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/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
isOpen_extChartAt_source
|
[] |
[
1046,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1045,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean
|
AffineIsometryEquiv.refl_trans
|
[] |
[
569,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
568,
1
] |
Mathlib/Algebra/CharP/Basic.lean
|
add_pow_prime_eq
|
[] |
[
80,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Data/PNat/Factors.lean
|
PrimeMultiset.prod_smul
|
[
{
"state_after": "case zero\nu : PrimeMultiset\n⊢ prod (Nat.zero • u) = Pow.pow (prod u) Nat.zero\n\ncase succ\nu : PrimeMultiset\nn : ℕ\nih : prod (n • u) = Pow.pow (prod u) n\n⊢ prod (Nat.succ n • u) = Pow.pow (prod u) (Nat.succ n)",
"state_before": "d : ℕ\nu : PrimeMultiset\n⊢ prod (d • u) = Pow.pow (prod u) d",
"tactic": "induction' d with n ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nu : PrimeMultiset\n⊢ prod (Nat.zero • u) = Pow.pow (prod u) Nat.zero",
"tactic": "rfl"
},
{
"state_after": "case succ\nu : PrimeMultiset\nn : ℕ\nih : prod (n • u) = Pow.pow (prod u) n\nthis : ∀ (n' : ℕ), Pow.pow (prod u) n' = Monoid.npow n' (prod u)\n⊢ prod (Nat.succ n • u) = Pow.pow (prod u) (Nat.succ n)",
"state_before": "case succ\nu : PrimeMultiset\nn : ℕ\nih : prod (n • u) = Pow.pow (prod u) n\n⊢ prod (Nat.succ n • u) = Pow.pow (prod u) (Nat.succ n)",
"tactic": "have : ∀ n' : ℕ, Pow.pow (prod u) n' = Monoid.npow n' (prod u) := fun _ ↦ rfl"
},
{
"state_after": "no goals",
"state_before": "case succ\nu : PrimeMultiset\nn : ℕ\nih : prod (n • u) = Pow.pow (prod u) n\nthis : ∀ (n' : ℕ), Pow.pow (prod u) n' = Monoid.npow n' (prod u)\n⊢ prod (Nat.succ n • u) = Pow.pow (prod u) (Nat.succ n)",
"tactic": "rw [succ_nsmul, prod_add, ih, this, this, Monoid.npow_succ, mul_comm]"
}
] |
[
234,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
230,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.normSq_I
|
[
{
"state_after": "no goals",
"state_before": "⊢ ↑normSq I = 1",
"tactic": "simp [normSq]"
}
] |
[
621,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
621,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.lsub_const
|
[] |
[
1683,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1682,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.coe_foldr
|
[] |
[
1406,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1404,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.iSup_of_empty
|
[] |
[
1012,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1011,
11
] |
Mathlib/Tactic/NormNum/Basic.lean
|
Mathlib.Meta.NormNum.isRat_inv_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DivisionRing α\n⊢ (↑0)⁻¹ = ↑0",
"tactic": "simp"
}
] |
[
497,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
495,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.coe_toAddSubgroup
