file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Topology/Basic.lean
|
IsOpen.inter
|
[] |
[
126,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
|
Real.cos_arcsin
|
[
{
"state_after": "case pos\nx : ℝ\nhx₁ : -1 ≤ x\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)\n\ncase neg\nx : ℝ\nhx₁ : ¬-1 ≤ x\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)",
"state_before": "x : ℝ\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)",
"tactic": "by_cases hx₁ : -1 ≤ x"
},
{
"state_after": "case neg\nx : ℝ\nhx₁ : ¬-1 ≤ x\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)\n\ncase pos\nx : ℝ\nhx₁ : -1 ≤ x\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)",
"state_before": "case pos\nx : ℝ\nhx₁ : -1 ≤ x\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)\n\ncase neg\nx : ℝ\nhx₁ : ¬-1 ≤ x\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)",
"tactic": "swap"
},
{
"state_after": "case pos\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : x ≤ 1\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)\n\ncase neg\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : ¬x ≤ 1\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)",
"state_before": "case pos\nx : ℝ\nhx₁ : -1 ≤ x\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)",
"tactic": "by_cases hx₂ : x ≤ 1"
},
{
"state_after": "case neg\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : ¬x ≤ 1\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)\n\ncase pos\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : x ≤ 1\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)",
"state_before": "case pos\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : x ≤ 1\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)\n\ncase neg\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : ¬x ≤ 1\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)",
"tactic": "swap"
},
{
"state_after": "case pos\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : x ≤ 1\nthis : sin (arcsin x) ^ 2 + cos (arcsin x) ^ 2 = 1\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)",
"state_before": "case pos\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : x ≤ 1\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)",
"tactic": "have : sin (arcsin x) ^ 2 + cos (arcsin x) ^ 2 = 1 := sin_sq_add_cos_sq (arcsin x)"
},
{
"state_after": "case pos\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : x ≤ 1\nthis : cos (arcsin x) = sqrt (1 - sin (arcsin x) ^ 2)\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)",
"state_before": "case pos\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : x ≤ 1\nthis : sin (arcsin x) ^ 2 + cos (arcsin x) ^ 2 = 1\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)",
"tactic": "rw [← eq_sub_iff_add_eq', ← sqrt_inj (sq_nonneg _) (sub_nonneg.2 (sin_sq_le_one (arcsin x))), sq,\n sqrt_mul_self (cos_arcsin_nonneg _)] at this"
},
{
"state_after": "no goals",
"state_before": "case pos\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : x ≤ 1\nthis : cos (arcsin x) = sqrt (1 - sin (arcsin x) ^ 2)\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)",
"tactic": "rw [this, sin_arcsin hx₁ hx₂]"
},
{
"state_after": "case neg\nx : ℝ\nhx₁ : x < -1\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)",
"state_before": "case neg\nx : ℝ\nhx₁ : ¬-1 ≤ x\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)",
"tactic": "rw [not_le] at hx₁"
},
{
"state_after": "case neg\nx : ℝ\nhx₁ : x < -1\n⊢ 1 - x ^ 2 ≤ 0",
"state_before": "case neg\nx : ℝ\nhx₁ : x < -1\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)",
"tactic": "rw [arcsin_of_le_neg_one hx₁.le, cos_neg, cos_pi_div_two, sqrt_eq_zero_of_nonpos]"
},
{
"state_after": "no goals",
"state_before": "case neg\nx : ℝ\nhx₁ : x < -1\n⊢ 1 - x ^ 2 ≤ 0",
"tactic": "nlinarith"
},
{
"state_after": "case neg\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : 1 < x\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)",
"state_before": "case neg\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : ¬x ≤ 1\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)",
"tactic": "rw [not_le] at hx₂"
},
{
"state_after": "case neg\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : 1 < x\n⊢ 1 - x ^ 2 ≤ 0",
"state_before": "case neg\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : 1 < x\n⊢ cos (arcsin x) = sqrt (1 - x ^ 2)",
"tactic": "rw [arcsin_of_one_le hx₂.le, cos_pi_div_two, sqrt_eq_zero_of_nonpos]"
},
{
"state_after": "no goals",
"state_before": "case neg\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : 1 < x\n⊢ 1 - x ^ 2 ≤ 0",
"tactic": "nlinarith"
}
] |
[
318,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
306,
1
] |
Mathlib/Algebra/Homology/ImageToKernel.lean
|
imageToKernel_arrow
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.5291\nV : Type u\ninst✝³ : Category V\ninst✝² : HasZeroMorphisms V\nA B C : V\nf : A ⟶ B\ninst✝¹ : HasImage f\ng : B ⟶ C\ninst✝ : HasKernel g\nw : f ≫ g = 0\n⊢ imageToKernel f g w ≫ Subobject.arrow (kernelSubobject g) = Subobject.arrow (imageSubobject f)",
"tactic": "simp [imageToKernel]"
}
] |
[
67,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/Analysis/Calculus/TangentCone.lean
|
UniqueDiffWithinAt.mono_nhds
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type ?u.112500\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.112590\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nx y : E\ns t : Set E\nst : 𝓝[s] x ≤ 𝓝[t] x\nh : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ∧ x ∈ closure s\n⊢ Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 t x)) ∧ x ∈ closure t",
"state_before": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type ?u.112500\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.112590\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nx y : E\ns t : Set E\nh : UniqueDiffWithinAt 𝕜 s x\nst : 𝓝[s] x ≤ 𝓝[t] x\n⊢ UniqueDiffWithinAt 𝕜 t x",
"tactic": "simp only [uniqueDiffWithinAt_iff] at *"
},
{
"state_after": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type ?u.112500\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.112590\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nx y : E\ns t : Set E\nst : 𝓝[s] x ≤ 𝓝[t] x\nh : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ∧ NeBot (𝓝[s] x)\n⊢ Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 t x)) ∧ NeBot (𝓝[t] x)",
"state_before": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type ?u.112500\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.112590\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nx y : E\ns t : Set E\nst : 𝓝[s] x ≤ 𝓝[t] x\nh : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ∧ x ∈ closure s\n⊢ Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 t x)) ∧ x ∈ closure t",
"tactic": "rw [mem_closure_iff_nhdsWithin_neBot] at h⊢"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type ?u.112500\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.112590\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nx y : E\ns t : Set E\nst : 𝓝[s] x ≤ 𝓝[t] x\nh : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ∧ NeBot (𝓝[s] x)\n⊢ Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 t x)) ∧ NeBot (𝓝[t] x)",
"tactic": "exact ⟨h.1.mono <| Submodule.span_mono <| tangentCone_mono_nhds st, h.2.mono st⟩"
}
] |
[
285,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
281,
1
] |
Mathlib/Data/Set/Intervals/Monotone.lean
|
Antitone.Ici
|
[] |
[
49,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
48,
11
] |
Mathlib/GroupTheory/Abelianization.lean
|
rank_commutator_le_card
|
[
{
"state_after": "G : Type u\ninst✝¹ : Group G\ninst✝ : Finite ↑(commutatorSet G)\n⊢ Group.rank { x // x ∈ Subgroup.closure (commutatorSet G) } ≤ Nat.card ↑(commutatorSet G)",
"state_before": "G : Type u\ninst✝¹ : Group G\ninst✝ : Finite ↑(commutatorSet G)\n⊢ Group.rank { x // x ∈ commutator G } ≤ Nat.card ↑(commutatorSet G)",
"tactic": "rw [Subgroup.rank_congr (commutator_eq_closure G)]"
},
{
"state_after": "no goals",
"state_before": "G : Type u\ninst✝¹ : Group G\ninst✝ : Finite ↑(commutatorSet G)\n⊢ Group.rank { x // x ∈ Subgroup.closure (commutatorSet G) } ≤ Nat.card ↑(commutatorSet G)",
"tactic": "apply Subgroup.rank_closure_finite_le_nat_card"
}
] |
[
69,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.toNNReal_inv
|
[
{
"state_after": "case top\nα : Type ?u.836153\nβ : Type ?u.836156\na b c d : ℝ≥0∞\nr p q : ℝ≥0\n⊢ ENNReal.toNNReal ⊤⁻¹ = (ENNReal.toNNReal ⊤)⁻¹\n\ncase coe\nα : Type ?u.836153\nβ : Type ?u.836156\na✝ b c d : ℝ≥0∞\nr p q a : ℝ≥0\n⊢ ENNReal.toNNReal (↑a)⁻¹ = (ENNReal.toNNReal ↑a)⁻¹",
"state_before": "α : Type ?u.836153\nβ : Type ?u.836156\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\na : ℝ≥0∞\n⊢ ENNReal.toNNReal a⁻¹ = (ENNReal.toNNReal a)⁻¹",
"tactic": "induction' a using recTopCoe with a"
},
{
"state_after": "case coe.inl\nα : Type ?u.836153\nβ : Type ?u.836156\na b c d : ℝ≥0∞\nr p q : ℝ≥0\n⊢ ENNReal.toNNReal (↑0)⁻¹ = (ENNReal.toNNReal ↑0)⁻¹\n\ncase coe.inr\nα : Type ?u.836153\nβ : Type ?u.836156\na✝ b c d : ℝ≥0∞\nr p q a : ℝ≥0\nha : a ≠ 0\n⊢ ENNReal.toNNReal (↑a)⁻¹ = (ENNReal.toNNReal ↑a)⁻¹",
"state_before": "case coe\nα : Type ?u.836153\nβ : Type ?u.836156\na✝ b c d : ℝ≥0∞\nr p q a : ℝ≥0\n⊢ ENNReal.toNNReal (↑a)⁻¹ = (ENNReal.toNNReal ↑a)⁻¹",
"tactic": "rcases eq_or_ne a 0 with (rfl | ha)"
},
{
"state_after": "no goals",
"state_before": "case coe.inr\nα : Type ?u.836153\nβ : Type ?u.836156\na✝ b c d : ℝ≥0∞\nr p q a : ℝ≥0\nha : a ≠ 0\n⊢ ENNReal.toNNReal (↑a)⁻¹ = (ENNReal.toNNReal ↑a)⁻¹",
"tactic": "rw [← coe_inv ha, toNNReal_coe, toNNReal_coe]"
},
{
"state_after": "no goals",
"state_before": "case top\nα : Type ?u.836153\nβ : Type ?u.836156\na b c d : ℝ≥0∞\nr p q : ℝ≥0\n⊢ ENNReal.toNNReal ⊤⁻¹ = (ENNReal.toNNReal ⊤)⁻¹",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case coe.inl\nα : Type ?u.836153\nβ : Type ?u.836156\na b c d : ℝ≥0∞\nr p q : ℝ≥0\n⊢ ENNReal.toNNReal (↑0)⁻¹ = (ENNReal.toNNReal ↑0)⁻¹",
