file_path
stringlengths 11
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stringclasses 4
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Mathlib/Data/Real/CauSeqCompletion.lean | CauSeq.lim_lt | []
| [
462,
75
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
461,
1
]
|
Mathlib/Topology/Homeomorph.lean | Homeomorph.coe_symm_toEquiv | []
| [
99,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
98,
1
]
|
Mathlib/Analysis/Quaternion.lean | Quaternion.norm_piLp_equiv_symm_equivTuple | [
{
"state_after": "x : ℍ\n⊢ Real.sqrt\n (inner (↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 0)\n (↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 0) +\n inner (↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 1)\n (↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 1) +\n inner (↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 2)\n (↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 2) +\n inner (↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 3)\n (↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 3)) =\n Real.sqrt (x.re ^ 2 + x.imI ^ 2 + x.imJ ^ 2 + x.imK ^ 2)",
"state_before": "x : ℍ\n⊢ ‖↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x)‖ = ‖x‖",
"tactic": "rw [norm_eq_sqrt_real_inner, norm_eq_sqrt_real_inner, inner_self, normSq_def', PiLp.inner_apply,\n Fin.sum_univ_four]"
},
{
"state_after": "x : ℍ\n⊢ Real.sqrt\n (↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 0 ^ 2 +\n ↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 1 ^ 2 +\n ↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 2 ^ 2 +\n ↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 3 ^ 2) =\n Real.sqrt (x.re ^ 2 + x.imI ^ 2 + x.imJ ^ 2 + x.imK ^ 2)",
"state_before": "x : ℍ\n⊢ Real.sqrt\n (inner (↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 0)\n (↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 0) +\n inner (↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 1)\n (↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 1) +\n inner (↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 2)\n (↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 2) +\n inner (↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 3)\n (↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 3)) =\n Real.sqrt (x.re ^ 2 + x.imI ^ 2 + x.imJ ^ 2 + x.imK ^ 2)",
"tactic": "simp_rw [IsROrC.inner_apply, starRingEnd_apply, star_trivial, ← sq]"
},
{
"state_after": "no goals",
"state_before": "x : ℍ\n⊢ Real.sqrt\n (↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 0 ^ 2 +\n ↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 1 ^ 2 +\n ↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 2 ^ 2 +\n ↑(PiLp.equiv 2 fun x => ℝ).symm (↑(equivTuple ℝ) x) 3 ^ 2) =\n Real.sqrt (x.re ^ 2 + x.imI ^ 2 + x.imJ ^ 2 + x.imK ^ 2)",
"tactic": "rfl"
}
]
| [
182,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
177,
1
]
|
Mathlib/CategoryTheory/Limits/HasLimits.lean | CategoryTheory.Limits.ι_colimMap | []
| [
777,
21
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
775,
1
]
|
Mathlib/MeasureTheory/Measure/VectorMeasure.lean | MeasureTheory.VectorMeasure.mapRange_zero | [
{
"state_after": "case h\nα : Type u_3\nβ : Type ?u.297551\nm inst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : TopologicalSpace M\nv : VectorMeasure α M\nN : Type u_2\ninst✝¹ : AddCommMonoid N\ninst✝ : TopologicalSpace N\nf : M →+ N\nhf : Continuous ↑f\ni✝ : Set α\na✝ : MeasurableSet i✝\n⊢ ↑(mapRange 0 f hf) i✝ = ↑0 i✝",
"state_before": "α : Type u_3\nβ : Type ?u.297551\nm inst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : TopologicalSpace M\nv : VectorMeasure α M\nN : Type u_2\ninst✝¹ : AddCommMonoid N\ninst✝ : TopologicalSpace N\nf : M →+ N\nhf : Continuous ↑f\n⊢ mapRange 0 f hf = 0",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_3\nβ : Type ?u.297551\nm inst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : TopologicalSpace M\nv : VectorMeasure α M\nN : Type u_2\ninst✝¹ : AddCommMonoid N\ninst✝ : TopologicalSpace N\nf : M →+ N\nhf : Continuous ↑f\ni✝ : Set α\na✝ : MeasurableSet i✝\n⊢ ↑(mapRange 0 f hf) i✝ = ↑0 i✝",
"tactic": "simp"
}
]
| [
634,
7
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
631,
1
]
|
Mathlib/Data/Fin/Basic.lean | Fin.val_strictMono | []
| [
338,
71
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
338,
1
]
|
Mathlib/Data/Finset/Pointwise.lean | Finset.image_smul_comm | []
| [
1935,
13
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1933,
1
]
|
Mathlib/Algebra/BigOperators/Basic.lean | Finset.prod_dite_of_true | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.428568\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\nhp : DecidablePred p\nh : ∀ (x : α), x ∈ s → p x\nf : (x : α) → p x → β\ng : (x : α) → ¬p x → β\nx : α\nhx : x ∈ s\n⊢ (fun x hx => { val := x, property := hx }) x hx ∈ univ",
"tactic": "simp"
},
{
"state_after": "ι : Type ?u.428568\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\nhp : DecidablePred p\nh : ∀ (x : α), x ∈ s → p x\nf : (x : α) → p x → β\ng : (x : α) → ¬p x → β\na : α\nha : a ∈ s\n⊢ (if hx : p a then f a hx else g a hx) = f a (_ : p a)",
"state_before": "ι : Type ?u.428568\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\nhp : DecidablePred p\nh : ∀ (x : α), x ∈ s → p x\nf : (x : α) → p x → β\ng : (x : α) → ¬p x → β\na : α\nha : a ∈ s\n⊢ (if hx : p a then f a hx else g a hx) =\n f ↑((fun x hx => { val := x, property := hx }) a ha) (_ : p ↑((fun x hx => { val := x, property := hx }) a ha))",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.428568\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\nhp : DecidablePred p\nh : ∀ (x : α), x ∈ s → p x\nf : (x : α) → p x → β\ng : (x : α) → ¬p x → β\na : α\nha : a ∈ s\n⊢ (if hx : p a then f a hx else g a hx) = f a (_ : p a)",
"tactic": "rw [dif_pos]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.428568\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\nhp : DecidablePred p\nh : ∀ (x : α), x ∈ s → p x\nf : (x : α) → p x → β\ng : (x : α) → ¬p x → β\nb : { x // x ∈ s }\n_hb : b ∈ univ\n⊢ b = (fun x hx => { val := x, property := hx }) ↑b (_ : ↑b ∈ s)",
"tactic": "simp"
}
]
| [
1191,
86
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1184,
1
]
|
Mathlib/Data/PFun.lean | PFun.mem_image | []
| [
409,
10
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
408,
1
]
|
Mathlib/SetTheory/Cardinal/Cofinality.lean | Ordinal.unbounded_of_unbounded_sUnion | [
{
"state_after": "α : Type u_1\nr✝ r : α → α → Prop\nwo : IsWellOrder α r\ns : Set (Set α)\nh₁ : Unbounded r (⋃₀ s)\nh₂ : (#↑s) < StrictOrder.cof r\nh : ∀ (x : Set α), x ∈ s → ¬Unbounded r x\n⊢ False",
"state_before": "α : Type u_1\nr✝ r : α → α → Prop\nwo : IsWellOrder α r\ns : Set (Set α)\nh₁ : Unbounded r (⋃₀ s)\nh₂ : (#↑s) < StrictOrder.cof r\n⊢ ∃ x, x ∈ s ∧ Unbounded r x",
"tactic": "by_contra' h"
},
{
"state_after": "α : Type u_1\nr✝ r : α → α → Prop\nwo : IsWellOrder α r\ns : Set (Set α)\nh₁ : Unbounded r (⋃₀ s)\nh₂ : (#↑s) < StrictOrder.cof r\nh : ∀ (x : Set α), x ∈ s → Bounded r x\n⊢ False",
"state_before": "α : Type u_1\nr✝ r : α → α → Prop\nwo : IsWellOrder α r\ns : Set (Set α)\nh₁ : Unbounded r (⋃₀ s)\nh₂ : (#↑s) < StrictOrder.cof r\nh : ∀ (x : Set α), x ∈ s → ¬Unbounded r x\n⊢ False",
"tactic": "simp_rw [not_unbounded_iff] at h"
},
{
"state_after": "α : Type u_1\nr✝ r : α → α → Prop\nwo : IsWellOrder α r\ns : Set (Set α)\nh₁ : Unbounded r (⋃₀ s)\nh₂ : (#↑s) < StrictOrder.cof r\nh : ∀ (x : Set α), x ∈ s → Bounded r x\nf : ↑s → α := fun x => WellFounded.sup (_ : WellFounded r) ↑x (_ : Bounded r ↑x)\n⊢ False",
"state_before": "α : Type u_1\nr✝ r : α → α → Prop\nwo : IsWellOrder α r\ns : Set (Set α)\nh₁ : Unbounded r (⋃₀ s)\nh₂ : (#↑s) < StrictOrder.cof r\nh : ∀ (x : Set α), x ∈ s → Bounded r x\n⊢ False",
"tactic": "let f : s → α := fun x : s => wo.wf.sup x (h x.1 x.2)"
},
{
"state_after": "α : Type u_1\nr✝ r : α → α → Prop\nwo : IsWellOrder α r\ns : Set (Set α)\nh₁ : Unbounded r (⋃₀ s)\nh₂ : (#↑s) < StrictOrder.cof r\nh : ∀ (x : Set α), x ∈ s → Bounded r x\nf : ↑s → α := fun x => WellFounded.sup (_ : WellFounded r) ↑x (_ : Bounded r ↑x)\nx : α\n⊢ ∃ b, b ∈ range f ∧ (swap rᶜ) x b",
"state_before": "α : Type u_1\nr✝ r : α → α → Prop\nwo : IsWellOrder α r\ns : Set (Set α)\nh₁ : Unbounded r (⋃₀ s)\nh₂ : (#↑s) < StrictOrder.cof r\nh : ∀ (x : Set α), x ∈ s → Bounded r x\nf : ↑s → α := fun x => WellFounded.sup (_ : WellFounded r) ↑x (_ : Bounded r ↑x)\n⊢ False",
"tactic": "refine' h₂.not_le (le_trans (csInf_le' ⟨range f, fun x => _, rfl⟩) mk_range_le)"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\nwo : IsWellOrder α r\ns : Set (Set α)\nh₁ : Unbounded r (⋃₀ s)\nh₂ : (#↑s) < StrictOrder.cof r\nh : ∀ (x : Set α), x ∈ s → Bounded r x\nf : ↑s → α := fun x => WellFounded.sup (_ : WellFounded r) ↑x (_ : Bounded r ↑x)\nx y : α\nhxy : ¬r y x\nc : Set α\nhc : c ∈ s\nhy : y ∈ c\n⊢ ∃ b, b ∈ range f ∧ (swap rᶜ) x b",
"state_before": "α : Type u_1\nr✝ r : α → α → Prop\nwo : IsWellOrder α r\ns : Set (Set α)\nh₁ : Unbounded r (⋃₀ s)\nh₂ : (#↑s) < StrictOrder.cof r\nh : ∀ (x : Set α), x ∈ s → Bounded r x\nf : ↑s → α := fun x => WellFounded.sup (_ : WellFounded r) ↑x (_ : Bounded r ↑x)\nx : α\n⊢ ∃ b, b ∈ range f ∧ (swap rᶜ) x b",
"tactic": "rcases h₁ x with ⟨y, ⟨c, hc, hy⟩, hxy⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\nwo : IsWellOrder α r\ns : Set (Set α)\nh₁ : Unbounded r (⋃₀ s)\nh₂ : (#↑s) < StrictOrder.cof r\nh : ∀ (x : Set α), x ∈ s → Bounded r x\nf : ↑s → α := fun x => WellFounded.sup (_ : WellFounded r) ↑x (_ : Bounded r ↑x)\nx y : α\nhxy : ¬r y x\nc : Set α\nhc : c ∈ s\nhy : y ∈ c\n⊢ ∃ b, b ∈ range f ∧ (swap rᶜ) x b",
"tactic": "exact ⟨f ⟨c, hc⟩, mem_range_self _, fun hxz => hxy (Trans.trans (wo.wf.lt_sup _ hy) hxz)⟩"
}
]
| [
790,
92
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
783,
1
]
|
Mathlib/Topology/Instances/ENNReal.lean | ENNReal.tsum_const_smul | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.256888\nγ : Type ?u.256891\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf g : α → ℝ≥0∞\nR : Type u_1\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\na : R\n⊢ (∑' (i : α), a • f i) = a • ∑' (i : α), f i",
"tactic": "simpa only [smul_one_mul] using @ENNReal.tsum_mul_left _ (a • (1 : ℝ≥0∞)) _"
}
]
| [
916,
78
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
914,
11
]
|
Mathlib/Analysis/NormedSpace/Exponential.lean | expSeries_sum_eq | []
| [
108,
45
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
107,
1
]
|
Mathlib/Order/GaloisConnection.lean | GaloisConnection.l_u_l_eq_l' | []
| [
213,
23
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
212,
1
]
|
Mathlib/Data/Nat/Log.lean | Nat.clog_of_two_le | [
{
"state_after": "no goals",
"state_before": "b n : ℕ\nhb : 1 < b\nhn : 2 ≤ n\n⊢ clog b n = clog b ((n + b - 1) / b) + 1",
"tactic": "rw [clog, dif_pos (⟨hb, hn⟩ : 1 < b ∧ 1 < n)]"
}
]
| [
279,
96
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
278,
1
]
|
Mathlib/Order/UpperLower/Basic.lean | LowerSet.symm_map | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.101991\nι : Sort ?u.101994\nκ : ι → Sort ?u.101999\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nf✝ : α ≃o β\ns✝ t : LowerSet α\na : α\nb : β\nf : α ≃o β\ns : LowerSet β\n⊢ ↑(↑(OrderIso.symm (map f)) s) = ↑(↑(map (OrderIso.symm f)) s)",
"tactic": "convert Set.preimage_equiv_eq_image_symm s f.toEquiv"
}
]
| [
1016,
90
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1015,
1
]
|
Mathlib/Analysis/SpecialFunctions/Log/Base.lean | Real.logb_le_logb_of_base_lt_one | [
{
"state_after": "no goals",
"state_before": "b x y : ℝ\nb_pos : 0 < b\nb_lt_one : b < 1\nh : 0 < x\nh₁ : 0 < y\n⊢ logb b x ≤ logb b y ↔ y ≤ x",
"tactic": "rw [logb, logb, div_le_div_right_of_neg (log_neg b_pos b_lt_one), log_le_log h₁ h]"
}
]
| [
257,
85
