den✝\n⊢ mk' num✝¹ den✝¹ * mk' num✝ den✝ = normalize (num✝¹ * ↑g₁ * n₂) (den✝¹ * g₁ * d₂)",
"state_before": "case mk'.intro.intro.intro\nn₂ : Int\nd₂ : Nat\nz₂ : d₂ ≠ 0\nnum✝ : Int\nden✝ : Nat\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs num✝) den✝\ng₁ : Nat\nzg₁ : g₁ ≠ 0\nz₁ : den✝ * g₁ ≠ 0\ne₁ : normalize (num✝ * ↑g₁) (den✝ * g₁) = mk' num✝ den✝\n⊢ mk' num✝ den✝ * normalize n₂ d₂ = normalize (num✝ * ↑g₁ * n₂) (den✝ * g₁ * d₂)",
"tactic": "cases e₂ : normalize n₂ d₂ z₂"
},
{
"state_after": "case mk'.intro.intro.intro.mk'.intro.intro.intro\nnum✝¹ : Int\nden✝¹ : Nat\nden_nz✝¹ : den✝¹ ≠ 0\nreduced✝¹ : Nat.coprime (Int.natAbs num✝¹) den✝¹\ng₁ : Nat\nzg₁ : g₁ ≠ 0\nz₁ : den✝¹ * g₁ ≠ 0\ne₁ : normalize (num✝¹ * ↑g₁) (den✝¹ * g₁) = mk' num✝¹ den✝¹\nnum✝ : Int\nden✝ : Nat\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs num✝) den✝\ng₂ : Nat\nzg₂ : g₂ ≠ 0\nz₂ : den✝ * g₂ ≠ 0\ne₂ : normalize (num✝ * ↑g₂) (den✝ * g₂) = mk' num✝ den✝\n⊢ mk' num✝¹ den✝¹ * mk' num✝ den✝ = normalize (num✝¹ * ↑g₁ * (num✝ * ↑g₂)) (den✝¹ * g₁ * (den✝ * g₂))",
"state_before": "case mk'.intro.intro.intro.mk'\nn₂ : Int\nd₂ : Nat\nz₂ : d₂ ≠ 0\nnum✝¹ : Int\nden✝¹ : Nat\nden_nz✝¹ : den✝¹ ≠ 0\nreduced✝¹ : Nat.coprime (Int.natAbs num✝¹) den✝¹\ng₁ : Nat\nzg₁ : g₁ ≠ 0\nz₁ : den✝¹ * g₁ ≠ 0\ne₁ : normalize (num✝¹ * ↑g₁) (den✝¹ * g₁) = mk' num✝¹ den✝¹\nnum✝ : Int\nden✝ : Nat\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs num✝) den✝\ne₂ : normalize n₂ d₂ = mk' num✝ den✝\n⊢ mk' num✝¹ den✝¹ * mk' num✝ den✝ = normalize (num✝¹ * ↑g₁ * n₂) (den✝¹ * g₁ * d₂)",
"tactic": "rcases normalize_num_den e₂ with ⟨g₂, zg₂, rfl, rfl⟩"
},
{
"state_after": "case mk'.intro.intro.intro.mk'.intro.intro.intro\nnum✝¹ : Int\nden✝¹ : Nat\nden_nz✝¹ : den✝¹ ≠ 0\nreduced✝¹ : Nat.coprime (Int.natAbs num✝¹) den✝¹\ng₁ : Nat\nzg₁ : g₁ ≠ 0\nz₁ : den✝¹ * g₁ ≠ 0\ne₁ : normalize (num✝¹ * ↑g₁) (den✝¹ * g₁) = mk' num✝¹ den✝¹\nnum✝ : Int\nden✝ : Nat\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs num✝) den✝\ng₂ : Nat\nzg₂ : g₂ ≠ 0\nz₂ : den✝ * g₂ ≠ 0\ne₂ : normalize (num✝ * ↑g₂) (den✝ * g₂) = mk' num✝ den✝\n⊢ normalize (num✝¹ * num✝) (den✝¹ * den✝) = normalize (num✝¹ * ↑g₁ * (num✝ * ↑g₂)) (den✝¹ * g₁ * (den✝ * g₂))",
"state_before": "case mk'.intro.intro.intro.mk'.intro.intro.intro\nnum✝¹ : Int\nden✝¹ : Nat\nden_nz✝¹ : den✝¹ ≠ 0\nreduced✝¹ : Nat.coprime (Int.natAbs num✝¹) den✝¹\ng₁ : Nat\nzg₁ : g₁ ≠ 0\nz₁ : den✝¹ * g₁ ≠ 0\ne₁ : normalize (num✝¹ * ↑g₁) (den✝¹ * g₁) = mk' num✝¹ den✝¹\nnum✝ : Int\nden✝ : Nat\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs num✝) den✝\ng₂ : Nat\nzg₂ : g₂ ≠ 0\nz₂ : den✝ * g₂ ≠ 0\ne₂ : normalize (num✝ * ↑g₂) (den✝ * g₂) = mk' num✝ den✝\n⊢ mk' num✝¹ den✝¹ * mk' num✝ den✝ = normalize (num✝¹ * ↑g₁ * (num✝ * ↑g₂)) (den✝¹ * g₁ * (den✝ * g₂))",
"tactic": "simp only [mul_def]"
},
{
"state_after": "case mk'.intro.intro.intro.mk'.intro.intro.intro\nnum✝¹ : Int\nden✝¹ : Nat\nden_nz✝¹ : den✝¹ ≠ 0\nreduced✝¹ : Nat.coprime (Int.natAbs num✝¹) den✝¹\ng₁ : Nat\nzg₁ : g₁ ≠ 0\nz₁ : den✝¹ * g₁ ≠ 0\ne₁ : normalize (num✝¹ * ↑g₁) (den✝¹ * g₁) = mk' num✝¹ den✝¹\nnum✝ : Int\nden✝ : Nat\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs num✝) den✝\ng₂ : Nat\nzg₂ : g₂ ≠ 0\nz₂ : den✝ * g₂ ≠ 0\ne₂ : normalize (num✝ * ↑g₂) (den✝ * g₂) = mk' num✝ den✝\n⊢ normalize (num✝¹ * num✝ * ↑(g₁ * g₂)) (den✝¹ * den✝ * (g₁ * g₂)) =\n normalize (num✝¹ * ↑g₁ * (num✝ * ↑g₂)) (den✝¹ * g₁ * (den✝ * g₂))",
"state_before": "case mk'.intro.intro.intro.mk'.intro.intro.intro\nnum✝¹ : Int\nden✝¹ : Nat\nden_nz✝¹ : den✝¹ ≠ 0\nreduced✝¹ : Nat.coprime (Int.natAbs num✝¹) den✝¹\ng₁ : Nat\nzg₁ : g₁ ≠ 0\nz₁ : den✝¹ * g₁ ≠ 0\ne₁ : normalize (num✝¹ * ↑g₁) (den✝¹ * g₁) = mk' num✝¹ den✝¹\nnum✝ : Int\nden✝ : Nat\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs num✝) den✝\ng₂ : Nat\nzg₂ : g₂ ≠ 0\nz₂ : den✝ * g₂ ≠ 0\ne₂ : normalize (num✝ * ↑g₂) (den✝ * g₂) = mk' num✝ den✝\n⊢ normalize (num✝¹ * num✝) (den✝¹ * den✝) = normalize (num✝¹ * ↑g₁ * (num✝ * ↑g₂)) (den✝¹ * g₁ * (den✝ * g₂))",
"tactic": "rw [← normalize_mul_right _ (Nat.mul_ne_zero zg₁ zg₂)]"
},
{
"state_after": "case mk'.intro.intro.intro.mk'.intro.intro.intro.e_num\nnum✝¹ : Int\nden✝¹ : Nat\nden_nz✝¹ : den✝¹ ≠ 0\nreduced✝¹ : Nat.coprime (Int.natAbs num✝¹) den✝¹\ng₁ : Nat\nzg₁ : g₁ ≠ 0\nz₁ : den✝¹ * g₁ ≠ 0\ne₁ : normalize (num✝¹ * ↑g₁) (den✝¹ * g₁) = mk' num✝¹ den✝¹\nnum✝ : Int\nden✝ : Nat\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs num✝) den✝\ng₂ : Nat\nzg₂ : g₂ ≠ 0\nz₂ : den✝ * g₂ ≠ 0\ne₂ : normalize (num✝ * ↑g₂) (den✝ * g₂) = mk' num✝ den✝\n⊢ num✝¹ * num✝ * ↑(g₁ * g₂) = num✝¹ * ↑g₁ * (num✝ * ↑g₂)\n\ncase mk'.intro.intro.intro.mk'.intro.intro.intro.e_den\nnum✝¹ : Int\nden✝¹ : Nat\nden_nz✝¹ : den✝¹ ≠ 0\nreduced✝¹ : Nat.coprime (Int.natAbs num✝¹) den✝¹\ng₁ : Nat\nzg₁ : g₁ ≠ 0\nz₁ : den✝¹ * g₁ ≠ 0\ne₁ : normalize (num✝¹ * ↑g₁) (den✝¹ * g₁) = mk' num✝¹ den✝¹\nnum✝ : Int\nden✝ : Nat\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs num✝) den✝\ng₂ : Nat\nzg₂ : g₂ ≠ 0\nz₂ : den✝ * g₂ ≠ 0\ne₂ : normalize (num✝ * ↑g₂) (den✝ * g₂) = mk' num✝ den✝\n⊢ den✝¹ * den✝ * (g₁ * g₂) = den✝¹ * g₁ * (den✝ * g₂)",
"state_before": "case mk'.intro.intro.intro.mk'.intro.intro.intro\nnum✝¹ : Int\nden✝¹ : Nat\nden_nz✝¹ : den✝¹ ≠ 0\nreduced✝¹ : Nat.coprime (Int.natAbs num✝¹) den✝¹\ng₁ : Nat\nzg₁ : g₁ ≠ 0\nz₁ : den✝¹ * g₁ ≠ 0\ne₁ : normalize (num✝¹ * ↑g₁) (den✝¹ * g₁) = mk' num✝¹ den✝¹\nnum✝ : Int\nden✝ : Nat\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs num✝) den✝\ng₂ : Nat\nzg₂ : g₂ ≠ 0\nz₂ : den✝ * g₂ ≠ 0\ne₂ : normalize (num✝ * ↑g₂) (den✝ * g₂) = mk' num✝ den✝\n⊢ normalize (num✝¹ * num✝ * ↑(g₁ * g₂)) (den✝¹ * den✝ * (g₁ * g₂)) =\n normalize (num✝¹ * ↑g₁ * (num✝ * ↑g₂)) (den✝¹ * g₁ * (den✝ * g₂))",
"tactic": "congr 1"
},
{
"state_after": "no goals",
"state_before": "case mk'.intro.intro.intro.mk'.intro.intro.intro.e_num\nnum✝¹ : Int\nden✝¹ : Nat\nden_nz✝¹ : den✝¹ ≠ 0\nreduced✝¹ : Nat.coprime (Int.natAbs num✝¹) den✝¹\ng₁ : Nat\nzg₁ : g₁ ≠ 0\nz₁ : den✝¹ * g₁ ≠ 0\ne₁ : normalize (num✝¹ * ↑g₁) (den✝¹ * g₁) = mk' num✝¹ den✝¹\nnum✝ : Int\nden✝ : Nat\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs num✝) den✝\ng₂ : Nat\nzg₂ : g₂ ≠ 0\nz₂ : den✝ * g₂ ≠ 0\ne₂ : normalize (num✝ * ↑g₂) (den✝ * g₂) = mk' num✝ den✝\n⊢ num✝¹ * num✝ * ↑(g₁ * g₂) = num✝¹ * ↑g₁ * (num✝ * ↑g₂)",
"tactic": "simp [Int.ofNat_mul, Int.mul_assoc, Int.mul_left_comm]"
},
{
"state_after": "no goals",
"state_before": "case mk'.intro.intro.intro.mk'.intro.intro.intro.e_den\nnum✝¹ : Int\nden✝¹ : Nat\nden_nz✝¹ : den✝¹ ≠ 0\nreduced✝¹ : Nat.coprime (Int.natAbs num✝¹) den✝¹\ng₁ : Nat\nzg₁ : g₁ ≠ 0\nz₁ : den✝¹ * g₁ ≠ 0\ne₁ : normalize (num✝¹ * ↑g₁) (den✝¹ * g₁) = mk' num✝¹ den✝¹\nnum✝ : Int\nden✝ : Nat\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs num✝) den✝\ng₂ : Nat\nzg₂ : g₂ ≠ 0\nz₂ : den✝ * g₂ ≠ 0\ne₂ : normalize (num✝ * ↑g₂) (den✝ * g₂) = mk' num✝ den✝\n⊢ den✝¹ * den✝ * (g₁ * g₂) = den✝¹ * g₁ * (den✝ * g₂)",
"tactic": "simp [Nat.mul_left_comm, Nat.mul_comm]"
}
] |
[
272,
43
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
265,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
|
CategoryTheory.Limits.BinaryBicone.toCone_π_app_left
|
[] |
[
1003,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1002,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.integral_mono
|
[] |
[
1305,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1303,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.succ_lt_of_not_succ
|
[] |
[
208,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Control/EquivFunctor.lean
|
EquivFunctor.mapEquiv_symm_apply
|
[] |
[
68,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Topology/StoneCech.lean
|
continuous_stoneCechExtend
|
[] |
[
284,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean
|
mul_le_of_le_one_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\na b c d : α\ninst✝³ : MulOneClass α\ninst✝² : Zero α\ninst✝¹ : Preorder α\ninst✝ : MulPosMono α\nhb : 0 ≤ b\nh : a ≤ 1\n⊢ a * b ≤ b",
"tactic": "simpa only [one_mul] using mul_le_mul_of_nonneg_right h hb"
}
] |
[
670,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
669,
1
] |
Std/Data/RBMap/WF.lean
|
Std.RBNode.Ordered.setBlack
|
[
{
"state_after": "α : Type u_1\ncmp : α → α → Ordering\nt : RBNode α\n⊢ Ordered cmp\n (match t with\n | nil => nil\n | node c l v r => node black l v r) ↔\n Ordered cmp t",
"state_before": "α : Type u_1\ncmp : α → α → Ordering\nt : RBNode α\n⊢ Ordered cmp (setBlack t) ↔ Ordered cmp t",
"tactic": "unfold setBlack"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ncmp : α → α → Ordering\nt : RBNode α\n⊢ Ordered cmp\n (match t with\n | nil => nil\n | node c l v r => node black l v r) ↔\n Ordered cmp t",
"tactic": "split <;> simp [Ordered]"
}
] |
[
80,
44
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
79,
11
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.bot_eq_top_of_rank_adjoin_eq_one
|
[
{
"state_after": "case h\nF : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nS : Set E\nK L : IntermediateField F E\nh : ∀ (x : E), Module.rank F { x_1 // x_1 ∈ F⟮x⟯ } = 1\ny : E\n⊢ y ∈ ⊥ ↔ y ∈ ⊤",
"state_before": "F : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nS : Set E\nK L : IntermediateField F E\nh : ∀ (x : E), Module.rank F { x_1 // x_1 ∈ F⟮x⟯ } = 1\n⊢ ⊥ = ⊤",
"tactic": "ext y"
},
{
"state_after": "case h\nF : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nS : Set E\nK L : IntermediateField F E\nh : ∀ (x : E), Module.rank F { x_1 // x_1 ∈ F⟮x⟯ } = 1\ny : E\n⊢ y ∈ ⊥",
"state_before": "case h\nF : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nS : Set E\nK L : IntermediateField F E\nh : ∀ (x : E), Module.rank F { x_1 // x_1 ∈ F⟮x⟯ } = 1\ny : E\n⊢ y ∈ ⊥ ↔ y ∈ ⊤",
"tactic": "rw [iff_true_right IntermediateField.mem_top]"
},
{
"state_after": "no goals",
"state_before": "case h\nF : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nS : Set E\nK L : IntermediateField F E\nh : ∀ (x : E), Module.rank F { x_1 // x_1 ∈ F⟮x⟯ } = 1\ny : E\n⊢ y ∈ ⊥",
"tactic": "exact rank_adjoin_simple_eq_one_iff.mp (h y)"
}
] |
[
753,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
749,
1
] |
Mathlib/Data/Part.lean
|
Part.left_dom_of_mod_dom
|
[] |
[
788,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
788,
1
] |
Mathlib/CategoryTheory/Adjunction/Basic.lean
|
CategoryTheory.Adjunction.eq_homEquiv_apply
|
[
{
"state_after": "case refl\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nA : C\nB : D\nf : F.obj A ⟶ B\n⊢ ↑(homEquiv adj A B).symm (↑(homEquiv adj A B) f) = f",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nA : C\nB : D\nf : F.obj A ⟶ B\ng : A ⟶ G.obj B\nh : g = ↑(homEquiv adj A B) f\n⊢ ↑(homEquiv adj A B).symm g = f",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case refl\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nA : C\nB : D\nf : F.obj A ⟶ B\n⊢ ↑(homEquiv adj A B).symm (↑(homEquiv adj A B) f) = f",
"tactic": "simp"
},
{
"state_after": "case refl\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nA : C\nB : D\ng : A ⟶ G.obj B\n⊢ g = ↑(homEquiv adj A B) (↑(homEquiv adj A B).symm g)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nA : C\nB : D\nf : F.obj A ⟶ B\ng : A ⟶ G.obj B\nh : ↑(homEquiv adj A B).symm g = f\n⊢ g = ↑(homEquiv adj A B) f",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case refl\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nA : C\nB : D\ng : A ⟶ G.obj B\n⊢ g = ↑(homEquiv adj A B) (↑(homEquiv adj A B).symm g)",
"tactic": "simp"
}
] |
[
232,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
226,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.natAbs_mul
|
[
{
"state_after": "no goals",
"state_before": "a b : Int\n⊢ natAbs (a * b) = natAbs a * natAbs b",
"tactic": "cases a <;> cases b <;>\n simp only [← Int.mul_def, Int.mul, natAbs_negOfNat] <;> simp only [natAbs]"
}
] |
[
173,
79
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
171,
1
] |
Mathlib/CategoryTheory/Sites/Sheaf.lean
|
CategoryTheory.Sheaf.Hom.mono_of_presheaf_mono
|
[] |
[
324,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
323,
1
] |
Mathlib/Order/Grade.lean
|
grade_lt_grade_iff
|
[] |
[
185,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.sum_disj_union_boxes
|
[] |
[
665,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
662,
1
] |
Mathlib/Data/Polynomial/UnitTrinomial.lean
|
Polynomial.IsUnitTrinomial.card_support_eq_three
|
[
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nq : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\n⊢ card (support (trinomial k m n ↑u ↑v ↑w)) = 3",
"state_before": "p q : ℤ[X]\nhp : IsUnitTrinomial p\n⊢ card (support p) = 3",
"tactic": "obtain ⟨k, m, n, hkm, hmn, u, v, w, rfl⟩ := hp"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nq : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\n⊢ card (support (trinomial k m n ↑u ↑v ↑w)) = 3",
"tactic": "exact card_support_trinomial hkm hmn u.ne_zero v.ne_zero w.ne_zero"
}
] |
[
151,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Analysis/Calculus/LocalExtr.lean
|
IsLocalMaxOn.hasFDerivWithinAt_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nhy : y ∈ posTangentConeAt s a\nhy' : -y ∈ posTangentConeAt s a\n⊢ 0 ≤ ↑f' y",
"tactic": "simpa using h.hasFDerivWithinAt_nonpos hf hy'"
}
] |
[
142,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.add_isLimit
|
[] |
[
523,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
522,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.succ_lt_succ_iff
|
[] |
[
894,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
893,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.continuous_coe_iff
|
[] |
[
85,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Data/Int/Bitwise.lean
|
Int.bit_val
|
[
{