"tactic": "simp"
}
] |
[
2314,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2311,
1
] |
Mathlib/Data/Nat/Basic.lean
|
Nat.pred_eq_of_eq_succ
|
[
{
"state_after": "no goals",
"state_before": "m✝ n✝ k m n : ℕ\nH : m = succ n\n⊢ pred m = n",
"tactic": "simp [H]"
}
] |
[
311,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
311,
1
] |
Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean
|
SL_reduction_mod_hom_val
|
[] |
[
45,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
ZFSet.sUnion_lem
|
[
{
"state_after": "α β : Type u\nA : α → PSet\nB : β → PSet\nαβ : ∀ (a : α), ∃ b, PSet.Equiv (A a) (B b)\na : Type (PSet.mk α A)\nc : Type (Func (PSet.mk α A) a)\nb : β\nhb : PSet.Equiv (A a) (B b)\n⊢ ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)",
"state_before": "α β : Type u\nA : α → PSet\nB : β → PSet\nαβ : ∀ (a : α), ∃ b, PSet.Equiv (A a) (B b)\na : Type (PSet.mk α A)\nc : Type (Func (PSet.mk α A) a)\n⊢ ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)",
"tactic": "let ⟨b, hb⟩ := αβ a"
},
{
"state_after": "case mk\nα β : Type u\nA : α → PSet\nB : β → PSet\nαβ : ∀ (a : α), ∃ b, PSet.Equiv (A a) (B b)\na : Type (PSet.mk α A)\nc : Type (Func (PSet.mk α A) a)\nb : β\nhb : PSet.Equiv (A a) (B b)\nx✝ : PSet\nea✝ : A a = x✝\nγ : Type u\nΓ : γ → PSet\nA_ih✝ :\n ∀ (a_1 : γ), A a = Γ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)\nea : A a = PSet.mk γ Γ\n⊢ ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)",
"state_before": "α β : Type u\nA : α → PSet\nB : β → PSet\nαβ : ∀ (a : α), ∃ b, PSet.Equiv (A a) (B b)\na : Type (PSet.mk α A)\nc : Type (Func (PSet.mk α A) a)\nb : β\nhb : PSet.Equiv (A a) (B b)\n⊢ ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)",
"tactic": "induction' ea : A a with γ Γ"
},
{
"state_after": "case mk.mk\nα β : Type u\nA : α → PSet\nB : β → PSet\nαβ : ∀ (a : α), ∃ b, PSet.Equiv (A a) (B b)\na : Type (PSet.mk α A)\nc : Type (Func (PSet.mk α A) a)\nb : β\nhb : PSet.Equiv (A a) (B b)\nx✝¹ : PSet\nea✝ : A a = x✝¹\nγ : Type u\nΓ : γ → PSet\nA_ih✝¹ :\n ∀ (a_1 : γ), A a = Γ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)\nea : A a = PSet.mk γ Γ\nx✝ : PSet\neb✝ : B b = x✝\nδ : Type u\nΔ : δ → PSet\nA_ih✝ :\n ∀ (a_1 : δ), B b = Δ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)\neb : B b = PSet.mk δ Δ\n⊢ ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)",
"state_before": "case mk\nα β : Type u\nA : α → PSet\nB : β → PSet\nαβ : ∀ (a : α), ∃ b, PSet.Equiv (A a) (B b)\na : Type (PSet.mk α A)\nc : Type (Func (PSet.mk α A) a)\nb : β\nhb : PSet.Equiv (A a) (B b)\nx✝ : PSet\nea✝ : A a = x✝\nγ : Type u\nΓ : γ → PSet\nA_ih✝ :\n ∀ (a_1 : γ), A a = Γ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)\nea : A a = PSet.mk γ Γ\n⊢ ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)",
"tactic": "induction' eb : B b with δ Δ"
},
{
"state_after": "case mk.mk\nα β : Type u\nA : α → PSet\nB : β → PSet\nαβ : ∀ (a : α), ∃ b, PSet.Equiv (A a) (B b)\na : Type (PSet.mk α A)\nc : Type (Func (PSet.mk α A) a)\nb : β\nx✝¹ : PSet\nea✝ : A a = x✝¹\nγ : Type u\nΓ : γ → PSet\nA_ih✝¹ :\n ∀ (a_1 : γ), A a = Γ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)\nea : A a = PSet.mk γ Γ\nx✝ : PSet\neb✝ : B b = x✝\nδ : Type u\nΔ : δ → PSet\nhb : PSet.Equiv (PSet.mk γ Γ) (PSet.mk δ Δ)\nA_ih✝ :\n ∀ (a_1 : δ), B b = Δ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)\neb : B b = PSet.mk δ Δ\n⊢ ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)",
"state_before": "case mk.mk\nα β : Type u\nA : α → PSet\nB : β → PSet\nαβ : ∀ (a : α), ∃ b, PSet.Equiv (A a) (B b)\na : Type (PSet.mk α A)\nc : Type (Func (PSet.mk α A) a)\nb : β\nhb : PSet.Equiv (A a) (B b)\nx✝¹ : PSet\nea✝ : A a = x✝¹\nγ : Type u\nΓ : γ → PSet\nA_ih✝¹ :\n ∀ (a_1 : γ), A a = Γ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)\nea : A a = PSet.mk γ Γ\nx✝ : PSet\neb✝ : B b = x✝\nδ : Type u\nΔ : δ → PSet\nA_ih✝ :\n ∀ (a_1 : δ), B b = Δ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)\neb : B b = PSet.mk δ Δ\n⊢ ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)",
"tactic": "rw [ea, eb] at hb"
},
{
"state_after": "case mk.mk.intro\nα β : Type u\nA : α → PSet\nB : β → PSet\nαβ : ∀ (a : α), ∃ b, PSet.Equiv (A a) (B b)\na : Type (PSet.mk α A)\nc : Type (Func (PSet.mk α A) a)\nb : β\nx✝¹ : PSet\nea✝ : A a = x✝¹\nγ : Type u\nΓ : γ → PSet\nA_ih✝¹ :\n ∀ (a_1 : γ), A a = Γ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)\nea : A a = PSet.mk γ Γ\nx✝ : PSet\neb✝ : B b = x✝\nδ : Type u\nΔ : δ → PSet\nA_ih✝ :\n ∀ (a_1 : δ), B b = Δ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)\neb : B b = PSet.mk δ Δ\nγδ : ∀ (a : γ), ∃ b, PSet.Equiv (Γ a) (Δ b)\nδγ : ∀ (b : δ), ∃ a, PSet.Equiv (Γ a) (Δ b)\n⊢ ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)",
"state_before": "case mk.mk\nα β : Type u\nA : α → PSet\nB : β → PSet\nαβ : ∀ (a : α), ∃ b, PSet.Equiv (A a) (B b)\na : Type (PSet.mk α A)\nc : Type (Func (PSet.mk α A) a)\nb : β\nx✝¹ : PSet\nea✝ : A a = x✝¹\nγ : Type u\nΓ : γ → PSet\nA_ih✝¹ :\n ∀ (a_1 : γ), A a = Γ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)\nea : A a = PSet.mk γ Γ\nx✝ : PSet\neb✝ : B b = x✝\nδ : Type u\nΔ : δ → PSet\nhb : PSet.Equiv (PSet.mk γ Γ) (PSet.mk δ Δ)\nA_ih✝ :\n ∀ (a_1 : δ), B b = Δ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)\neb : B b = PSet.mk δ Δ\n⊢ ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)",
"tactic": "cases' hb with γδ δγ"
},
{
"state_after": "case mk.mk.intro\nα β : Type u\nA : α → PSet\nB : β → PSet\nαβ : ∀ (a : α), ∃ b, PSet.Equiv (A a) (B b)\na : Type (PSet.mk α A)\nc✝ : Type (Func (PSet.mk α A) a)\nb : β\nx✝¹ : PSet\nea✝ : A a = x✝¹\nγ : Type u\nΓ : γ → PSet\nA_ih✝¹ :\n ∀ (a_1 : γ), A a = Γ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)\nea : A a = PSet.mk γ Γ\nx✝ : PSet\neb✝ : B b = x✝\nδ : Type u\nΔ : δ → PSet\nA_ih✝ :\n ∀ (a_1 : δ), B b = Δ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)\neb : B b = PSet.mk δ Δ\nγδ : ∀ (a : γ), ∃ b, PSet.Equiv (Γ a) (Δ b)\nδγ : ∀ (b : δ), ∃ a, PSet.Equiv (Γ a) (Δ b)\nc : Type (A a) := c✝\n⊢ ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)",
"state_before": "case mk.mk.intro\nα β : Type u\nA : α → PSet\nB : β → PSet\nαβ : ∀ (a : α), ∃ b, PSet.Equiv (A a) (B b)\na : Type (PSet.mk α A)\nc : Type (Func (PSet.mk α A) a)\nb : β\nx✝¹ : PSet\nea✝ : A a = x✝¹\nγ : Type u\nΓ : γ → PSet\nA_ih✝¹ :\n ∀ (a_1 : γ), A a = Γ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)\nea : A a = PSet.mk γ Γ\nx✝ : PSet\neb✝ : B b = x✝\nδ : Type u\nΔ : δ → PSet\nA_ih✝ :\n ∀ (a_1 : δ), B b = Δ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)\neb : B b = PSet.mk δ Δ\nγδ : ∀ (a : γ), ∃ b, PSet.Equiv (Γ a) (Δ b)\nδγ : ∀ (b : δ), ∃ a, PSet.Equiv (Γ a) (Δ b)\n⊢ ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c }) (Func (⋃₀ PSet.mk β B) b)",
"tactic": "let c : (A a).Type := c"
},
{
"state_after": "case mk.mk.intro\nα β : Type u\nA : α → PSet\nB : β → PSet\nαβ : ∀ (a : α), ∃ b, PSet.Equiv (A a) (B b)\na : Type (PSet.mk α A)\nc✝ : Type (Func (PSet.mk α A) a)\nb : β\nx✝¹ : PSet\nea✝ : A a = x✝¹\nγ : Type u\nΓ : γ → PSet\nA_ih✝¹ :\n ∀ (a_1 : γ), A a = Γ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)\nea : A a = PSet.mk γ Γ\nx✝ : PSet\neb✝ : B b = x✝\nδ : Type u\nΔ : δ → PSet\nA_ih✝ :\n ∀ (a_1 : δ), B b = Δ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)\neb : B b = PSet.mk δ Δ\nγδ : ∀ (a : γ), ∃ b, PSet.Equiv (Γ a) (Δ b)\nδγ : ∀ (b : δ), ∃ a, PSet.Equiv (Γ a) (Δ b)\nc : Type (A a) := c✝\nd : δ\nhd : PSet.Equiv (Γ (Eq.mp (_ : Type (A a) = Type (PSet.mk γ Γ)) c)) (Δ d)\n⊢ ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)",
"state_before": "case mk.mk.intro\nα β : Type u\nA : α → PSet\nB : β → PSet\nαβ : ∀ (a : α), ∃ b, PSet.Equiv (A a) (B b)\na : Type (PSet.mk α A)\nc✝ : Type (Func (PSet.mk α A) a)\nb : β\nx✝¹ : PSet\nea✝ : A a = x✝¹\nγ : Type u\nΓ : γ → PSet\nA_ih✝¹ :\n ∀ (a_1 : γ), A a = Γ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)\nea : A a = PSet.mk γ Γ\nx✝ : PSet\neb✝ : B b = x✝\nδ : Type u\nΔ : δ → PSet\nA_ih✝ :\n ∀ (a_1 : δ), B b = Δ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)\neb : B b = PSet.mk δ Δ\nγδ : ∀ (a : γ), ∃ b, PSet.Equiv (Γ a) (Δ b)\nδγ : ∀ (b : δ), ∃ a, PSet.Equiv (Γ a) (Δ b)\nc : Type (A a) := c✝\n⊢ ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)",
"tactic": "let ⟨d, hd⟩ := γδ (by rwa [ea] at c)"
},
{
"state_after": "case mk.mk.intro\nα β : Type u\nA : α → PSet\nB : β → PSet\nαβ : ∀ (a : α), ∃ b, PSet.Equiv (A a) (B b)\na : Type (PSet.mk α A)\nc✝ : Type (Func (PSet.mk α A) a)\nb : β\nx✝¹ : PSet\nea✝ : A a = x✝¹\nγ : Type u\nΓ : γ → PSet\nA_ih✝¹ :\n ∀ (a_1 : γ), A a = Γ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)\nea : A a = PSet.mk γ Γ\nx✝ : PSet\neb✝ : B b = x✝\nδ : Type u\nΔ : δ → PSet\nA_ih✝ :\n ∀ (a_1 : δ), B b = Δ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)\neb : B b = PSet.mk δ Δ\nγδ : ∀ (a : γ), ∃ b, PSet.Equiv (Γ a) (Δ b)\nδγ : ∀ (b : δ), ∃ a, PSet.Equiv (Γ a) (Δ b)\nc : Type (A a) := c✝\nd : δ\nhd : PSet.Equiv (Γ (Eq.mp (_ : Type (A a) = Type (PSet.mk γ Γ)) c)) (Δ d)\n⊢ PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ })\n (Func (⋃₀ PSet.mk β B) { fst := b, snd := (_ : PSet.mk δ Δ = Func (PSet.mk β B) b) ▸ d })",
"state_before": "case mk.mk.intro\nα β : Type u\nA : α → PSet\nB : β → PSet\nαβ : ∀ (a : α), ∃ b, PSet.Equiv (A a) (B b)\na : Type (PSet.mk α A)\nc✝ : Type (Func (PSet.mk α A) a)\nb : β\nx✝¹ : PSet\nea✝ : A a = x✝¹\nγ : Type u\nΓ : γ → PSet\nA_ih✝¹ :\n ∀ (a_1 : γ), A a = Γ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)\nea : A a = PSet.mk γ Γ\nx✝ : PSet\neb✝ : B b = x✝\nδ : Type u\nΔ : δ → PSet\nA_ih✝ :\n ∀ (a_1 : δ), B b = Δ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)\neb : B b = PSet.mk δ Δ\nγδ : ∀ (a : γ), ∃ b, PSet.Equiv (Γ a) (Δ b)\nδγ : ∀ (b : δ), ∃ a, PSet.Equiv (Γ a) (Δ b)\nc : Type (A a) := c✝\nd : δ\nhd : PSet.Equiv (Γ (Eq.mp (_ : Type (A a) = Type (PSet.mk γ Γ)) c)) (Δ d)\n⊢ ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)",