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
256,
1
]
|
Mathlib/Analysis/Normed/Group/Quotient.lean | AddSubgroup.norm_normedMk_le | [
{
"state_after": "no goals",
"state_before": "M : Type u_1\nN : Type ?u.424521\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nm : M\n⊢ ‖↑(normedMk S) m‖ ≤ 1 * ‖m‖",
"tactic": "simp [quotient_norm_mk_le']"
}
]
| [
305,
90
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
304,
1
]
|
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tsum_dite_right | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.326054\nδ : Type ?u.326057\ninst✝³ : AddCommMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : T2Space α\nf g : β → α\na a₁ a₂ : α\nP : Prop\ninst✝ : Decidable P\nx : β → ¬P → α\n⊢ (∑' (b : β), if h : P then 0 else x b h) = if h : P then 0 else ∑' (b : β), x b h",
"tactic": "by_cases hP : P <;> simp [hP]"
}
]
| [
547,
32
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
545,
1
]
|
Mathlib/Data/Sigma/Order.lean | Sigma.le_def | [
{
"state_after": "case mp\nι : Type u_2\nα : ι → Type u_1\ninst✝ : (i : ι) → LE (α i)\na b : (i : ι) × α i\n⊢ a ≤ b → ∃ h, h ▸ a.snd ≤ b.snd\n\ncase mpr\nι : Type u_2\nα : ι → Type u_1\ninst✝ : (i : ι) → LE (α i)\na b : (i : ι) × α i\n⊢ (∃ h, h ▸ a.snd ≤ b.snd) → a ≤ b",
"state_before": "ι : Type u_2\nα : ι → Type u_1\ninst✝ : (i : ι) → LE (α i)\na b : (i : ι) × α i\n⊢ a ≤ b ↔ ∃ h, h ▸ a.snd ≤ b.snd",
"tactic": "constructor"
},
{
"state_after": "case mp.fiber\nι : Type u_2\nα : ι → Type u_1\ninst✝ : (i : ι) → LE (α i)\ni : ι\na b : α i\nh : a ≤ b\n⊢ ∃ h, h ▸ { fst := i, snd := a }.snd ≤ { fst := i, snd := b }.snd",
"state_before": "case mp\nι : Type u_2\nα : ι → Type u_1\ninst✝ : (i : ι) → LE (α i)\na b : (i : ι) × α i\n⊢ a ≤ b → ∃ h, h ▸ a.snd ≤ b.snd",
"tactic": "rintro ⟨i, a, b, h⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.fiber\nι : Type u_2\nα : ι → Type u_1\ninst✝ : (i : ι) → LE (α i)\ni : ι\na b : α i\nh : a ≤ b\n⊢ ∃ h, h ▸ { fst := i, snd := a }.snd ≤ { fst := i, snd := b }.snd",
"tactic": "exact ⟨rfl, h⟩"
},
{
"state_after": "case mpr.mk\nι : Type u_2\nα : ι → Type u_1\ninst✝ : (i : ι) → LE (α i)\nb : (i : ι) × α i\ni : ι\na : α i\n⊢ (∃ h, h ▸ { fst := i, snd := a }.snd ≤ b.snd) → { fst := i, snd := a } ≤ b",
"state_before": "case mpr\nι : Type u_2\nα : ι → Type u_1\ninst✝ : (i : ι) → LE (α i)\na b : (i : ι) × α i\n⊢ (∃ h, h ▸ a.snd ≤ b.snd) → a ≤ b",
"tactic": "obtain ⟨i, a⟩ := a"
},
{
"state_after": "case mpr.mk.mk\nι : Type u_2\nα : ι → Type u_1\ninst✝ : (i : ι) → LE (α i)\ni : ι\na : α i\nj : ι\nb : α j\n⊢ (∃ h, h ▸ { fst := i, snd := a }.snd ≤ { fst := j, snd := b }.snd) → { fst := i, snd := a } ≤ { fst := j, snd := b }",
"state_before": "case mpr.mk\nι : Type u_2\nα : ι → Type u_1\ninst✝ : (i : ι) → LE (α i)\nb : (i : ι) × α i\ni : ι\na : α i\n⊢ (∃ h, h ▸ { fst := i, snd := a }.snd ≤ b.snd) → { fst := i, snd := a } ≤ b",
"tactic": "obtain ⟨j, b⟩ := b"
},
{
"state_after": "case mpr.mk.mk.intro\nι : Type u_2\nα : ι → Type u_1\ninst✝ : (i : ι) → LE (α i)\ni : ι\na b : α i\nh : (_ : i = i) ▸ { fst := i, snd := a }.snd ≤ { fst := i, snd := b }.snd\n⊢ { fst := i, snd := a } ≤ { fst := i, snd := b }",
"state_before": "case mpr.mk.mk\nι : Type u_2\nα : ι → Type u_1\ninst✝ : (i : ι) → LE (α i)\ni : ι\na : α i\nj : ι\nb : α j\n⊢ (∃ h, h ▸ { fst := i, snd := a }.snd ≤ { fst := j, snd := b }.snd) → { fst := i, snd := a } ≤ { fst := j, snd := b }",
"tactic": "rintro ⟨rfl : i = j, h⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.mk.mk.intro\nι : Type u_2\nα : ι → Type u_1\ninst✝ : (i : ι) → LE (α i)\ni : ι\na b : α i\nh : (_ : i = i) ▸ { fst := i, snd := a }.snd ≤ { fst := i, snd := b }.snd\n⊢ { fst := i, snd := a } ≤ { fst := i, snd := b }",
"tactic": "exact le.fiber _ _ _ h"
}
]
| [
88,
27
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
81,
1
]
|
Mathlib/CategoryTheory/Functor/Category.lean | CategoryTheory.NatTrans.comp_app | []
| [
80,
45
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
79,
1
]
|
Mathlib/Data/Set/Intervals/WithBotTop.lean | WithTop.preimage_coe_Ioo_top | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : PartialOrder α\na b : α\n⊢ some ⁻¹' Ioo ↑a ⊤ = Ioi a",
"tactic": "simp [← Ioi_inter_Iio]"
}
]
| [
87,
25
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
86,
1
]
|
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | CategoryTheory.Limits.spanCompIso_inv_app_left | []
| [
368,
93
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
368,
1
]
|
Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous_id | []
| [
1127,
71
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1127,
1
]
|
Mathlib/Algebra/BigOperators/Basic.lean | Finset.eventually_constant_prod | [
{
"state_after": "case intro\nι : Type ?u.436010\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nu : ℕ → β\nN : ℕ\nhu : ∀ (n : ℕ), n ≥ N → u n = 1\nm : ℕ\nhn : N ≤ N + m\n⊢ ∏ k in range (N + m + 1), u k = ∏ k in range (N + 1), u k",
"state_before": "ι : Type ?u.436010\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nu : ℕ → β\nN : ℕ\nhu : ∀ (n : ℕ), n ≥ N → u n = 1\nn : ℕ\nhn : N ≤ n\n⊢ ∏ k in range (n + 1), u k = ∏ k in range (N + 1), u k",
"tactic": "obtain ⟨m, rfl : n = N + m⟩ := le_iff_exists_add.mp hn"
},
{
"state_after": "case intro\nι : Type ?u.436010\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nu : ℕ → β\nN : ℕ\nhu : ∀ (n : ℕ), n ≥ N → u n = 1\nm : ℕ\n⊢ ∏ k in range (N + m + 1), u k = ∏ k in range (N + 1), u k",
"state_before": "case intro\nι : Type ?u.436010\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nu : ℕ → β\nN : ℕ\nhu : ∀ (n : ℕ), n ≥ N → u n = 1\nm : ℕ\nhn : N ≤ N + m\n⊢ ∏ k in range (N + m + 1), u k = ∏ k in range (N + 1), u k",
"tactic": "clear hn"
},
{
"state_after": "case intro.zero\nι : Type ?u.436010\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nu : ℕ → β\nN : ℕ\nhu : ∀ (n : ℕ), n ≥ N → u n = 1\n⊢ ∏ k in range (N + Nat.zero + 1), u k = ∏ k in range (N + 1), u k\n\ncase intro.succ\nι : Type ?u.436010\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nu : ℕ → β\nN : ℕ\nhu : ∀ (n : ℕ), n ≥ N → u n = 1\nm : ℕ\nhm : ∏ k in range (N + m + 1), u k = ∏ k in range (N + 1), u k\n⊢ ∏ k in range (N + Nat.succ m + 1), u k = ∏ k in range (N + 1), u k",
"state_before": "case intro\nι : Type ?u.436010\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nu : ℕ → β\nN : ℕ\nhu : ∀ (n : ℕ), n ≥ N → u n = 1\nm : ℕ\n⊢ ∏ k in range (N + m + 1), u k = ∏ k in range (N + 1), u k",
"tactic": "induction' m with m hm"
},
{
"state_after": "case intro.succ\nι : Type ?u.436010\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nu : ℕ → β\nN : ℕ\nhu : ∀ (n : ℕ), n ≥ N → u n = 1\nm : ℕ\nhm : ∏ k in range (N + m + 1), u k = ∏ k in range (N + 1), u k\n⊢ (∏ k in range (N + 1), u k) * u (N + Nat.succ m) = ∏ k in range (N + 1), u k",
"state_before": "case intro.succ\nι : Type ?u.436010\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nu : ℕ → β\nN : ℕ\nhu : ∀ (n : ℕ), n ≥ N → u n = 1\nm : ℕ\nhm : ∏ k in range (N + m + 1), u k = ∏ k in range (N + 1), u k\n⊢ ∏ k in range (N + Nat.succ m + 1), u k = ∏ k in range (N + 1), u k",
"tactic": "erw [prod_range_succ, hm]"
},
{
"state_after": "no goals",
"state_before": "case intro.succ\nι : Type ?u.436010\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nu : ℕ → β\nN : ℕ\nhu : ∀ (n : ℕ), n ≥ N → u n = 1\nm : ℕ\nhm : ∏ k in range (N + m + 1), u k = ∏ k in range (N + 1), u k\n⊢ (∏ k in range (N + 1), u k) * u (N + Nat.succ m) = ∏ k in range (N + 1), u k",
"tactic": "simp [hu, @zero_le' ℕ]"
},
{
"state_after": "no goals",
"state_before": "case intro.zero\nι : Type ?u.436010\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nu : ℕ → β\nN : ℕ\nhu : ∀ (n : ℕ), n ≥ N → u n = 1\n⊢ ∏ k in range (N + Nat.zero + 1), u k = ∏ k in range (N + 1), u k",
"tactic": "simp"
}
]
| [
1240,
25
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1233,
1
]
|
Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean | Polynomial.exists_eq_polynomial | [
{
"state_after": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : natDegree b ≤ d\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) ↑j\n⊢ ∃ i₀ i₁, i₀ ≠ i₁ ∧ A i₁ = A i₀",
"state_before": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : natDegree b ≤ d\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\n⊢ ∃ i₀ i₁, i₀ ≠ i₁ ∧ A i₁ = A i₀",
"tactic": "set f : Fin m.succ → Fin d → Fq := fun i j => (A i).coeff j"
},
{
"state_after": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : natDegree b ≤ d\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) ↑j\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\n⊢ ∃ i₀ i₁, i₀ ≠ i₁ ∧ A i₁ = A i₀",
"state_before": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : natDegree b ≤ d\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) ↑j\n⊢ ∃ i₀ i₁, i₀ ≠ i₁ ∧ A i₁ = A i₀",
"tactic": "have : Fintype.card (Fin d → Fq) < Fintype.card (Fin m.succ) := by\n simpa using lt_of_le_of_lt hm (Nat.lt_succ_self m)"
},
{
"state_after": "case intro.intro.intro\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : natDegree b ≤ d\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) ↑j\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\n⊢ ∃ i₀ i₁, i₀ ≠ i₁ ∧ A i₁ = A i₀",
"state_before": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : natDegree b ≤ d\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) ↑j\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\n⊢ ∃ i₀ i₁, i₀ ≠ i₁ ∧ A i₁ = A i₀",
"tactic": "obtain ⟨i₀, i₁, i_ne, i_eq⟩ := Fintype.exists_ne_map_eq_of_card_lt f this"
},
{
"state_after": "case intro.intro.intro\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : natDegree b ≤ d\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) ↑j\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\n⊢ A i₁ = A i₀",
"state_before": "case intro.intro.intro\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : natDegree b ≤ d\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) ↑j\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\n⊢ ∃ i₀ i₁, i₀ ≠ i₁ ∧ A i₁ = A i₀",
"tactic": "use i₀, i₁, i_ne"
},
{
"state_after": "case intro.intro.intro.a\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : natDegree b ≤ d\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) ↑j\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\n⊢ coeff (A i₁) j = coeff (A i₀) j",
"state_before": "case intro.intro.intro\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : natDegree b ≤ d\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) ↑j\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\n⊢ A i₁ = A i₀",
"tactic": "ext j"
},
{
"state_after": "case pos\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : natDegree b ≤ d\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) ↑j\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhbj : degree b ≤ ↑j\n⊢ coeff (A i₁) j = coeff (A i₀) j\n\ncase neg\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : natDegree b ≤ d\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) ↑j\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhbj : ¬degree b ≤ ↑j\n⊢ coeff (A i₁) j = coeff (A i₀) j",
"state_before": "case intro.intro.intro.a\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : natDegree b ≤ d\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) ↑j\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\n⊢ coeff (A i₁) j = coeff (A i₀) j",
"tactic": "by_cases hbj : degree b ≤ j"
},
{
"state_after": "case neg\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : natDegree b ≤ d\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) ↑j\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhbj : ↑j < degree b\n⊢ coeff (A i₁) j = coeff (A i₀) j",
"state_before": "case neg\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : natDegree b ≤ d\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) ↑j\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhbj : ¬degree b ≤ ↑j\n⊢ coeff (A i₁) j = coeff (A i₀) j",