"state_after": "case false\nn : ℤ\n⊢ bit false n = 2 * n + bif false then 1 else 0\n\ncase true\nn : ℤ\n⊢ bit true n = 2 * n + bif true then 1 else 0",
"state_before": "b : Bool\nn : ℤ\n⊢ bit b n = 2 * n + bif b then 1 else 0",
"tactic": "cases b"
},
{
"state_after": "case true\nn : ℤ\n⊢ bit true n = 2 * n + bif true then 1 else 0",
"state_before": "case false\nn : ℤ\n⊢ bit false n = 2 * n + bif false then 1 else 0\n\ncase true\nn : ℤ\n⊢ bit true n = 2 * n + bif true then 1 else 0",
"tactic": "apply (bit0_val n).trans (add_zero _).symm"
},
{
"state_after": "no goals",
"state_before": "case true\nn : ℤ\n⊢ bit true n = 2 * n + bif true then 1 else 0",
"tactic": "apply bit1_val"
}
] |
[
135,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/CategoryTheory/Iso.lean
|
CategoryTheory.isIso_of_hom_comp_eq_id
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\nX Y Z : C\ng : X ⟶ Y\ninst✝ : IsIso g\nf : Y ⟶ X\nh : g ≫ f = 𝟙 X\n⊢ IsIso (inv g)",
"state_before": "C : Type u\ninst✝¹ : Category C\nX Y Z : C\ng : X ⟶ Y\ninst✝ : IsIso g\nf : Y ⟶ X\nh : g ≫ f = 𝟙 X\n⊢ IsIso f",
"tactic": "rw [(hom_comp_eq_id _).mp h]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nX Y Z : C\ng : X ⟶ Y\ninst✝ : IsIso g\nf : Y ⟶ X\nh : g ≫ f = 𝟙 X\n⊢ IsIso (inv g)",
"tactic": "infer_instance"
}
] |
[
499,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.normSq_mul
|
[] |
[
508,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
507,
1
] |
Mathlib/Data/List/Nodup.lean
|
List.nodup_singleton
|
[] |
[
64,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/Order/Filter/Archimedean.lean
|
Rat.comap_cast_atBot
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.9070\nR : Type u_1\ninst✝¹ : LinearOrderedField R\ninst✝ : Archimedean R\nr : R\nn : ℕ\nhn : -r ≤ ↑n\n⊢ ↑(-↑n) ≤ r",
"tactic": "simpa [neg_le]"
}
] |
[
85,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
9
] |
Std/Logic.lean
|
Exists.nonempty
|
[] |
[
435,
66
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
435,
1
] |
Mathlib/Order/Lattice.lean
|
Subtype.mk_inf_mk
|
[] |
[
1397,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1393,
1
] |
Mathlib/Dynamics/PeriodicPts.lean
|
Function.minimalPeriod_iterate_eq_div_gcd
|
[] |
[
471,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
469,
1
] |
Mathlib/Algebra/Lie/Basic.lean
|
LieModuleHom.coe_id
|
[] |
[
770,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
769,
1
] |
Mathlib/Tactic/Lift.lean
|
Subtype.exists_pi_extension
|
[
{
"state_after": "ι : Sort u_1\nα : ι → Sort u_2\nne : ∀ (i : ι), Nonempty (α i)\np : ι → Prop\nf : (i : Subtype p) → α i.val\nthis : DecidablePred p\n⊢ ∃ g, (fun i => g i.val) = f",
"state_before": "ι : Sort u_1\nα : ι → Sort u_2\nne : ∀ (i : ι), Nonempty (α i)\np : ι → Prop\nf : (i : Subtype p) → α i.val\n⊢ ∃ g, (fun i => g i.val) = f",
"tactic": "haveI : DecidablePred p := fun i ↦ Classical.propDecidable (p i)"
},
{
"state_after": "no goals",
"state_before": "ι : Sort u_1\nα : ι → Sort u_2\nne : ∀ (i : ι), Nonempty (α i)\np : ι → Prop\nf : (i : Subtype p) → α i.val\nthis : DecidablePred p\n⊢ ∃ g, (fun i => g i.val) = f",
"tactic": "exact ⟨fun i => if hi : p i then f ⟨i, hi⟩ else Classical.choice (ne i),\n funext fun i ↦ dif_pos i.2⟩"
}
] |
[
45,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Mathlib/Analysis/Convex/Extreme.lean
|
extremePoints_subset
|
[] |
[
152,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/Data/Set/Ncard.lean
|
Set.exists_mem_not_mem_of_ncard_lt_ncard
|
[] |
[
537,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
535,
1
] |
Mathlib/Computability/TMToPartrec.lean
|
Turing.ToPartrec.code_is_ok
|
[
{
"state_after": "case zero'\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => Cfg.ret k (0 :: v)) k v) = do\n let v ← Code.eval Code.zero' v\n eval step (Cfg.ret k v)\n\ncase succ\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => Cfg.ret k [Nat.succ (List.headI v)]) k v) = do\n let v ← Code.eval Code.succ v\n eval step (Cfg.ret k v)\n\ncase tail\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => Cfg.ret k (List.tail v)) k v) = do\n let v ← Code.eval Code.tail v\n eval step (Cfg.ret k v)\n\ncase cons\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝¹ (Cont.cons₁ a✝ v k) v) k v) = do\n let v ← Code.eval (Code.cons a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase comp\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝ (Cont.comp a✝¹ k) v) k v) = do\n let v ← Code.eval (Code.comp a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase case\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step\n ((fun k v =>\n Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v))\n k v) =\n do\n let v ← Code.eval (Code.case a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase fix\na✝ : Code\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do\n let v ← Code.eval (Code.fix a✝) v\n eval step (Cfg.ret k v)",
"state_before": "c : Code\n⊢ Code.Ok c",
"tactic": "induction c <;> intro k v <;> rw [stepNormal]"
},
{
"state_after": "case cons\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝¹ (Cont.cons₁ a✝ v k) v) k v) = do\n let v ← Code.eval (Code.cons a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase comp\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝ (Cont.comp a✝¹ k) v) k v) = do\n let v ← Code.eval (Code.comp a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase case\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step\n ((fun k v =>\n Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v))\n k v) =\n do\n let v ← Code.eval (Code.case a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase fix\na✝ : Code\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do\n let v ← Code.eval (Code.fix a✝) v\n eval step (Cfg.ret k v)",
"state_before": "case zero'\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => Cfg.ret k (0 :: v)) k v) = do\n let v ← Code.eval Code.zero' v\n eval step (Cfg.ret k v)\n\ncase succ\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => Cfg.ret k [Nat.succ (List.headI v)]) k v) = do\n let v ← Code.eval Code.succ v\n eval step (Cfg.ret k v)\n\ncase tail\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => Cfg.ret k (List.tail v)) k v) = do\n let v ← Code.eval Code.tail v\n eval step (Cfg.ret k v)\n\ncase cons\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝¹ (Cont.cons₁ a✝ v k) v) k v) = do\n let v ← Code.eval (Code.cons a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase comp\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝ (Cont.comp a✝¹ k) v) k v) = do\n let v ← Code.eval (Code.comp a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase case\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step\n ((fun k v =>\n Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v))\n k v) =\n do\n let v ← Code.eval (Code.case a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase fix\na✝ : Code\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do\n let v ← Code.eval (Code.fix a✝) v\n eval step (Cfg.ret k v)",
"tactic": "iterate 3 simp only [Code.eval, pure_bind]"
},
{
"state_after": "case comp\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝ (Cont.comp a✝¹ k) v) k v) = do\n let v ← Code.eval (Code.comp a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase case\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step\n ((fun k v =>\n Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v))\n k v) =\n do\n let v ← Code.eval (Code.case a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase fix\na✝ : Code\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do\n let v ← Code.eval (Code.fix a✝) v\n eval step (Cfg.ret k v)",
"state_before": "case cons\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝¹ (Cont.cons₁ a✝ v k) v) k v) = do\n let v ← Code.eval (Code.cons a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase comp\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝ (Cont.comp a✝¹ k) v) k v) = do\n let v ← Code.eval (Code.comp a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase case\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step\n ((fun k v =>\n Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v))\n k v) =\n do\n let v ← Code.eval (Code.case a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase fix\na✝ : Code\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do\n let v ← Code.eval (Code.fix a✝) v\n eval step (Cfg.ret k v)",
"tactic": "case cons f fs IHf IHfs =>\n rw [Code.eval, IHf]\n simp only [bind_assoc, Cont.eval, pure_bind]; congr ; funext v\n rw [reaches_eval]; swap; exact ReflTransGen.single rfl\n rw [stepRet, IHfs]; congr ; funext v'\n refine' Eq.trans _ (Eq.symm _) <;> try exact reaches_eval (ReflTransGen.single rfl)"
},
{
"state_after": "case case\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step\n ((fun k v =>\n Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v))\n k v) =\n do\n let v ← Code.eval (Code.case a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase fix\na✝ : Code\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do\n let v ← Code.eval (Code.fix a✝) v\n eval step (Cfg.ret k v)",
"state_before": "case comp\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝ (Cont.comp a✝¹ k) v) k v) = do\n let v ← Code.eval (Code.comp a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase case\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step\n ((fun k v =>\n Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v))\n k v) =\n do\n let v ← Code.eval (Code.case a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase fix\na✝ : Code\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do\n let v ← Code.eval (Code.fix a✝) v\n eval step (Cfg.ret k v)",
"tactic": "case comp f g IHf IHg =>\n rw [Code.eval, IHg]\n simp only [bind_assoc, Cont.eval, pure_bind]; congr ; funext v\n rw [reaches_eval]; swap; exact ReflTransGen.single rfl\n rw [stepRet, IHf]"
},
{
"state_after": "case fix\na✝ : Code\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do\n let v ← Code.eval (Code.fix a✝) v\n eval step (Cfg.ret k v)",
"state_before": "case case\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step\n ((fun k v =>\n Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v))\n k v) =\n do\n let v ← Code.eval (Code.case a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase fix\na✝ : Code\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do\n let v ← Code.eval (Code.fix a✝) v\n eval step (Cfg.ret k v)",
"tactic": "case case f g IHf IHg =>\n simp only [Code.eval]\n cases v.headI <;> simp only [Code.eval] <;> [apply IHf; apply IHg]"
},
{
"state_after": "no goals",
"state_before": "case fix\na✝ : Code\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do\n let v ← Code.eval (Code.fix a✝) v\n eval step (Cfg.ret k v)",
"tactic": "case fix f IHf => rw [cont_eval_fix IHf]"
},
{
"state_after": "case cons\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝¹ (Cont.cons₁ a✝ v k) v) k v) = do\n let v ← Code.eval (Code.cons a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase comp\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝ (Cont.comp a✝¹ k) v) k v) = do\n let v ← Code.eval (Code.comp a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase case\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step\n ((fun k v =>\n Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v))\n k v) =\n do\n let v ← Code.eval (Code.case a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase fix\na✝ : Code\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do\n let v ← Code.eval (Code.fix a✝) v\n eval step (Cfg.ret k v)",
"state_before": "case tail\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => Cfg.ret k (List.tail v)) k v) = do\n let v ← Code.eval Code.tail v\n eval step (Cfg.ret k v)\n\ncase cons\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝¹ (Cont.cons₁ a✝ v k) v) k v) = do\n let v ← Code.eval (Code.cons a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase comp\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝ (Cont.comp a✝¹ k) v) k v) = do\n let v ← Code.eval (Code.comp a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase case\na✝¹ a✝ : Code\na_ih✝¹ : Code.Ok a✝¹\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step\n ((fun k v =>\n Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v))\n k v) =\n do\n let v ← Code.eval (Code.case a✝¹ a✝) v\n eval step (Cfg.ret k v)\n\ncase fix\na✝ : Code\na_ih✝ : Code.Ok a✝\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do\n let v ← Code.eval (Code.fix a✝) v\n eval step (Cfg.ret k v)",
"tactic": "simp only [Code.eval, pure_bind]"
},
{
"state_after": "f fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv : List ℕ\n⊢ (do\n let v_1 ← Code.eval f v\n eval step (Cfg.ret (Cont.cons₁ fs v k) v_1)) =\n do\n let v ←\n (fun v => do\n let n ← Code.eval f v\n let ns ← Code.eval fs v\n pure (List.headI n :: ns))\n v\n eval step (Cfg.ret k v)",
"state_before": "f fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal f (Cont.cons₁ fs v k) v) k v) = do\n let v ← Code.eval (Code.cons f fs) v\n eval step (Cfg.ret k v)",
"tactic": "rw [Code.eval, IHf]"
},
{
"state_after": "f fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv : List ℕ\n⊢ (do\n let v_1 ← Code.eval f v\n eval step (Cfg.ret (Cont.cons₁ fs v k) v_1)) =\n do\n let x ← Code.eval f v\n let x_1 ← Code.eval fs v\n eval step (Cfg.ret k (List.headI x :: x_1))",
"state_before": "f fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv : List ℕ\n⊢ (do\n let v_1 ← Code.eval f v\n eval step (Cfg.ret (Cont.cons₁ fs v k) v_1)) =\n do\n let v ←\n (fun v => do\n let n ← Code.eval f v\n let ns ← Code.eval fs v\n pure (List.headI n :: ns))\n v\n eval step (Cfg.ret k v)",