|
[] |
[
538,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
537,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isLittleO_const_left_of_ne
|
[
{
"state_after": "α : Type u_2\nβ : Type ?u.607645\nE : Type ?u.607648\nF : Type u_3\nG : Type ?u.607654\nE' : Type ?u.607657\nF' : Type ?u.607660\nG' : Type ?u.607663\nE'' : Type u_1\nF'' : Type ?u.607669\nG'' : Type ?u.607672\nR : Type ?u.607675\nR' : Type ?u.607678\n𝕜 : Type ?u.607681\n𝕜' : Type ?u.607684\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc✝ c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nc : E''\nhc : c ≠ 0\n⊢ (fun _x => c) =o[l] g ↔ (fun _x => 1) =o[l] fun x => g x",
"state_before": "α : Type u_2\nβ : Type ?u.607645\nE : Type ?u.607648\nF : Type u_3\nG : Type ?u.607654\nE' : Type ?u.607657\nF' : Type ?u.607660\nG' : Type ?u.607663\nE'' : Type u_1\nF'' : Type ?u.607669\nG'' : Type ?u.607672\nR : Type ?u.607675\nR' : Type ?u.607678\n𝕜 : Type ?u.607681\n𝕜' : Type ?u.607684\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc✝ c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nc : E''\nhc : c ≠ 0\n⊢ (fun _x => c) =o[l] g ↔ Tendsto (fun x => ‖g x‖) l atTop",
"tactic": "simp only [← isLittleO_one_left_iff ℝ]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.607645\nE : Type ?u.607648\nF : Type u_3\nG : Type ?u.607654\nE' : Type ?u.607657\nF' : Type ?u.607660\nG' : Type ?u.607663\nE'' : Type u_1\nF'' : Type ?u.607669\nG'' : Type ?u.607672\nR : Type ?u.607675\nR' : Type ?u.607678\n𝕜 : Type ?u.607681\n𝕜' : Type ?u.607684\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc✝ c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nc : E''\nhc : c ≠ 0\n⊢ (fun _x => c) =o[l] g ↔ (fun _x => 1) =o[l] fun x => g x",
"tactic": "exact ⟨(isBigO_const_const (1 : ℝ) hc l).trans_isLittleO,\n (isBigO_const_one ℝ c l).trans_isLittleO⟩"
}
] |
[
1852,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1848,
1
] |
Mathlib/ModelTheory/Semantics.lean
|
FirstOrder.Language.LHom.onTheory_model
|
[
{
"state_after": "no goals",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.330070\nP : Type ?u.330073\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT✝ : Theory L\ninst✝¹ : Structure L' M\nφ : L →ᴸ L'\ninst✝ : IsExpansionOn φ M\nT : Theory L\n⊢ M ⊨ onTheory φ T ↔ M ⊨ T",
"tactic": "simp [Theory.model_iff, LHom.onTheory]"
}
] |
[
828,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
827,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
toDual_sdiff
|
[] |
[
1152,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1151,
1
] |
Mathlib/Algebra/Group/Units.lean
|
Units.inv_mul_eq_iff_eq_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : Monoid α\na b✝ c✝ u : αˣ\nb c : α\nh : ↑a⁻¹ * b = c\n⊢ b = ↑a * c",
"tactic": "rw [← h, mul_inv_cancel_left]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : Monoid α\na b✝ c✝ u : αˣ\nb c : α\nh : b = ↑a * c\n⊢ ↑a⁻¹ * b = c",