"tactic": "rw [not_le] at hbj"
},
{
"state_after": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : natDegree b ≤ d\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) ↑j\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhbj : ↑j < degree b\n⊢ j < d",
"state_before": "case neg\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : natDegree b ≤ d\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) ↑j\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhbj : ↑j < degree b\n⊢ coeff (A i₁) j = coeff (A i₀) j",
"tactic": "apply congr_fun i_eq.symm ⟨j, _⟩"
},
{
"state_after": "no goals",
"state_before": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : natDegree b ≤ d\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) ↑j\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhbj : ↑j < degree b\n⊢ j < d",
"tactic": "exact lt_of_lt_of_le (coe_lt_degree.mp hbj) hb"
},
{
"state_after": "no goals",
"state_before": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : natDegree b ≤ d\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) ↑j\n⊢ Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))",
"tactic": "simpa using lt_of_le_of_lt hm (Nat.lt_succ_self m)"
},
{
"state_after": "no goals",
"state_before": "case pos\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : natDegree b ≤ d\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) ↑j\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhbj : degree b ≤ ↑j\n⊢ coeff (A i₁) j = coeff (A i₀) j",
"tactic": "rw [coeff_eq_zero_of_degree_lt (lt_of_lt_of_le (hA _) hbj),\n coeff_eq_zero_of_degree_lt (lt_of_lt_of_le (hA _) hbj)]"
}
]
| [
60,
49
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
39,
1
]
|
Mathlib/CategoryTheory/Limits/Shapes/Types.lean | CategoryTheory.Limits.Types.binaryProductIso_hom_comp_fst | []
| [
157,
75
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
155,
1
]
|
Mathlib/LinearAlgebra/Matrix/Charpoly/Minpoly.lean | LinearMap.minpoly_toMatrix' | []
| [
67,
72
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
66,
1
]
|
Mathlib/Data/Finset/Sum.lean | Finset.inl_mem_disjSum | []
| [
77,
27
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
76,
1
]
|
Mathlib/Data/Rat/NNRat.lean | NNRat.coe_sub | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.13919\np q : ℚ≥0\nh : q ≤ p\n⊢ ↑q ≤ ↑p - 0",
"tactic": "rwa [sub_zero]"
}
]
| [
154,
52
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
153,
1
]
|
Mathlib/CategoryTheory/Sites/Sieves.lean | CategoryTheory.Sieve.inter_apply | []
| [
369,
10
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
368,
1
]
|
Mathlib/Topology/Order/Priestley.lean | exists_clopen_upper_of_not_le | []
| [
55,
27
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
53,
1
]
|
Mathlib/Order/Hom/Lattice.lean | SupBotHom.ext | []
| [
746,
20
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
745,
1
]
|
Mathlib/Combinatorics/Partition.lean | Nat.Partition.count_ofSums_of_ne_zero | []
| [
126,
25
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
124,
1
]
|
Mathlib/Order/OmegaCompletePartialOrder.lean | OmegaCompletePartialOrder.const_continuous' | []
| [
301,
67
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
300,
1
]
|
Mathlib/Topology/Order/Basic.lean | eventually_ge_nhds | []
| [
1197,
82
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1197,
1
]
|
Mathlib/Topology/Separation.lean | t1Space_of_injective_of_continuous | []
| [
591,
66
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
589,
1
]
|
Mathlib/Topology/Algebra/Ring/Basic.lean | Subsemiring.isClosed_topologicalClosure | []
| [
122,
64
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
121,
1
]
|
Mathlib/Algebra/GradedMonoid.lean | SetLike.list_dProd_eq | []
| [
668,
48
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
662,
1
]
|
Mathlib/Topology/TietzeExtension.lean | BoundedContinuousFunction.exists_extension_forall_mem_of_closedEmbedding | [
{
"state_after": "case inl\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\nt : Set ℝ\ne : X → Y\nhs : OrdConnected t\nhf : ∀ (x : X), ↑f x ∈ t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh✝ : IsEmpty X\n⊢ ∃ g, (∀ (y : Y), ↑g y ∈ t) ∧ ↑g ∘ e = ↑f\n\ncase inr\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\nt : Set ℝ\ne : X → Y\nhs : OrdConnected t\nhf : ∀ (x : X), ↑f x ∈ t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh✝ : Nonempty X\n⊢ ∃ g, (∀ (y : Y), ↑g y ∈ t) ∧ ↑g ∘ e = ↑f",
"state_before": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\nt : Set ℝ\ne : X → Y\nhs : OrdConnected t\nhf : ∀ (x : X), ↑f x ∈ t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\n⊢ ∃ g, (∀ (y : Y), ↑g y ∈ t) ∧ ↑g ∘ e = ↑f",
"tactic": "cases isEmpty_or_nonempty X"
},
{
"state_after": "case inr.intro.intro\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\nt : Set ℝ\ne : X → Y\nhs : OrdConnected t\nhf : ∀ (x : X), ↑f x ∈ t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh✝ : Nonempty X\ng : Y →ᵇ ℝ\nhg : ∀ (y : Y), ∃ x₁ x₂, ↑g y ∈ Icc (↑f x₁) (↑f x₂)\nhgf : ↑g ∘ e = ↑f\n⊢ ∃ g, (∀ (y : Y), ↑g y ∈ t) ∧ ↑g ∘ e = ↑f",
"state_before": "case inr\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\nt : Set ℝ\ne : X → Y\nhs : OrdConnected t\nhf : ∀ (x : X), ↑f x ∈ t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh✝ : Nonempty X\n⊢ ∃ g, (∀ (y : Y), ↑g y ∈ t) ∧ ↑g ∘ e = ↑f",
"tactic": "rcases exists_extension_forall_exists_le_ge_of_closedEmbedding f he with ⟨g, hg, hgf⟩"
},
{
"state_after": "case inr.intro.intro\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\nt : Set ℝ\ne : X → Y\nhs : OrdConnected t\nhf : ∀ (x : X), ↑f x ∈ t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh✝ : Nonempty X\ng : Y →ᵇ ℝ\nhg : ∀ (y : Y), ∃ x₁ x₂, ↑g y ∈ Icc (↑f x₁) (↑f x₂)\nhgf : ↑g ∘ e = ↑f\ny : Y\n⊢ ↑g y ∈ t",
"state_before": "case inr.intro.intro\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\nt : Set ℝ\ne : X → Y\nhs : OrdConnected t\nhf : ∀ (x : X), ↑f x ∈ t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh✝ : Nonempty X\ng : Y →ᵇ ℝ\nhg : ∀ (y : Y), ∃ x₁ x₂, ↑g y ∈ Icc (↑f x₁) (↑f x₂)\nhgf : ↑g ∘ e = ↑f\n⊢ ∃ g, (∀ (y : Y), ↑g y ∈ t) ∧ ↑g ∘ e = ↑f",
"tactic": "refine' ⟨g, fun y => _, hgf⟩"
},
{
"state_after": "case inr.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\nt : Set ℝ\ne : X → Y\nhs : OrdConnected t\nhf : ∀ (x : X), ↑f x ∈ t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh✝ : Nonempty X\ng : Y →ᵇ ℝ\nhg : ∀ (y : Y), ∃ x₁ x₂, ↑g y ∈ Icc (↑f x₁) (↑f x₂)\nhgf : ↑g ∘ e = ↑f\ny : Y\nxl xu : X\nh : ↑g y ∈ Icc (↑f xl) (↑f xu)\n⊢ ↑g y ∈ t",
"state_before": "case inr.intro.intro\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\nt : Set ℝ\ne : X → Y\nhs : OrdConnected t\nhf : ∀ (x : X), ↑f x ∈ t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh✝ : Nonempty X\ng : Y →ᵇ ℝ\nhg : ∀ (y : Y), ∃ x₁ x₂, ↑g y ∈ Icc (↑f x₁) (↑f x₂)\nhgf : ↑g ∘ e = ↑f\ny : Y\n⊢ ↑g y ∈ t",
"tactic": "rcases hg y with ⟨xl, xu, h⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\nt : Set ℝ\ne : X → Y\nhs : OrdConnected t\nhf : ∀ (x : X), ↑f x ∈ t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh✝ : Nonempty X\ng : Y →ᵇ ℝ\nhg : ∀ (y : Y), ∃ x₁ x₂, ↑g y ∈ Icc (↑f x₁) (↑f x₂)\nhgf : ↑g ∘ e = ↑f\ny : Y\nxl xu : X\nh : ↑g y ∈ Icc (↑f xl) (↑f xu)\n⊢ ↑g y ∈ t",
"tactic": "exact hs.out (hf _) (hf _) h"
},
{
"state_after": "case inl.intro\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\nt : Set ℝ\ne : X → Y\nhs : OrdConnected t\nhf : ∀ (x : X), ↑f x ∈ t\nhe : ClosedEmbedding e\nh✝ : IsEmpty X\nc : ℝ\nhc : c ∈ t\n⊢ ∃ g, (∀ (y : Y), ↑g y ∈ t) ∧ ↑g ∘ e = ↑f",
"state_before": "case inl\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\nt : Set ℝ\ne : X → Y\nhs : OrdConnected t\nhf : ∀ (x : X), ↑f x ∈ t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh✝ : IsEmpty X\n⊢ ∃ g, (∀ (y : Y), ↑g y ∈ t) ∧ ↑g ∘ e = ↑f",
"tactic": "rcases hne with ⟨c, hc⟩"
},
{
"state_after": "no goals",
"state_before": "case inl.intro\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\nt : Set ℝ\ne : X → Y\nhs : OrdConnected t\nhf : ∀ (x : X), ↑f x ∈ t\nhe : ClosedEmbedding e\nh✝ : IsEmpty X\nc : ℝ\nhc : c ∈ t\n⊢ ∃ g, (∀ (y : Y), ↑g y ∈ t) ∧ ↑g ∘ e = ↑f",
"tactic": "refine' ⟨const Y c, fun _ => hc, funext fun x => isEmptyElim x⟩"
}
]
| [
324,
31
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
315,
1
]
|
Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | MeasureTheory.FiniteMeasure.tendsto_zero_testAgainstNN_of_tendsto_zero_mass | [
{
"state_after": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.143739\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\n⊢ Tendsto (fun b => dist (testAgainstNN (μs b) f) 0) F (𝓝 0)",
"state_before": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.143739\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\n⊢ Tendsto (fun i => testAgainstNN (μs i) f) F (𝓝 0)",
"tactic": "apply tendsto_iff_dist_tendsto_zero.mpr"
},
{
"state_after": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.143739\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\nobs : ∀ (i : γ), testAgainstNN (μs i) f ≤ testAgainstNN (μs i) 0 + nndist f 0 * mass (μs i)\n⊢ Tendsto (fun b => dist (testAgainstNN (μs b) f) 0) F (𝓝 0)",
"state_before": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.143739\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\n⊢ Tendsto (fun b => dist (testAgainstNN (μs b) f) 0) F (𝓝 0)",
"tactic": "have obs := fun i => (μs i).testAgainstNN_lipschitz_estimate f 0"
},
{
"state_after": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.143739\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\nobs : ∀ (i : γ), testAgainstNN (μs i) f ≤ nndist f 0 * mass (μs i)\n⊢ Tendsto (fun b => dist (testAgainstNN (μs b) f) 0) F (𝓝 0)",
"state_before": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.143739\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\nobs : ∀ (i : γ), testAgainstNN (μs i) f ≤ testAgainstNN (μs i) 0 + nndist f 0 * mass (μs i)\n⊢ Tendsto (fun b => dist (testAgainstNN (μs b) f) 0) F (𝓝 0)",
"tactic": "simp_rw [testAgainstNN_zero, zero_add] at obs"
},
{
"state_after": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.143739\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\nobs : ∀ (i : γ), testAgainstNN (μs i) f ≤ nndist f 0 * mass (μs i)\n⊢ Tendsto (fun b => ↑(testAgainstNN (μs b) f)) F (𝓝 0)",
"state_before": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.143739\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\nobs : ∀ (i : γ), testAgainstNN (μs i) f ≤ nndist f 0 * mass (μs i)\n⊢ Tendsto (fun b => dist (testAgainstNN (μs b) f) 0) F (𝓝 0)",
"tactic": "simp_rw [show ∀ i, dist ((μs i).testAgainstNN f) 0 = (μs i).testAgainstNN f by\n simp only [dist_nndist, NNReal.nndist_zero_eq_val', eq_self_iff_true, imp_true_iff]]"
},
{
"state_after": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.143739\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\nobs : ∀ (i : γ), testAgainstNN (μs i) f ≤ nndist f 0 * mass (μs i)\n⊢ Tendsto (fun t => (fun a => ↑a) (nndist f 0 * mass (μs t))) F (𝓝 0)",
"state_before": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.143739\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\nobs : ∀ (i : γ), testAgainstNN (μs i) f ≤ nndist f 0 * mass (μs i)\n⊢ Tendsto (fun b => ↑(testAgainstNN (μs b) f)) F (𝓝 0)",
"tactic": "refine' squeeze_zero (fun i => NNReal.coe_nonneg _) obs _"
},
{
"state_after": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.143739\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\nobs : ∀ (i : γ), testAgainstNN (μs i) f ≤ nndist f 0 * mass (μs i)\n⊢ Tendsto (fun t => ↑(nndist f 0 * mass (μs t))) F (𝓝 0)",