"tactic": "simp only [bind_assoc, Cont.eval, pure_bind]"
},
{
"state_after": "case e_a\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv : List ℕ\n⊢ (fun v_1 => eval step (Cfg.ret (Cont.cons₁ fs v k) v_1)) = fun x => do\n let x_1 ← Code.eval fs v\n eval step (Cfg.ret k (List.headI x :: x_1))",
"state_before": "f fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv : List ℕ\n⊢ (do\n let v_1 ← Code.eval f v\n eval step (Cfg.ret (Cont.cons₁ fs v k) v_1)) =\n do\n let x ← Code.eval f v\n let x_1 ← Code.eval fs v\n eval step (Cfg.ret k (List.headI x :: x_1))",
"tactic": "congr"
},
{
"state_after": "case e_a.h\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v : List ℕ\n⊢ eval step (Cfg.ret (Cont.cons₁ fs v✝ k) v) = do\n let x ← Code.eval fs v✝\n eval step (Cfg.ret k (List.headI v :: x))",
"state_before": "case e_a\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv : List ℕ\n⊢ (fun v_1 => eval step (Cfg.ret (Cont.cons₁ fs v k) v_1)) = fun x => do\n let x_1 ← Code.eval fs v\n eval step (Cfg.ret k (List.headI x :: x_1))",
"tactic": "funext v"
},
{
"state_after": "case e_a.h\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v : List ℕ\n⊢ eval step ?m.196952 = do\n let x ← Code.eval fs v✝\n eval step (Cfg.ret k (List.headI v :: x))\n\ncase e_a.h\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v : List ℕ\n⊢ Reaches step (Cfg.ret (Cont.cons₁ fs v✝ k) v) ?m.196952\n\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v : List ℕ\n⊢ Cfg",
"state_before": "case e_a.h\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v : List ℕ\n⊢ eval step (Cfg.ret (Cont.cons₁ fs v✝ k) v) = do\n let x ← Code.eval fs v✝\n eval step (Cfg.ret k (List.headI v :: x))",
"tactic": "rw [reaches_eval]"
},
{
"state_after": "case e_a.h\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v : List ℕ\n⊢ Reaches step (Cfg.ret (Cont.cons₁ fs v✝ k) v) ?m.196952\n\ncase e_a.h\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v : List ℕ\n⊢ eval step ?m.196952 = do\n let x ← Code.eval fs v✝\n eval step (Cfg.ret k (List.headI v :: x))\n\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v : List ℕ\n⊢ Cfg",
"state_before": "case e_a.h\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v : List ℕ\n⊢ eval step ?m.196952 = do\n let x ← Code.eval fs v✝\n eval step (Cfg.ret k (List.headI v :: x))\n\ncase e_a.h\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v : List ℕ\n⊢ Reaches step (Cfg.ret (Cont.cons₁ fs v✝ k) v) ?m.196952\n\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v : List ℕ\n⊢ Cfg",
"tactic": "swap"
},
{
"state_after": "case e_a.h\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v : List ℕ\n⊢ eval step (stepRet (Cont.cons₁ fs v✝ k) v) = do\n let x ← Code.eval fs v✝\n eval step (Cfg.ret k (List.headI v :: x))",
"state_before": "case e_a.h\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v : List ℕ\n⊢ Reaches step (Cfg.ret (Cont.cons₁ fs v✝ k) v) ?m.196952\n\ncase e_a.h\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v : List ℕ\n⊢ eval step ?m.196952 = do\n let x ← Code.eval fs v✝\n eval step (Cfg.ret k (List.headI v :: x))\n\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v : List ℕ\n⊢ Cfg",
"tactic": "exact ReflTransGen.single rfl"
},
{
"state_after": "case e_a.h\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v : List ℕ\n⊢ (do\n let v_1 ← Code.eval fs v✝\n eval step (Cfg.ret (Cont.cons₂ v k) v_1)) =\n do\n let x ← Code.eval fs v✝\n eval step (Cfg.ret k (List.headI v :: x))",
"state_before": "case e_a.h\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v : List ℕ\n⊢ eval step (stepRet (Cont.cons₁ fs v✝ k) v) = do\n let x ← Code.eval fs v✝\n eval step (Cfg.ret k (List.headI v :: x))",
"tactic": "rw [stepRet, IHfs]"
},
{
"state_after": "case e_a.h.e_a\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v : List ℕ\n⊢ (fun v_1 => eval step (Cfg.ret (Cont.cons₂ v k) v_1)) = fun x => eval step (Cfg.ret k (List.headI v :: x))",
"state_before": "case e_a.h\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v : List ℕ\n⊢ (do\n let v_1 ← Code.eval fs v✝\n eval step (Cfg.ret (Cont.cons₂ v k) v_1)) =\n do\n let x ← Code.eval fs v✝\n eval step (Cfg.ret k (List.headI v :: x))",
"tactic": "congr"
},
{
"state_after": "case e_a.h.e_a.h\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v v' : List ℕ\n⊢ eval step (Cfg.ret (Cont.cons₂ v k) v') = eval step (Cfg.ret k (List.headI v :: v'))",
"state_before": "case e_a.h.e_a\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v : List ℕ\n⊢ (fun v_1 => eval step (Cfg.ret (Cont.cons₂ v k) v_1)) = fun x => eval step (Cfg.ret k (List.headI v :: x))",
"tactic": "funext v'"
},
{
"state_after": "no goals",
"state_before": "case e_a.h.e_a.h\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v v' : List ℕ\n⊢ eval step (Cfg.ret (Cont.cons₂ v k) v') = eval step (Cfg.ret k (List.headI v :: v'))",
"tactic": "refine' Eq.trans _ (Eq.symm _) <;> try exact reaches_eval (ReflTransGen.single rfl)"
},
{
"state_after": "no goals",
"state_before": "case e_a.h.e_a.h.refine'_3\nf fs : Code\nIHf : Code.Ok f\nIHfs : Code.Ok fs\nk : Cont\nv✝ v v' : List ℕ\n⊢ eval step (Cfg.ret k (List.headI v :: v')) = eval step (stepRet (Cont.cons₂ v k) v')",
"tactic": "exact reaches_eval (ReflTransGen.single rfl)"
},
{
"state_after": "f g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv : List ℕ\n⊢ (do\n let v ← Code.eval g v\n eval step (Cfg.ret (Cont.comp f k) v)) =\n do\n let v ← (fun v => Code.eval g v >>= Code.eval f) v\n eval step (Cfg.ret k v)",
"state_before": "f g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal g (Cont.comp f k) v) k v) = do\n let v ← Code.eval (Code.comp f g) v\n eval step (Cfg.ret k v)",
"tactic": "rw [Code.eval, IHg]"
},
{
"state_after": "f g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv : List ℕ\n⊢ (do\n let v ← Code.eval g v\n eval step (Cfg.ret (Cont.comp f k) v)) =\n do\n let x ← Code.eval g v\n let v ← Code.eval f x\n eval step (Cfg.ret k v)",
"state_before": "f g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv : List ℕ\n⊢ (do\n let v ← Code.eval g v\n eval step (Cfg.ret (Cont.comp f k) v)) =\n do\n let v ← (fun v => Code.eval g v >>= Code.eval f) v\n eval step (Cfg.ret k v)",
"tactic": "simp only [bind_assoc, Cont.eval, pure_bind]"
},
{
"state_after": "case e_a\nf g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv : List ℕ\n⊢ (fun v => eval step (Cfg.ret (Cont.comp f k) v)) = fun x => do\n let v ← Code.eval f x\n eval step (Cfg.ret k v)",
"state_before": "f g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv : List ℕ\n⊢ (do\n let v ← Code.eval g v\n eval step (Cfg.ret (Cont.comp f k) v)) =\n do\n let x ← Code.eval g v\n let v ← Code.eval f x\n eval step (Cfg.ret k v)",
"tactic": "congr"
},
{
"state_after": "case e_a.h\nf g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv✝ v : List ℕ\n⊢ eval step (Cfg.ret (Cont.comp f k) v) = do\n let v ← Code.eval f v\n eval step (Cfg.ret k v)",
"state_before": "case e_a\nf g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv : List ℕ\n⊢ (fun v => eval step (Cfg.ret (Cont.comp f k) v)) = fun x => do\n let v ← Code.eval f x\n eval step (Cfg.ret k v)",
"tactic": "funext v"
},
{
"state_after": "case e_a.h\nf g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv✝ v : List ℕ\n⊢ eval step ?m.197726 = do\n let v ← Code.eval f v\n eval step (Cfg.ret k v)\n\ncase e_a.h\nf g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv✝ v : List ℕ\n⊢ Reaches step (Cfg.ret (Cont.comp f k) v) ?m.197726\n\nf g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv✝ v : List ℕ\n⊢ Cfg",
"state_before": "case e_a.h\nf g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv✝ v : List ℕ\n⊢ eval step (Cfg.ret (Cont.comp f k) v) = do\n let v ← Code.eval f v\n eval step (Cfg.ret k v)",
"tactic": "rw [reaches_eval]"
},
{
"state_after": "case e_a.h\nf g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv✝ v : List ℕ\n⊢ Reaches step (Cfg.ret (Cont.comp f k) v) ?m.197726\n\ncase e_a.h\nf g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv✝ v : List ℕ\n⊢ eval step ?m.197726 = do\n let v ← Code.eval f v\n eval step (Cfg.ret k v)\n\nf g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv✝ v : List ℕ\n⊢ Cfg",
"state_before": "case e_a.h\nf g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv✝ v : List ℕ\n⊢ eval step ?m.197726 = do\n let v ← Code.eval f v\n eval step (Cfg.ret k v)\n\ncase e_a.h\nf g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv✝ v : List ℕ\n⊢ Reaches step (Cfg.ret (Cont.comp f k) v) ?m.197726\n\nf g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv✝ v : List ℕ\n⊢ Cfg",
"tactic": "swap"
},
{
"state_after": "case e_a.h\nf g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv✝ v : List ℕ\n⊢ eval step (stepRet (Cont.comp f k) v) = do\n let v ← Code.eval f v\n eval step (Cfg.ret k v)",
"state_before": "case e_a.h\nf g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv✝ v : List ℕ\n⊢ Reaches step (Cfg.ret (Cont.comp f k) v) ?m.197726\n\ncase e_a.h\nf g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv✝ v : List ℕ\n⊢ eval step ?m.197726 = do\n let v ← Code.eval f v\n eval step (Cfg.ret k v)\n\nf g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv✝ v : List ℕ\n⊢ Cfg",
"tactic": "exact ReflTransGen.single rfl"
},
{
"state_after": "no goals",
"state_before": "case e_a.h\nf g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv✝ v : List ℕ\n⊢ eval step (stepRet (Cont.comp f k) v) = do\n let v ← Code.eval f v\n eval step (Cfg.ret k v)",
"tactic": "rw [stepRet, IHf]"
},
{
"state_after": "f g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv : List ℕ\n⊢ eval step (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v)) = do\n let v ← Nat.rec (Code.eval f (List.tail v)) (fun y x => Code.eval g (y :: List.tail v)) (List.headI v)\n eval step (Cfg.ret k v)",
"state_before": "f g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv : List ℕ\n⊢ eval step\n ((fun k v => Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v))\n k v) =\n do\n let v ← Code.eval (Code.case f g) v\n eval step (Cfg.ret k v)",
"tactic": "simp only [Code.eval]"
},
{
"state_after": "no goals",
"state_before": "f g : Code\nIHf : Code.Ok f\nIHg : Code.Ok g\nk : Cont\nv : List ℕ\n⊢ eval step (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v)) = do\n let v ← Nat.rec (Code.eval f (List.tail v)) (fun y x => Code.eval g (y :: List.tail v)) (List.headI v)\n eval step (Cfg.ret k v)",
"tactic": "cases v.headI <;> simp only [Code.eval] <;> [apply IHf; apply IHg]"
},
{
"state_after": "no goals",
"state_before": "f : Code\nIHf : Code.Ok f\nk : Cont\nv : List ℕ\n⊢ eval step ((fun k v => stepNormal f (Cont.fix f k) v) k v) = do\n let v ← Code.eval (Code.fix f) v\n eval step (Cfg.ret k v)",
"tactic": "rw [cont_eval_fix IHf]"
}
] |
[
718,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
701,
1
] |
Mathlib/Algebra/BigOperators/Fin.lean
|
Fin.prod_univ_five
|
[
{
"state_after": "α : Type ?u.22461\nβ : Type u_1\ninst✝ : CommMonoid β\nf : Fin 5 → β\n⊢ f (↑castSucc 0) * f (↑castSucc 1) * f (↑castSucc 2) * f (↑castSucc 3) * f (last 4) = f 0 * f 1 * f 2 * f 3 * f 4",
"state_before": "α : Type ?u.22461\nβ : Type u_1\ninst✝ : CommMonoid β\nf : Fin 5 → β\n⊢ ∏ i : Fin 5, f i = f 0 * f 1 * f 2 * f 3 * f 4",
"tactic": "rw [prod_univ_castSucc, prod_univ_four]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.22461\nβ : Type u_1\ninst✝ : CommMonoid β\nf : Fin 5 → β\n⊢ f (↑castSucc 0) * f (↑castSucc 1) * f (↑castSucc 2) * f (↑castSucc 3) * f (last 4) = f 0 * f 1 * f 2 * f 3 * f 4",
"tactic": "rfl"
}
] |
[
135,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/CategoryTheory/Extensive.lean
|
CategoryTheory.BinaryCofan.isVanKampen_mk
|
[
{
"state_after": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\n⊢ ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : c'.pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f →\n (Nonempty (IsColimit c') ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c))",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\n⊢ IsVanKampenColimit c",
"tactic": "rw [BinaryCofan.isVanKampen_iff]"
},
{
"state_after": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\n⊢ Nonempty (IsColimit c') ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧ IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\n⊢ ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : c'.pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f →\n (Nonempty (IsColimit c') ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c))",
"tactic": "introv hX hY"
},
{
"state_after": "case mp\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\n⊢ Nonempty (IsColimit c') →\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧ IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)\n\ncase mpr\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\n⊢ IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧ IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c) →\n Nonempty (IsColimit c')",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\n⊢ Nonempty (IsColimit c') ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧ IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)",
"tactic": "constructor"
},
{
"state_after": "case mp.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\n⊢ IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧ IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)",
"state_before": "case mp\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\n⊢ Nonempty (IsColimit c') →\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧ IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)",
"tactic": "rintro ⟨h⟩"
},
{