"tactic": "rw [h, inv_mul_cancel_left]"
}
] |
[
335,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
1
] |
Mathlib/Algebra/Module/GradedModule.lean
|
DirectSum.Gmodule.mul_smul'
|
[
{
"state_after": "ι : Type u_1\nA : ι → Type u_2\nM : ι → Type u_3\ninst✝⁵ : AddMonoid ι\ninst✝⁴ : (i : ι) → AddCommMonoid (A i)\ninst✝³ : (i : ι) → AddCommMonoid (M i)\ninst✝² : DecidableEq ι\ninst✝¹ : GSemiring A\ninst✝ : Gmodule A M\na b : ⨁ (i : ι), A i\nc : ⨁ (i : ι), M i\n⊢ comp (↑compHom (smulAddMonoidHom A M)) (mulHom A) =\n AddMonoidHom.flip\n (↑(↑compHom flipHom) (comp (↑compHom (AddMonoidHom.flip (smulAddMonoidHom A M))) (smulAddMonoidHom A M)))",
"state_before": "ι : Type u_1\nA : ι → Type u_2\nM : ι → Type u_3\ninst✝⁵ : AddMonoid ι\ninst✝⁴ : (i : ι) → AddCommMonoid (A i)\ninst✝³ : (i : ι) → AddCommMonoid (M i)\ninst✝² : DecidableEq ι\ninst✝¹ : GSemiring A\ninst✝ : Gmodule A M\na b : ⨁ (i : ι), A i\nc : ⨁ (i : ι), M i\n⊢ (a * b) • c = a • b • c",
"tactic": "suffices\n (smulAddMonoidHom\n A M).compHom.comp\n (DirectSum.mulHom A) =\n (AddMonoidHom.compHom AddMonoidHom.flipHom <|\n (smulAddMonoidHom A M).flip.compHom.comp <| smulAddMonoidHom A M).flip\n fromFunLike.congr_fun (FunLike.congr_fun (FunLike.congr_fun this a) b) c"
},
{
"state_after": "case H.h.H.h.H.h\nι : Type u_1\nA : ι → Type u_2\nM : ι → Type u_3\ninst✝⁵ : AddMonoid ι\ninst✝⁴ : (i : ι) → AddCommMonoid (A i)\ninst✝³ : (i : ι) → AddCommMonoid (M i)\ninst✝² : DecidableEq ι\ninst✝¹ : GSemiring A\ninst✝ : Gmodule A M\na b : ⨁ (i : ι), A i\nc : ⨁ (i : ι), M i\nai : ι\nax : A ai\nbi : ι\nbx : A bi\nci : ι\ncx : M ci\n⊢ ↑(comp\n (↑(comp (↑(comp (comp (↑compHom (smulAddMonoidHom A M)) (mulHom A)) (of (fun i => A i) ai)) ax)\n (of (fun i => A i) bi))\n bx)\n (of (fun i => M i) ci))\n cx =\n ↑(comp\n (↑(comp\n (↑(comp\n (AddMonoidHom.flip\n (↑(↑compHom flipHom)\n (comp (↑compHom (AddMonoidHom.flip (smulAddMonoidHom A M))) (smulAddMonoidHom A M))))\n (of (fun i => A i) ai))\n ax)\n (of (fun i => A i) bi))\n bx)\n (of (fun i => M i) ci))\n cx",
"state_before": "ι : Type u_1\nA : ι → Type u_2\nM : ι → Type u_3\ninst✝⁵ : AddMonoid ι\ninst✝⁴ : (i : ι) → AddCommMonoid (A i)\ninst✝³ : (i : ι) → AddCommMonoid (M i)\ninst✝² : DecidableEq ι\ninst✝¹ : GSemiring A\ninst✝ : Gmodule A M\na b : ⨁ (i : ι), A i\nc : ⨁ (i : ι), M i\n⊢ comp (↑compHom (smulAddMonoidHom A M)) (mulHom A) =\n AddMonoidHom.flip\n (↑(↑compHom flipHom) (comp (↑compHom (AddMonoidHom.flip (smulAddMonoidHom A M))) (smulAddMonoidHom A M)))",
"tactic": "ext (ai ax bi bx ci cx) : 6"
},
{