"tactic": "use ⟨b, Eq.ndrec d (Eq.symm eb)⟩"
},
{
"state_after": "case mk.mk.intro\nα β : Type u\nA : α → PSet\nB : β → PSet\nαβ : ∀ (a : α), ∃ b, PSet.Equiv (A a) (B b)\na : Type (PSet.mk α A)\nc✝ : Type (Func (PSet.mk α A) a)\nb : β\nx✝¹ : PSet\nea✝ : A a = x✝¹\nγ : Type u\nΓ : γ → PSet\nA_ih✝¹ :\n ∀ (a_1 : γ), A a = Γ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)\nea : A a = PSet.mk γ Γ\nx✝ : PSet\neb✝ : B b = x✝\nδ : Type u\nΔ : δ → PSet\nA_ih✝ :\n ∀ (a_1 : δ), B b = Δ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)\neb : B b = PSet.mk δ Δ\nγδ : ∀ (a : γ), ∃ b, PSet.Equiv (Γ a) (Δ b)\nδγ : ∀ (b : δ), ∃ a, PSet.Equiv (Γ a) (Δ b)\nc : Type (A a) := c✝\nd : δ\nhd : PSet.Equiv (Γ (Eq.mp (_ : Type (A a) = Type (PSet.mk γ Γ)) c)) (Δ d)\n⊢ PSet.Equiv (Func (A a) c) (Func (B b) ((_ : PSet.mk δ Δ = B b) ▸ d))",
"state_before": "case mk.mk.intro\nα β : Type u\nA : α → PSet\nB : β → PSet\nαβ : ∀ (a : α), ∃ b, PSet.Equiv (A a) (B b)\na : Type (PSet.mk α A)\nc✝ : Type (Func (PSet.mk α A) a)\nb : β\nx✝¹ : PSet\nea✝ : A a = x✝¹\nγ : Type u\nΓ : γ → PSet\nA_ih✝¹ :\n ∀ (a_1 : γ), A a = Γ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)\nea : A a = PSet.mk γ Γ\nx✝ : PSet\neb✝ : B b = x✝\nδ : Type u\nΔ : δ → PSet\nA_ih✝ :\n ∀ (a_1 : δ), B b = Δ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)\neb : B b = PSet.mk δ Δ\nγδ : ∀ (a : γ), ∃ b, PSet.Equiv (Γ a) (Δ b)\nδγ : ∀ (b : δ), ∃ a, PSet.Equiv (Γ a) (Δ b)\nc : Type (A a) := c✝\nd : δ\nhd : PSet.Equiv (Γ (Eq.mp (_ : Type (A a) = Type (PSet.mk γ Γ)) c)) (Δ d)\n⊢ PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ })\n (Func (⋃₀ PSet.mk β B) { fst := b, snd := (_ : PSet.mk δ Δ = Func (PSet.mk β B) b) ▸ d })",
"tactic": "change PSet.Equiv ((A a).Func c) ((B b).Func (Eq.ndrec d eb.symm))"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.intro\nα β : Type u\nA : α → PSet\nB : β → PSet\nαβ : ∀ (a : α), ∃ b, PSet.Equiv (A a) (B b)\na : Type (PSet.mk α A)\nc✝ : Type (Func (PSet.mk α A) a)\nb : β\nx✝¹ : PSet\nea✝ : A a = x✝¹\nγ : Type u\nΓ : γ → PSet\nA_ih✝¹ :\n ∀ (a_1 : γ), A a = Γ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)\nea : A a = PSet.mk γ Γ\nx✝ : PSet\neb✝ : B b = x✝\nδ : Type u\nΔ : δ → PSet\nA_ih✝ :\n ∀ (a_1 : δ), B b = Δ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)\neb : B b = PSet.mk δ Δ\nγδ : ∀ (a : γ), ∃ b, PSet.Equiv (Γ a) (Δ b)\nδγ : ∀ (b : δ), ∃ a, PSet.Equiv (Γ a) (Δ b)\nc : Type (A a) := c✝\nd : δ\nhd : PSet.Equiv (Γ (Eq.mp (_ : Type (A a) = Type (PSet.mk γ Γ)) c)) (Δ d)\n⊢ PSet.Equiv (Func (A a) c) (Func (B b) ((_ : PSet.mk δ Δ = B b) ▸ d))",
"tactic": "match A a, B b, ea, eb, c, d, hd with\n| _, _, rfl, rfl, _, _, hd => exact hd"
},
{
"state_after": "no goals",
"state_before": "α β : Type u\nA : α → PSet\nB : β → PSet\nαβ : ∀ (a : α), ∃ b, PSet.Equiv (A a) (B b)\na : Type (PSet.mk α A)\nc✝ : Type (Func (PSet.mk α A) a)\nb : β\nx✝¹ : PSet\nea✝ : A a = x✝¹\nγ : Type u\nΓ : γ → PSet\nA_ih✝¹ :\n ∀ (a_1 : γ), A a = Γ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)\nea : A a = PSet.mk γ Γ\nx✝ : PSet\neb✝ : B b = x✝\nδ : Type u\nΔ : δ → PSet\nA_ih✝ :\n ∀ (a_1 : δ), B b = Δ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)\neb : B b = PSet.mk δ Δ\nγδ : ∀ (a : γ), ∃ b, PSet.Equiv (Γ a) (Δ b)\nδγ : ∀ (b : δ), ∃ a, PSet.Equiv (Γ a) (Δ b)\nc : Type (A a) := c✝\n⊢ γ",
"tactic": "rwa [ea] at c"
},
{
"state_after": "no goals",
"state_before": "α β : Type u\nA : α → PSet\nB : β → PSet\nαβ : ∀ (a : α), ∃ b, PSet.Equiv (A a) (B b)\na : Type (PSet.mk α A)\nc✝ : Type (Func (PSet.mk α A) a)\nb : β\nx✝³ : PSet\nea✝ : A a = x✝³\nγ : Type u\nΓ : γ → PSet\nA_ih✝¹ :\n ∀ (a_1 : γ), A a = Γ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)\nea : A a = PSet.mk γ Γ\nx✝² : PSet\neb✝ : B b = x✝²\nδ : Type u\nΔ : δ → PSet\nA_ih✝ :\n ∀ (a_1 : δ), B b = Δ a_1 → ∃ b, PSet.Equiv (Func (⋃₀ PSet.mk α A) { fst := a, snd := c✝ }) (Func (⋃₀ PSet.mk β B) b)\neb : B b = PSet.mk δ Δ\nγδ : ∀ (a : γ), ∃ b, PSet.Equiv (Γ a) (Δ b)\nδγ : ∀ (b : δ), ∃ a, PSet.Equiv (Γ a) (Δ b)\nc : Type (A a) := c✝\nd : δ\nhd✝ : PSet.Equiv (Γ (Eq.mp (_ : Type (A a) = Type (PSet.mk γ Γ)) c)) (Δ d)\nx✝¹ : Type (PSet.mk γ Γ)\nx✝ : δ\nhd : PSet.Equiv (Γ (Eq.mp (_ : Type (PSet.mk γ Γ) = Type (PSet.mk γ Γ)) x✝¹)) (Δ x✝)\n⊢ PSet.Equiv (Func (PSet.mk γ Γ) x✝¹) (Func (PSet.mk δ Δ) ((_ : PSet.mk δ Δ = PSet.mk δ Δ) ▸ x✝))",
"tactic": "exact hd"
}
] |
[
1029,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1016,
1
] |
Mathlib/Data/Nat/Factorial/Basic.lean
|
Nat.factorial_mul_ascFactorial
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ n ! * ascFactorial n 0 = (n + 0)!",
"tactic": "rw [ascFactorial, add_zero, mul_one]"
},
{
"state_after": "no goals",
"state_before": "n k : ℕ\n⊢ n ! * ascFactorial n (k + 1) = (n + (k + 1))!",
"tactic": "rw [ascFactorial_succ, mul_left_comm, factorial_mul_ascFactorial n k,\n ← add_assoc, ← Nat.succ_eq_add_one (n + k), factorial]"
}
] |
[
267,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.coe_comp
|
[] |
[
783,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
781,
1
] |
Mathlib/Data/Set/Prod.lean
|
Set.prod_inter
|
[
{
"state_after": "case h.mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.13035\nδ : Type ?u.13038\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx : α\ny : β\n⊢ (x, y) ∈ s ×ˢ (t₁ ∩ t₂) ↔ (x, y) ∈ s ×ˢ t₁ ∩ s ×ˢ t₂",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.13035\nδ : Type ?u.13038\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\n⊢ s ×ˢ (t₁ ∩ t₂) = s ×ˢ t₁ ∩ s ×ˢ t₂",
"tactic": "ext ⟨x, y⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.13035\nδ : Type ?u.13038\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx : α\ny : β\n⊢ (x, y) ∈ s ×ˢ (t₁ ∩ t₂) ↔ (x, y) ∈ s ×ˢ t₁ ∩ s ×ˢ t₂",
"tactic": "simp only [← and_and_left, mem_inter_iff, mem_prod]"
}
] |
[
168,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/Algebra/Module/Projective.lean
|
Module.Projective.of_lifting_property
|
[] |
[
202,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/ModelTheory/Satisfiability.lean
|
FirstOrder.Language.completeTheory.isMaximal
|
[] |
[
509,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
508,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.sin_eq_sqrt_one_sub_cos_sq
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\nhl : 0 ≤ x\nhu : x ≤ π\n⊢ sin x = sqrt (1 - cos x ^ 2)",
"tactic": "rw [← abs_sin_eq_sqrt_one_sub_cos_sq, abs_of_nonneg (sin_nonneg_of_nonneg_of_le_pi hl hu)]"
}
] |
[
489,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
487,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
UniqueFactorizationMonoid.normalizedFactors_prod_of_prime
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : CancelCommMonoidWithZero α\ninst✝⁴ : DecidableEq α\ninst✝³ : NormalizationMonoid α\ninst✝² : UniqueFactorizationMonoid α\ninst✝¹ : Nontrivial α\ninst✝ : Unique αˣ\nm : Multiset α\nh : ∀ (p : α), p ∈ m → Prime p\n⊢ normalizedFactors (Multiset.prod m) = m",
"tactic": "simpa only [← Multiset.rel_eq, ← associated_eq_eq] using\n prime_factors_unique prime_of_normalized_factor h\n (normalizedFactors_prod (m.prod_ne_zero_of_prime h))"
}
] |
[
767,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
763,
1
] |
Mathlib/Topology/Covering.lean
|
IsEvenlyCovered.continuousAt
|
[] |
[
69,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
11
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isBigO_zero
|
[] |
[
1208,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1207,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Subalgebra.copy_eq
|
[] |
[
102,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Algebra/Periodic.lean
|
Function.Antiperiodic.smul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nf g : α → β\nc c₁ c₂ x : α\ninst✝³ : Add α\ninst✝² : Monoid γ\ninst✝¹ : AddGroup β\ninst✝ : DistribMulAction γ β\nh : Antiperiodic f c\na : γ\n⊢ Antiperiodic (a • f) c",
"tactic": "simp_all"
}
] |
[
455,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
454,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Group.subset_conjugatesOfSet
|
[] |
[
2419,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2418,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.power_one
|
[] |
[
514,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
513,
1
] |
Mathlib/LinearAlgebra/Matrix/IsDiag.lean
|
Matrix.IsDiag.fromBlocks_of_isSymm
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.26056\nR : Type ?u.26059\nn : Type u_3\nm : Type u_2\ninst✝ : Zero α\nA : Matrix m m α\nC : Matrix n m α\nD : Matrix n n α\nh : IsSymm (Matrix.fromBlocks A 0 C D)\nha : IsDiag A\nhd : IsDiag D\n⊢ IsDiag (Matrix.fromBlocks A 0 0ᵀ D)",
"state_before": "α : Type u_1\nβ : Type ?u.26056\nR : Type ?u.26059\nn : Type u_3\nm : Type u_2\ninst✝ : Zero α\nA : Matrix m m α\nC : Matrix n m α\nD : Matrix n n α\nh : IsSymm (Matrix.fromBlocks A 0 C D)\nha : IsDiag A\nhd : IsDiag D\n⊢ IsDiag (Matrix.fromBlocks A 0 C D)",
"tactic": "rw [← (isSymm_fromBlocks_iff.1 h).2.1]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.26056\nR : Type ?u.26059\nn : Type u_3\nm : Type u_2\ninst✝ : Zero α\nA : Matrix m m α\nC : Matrix n m α\nD : Matrix n n α\nh : IsSymm (Matrix.fromBlocks A 0 C D)\nha : IsDiag A\nhd : IsDiag D\n⊢ IsDiag (Matrix.fromBlocks A 0 0ᵀ D)",
"tactic": "exact ha.fromBlocks hd"
}
] |
[
191,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean
|
Zsqrtd.le_antisymm
|
[
{
"state_after": "no goals",
"state_before": "d : ℕ\ndnsq : Nonsquare d\na b : ℤ√↑d\nab : a ≤ b\nba : b ≤ a\n⊢ Nonneg (-(a - b))",
"tactic": "rwa [neg_sub]"
}
] |
[
947,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