"state_after": "case mp.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\n⊢ IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧ IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)",
"state_before": "case mp.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\n⊢ IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧ IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)",
"tactic": "let e := h.coconePointUniqueUpToIso (colimits _ _)"
},
{
"state_after": "case mp.intro.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\n⊢ IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧ IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)",
"state_before": "case mp.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\n⊢ IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧ IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)",
"tactic": "obtain ⟨hl, hr⟩ := h₁ αX αY (e.inv ≫ f) (by simp [hX]) (by simp [hY])"
},
{
"state_after": "case mp.intro.intro.left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\n⊢ IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c)\n\ncase mp.intro.intro.right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\n⊢ IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)",
"state_before": "case mp.intro.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\n⊢ IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧ IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\n⊢ αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ e.inv ≫ f",
"tactic": "simp [hX]"
},
{
"state_after": "no goals",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\n⊢ αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ e.inv ≫ f",
"tactic": "simp [hY]"
},
{
"state_after": "case mp.intro.intro.left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\n⊢ IsPullback (BinaryCofan.inl c') (𝟙 X' ≫ αX) (e.hom ≫ e.inv ≫ f) (BinaryCofan.inl c)",
"state_before": "case mp.intro.intro.left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\n⊢ IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c)",
"tactic": "rw [← Category.id_comp αX, ← Iso.hom_inv_id_assoc e f]"
},
{
"state_after": "case mp.intro.intro.left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\nthis : IsIso (𝟙 X')\n⊢ IsPullback (BinaryCofan.inl c') (𝟙 X' ≫ αX) (e.hom ≫ e.inv ≫ f) (BinaryCofan.inl c)",
"state_before": "case mp.intro.intro.left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\n⊢ IsPullback (BinaryCofan.inl c') (𝟙 X' ≫ αX) (e.hom ≫ e.inv ≫ f) (BinaryCofan.inl c)",
"tactic": "haveI : IsIso (𝟙 X') := inferInstance"
},
{
"state_after": "case mp.intro.intro.left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\nthis✝ : IsIso (𝟙 X')\nthis : BinaryCofan.inl c' ≫ e.hom = 𝟙 X' ≫ BinaryCofan.inl (cofans X' Y')\n⊢ IsPullback (BinaryCofan.inl c') (𝟙 X' ≫ αX) (e.hom ≫ e.inv ≫ f) (BinaryCofan.inl c)",
"state_before": "case mp.intro.intro.left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\nthis : IsIso (𝟙 X')\n⊢ IsPullback (BinaryCofan.inl c') (𝟙 X' ≫ αX) (e.hom ≫ e.inv ≫ f) (BinaryCofan.inl c)",
"tactic": "have : c'.inl ≫ e.hom = 𝟙 X' ≫ (cofans X' Y').inl := by\n dsimp\n simp"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\nthis✝ : IsIso (𝟙 X')\nthis : BinaryCofan.inl c' ≫ e.hom = 𝟙 X' ≫ BinaryCofan.inl (cofans X' Y')\n⊢ IsPullback (BinaryCofan.inl c') (𝟙 X' ≫ αX) (e.hom ≫ e.inv ≫ f) (BinaryCofan.inl c)",
"tactic": "exact (IsPullback.of_vert_isIso ⟨this⟩).paste_vert hl"
},
{
"state_after": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\nthis : IsIso (𝟙 X')\n⊢ BinaryCofan.inl c' ≫ (IsColimit.coconePointUniqueUpToIso h (colimits X' Y')).hom =\n 𝟙 X' ≫ BinaryCofan.inl (cofans X' Y')",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\nthis : IsIso (𝟙 X')\n⊢ BinaryCofan.inl c' ≫ e.hom = 𝟙 X' ≫ BinaryCofan.inl (cofans X' Y')",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\nthis : IsIso (𝟙 X')\n⊢ BinaryCofan.inl c' ≫ (IsColimit.coconePointUniqueUpToIso h (colimits X' Y')).hom =\n 𝟙 X' ≫ BinaryCofan.inl (cofans X' Y')",
"tactic": "simp"
},
{
"state_after": "case mp.intro.intro.right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\n⊢ IsPullback (BinaryCofan.inr c') (𝟙 Y' ≫ αY) (e.hom ≫ e.inv ≫ f) (BinaryCofan.inr c)",
"state_before": "case mp.intro.intro.right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\n⊢ IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)",
"tactic": "rw [← Category.id_comp αY, ← Iso.hom_inv_id_assoc e f]"
},
{
"state_after": "case mp.intro.intro.right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\nthis : IsIso (𝟙 Y')\n⊢ IsPullback (BinaryCofan.inr c') (𝟙 Y' ≫ αY) (e.hom ≫ e.inv ≫ f) (BinaryCofan.inr c)",
"state_before": "case mp.intro.intro.right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\n⊢ IsPullback (BinaryCofan.inr c') (𝟙 Y' ≫ αY) (e.hom ≫ e.inv ≫ f) (BinaryCofan.inr c)",
"tactic": "haveI : IsIso (𝟙 Y') := inferInstance"
},
{
"state_after": "case mp.intro.intro.right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\nthis✝ : IsIso (𝟙 Y')\nthis : BinaryCofan.inr c' ≫ e.hom = 𝟙 Y' ≫ BinaryCofan.inr (cofans X' Y')\n⊢ IsPullback (BinaryCofan.inr c') (𝟙 Y' ≫ αY) (e.hom ≫ e.inv ≫ f) (BinaryCofan.inr c)",
"state_before": "case mp.intro.intro.right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\nthis : IsIso (𝟙 Y')\n⊢ IsPullback (BinaryCofan.inr c') (𝟙 Y' ≫ αY) (e.hom ≫ e.inv ≫ f) (BinaryCofan.inr c)",
"tactic": "have : c'.inr ≫ e.hom = 𝟙 Y' ≫ (cofans X' Y').inr := by\n dsimp\n simp"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\nthis✝ : IsIso (𝟙 Y')\nthis : BinaryCofan.inr c' ≫ e.hom = 𝟙 Y' ≫ BinaryCofan.inr (cofans X' Y')\n⊢ IsPullback (BinaryCofan.inr c') (𝟙 Y' ≫ αY) (e.hom ≫ e.inv ≫ f) (BinaryCofan.inr c)",
"tactic": "exact (IsPullback.of_vert_isIso ⟨this⟩).paste_vert hr"
},
{
"state_after": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\nthis : IsIso (𝟙 Y')\n⊢ BinaryCofan.inr c' ≫ (IsColimit.coconePointUniqueUpToIso h (colimits X' Y')).hom =\n 𝟙 Y' ≫ BinaryCofan.inr (cofans X' Y')",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\nthis : IsIso (𝟙 Y')\n⊢ BinaryCofan.inr c' ≫ e.hom = 𝟙 Y' ≫ BinaryCofan.inr (cofans X' Y')",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nh : IsColimit c'\ne : c'.pt ≅ (cofans X' Y').pt := IsColimit.coconePointUniqueUpToIso h (colimits X' Y')\nhl : IsPullback (BinaryCofan.inl (cofans X' Y')) αX (e.inv ≫ f) (BinaryCofan.inl c)\nhr : IsPullback (BinaryCofan.inr (cofans X' Y')) αY (e.inv ≫ f) (BinaryCofan.inr c)\nthis : IsIso (𝟙 Y')\n⊢ BinaryCofan.inr c' ≫ (IsColimit.coconePointUniqueUpToIso h (colimits X' Y')).hom =\n 𝟙 Y' ≫ BinaryCofan.inr (cofans X' Y')",
"tactic": "simp"
},
{
"state_after": "case mpr.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nH₁ : IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c)\nH₂ : IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)\n⊢ Nonempty (IsColimit c')",
"state_before": "case mpr\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\n⊢ IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧ IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c) →\n Nonempty (IsColimit c')",
"tactic": "rintro ⟨H₁, H₂⟩"
},
{
"state_after": "case mpr.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nH₁ : IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c)\nH₂ : IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)\n⊢ IsColimit (BinaryCofan.mk (BinaryCofan.inl c') (BinaryCofan.inr c'))",
"state_before": "case mpr.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nH₁ : IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c)\nH₂ : IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)\n⊢ Nonempty (IsColimit c')",
"tactic": "refine' ⟨IsColimit.ofIsoColimit _ <| (isoBinaryCofanMk _).symm⟩"
},
{
"state_after": "case mpr.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nH₁ : IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c)\nH₂ : IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)\ne₁ : X' ≅ (cones f (BinaryCofan.inl c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₁) (limits f (BinaryCofan.inl c))\n⊢ IsColimit (BinaryCofan.mk (BinaryCofan.inl c') (BinaryCofan.inr c'))",
"state_before": "case mpr.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nH₁ : IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c)\nH₂ : IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)\n⊢ IsColimit (BinaryCofan.mk (BinaryCofan.inl c') (BinaryCofan.inr c'))",
"tactic": "let e₁ : X' ≅ _ := H₁.isLimit.conePointUniqueUpToIso (limits _ _)"
},
{
"state_after": "case mpr.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nH₁ : IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c)\nH₂ : IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)\ne₁ : X' ≅ (cones f (BinaryCofan.inl c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₁) (limits f (BinaryCofan.inl c))\ne₂ : Y' ≅ (cones f (BinaryCofan.inr c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₂) (limits f (BinaryCofan.inr c))\n⊢ IsColimit (BinaryCofan.mk (BinaryCofan.inl c') (BinaryCofan.inr c'))",
"state_before": "case mpr.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nH₁ : IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c)\nH₂ : IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)\ne₁ : X' ≅ (cones f (BinaryCofan.inl c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₁) (limits f (BinaryCofan.inl c))\n⊢ IsColimit (BinaryCofan.mk (BinaryCofan.inl c') (BinaryCofan.inr c'))",
"tactic": "let e₂ : Y' ≅ _ := H₂.isLimit.conePointUniqueUpToIso (limits _ _)"
},
{
"state_after": "case mpr.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nH₁ : IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c)\nH₂ : IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)\ne₁ : X' ≅ (cones f (BinaryCofan.inl c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₁) (limits f (BinaryCofan.inl c))\ne₂ : Y' ≅ (cones f (BinaryCofan.inr c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₂) (limits f (BinaryCofan.inr c))\nhe₁ : BinaryCofan.inl c' = e₁.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inl c))\n⊢ IsColimit (BinaryCofan.mk (BinaryCofan.inl c') (BinaryCofan.inr c'))",
"state_before": "case mpr.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nH₁ : IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c)\nH₂ : IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)\ne₁ : X' ≅ (cones f (BinaryCofan.inl c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₁) (limits f (BinaryCofan.inl c))\ne₂ : Y' ≅ (cones f (BinaryCofan.inr c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₂) (limits f (BinaryCofan.inr c))\n⊢ IsColimit (BinaryCofan.mk (BinaryCofan.inl c') (BinaryCofan.inr c'))",
"tactic": "have he₁ : c'.inl = e₁.hom ≫ (cones f c.inl).fst := by simp"
},
{
"state_after": "case mpr.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nH₁ : IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c)\nH₂ : IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)\ne₁ : X' ≅ (cones f (BinaryCofan.inl c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₁) (limits f (BinaryCofan.inl c))\ne₂ : Y' ≅ (cones f (BinaryCofan.inr c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₂) (limits f (BinaryCofan.inr c))\nhe₁ : BinaryCofan.inl c' = e₁.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inl c))\nhe₂ : BinaryCofan.inr c' = e₂.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inr c))\n⊢ IsColimit (BinaryCofan.mk (BinaryCofan.inl c') (BinaryCofan.inr c'))",
"state_before": "case mpr.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nH₁ : IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c)\nH₂ : IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)\ne₁ : X' ≅ (cones f (BinaryCofan.inl c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₁) (limits f (BinaryCofan.inl c))\ne₂ : Y' ≅ (cones f (BinaryCofan.inr c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₂) (limits f (BinaryCofan.inr c))\nhe₁ : BinaryCofan.inl c' = e₁.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inl c))\n⊢ IsColimit (BinaryCofan.mk (BinaryCofan.inl c') (BinaryCofan.inr c'))",
"tactic": "have he₂ : c'.inr = e₂.hom ≫ (cones f c.inr).fst := by simp"
},
{
"state_after": "case mpr.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nH₁ : IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c)\nH₂ : IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)\ne₁ : X' ≅ (cones f (BinaryCofan.inl c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₁) (limits f (BinaryCofan.inl c))\ne₂ : Y' ≅ (cones f (BinaryCofan.inr c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₂) (limits f (BinaryCofan.inr c))\nhe₁ : BinaryCofan.inl c' = e₁.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inl c))\nhe₂ : BinaryCofan.inr c' = e₂.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inr c))\n⊢ IsColimit\n (BinaryCofan.mk (e₁.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (e₂.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inr c))))",
"state_before": "case mpr.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nH₁ : IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c)\nH₂ : IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)\ne₁ : X' ≅ (cones f (BinaryCofan.inl c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₁) (limits f (BinaryCofan.inl c))\ne₂ : Y' ≅ (cones f (BinaryCofan.inr c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₂) (limits f (BinaryCofan.inr c))\nhe₁ : BinaryCofan.inl c' = e₁.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inl c))\nhe₂ : BinaryCofan.inr c' = e₂.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inr c))\n⊢ IsColimit (BinaryCofan.mk (BinaryCofan.inl c') (BinaryCofan.inr c'))",
"tactic": "rw [he₁, he₂]"
},
{