"state_before": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.143739\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\nobs : ∀ (i : γ), testAgainstNN (μs i) f ≤ nndist f 0 * mass (μs i)\n⊢ Tendsto (fun t => (fun a => ↑a) (nndist f 0 * mass (μs t))) F (𝓝 0)",
"tactic": "simp_rw [NNReal.coe_mul]"
},
{
"state_after": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.143739\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\nobs : ∀ (i : γ), testAgainstNN (μs i) f ≤ nndist f 0 * mass (μs i)\nlim_pair : Tendsto (fun i => (↑(nndist f 0), ↑(mass (μs i)))) F (𝓝 (↑(nndist f 0), 0))\n⊢ Tendsto (fun t => ↑(nndist f 0 * mass (μs t))) F (𝓝 0)",
"state_before": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.143739\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\nobs : ∀ (i : γ), testAgainstNN (μs i) f ≤ nndist f 0 * mass (μs i)\n⊢ Tendsto (fun t => ↑(nndist f 0 * mass (μs t))) F (𝓝 0)",
"tactic": "have lim_pair : Tendsto (fun i => (⟨nndist f 0, (μs i).mass⟩ : ℝ × ℝ)) F (𝓝 ⟨nndist f 0, 0⟩) := by\n refine' (Prod.tendsto_iff _ _).mpr ⟨tendsto_const_nhds, _⟩\n exact (NNReal.continuous_coe.tendsto 0).comp mass_lim"
},
{
"state_after": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.143739\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\nobs : ∀ (i : γ), testAgainstNN (μs i) f ≤ nndist f 0 * mass (μs i)\nlim_pair : Tendsto (fun i => (↑(nndist f 0), ↑(mass (μs i)))) F (𝓝 (↑(nndist f 0), 0))\nkey : Tendsto ((fun p => p.fst * p.snd) ∘ fun i => (↑(nndist f 0), ↑(mass (μs i)))) F (𝓝 (↑(nndist f 0) * 0))\n⊢ Tendsto (fun t => ↑(nndist f 0 * mass (μs t))) F (𝓝 0)",
"state_before": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.143739\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\nobs : ∀ (i : γ), testAgainstNN (μs i) f ≤ nndist f 0 * mass (μs i)\nlim_pair : Tendsto (fun i => (↑(nndist f 0), ↑(mass (μs i)))) F (𝓝 (↑(nndist f 0), 0))\n⊢ Tendsto (fun t => ↑(nndist f 0 * mass (μs t))) F (𝓝 0)",
"tactic": "have key := tendsto_mul.comp lim_pair"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.143739\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\nobs : ∀ (i : γ), testAgainstNN (μs i) f ≤ nndist f 0 * mass (μs i)\nlim_pair : Tendsto (fun i => (↑(nndist f 0), ↑(mass (μs i)))) F (𝓝 (↑(nndist f 0), 0))\nkey : Tendsto ((fun p => p.fst * p.snd) ∘ fun i => (↑(nndist f 0), ↑(mass (μs i)))) F (𝓝 (↑(nndist f 0) * 0))\n⊢ Tendsto (fun t => ↑(nndist f 0 * mass (μs t))) F (𝓝 0)",
"tactic": "rwa [MulZeroClass.mul_zero] at key"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.143739\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\nobs : ∀ (i : γ), testAgainstNN (μs i) f ≤ nndist f 0 * mass (μs i)\n⊢ ∀ (i : γ), dist (testAgainstNN (μs i) f) 0 = ↑(testAgainstNN (μs i) f)",
"tactic": "simp only [dist_nndist, NNReal.nndist_zero_eq_val', eq_self_iff_true, imp_true_iff]"
},
{
"state_after": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.143739\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\nobs : ∀ (i : γ), testAgainstNN (μs i) f ≤ nndist f 0 * mass (μs i)\n⊢ Tendsto (fun n => (↑(nndist f 0), ↑(mass (μs n))).snd) F (𝓝 (↑(nndist f 0), 0).snd)",
"state_before": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.143739\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\nobs : ∀ (i : γ), testAgainstNN (μs i) f ≤ nndist f 0 * mass (μs i)\n⊢ Tendsto (fun i => (↑(nndist f 0), ↑(mass (μs i)))) F (𝓝 (↑(nndist f 0), 0))",
"tactic": "refine' (Prod.tendsto_iff _ _).mpr ⟨tendsto_const_nhds, _⟩"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.143739\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\nobs : ∀ (i : γ), testAgainstNN (μs i) f ≤ nndist f 0 * mass (μs i)\n⊢ Tendsto (fun n => (↑(nndist f 0), ↑(mass (μs n))).snd) F (𝓝 (↑(nndist f 0), 0).snd)",
"tactic": "exact (NNReal.continuous_coe.tendsto 0).comp mass_lim"
}
]
| [
531,
37
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
517,
1
]
|
Mathlib/Topology/Connected.lean | IsConnected.image | []
| [
340,
67
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
338,
11
]
|
Mathlib/Data/Polynomial/RingDivision.lean | Polynomial.natDegree_mul | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np q : R[X]\nhp : p ≠ 0\nhq : q ≠ 0\n⊢ natDegree (p * q) = natDegree p + natDegree q",
"tactic": "rw [← WithBot.coe_eq_coe, ← Nat.cast_withBot, ←degree_eq_natDegree (mul_ne_zero hp hq),\n WithBot.coe_add, ← Nat.cast_withBot, ←degree_eq_natDegree hp, ← Nat.cast_withBot,\n ← degree_eq_natDegree hq, degree_mul]"
}
]
| [
139,
42
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
136,
1
]
|
Std/Data/List/Lemmas.lean | List.pair_mem_product | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nxs : List α\nys : List β\nx : α\ny : β\n⊢ (x, y) ∈ product xs ys ↔ x ∈ xs ∧ y ∈ ys",
"tactic": "simp only [product, and_imp, exists_prop, mem_map, Prod.mk.injEq,\n exists_eq_right_right, mem_bind, iff_self]"
}
]
| [
1453,
47
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1450,
1
]
|
Mathlib/Algebra/GroupPower/Lemmas.lean | powersMulHom_apply | []
| [
975,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
973,
1
]
|
Mathlib/Data/Nat/Choose/Cast.lean | Nat.cast_choose_two | [
{
"state_after": "no goals",
"state_before": "K : Type u_1\ninst✝¹ : DivisionRing K\ninst✝ : CharZero K\na : ℕ\n⊢ ↑(choose a 2) = ↑a * (↑a - 1) / 2",
"tactic": "rw [← cast_descFactorial_two, descFactorial_eq_factorial_mul_choose, factorial_two, mul_comm,\n cast_mul, cast_two, eq_div_iff_mul_eq (two_ne_zero : (2 : K) ≠ 0)]"
}
]
| [
46,
71
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
44,
1
]
|
Mathlib/RingTheory/Subring/Basic.lean | Subring.range_subtype | []
| [
1275,
66
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1274,
1
]
|
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | ENNReal.le_rpow_self_of_one_le | [
{
"state_after": "x : ℝ≥0∞\nz : ℝ\nhx : 1 ≤ x\nh_one_le : 1 ≤ z\n⊢ x ^ 1 ≤ x ^ z",
"state_before": "x : ℝ≥0∞\nz : ℝ\nhx : 1 ≤ x\nh_one_le : 1 ≤ z\n⊢ x ≤ x ^ z",
"tactic": "nth_rw 1 [← ENNReal.rpow_one x]"
},
{
"state_after": "no goals",
"state_before": "x : ℝ≥0∞\nz : ℝ\nhx : 1 ≤ x\nh_one_le : 1 ≤ z\n⊢ x ^ 1 ≤ x ^ z",
"tactic": "exact ENNReal.rpow_le_rpow_of_exponent_le hx h_one_le"
}
]
| [
646,
56
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
644,
1
]
|
Mathlib/Logic/Basic.lean | exists_eq_right' | [
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.20626\nα : Sort u_1\nκ : ι → Sort ?u.20628\np q : α → Prop\na' : α\n⊢ (∃ a, p a ∧ a' = a) ↔ p a'",
"tactic": "simp [@eq_comm _ a']"
}
]
| [
823,
98
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
823,
9
]
|
Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean | mul_div_cancel_left | []
| [
168,
41
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
167,
1
]
|
Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean | Finpartition.mk_mem_nonUniforms_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\n𝕜 : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε✝ : 𝕜\nu v : Finset α\nε : 𝕜\n⊢ (u, v) ∈ nonUniforms P G ε ↔ u ∈ P.parts ∧ v ∈ P.parts ∧ u ≠ v ∧ ¬SimpleGraph.IsUniform G ε u v",
"tactic": "rw [nonUniforms, mem_filter, mem_offDiag, and_assoc, and_assoc]"
}
]
| [
213,
66
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
211,
1
]
|
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | Real.exp_log_eq_abs | [
{
"state_after": "no goals",
"state_before": "x y : ℝ\nhx : x ≠ 0\n⊢ exp (log x) = abs x",
"tactic": "rw [log_of_ne_zero hx, ← coe_expOrderIso_apply, OrderIso.apply_symm_apply, Subtype.coe_mk]"
}
]
| [
59,
93
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
58,
1
]
|
Mathlib/Order/Filter/Ultrafilter.lean | Filter.mem_hyperfilter_of_finite_compl | []
| [
506,
43
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
505,
1
]
|
Mathlib/Data/Multiset/Basic.lean | Multiset.count_filter_of_pos | [
{
"state_after": "α : Type u_1\nβ : Type ?u.396498\nγ : Type ?u.396501\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\na : α\ns : Multiset α\nh : p a\n_l : List α\n⊢ List.count a (List.filter (fun b => decide (p b)) _l) = List.count a _l",
"state_before": "α : Type u_1\nβ : Type ?u.396498\nγ : Type ?u.396501\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\na : α\ns : Multiset α\nh : p a\n_l : List α\n⊢ count a (filter p (Quot.mk Setoid.r _l)) = count a (Quot.mk Setoid.r _l)",
"tactic": "simp only [quot_mk_to_coe'', coe_filter, mem_coe, coe_count, decide_eq_true_eq]"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.396498\nγ : Type ?u.396501\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\na : α\ns : Multiset α\nh : p a\n_l : List α\n⊢ decide (p a) = true",
"state_before": "α : Type u_1\nβ : Type ?u.396498\nγ : Type ?u.396501\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\na : α\ns : Multiset α\nh : p a\n_l : List α\n⊢ List.count a (List.filter (fun b => decide (p b)) _l) = List.count a _l",
"tactic": "apply count_filter"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.396498\nγ : Type ?u.396501\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\na : α\ns : Multiset α\nh : p a\n_l : List α\n⊢ decide (p a) = true",
"tactic": "simpa using h"
}
]
| [
2489,
18
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2484,
1
]
|
Mathlib/Algebra/Group/Units.lean | Units.inv_eq_of_mul_eq_one_right | [
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : Monoid α\na✝ b c u : αˣ\na : α\nh : ↑u * a = 1\n⊢ ↑u⁻¹ = ↑u⁻¹ * 1",
"tactic": "rw [mul_one]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : Monoid α\na✝ b c u : αˣ\na : α\nh : ↑u * a = 1\n⊢ ↑u⁻¹ * 1 = a",
"tactic": "rw [← h, inv_mul_cancel_left]"
}
]
| [
360,
46
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
357,
11
]
|
Mathlib/Data/Seq/WSeq.lean | Stream'.WSeq.toList'_cons | [
{
"state_after": "no goals",
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"tactic": "simp [toList, cons]"
}
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| [
1288,
46
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1277,
1
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|
Mathlib/SetTheory/Ordinal/Topology.lean | Ordinal.isClosed_iff_sup | [
{
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"tactic": "use fun hs ι hι f hf => (mem_closed_iff_sup hs).2 ⟨ι, hι, f, hf, rfl⟩"
},
{
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"tactic": "rw [← closure_subset_iff_isClosed]"
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{
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"tactic": "intro h x hx"
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{
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"tactic": "rcases mem_closure_iff_sup.1 hx with ⟨ι, hι, f, hf, rfl⟩"
},
{
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"tactic": "exact h hι f hf"
}
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162,
18
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
155,
1
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|
Mathlib/Topology/MetricSpace/Isometry.lean | IsometryEquiv.symm_symm | []
| [
423,
56
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
423,
1
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|
Mathlib/Algebra/Category/GroupCat/FilteredColimits.lean | GroupCat.FilteredColimits.colimitInvAux_eq_of_rel | [
{
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"tactic": "obtain ⟨k, f, g, hfg⟩ := h"
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{
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"tactic": "rw [MonoidHom.map_inv, MonoidHom.map_inv, inv_inj]"
},
{
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"tactic": "exact hfg"
}
]
| [
101,
12
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
94,
1
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|
Mathlib/Algebra/BigOperators/Finprod.lean | finprod_mem_insert_of_eq_one_if_not_mem | [
{
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},
{