"state_after": "case H.h.H.h.H.h\nι : Type u_1\nA : ι → Type u_2\nM : ι → Type u_3\ninst✝⁵ : AddMonoid ι\ninst✝⁴ : (i : ι) → AddCommMonoid (A i)\ninst✝³ : (i : ι) → AddCommMonoid (M i)\ninst✝² : DecidableEq ι\ninst✝¹ : GSemiring A\ninst✝ : Gmodule A M\na b : ⨁ (i : ι), A i\nc : ⨁ (i : ι), M i\nai : ι\nax : A ai\nbi : ι\nbx : A bi\nci : ι\ncx : M ci\n⊢ ↑(↑(smulAddMonoidHom A M) (↑(↑(mulHom A) (↑(of (fun i => A i) ai) ax)) (↑(of (fun i => A i) bi) bx)))\n (↑(of (fun i => M i) ci) cx) =\n ↑(↑(smulAddMonoidHom A M) (↑(of (fun i => A i) ai) ax))\n (↑(↑(smulAddMonoidHom A M) (↑(of (fun i => A i) bi) bx)) (↑(of (fun i => M i) ci) cx))",
"state_before": "case H.h.H.h.H.h\nι : Type u_1\nA : ι → Type u_2\nM : ι → Type u_3\ninst✝⁵ : AddMonoid ι\ninst✝⁴ : (i : ι) → AddCommMonoid (A i)\ninst✝³ : (i : ι) → AddCommMonoid (M i)\ninst✝² : DecidableEq ι\ninst✝¹ : GSemiring A\ninst✝ : Gmodule A M\na b : ⨁ (i : ι), A i\nc : ⨁ (i : ι), M i\nai : ι\nax : A ai\nbi : ι\nbx : A bi\nci : ι\ncx : M ci\n⊢ ↑(comp\n (↑(comp (↑(comp (comp (↑compHom (smulAddMonoidHom A M)) (mulHom A)) (of (fun i => A i) ai)) ax)\n (of (fun i => A i) bi))\n bx)\n (of (fun i => M i) ci))\n cx =\n ↑(comp\n (↑(comp\n (↑(comp\n (AddMonoidHom.flip\n (↑(↑compHom flipHom)\n (comp (↑compHom (AddMonoidHom.flip (smulAddMonoidHom A M))) (smulAddMonoidHom A M))))\n (of (fun i => A i) ai))\n ax)\n (of (fun i => A i) bi))\n bx)\n (of (fun i => M i) ci))\n cx",
"tactic": "dsimp only [coe_comp, Function.comp_apply, compHom_apply_apply, flip_apply, flipHom_apply]"
},
{
"state_after": "case H.h.H.h.H.h\nι : Type u_1\nA : ι → Type u_2\nM : ι → Type u_3\ninst✝⁵ : AddMonoid ι\ninst✝⁴ : (i : ι) → AddCommMonoid (A i)\ninst✝³ : (i : ι) → AddCommMonoid (M i)\ninst✝² : DecidableEq ι\ninst✝¹ : GSemiring A\ninst✝ : Gmodule A M\na b : ⨁ (i : ι), A i\nc : ⨁ (i : ι), M i\nai : ι\nax : A ai\nbi : ι\nbx : A bi\nci : ι\ncx : M ci\n⊢ ↑(of M (ai + bi + ci)) (GSmul.smul (GMul.mul ax bx) cx) = ↑(of M (ai + (bi + ci))) (GSmul.smul ax (GSmul.smul bx cx))",
"state_before": "case H.h.H.h.H.h\nι : Type u_1\nA : ι → Type u_2\nM : ι → Type u_3\ninst✝⁵ : AddMonoid ι\ninst✝⁴ : (i : ι) → AddCommMonoid (A i)\ninst✝³ : (i : ι) → AddCommMonoid (M i)\ninst✝² : DecidableEq ι\ninst✝¹ : GSemiring A\ninst✝ : Gmodule A M\na b : ⨁ (i : ι), A i\nc : ⨁ (i : ι), M i\nai : ι\nax : A ai\nbi : ι\nbx : A bi\nci : ι\ncx : M ci\n⊢ ↑(↑(smulAddMonoidHom A M) (↑(↑(mulHom A) (↑(of (fun i => A i) ai) ax)) (↑(of (fun i => A i) bi) bx)))\n (↑(of (fun i => M i) ci) cx) =\n ↑(↑(smulAddMonoidHom A M) (↑(of (fun i => A i) ai) ax))\n (↑(↑(smulAddMonoidHom A M) (↑(of (fun i => A i) bi) bx)) (↑(of (fun i => M i) ci) cx))",
"tactic": "rw [smulAddMonoidHom_apply_of_of, smulAddMonoidHom_apply_of_of, DirectSum.mulHom_of_of,\n smulAddMonoidHom_apply_of_of]"
},
{
"state_after": "no goals",
"state_before": "case H.h.H.h.H.h\nι : Type u_1\nA : ι → Type u_2\nM : ι → Type u_3\ninst✝⁵ : AddMonoid ι\ninst✝⁴ : (i : ι) → AddCommMonoid (A i)\ninst✝³ : (i : ι) → AddCommMonoid (M i)\ninst✝² : DecidableEq ι\ninst✝¹ : GSemiring A\ninst✝ : Gmodule A M\na b : ⨁ (i : ι), A i\nc : ⨁ (i : ι), M i\nai : ι\nax : A ai\nbi : ι\nbx : A bi\nci : ι\ncx : M ci\n⊢ ↑(of M (ai + bi + ci)) (GSmul.smul (GMul.mul ax bx) cx) = ↑(of M (ai + (bi + ci))) (GSmul.smul ax (GSmul.smul bx cx))",