"state_after": "case mpr.intro.h\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nH₁ : IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c)\nH₂ : IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)\ne₁ : X' ≅ (cones f (BinaryCofan.inl c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₁) (limits f (BinaryCofan.inl c))\ne₂ : Y' ≅ (cones f (BinaryCofan.inr c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₂) (limits f (BinaryCofan.inr c))\nhe₁ : BinaryCofan.inl c' = e₁.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inl c))\nhe₂ : BinaryCofan.inr c' = e₂.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inr c))\n⊢ IsColimit\n (BinaryCofan.mk (e₁.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inl c)))\n ((cones f (BinaryCofan.inr c)).π.1 WalkingCospan.left))",
"state_before": "case mpr.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nH₁ : IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c)\nH₂ : IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)\ne₁ : X' ≅ (cones f (BinaryCofan.inl c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₁) (limits f (BinaryCofan.inl c))\ne₂ : Y' ≅ (cones f (BinaryCofan.inr c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₂) (limits f (BinaryCofan.inr c))\nhe₁ : BinaryCofan.inl c' = e₁.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inl c))\nhe₂ : BinaryCofan.inr c' = e₂.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inr c))\n⊢ IsColimit\n (BinaryCofan.mk (e₁.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (e₂.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inr c))))",
"tactic": "apply BinaryCofan.isColimitCompRightIso (BinaryCofan.mk _ _)"
},
{
"state_after": "case mpr.intro.h.h\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nH₁ : IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c)\nH₂ : IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)\ne₁ : X' ≅ (cones f (BinaryCofan.inl c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₁) (limits f (BinaryCofan.inl c))\ne₂ : Y' ≅ (cones f (BinaryCofan.inr c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₂) (limits f (BinaryCofan.inr c))\nhe₁ : BinaryCofan.inl c' = e₁.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inl c))\nhe₂ : BinaryCofan.inr c' = e₂.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inr c))\n⊢ IsColimit\n (BinaryCofan.mk ((cones f (BinaryCofan.inl c)).π.1 WalkingCospan.left)\n ((cones f (BinaryCofan.inr c)).π.1 WalkingCospan.left))",
"state_before": "case mpr.intro.h\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nH₁ : IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c)\nH₂ : IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)\ne₁ : X' ≅ (cones f (BinaryCofan.inl c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₁) (limits f (BinaryCofan.inl c))\ne₂ : Y' ≅ (cones f (BinaryCofan.inr c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₂) (limits f (BinaryCofan.inr c))\nhe₁ : BinaryCofan.inl c' = e₁.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inl c))\nhe₂ : BinaryCofan.inr c' = e₂.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inr c))\n⊢ IsColimit\n (BinaryCofan.mk (e₁.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inl c)))\n ((cones f (BinaryCofan.inr c)).π.1 WalkingCospan.left))",
"tactic": "apply BinaryCofan.isColimitCompLeftIso (BinaryCofan.mk _ _)"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.h.h\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nH₁ : IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c)\nH₂ : IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)\ne₁ : X' ≅ (cones f (BinaryCofan.inl c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₁) (limits f (BinaryCofan.inl c))\ne₂ : Y' ≅ (cones f (BinaryCofan.inr c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₂) (limits f (BinaryCofan.inr c))\nhe₁ : BinaryCofan.inl c' = e₁.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inl c))\nhe₂ : BinaryCofan.inr c' = e₂.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inr c))\n⊢ IsColimit\n (BinaryCofan.mk ((cones f (BinaryCofan.inl c)).π.1 WalkingCospan.left)\n ((cones f (BinaryCofan.inr c)).π.1 WalkingCospan.left))",
"tactic": "exact h₂ f"
},
{
"state_after": "no goals",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nH₁ : IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c)\nH₂ : IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)\ne₁ : X' ≅ (cones f (BinaryCofan.inl c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₁) (limits f (BinaryCofan.inl c))\ne₂ : Y' ≅ (cones f (BinaryCofan.inr c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₂) (limits f (BinaryCofan.inr c))\n⊢ BinaryCofan.inl c' = e₁.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inl c))",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX✝ Y✝ X Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\nh₁ :\n ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl (cofans X' Y') ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr (cofans X' Y') ≫ f →\n IsPullback (BinaryCofan.inl (cofans X' Y')) αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr (cofans X' Y')) αY f (BinaryCofan.inr c)\nh₂ :\n {Z : C} →\n (f : Z ⟶ c.pt) →\n IsColimit\n (BinaryCofan.mk (PullbackCone.fst (cones f (BinaryCofan.inl c)))\n (PullbackCone.fst (cones f (BinaryCofan.inr c))))\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nH₁ : IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c)\nH₂ : IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)\ne₁ : X' ≅ (cones f (BinaryCofan.inl c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₁) (limits f (BinaryCofan.inl c))\ne₂ : Y' ≅ (cones f (BinaryCofan.inr c)).pt :=\n IsLimit.conePointUniqueUpToIso (IsPullback.isLimit H₂) (limits f (BinaryCofan.inr c))\nhe₁ : BinaryCofan.inl c' = e₁.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inl c))\n⊢ BinaryCofan.inr c' = e₂.hom ≫ PullbackCone.fst (cones f (BinaryCofan.inr c))",
"tactic": "simp"
}
] |
[
230,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Topology/PathConnected.lean
|
Continuous.path_trans
|
[
{
"state_after": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.347696\nγ : Path x y\nf : Y → Path x y\ng : Y → Path y z\nhf : Continuous f\nhg : Continuous g\n⊢ Continuous fun t => trans (f t) (g t)",
"state_before": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.347696\nγ : Path x y\nf : Y → Path x y\ng : Y → Path y z\n⊢ Continuous f → Continuous g → Continuous fun t => trans (f t) (g t)",
"tactic": "intro hf hg"
},
{
"state_after": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.347696\nγ : Path x y\nf : Y → Path x y\ng : Y → Path y z\nhf : Continuous f\nhg : Continuous g\n⊢ Continuous ↿fun t => trans (f t) (g t)",
"state_before": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.347696\nγ : Path x y\nf : Y → Path x y\ng : Y → Path y z\nhf : Continuous f\nhg : Continuous g\n⊢ Continuous fun t => trans (f t) (g t)",
"tactic": "apply continuous_uncurry_iff.mp"
},
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.347696\nγ : Path x y\nf : Y → Path x y\ng : Y → Path y z\nhf : Continuous f\nhg : Continuous g\n⊢ Continuous ↿fun t => trans (f t) (g t)",
"tactic": "exact trans_continuous_family _ (continuous_uncurry_iff.mpr hf) _ (continuous_uncurry_iff.mpr hg)"
}
] |
[
533,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
529,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.tendsto_const_mul_pow_atTop
|
[] |
[
1057,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1055,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.fuzzy_congr_right
|
[] |
[
971,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
970,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
EMetric.tendstoLocallyUniformly_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.31390\ninst✝¹ : PseudoEMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nf : β → α\np : Filter ι\n⊢ TendstoLocallyUniformly F f p ↔\n ∀ (ε : ℝ≥0∞), ε > 0 → ∀ (x : β), ∃ t, t ∈ 𝓝 x ∧ ∀ᶠ (n : ι) in p, ∀ (y : β), y ∈ t → edist (f y) (F n y) < ε",
"tactic": "simp only [← tendstoLocallyUniformlyOn_univ, tendstoLocallyUniformlyOn_iff, mem_univ,\n forall_const, exists_prop, nhdsWithin_univ]"
}
] |
[
366,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/LinearAlgebra/Quotient.lean
|
Submodule.coe_quotEquivOfEqBot_symm
|
[] |
[
636,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
634,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.mul_inv_of_unit
|
[
{
"state_after": "case intro\nn : ℕ\nu : (ZMod n)ˣ\n⊢ ↑u * (↑u)⁻¹ = 1",
"state_before": "n : ℕ\na : ZMod n\nh : IsUnit a\n⊢ a * a⁻¹ = 1",
"tactic": "rcases h with ⟨u, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nn : ℕ\nu : (ZMod n)ˣ\n⊢ ↑u * (↑u)⁻¹ = 1",
"tactic": "rw [inv_coe_unit, u.mul_inv]"
}
] |
[
720,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
718,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.trans_source''
|
[] |
[
830,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
829,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.coe_ofEquivFun
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type ?u.687118\nR : Type u_2\nR₂ : Type ?u.687124\nK : Type ?u.687127\nM : Type u_3\nM' : Type ?u.687133\nM'' : Type ?u.687136\nV : Type u\nV' : Type ?u.687141\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M'\ninst✝¹ : Fintype ι\nb : Basis ι R M\ninst✝ : DecidableEq ι\ne : M ≃ₗ[R] ι → R\ni j : ι\n⊢ ↑e (↑(ofEquivFun e) i) j = ↑e (↑(LinearEquiv.symm e) (update 0 i 1)) j",
"tactic": "simp [Basis.ofEquivFun, ← Finsupp.single_eq_pi_single, Finsupp.single_eq_update]"
}
] |
[
971,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
966,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.isIso_colimit_cocone_parallelPair_of_self
|
[] |
[
1061,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1060,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.id_def
|
[] |
[
64,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/Algebra/Module/Zlattice.lean
|
Zspan.norm_fract_le
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_2\nι : Type u_3\nK : Type u_1\ninst✝⁵ : NormedLinearOrderedField K\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\nb : Basis ι K E\ninst✝² : FloorRing K\ninst✝¹ : Fintype ι\ninst✝ : HasSolidNorm K\nm : E\n⊢ ‖fract b m‖ ≤ ∑ i : ι, ‖↑b i‖",
"tactic": "classical\ncalc\n ‖fract b m‖ = ‖∑ i, b.repr (fract b m) i • b i‖ := by rw [b.sum_repr]\n _ = ‖∑ i, Int.fract (b.repr m i) • b i‖ := by simp_rw [repr_fract_apply]\n _ ≤ ∑ i, ‖Int.fract (b.repr m i) • b i‖ := (norm_sum_le _ _)\n _ = ∑ i, ‖Int.fract (b.repr m i)‖ * ‖b i‖ := by simp_rw [norm_smul]\n _ ≤ ∑ i, ‖b i‖ := Finset.sum_le_sum fun i _ => ?_\nsuffices ‖Int.fract ((b.repr m) i)‖ ≤ 1 by\n convert mul_le_mul_of_nonneg_right this (norm_nonneg _ : 0 ≤ ‖b i‖)\n exact (one_mul _).symm\nrw [(norm_one.symm : 1 = ‖(1 : K)‖)]\napply norm_le_norm_of_abs_le_abs\nrw [abs_one, Int.abs_fract]\nexact le_of_lt (Int.fract_lt_one _)"
},
{
"state_after": "E : Type u_2\nι : Type u_3\nK : Type u_1\ninst✝⁵ : NormedLinearOrderedField K\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\nb : Basis ι K E\ninst✝² : FloorRing K\ninst✝¹ : Fintype ι\ninst✝ : HasSolidNorm K\nm : E\ni : ι\nx✝ : i ∈ Finset.univ\n⊢ ‖Int.fract (↑(↑b.repr m) i)‖ * ‖↑b i‖ ≤ ‖↑b i‖",
"state_before": "E : Type u_2\nι : Type u_3\nK : Type u_1\ninst✝⁵ : NormedLinearOrderedField K\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\nb : Basis ι K E\ninst✝² : FloorRing K\ninst✝¹ : Fintype ι\ninst✝ : HasSolidNorm K\nm : E\n⊢ ‖fract b m‖ ≤ ∑ i : ι, ‖↑b i‖",
"tactic": "calc\n ‖fract b m‖ = ‖∑ i, b.repr (fract b m) i • b i‖ := by rw [b.sum_repr]\n _ = ‖∑ i, Int.fract (b.repr m i) • b i‖ := by simp_rw [repr_fract_apply]\n _ ≤ ∑ i, ‖Int.fract (b.repr m i) • b i‖ := (norm_sum_le _ _)\n _ = ∑ i, ‖Int.fract (b.repr m i)‖ * ‖b i‖ := by simp_rw [norm_smul]\n _ ≤ ∑ i, ‖b i‖ := Finset.sum_le_sum fun i _ => ?_"
},
{
"state_after": "E : Type u_2\nι : Type u_3\nK : Type u_1\ninst✝⁵ : NormedLinearOrderedField K\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\nb : Basis ι K E\ninst✝² : FloorRing K\ninst✝¹ : Fintype ι\ninst✝ : HasSolidNorm K\nm : E\ni : ι\nx✝ : i ∈ Finset.univ\n⊢ ‖Int.fract (↑(↑b.repr m) i)‖ ≤ 1",
"state_before": "E : Type u_2\nι : Type u_3\nK : Type u_1\ninst✝⁵ : NormedLinearOrderedField K\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\nb : Basis ι K E\ninst✝² : FloorRing K\ninst✝¹ : Fintype ι\ninst✝ : HasSolidNorm K\nm : E\ni : ι\nx✝ : i ∈ Finset.univ\n⊢ ‖Int.fract (↑(↑b.repr m) i)‖ * ‖↑b i‖ ≤ ‖↑b i‖",
"tactic": "suffices ‖Int.fract ((b.repr m) i)‖ ≤ 1 by\n convert mul_le_mul_of_nonneg_right this (norm_nonneg _ : 0 ≤ ‖b i‖)\n exact (one_mul _).symm"
},
{
"state_after": "E : Type u_2\nι : Type u_3\nK : Type u_1\ninst✝⁵ : NormedLinearOrderedField K\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\nb : Basis ι K E\ninst✝² : FloorRing K\ninst✝¹ : Fintype ι\ninst✝ : HasSolidNorm K\nm : E\ni : ι\nx✝ : i ∈ Finset.univ\n⊢ ‖Int.fract (↑(↑b.repr m) i)‖ ≤ ‖1‖",
"state_before": "E : Type u_2\nι : Type u_3\nK : Type u_1\ninst✝⁵ : NormedLinearOrderedField K\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\nb : Basis ι K E\ninst✝² : FloorRing K\ninst✝¹ : Fintype ι\ninst✝ : HasSolidNorm K\nm : E\ni : ι\nx✝ : i ∈ Finset.univ\n⊢ ‖Int.fract (↑(↑b.repr m) i)‖ ≤ 1",
"tactic": "rw [(norm_one.symm : 1 = ‖(1 : K)‖)]"
},
{
"state_after": "case h\nE : Type u_2\nι : Type u_3\nK : Type u_1\ninst✝⁵ : NormedLinearOrderedField K\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\nb : Basis ι K E\ninst✝² : FloorRing K\ninst✝¹ : Fintype ι\ninst✝ : HasSolidNorm K\nm : E\ni : ι\nx✝ : i ∈ Finset.univ\n⊢ abs (Int.fract (↑(↑b.repr m) i)) ≤ abs 1",