946,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean
|
CategoryTheory.Limits.diagonalObjPullbackFstIso_inv_snd_fst
|
[
{
"state_after": "C : Type u_2\ninst✝⁵ : Category C\nX✝ Y✝ Z✝ : C\ninst✝⁴ : HasPullbacks C\nS T : C\nf✝ : X✝ ⟶ T\ng✝ : Y✝ ⟶ T\ni : T ⟶ S\ninst✝³ : HasPullback i i\ninst✝² : HasPullback f✝ g✝\ninst✝¹ : HasPullback (f✝ ≫ i) (g✝ ≫ i)\ninst✝ :\n HasPullback (diagonal i)\n (map (f✝ ≫ i) (g✝ ≫ i) i i f✝ g✝ (𝟙 S) (_ : (f✝ ≫ i) ≫ 𝟙 S = f✝ ≫ i) (_ : (g✝ ≫ i) ≫ 𝟙 S = g✝ ≫ i))\nX Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ (pullbackRightPullbackFstIso f g fst ≪≫\n congrHom (_ : fst ≫ f = snd ≫ g) (_ : g = g) ≪≫\n pullbackAssoc f g g g ≪≫ pullbackSymmetry f (fst ≫ g) ≪≫ congrHom (_ : fst ≫ g = snd ≫ g) (_ : f = f)).inv ≫\n snd ≫ fst =\n snd",
"state_before": "C : Type u_2\ninst✝⁵ : Category C\nX✝ Y✝ Z✝ : C\ninst✝⁴ : HasPullbacks C\nS T : C\nf✝ : X✝ ⟶ T\ng✝ : Y✝ ⟶ T\ni : T ⟶ S\ninst✝³ : HasPullback i i\ninst✝² : HasPullback f✝ g✝\ninst✝¹ : HasPullback (f✝ ≫ i) (g✝ ≫ i)\ninst✝ :\n HasPullback (diagonal i)\n (map (f✝ ≫ i) (g✝ ≫ i) i i f✝ g✝ (𝟙 S) (_ : (f✝ ≫ i) ≫ 𝟙 S = f✝ ≫ i) (_ : (g✝ ≫ i) ≫ 𝟙 S = g✝ ≫ i))\nX Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ (diagonalObjPullbackFstIso f g).inv ≫ snd ≫ fst = snd",
"tactic": "delta diagonalObjPullbackFstIso"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝⁵ : Category C\nX✝ Y✝ Z✝ : C\ninst✝⁴ : HasPullbacks C\nS T : C\nf✝ : X✝ ⟶ T\ng✝ : Y✝ ⟶ T\ni : T ⟶ S\ninst✝³ : HasPullback i i\ninst✝² : HasPullback f✝ g✝\ninst✝¹ : HasPullback (f✝ ≫ i) (g✝ ≫ i)\ninst✝ :\n HasPullback (diagonal i)\n (map (f✝ ≫ i) (g✝ ≫ i) i i f✝ g✝ (𝟙 S) (_ : (f✝ ≫ i) ≫ 𝟙 S = f✝ ≫ i) (_ : (g✝ ≫ i) ≫ 𝟙 S = g✝ ≫ i))\nX Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ (pullbackRightPullbackFstIso f g fst ≪≫\n congrHom (_ : fst ≫ f = snd ≫ g) (_ : g = g) ≪≫\n pullbackAssoc f g g g ≪≫ pullbackSymmetry f (fst ≫ g) ≪≫ congrHom (_ : fst ≫ g = snd ≫ g) (_ : f = f)).inv ≫\n snd ≫ fst =\n snd",
"tactic": "simp"
}
] |
[
346,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
343,
1
] |
Mathlib/CategoryTheory/Sites/Sheafification.lean
|
CategoryTheory.Meq.refine_apply
|
[] |
[
85,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.span_pair_mul_span_pair
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nι : Type ?u.306186\ninst✝ : CommSemiring R\nI J K L : Ideal R\nw x y z : R\n⊢ span {w, x} * span {y, z} = span {w * y, w * z, x * y, x * z}",
"tactic": "simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc]"
}
] |
[
828,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
826,
1
] |
Mathlib/Data/Part.lean
|
Part.none_toOption
|
[] |
[
249,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
248,
1
] |
Mathlib/Algebra/GroupPower/Ring.lean
|
pow_eq_zero_iff'
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.25956\nS : Type ?u.25959\nM : Type u_1\ninst✝² : MonoidWithZero M\ninst✝¹ : NoZeroDivisors M\ninst✝ : Nontrivial M\na : M\nn : ℕ\n⊢ a ^ n = 0 ↔ a = 0 ∧ n ≠ 0",
"tactic": "cases (zero_le n).eq_or_gt <;> simp [*, ne_of_gt]"
}
] |
[
73,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Algebra/Algebra/Bilinear.lean
|
LinearMap.mulRight_mul
|
[
{
"state_after": "case h\nR : Type u_2\nA : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalSemiring A\ninst✝² : Module R A\ninst✝¹ : SMulCommClass R A A\ninst✝ : IsScalarTower R A A\na b x✝ : A\n⊢ ↑(mulRight R (a * b)) x✝ = ↑(comp (mulRight R b) (mulRight R a)) x✝",
"state_before": "R : Type u_2\nA : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalSemiring A\ninst✝² : Module R A\ninst✝¹ : SMulCommClass R A A\ninst✝ : IsScalarTower R A A\na b : A\n⊢ mulRight R (a * b) = comp (mulRight R b) (mulRight R a)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_2\nA : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalSemiring A\ninst✝² : Module R A\ninst✝¹ : SMulCommClass R A A\ninst✝ : IsScalarTower R A A\na b x✝ : A\n⊢ ↑(mulRight R (a * b)) x✝ = ↑(comp (mulRight R b) (mulRight R a)) x✝",
"tactic": "simp only [mulRight_apply, comp_apply, mul_assoc]"
}
] |
[
152,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Analysis/Convex/Combination.lean
|
Finset.inf_le_centerMass
|
[
{
"state_after": "R : Type ?u.76609\nE : Type ?u.76612\nF : Type ?u.76615\nι : Type ?u.76618\nι' : Type ?u.76621\nα : Type ?u.76624\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw✝ : ι → R\nz : ι → E\nf : ι → α\nw : ι → R\nhw₀ : ∀ (i : ι), i ∈ ∅ → 0 ≤ w i\nhw₁ : 0 < ∑ i in ∅, w i\n⊢ False",
"state_before": "R : Type ?u.76609\nE : Type ?u.76612\nF : Type ?u.76615\nι : Type ?u.76618\nι' : Type ?u.76621\nα : Type ?u.76624\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι\nc : R\nt : Finset ι\nw✝ : ι → R\nz : ι → E\ns : Finset ι\nf : ι → α\nw : ι → R\nhw₀ : ∀ (i : ι), i ∈ s → 0 ≤ w i\nhw₁ : 0 < ∑ i in s, w i\n⊢ s ≠ ∅",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.76609\nE : Type ?u.76612\nF : Type ?u.76615\nι : Type ?u.76618\nι' : Type ?u.76621\nα : Type ?u.76624\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw✝ : ι → R\nz : ι → E\nf : ι → α\nw : ι → R\nhw₀ : ∀ (i : ι), i ∈ ∅ → 0 ≤ w i\nhw₁ : 0 < ∑ i in ∅, w i\n⊢ False",
"tactic": "simp at hw₁"
}
] |
[
147,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.eq_preimage_iff_image_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nf✝ f : α → β\nhf : Bijective f\ns : Set α\nt : Set β\n⊢ s = f ⁻¹' t ↔ f '' s = t",
"tactic": "rw [← image_eq_image hf.1, hf.2.image_preimage]"
}
] |
[
1567,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1566,
1
] |
Mathlib/Algebra/Algebra/Tower.lean
|
IsScalarTower.of_algebraMap_eq'
|
[] |
[
94,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean
|
ContinuousLinearMap.map_add₂
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_6\ninst✝¹⁶ : NontriviallyNormedField 𝕜\nE : Type ?u.134067\ninst✝¹⁵ : NormedAddCommGroup E\ninst✝¹⁴ : NormedSpace 𝕜 E\nF : Type u_5\ninst✝¹³ : NormedAddCommGroup F\ninst✝¹² : NormedSpace 𝕜 F\nG : Type ?u.134252\ninst✝¹¹ : NormedAddCommGroup G\ninst✝¹⁰ : NormedSpace 𝕜 G\nR : Type u_1\n𝕜₂ : Type u_7\n𝕜' : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NontriviallyNormedField 𝕜₂\nM : Type u_3\ninst✝⁷ : TopologicalSpace M\nσ₁₂ : 𝕜 →+* 𝕜₂\nG' : Type u_4\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace 𝕜₂ G'\ninst✝⁴ : NormedSpace 𝕜' G'\ninst✝³ : SMulCommClass 𝕜₂ 𝕜' G'\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nρ₁₂ : R →+* 𝕜'\nf : M →SL[ρ₁₂] F →SL[σ₁₂] G'\nx x' : M\ny : F\n⊢ ↑(↑f (x + x')) y = ↑(↑f x) y + ↑(↑f x') y",
"tactic": "rw [f.map_add, add_apply]"
}
] |
[
297,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
296,
1
] |
Mathlib/Order/Filter/Archimedean.lean
|
Rat.comap_cast_atTop
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.7888\nR : Type u_1\ninst✝¹ : LinearOrderedField R\ninst✝ : Archimedean R\nr : R\nn : ℕ\nhn : r ≤ ↑n\n⊢ r ≤ ↑↑n",
"tactic": "simpa"
}
] |
[
78,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Analysis/Calculus/MeanValue.lean
|
is_const_of_deriv_eq_zero
|
[
{
"state_after": "case h\nE : Type ?u.290841\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nF : Type ?u.290937\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\n𝕜 : Type u_1\nG : Type u_2\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf f' : 𝕜 → G\ns : Set 𝕜\nx✝ y✝ : 𝕜\nhf : Differentiable 𝕜 f\nhf' : ∀ (x : 𝕜), deriv f x = 0\nx y z : 𝕜\n⊢ ↑(fderiv 𝕜 f z) 1 = ↑0 1",
"state_before": "E : Type ?u.290841\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nF : Type ?u.290937\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\n𝕜 : Type u_1\nG : Type u_2\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf f' : 𝕜 → G\ns : Set 𝕜\nx✝ y✝ : 𝕜\nhf : Differentiable 𝕜 f\nhf' : ∀ (x : 𝕜), deriv f x = 0\nx y z : 𝕜\n⊢ fderiv 𝕜 f z = 0",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nE : Type ?u.290841\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nF : Type ?u.290937\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\n𝕜 : Type u_1\nG : Type u_2\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf f' : 𝕜 → G\ns : Set 𝕜\nx✝ y✝ : 𝕜\nhf : Differentiable 𝕜 f\nhf' : ∀ (x : 𝕜), deriv f x = 0\nx y z : 𝕜\n⊢ ↑(fderiv 𝕜 f z) 1 = ↑0 1",
"tactic": "simp [← deriv_fderiv, hf']"
}
] |
[
696,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
694,
1
] |
Mathlib/Algebra/EuclideanDomain/Basic.lean
|
EuclideanDomain.gcd_val
|
[
{
"state_after": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\n⊢ (fun b =>\n if a0 : a = 0 then b\n else\n let_fun x := (_ : EuclideanDomain.r (b % a) a);\n gcd (b % a) a)\n b =\n gcd (b % a) a",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\n⊢ gcd a b = gcd (b % a) a",
"tactic": "rw [gcd]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\n⊢ (fun b =>\n if a0 : a = 0 then b\n else\n let_fun x := (_ : EuclideanDomain.r (b % a) a);\n gcd (b % a) a)\n b =\n gcd (b % a) a",
"tactic": "split_ifs with h <;> [simp only [h, mod_zero, gcd_zero_right]; rfl]"
}
] |
[
146,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/MeasureTheory/Integral/Average.lean
|
MeasureTheory.set_average_const
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.235504\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ : Measure α\ns✝ : Set E\ns : Set α\nhs₀ : ↑↑μ s ≠ 0\nhs : ↑↑μ s ≠ ⊤\nc : E\n⊢ (⨍ (x : α) in s, c ∂μ) = c",
"tactic": "simp only [set_average_eq, integral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter,\n smul_smul, ← ENNReal.toReal_inv, ← ENNReal.toReal_mul, ENNReal.inv_mul_cancel hs₀ hs,\n ENNReal.one_toReal, one_smul]"
}
] |
[
207,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
TrivSqZeroExt.inl_pow
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nM : Type v\ninst✝³ : Monoid R\ninst✝² : AddMonoid M\ninst✝¹ : DistribMulAction R M\ninst✝ : DistribMulAction Rᵐᵒᵖ M\nr : R\nn : ℕ\n⊢ snd (inl r ^ n) = snd (inl (r ^ n))",
"tactic": "simp [snd_pow_eq_sum]"