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"tactic": "rintro (rfl | hxs)"
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{
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"tactic": "exacts [not_imp_comm.1 h hx, hxs]"
}
]
| [
878,
36
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
874,
1
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|
Mathlib/RingTheory/MvPolynomial/Homogeneous.lean | MvPolynomial.coeff_homogeneousComponent | []
| [
258,
75
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
256,
1
]
|
Mathlib/Analysis/Convex/Function.lean | ConcaveOn.translate_right | []
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292,
28
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
290,
1
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|
Mathlib/Analysis/ODE/PicardLindelof.lean | PicardLindelof.nonempty_Icc | []
| [
102,
32
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
101,
11
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|
Mathlib/Order/Monotone/Basic.lean | strictAntiOn_comp_ofDual_iff | []
| [
207,
15
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
206,
1
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|
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean | CategoryTheory.IsPushout.of_horiz_isIso | [
{
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"tactic": "refine'\n PushoutCocone.IsColimit.mk _ (fun s => inv inr ≫ s.inr) (fun s => _)\n (by aesop_cat) (by aesop_cat)"
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{
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{
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873,
83
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
867,
1
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|
Mathlib/Computability/TMToPartrec.lean | Turing.PartrecToTM2.codeSupp_case | [
{
"state_after": "no goals",
"state_before": "f g : Code\nk : Cont'\n⊢ codeSupp (Code.case f g) k = trStmts₁ (trNormal (Code.case f g) k) ∪ (codeSupp f k ∪ codeSupp g k)",
"tactic": "simp [codeSupp, codeSupp', contSupp, Finset.union_assoc, Finset.union_left_comm]"
}
]
| [
1852,
86
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1849,
1
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Mathlib/LinearAlgebra/Multilinear/Basic.lean | MultilinearMap.sum_apply | [
{
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},
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{
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253,
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
245,
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|
Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | StructureGroupoid.LocalInvariantProp.liftPropWithinAt_congr_of_eventuallyEq | [
{
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"tactic": "refine' ⟨h.1.congr_of_eventuallyEq h₁ hx, _⟩"
},
{
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"tactic": "refine' hG.congr_nhdsWithin' _\n (by simp_rw [Function.comp_apply, (chartAt H x).left_inv (mem_chart_source H x), hx]) h.2"
},
{
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"state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type ?u.41226\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : LocalInvariantProp G G' P\nh : LiftPropWithinAt P g s x\nh₁ : g' =ᶠ[𝓝[s] x] g\nhx : g' x = g x\n⊢ ↑(chartAt H' (g' x)) ∘\n g' ∘ ↑(LocalHomeomorph.symm (chartAt H x)) =ᶠ[𝓝[↑(LocalHomeomorph.symm (chartAt H x)) ⁻¹' s] ↑(chartAt H x) x]\n ↑(chartAt H' (g x)) ∘ g ∘ ↑(LocalHomeomorph.symm (chartAt H x))",
"tactic": "simp_rw [EventuallyEq, Function.comp_apply]"
},
{
"state_after": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type ?u.41226\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : LocalInvariantProp G G' P\nh : LiftPropWithinAt P g s x\nh₁ : g' =ᶠ[𝓝[s] x] g\nhx : g' x = g x\n⊢ ∀ᶠ (x_1 : M) in 𝓝[s] x, ↑(chartAt H' (g' x)) (g' x_1) = ↑(chartAt H' (g x)) (g x_1)",
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"tactic": "rw [(chartAt H x).eventually_nhdsWithin'\n (fun y ↦ chartAt H' (g' x) (g' y) = chartAt H' (g x) (g y)) (mem_chart_source H x)]"
},
{
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"tactic": "exact h₁.mono fun y hy ↦ by rw [hx, hy]"
},
{
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"tactic": "simp_rw [Function.comp_apply, (chartAt H x).left_inv (mem_chart_source H x), hx]"
},
{
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"state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type ?u.41226\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : LocalInvariantProp G G' P\nh : LiftPropWithinAt P g s x\nh₁ : g' =ᶠ[𝓝[s] x] g\nhx : g' x = g x\ny : M\nhy : g' y = g y\n⊢ ↑(chartAt H' (g' x)) (g' y) = ↑(chartAt H' (g x)) (g y)",
"tactic": "rw [hx, hy]"
}
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410,
42
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
402,
1
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|
Mathlib/Data/Bool/Basic.lean | Bool.and_elim_left | [
{
"state_after": "no goals",
"state_before": "⊢ ∀ {a b : Bool}, (a && b) = true → a = true",
"tactic": "decide"
}
]
| [
173,
64
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
173,
1
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|
Mathlib/Algebra/Order/Floor.lean | round_add_nat | [
{
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"tactic": "rw [round, round, fract_add_nat, Int.floor_add_nat, Int.ceil_add_nat, ← apply_ite₂, ite_self]"
}
]
| [
1380,
96
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1379,
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|
Mathlib/Algebra/Hom/Group.lean | MonoidHom.coe_inj | []
| [
738,
35
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
737,
1
]
|
Mathlib/Algebra/Algebra/Basic.lean | Algebra.mul_sub_algebraMap_commutes | [
{
"state_after": "no goals",
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"tactic": "rw [mul_sub, ← commutes, sub_mul]"
}
]
| [
568,
100
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
567,
1
]
|
Mathlib/AlgebraicTopology/DoldKan/PInfty.lean | AlgebraicTopology.DoldKan.PInfty_f_naturality | []
| [
94,
23
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
92,
1
]
|
Mathlib/MeasureTheory/Function/SimpleFunc.lean | MeasureTheory.SimpleFunc.map_coe_ennreal_restrict | []
| [
775,
44
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
773,
1
]
|
Mathlib/GroupTheory/Perm/Cycle/Type.lean | Equiv.Perm.IsCycle.cycleType | []
| [
109,
41
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
107,
1
]
|
Mathlib/Data/Polynomial/Splits.lean | Polynomial.degree_eq_one_of_irreducible_of_splits | [
{
"state_after": "case inl\nF : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\np : K[X]\nhp : Irreducible p\nh✝ : map (RingHom.id K) p = 0\n⊢ degree p = 1\n\ncase inr\nF : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\np : K[X]\nhp : Irreducible p\nhp_splits : ∀ {g : K[X]}, Irreducible g → g ∣ map (RingHom.id K) p → degree g = 1\n⊢ degree p = 1",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\np : K[X]\nhp : Irreducible p\nhp_splits : Splits (RingHom.id K) p\n⊢ degree p = 1",
"tactic": "rcases hp_splits with ⟨⟩ | hp_splits"
},
{
"state_after": "case inl.h\nF : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\np : K[X]\nhp : Irreducible p\nh✝ : map (RingHom.id K) p = 0\n⊢ False",
"state_before": "case inl\nF : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\np : K[X]\nhp : Irreducible p\nh✝ : map (RingHom.id K) p = 0\n⊢ degree p = 1",
"tactic": "exfalso"
},
{
"state_after": "no goals",
"state_before": "case inl.h\nF : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\np : K[X]\nhp : Irreducible p\nh✝ : map (RingHom.id K) p = 0\n⊢ False",
"tactic": "simp_all"
},
{
"state_after": "case inr\nF : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\np : K[X]\nhp : Irreducible p\nhp_splits : ∀ {g : K[X]}, Irreducible g → g ∣ map (RingHom.id K) p → degree g = 1\n⊢ p ∣ map (RingHom.id K) p",
"state_before": "case inr\nF : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\np : K[X]\nhp : Irreducible p\nhp_splits : ∀ {g : K[X]}, Irreducible g → g ∣ map (RingHom.id K) p → degree g = 1\n⊢ degree p = 1",
"tactic": "apply hp_splits hp"
},
{
"state_after": "no goals",
"state_before": "case inr\nF : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\np : K[X]\nhp : Irreducible p\nhp_splits : ∀ {g : K[X]}, Irreducible g → g ∣ map (RingHom.id K) p → degree g = 1\n⊢ p ∣ map (RingHom.id K) p",
"tactic": "simp"
}
]
| [
290,
9
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
284,
1
]
|
Mathlib/Algebra/Algebra/Equiv.lean | AlgEquiv.mk_coe' | []
| [
360,
47
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
358,
1
]
|
Std/Data/List/Lemmas.lean | List.append_ne_nil_of_ne_nil_right | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ns t : List α\n⊢ t ≠ [] → s ++ t ≠ []",
"tactic": "simp_all"
}
]
| [
115,
91
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
115,
1
]
|
Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | IsBoundedLinearMap.comp | []
| [
161,
78
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
159,
1
]
|
Mathlib/Algebra/Order/Sub/Canonical.lean | tsub_lt_tsub_iff_left_of_le_of_le | []
| [
289,
7
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
286,
1
]
|
Mathlib/Algebra/CharP/Basic.lean | NeZero.of_not_dvd | []
| [
746,
45
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
745,
1
]
|
Mathlib/LinearAlgebra/Determinant.lean | LinearMap.det_eq_det_toMatrix_of_finset | [
{
"state_after": "R : Type ?u.374035\ninst✝¹⁰ : CommRing R\nM : Type u_1\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nM' : Type ?u.374626\ninst✝⁷ : AddCommGroup M'\ninst✝⁶ : Module R M'\nι : Type ?u.375168\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Fintype ι\ne : Basis ι R M\nA : Type u_2\ninst✝³ : CommRing A\ninst✝² : Module A M\nκ : Type ?u.376144\ninst✝¹ : Fintype κ\ninst✝ : DecidableEq M\ns : Finset M\nb : Basis { x // x ∈ s } A M\nf : M →ₗ[A] M\nthis : ∃ s, Nonempty (Basis { x // x ∈ s } A M)\n⊢ ↑LinearMap.det f = det (↑(toMatrix b b) f)",
"state_before": "R : Type ?u.374035\ninst✝¹⁰ : CommRing R\nM : Type u_1\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nM' : Type ?u.374626\ninst✝⁷ : AddCommGroup M'\ninst✝⁶ : Module R M'\nι : Type ?u.375168\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Fintype ι\ne : Basis ι R M\nA : Type u_2\ninst✝³ : CommRing A\ninst✝² : Module A M\nκ : Type ?u.376144\ninst✝¹ : Fintype κ\ninst✝ : DecidableEq M\ns : Finset M\nb : Basis { x // x ∈ s } A M\nf : M →ₗ[A] M\n⊢ ↑LinearMap.det f = det (↑(toMatrix b b) f)",
"tactic": "have : ∃ s : Finset M, Nonempty (Basis s A M) := ⟨s, ⟨b⟩⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.374035\ninst✝¹⁰ : CommRing R\nM : Type u_1\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nM' : Type ?u.374626\ninst✝⁷ : AddCommGroup M'\ninst✝⁶ : Module R M'\nι : Type ?u.375168\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Fintype ι\ne : Basis ι R M\nA : Type u_2\ninst✝³ : CommRing A\ninst✝² : Module A M\nκ : Type ?u.376144\ninst✝¹ : Fintype κ\ninst✝ : DecidableEq M\ns : Finset M\nb : Basis { x // x ∈ s } A M\nf : M →ₗ[A] M\nthis : ∃ s, Nonempty (Basis { x // x ∈ s } A M)\n⊢ ↑LinearMap.det f = det (↑(toMatrix b b) f)",
"tactic": "rw [LinearMap.coe_det, dif_pos, detAux_def'' _ b] <;> assumption"
}
]
| [
206,
67
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
203,
1
]
|
Mathlib/Analysis/Calculus/FDeriv/Mul.lean | Differentiable.pow | []
| [
360,
24
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
359,
1
]
|
Mathlib/LinearAlgebra/Matrix/Transvection.lean | Matrix.Pivot.mul_listTransvecRow_last_col_take | [
{
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"state_before": "n : Type ?u.175750\np : Type ?u.175753\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk : ℕ\nhk : k ≤ r\n⊢ (M ⬝ List.prod (List.take k (listTransvecRow M))) i (inr ()) = M i (inr ())",