"state_before": "E : Type u_2\nι : Type u_3\nK : Type u_1\ninst✝⁵ : NormedLinearOrderedField K\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\nb : Basis ι K E\ninst✝² : FloorRing K\ninst✝¹ : Fintype ι\ninst✝ : HasSolidNorm K\nm : E\ni : ι\nx✝ : i ∈ Finset.univ\n⊢ ‖Int.fract (↑(↑b.repr m) i)‖ ≤ ‖1‖",
"tactic": "apply norm_le_norm_of_abs_le_abs"
},
{
"state_after": "case h\nE : Type u_2\nι : Type u_3\nK : Type u_1\ninst✝⁵ : NormedLinearOrderedField K\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\nb : Basis ι K E\ninst✝² : FloorRing K\ninst✝¹ : Fintype ι\ninst✝ : HasSolidNorm K\nm : E\ni : ι\nx✝ : i ∈ Finset.univ\n⊢ Int.fract (↑(↑b.repr m) i) ≤ 1",
"state_before": "case h\nE : Type u_2\nι : Type u_3\nK : Type u_1\ninst✝⁵ : NormedLinearOrderedField K\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\nb : Basis ι K E\ninst✝² : FloorRing K\ninst✝¹ : Fintype ι\ninst✝ : HasSolidNorm K\nm : E\ni : ι\nx✝ : i ∈ Finset.univ\n⊢ abs (Int.fract (↑(↑b.repr m) i)) ≤ abs 1",
"tactic": "rw [abs_one, Int.abs_fract]"
},
{
"state_after": "no goals",
"state_before": "case h\nE : Type u_2\nι : Type u_3\nK : Type u_1\ninst✝⁵ : NormedLinearOrderedField K\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\nb : Basis ι K E\ninst✝² : FloorRing K\ninst✝¹ : Fintype ι\ninst✝ : HasSolidNorm K\nm : E\ni : ι\nx✝ : i ∈ Finset.univ\n⊢ Int.fract (↑(↑b.repr m) i) ≤ 1",
"tactic": "exact le_of_lt (Int.fract_lt_one _)"
},
{
"state_after": "no goals",
"state_before": "E : Type u_2\nι : Type u_3\nK : Type u_1\ninst✝⁵ : NormedLinearOrderedField K\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\nb : Basis ι K E\ninst✝² : FloorRing K\ninst✝¹ : Fintype ι\ninst✝ : HasSolidNorm K\nm : E\n⊢ ‖fract b m‖ = ‖∑ i : ι, ↑(↑b.repr (fract b m)) i • ↑b i‖",
"tactic": "rw [b.sum_repr]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_2\nι : Type u_3\nK : Type u_1\ninst✝⁵ : NormedLinearOrderedField K\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\nb : Basis ι K E\ninst✝² : FloorRing K\ninst✝¹ : Fintype ι\ninst✝ : HasSolidNorm K\nm : E\n⊢ ‖∑ i : ι, ↑(↑b.repr (fract b m)) i • ↑b i‖ = ‖∑ i : ι, Int.fract (↑(↑b.repr m) i) • ↑b i‖",
"tactic": "simp_rw [repr_fract_apply]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_2\nι : Type u_3\nK : Type u_1\ninst✝⁵ : NormedLinearOrderedField K\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\nb : Basis ι K E\ninst✝² : FloorRing K\ninst✝¹ : Fintype ι\ninst✝ : HasSolidNorm K\nm : E\n⊢ ∑ i : ι, ‖Int.fract (↑(↑b.repr m) i) • ↑b i‖ = ∑ i : ι, ‖Int.fract (↑(↑b.repr m) i)‖ * ‖↑b i‖",
"tactic": "simp_rw [norm_smul]"
},
{
"state_after": "case h.e'_4\nE : Type u_2\nι : Type u_3\nK : Type u_1\ninst✝⁵ : NormedLinearOrderedField K\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\nb : Basis ι K E\ninst✝² : FloorRing K\ninst✝¹ : Fintype ι\ninst✝ : HasSolidNorm K\nm : E\ni : ι\nx✝ : i ∈ Finset.univ\nthis : ‖Int.fract (↑(↑b.repr m) i)‖ ≤ 1\n⊢ ‖↑b i‖ = 1 * ‖↑b i‖",
"state_before": "E : Type u_2\nι : Type u_3\nK : Type u_1\ninst✝⁵ : NormedLinearOrderedField K\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\nb : Basis ι K E\ninst✝² : FloorRing K\ninst✝¹ : Fintype ι\ninst✝ : HasSolidNorm K\nm : E\ni : ι\nx✝ : i ∈ Finset.univ\nthis : ‖Int.fract (↑(↑b.repr m) i)‖ ≤ 1\n⊢ ‖Int.fract (↑(↑b.repr m) i)‖ * ‖↑b i‖ ≤ ‖↑b i‖",
"tactic": "convert mul_le_mul_of_nonneg_right this (norm_nonneg _ : 0 ≤ ‖b i‖)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nE : Type u_2\nι : Type u_3\nK : Type u_1\ninst✝⁵ : NormedLinearOrderedField K\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\nb : Basis ι K E\ninst✝² : FloorRing K\ninst✝¹ : Fintype ι\ninst✝ : HasSolidNorm K\nm : E\ni : ι\nx✝ : i ∈ Finset.univ\nthis : ‖Int.fract (↑(↑b.repr m) i)‖ ≤ 1\n⊢ ‖↑b i‖ = 1 * ‖↑b i‖",
"tactic": "exact (one_mul _).symm"
}
] |
[
181,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.LiftRelO.imp
|
[] |
[
463,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
458,
1
] |
Mathlib/Data/Nat/WithBot.lean
|
Nat.WithBot.coe_nonneg
|
[
{
"state_after": "n : ℕ\n⊢ ↑0 ≤ ↑n",
"state_before": "n : ℕ\n⊢ 0 ≤ ↑n",
"tactic": "rw [← WithBot.coe_zero]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ ↑0 ≤ ↑n",
"tactic": "exact WithBot.coe_le_coe.mpr (Nat.zero_le n)"
}
] |
[
59,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/Order/Interval.lean
|
NonemptyInterval.fst_dual
|
[] |
[
113,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.tanh_neg
|
[
{
"state_after": "no goals",
"state_before": "x y : ℂ\n⊢ tanh (-x) = -tanh x",
"tactic": "simp [tanh, neg_div]"
}
] |
[
701,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
701,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.iUnion_top
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\n⊢ Prepartition.iUnion ⊤ = ↑I",
"tactic": "simp [Prepartition.iUnion]"
}
] |
[
234,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.tendsto_neg_const_mul_pow_atTop
|
[] |
[
1241,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1239,
1
] |
Mathlib/GroupTheory/Submonoid/Operations.lean
|
Submonoid.map_injective_of_injective
|
[] |
[
396,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
395,
1
] |
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
|
WithSeminorms.hasBasis_ball
|
[
{
"state_after": "𝕜 : Type u_1\n𝕜₂ : Type ?u.189437\n𝕝 : Type ?u.189440\n𝕝₂ : Type ?u.189443\nE : Type u_2\nF : Type ?u.189449\nG : Type ?u.189452\nι : Type u_3\nι' : Type ?u.189458\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\ninst✝ : TopologicalSpace E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\nx : E\nthis : TopologicalAddGroup E\n⊢ Filter.HasBasis (𝓝 x) (fun sr => 0 < sr.snd) fun sr => ball (Finset.sup sr.fst p) x sr.snd",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type ?u.189437\n𝕝 : Type ?u.189440\n𝕝₂ : Type ?u.189443\nE : Type u_2\nF : Type ?u.189449\nG : Type ?u.189452\nι : Type u_3\nι' : Type ?u.189458\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\ninst✝ : TopologicalSpace E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\nx : E\n⊢ Filter.HasBasis (𝓝 x) (fun sr => 0 < sr.snd) fun sr => ball (Finset.sup sr.fst p) x sr.snd",
"tactic": "haveI : TopologicalAddGroup E := hp.topologicalAddGroup"
},
{
"state_after": "𝕜 : Type u_1\n𝕜₂ : Type ?u.189437\n𝕝 : Type ?u.189440\n𝕝₂ : Type ?u.189443\nE : Type u_2\nF : Type ?u.189449\nG : Type ?u.189452\nι : Type u_3\nι' : Type ?u.189458\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\ninst✝ : TopologicalSpace E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\nx : E\nthis : TopologicalAddGroup E\n⊢ Filter.HasBasis (Filter.map ((fun x x_1 => x + x_1) x) (𝓝 0)) (fun sr => 0 < sr.snd) fun sr =>\n ball (Finset.sup sr.fst p) x sr.snd",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type ?u.189437\n𝕝 : Type ?u.189440\n𝕝₂ : Type ?u.189443\nE : Type u_2\nF : Type ?u.189449\nG : Type ?u.189452\nι : Type u_3\nι' : Type ?u.189458\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\ninst✝ : TopologicalSpace E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\nx : E\nthis : TopologicalAddGroup E\n⊢ Filter.HasBasis (𝓝 x) (fun sr => 0 < sr.snd) fun sr => ball (Finset.sup sr.fst p) x sr.snd",
"tactic": "rw [← map_add_left_nhds_zero]"
},
{
"state_after": "case h.e'_5.h\n𝕜 : Type u_1\n𝕜₂ : Type ?u.189437\n𝕝 : Type ?u.189440\n𝕝₂ : Type ?u.189443\nE : Type u_2\nF : Type ?u.189449\nG : Type ?u.189452\nι : Type u_3\nι' : Type ?u.189458\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\ninst✝ : TopologicalSpace E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\nx : E\nthis : TopologicalAddGroup E\nsr : Finset ι × ℝ\n⊢ ball (Finset.sup sr.fst p) x sr.snd = (fun x x_1 => x + x_1) x '' ball (Finset.sup sr.fst p) 0 sr.snd",
"state_before": "case h.e'_5\n𝕜 : Type u_1\n𝕜₂ : Type ?u.189437\n𝕝 : Type ?u.189440\n𝕝₂ : Type ?u.189443\nE : Type u_2\nF : Type ?u.189449\nG : Type ?u.189452\nι : Type u_3\nι' : Type ?u.189458\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\ninst✝ : TopologicalSpace E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\nx : E\nthis : TopologicalAddGroup E\n⊢ (fun sr => ball (Finset.sup sr.fst p) x sr.snd) = fun i =>\n (fun x x_1 => x + x_1) x '' ball (Finset.sup i.fst p) 0 i.snd",
"tactic": "ext sr : 1"
},
{
"state_after": "case h.e'_5.h\n𝕜 : Type u_1\n𝕜₂ : Type ?u.189437\n𝕝 : Type ?u.189440\n𝕝₂ : Type ?u.189443\nE : Type u_2\nF : Type ?u.189449\nG : Type ?u.189452\nι : Type u_3\nι' : Type ?u.189458\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\ninst✝ : TopologicalSpace E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\nx : E\nthis✝ : TopologicalAddGroup E\nsr : Finset ι × ℝ\nthis : ball (Finset.sup sr.fst p) (x +ᵥ 0) sr.snd = x +ᵥ ball (Finset.sup sr.fst p) 0 sr.snd\n⊢ ball (Finset.sup sr.fst p) x sr.snd = (fun x x_1 => x + x_1) x '' ball (Finset.sup sr.fst p) 0 sr.snd",
"state_before": "case h.e'_5.h\n𝕜 : Type u_1\n𝕜₂ : Type ?u.189437\n𝕝 : Type ?u.189440\n𝕝₂ : Type ?u.189443\nE : Type u_2\nF : Type ?u.189449\nG : Type ?u.189452\nι : Type u_3\nι' : Type ?u.189458\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\ninst✝ : TopologicalSpace E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\nx : E\nthis : TopologicalAddGroup E\nsr : Finset ι × ℝ\n⊢ ball (Finset.sup sr.fst p) x sr.snd = (fun x x_1 => x + x_1) x '' ball (Finset.sup sr.fst p) 0 sr.snd",
"tactic": "have : (sr.fst.sup p).ball (x +ᵥ (0 : E)) sr.snd = x +ᵥ (sr.fst.sup p).ball 0 sr.snd :=\n Eq.symm (Seminorm.vadd_ball (sr.fst.sup p))"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h\n𝕜 : Type u_1\n𝕜₂ : Type ?u.189437\n𝕝 : Type ?u.189440\n𝕝₂ : Type ?u.189443\nE : Type u_2\nF : Type ?u.189449\nG : Type ?u.189452\nι : Type u_3\nι' : Type ?u.189458\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\ninst✝ : TopologicalSpace E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\nx : E\nthis✝ : TopologicalAddGroup E\nsr : Finset ι × ℝ\nthis : ball (Finset.sup sr.fst p) (x +ᵥ 0) sr.snd = x +ᵥ ball (Finset.sup sr.fst p) 0 sr.snd\n⊢ ball (Finset.sup sr.fst p) x sr.snd = (fun x x_1 => x + x_1) x '' ball (Finset.sup sr.fst p) 0 sr.snd",
"tactic": "rwa [vadd_eq_add, add_zero] at this"
}
] |
[
319,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
309,
1
] |
Std/Data/RBMap/Lemmas.lean
|
Std.RBNode.Ordered.unique
|
[
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\ncmp : α → α → Ordering\nx y : α\ninst✝ : TransCmp cmp\ne : cmp x y = Ordering.eq\nht : Ordered cmp nil\nhx : x ∈ nil\nhy : y ∈ nil\n⊢ x = y",
"tactic": "cases hx"
},
{
"state_after": "case node\nα : Type u_1\ncmp : α → α → Ordering\nx y : α\ninst✝ : TransCmp cmp\ne : cmp x y = Ordering.eq\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → x ∈ l → y ∈ l → x = y\nihr : Ordered cmp r → x ∈ r → y ∈ r → x = y\nht : Ordered cmp (node c✝ l v✝ r)\nhx : x ∈ node c✝ l v✝ r\nhy : y ∈ node c✝ l v✝ r\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\n⊢ x = y",
"state_before": "case node\nα : Type u_1\ncmp : α → α → Ordering\nx y : α\ninst✝ : TransCmp cmp\ne : cmp x y = Ordering.eq\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → x ∈ l → y ∈ l → x = y\nihr : Ordered cmp r → x ∈ r → y ∈ r → x = y\nht : Ordered cmp (node c✝ l v✝ r)\nhx : x ∈ node c✝ l v✝ r\nhy : y ∈ node c✝ l v✝ r\n⊢ x = y",
"tactic": "let ⟨lx, xr, hl, hr⟩ := ht"
},
{
"state_after": "case node.inl.inl\nα : Type u_1\ncmp : α → α → Ordering\ny : α\ninst✝ : TransCmp cmp\nc✝ : RBColor\nl r : RBNode α\nhl : Ordered cmp l\nhr : Ordered cmp r\ne : cmp y y = Ordering.eq\nihl : Ordered cmp l → y ∈ l → y ∈ l → y = y\nihr : Ordered cmp r → y ∈ r → y ∈ r → y = y\nht : Ordered cmp (node c✝ l y r)\nlx : All (fun x => cmpLT cmp x y) l\nxr : All (fun x => cmpLT cmp y x) r\n⊢ y = y\n\ncase node.inl.inr.inl\nα : Type u_1\ncmp : α → α → Ordering\nx y : α\ninst✝ : TransCmp cmp\ne : cmp x y = Ordering.eq\nc✝ : RBColor\nl r : RBNode α\nihl : Ordered cmp l → x ∈ l → y ∈ l → x = y\nihr : Ordered cmp r → x ∈ r → y ∈ r → x = y\nhl : Ordered cmp l\nhr : Ordered cmp r\nht : Ordered cmp (node c✝ l x r)\nlx : All (fun x_1 => cmpLT cmp x_1 x) l\nxr : All (fun x_1 => cmpLT cmp x x_1) r\nhy : Any (fun x => y = x) l\n⊢ x = y\n\ncase node.inl.inr.inr\nα : Type u_1\ncmp : α → α → Ordering\nx y : α\ninst✝ : TransCmp cmp\ne : cmp x y = Ordering.eq\nc✝ : RBColor\nl r : RBNode α\nihl : Ordered cmp l → x ∈ l → y ∈ l → x = y\nihr : Ordered cmp r → x ∈ r → y ∈ r → x = y\nhl : Ordered cmp l\nhr : Ordered cmp r\nht : Ordered cmp (node c✝ l x r)\nlx : All (fun x_1 => cmpLT cmp x_1 x) l\nxr : All (fun x_1 => cmpLT cmp x x_1) r\nhy : Any (fun x => y = x) r\n⊢ x = y\n\ncase node.inr.inl.inl\nα : Type u_1\ncmp : α → α → Ordering\nx y : α\ninst✝ : TransCmp cmp\ne : cmp x y = Ordering.eq\nc✝ : RBColor\nl r : RBNode α\nihl : Ordered cmp l → x ∈ l → y ∈ l → x = y\nihr : Ordered cmp r → x ∈ r → y ∈ r → x = y\nhl : Ordered cmp l\nhr : Ordered cmp r\nhx : Any (fun x_1 => x = x_1) l\nht : Ordered cmp (node c✝ l y r)\nlx : All (fun x => cmpLT cmp x y) l\nxr : All (fun x => cmpLT cmp y x) r\n⊢ x = y\n\ncase node.inr.inl.inr.inl\nα : Type u_1\ncmp : α → α → Ordering\nx y : α\ninst✝ : TransCmp cmp\ne : cmp x y = Ordering.eq\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → x ∈ l → y ∈ l → x = y\nihr : Ordered cmp r → x ∈ r → y ∈ r → x = y\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nhx : Any (fun x_1 => x = x_1) l\nhy : Any (fun x => y = x) l\n⊢ x = y\n\ncase node.inr.inl.inr.inr\nα : Type u_1\ncmp : α → α → Ordering\nx y : α\ninst✝ : TransCmp cmp\ne : cmp x y = Ordering.eq\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → x ∈ l → y ∈ l → x = y\nihr : Ordered cmp r → x ∈ r → y ∈ r → x = y\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nhx : Any (fun x_1 => x = x_1) l\nhy : Any (fun x => y = x) r\n⊢ x = y\n\ncase node.inr.inr.inl\nα : Type u_1\ncmp : α → α → Ordering\nx y : α\ninst✝ : TransCmp cmp\ne : cmp x y = Ordering.eq\nc✝ : RBColor\nl r : RBNode α\nihl : Ordered cmp l → x ∈ l → y ∈ l → x = y\nihr : Ordered cmp r → x ∈ r → y ∈ r → x = y\nhl : Ordered cmp l\nhr : Ordered cmp r\nhx : Any (fun x_1 => x = x_1) r\nht : Ordered cmp (node c✝ l y r)\nlx : All (fun x => cmpLT cmp x y) l\nxr : All (fun x => cmpLT cmp y x) r\n⊢ x = y\n\ncase node.inr.inr.inr.inl\nα : Type u_1\ncmp : α → α → Ordering\nx y : α\ninst✝ : TransCmp cmp\ne : cmp x y = Ordering.eq\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → x ∈ l → y ∈ l → x = y\nihr : Ordered cmp r → x ∈ r → y ∈ r → x = y\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nhx : Any (fun x_1 => x = x_1) r\nhy : Any (fun x => y = x) l\n⊢ x = y\n\ncase node.inr.inr.inr.inr\nα : Type u_1\ncmp : α → α → Ordering\nx y : α\ninst✝ : TransCmp cmp\ne : cmp x y = Ordering.eq\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → x ∈ l → y ∈ l → x = y\nihr : Ordered cmp r → x ∈ r → y ∈ r → x = y\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nhx : Any (fun x_1 => x = x_1) r\nhy : Any (fun x => y = x) r\n⊢ x = y",