}
] |
[
640,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
638,
1
] |
src/lean/Init/SimpLemmas.lean
|
eq_false'
|
[] |
[
20,
60
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
20,
1
] |
Mathlib/MeasureTheory/Group/FundamentalDomain.lean
|
MeasureTheory.IsFundamentalDomain.set_integral_eq_tsum
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\nH : Type ?u.379290\nα : Type u_2\nβ : Type ?u.379296\nE : Type u_3\ninst✝¹² : Group G\ninst✝¹¹ : Group H\ninst✝¹⁰ : MulAction G α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : MulAction H β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : NormedAddCommGroup E\ns t✝ : Set α\nμ : Measure α\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : MeasurableSMul G α\ninst✝³ : SMulInvariantMeasure G α μ\ninst✝² : Countable G\nν : Measure α\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nh : IsFundamentalDomain G s\nf : α → E\nt : Set α\nhf : IntegrableOn f t\n⊢ (∑' (g : G), ∫ (x : α) in g • s, f x ∂Measure.restrict μ t) = ∑' (g : G), ∫ (x : α) in t ∩ g • s, f x ∂μ",
"tactic": "simp only [h.restrict_restrict, measure_smul, inter_comm]"
}
] |
[
439,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
433,
1
] |
Std/Data/Array/Lemmas.lean
|
Array.fin_cast_val
|
[
{
"state_after": "case refl\nn : Nat\ni : Fin n\n⊢ (_ : n = n) ▸ i = { val := i.val, isLt := (_ : i.val < n) }",
"state_before": "n n' : Nat\ne : n = n'\ni : Fin n\n⊢ e ▸ i = { val := i.val, isLt := (_ : i.val < n') }",
"tactic": "cases e"
},
{
"state_after": "no goals",
"state_before": "case refl\nn : Nat\ni : Fin n\n⊢ (_ : n = n) ▸ i = { val := i.val, isLt := (_ : i.val < n) }",
"tactic": "rfl"
}
] |
[
113,
98
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
113,
9
] |
Mathlib/Topology/Sheaves/Functors.lean
|
TopCat.Presheaf.SheafConditionPairwiseIntersections.pushforward_sheaf_of_sheaf
|
[
{
"state_after": "case h.e'_1.h.e'_5\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\n⊢ Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)) ⋙ F\n\ncase h.e'_1.h.e'_6\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\ne_5✝ : Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)) ⋙ F\n⊢ HEq ((f _* F).mapCone (Cocone.op (Pairwise.cocone U)))\n (F.mapCone (Cocone.op (Pairwise.cocone ((Opens.map f).toPrefunctor.obj ∘ U))))",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\n⊢ Nonempty (IsLimit ((f _* F).mapCone (Cocone.op (Pairwise.cocone U))))",
"tactic": "convert h ((Opens.map f).obj ∘ U) using 2"
},
{
"state_after": "case h.e'_1.h.e'_5\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\n⊢ Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram U ⋙ Opens.map f) ⋙ F\n\ncase h.e'_1.h.e'_6\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\ne_5✝ : Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)) ⋙ F\n⊢ HEq ((f _* F).mapCone (Cocone.op (Pairwise.cocone U)))\n (F.mapCone (Cocone.op (Pairwise.cocone ((Opens.map f).toPrefunctor.obj ∘ U))))",
"state_before": "case h.e'_1.h.e'_5\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\n⊢ Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)) ⋙ F\n\ncase h.e'_1.h.e'_6\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\ne_5✝ : Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)) ⋙ F\n⊢ HEq ((f _* F).mapCone (Cocone.op (Pairwise.cocone U)))\n (F.mapCone (Cocone.op (Pairwise.cocone ((Opens.map f).toPrefunctor.obj ∘ U))))",
"tactic": "rw [← map_diagram]"
},
{
"state_after": "case h.e'_1.h.e'_6\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\ne_5✝ : Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)) ⋙ F\n⊢ HEq ((f _* F).mapCone (Cocone.op (Pairwise.cocone U)))\n (F.mapCone (Cocone.op (Pairwise.cocone ((Opens.map f).toPrefunctor.obj ∘ U))))",
"state_before": "case h.e'_1.h.e'_5\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\n⊢ Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram U ⋙ Opens.map f) ⋙ F\n\ncase h.e'_1.h.e'_6\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\ne_5✝ : Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)) ⋙ F\n⊢ HEq ((f _* F).mapCone (Cocone.op (Pairwise.cocone U)))\n (F.mapCone (Cocone.op (Pairwise.cocone ((Opens.map f).toPrefunctor.obj ∘ U))))",
"tactic": "rfl"
},
{
"state_after": "case h.e'_1.h.e'_6\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\ne_5✝ : Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)) ⋙ F\n⊢ HEq (F.mapCone (Cocone.op ((Opens.map f).mapCocone (Pairwise.cocone U))))\n (F.mapCone (Cocone.op (Pairwise.cocone ((Opens.map f).toPrefunctor.obj ∘ U))))",
"state_before": "case h.e'_1.h.e'_6\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\ne_5✝ : Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)) ⋙ F\n⊢ HEq ((f _* F).mapCone (Cocone.op (Pairwise.cocone U)))\n (F.mapCone (Cocone.op (Pairwise.cocone ((Opens.map f).toPrefunctor.obj ∘ U))))",
"tactic": "change HEq (Functor.mapCone F ((Opens.map f).mapCocone (Pairwise.cocone U)).op) _"
},
{
"state_after": "case h.e'_1.h.e'_6.e_8.h.e_F\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\ne_5✝ : Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)) ⋙ F\n⊢ Pairwise.diagram U ⋙ Opens.map f = Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)\n\ncase h.e'_1.h.e'_6.e_9.e_5.h\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\ne_5✝ : Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)) ⋙ F\n⊢ Pairwise.diagram U ⋙ Opens.map f = Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)\n\ncase h.e'_1.h.e'_6.e_9.e_6\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\ne_5✝ : Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)) ⋙ F\n⊢ HEq ((Opens.map f).mapCocone (Pairwise.cocone U)) (Pairwise.cocone ((Opens.map f).toPrefunctor.obj ∘ U))",
"state_before": "case h.e'_1.h.e'_6\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\ne_5✝ : Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)) ⋙ F\n⊢ HEq (F.mapCone (Cocone.op ((Opens.map f).mapCocone (Pairwise.cocone U))))\n (F.mapCone (Cocone.op (Pairwise.cocone ((Opens.map f).toPrefunctor.obj ∘ U))))",
"tactic": "congr"
},
{
"state_after": "case h.e'_1.h.e'_6.e_9.e_6\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\ne_5✝ : Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)) ⋙ F\n⊢ HEq ((Opens.map f).mapCocone (Pairwise.cocone U)) (Pairwise.cocone ((Opens.map f).toPrefunctor.obj ∘ U))",
"state_before": "case h.e'_1.h.e'_6.e_8.h.e_F\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\ne_5✝ : Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)) ⋙ F\n⊢ Pairwise.diagram U ⋙ Opens.map f = Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)\n\ncase h.e'_1.h.e'_6.e_9.e_5.h\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\ne_5✝ : Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)) ⋙ F\n⊢ Pairwise.diagram U ⋙ Opens.map f = Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)\n\ncase h.e'_1.h.e'_6.e_9.e_6\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\ne_5✝ : Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)) ⋙ F\n⊢ HEq ((Opens.map f).mapCocone (Pairwise.cocone U)) (Pairwise.cocone ((Opens.map f).toPrefunctor.obj ∘ U))",
"tactic": "iterate 2 rw [map_diagram]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_1.h.e'_6.e_9.e_6\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\ne_5✝ : Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)) ⋙ F\n⊢ HEq ((Opens.map f).mapCocone (Pairwise.cocone U)) (Pairwise.cocone ((Opens.map f).toPrefunctor.obj ∘ U))",
"tactic": "apply mapCocone"
},
{
"state_after": "case h.e'_1.h.e'_6.e_9.e_6\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\ne_5✝ : Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)) ⋙ F\n⊢ HEq ((Opens.map f).mapCocone (Pairwise.cocone U)) (Pairwise.cocone ((Opens.map f).toPrefunctor.obj ∘ U))",
"state_before": "case h.e'_1.h.e'_6.e_9.e_5.h\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\ne_5✝ : Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)) ⋙ F\n⊢ Pairwise.diagram U ⋙ Opens.map f = Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)\n\ncase h.e'_1.h.e'_6.e_9.e_6\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nι✝ : Type w\nU✝ : ι✝ → Opens ↑Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\nι : Type w\nU : ι → Opens ↑Y\ne_5✝ : Functor.op (Pairwise.diagram U) ⋙ f _* F = Functor.op (Pairwise.diagram ((Opens.map f).toPrefunctor.obj ∘ U)) ⋙ F\n⊢ HEq ((Opens.map f).mapCocone (Pairwise.cocone U)) (Pairwise.cocone ((Opens.map f).toPrefunctor.obj ∘ U))",
"tactic": "rw [map_diagram]"
}
] |
[
73,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Logic/Basic.lean
|
imp_iff_or_not
|
[] |
[
375,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
375,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Equiv.lean
|
ContinuousLinearEquiv.differentiableOn
|
[] |
[
94,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
11
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
deriv_csinh
|
[] |
[
288,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
286,
1
] |
Mathlib/RingTheory/MatrixAlgebra.lean
|
matrixEquivTensor_apply
|
[] |
[
166,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.smul_finset_inter
|
[] |
[
1992,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1991,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean
|
CategoryTheory.Limits.kernelIsoOfEq_inv_comp_ι
|
[
{
"state_after": "case refl\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf✝ : X ⟶ Y\ninst✝² : HasKernel f✝\nf : X ⟶ Y\ninst✝¹ inst✝ : HasKernel f\n⊢ (kernelIsoOfEq (_ : f = f)).inv ≫ kernel.ι f = kernel.ι f",
"state_before": "C : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf✝ : X ⟶ Y\ninst✝² : HasKernel f✝\nf g : X ⟶ Y\ninst✝¹ : HasKernel f\ninst✝ : HasKernel g\nh : f = g\n⊢ (kernelIsoOfEq h).inv ≫ kernel.ι f = kernel.ι g",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case refl\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf✝ : X ⟶ Y\ninst✝² : HasKernel f✝\nf : X ⟶ Y\ninst✝¹ inst✝ : HasKernel f\n⊢ (kernelIsoOfEq (_ : f = f)).inv ≫ kernel.ι f = kernel.ι f",