"tactic": "induction' k with k IH"
},
{
"state_after": "no goals",
"state_before": "case zero\nn : Type ?u.175750\np : Type ?u.175753\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk : ℕ\nhk✝ : k ≤ r\nhk : Nat.zero ≤ r\n⊢ (M ⬝ List.prod (List.take Nat.zero (listTransvecRow M))) i (inr ()) = M i (inr ())",
"tactic": "simp only [Matrix.mul_one, List.take_zero, List.prod_nil, List.take, Matrix.mul_one]"
},
{
"state_after": "case succ\nn : Type ?u.175750\np : Type ?u.175753\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk✝ : ℕ\nhk✝ : k✝ ≤ r\nk : ℕ\nIH : k ≤ r → (M ⬝ List.prod (List.take k (listTransvecRow M))) i (inr ()) = M i (inr ())\nhk : Nat.succ k ≤ r\nhkr : k < r\n⊢ (M ⬝ List.prod (List.take (Nat.succ k) (listTransvecRow M))) i (inr ()) = M i (inr ())",
"state_before": "case succ\nn : Type ?u.175750\np : Type ?u.175753\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk✝ : ℕ\nhk✝ : k✝ ≤ r\nk : ℕ\nIH : k ≤ r → (M ⬝ List.prod (List.take k (listTransvecRow M))) i (inr ()) = M i (inr ())\nhk : Nat.succ k ≤ r\n⊢ (M ⬝ List.prod (List.take (Nat.succ k) (listTransvecRow M))) i (inr ()) = M i (inr ())",
"tactic": "have hkr : k < r := hk"
},
{
"state_after": "case succ\nn : Type ?u.175750\np : Type ?u.175753\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk✝ : ℕ\nhk✝ : k✝ ≤ r\nk : ℕ\nIH : k ≤ r → (M ⬝ List.prod (List.take k (listTransvecRow M))) i (inr ()) = M i (inr ())\nhk : Nat.succ k ≤ r\nhkr : k < r\nk' : Fin r := { val := k, isLt := hkr }\n⊢ (M ⬝ List.prod (List.take (Nat.succ k) (listTransvecRow M))) i (inr ()) = M i (inr ())",
"state_before": "case succ\nn : Type ?u.175750\np : Type ?u.175753\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk✝ : ℕ\nhk✝ : k✝ ≤ r\nk : ℕ\nIH : k ≤ r → (M ⬝ List.prod (List.take k (listTransvecRow M))) i (inr ()) = M i (inr ())\nhk : Nat.succ k ≤ r\nhkr : k < r\n⊢ (M ⬝ List.prod (List.take (Nat.succ k) (listTransvecRow M))) i (inr ()) = M i (inr ())",
"tactic": "let k' : Fin r := ⟨k, hkr⟩"
},
{
"state_after": "case succ\nn : Type ?u.175750\np : Type ?u.175753\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk✝ : ℕ\nhk✝ : k✝ ≤ r\nk : ℕ\nIH : k ≤ r → (M ⬝ List.prod (List.take k (listTransvecRow M))) i (inr ()) = M i (inr ())\nhk : Nat.succ k ≤ r\nhkr : k < r\nk' : Fin r := { val := k, isLt := hkr }\nthis :\n List.get? (listTransvecRow M) k = some (transvection (inr ()) (inl k') (-M (inr ()) (inl k') / M (inr ()) (inr ())))\n⊢ (M ⬝ List.prod (List.take (Nat.succ k) (listTransvecRow M))) i (inr ()) = M i (inr ())",
"state_before": "case succ\nn : Type ?u.175750\np : Type ?u.175753\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk✝ : ℕ\nhk✝ : k✝ ≤ r\nk : ℕ\nIH : k ≤ r → (M ⬝ List.prod (List.take k (listTransvecRow M))) i (inr ()) = M i (inr ())\nhk : Nat.succ k ≤ r\nhkr : k < r\nk' : Fin r := { val := k, isLt := hkr }\n⊢ (M ⬝ List.prod (List.take (Nat.succ k) (listTransvecRow M))) i (inr ()) = M i (inr ())",
"tactic": "have :\n (listTransvecRow M).get? k =\n ↑(transvection (inr Unit.unit) (inl k')\n (-M (inr Unit.unit) (inl k') / M (inr Unit.unit) (inr Unit.unit))) := by\n simp only [listTransvecRow, List.ofFnNthVal, hkr, dif_pos, List.get?_ofFn]"
},
{
"state_after": "case succ\nn : Type ?u.175750\np : Type ?u.175753\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk✝ : ℕ\nhk✝ : k✝ ≤ r\nk : ℕ\nIH : k ≤ r → (M ⬝ List.prod (List.take k (listTransvecRow M))) i (inr ()) = M i (inr ())\nhk : Nat.succ k ≤ r\nhkr : k < r\nk' : Fin r := { val := k, isLt := hkr }\nthis :\n List.get? (listTransvecRow M) k = some (transvection (inr ()) (inl k') (-M (inr ()) (inl k') / M (inr ()) (inr ())))\n⊢ (M ⬝ List.prod (List.take k (listTransvecRow M)) ⬝\n transvection (inr ()) (inl { val := k, isLt := hkr })\n (-M (inr ()) (inl { val := k, isLt := hkr }) / M (inr ()) (inr ())))\n i (inr ()) =\n M i (inr ())",
"state_before": "case succ\nn : Type ?u.175750\np : Type ?u.175753\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk✝ : ℕ\nhk✝ : k✝ ≤ r\nk : ℕ\nIH : k ≤ r → (M ⬝ List.prod (List.take k (listTransvecRow M))) i (inr ()) = M i (inr ())\nhk : Nat.succ k ≤ r\nhkr : k < r\nk' : Fin r := { val := k, isLt := hkr }\nthis :\n List.get? (listTransvecRow M) k = some (transvection (inr ()) (inl k') (-M (inr ()) (inl k') / M (inr ()) (inr ())))\n⊢ (M ⬝ List.prod (List.take (Nat.succ k) (listTransvecRow M))) i (inr ()) = M i (inr ())",
"tactic": "simp only [List.take_succ, ← Matrix.mul_assoc, this, List.prod_append, Matrix.mul_one,\n Matrix.mul_eq_mul, List.prod_cons, List.prod_nil, Option.to_list_some]"
},
{
"state_after": "case succ.hb\nn : Type ?u.175750\np : Type ?u.175753\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk✝ : ℕ\nhk✝ : k✝ ≤ r\nk : ℕ\nIH : k ≤ r → (M ⬝ List.prod (List.take k (listTransvecRow M))) i (inr ()) = M i (inr ())\nhk : Nat.succ k ≤ r\nhkr : k < r\nk' : Fin r := { val := k, isLt := hkr }\nthis :\n List.get? (listTransvecRow M) k = some (transvection (inr ()) (inl k') (-M (inr ()) (inl k') / M (inr ()) (inr ())))\n⊢ inr () ≠ inl { val := k, isLt := hkr }",
"state_before": "case succ\nn : Type ?u.175750\np : Type ?u.175753\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk✝ : ℕ\nhk✝ : k✝ ≤ r\nk : ℕ\nIH : k ≤ r → (M ⬝ List.prod (List.take k (listTransvecRow M))) i (inr ()) = M i (inr ())\nhk : Nat.succ k ≤ r\nhkr : k < r\nk' : Fin r := { val := k, isLt := hkr }\nthis :\n List.get? (listTransvecRow M) k = some (transvection (inr ()) (inl k') (-M (inr ()) (inl k') / M (inr ()) (inr ())))\n⊢ (M ⬝ List.prod (List.take k (listTransvecRow M)) ⬝\n transvection (inr ()) (inl { val := k, isLt := hkr })\n (-M (inr ()) (inl { val := k, isLt := hkr }) / M (inr ()) (inr ())))\n i (inr ()) =\n M i (inr ())",
"tactic": "rw [mul_transvection_apply_of_ne, IH hkr.le]"
},
{
"state_after": "no goals",
"state_before": "case succ.hb\nn : Type ?u.175750\np : Type ?u.175753\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk✝ : ℕ\nhk✝ : k✝ ≤ r\nk : ℕ\nIH : k ≤ r → (M ⬝ List.prod (List.take k (listTransvecRow M))) i (inr ()) = M i (inr ())\nhk : Nat.succ k ≤ r\nhkr : k < r\nk' : Fin r := { val := k, isLt := hkr }\nthis :\n List.get? (listTransvecRow M) k = some (transvection (inr ()) (inl k') (-M (inr ()) (inl k') / M (inr ()) (inr ())))\n⊢ inr () ≠ inl { val := k, isLt := hkr }",
"tactic": "simp only [Ne.def, not_false_iff]"
},
{
"state_after": "no goals",
"state_before": "n : Type ?u.175750\np : Type ?u.175753\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk✝ : ℕ\nhk✝ : k✝ ≤ r\nk : ℕ\nIH : k ≤ r → (M ⬝ List.prod (List.take k (listTransvecRow M))) i (inr ()) = M i (inr ())\nhk : Nat.succ k ≤ r\nhkr : k < r\nk' : Fin r := { val := k, isLt := hkr }\n⊢ List.get? (listTransvecRow M) k = some (transvection (inr ()) (inl k') (-M (inr ()) (inl k') / M (inr ()) (inr ())))",
"tactic": "simp only [listTransvecRow, List.ofFnNthVal, hkr, dif_pos, List.get?_ofFn]"
}
]
| [
441,
38
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
427,
1
]
|
Mathlib/Data/MvPolynomial/Equiv.lean | MvPolynomial.support_finSuccEquiv_nonempty | [
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1276068\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nh : f ≠ 0\n⊢ ¬↑(finSuccEquiv R n) f = 0",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1276068\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nh : f ≠ 0\n⊢ Finset.Nonempty (Polynomial.support (↑(finSuccEquiv R n) f))",
"tactic": "simp only [Finset.nonempty_iff_ne_empty, Ne, Polynomial.support_eq_empty]"
},
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1276068\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nh : f ≠ 0\nc : ↑(finSuccEquiv R n) f = 0\n⊢ f = 0",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1276068\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nh : f ≠ 0\n⊢ ¬↑(finSuccEquiv R n) f = 0",
"tactic": "refine fun c => h ?_"
},
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1276068\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nh : f ≠ 0\nc : ↑(finSuccEquiv R n) f = 0\nii : (MvPolynomial (Fin n) R)[X] ≃ₐ[R] MvPolynomial (Fin (n + 1)) R := AlgEquiv.symm (finSuccEquiv R n)\n⊢ f = 0",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1276068\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nh : f ≠ 0\nc : ↑(finSuccEquiv R n) f = 0\n⊢ f = 0",
"tactic": "let ii := (finSuccEquiv R n).symm"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1276068\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nh : f ≠ 0\nc : ↑(finSuccEquiv R n) f = 0\nii : (MvPolynomial (Fin n) R)[X] ≃ₐ[R] MvPolynomial (Fin (n + 1)) R := AlgEquiv.symm (finSuccEquiv R n)\n⊢ f = 0",
"tactic": "calc\n f = ii (finSuccEquiv R n f) := by\n simpa only [← AlgEquiv.invFun_eq_symm] using ((finSuccEquiv R n).left_inv f).symm\n _ = ii 0 := by rw [c]\n _ = 0 := by simp"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1276068\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nh : f ≠ 0\nc : ↑(finSuccEquiv R n) f = 0\nii : (MvPolynomial (Fin n) R)[X] ≃ₐ[R] MvPolynomial (Fin (n + 1)) R := AlgEquiv.symm (finSuccEquiv R n)\n⊢ f = ↑ii (↑(finSuccEquiv R n) f)",
"tactic": "simpa only [← AlgEquiv.invFun_eq_symm] using ((finSuccEquiv R n).left_inv f).symm"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1276068\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nh : f ≠ 0\nc : ↑(finSuccEquiv R n) f = 0\nii : (MvPolynomial (Fin n) R)[X] ≃ₐ[R] MvPolynomial (Fin (n + 1)) R := AlgEquiv.symm (finSuccEquiv R n)\n⊢ ↑ii (↑(finSuccEquiv R n) f) = ↑ii 0",
"tactic": "rw [c]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1276068\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nh : f ≠ 0\nc : ↑(finSuccEquiv R n) f = 0\nii : (MvPolynomial (Fin n) R)[X] ≃ₐ[R] MvPolynomial (Fin (n + 1)) R := AlgEquiv.symm (finSuccEquiv R n)\n⊢ ↑ii 0 = 0",
"tactic": "simp"
}
]
| [
472,
21
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
463,
1
]
|
Mathlib/Data/Complex/Exponential.lean | Complex.ofReal_sin | []
| [
939,
25
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
938,
1
]
|
Mathlib/Data/Finset/LocallyFinite.lean | Finset.Icc_self | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.71479\nα : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : LocallyFiniteOrder α\na✝ b c a : α\n⊢ Icc a a = {a}",
"tactic": "rw [← coe_eq_singleton, coe_Icc, Set.Icc_self]"
}
]
| [
528,
94
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
528,
1
]
|
Mathlib/LinearAlgebra/Matrix/Reindex.lean | Matrix.det_reindexAlgEquiv | []
| [
172,
23
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
170,
1
]
|
Mathlib/RingTheory/Polynomial/Content.lean | Polynomial.isPrimitive_one | []
| [
56,
36
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
55,
1
]
|
Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | Pmf.bernoulli_apply | []
| [
312,
68
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
312,
1
]
|
Mathlib/Topology/Instances/ENNReal.lean | NNReal.tsum_lt_tsum | []
| [
1216,
56
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1214,
1
]
|
Mathlib/Order/Filter/NAry.lean | Filter.image2_mem_map₂ | []
| [
64,
29
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
63,
1
]
|
Mathlib/Data/Set/Intervals/Basic.lean | IsMin.Iic_eq | []
| [
957,
18
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
956,
1
]
|
Mathlib/Algebra/Group/Units.lean | Units.mul_inv_eq_one | []
| [
379,
80
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
378,
1
]
|
Mathlib/Algebra/Ring/Equiv.lean | RingEquiv.toRingHom_trans | []
| [
762,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
760,
1
]
|
Mathlib/Algebra/BigOperators/Basic.lean | Finset.prod_bij_ne_one | [
{