"state_before": "case node\nα : Type u_1\ncmp : α → α → Ordering\nx y : α\ninst✝ : TransCmp cmp\ne : cmp x y = Ordering.eq\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → x ∈ l → y ∈ l → x = y\nihr : Ordered cmp r → x ∈ r → y ∈ r → x = y\nht : Ordered cmp (node c✝ l v✝ r)\nhx : x ∈ node c✝ l v✝ r\nhy : y ∈ node c✝ l v✝ r\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\n⊢ x = y",
"tactic": "rcases hx, hy with ⟨rfl | hx | hx, rfl | hy | hy⟩"
},
{
"state_after": "no goals",
"state_before": "case node.inl.inl\nα : Type u_1\ncmp : α → α → Ordering\ny : α\ninst✝ : TransCmp cmp\nc✝ : RBColor\nl r : RBNode α\nhl : Ordered cmp l\nhr : Ordered cmp r\ne : cmp y y = Ordering.eq\nihl : Ordered cmp l → y ∈ l → y ∈ l → y = y\nihr : Ordered cmp r → y ∈ r → y ∈ r → y = y\nht : Ordered cmp (node c✝ l y r)\nlx : All (fun x => cmpLT cmp x y) l\nxr : All (fun x => cmpLT cmp y x) r\n⊢ y = y",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case node.inl.inr.inl\nα : Type u_1\ncmp : α → α → Ordering\nx y : α\ninst✝ : TransCmp cmp\ne : cmp x y = Ordering.eq\nc✝ : RBColor\nl r : RBNode α\nihl : Ordered cmp l → x ∈ l → y ∈ l → x = y\nihr : Ordered cmp r → x ∈ r → y ∈ r → x = y\nhl : Ordered cmp l\nhr : Ordered cmp r\nht : Ordered cmp (node c✝ l x r)\nlx : All (fun x_1 => cmpLT cmp x_1 x) l\nxr : All (fun x_1 => cmpLT cmp x x_1) r\nhy : Any (fun x => y = x) l\n⊢ x = y",
"tactic": "cases e.symm.trans <| OrientedCmp.cmp_eq_gt.2 (All_def.1 lx _ hy).1"
},
{
"state_after": "no goals",
"state_before": "case node.inl.inr.inr\nα : Type u_1\ncmp : α → α → Ordering\nx y : α\ninst✝ : TransCmp cmp\ne : cmp x y = Ordering.eq\nc✝ : RBColor\nl r : RBNode α\nihl : Ordered cmp l → x ∈ l → y ∈ l → x = y\nihr : Ordered cmp r → x ∈ r → y ∈ r → x = y\nhl : Ordered cmp l\nhr : Ordered cmp r\nht : Ordered cmp (node c✝ l x r)\nlx : All (fun x_1 => cmpLT cmp x_1 x) l\nxr : All (fun x_1 => cmpLT cmp x x_1) r\nhy : Any (fun x => y = x) r\n⊢ x = y",
"tactic": "cases e.symm.trans (All_def.1 xr _ hy).1"
},
{
"state_after": "no goals",
"state_before": "case node.inr.inl.inl\nα : Type u_1\ncmp : α → α → Ordering\nx y : α\ninst✝ : TransCmp cmp\ne : cmp x y = Ordering.eq\nc✝ : RBColor\nl r : RBNode α\nihl : Ordered cmp l → x ∈ l → y ∈ l → x = y\nihr : Ordered cmp r → x ∈ r → y ∈ r → x = y\nhl : Ordered cmp l\nhr : Ordered cmp r\nhx : Any (fun x_1 => x = x_1) l\nht : Ordered cmp (node c✝ l y r)\nlx : All (fun x => cmpLT cmp x y) l\nxr : All (fun x => cmpLT cmp y x) r\n⊢ x = y",
"tactic": "cases e.symm.trans (All_def.1 lx _ hx).1"
},
{
"state_after": "no goals",
"state_before": "case node.inr.inl.inr.inl\nα : Type u_1\ncmp : α → α → Ordering\nx y : α\ninst✝ : TransCmp cmp\ne : cmp x y = Ordering.eq\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → x ∈ l → y ∈ l → x = y\nihr : Ordered cmp r → x ∈ r → y ∈ r → x = y\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nhx : Any (fun x_1 => x = x_1) l\nhy : Any (fun x => y = x) l\n⊢ x = y",
"tactic": "exact ihl hl hx hy"
},
{
"state_after": "no goals",
"state_before": "case node.inr.inl.inr.inr\nα : Type u_1\ncmp : α → α → Ordering\nx y : α\ninst✝ : TransCmp cmp\ne : cmp x y = Ordering.eq\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → x ∈ l → y ∈ l → x = y\nihr : Ordered cmp r → x ∈ r → y ∈ r → x = y\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nhx : Any (fun x_1 => x = x_1) l\nhy : Any (fun x => y = x) r\n⊢ x = y",
"tactic": "cases e.symm.trans ((All_def.1 lx _ hx).trans (All_def.1 xr _ hy)).1"
},
{
"state_after": "no goals",
"state_before": "case node.inr.inr.inl\nα : Type u_1\ncmp : α → α → Ordering\nx y : α\ninst✝ : TransCmp cmp\ne : cmp x y = Ordering.eq\nc✝ : RBColor\nl r : RBNode α\nihl : Ordered cmp l → x ∈ l → y ∈ l → x = y\nihr : Ordered cmp r → x ∈ r → y ∈ r → x = y\nhl : Ordered cmp l\nhr : Ordered cmp r\nhx : Any (fun x_1 => x = x_1) r\nht : Ordered cmp (node c✝ l y r)\nlx : All (fun x => cmpLT cmp x y) l\nxr : All (fun x => cmpLT cmp y x) r\n⊢ x = y",
"tactic": "cases e.symm.trans <| OrientedCmp.cmp_eq_gt.2 (All_def.1 xr _ hx).1"
},
{
"state_after": "no goals",
"state_before": "case node.inr.inr.inr.inl\nα : Type u_1\ncmp : α → α → Ordering\nx y : α\ninst✝ : TransCmp cmp\ne : cmp x y = Ordering.eq\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → x ∈ l → y ∈ l → x = y\nihr : Ordered cmp r → x ∈ r → y ∈ r → x = y\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nhx : Any (fun x_1 => x = x_1) r\nhy : Any (fun x => y = x) l\n⊢ x = y",
"tactic": "cases e.symm.trans <| OrientedCmp.cmp_eq_gt.2\n ((All_def.1 lx _ hy).trans (All_def.1 xr _ hx)).1"
},
{
"state_after": "no goals",
"state_before": "case node.inr.inr.inr.inr\nα : Type u_1\ncmp : α → α → Ordering\nx y : α\ninst✝ : TransCmp cmp\ne : cmp x y = Ordering.eq\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → x ∈ l → y ∈ l → x = y\nihr : Ordered cmp r → x ∈ r → y ∈ r → x = y\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nhx : Any (fun x_1 => x = x_1) r\nhy : Any (fun x => y = x) r\n⊢ x = y",
"tactic": "exact ihr hr hx hy"
}
] |
[
206,
25
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
190,
1
] |
Mathlib/LinearAlgebra/Matrix/BilinearForm.lean
|
Matrix.nondegenerate_toBilin'_iff
|
[] |
[
560,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
558,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Equiv.prodCongr_refl_right
|
[
{
"state_after": "case H.mk\nα₁ : Type u_3\nβ₁ : Type u_1\nβ₂ : Type u_2\ne✝ : α₁ → β₁ ≃ β₂\ne : β₁ ≃ β₂\na : β₁\nb : α₁\n⊢ ↑(prodCongr e (Equiv.refl α₁)) (a, b) = ↑(prodCongrLeft fun x => e) (a, b)",
"state_before": "α₁ : Type u_3\nβ₁ : Type u_1\nβ₂ : Type u_2\ne✝ : α₁ → β₁ ≃ β₂\ne : β₁ ≃ β₂\n⊢ prodCongr e (Equiv.refl α₁) = prodCongrLeft fun x => e",
"tactic": "ext ⟨a, b⟩ : 1"
},
{
"state_after": "no goals",
"state_before": "case H.mk\nα₁ : Type u_3\nβ₁ : Type u_1\nβ₂ : Type u_2\ne✝ : α₁ → β₁ ≃ β₂\ne : β₁ ≃ β₂\na : β₁\nb : α₁\n⊢ ↑(prodCongr e (Equiv.refl α₁)) (a, b) = ↑(prodCongrLeft fun x => e) (a, b)",
"tactic": "simp"
}
] |
[
715,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
712,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
exists_mem_nhds_ball_subset_of_mem_nhds
|
[] |
[
807,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
804,
1
] |
Mathlib/Order/Hom/Lattice.lean
|
BoundedLatticeHom.cancel_left
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.205169\nι : Type ?u.205172\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.205184\ninst✝⁷ : Lattice α\ninst✝⁶ : Lattice β\ninst✝⁵ : Lattice γ\ninst✝⁴ : Lattice δ\ninst✝³ : BoundedOrder α\ninst✝² : BoundedOrder β\ninst✝¹ : BoundedOrder γ\ninst✝ : BoundedOrder δ\ng : BoundedLatticeHom β γ\nf₁ f₂ : BoundedLatticeHom α β\nhg : Injective ↑g\nh : comp g f₁ = comp g f₂\na : α\n⊢ ↑g (↑f₁ a) = ↑g (↑f₂ a)",
"tactic": "rw [← comp_apply, h, comp_apply]"
}
] |
[
1356,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1354,
1
] |
Mathlib/Data/List/Lattice.lean
|
List.mem_of_mem_inter_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl l₁ l₂ : List α\np : α → Prop\na : α\ninst✝ : DecidableEq α\nh : a ∈ l₁ ∩ l₂\n⊢ a ∈ l₂",
"tactic": "simpa using of_mem_filter h"
}
] |
[
159,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Std/Data/List/Lemmas.lean
|
List.not_mem_append
|
[] |
[
141,
40
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
140,
1
] |
Mathlib/Topology/Algebra/Semigroup.lean
|
exists_idempotent_of_compact_t2_of_continuous_mul_left
|
[
{
"state_after": "M : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\n⊢ ∃ m, m * m = m",
"state_before": "M : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\n⊢ ∃ m, m * m = m",
"tactic": "let S : Set (Set M) :=\n { N | IsClosed N ∧ N.Nonempty ∧ ∀ (m) (_ : m ∈ N) (m') (_ : m' ∈ N), m * m' ∈ N }"
},
{
"state_after": "M : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\n⊢ ∃ N, N ∈ S ∧ ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\n\ncase intro.intro.intro.intro.intro\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\n⊢ ∃ m, m * m = m",
"state_before": "M : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\n⊢ ∃ m, m * m = m",
"tactic": "obtain ⟨N, ⟨N_closed, ⟨m, hm⟩, N_mul⟩, N_minimal⟩ : ∃ N ∈ S, ∀ N' ∈ S, N' ⊆ N → N' = N"
},
{
"state_after": "case intro.intro.intro.intro.intro\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\n⊢ ∃ m, m * m = m\n\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\n⊢ ∃ N, N ∈ S ∧ ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N",
"state_before": "M : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\n⊢ ∃ N, N ∈ S ∧ ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\n\ncase intro.intro.intro.intro.intro\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\n⊢ ∃ m, m * m = m",
"tactic": "rotate_left"
},
{
"state_after": "M : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nc : Set (Set M)\nhcs : c ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) c\n⊢ ∃ lb, lb ∈ S ∧ ∀ (s : Set M), s ∈ c → lb ⊆ s",
"state_before": "M : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\n⊢ ∃ N, N ∈ S ∧ ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N",
"tactic": "refine' zorn_superset _ fun c hcs hc => _"
},
{
"state_after": "case refine'_1\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nc : Set (Set M)\nhcs : c ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) c\n⊢ Set.Nonempty (⋂₀ c)\n\ncase refine'_2\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nc : Set (Set M)\nhcs : c ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) c\nm : M\nhm : m ∈ ⋂₀ c\nm' : M\nhm' : m' ∈ ⋂₀ c\n⊢ m * m' ∈ ⋂₀ c",
"state_before": "M : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nc : Set (Set M)\nhcs : c ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) c\n⊢ ∃ lb, lb ∈ S ∧ ∀ (s : Set M), s ∈ c → lb ⊆ s",
"tactic": "refine'\n ⟨⋂₀ c, ⟨isClosed_sInter fun t ht => (hcs ht).1, _, fun m hm m' hm' => _⟩, fun s hs =>\n Set.sInter_subset_of_mem hs⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\n⊢ m * m = m",
"state_before": "case intro.intro.intro.intro.intro\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\n⊢ ∃ m, m * m = m",
"tactic": "use m"
},
{
"state_after": "case intro.intro.intro.intro.intro\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N ∩ {m' | m' * m = m}\nscaling_eq_self : (fun x => x * m) '' N = N\nabsorbing_eq_self : N ∩ {m' | m' * m = m} = N\n⊢ m * m = m",
"state_before": "case intro.intro.intro.intro.intro\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\nscaling_eq_self : (fun x => x * m) '' N = N\nabsorbing_eq_self : N ∩ {m' | m' * m = m} = N\n⊢ m * m = m",
"tactic": "rw [← absorbing_eq_self] at hm"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N ∩ {m' | m' * m = m}\nscaling_eq_self : (fun x => x * m) '' N = N\nabsorbing_eq_self : N ∩ {m' | m' * m = m} = N\n⊢ m * m = m",
"tactic": "exact hm.2"
},
{
"state_after": "case a\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\n⊢ (fun x => x * m) '' N ∈ S\n\ncase a\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\n⊢ (fun x => x * m) '' N ⊆ N",
"state_before": "M : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\n⊢ (fun x => x * m) '' N = N",
"tactic": "apply N_minimal"
},
{
"state_after": "case a\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\n⊢ ∀ (m_1 : M), m_1 ∈ (fun x => x * m) '' N → ∀ (m' : M), m' ∈ (fun x => x * m) '' N → m_1 * m' ∈ (fun x => x * m) '' N",
"state_before": "case a\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\n⊢ (fun x => x * m) '' N ∈ S",
"tactic": "refine' ⟨(continuous_mul_left m).isClosedMap _ N_closed, ⟨_, ⟨m, hm, rfl⟩⟩, _⟩"
},
{
"state_after": "case a.intro.intro.intro.intro\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\nm'' : M\nhm'' : m'' ∈ N\nm' : M\nhm' : m' ∈ N\n⊢ (fun x => x * m) m'' * (fun x => x * m) m' ∈ (fun x => x * m) '' N",
"state_before": "case a\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\n⊢ ∀ (m_1 : M), m_1 ∈ (fun x => x * m) '' N → ∀ (m' : M), m' ∈ (fun x => x * m) '' N → m_1 * m' ∈ (fun x => x * m) '' N",
"tactic": "rintro _ ⟨m'', hm'', rfl⟩ _ ⟨m', hm', rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case a.intro.intro.intro.intro\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\nm'' : M\nhm'' : m'' ∈ N\nm' : M\nhm' : m' ∈ N\n⊢ (fun x => x * m) m'' * (fun x => x * m) m' ∈ (fun x => x * m) '' N",
"tactic": "refine' ⟨m'' * m * m', N_mul _ (N_mul _ hm'' _ hm) _ hm', mul_assoc _ _ _⟩"
},
{
"state_after": "case a.intro.intro\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\nm' : M\nhm' : m' ∈ N\n⊢ (fun x => x * m) m' ∈ N",
"state_before": "case a\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\n⊢ (fun x => x * m) '' N ⊆ N",