"tactic": "simp"
}
] |
[
365,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
363,
1
] |
Mathlib/Analysis/Normed/Group/Seminorm.lean
|
GroupSeminorm.mul_bddBelow_range_add
|
[
{
"state_after": "case intro\nι : Type ?u.89294\nR : Type ?u.89297\nR' : Type ?u.89300\nE : Type u_1\nF : Type ?u.89306\nG : Type ?u.89309\ninst✝¹ : CommGroup E\ninst✝ : CommGroup F\np✝ q✝ : GroupSeminorm E\nx✝¹ y : E\np q : GroupSeminorm E\nx✝ x : E\n⊢ 0 ≤ (fun y => ↑p y + ↑q (x✝ / y)) x",
"state_before": "ι : Type ?u.89294\nR : Type ?u.89297\nR' : Type ?u.89300\nE : Type u_1\nF : Type ?u.89306\nG : Type ?u.89309\ninst✝¹ : CommGroup E\ninst✝ : CommGroup F\np✝ q✝ : GroupSeminorm E\nx✝ y : E\np q : GroupSeminorm E\nx : E\n⊢ 0 ∈ lowerBounds (range fun y => ↑p y + ↑q (x / y))",
"tactic": "rintro _ ⟨x, rfl⟩"
},
{
"state_after": "case intro\nι : Type ?u.89294\nR : Type ?u.89297\nR' : Type ?u.89300\nE : Type u_1\nF : Type ?u.89306\nG : Type ?u.89309\ninst✝¹ : CommGroup E\ninst✝ : CommGroup F\np✝ q✝ : GroupSeminorm E\nx✝¹ y : E\np q : GroupSeminorm E\nx✝ x : E\n⊢ 0 ≤ ↑p x + ↑q (x✝ / x)",
"state_before": "case intro\nι : Type ?u.89294\nR : Type ?u.89297\nR' : Type ?u.89300\nE : Type u_1\nF : Type ?u.89306\nG : Type ?u.89309\ninst✝¹ : CommGroup E\ninst✝ : CommGroup F\np✝ q✝ : GroupSeminorm E\nx✝¹ y : E\np q : GroupSeminorm E\nx✝ x : E\n⊢ 0 ≤ (fun y => ↑p y + ↑q (x✝ / y)) x",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case intro\nι : Type ?u.89294\nR : Type ?u.89297\nR' : Type ?u.89300\nE : Type u_1\nF : Type ?u.89306\nG : Type ?u.89309\ninst✝¹ : CommGroup E\ninst✝ : CommGroup F\np✝ q✝ : GroupSeminorm E\nx✝¹ y : E\np q : GroupSeminorm E\nx✝ x : E\n⊢ 0 ≤ ↑p x + ↑q (x✝ / x)",
"tactic": "positivity"
}
] |
[
402,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
397,
1
] |
Mathlib/Order/Lattice.lean
|
le_inf
|
[] |
[
408,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
407,
1
] |
Mathlib/Algebra/DirectSum/Ring.lean
|
DirectSum.toSemiring_coe_addMonoidHom
|
[] |
[
640,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
638,
1
] |
Mathlib/CategoryTheory/Preadditive/Biproducts.lean
|
CategoryTheory.Limits.biproduct.map_eq
|
[
{
"state_after": "case w.w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : Preadditive C\nJ : Type\ninst✝² : Fintype J\nf✝ : J → C\ninst✝¹ : HasBiproduct f✝\ninst✝ : HasFiniteBiproducts C\nf g : J → C\nh : (j : J) → f j ⟶ g j\nj✝¹ j✝ : J\n⊢ (ι (fun b => f b) j✝¹ ≫ map h) ≫ π (fun b => g b) j✝ =\n (ι (fun b => f b) j✝¹ ≫ ∑ j : J, π f j ≫ h j ≫ ι g j) ≫ π (fun b => g b) j✝",
"state_before": "C : Type u\ninst✝⁴ : Category C\ninst✝³ : Preadditive C\nJ : Type\ninst✝² : Fintype J\nf✝ : J → C\ninst✝¹ : HasBiproduct f✝\ninst✝ : HasFiniteBiproducts C\nf g : J → C\nh : (j : J) → f j ⟶ g j\n⊢ map h = ∑ j : J, π f j ≫ h j ≫ ι g j",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case w.w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : Preadditive C\nJ : Type\ninst✝² : Fintype J\nf✝ : J → C\ninst✝¹ : HasBiproduct f✝\ninst✝ : HasFiniteBiproducts C\nf g : J → C\nh : (j : J) → f j ⟶ g j\nj✝¹ j✝ : J\n⊢ (ι (fun b => f b) j✝¹ ≫ map h) ≫ π (fun b => g b) j✝ =\n (ι (fun b => f b) j✝¹ ≫ ∑ j : J, π f j ≫ h j ≫ ι g j) ≫ π (fun b => g b) j✝",
"tactic": "simp [biproduct.ι_π, biproduct.ι_π_assoc, comp_sum, sum_comp, comp_dite, dite_comp]"
}
] |
[
238,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
235,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.max'_image
|
[
{
"state_after": "F : Type ?u.364381\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.364390\nι : Type ?u.364393\nκ : Type ?u.364396\ninst✝¹ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\ns : Finset α\nh : Finset.Nonempty (image f s)\ny : β\nhy : y ∈ image f s\n⊢ y ≤ f (max' s (_ : Finset.Nonempty s))",
"state_before": "F : Type ?u.364381\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.364390\nι : Type ?u.364393\nκ : Type ?u.364396\ninst✝¹ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\ns : Finset α\nh : Finset.Nonempty (image f s)\n⊢ max' (image f s) h = f (max' s (_ : Finset.Nonempty s))",
"tactic": "refine'\n le_antisymm (max'_le _ _ _ fun y hy => _) (le_max' _ _ (mem_image.mpr ⟨_, max'_mem _ _, rfl⟩))"
},
{
"state_after": "case intro.intro\nF : Type ?u.364381\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.364390\nι : Type ?u.364393\nκ : Type ?u.364396\ninst✝¹ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx✝ : α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\ns : Finset α\nh : Finset.Nonempty (image f s)\nx : α\nhx : x ∈ s\nhy : f x ∈ image f s\n⊢ f x ≤ f (max' s (_ : Finset.Nonempty s))",
"state_before": "F : Type ?u.364381\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.364390\nι : Type ?u.364393\nκ : Type ?u.364396\ninst✝¹ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\ns : Finset α\nh : Finset.Nonempty (image f s)\ny : β\nhy : y ∈ image f s\n⊢ y ≤ f (max' s (_ : Finset.Nonempty s))",
"tactic": "obtain ⟨x, hx, rfl⟩ := mem_image.mp hy"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nF : Type ?u.364381\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.364390\nι : Type ?u.364393\nκ : Type ?u.364396\ninst✝¹ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx✝ : α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\ns : Finset α\nh : Finset.Nonempty (image f s)\nx : α\nhx : x ∈ s\nhy : f x ∈ image f s\n⊢ f x ≤ f (max' s (_ : Finset.Nonempty s))",
"tactic": "exact hf (le_max' _ _ hx)"
}
] |
[
1503,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1498,
1
] |
Mathlib/Data/Nat/Prime.lean
|
Nat.Prime.dvd_iff_eq
|
[
{
"state_after": "p a : ℕ\nhp : Prime p\na1 : a ≠ 1\n⊢ a ∣ p → p = a",
"state_before": "p a : ℕ\nhp : Prime p\na1 : a ≠ 1\n⊢ a ∣ p ↔ p = a",
"tactic": "refine'\n ⟨_, by\n rintro rfl\n rfl⟩"
},
{
"state_after": "case intro\na : ℕ\na1 : a ≠ 1\nj : ℕ\nhp : Prime (a * j)\n⊢ a * j = a",
"state_before": "p a : ℕ\nhp : Prime p\na1 : a ≠ 1\n⊢ a ∣ p → p = a",
"tactic": "rintro ⟨j, rfl⟩"
},
{
"state_after": "case intro.inl.intro\na : ℕ\na1 : a ≠ 1\nleft✝ : Prime a\nhp : Prime (a * 1)\n⊢ a * 1 = a\n\ncase intro.inr.intro\nj : ℕ\nleft✝ : Prime j\na1 : 1 ≠ 1\nhp : Prime (1 * j)\n⊢ 1 * j = 1",
"state_before": "case intro\na : ℕ\na1 : a ≠ 1\nj : ℕ\nhp : Prime (a * j)\n⊢ a * j = a",
"tactic": "rcases prime_mul_iff.mp hp with (⟨_, rfl⟩ | ⟨_, rfl⟩)"
},
{
"state_after": "p : ℕ\nhp : Prime p\na1 : p ≠ 1\n⊢ p ∣ p",
"state_before": "p a : ℕ\nhp : Prime p\na1 : a ≠ 1\n⊢ p = a → a ∣ p",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Prime p\na1 : p ≠ 1\n⊢ p ∣ p",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case intro.inl.intro\na : ℕ\na1 : a ≠ 1\nleft✝ : Prime a\nhp : Prime (a * 1)\n⊢ a * 1 = a",
"tactic": "exact mul_one _"
},
{
"state_after": "no goals",
"state_before": "case intro.inr.intro\nj : ℕ\nleft✝ : Prime j\na1 : 1 ≠ 1\nhp : Prime (1 * j)\n⊢ 1 * j = 1",
"tactic": "exact (a1 rfl).elim"
}
] |
[
241,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
edist_le_tsum_of_edist_le_of_tendsto
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.567007\nγ : Type ?u.567010\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0∞\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ d n\na : α\nha : Tendsto f atTop (𝓝 a)\nn m : ℕ\nhnm : m ≥ n\n⊢ m ∈ {x | (fun c => edist (f n) (f c) ≤ ∑' (m : ℕ), d (n + m)) x}",
"state_before": "α : Type u_1\nβ : Type ?u.567007\nγ : Type ?u.567010\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0∞\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ d n\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\n⊢ edist (f n) a ≤ ∑' (m : ℕ), d (n + m)",
"tactic": "refine' le_of_tendsto (tendsto_const_nhds.edist ha) (mem_atTop_sets.2 ⟨n, fun m hnm => _⟩)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.567007\nγ : Type ?u.567010\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0∞\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ d n\na : α\nha : Tendsto f atTop (𝓝 a)\nn m : ℕ\nhnm : m ≥ n\n⊢ ∑ i in Finset.Ico n m, d i ≤ ∑' (m : ℕ), d (n + m)",
"state_before": "α : Type u_1\nβ : Type ?u.567007\nγ : Type ?u.567010\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0∞\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ d n\na : α\nha : Tendsto f atTop (𝓝 a)\nn m : ℕ\nhnm : m ≥ n\n⊢ m ∈ {x | (fun c => edist (f n) (f c) ≤ ∑' (m : ℕ), d (n + m)) x}",
"tactic": "refine' le_trans (edist_le_Ico_sum_of_edist_le hnm fun _ _ => hf _) _"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.567007\nγ : Type ?u.567010\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0∞\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ d n\na : α\nha : Tendsto f atTop (𝓝 a)\nn m : ℕ\nhnm : m ≥ n\n⊢ ∑ k in Finset.range (m - n), d (n + k) ≤ ∑' (m : ℕ), d (n + m)",
"state_before": "α : Type u_1\nβ : Type ?u.567007\nγ : Type ?u.567010\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0∞\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ d n\na : α\nha : Tendsto f atTop (𝓝 a)\nn m : ℕ\nhnm : m ≥ n\n⊢ ∑ i in Finset.Ico n m, d i ≤ ∑' (m : ℕ), d (n + m)",
"tactic": "rw [Finset.sum_Ico_eq_sum_range]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.567007\nγ : Type ?u.567010\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0∞\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ d n\na : α\nha : Tendsto f atTop (𝓝 a)\nn m : ℕ\nhnm : m ≥ n\n⊢ ∑ k in Finset.range (m - n), d (n + k) ≤ ∑' (m : ℕ), d (n + m)",
"tactic": "exact sum_le_tsum _ (fun _ _ => zero_le _) ENNReal.summable"
}
] |
[
1581,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1575,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean
|
NormedAddGroupHom.isClosed_ker
|
[] |
[
776,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
774,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.tendsto_atTop_add_left_of_le'