"tactic": "exact\n DirectSum.of_eq_of_gradedMonoid_eq\n (mul_smul (GradedMonoid.mk ai ax) (GradedMonoid.mk bi bx) (GradedMonoid.mk ci cx))"
}
] |
[
143,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
9
] |
Mathlib/LinearAlgebra/QuadraticForm/Prod.lean
|
QuadraticForm.anisotropic_of_prod
|
[
{
"state_after": "ι : Type ?u.37629\nR✝ : Type ?u.37632\nM₁ : Type u_2\nM₂ : Type u_3\nN₁ : Type ?u.37641\nN₂ : Type ?u.37644\nMᵢ : ι → Type ?u.37649\nNᵢ : ι → Type ?u.37654\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Module R✝ M₁\ninst✝⁹ : Module R✝ M₂\ninst✝⁸ : Module R✝ N₁\ninst✝⁷ : Module R✝ N₂\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁴ : (i : ι) → Module R✝ (Mᵢ i)\ninst✝³ : (i : ι) → Module R✝ (Nᵢ i)\nR : Type u_1\ninst✝² : OrderedRing R\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nh : ∀ (a : M₁) (b : M₂), ↑Q₁ a + ↑Q₂ b = 0 → a = 0 ∧ b = 0\n⊢ Anisotropic Q₁ ∧ Anisotropic Q₂",
"state_before": "ι : Type ?u.37629\nR✝ : Type ?u.37632\nM₁ : Type u_2\nM₂ : Type u_3\nN₁ : Type ?u.37641\nN₂ : Type ?u.37644\nMᵢ : ι → Type ?u.37649\nNᵢ : ι → Type ?u.37654\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Module R✝ M₁\ninst✝⁹ : Module R✝ M₂\ninst✝⁸ : Module R✝ N₁\ninst✝⁷ : Module R✝ N₂\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁴ : (i : ι) → Module R✝ (Mᵢ i)\ninst✝³ : (i : ι) → Module R✝ (Nᵢ i)\nR : Type u_1\ninst✝² : OrderedRing R\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nh : Anisotropic (prod Q₁ Q₂)\n⊢ Anisotropic Q₁ ∧ Anisotropic Q₂",
"tactic": "simp_rw [Anisotropic, prod_apply, Prod.forall, Prod.mk_eq_zero] at h"
},
{
"state_after": "case left\nι : Type ?u.37629\nR✝ : Type ?u.37632\nM₁ : Type u_2\nM₂ : Type u_3\nN₁ : Type ?u.37641\nN₂ : Type ?u.37644\nMᵢ : ι → Type ?u.37649\nNᵢ : ι → Type ?u.37654\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Module R✝ M₁\ninst✝⁹ : Module R✝ M₂\ninst✝⁸ : Module R✝ N₁\ninst✝⁷ : Module R✝ N₂\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁴ : (i : ι) → Module R✝ (Mᵢ i)\ninst✝³ : (i : ι) → Module R✝ (Nᵢ i)\nR : Type u_1\ninst✝² : OrderedRing R\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nh : ∀ (a : M₁) (b : M₂), ↑Q₁ a + ↑Q₂ b = 0 → a = 0 ∧ b = 0\n⊢ Anisotropic Q₁\n\ncase right\nι : Type ?u.37629\nR✝ : Type ?u.37632\nM₁ : Type u_2\nM₂ : Type u_3\nN₁ : Type ?u.37641\nN₂ : Type ?u.37644\nMᵢ : ι → Type ?u.37649\nNᵢ : ι → Type ?u.37654\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Module R✝ M₁\ninst✝⁹ : Module R✝ M₂\ninst✝⁸ : Module R✝ N₁\ninst✝⁷ : Module R✝ N₂\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁴ : (i : ι) → Module R✝ (Mᵢ i)\ninst✝³ : (i : ι) → Module R✝ (Nᵢ i)\nR : Type u_1\ninst✝² : OrderedRing R\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nh : ∀ (a : M₁) (b : M₂), ↑Q₁ a + ↑Q₂ b = 0 → a = 0 ∧ b = 0\n⊢ Anisotropic Q₂",