"state_after": "case calc_1\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\n⊢ ∀ (a : α) (ha : a ∈ filter (fun x => f x ≠ 1) s),\n (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha ∈ filter (fun x => g x ≠ 1) t\n\ncase calc_2\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\n⊢ ∀ (a : α) (ha : a ∈ filter (fun x => f x ≠ 1) s), f a = g ((fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha)\n\ncase calc_3\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\n⊢ ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ filter (fun x => f x ≠ 1) s) (ha₂ : a₂ ∈ filter (fun x => f x ≠ 1) s),\n (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a₁ ha₁ = (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a₂ ha₂ → a₁ = a₂\n\ncase calc_4\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\n⊢ ∀ (b : γ), b ∈ filter (fun x => g x ≠ 1) t → ∃ a ha, b = (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha",
"state_before": "ι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\n⊢ ∏ x in s, f x = ∏ x in t, g x",
"tactic": "calc\n (∏ x in s, f x) = ∏ x in s.filter fun x => f x ≠ 1, f x := prod_filter_ne_one.symm\n _ = ∏ x in t.filter fun x => g x ≠ 1, g x :=\n prod_bij (fun a ha => i a (mem_filter.mp ha).1 $ by simpa using (mem_filter.mp ha).2)\n ?_ ?_ ?_ ?_\n _ = ∏ x in t, g x := prod_filter_ne_one"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na : α\nha : a ∈ filter (fun x => f x ≠ 1) s\n⊢ f a ≠ 1",
"tactic": "simpa using (mem_filter.mp ha).2"
},
{
"state_after": "case calc_1\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na : α\nha : a ∈ filter (fun x => f x ≠ 1) s\n⊢ (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha ∈ filter (fun x => g x ≠ 1) t",
"state_before": "case calc_1\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\n⊢ ∀ (a : α) (ha : a ∈ filter (fun x => f x ≠ 1) s),\n (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha ∈ filter (fun x => g x ≠ 1) t",
"tactic": "intros a ha"
},
{
"state_after": "case calc_1\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na : α\nha : a ∈ filter (fun x => f x ≠ 1) s\n⊢ a ∈ s → f a ≠ 1 → (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha ∈ filter (fun x => g x ≠ 1) t",
"state_before": "case calc_1\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na : α\nha : a ∈ filter (fun x => f x ≠ 1) s\n⊢ (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha ∈ filter (fun x => g x ≠ 1) t",
"tactic": "refine' (mem_filter.mp ha).elim _"
},
{
"state_after": "case calc_1\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na : α\nha : a ∈ filter (fun x => f x ≠ 1) s\nh₁ : a ∈ s\nh₂ : f a ≠ 1\n⊢ (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha ∈ filter (fun x => g x ≠ 1) t",
"state_before": "case calc_1\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na : α\nha : a ∈ filter (fun x => f x ≠ 1) s\n⊢ a ∈ s → f a ≠ 1 → (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha ∈ filter (fun x => g x ≠ 1) t",
"tactic": "intros h₁ h₂"
},
{
"state_after": "case calc_1\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na : α\nha : a ∈ filter (fun x => f x ≠ 1) s\nh₁ : a ∈ s\nh₂ : f a ≠ 1\n⊢ g ((fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha) ≠ 1",
"state_before": "case calc_1\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na : α\nha : a ∈ filter (fun x => f x ≠ 1) s\nh₁ : a ∈ s\nh₂ : f a ≠ 1\n⊢ (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha ∈ filter (fun x => g x ≠ 1) t",
"tactic": "refine (mem_filter.mpr ⟨hi a h₁ _, ?_⟩)"
},
{
"state_after": "case calc_1\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\na : α\nha : a ∈ filter (fun x => f x ≠ 1) s\nh₁ : a ∈ s\nh₂ : f a ≠ 1\nh : f a = g (i a h₁ (_ : f a = 1 → False))\n⊢ g ((fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha) ≠ 1",
"state_before": "case calc_1\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na : α\nha : a ∈ filter (fun x => f x ≠ 1) s\nh₁ : a ∈ s\nh₂ : f a ≠ 1\n⊢ g ((fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha) ≠ 1",
"tactic": "specialize h a h₁ fun H ↦ by rw [H] at h₂; simp at h₂"
},
{
"state_after": "no goals",
"state_before": "case calc_1\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\na : α\nha : a ∈ filter (fun x => f x ≠ 1) s\nh₁ : a ∈ s\nh₂ : f a ≠ 1\nh : f a = g (i a h₁ (_ : f a = 1 → False))\n⊢ g ((fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha) ≠ 1",
"tactic": "rwa [← h]"
},
{
"state_after": "ι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na : α\nha : a ∈ filter (fun x => f x ≠ 1) s\nh₁ : a ∈ s\nh₂ : 1 ≠ 1\nH : f a = 1\n⊢ False",
"state_before": "ι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na : α\nha : a ∈ filter (fun x => f x ≠ 1) s\nh₁ : a ∈ s\nh₂ : f a ≠ 1\nH : f a = 1\n⊢ False",
"tactic": "rw [H] at h₂"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na : α\nha : a ∈ filter (fun x => f x ≠ 1) s\nh₁ : a ∈ s\nh₂ : 1 ≠ 1\nH : f a = 1\n⊢ False",
"tactic": "simp at h₂"
},
{
"state_after": "case calc_2\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na : α\nha : a ∈ filter (fun x => f x ≠ 1) s\nh₁ : a ∈ s\nh₂ : f a ≠ 1\n⊢ f a = g ((fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha)",
"state_before": "case calc_2\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\n⊢ ∀ (a : α) (ha : a ∈ filter (fun x => f x ≠ 1) s), f a = g ((fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha)",
"tactic": "refine' (fun a ha => (mem_filter.mp ha).elim fun h₁ h₂ ↦ _)"
},
{
"state_after": "no goals",
"state_before": "case calc_2\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na : α\nha : a ∈ filter (fun x => f x ≠ 1) s\nh₁ : a ∈ s\nh₂ : f a ≠ 1\n⊢ f a = g ((fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha)",
"tactic": "exact h a h₁ fun H ↦ by rw [H] at h₂; simp at h₂"
},
{
"state_after": "ι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na : α\nha : a ∈ filter (fun x => f x ≠ 1) s\nh₁ : a ∈ s\nh₂ : 1 ≠ 1\nH : f a = 1\n⊢ False",
"state_before": "ι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na : α\nha : a ∈ filter (fun x => f x ≠ 1) s\nh₁ : a ∈ s\nh₂ : f a ≠ 1\nH : f a = 1\n⊢ False",
"tactic": "rw [H] at h₂"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na : α\nha : a ∈ filter (fun x => f x ≠ 1) s\nh₁ : a ∈ s\nh₂ : 1 ≠ 1\nH : f a = 1\n⊢ False",
"tactic": "simp at h₂"
},
{
"state_after": "case calc_3\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na₁ a₂ : α\nha₁ : a₁ ∈ filter (fun x => f x ≠ 1) s\nha₂ : a₂ ∈ filter (fun x => f x ≠ 1) s\n⊢ (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a₁ ha₁ = (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a₂ ha₂ → a₁ = a₂",
"state_before": "case calc_3\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\n⊢ ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ filter (fun x => f x ≠ 1) s) (ha₂ : a₂ ∈ filter (fun x => f x ≠ 1) s),\n (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a₁ ha₁ = (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a₂ ha₂ → a₁ = a₂",
"tactic": "intros a₁ a₂ ha₁ ha₂"
},
{
"state_after": "case calc_3\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na₁ a₂ : α\nha₁ : a₁ ∈ filter (fun x => f x ≠ 1) s\nha₂ : a₂ ∈ filter (fun x => f x ≠ 1) s\n_ha₁₁ : a₁ ∈ s\n_ha₁₂ : f a₁ ≠ 1\n⊢ (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a₁ ha₁ = (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a₂ ha₂ → a₁ = a₂",
"state_before": "case calc_3\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na₁ a₂ : α\nha₁ : a₁ ∈ filter (fun x => f x ≠ 1) s\nha₂ : a₂ ∈ filter (fun x => f x ≠ 1) s\n⊢ (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a₁ ha₁ = (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a₂ ha₂ → a₁ = a₂",
"tactic": "refine' (mem_filter.mp ha₁).elim fun _ha₁₁ _ha₁₂ ↦ _"
},
{
"state_after": "case calc_3\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na₁ a₂ : α\nha₁ : a₁ ∈ filter (fun x => f x ≠ 1) s\nha₂ : a₂ ∈ filter (fun x => f x ≠ 1) s\n_ha₁₁ : a₁ ∈ s\n_ha₁₂ : f a₁ ≠ 1\n_ha₂₁ : a₂ ∈ s\n_ha₂₂ : f a₂ ≠ 1\n⊢ (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a₁ ha₁ = (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a₂ ha₂ → a₁ = a₂",
"state_before": "case calc_3\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na₁ a₂ : α\nha₁ : a₁ ∈ filter (fun x => f x ≠ 1) s\nha₂ : a₂ ∈ filter (fun x => f x ≠ 1) s\n_ha₁₁ : a₁ ∈ s\n_ha₁₂ : f a₁ ≠ 1\n⊢ (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a₁ ha₁ = (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a₂ ha₂ → a₁ = a₂",
"tactic": "refine' (mem_filter.mp ha₂).elim fun _ha₂₁ _ha₂₂ ↦ _"
},
{
"state_after": "no goals",
"state_before": "case calc_3\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\na₁ a₂ : α\nha₁ : a₁ ∈ filter (fun x => f x ≠ 1) s\nha₂ : a₂ ∈ filter (fun x => f x ≠ 1) s\n_ha₁₁ : a₁ ∈ s\n_ha₁₂ : f a₁ ≠ 1\n_ha₂₁ : a₂ ∈ s\n_ha₂₂ : f a₂ ≠ 1\n⊢ (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a₁ ha₁ = (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a₂ ha₂ → a₁ = a₂",
"tactic": "apply i_inj"
},
{
"state_after": "case calc_4\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\nb : γ\nhb : b ∈ filter (fun x => g x ≠ 1) t\n⊢ ∃ a ha, b = (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha",
"state_before": "case calc_4\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\n⊢ ∀ (b : γ), b ∈ filter (fun x => g x ≠ 1) t → ∃ a ha, b = (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha",
"tactic": "intros b hb"
},
{
"state_after": "case calc_4\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\nb : γ\nhb : b ∈ filter (fun x => g x ≠ 1) t\nh₁ : b ∈ t\nh₂ : g b ≠ 1\n⊢ ∃ a ha, b = (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha",
"state_before": "case calc_4\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\nb : γ\nhb : b ∈ filter (fun x => g x ≠ 1) t\n⊢ ∃ a ha, b = (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha",
"tactic": "refine' (mem_filter.mp hb).elim fun h₁ h₂ ↦ _"
},
{
"state_after": "case calc_4.intro.intro.intro\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\nb : γ\nhb : b ∈ filter (fun x => g x ≠ 1) t\nh₁ : b ∈ t\nh₂ : g b ≠ 1\na : α\nha₁ : a ∈ s\nha₂ : f a ≠ 1\neq : b = i a ha₁ ha₂\n⊢ ∃ a ha, b = (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha",
"state_before": "case calc_4\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\nb : γ\nhb : b ∈ filter (fun x => g x ≠ 1) t\nh₁ : b ∈ t\nh₂ : g b ≠ 1\n⊢ ∃ a ha, b = (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha",
"tactic": "obtain ⟨a, ha₁, ha₂, eq⟩ := i_surj b h₁ fun H ↦ by rw [H] at h₂; simp at h₂"
},
{
"state_after": "no goals",
"state_before": "case calc_4.intro.intro.intro\nι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\nb : γ\nhb : b ∈ filter (fun x => g x ≠ 1) t\nh₁ : b ∈ t\nh₂ : g b ≠ 1\na : α\nha₁ : a ∈ s\nha₂ : f a ≠ 1\neq : b = i a ha₁ ha₂\n⊢ ∃ a ha, b = (fun a ha => i a (_ : a ∈ s) (_ : f a ≠ 1)) a ha",
"tactic": "exact ⟨a, mem_filter.mpr ⟨ha₁, ha₂⟩, eq⟩"
},
{
"state_after": "ι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\nb : γ\nhb : b ∈ filter (fun x => g x ≠ 1) t\nh₁ : b ∈ t\nh₂ : 1 ≠ 1\nH : g b = 1\n⊢ False",
"state_before": "ι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\nb : γ\nhb : b ∈ filter (fun x => g x ≠ 1) t\nh₁ : b ∈ t\nh₂ : g b ≠ 1\nH : g b = 1\n⊢ False",
"tactic": "rw [H] at h₂"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.422672\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α → β\ng : γ → β\ni : (a : α) → a ∈ s → f a ≠ 1 → γ\nhi : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ a₂ : α) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂\ni_surj : ∀ (b : γ), b ∈ t → g b ≠ 1 → ∃ a h₁ h₂, b = i a h₁ h₂\nh : ∀ (a : α) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)\nb : γ\nhb : b ∈ filter (fun x => g x ≠ 1) t\nh₁ : b ∈ t\nh₂ : 1 ≠ 1\nH : g b = 1\n⊢ False",
"tactic": "simp at h₂"
}
]
| [
1167,
45
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1140,
1
]
|
Mathlib/LinearAlgebra/SesquilinearForm.lean | LinearMap.isSymm_iff_eq_flip | [
{