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.145656\nι' : Type ?u.145659\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.145668\ninst✝ : OrderedAddCommGroup β\nl : Filter α\nf g : α → β\nC : β\nhf : ∀ᶠ (x : α) in l, C ≤ f x\nhg : Tendsto g l atTop\n⊢ ∀ᶠ (x : α) in l, (fun x => -f x) x ≤ -C",
"tactic": "simpa"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.145656\nι' : Type ?u.145659\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.145668\ninst✝ : OrderedAddCommGroup β\nl : Filter α\nf g : α → β\nC : β\nhf : ∀ᶠ (x : α) in l, C ≤ f x\nhg : Tendsto g l atTop\n⊢ Tendsto (fun x => (fun x => -f x) x + (fun x => f x + g x) x) l atTop",
"tactic": "simpa"
}
] |
[
770,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
767,
1
] |
Mathlib/CategoryTheory/Sites/Sheaf.lean
|
CategoryTheory.Presheaf.isSheaf_of_isTerminal
|
[] |
[
255,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/RingTheory/Noetherian.lean
|
fg_of_ker_bot
|
[] |
[
198,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Std/Data/Array/Init/Lemmas.lean
|
Array.foldl_eq_foldl_data
|
[] |
[
49,
53
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
47,
1
] |
Mathlib/Topology/Inseparable.lean
|
specializes_iff_forall_open
|
[] |
[
116,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.Nonempty.of_div_left
|
[] |
[
653,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
652,
1
] |
Mathlib/RingTheory/PrincipalIdealDomain.lean
|
PrincipalIdealRing.irreducible_iff_prime
|
[] |
[
267,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/Order/Circular.lean
|
sbtw_irrefl_left
|
[] |
[
260,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.coeff_inv_aux
|
[
{
"state_after": "σ : Type u_1\nR : Type u_2\ninst✝¹ : Ring R\ninst✝ : DecidableEq σ\nn : σ →₀ ℕ\na : R\nφ : MvPowerSeries σ R\n⊢ (if n = 0 then a\n else -a * ∑ x in antidiagonal n, if x_1 : x.snd < n then ↑(coeff R x.fst) φ * inv.aux a φ x.snd else 0) =\n if n = 0 then a\n else -a * ∑ x in antidiagonal n, if x.snd < n then ↑(coeff R x.fst) φ * ↑(coeff R x.snd) (inv.aux a φ) else 0",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝¹ : Ring R\ninst✝ : DecidableEq σ\nn : σ →₀ ℕ\na : R\nφ : MvPowerSeries σ R\n⊢ inv.aux a φ n =\n if n = 0 then a\n else -a * ∑ x in antidiagonal n, if x.snd < n then ↑(coeff R x.fst) φ * ↑(coeff R x.snd) (inv.aux a φ) else 0",
"tactic": "rw [inv.aux]"
},
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝¹ : Ring R\ninst✝ : DecidableEq σ\nn : σ →₀ ℕ\na : R\nφ : MvPowerSeries σ R\n⊢ (if n = 0 then a\n else -a * ∑ x in antidiagonal n, if x_1 : x.snd < n then ↑(coeff R x.fst) φ * inv.aux a φ x.snd else 0) =\n if n = 0 then a\n else -a * ∑ x in antidiagonal n, if x.snd < n then ↑(coeff R x.fst) φ * ↑(coeff R x.snd) (inv.aux a φ) else 0",
"tactic": "convert rfl"
}
] |
[
835,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
826,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Type.lean
|
Equiv.Perm.one_lt_of_mem_cycleType
|
[] |
[
104,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/Order/BoundedOrder.lean
|
Pi.top_def
|
[] |
[
650,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
649,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.mem_biUnionIndex
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nhJ : J ∈ biUnion π πi\n⊢ J ∈ πi (biUnionIndex π πi J)",
"tactic": "convert (π.mem_biUnion.1 hJ).choose_spec.2 <;> exact dif_pos hJ"
}
] |
[
375,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
374,
1
] |
src/lean/Init/Control/Lawful.lean
|
ExceptT.run_throw
|
[] |
[
112,
99
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
112,
9
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
AffineSubspace.mem_map_iff_mem_of_injective
|
[] |
[
1531,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1529,
1
] |
Mathlib/Data/Nat/Basic.lean
|
Nat.mul_ne_mul_right
|
[] |
[
350,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
349,
1
] |
Mathlib/Order/RelIso/Group.lean
|
RelIso.coe_mul
|
[] |
[
39,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
38,
1
] |
Mathlib/Data/Analysis/Filter.lean
|
Filter.Realizer.ofEquiv_F
|
[] |
[
151,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.comap_iInf
|
[] |
[
1481,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1480,
1
] |
Mathlib/Analysis/Calculus/BumpFunctionInner.lean
|
expNegInvGlue.pos_of_pos
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : 0 < x\n⊢ 0 < expNegInvGlue x",
"tactic": "simp [expNegInvGlue, not_le.2 hx, exp_pos]"
}
] |
[
75,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.castSucc_fin_succ
|
[
{
"state_after": "no goals",
"state_before": "n✝ m n : ℕ\nj : Fin n\n⊢ ↑castSucc (succ j) = succ (↑castSucc j)",
"tactic": "simp [Fin.ext_iff]"
}
] |
[
1299,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1298,
1
] |
Mathlib/Init/Data/Bool/Lemmas.lean
|
Bool.decide_false
|
[] |
[
148,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
ModelWithCorners.unique_diff
|
[] |
[
258,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
11
] |
Mathlib/Order/SuccPred/Basic.lean
|
Order.Ioo_succ_right_of_not_isMax
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : SuccOrder α\na b : α\nhb : ¬IsMax b\n⊢ Ioo a (succ b) = Ioc a b",
"tactic": "rw [← Ioi_inter_Iio, Iio_succ_of_not_isMax hb, Ioi_inter_Iic]"
}
] |
[
316,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
Class.eq_univ_of_forall
|
[] |
[
1528,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1527,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
|
AffineMap.map_vadd
|
[] |
[
132,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.lift_aleph0
|
[] |
[
1251,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1250,
1
] |
Mathlib/GroupTheory/Perm/List.lean
|
List.formPerm_eq_formPerm_iff
|
[
{
"state_after": "case nil\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx : α\nl' : List α\nhl' : Nodup l'\nhl : Nodup []\n⊢ formPerm [] = formPerm l' ↔ [] ~r l' ∨ length [] ≤ 1 ∧ length l' ≤ 1\n\ncase cons.nil\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nl' : List α\nhl' : Nodup l'\nx : α\nhl : Nodup [x]\n⊢ formPerm [x] = formPerm l' ↔ [x] ~r l' ∨ length [x] ≤ 1 ∧ length l' ≤ 1\n\ncase cons.cons\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ : α\nl' : List α\nhl' : Nodup l'\nx y : α\nl : List α\nhl : Nodup (x :: y :: l)\n⊢ formPerm (x :: y :: l) = formPerm l' ↔ (x :: y :: l) ~r l' ∨ length (x :: y :: l) ≤ 1 ∧ length l' ≤ 1",
"state_before": "α : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl l' : List α\nhl : Nodup l\nhl' : Nodup l'\n⊢ formPerm l = formPerm l' ↔ l ~r l' ∨ length l ≤ 1 ∧ length l' ≤ 1",
"tactic": "rcases l with (_ | ⟨x, _ | ⟨y, l⟩⟩)"
},
{
"state_after": "case nil\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx : α\nl' : List α\nhl' : Nodup l'\nhl : Nodup []\n⊢ length l' ≤ 1 ↔ l' = [] ∨ length l' ≤ 1",
"state_before": "case nil\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx : α\nl' : List α\nhl' : Nodup l'\nhl : Nodup []\n⊢ formPerm [] = formPerm l' ↔ [] ~r l' ∨ length [] ≤ 1 ∧ length l' ≤ 1",
"tactic": "suffices l'.length ≤ 1 ↔ l' = nil ∨ l'.length ≤ 1 by\n simpa [eq_comm, formPerm_eq_one_iff, hl, hl', length_eq_zero]"
},
{
"state_after": "case nil\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx : α\nl' : List α\nhl' : Nodup l'\nhl : Nodup []\n⊢ l' = [] ∨ length l' ≤ 1 → length l' ≤ 1",
"state_before": "case nil\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx : α\nl' : List α\nhl' : Nodup l'\nhl : Nodup []\n⊢ length l' ≤ 1 ↔ l' = [] ∨ length l' ≤ 1",
"tactic": "refine' ⟨fun h => Or.inr h, _⟩"
},
{
"state_after": "case nil.inl\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx : α\nhl hl' : Nodup []\n⊢ length [] ≤ 1\n\ncase nil.inr\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx : α\nl' : List α\nhl' : Nodup l'\nhl : Nodup []\nh : length l' ≤ 1\n⊢ length l' ≤ 1",
"state_before": "case nil\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx : α\nl' : List α\nhl' : Nodup l'\nhl : Nodup []\n⊢ l' = [] ∨ length l' ≤ 1 → length l' ≤ 1",
"tactic": "rintro (rfl | h)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx : α\nl' : List α\nhl' : Nodup l'\nhl : Nodup []\nthis : length l' ≤ 1 ↔ l' = [] ∨ length l' ≤ 1\n⊢ formPerm [] = formPerm l' ↔ [] ~r l' ∨ length [] ≤ 1 ∧ length l' ≤ 1",
"tactic": "simpa [eq_comm, formPerm_eq_one_iff, hl, hl', length_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "case nil.inl\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx : α\nhl hl' : Nodup []\n⊢ length [] ≤ 1",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case nil.inr\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx : α\nl' : List α\nhl' : Nodup l'\nhl : Nodup []\nh : length l' ≤ 1\n⊢ length l' ≤ 1",
"tactic": "exact h"
},
{
"state_after": "case cons.nil\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nl' : List α\nhl' : Nodup l'\nx : α\nhl : Nodup [x]\n⊢ length l' ≤ 1 ↔ [x] ~r l' ∨ length l' ≤ 1",
"state_before": "case cons.nil\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nl' : List α\nhl' : Nodup l'\nx : α\nhl : Nodup [x]\n⊢ formPerm [x] = formPerm l' ↔ [x] ~r l' ∨ length [x] ≤ 1 ∧ length l' ≤ 1",
"tactic": "suffices l'.length ≤ 1 ↔ [x] ~r l' ∨ l'.length ≤ 1 by\n simpa [eq_comm, formPerm_eq_one_iff, hl, hl', length_eq_zero, le_rfl]"
},
{
"state_after": "case cons.nil\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nl' : List α\nhl' : Nodup l'\nx : α\nhl : Nodup [x]\n⊢ [x] ~r l' ∨ length l' ≤ 1 → length l' ≤ 1",
"state_before": "case cons.nil\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nl' : List α\nhl' : Nodup l'\nx : α\nhl : Nodup [x]\n⊢ length l' ≤ 1 ↔ [x] ~r l' ∨ length l' ≤ 1",
"tactic": "refine' ⟨fun h => Or.inr h, _⟩"
},
{