"state_before": "ι : Type ?u.37629\nR✝ : Type ?u.37632\nM₁ : Type u_2\nM₂ : Type u_3\nN₁ : Type ?u.37641\nN₂ : Type ?u.37644\nMᵢ : ι → Type ?u.37649\nNᵢ : ι → Type ?u.37654\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Module R✝ M₁\ninst✝⁹ : Module R✝ M₂\ninst✝⁸ : Module R✝ N₁\ninst✝⁷ : Module R✝ N₂\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁴ : (i : ι) → Module R✝ (Mᵢ i)\ninst✝³ : (i : ι) → Module R✝ (Nᵢ i)\nR : Type u_1\ninst✝² : OrderedRing R\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nh : ∀ (a : M₁) (b : M₂), ↑Q₁ a + ↑Q₂ b = 0 → a = 0 ∧ b = 0\n⊢ Anisotropic Q₁ ∧ Anisotropic Q₂",
"tactic": "constructor"
},
{
"state_after": "case left\nι : Type ?u.37629\nR✝ : Type ?u.37632\nM₁ : Type u_2\nM₂ : Type u_3\nN₁ : Type ?u.37641\nN₂ : Type ?u.37644\nMᵢ : ι → Type ?u.37649\nNᵢ : ι → Type ?u.37654\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Module R✝ M₁\ninst✝⁹ : Module R✝ M₂\ninst✝⁸ : Module R✝ N₁\ninst✝⁷ : Module R✝ N₂\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁴ : (i : ι) → Module R✝ (Mᵢ i)\ninst✝³ : (i : ι) → Module R✝ (Nᵢ i)\nR : Type u_1\ninst✝² : OrderedRing R\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nh : ∀ (a : M₁) (b : M₂), ↑Q₁ a + ↑Q₂ b = 0 → a = 0 ∧ b = 0\nx : M₁\nhx : ↑Q₁ x = 0\n⊢ x = 0",
"state_before": "case left\nι : Type ?u.37629\nR✝ : Type ?u.37632\nM₁ : Type u_2\nM₂ : Type u_3\nN₁ : Type ?u.37641\nN₂ : Type ?u.37644\nMᵢ : ι → Type ?u.37649\nNᵢ : ι → Type ?u.37654\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Module R✝ M₁\ninst✝⁹ : Module R✝ M₂\ninst✝⁸ : Module R✝ N₁\ninst✝⁷ : Module R✝ N₂\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁴ : (i : ι) → Module R✝ (Mᵢ i)\ninst✝³ : (i : ι) → Module R✝ (Nᵢ i)\nR : Type u_1\ninst✝² : OrderedRing R\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nh : ∀ (a : M₁) (b : M₂), ↑Q₁ a + ↑Q₂ b = 0 → a = 0 ∧ b = 0\n⊢ Anisotropic Q₁",
"tactic": "intro x hx"
},
{
"state_after": "case left\nι : Type ?u.37629\nR✝ : Type ?u.37632\nM₁ : Type u_2\nM₂ : Type u_3\nN₁ : Type ?u.37641\nN₂ : Type ?u.37644\nMᵢ : ι → Type ?u.37649\nNᵢ : ι → Type ?u.37654\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Module R✝ M₁\ninst✝⁹ : Module R✝ M₂\ninst✝⁸ : Module R✝ N₁\ninst✝⁷ : Module R✝ N₂\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁴ : (i : ι) → Module R✝ (Mᵢ i)\ninst✝³ : (i : ι) → Module R✝ (Nᵢ i)\nR : Type u_1\ninst✝² : OrderedRing R\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nh : ∀ (a : M₁) (b : M₂), ↑Q₁ a + ↑Q₂ b = 0 → a = 0 ∧ b = 0\nx : M₁\nhx : ↑Q₁ x = 0\n⊢ ↑Q₁ x + ↑Q₂ 0 = 0",