"state_after": "case mp\nR : Type u_1\nR₁ : Type ?u.168800\nR₂ : Type ?u.168803\nR₃ : Type ?u.168806\nM : Type u_2\nM₁ : Type ?u.168812\nM₂ : Type ?u.168815\nMₗ₁ : Type ?u.168818\nMₗ₁' : Type ?u.168821\nMₗ₂ : Type ?u.168824\nMₗ₂' : Type ?u.168827\nK : Type ?u.168830\nK₁ : Type ?u.168833\nK₂ : Type ?u.168836\nV : Type ?u.168839\nV₁ : Type ?u.168842\nV₂ : Type ?u.168845\nn : Type ?u.168848\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nI : R →+* R\nB✝ : M →ₛₗ[I] M →ₗ[R] R\nB : M →ₗ[R] M →ₗ[R] R\nh : IsSymm B\n⊢ B = flip B\n\ncase mpr\nR : Type u_1\nR₁ : Type ?u.168800\nR₂ : Type ?u.168803\nR₃ : Type ?u.168806\nM : Type u_2\nM₁ : Type ?u.168812\nM₂ : Type ?u.168815\nMₗ₁ : Type ?u.168818\nMₗ₁' : Type ?u.168821\nMₗ₂ : Type ?u.168824\nMₗ₂' : Type ?u.168827\nK : Type ?u.168830\nK₁ : Type ?u.168833\nK₂ : Type ?u.168836\nV : Type ?u.168839\nV₁ : Type ?u.168842\nV₂ : Type ?u.168845\nn : Type ?u.168848\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nI : R →+* R\nB✝ : M →ₛₗ[I] M →ₗ[R] R\nB : M →ₗ[R] M →ₗ[R] R\nh : B = flip B\n⊢ IsSymm B",
"state_before": "R : Type u_1\nR₁ : Type ?u.168800\nR₂ : Type ?u.168803\nR₃ : Type ?u.168806\nM : Type u_2\nM₁ : Type ?u.168812\nM₂ : Type ?u.168815\nMₗ₁ : Type ?u.168818\nMₗ₁' : Type ?u.168821\nMₗ₂ : Type ?u.168824\nMₗ₂' : Type ?u.168827\nK : Type ?u.168830\nK₁ : Type ?u.168833\nK₂ : Type ?u.168836\nV : Type ?u.168839\nV₁ : Type ?u.168842\nV₂ : Type ?u.168845\nn : Type ?u.168848\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nI : R →+* R\nB✝ : M →ₛₗ[I] M →ₗ[R] R\nB : M →ₗ[R] M →ₗ[R] R\n⊢ IsSymm B ↔ B = flip B",
"tactic": "constructor <;> intro h"
},
{
"state_after": "case mpr\nR : Type u_1\nR₁ : Type ?u.168800\nR₂ : Type ?u.168803\nR₃ : Type ?u.168806\nM : Type u_2\nM₁ : Type ?u.168812\nM₂ : Type ?u.168815\nMₗ₁ : Type ?u.168818\nMₗ₁' : Type ?u.168821\nMₗ₂ : Type ?u.168824\nMₗ₂' : Type ?u.168827\nK : Type ?u.168830\nK₁ : Type ?u.168833\nK₂ : Type ?u.168836\nV : Type ?u.168839\nV₁ : Type ?u.168842\nV₂ : Type ?u.168845\nn : Type ?u.168848\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nI : R →+* R\nB✝ : M →ₛₗ[I] M →ₗ[R] R\nB : M →ₗ[R] M →ₗ[R] R\nh : B = flip B\nx y : M\n⊢ ↑(RingHom.id R) (↑(↑B x) y) = ↑(↑B y) x",
"state_before": "case mpr\nR : Type u_1\nR₁ : Type ?u.168800\nR₂ : Type ?u.168803\nR₃ : Type ?u.168806\nM : Type u_2\nM₁ : Type ?u.168812\nM₂ : Type ?u.168815\nMₗ₁ : Type ?u.168818\nMₗ₁' : Type ?u.168821\nMₗ₂ : Type ?u.168824\nMₗ₂' : Type ?u.168827\nK : Type ?u.168830\nK₁ : Type ?u.168833\nK₂ : Type ?u.168836\nV : Type ?u.168839\nV₁ : Type ?u.168842\nV₂ : Type ?u.168845\nn : Type ?u.168848\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nI : R →+* R\nB✝ : M →ₛₗ[I] M →ₗ[R] R\nB : M →ₗ[R] M →ₗ[R] R\nh : B = flip B\n⊢ IsSymm B",
"tactic": "intro x y"
},
{
"state_after": "no goals",
"state_before": "case mpr\nR : Type u_1\nR₁ : Type ?u.168800\nR₂ : Type ?u.168803\nR₃ : Type ?u.168806\nM : Type u_2\nM₁ : Type ?u.168812\nM₂ : Type ?u.168815\nMₗ₁ : Type ?u.168818\nMₗ₁' : Type ?u.168821\nMₗ₂ : Type ?u.168824\nMₗ₂' : Type ?u.168827\nK : Type ?u.168830\nK₁ : Type ?u.168833\nK₂ : Type ?u.168836\nV : Type ?u.168839\nV₁ : Type ?u.168842\nV₂ : Type ?u.168845\nn : Type ?u.168848\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nI : R →+* R\nB✝ : M →ₛₗ[I] M →ₗ[R] R\nB : M →ₗ[R] M →ₗ[R] R\nh : B = flip B\nx y : M\n⊢ ↑(RingHom.id R) (↑(↑B x) y) = ↑(↑B y) x",
"tactic": "conv_lhs => rw [h]"
},
{
"state_after": "case mp.h.h\nR : Type u_1\nR₁ : Type ?u.168800\nR₂ : Type ?u.168803\nR₃ : Type ?u.168806\nM : Type u_2\nM₁ : Type ?u.168812\nM₂ : Type ?u.168815\nMₗ₁ : Type ?u.168818\nMₗ₁' : Type ?u.168821\nMₗ₂ : Type ?u.168824\nMₗ₂' : Type ?u.168827\nK : Type ?u.168830\nK₁ : Type ?u.168833\nK₂ : Type ?u.168836\nV : Type ?u.168839\nV₁ : Type ?u.168842\nV₂ : Type ?u.168845\nn : Type ?u.168848\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nI : R →+* R\nB✝ : M →ₛₗ[I] M →ₗ[R] R\nB : M →ₗ[R] M →ₗ[R] R\nh : IsSymm B\nx✝¹ x✝ : M\n⊢ ↑(↑B x✝¹) x✝ = ↑(↑(flip B) x✝¹) x✝",
"state_before": "case mp\nR : Type u_1\nR₁ : Type ?u.168800\nR₂ : Type ?u.168803\nR₃ : Type ?u.168806\nM : Type u_2\nM₁ : Type ?u.168812\nM₂ : Type ?u.168815\nMₗ₁ : Type ?u.168818\nMₗ₁' : Type ?u.168821\nMₗ₂ : Type ?u.168824\nMₗ₂' : Type ?u.168827\nK : Type ?u.168830\nK₁ : Type ?u.168833\nK₂ : Type ?u.168836\nV : Type ?u.168839\nV₁ : Type ?u.168842\nV₂ : Type ?u.168845\nn : Type ?u.168848\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nI : R →+* R\nB✝ : M →ₛₗ[I] M →ₗ[R] R\nB : M →ₗ[R] M →ₗ[R] R\nh : IsSymm B\n⊢ B = flip B",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case mp.h.h\nR : Type u_1\nR₁ : Type ?u.168800\nR₂ : Type ?u.168803\nR₃ : Type ?u.168806\nM : Type u_2\nM₁ : Type ?u.168812\nM₂ : Type ?u.168815\nMₗ₁ : Type ?u.168818\nMₗ₁' : Type ?u.168821\nMₗ₂ : Type ?u.168824\nMₗ₂' : Type ?u.168827\nK : Type ?u.168830\nK₁ : Type ?u.168833\nK₂ : Type ?u.168836\nV : Type ?u.168839\nV₁ : Type ?u.168842\nV₂ : Type ?u.168845\nn : Type ?u.168848\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nI : R →+* R\nB✝ : M →ₛₗ[I] M →ₗ[R] R\nB : M →ₗ[R] M →ₗ[R] R\nh : IsSymm B\nx✝¹ x✝ : M\n⊢ ↑(↑B x✝¹) x✝ = ↑(↑(flip B) x✝¹) x✝",
"tactic": "rw [← h, flip_apply, RingHom.id_apply]"
}
]
| [
252,
21
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
247,
1
]
|
Mathlib/LinearAlgebra/AffineSpace/Basis.lean | AffineBasis.affineCombination_coord_eq_self | [
{
"state_after": "ι : Type u_1\nι' : Type ?u.118805\nk : Type u_4\nV : Type u_3\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni j : ι\ne : ι ≃ ι'\ninst✝ : Fintype ι\nq : P\nhq : q ∈ affineSpan k (range ↑b)\n⊢ (↑(Finset.affineCombination k Finset.univ ↑b) fun i => ↑(coord b i) q) = q",
"state_before": "ι : Type u_1\nι' : Type ?u.118805\nk : Type u_4\nV : Type u_3\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni j : ι\ne : ι ≃ ι'\ninst✝ : Fintype ι\nq : P\n⊢ (↑(Finset.affineCombination k Finset.univ ↑b) fun i => ↑(coord b i) q) = q",
"tactic": "have hq : q ∈ affineSpan k (range b) := by\n rw [b.tot]\n exact AffineSubspace.mem_top k V q"
},
{
"state_after": "case intro.intro\nι : Type u_1\nι' : Type ?u.118805\nk : Type u_4\nV : Type u_3\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni j : ι\ne : ι ≃ ι'\ninst✝ : Fintype ι\nw : ι → k\nhw : ∑ i : ι, w i = 1\nhq : ↑(Finset.affineCombination k Finset.univ ↑b) w ∈ affineSpan k (range ↑b)\n⊢ (↑(Finset.affineCombination k Finset.univ ↑b) fun i =>\n ↑(coord b i) (↑(Finset.affineCombination k Finset.univ ↑b) w)) =\n ↑(Finset.affineCombination k Finset.univ ↑b) w",
"state_before": "ι : Type u_1\nι' : Type ?u.118805\nk : Type u_4\nV : Type u_3\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni j : ι\ne : ι ≃ ι'\ninst✝ : Fintype ι\nq : P\nhq : q ∈ affineSpan k (range ↑b)\n⊢ (↑(Finset.affineCombination k Finset.univ ↑b) fun i => ↑(coord b i) q) = q",
"tactic": "obtain ⟨w, hw, rfl⟩ := eq_affineCombination_of_mem_affineSpan_of_fintype hq"
},
{
"state_after": "case intro.intro.h.e_6.h\nι : Type u_1\nι' : Type ?u.118805\nk : Type u_4\nV : Type u_3\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni j : ι\ne : ι ≃ ι'\ninst✝ : Fintype ι\nw : ι → k\nhw : ∑ i : ι, w i = 1\nhq : ↑(Finset.affineCombination k Finset.univ ↑b) w ∈ affineSpan k (range ↑b)\n⊢ (fun i => ↑(coord b i) (↑(Finset.affineCombination k Finset.univ ↑b) w)) = w",
"state_before": "case intro.intro\nι : Type u_1\nι' : Type ?u.118805\nk : Type u_4\nV : Type u_3\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni j : ι\ne : ι ≃ ι'\ninst✝ : Fintype ι\nw : ι → k\nhw : ∑ i : ι, w i = 1\nhq : ↑(Finset.affineCombination k Finset.univ ↑b) w ∈ affineSpan k (range ↑b)\n⊢ (↑(Finset.affineCombination k Finset.univ ↑b) fun i =>\n ↑(coord b i) (↑(Finset.affineCombination k Finset.univ ↑b) w)) =\n ↑(Finset.affineCombination k Finset.univ ↑b) w",
"tactic": "congr"
},
{
"state_after": "case intro.intro.h.e_6.h.h\nι : Type u_1\nι' : Type ?u.118805\nk : Type u_4\nV : Type u_3\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Fintype ι\nw : ι → k\nhw : ∑ i : ι, w i = 1\nhq : ↑(Finset.affineCombination k Finset.univ ↑b) w ∈ affineSpan k (range ↑b)\ni : ι\n⊢ ↑(coord b i) (↑(Finset.affineCombination k Finset.univ ↑b) w) = w i",
"state_before": "case intro.intro.h.e_6.h\nι : Type u_1\nι' : Type ?u.118805\nk : Type u_4\nV : Type u_3\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni j : ι\ne : ι ≃ ι'\ninst✝ : Fintype ι\nw : ι → k\nhw : ∑ i : ι, w i = 1\nhq : ↑(Finset.affineCombination k Finset.univ ↑b) w ∈ affineSpan k (range ↑b)\n⊢ (fun i => ↑(coord b i) (↑(Finset.affineCombination k Finset.univ ↑b) w)) = w",
"tactic": "ext i"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.h.e_6.h.h\nι : Type u_1\nι' : Type ?u.118805\nk : Type u_4\nV : Type u_3\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Fintype ι\nw : ι → k\nhw : ∑ i : ι, w i = 1\nhq : ↑(Finset.affineCombination k Finset.univ ↑b) w ∈ affineSpan k (range ↑b)\ni : ι\n⊢ ↑(coord b i) (↑(Finset.affineCombination k Finset.univ ↑b) w) = w i",
"tactic": "exact b.coord_apply_combination_of_mem (Finset.mem_univ i) hw"
},
{
"state_after": "ι : Type u_1\nι' : Type ?u.118805\nk : Type u_4\nV : Type u_3\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni j : ι\ne : ι ≃ ι'\ninst✝ : Fintype ι\nq : P\n⊢ q ∈ ⊤",
"state_before": "ι : Type u_1\nι' : Type ?u.118805\nk : Type u_4\nV : Type u_3\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni j : ι\ne : ι ≃ ι'\ninst✝ : Fintype ι\nq : P\n⊢ q ∈ affineSpan k (range ↑b)",
"tactic": "rw [b.tot]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type ?u.118805\nk : Type u_4\nV : Type u_3\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni j : ι\ne : ι ≃ ι'\ninst✝ : Fintype ι\nq : P\n⊢ q ∈ ⊤",
"tactic": "exact AffineSubspace.mem_top k V q"
}
]
| [
224,
64
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
216,
1
]
|
Mathlib/Order/Bounds/Basic.lean | bot_mem_lowerBounds | []
| [
102,
94
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
102,
1
]
|
Mathlib/Algebra/Module/Basic.lean | int_smul_eq_zsmul | [
{
"state_after": "α : Type ?u.158540\nR : Type ?u.158543\nk : Type ?u.158546\nS : Type ?u.158549\nM : Type u_1\nM₂ : Type ?u.158555\nM₃ : Type ?u.158558\nι : Type ?u.158561\ninst✝⁴ : Semiring S\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module S M\ninst✝ : Module R M\nh : Module ℤ M\nn : ℤ\nx : M\n⊢ SMul.smul n x = n • x",
"state_before": "α : Type ?u.158540\nR : Type ?u.158543\nk : Type ?u.158546\nS : Type ?u.158549\nM : Type u_1\nM₂ : Type ?u.158555\nM₃ : Type ?u.158558\nι : Type ?u.158561\ninst✝⁴ : Semiring S\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module S M\ninst✝ : Module R M\nh : Module ℤ M\nn : ℤ\nx : M\n⊢ SMul.smul n x = n • x",
"tactic": "rw [zsmul_eq_smul_cast ℤ n x, Int.cast_id]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.158540\nR : Type ?u.158543\nk : Type ?u.158546\nS : Type ?u.158549\nM : Type u_1\nM₂ : Type ?u.158555\nM₃ : Type ?u.158558\nι : Type ?u.158561\ninst✝⁴ : Semiring S\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module S M\ninst✝ : Module R M\nh : Module ℤ M\nn : ℤ\nx : M\n⊢ SMul.smul n x = n • x",
"tactic": "rfl"
}
]
| [
443,
94
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
442,
1
]
|
Std/Data/Rat/Lemmas.lean | Rat.neg_mkRat | [
{
"state_after": "no goals",
"state_before": "n : Int\nd : Nat\n⊢ -mkRat n d = mkRat (-n) d",
"tactic": "if z : d = 0 then simp [z] else simp [← normalize_eq_mkRat z, neg_normalize]"
},
{
"state_after": "no goals",
"state_before": "n : Int\nd : Nat\nz : d = 0\n⊢ -mkRat n d = mkRat (-n) d",
"tactic": "simp [z]"
},
{
"state_after": "no goals",
"state_before": "n : Int\nd : Nat\nz : ¬d = 0\n⊢ -mkRat n d = mkRat (-n) d",
"tactic": "simp [← normalize_eq_mkRat z, neg_normalize]"
}
]
| [
213,
79
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
212,
1
]
|
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