"state_after": "case cons.nil.inl\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nl' : List α\nhl' : Nodup l'\nx : α\nhl : Nodup [x]\nh : [x] ~r l'\n⊢ length l' ≤ 1\n\ncase cons.nil.inr\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nl' : List α\nhl' : Nodup l'\nx : α\nhl : Nodup [x]\nh : length l' ≤ 1\n⊢ length l' ≤ 1",
"state_before": "case cons.nil\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nl' : List α\nhl' : Nodup l'\nx : α\nhl : Nodup [x]\n⊢ [x] ~r l' ∨ length l' ≤ 1 → length l' ≤ 1",
"tactic": "rintro (h | h)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nl' : List α\nhl' : Nodup l'\nx : α\nhl : Nodup [x]\nthis : length l' ≤ 1 ↔ [x] ~r l' ∨ length l' ≤ 1\n⊢ formPerm [x] = formPerm l' ↔ [x] ~r l' ∨ length [x] ≤ 1 ∧ length l' ≤ 1",
"tactic": "simpa [eq_comm, formPerm_eq_one_iff, hl, hl', length_eq_zero, le_rfl]"
},
{
"state_after": "no goals",
"state_before": "case cons.nil.inl\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nl' : List α\nhl' : Nodup l'\nx : α\nhl : Nodup [x]\nh : [x] ~r l'\n⊢ length l' ≤ 1",
"tactic": "simp [← h.perm.length_eq]"
},
{
"state_after": "no goals",
"state_before": "case cons.nil.inr\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nl' : List α\nhl' : Nodup l'\nx : α\nhl : Nodup [x]\nh : length l' ≤ 1\n⊢ length l' ≤ 1",
"tactic": "exact h"
},
{
"state_after": "case cons.cons.nil\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ x y : α\nl : List α\nhl : Nodup (x :: y :: l)\nhl' : Nodup []\n⊢ formPerm (x :: y :: l) = formPerm [] ↔ (x :: y :: l) ~r [] ∨ length (x :: y :: l) ≤ 1 ∧ length [] ≤ 1\n\ncase cons.cons.cons.nil\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ x y : α\nl : List α\nhl : Nodup (x :: y :: l)\nx' : α\nhl' : Nodup [x']\n⊢ formPerm (x :: y :: l) = formPerm [x'] ↔ (x :: y :: l) ~r [x'] ∨ length (x :: y :: l) ≤ 1 ∧ length [x'] ≤ 1\n\ncase cons.cons.cons.cons\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ x y : α\nl : List α\nhl : Nodup (x :: y :: l)\nx' y' : α\nl' : List α\nhl' : Nodup (x' :: y' :: l')\n⊢ formPerm (x :: y :: l) = formPerm (x' :: y' :: l') ↔\n (x :: y :: l) ~r (x' :: y' :: l') ∨ length (x :: y :: l) ≤ 1 ∧ length (x' :: y' :: l') ≤ 1",
"state_before": "case cons.cons\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ : α\nl' : List α\nhl' : Nodup l'\nx y : α\nl : List α\nhl : Nodup (x :: y :: l)\n⊢ formPerm (x :: y :: l) = formPerm l' ↔ (x :: y :: l) ~r l' ∨ length (x :: y :: l) ≤ 1 ∧ length l' ≤ 1",
"tactic": "rcases l' with (_ | ⟨x', _ | ⟨y', l'⟩⟩)"
},
{
"state_after": "no goals",
"state_before": "case cons.cons.nil\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ x y : α\nl : List α\nhl : Nodup (x :: y :: l)\nhl' : Nodup []\n⊢ formPerm (x :: y :: l) = formPerm [] ↔ (x :: y :: l) ~r [] ∨ length (x :: y :: l) ≤ 1 ∧ length [] ≤ 1",
"tactic": "simp [formPerm_eq_one_iff _ hl, -formPerm_cons_cons]"
},
{
"state_after": "no goals",
"state_before": "case cons.cons.cons.nil\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ x y : α\nl : List α\nhl : Nodup (x :: y :: l)\nx' : α\nhl' : Nodup [x']\n⊢ formPerm (x :: y :: l) = formPerm [x'] ↔ (x :: y :: l) ~r [x'] ∨ length (x :: y :: l) ≤ 1 ∧ length [x'] ≤ 1",
"tactic": "simp [formPerm_eq_one_iff _ hl, -formPerm_cons_cons]"
},
{
"state_after": "no goals",
"state_before": "case cons.cons.cons.cons\nα : Type u_1\nβ : Type ?u.820291\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ x y : α\nl : List α\nhl : Nodup (x :: y :: l)\nx' y' : α\nl' : List α\nhl' : Nodup (x' :: y' :: l')\n⊢ formPerm (x :: y :: l) = formPerm (x' :: y' :: l') ↔\n (x :: y :: l) ~r (x' :: y' :: l') ∨ length (x :: y :: l) ≤ 1 ∧ length (x' :: y' :: l') ≤ 1",
"tactic": "simp [-formPerm_cons_cons, formPerm_ext_iff hl hl', Nat.succ_le_succ_iff]"
}
] |
[
444,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
426,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.mem_range
|
[] |
[
2985,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2984,
1
] |
Mathlib/Data/Set/NAry.lean
|
Set.image2_empty_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nα' : Type ?u.16359\nβ : Type u_3\nβ' : Type ?u.16365\nγ : Type u_1\nγ' : Type ?u.16371\nδ : Type ?u.16374\nδ' : Type ?u.16377\nε : Type ?u.16380\nε' : Type ?u.16383\nζ : Type ?u.16386\nζ' : Type ?u.16389\nν : Type ?u.16392\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na a' : α\nb b' : β\nc c' : γ\nd d' : δ\n⊢ ∀ (x : γ), x ∈ image2 f ∅ t ↔ x ∈ ∅",
"tactic": "simp"
}
] |
[
149,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/Topology/MetricSpace/Isometry.lean
|
Isometry.mapsTo_emetric_ball
|
[] |
[
167,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/Topology/PathConnected.lean
|
pathComponentIn_univ
|
[
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type ?u.633769\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type ?u.633784\nF : Set X\nx : X\n⊢ pathComponentIn x univ = pathComponent x",
"tactic": "simp [pathComponentIn, pathComponent, JoinedIn, Joined, exists_true_iff_nonempty]"
}
] |
[
943,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
942,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean
|
MeasureTheory.AEEqFun.lintegral_zero
|
[] |
[
859,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
858,
8
] |
Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean
|
Continuous.rpow_const
|
[] |
[
327,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
325,
1
] |
Mathlib/Data/Multiset/Fintype.lean
|
Multiset.image_toEnumFinset_fst
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nm✝ m : Multiset α\n⊢ Finset.image Prod.fst (toEnumFinset m) = toFinset m",
"tactic": "rw [Finset.image, Multiset.map_toEnumFinset_fst]"
}
] |
[
230,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
inv_lt_one
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.50280\nα : Type u_1\nβ : Type ?u.50286\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nha : 1 < a\n⊢ a⁻¹ < 1",
"tactic": "rwa [inv_lt (zero_lt_one.trans ha) zero_lt_one, inv_one]"
}
] |
[
300,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
299,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_mul_const
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.944622\nγ : Type ?u.944625\nδ : Type ?u.944628\nm : MeasurableSpace α\nμ ν : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ (∫⁻ (a : α), f a * r ∂μ) = (∫⁻ (a : α), f a ∂μ) * r",
"tactic": "simp_rw [mul_comm, lintegral_const_mul r hf]"
}
] |
[
734,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
733,
1
] |
Mathlib/MeasureTheory/Measure/Stieltjes.lean
|
StieltjesFunction.borel_le_measurable
|
[
{
"state_after": "f : StieltjesFunction\n⊢ MeasurableSpace.generateFrom (range Ioi) ≤ OuterMeasure.caratheodory (StieltjesFunction.outer f)",
"state_before": "f : StieltjesFunction\n⊢ borel ℝ ≤ OuterMeasure.caratheodory (StieltjesFunction.outer f)",
"tactic": "rw [borel_eq_generateFrom_Ioi]"
},
{
"state_after": "f : StieltjesFunction\n⊢ ∀ (t : Set ℝ), t ∈ range Ioi → MeasurableSet t",
"state_before": "f : StieltjesFunction\n⊢ MeasurableSpace.generateFrom (range Ioi) ≤ OuterMeasure.caratheodory (StieltjesFunction.outer f)",
"tactic": "refine' MeasurableSpace.generateFrom_le _"
},
{
"state_after": "no goals",
"state_before": "f : StieltjesFunction\n⊢ ∀ (t : Set ℝ), t ∈ range Ioi → MeasurableSet t",
"tactic": "simp (config := { contextual := true }) [f.measurableSet_Ioi]"
}
] |
[
499,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
496,
1
] |
Mathlib/Topology/LocalExtr.lean
|
IsLocalExtrOn.neg
|
[] |
[
417,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
416,
8
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.of_injective
|
[
{
"state_after": "α : Type u\nL L₁✝ L₂ L₃ L₄ : List (α × Bool)\nx✝¹ x✝ : α\nH : of x✝¹ = of x✝\nL₁ : List (α × Bool)\nhx : Red [(x✝¹, true)] L₁\nhy : Red [(x✝, true)] L₁\n⊢ x✝¹ = x✝",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nx✝¹ x✝ : α\nH : of x✝¹ = of x✝\n⊢ x✝¹ = x✝",
"tactic": "let ⟨L₁, hx, hy⟩ := Red.exact.1 H"
},
{
"state_after": "α : Type u\nL L₁✝ L₂ L₃ L₄ : List (α × Bool)\nx✝¹ x✝ : α\nH : of x✝¹ = of x✝\nL₁ : List (α × Bool)\nhx : L₁ = [(x✝¹, true)]\nhy : L₁ = [(x✝, true)]\n⊢ x✝¹ = x✝",
"state_before": "α : Type u\nL L₁✝ L₂ L₃ L₄ : List (α × Bool)\nx✝¹ x✝ : α\nH : of x✝¹ = of x✝\nL₁ : List (α × Bool)\nhx : Red [(x✝¹, true)] L₁\nhy : Red [(x✝, true)] L₁\n⊢ x✝¹ = x✝",
"tactic": "simp [Red.singleton_iff] at hx hy"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nL L₁✝ L₂ L₃ L₄ : List (α × Bool)\nx✝¹ x✝ : α\nH : of x✝¹ = of x✝\nL₁ : List (α × Bool)\nhx : L₁ = [(x✝¹, true)]\nhy : L₁ = [(x✝, true)]\n⊢ x✝¹ = x✝",
"tactic": "aesop"
}
] |
[
676,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
674,
1
] |
src/lean/Init/Data/Fin/Basic.lean
|
Fin.val_lt_of_le
|
[] |
[
116,
30
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
115,
1
] |
Mathlib/Logic/Function/Conjugate.lean
|
Function.Semiconj.option_map
|
[] |
[
77,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/LinearAlgebra/Pi.lean
|
LinearMap.pi_eq_zero
|
[
{
"state_after": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M₃\nφ : ι → Type i\ninst✝¹ : (i : ι) → AddCommMonoid (φ i)\ninst✝ : (i : ι) → Module R (φ i)\nf : (i : ι) → M₂ →ₗ[R] φ i\n⊢ (∀ (x : M₂) (a : ι), ↑(f a) x = ↑0 x a) ↔ ∀ (i : ι) (x : M₂), ↑(f i) x = ↑0 x",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M₃\nφ : ι → Type i\ninst✝¹ : (i : ι) → AddCommMonoid (φ i)\ninst✝ : (i : ι) → Module R (φ i)\nf : (i : ι) → M₂ →ₗ[R] φ i\n⊢ pi f = 0 ↔ ∀ (i : ι), f i = 0",
"tactic": "simp only [LinearMap.ext_iff, pi_apply, funext_iff]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M₃\nφ : ι → Type i\ninst✝¹ : (i : ι) → AddCommMonoid (φ i)\ninst✝ : (i : ι) → Module R (φ i)\nf : (i : ι) → M₂ →ₗ[R] φ i\n⊢ (∀ (x : M₂) (a : ι), ↑(f a) x = ↑0 x a) ↔ ∀ (i : ι) (x : M₂), ↑(f i) x = ↑0 x",
"tactic": "exact ⟨fun h a b => h b a, fun h a b => h b a⟩"
}
] |
[
69,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
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