"tactic": "rintro _ ⟨m', hm', rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case a.intro.intro\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\nm' : M\nhm' : m' ∈ N\n⊢ (fun x => x * m) m' ∈ N",
"tactic": "exact N_mul _ hm' _ hm"
},
{
"state_after": "case a\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\nscaling_eq_self : (fun x => x * m) '' N = N\n⊢ N ∩ {m' | m' * m = m} ∈ S\n\ncase a\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\nscaling_eq_self : (fun x => x * m) '' N = N\n⊢ N ∩ {m' | m' * m = m} ⊆ N",
"state_before": "M : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\nscaling_eq_self : (fun x => x * m) '' N = N\n⊢ N ∩ {m' | m' * m = m} = N",
"tactic": "apply N_minimal"
},
{
"state_after": "no goals",
"state_before": "case a\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\nscaling_eq_self : (fun x => x * m) '' N = N\n⊢ N ∩ {m' | m' * m = m} ⊆ N",
"tactic": "apply Set.inter_subset_left"
},
{
"state_after": "case a.refine'_1\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\nscaling_eq_self : (fun x => x * m) '' N = N\n⊢ Set.Nonempty (N ∩ {m' | m' * m = m})\n\ncase a.refine'_2\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\nscaling_eq_self : (fun x => x * m) '' N = N\n⊢ ∀ (m_1 : M), m_1 ∈ N ∩ {m' | m' * m = m} → ∀ (m' : M), m' ∈ N ∩ {m' | m' * m = m} → m_1 * m' ∈ N ∩ {m' | m' * m = m}",
"state_before": "case a\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\nscaling_eq_self : (fun x => x * m) '' N = N\n⊢ N ∩ {m' | m' * m = m} ∈ S",
"tactic": "refine' ⟨N_closed.inter ((T1Space.t1 m).preimage (continuous_mul_left m)), _, _⟩"
},
{
"state_after": "no goals",
"state_before": "case a.refine'_1\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\nscaling_eq_self : (fun x => x * m) '' N = N\n⊢ Set.Nonempty (N ∩ {m' | m' * m = m})",
"tactic": "rwa [← scaling_eq_self] at hm"
},
{
"state_after": "case a.refine'_2.intro.intro\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\nscaling_eq_self : (fun x => x * m) '' N = N\nm'' : M\nmem'' : m'' ∈ N\neq'' : m'' * m = m\nm' : M\nmem' : m' ∈ N\neq' : m' * m = m\n⊢ m'' * m' ∈ N ∩ {m' | m' * m = m}",
"state_before": "case a.refine'_2\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\nscaling_eq_self : (fun x => x * m) '' N = N\n⊢ ∀ (m_1 : M), m_1 ∈ N ∩ {m' | m' * m = m} → ∀ (m' : M), m' ∈ N ∩ {m' | m' * m = m} → m_1 * m' ∈ N ∩ {m' | m' * m = m}",
"tactic": "rintro m'' ⟨mem'', eq'' : _ = m⟩ m' ⟨mem', eq' : _ = m⟩"
},
{
"state_after": "case a.refine'_2.intro.intro\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\nscaling_eq_self : (fun x => x * m) '' N = N\nm'' : M\nmem'' : m'' ∈ N\neq'' : m'' * m = m\nm' : M\nmem' : m' ∈ N\neq' : m' * m = m\n⊢ m'' * m' ∈ {m' | m' * m = m}",
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"tactic": "refine' ⟨N_mul _ mem'' _ mem', _⟩"
},
{
"state_after": "no goals",
"state_before": "case a.refine'_2.intro.intro\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nN : Set M\nN_minimal : ∀ (N' : Set M), N' ∈ S → N' ⊆ N → N' = N\nN_closed : IsClosed N\nN_mul : ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N\nm : M\nhm : m ∈ N\nscaling_eq_self : (fun x => x * m) '' N = N\nm'' : M\nmem'' : m'' ∈ N\neq'' : m'' * m = m\nm' : M\nmem' : m' ∈ N\neq' : m' * m = m\n⊢ m'' * m' ∈ {m' | m' * m = m}",
"tactic": "rw [Set.mem_setOf_eq, mul_assoc, eq', eq'']"
},
{
"state_after": "case refine'_1.inl\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nhcs : ∅ ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) ∅\n⊢ Set.Nonempty (⋂₀ ∅)\n\ncase refine'_1.inr\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nc : Set (Set M)\nhcs : c ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) c\nhcnemp : Set.Nonempty c\n⊢ Set.Nonempty (⋂₀ c)",
"state_before": "case refine'_1\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nc : Set (Set M)\nhcs : c ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) c\n⊢ Set.Nonempty (⋂₀ c)",
"tactic": "obtain rfl | hcnemp := c.eq_empty_or_nonempty"
},
{
"state_after": "case h.e'_2\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nc : Set (Set M)\nhcs : c ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) c\nhcnemp : Set.Nonempty c\n⊢ ⋂₀ c = ⋂ (i : ↑c), ↑i\n\ncase refine'_1.inr.convert_1\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nc : Set (Set M)\nhcs : c ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) c\nhcnemp : Set.Nonempty c\n⊢ Directed (fun x x_1 => x ⊇ x_1) Subtype.val\n\ncase refine'_1.inr.convert_2\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nc : Set (Set M)\nhcs : c ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) c\nhcnemp : Set.Nonempty c\n⊢ ∀ (i : ↑c), Set.Nonempty ↑i\n\ncase refine'_1.inr.convert_3\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nc : Set (Set M)\nhcs : c ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) c\nhcnemp : Set.Nonempty c\n⊢ ∀ (i : ↑c), IsCompact ↑i\n\ncase refine'_1.inr.convert_4\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nc : Set (Set M)\nhcs : c ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) c\nhcnemp : Set.Nonempty c\n⊢ ∀ (i : ↑c), IsClosed ↑i",
"state_before": "case refine'_1.inr\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nc : Set (Set M)\nhcs : c ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) c\nhcnemp : Set.Nonempty c\n⊢ Set.Nonempty (⋂₀ c)",
"tactic": "convert\n @IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed _ _ _ hcnemp.coe_sort\n ((↑) : c → Set M) ?_ ?_ ?_ ?_"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.inr.convert_2\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nc : Set (Set M)\nhcs : c ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) c\nhcnemp : Set.Nonempty c\n⊢ ∀ (i : ↑c), Set.Nonempty ↑i\n\ncase refine'_1.inr.convert_3\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nc : Set (Set M)\nhcs : c ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) c\nhcnemp : Set.Nonempty c\n⊢ ∀ (i : ↑c), IsCompact ↑i\n\ncase refine'_1.inr.convert_4\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nc : Set (Set M)\nhcs : c ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) c\nhcnemp : Set.Nonempty c\n⊢ ∀ (i : ↑c), IsClosed ↑i",
"tactic": "exacts [fun i => (hcs i.prop).2.1, fun i => (hcs i.prop).1.isCompact, fun i => (hcs i.prop).1]"
},
{
"state_after": "case refine'_1.inl\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nhcs : ∅ ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) ∅\n⊢ Set.Nonempty Set.univ",
"state_before": "case refine'_1.inl\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nhcs : ∅ ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) ∅\n⊢ Set.Nonempty (⋂₀ ∅)",
"tactic": "rw [Set.sInter_empty]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.inl\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nhcs : ∅ ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) ∅\n⊢ Set.Nonempty Set.univ",
"tactic": "apply Set.univ_nonempty"
},
{
"state_after": "case h.e'_2\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nc : Set (Set M)\nhcs : c ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) c\nhcnemp : Set.Nonempty c\n⊢ ⋂₀ c = ⋂ (i : ↑c), ↑i",
"state_before": "case h.e'_2\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nc : Set (Set M)\nhcs : c ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) c\nhcnemp : Set.Nonempty c\n⊢ ⋂₀ c = ⋂ (i : ↑c), ↑i",
"tactic": "simp only [Subtype.range_coe_subtype, Set.setOf_mem_eq]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nc : Set (Set M)\nhcs : c ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) c\nhcnemp : Set.Nonempty c\n⊢ ⋂₀ c = ⋂ (i : ↑c), ↑i",
"tactic": "exact Set.sInter_eq_iInter"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.inr.convert_1\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nc : Set (Set M)\nhcs : c ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) c\nhcnemp : Set.Nonempty c\n⊢ Directed (fun x x_1 => x ⊇ x_1) Subtype.val",
"tactic": "refine' DirectedOn.directed_val (IsChain.directedOn hc.symm)"
},
{
"state_after": "case refine'_2\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nc : Set (Set M)\nhcs : c ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) c\nm : M\nhm : m ∈ ⋂₀ c\nm' : M\nhm' : m' ∈ ⋂₀ c\n⊢ ∀ (t : Set M), t ∈ c → m * m' ∈ t",
"state_before": "case refine'_2\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nc : Set (Set M)\nhcs : c ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) c\nm : M\nhm : m ∈ ⋂₀ c\nm' : M\nhm' : m' ∈ ⋂₀ c\n⊢ m * m' ∈ ⋂₀ c",
"tactic": "rw [Set.mem_sInter]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_mul_left : ∀ (r : M), Continuous fun x => x * r\nS : Set (Set M) := {N | IsClosed N ∧ Set.Nonempty N ∧ ∀ (m : M), m ∈ N → ∀ (m' : M), m' ∈ N → m * m' ∈ N}\nc : Set (Set M)\nhcs : c ⊆ S\nhc : IsChain (fun x x_1 => x ⊆ x_1) c\nm : M\nhm : m ∈ ⋂₀ c\nm' : M\nhm' : m' ∈ ⋂₀ c\n⊢ ∀ (t : Set M), t ∈ c → m * m' ∈ t",
"tactic": "exact fun t ht => (hcs ht).2.2 m (Set.mem_sInter.mp hm t ht) m' (Set.mem_sInter.mp hm' t ht)"
}
] |
[
78,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
31,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.HasBasis.comap
|
[
{
"state_after": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.65846\nι : Sort u_2\nι' : Sort ?u.65852\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt✝ : Set α\ni : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nf : β → α\nhl : HasBasis l p s\nt : Set β\n⊢ (∃ i, p i ∧ s i ⊆ {y | ∀ ⦃x : β⦄, f x = y → x ∈ t}) ↔ ∃ i, p i ∧ f ⁻¹' s i ⊆ t",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.65846\nι : Sort u_2\nι' : Sort ?u.65852\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt✝ : Set α\ni : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nf : β → α\nhl : HasBasis l p s\nt : Set β\n⊢ t ∈ Filter.comap f l ↔ ∃ i, p i ∧ f ⁻¹' s i ⊆ t",
"tactic": "simp only [mem_comap', hl.mem_iff]"
},
{
"state_after": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.65846\nι : Sort u_2\nι' : Sort ?u.65852\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt✝ : Set α\ni✝ : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nf : β → α\nhl : HasBasis l p s\nt : Set β\ni : ι\n⊢ s i ⊆ {y | ∀ ⦃x : β⦄, f x = y → x ∈ t} ↔ f ⁻¹' s i ⊆ t",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.65846\nι : Sort u_2\nι' : Sort ?u.65852\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt✝ : Set α\ni : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nf : β → α\nhl : HasBasis l p s\nt : Set β\n⊢ (∃ i, p i ∧ s i ⊆ {y | ∀ ⦃x : β⦄, f x = y → x ∈ t}) ↔ ∃ i, p i ∧ f ⁻¹' s i ⊆ t",
"tactic": "refine exists_congr (fun i => Iff.rfl.and ?_)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.65846\nι : Sort u_2\nι' : Sort ?u.65852\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt✝ : Set α\ni✝ : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nf : β → α\nhl : HasBasis l p s\nt : Set β\ni : ι\n⊢ s i ⊆ {y | ∀ ⦃x : β⦄, f x = y → x ∈ t} ↔ f ⁻¹' s i ⊆ t",
"tactic": "exact ⟨fun h x hx => h hx rfl, fun h y hy x hx => h <| by rwa [mem_preimage, hx]⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.65846\nι : Sort u_2\nι' : Sort ?u.65852\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt✝ : Set α\ni✝ : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nf : β → α\nhl : HasBasis l p s\nt : Set β\ni : ι\nh : f ⁻¹' s i ⊆ t\ny : α\nhy : y ∈ s i\nx : β\nhx : f x = y\n⊢ x ∈ f ⁻¹' s i",
"tactic": "rwa [mem_preimage, hx]"
}
] |
[
800,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
795,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.mem_singleton
|
[] |
[
1273,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1272,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.exists_ne_one_of_prod_ne_one
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.430162\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nh : ∏ x in s, f x ≠ 1\n⊢ ∃ a, a ∈ s ∧ f a ≠ 1",
"tactic": "classical\n rw [← prod_filter_ne_one] at h\n rcases nonempty_of_prod_ne_one h with ⟨x, hx⟩\n exact ⟨x, (mem_filter.1 hx).1, by simpa using (mem_filter.1 hx).2⟩"
},
{
"state_after": "ι : Type ?u.430162\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nh✝ : ∏ x in s, f x ≠ 1\nh : ∏ x in filter (fun x => f x ≠ 1) s, f x ≠ 1\n⊢ ∃ a, a ∈ s ∧ f a ≠ 1",
"state_before": "ι : Type ?u.430162\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nh : ∏ x in s, f x ≠ 1\n⊢ ∃ a, a ∈ s ∧ f a ≠ 1",
"tactic": "rw [← prod_filter_ne_one] at h"
},
{
"state_after": "case intro\nι : Type ?u.430162\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nh✝ : ∏ x in s, f x ≠ 1\nh : ∏ x in filter (fun x => f x ≠ 1) s, f x ≠ 1\nx : α\nhx : x ∈ filter (fun x => f x ≠ 1) s\n⊢ ∃ a, a ∈ s ∧ f a ≠ 1",
"state_before": "ι : Type ?u.430162\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nh✝ : ∏ x in s, f x ≠ 1\nh : ∏ x in filter (fun x => f x ≠ 1) s, f x ≠ 1\n⊢ ∃ a, a ∈ s ∧ f a ≠ 1",
"tactic": "rcases nonempty_of_prod_ne_one h with ⟨x, hx⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nι : Type ?u.430162\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nh✝ : ∏ x in s, f x ≠ 1\nh : ∏ x in filter (fun x => f x ≠ 1) s, f x ≠ 1\nx : α\nhx : x ∈ filter (fun x => f x ≠ 1) s\n⊢ ∃ a, a ∈ s ∧ f a ≠ 1",
"tactic": "exact ⟨x, (mem_filter.1 hx).1, by simpa using (mem_filter.1 hx).2⟩"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.430162\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nh✝ : ∏ x in s, f x ≠ 1\nh : ∏ x in filter (fun x => f x ≠ 1) s, f x ≠ 1\nx : α\nhx : x ∈ filter (fun x => f x ≠ 1) s\n⊢ f x ≠ 1",
"tactic": "simpa using (mem_filter.1 hx).2"
}
] |
[
1206,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1202,
1
] |
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