"state_before": "case left\nι : Type ?u.37629\nR✝ : Type ?u.37632\nM₁ : Type u_2\nM₂ : Type u_3\nN₁ : Type ?u.37641\nN₂ : Type ?u.37644\nMᵢ : ι → Type ?u.37649\nNᵢ : ι → Type ?u.37654\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Module R✝ M₁\ninst✝⁹ : Module R✝ M₂\ninst✝⁸ : Module R✝ N₁\ninst✝⁷ : Module R✝ N₂\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁴ : (i : ι) → Module R✝ (Mᵢ i)\ninst✝³ : (i : ι) → Module R✝ (Nᵢ i)\nR : Type u_1\ninst✝² : OrderedRing R\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nh : ∀ (a : M₁) (b : M₂), ↑Q₁ a + ↑Q₂ b = 0 → a = 0 ∧ b = 0\nx : M₁\nhx : ↑Q₁ x = 0\n⊢ x = 0",
"tactic": "refine' (h x 0 _).1"
},
{
"state_after": "no goals",
"state_before": "case left\nι : Type ?u.37629\nR✝ : Type ?u.37632\nM₁ : Type u_2\nM₂ : Type u_3\nN₁ : Type ?u.37641\nN₂ : Type ?u.37644\nMᵢ : ι → Type ?u.37649\nNᵢ : ι → Type ?u.37654\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Module R✝ M₁\ninst✝⁹ : Module R✝ M₂\ninst✝⁸ : Module R✝ N₁\ninst✝⁷ : Module R✝ N₂\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁴ : (i : ι) → Module R✝ (Mᵢ i)\ninst✝³ : (i : ι) → Module R✝ (Nᵢ i)\nR : Type u_1\ninst✝² : OrderedRing R\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nh : ∀ (a : M₁) (b : M₂), ↑Q₁ a + ↑Q₂ b = 0 → a = 0 ∧ b = 0\nx : M₁\nhx : ↑Q₁ x = 0\n⊢ ↑Q₁ x + ↑Q₂ 0 = 0",
"tactic": "rw [hx, zero_add, map_zero]"
},
{
"state_after": "case right\nι : Type ?u.37629\nR✝ : Type ?u.37632\nM₁ : Type u_2\nM₂ : Type u_3\nN₁ : Type ?u.37641\nN₂ : Type ?u.37644\nMᵢ : ι → Type ?u.37649\nNᵢ : ι → Type ?u.37654\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Module R✝ M₁\ninst✝⁹ : Module R✝ M₂\ninst✝⁸ : Module R✝ N₁\ninst✝⁷ : Module R✝ N₂\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁴ : (i : ι) → Module R✝ (Mᵢ i)\ninst✝³ : (i : ι) → Module R✝ (Nᵢ i)\nR : Type u_1\ninst✝² : OrderedRing R\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nh : ∀ (a : M₁) (b : M₂), ↑Q₁ a + ↑Q₂ b = 0 → a = 0 ∧ b = 0\nx : M₂\nhx : ↑Q₂ x = 0\n⊢ x = 0",
"state_before": "case right\nι : Type ?u.37629\nR✝ : Type ?u.37632\nM₁ : Type u_2\nM₂ : Type u_3\nN₁ : Type ?u.37641\nN₂ : Type ?u.37644\nMᵢ : ι → Type ?u.37649\nNᵢ : ι → Type ?u.37654\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Module R✝ M₁\ninst✝⁹ : Module R✝ M₂\ninst✝⁸ : Module R✝ N₁\ninst✝⁷ : Module R✝ N₂\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁴ : (i : ι) → Module R✝ (Mᵢ i)\ninst✝³ : (i : ι) → Module R✝ (Nᵢ i)\nR : Type u_1\ninst✝² : OrderedRing R\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nh : ∀ (a : M₁) (b : M₂), ↑Q₁ a + ↑Q₂ b = 0 → a = 0 ∧ b = 0\n⊢ Anisotropic Q₂",
"tactic": "intro x hx"
},
{
"state_after": "case right\nι : Type ?u.37629\nR✝ : Type ?u.37632\nM₁ : Type u_2\nM₂ : Type u_3\nN₁ : Type ?u.37641\nN₂ : Type ?u.37644\nMᵢ : ι → Type ?u.37649\nNᵢ : ι → Type ?u.37654\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Module R✝ M₁\ninst✝⁹ : Module R✝ M₂\ninst✝⁸ : Module R✝ N₁\ninst✝⁷ : Module R✝ N₂\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁴ : (i : ι) → Module R✝ (Mᵢ i)\ninst✝³ : (i : ι) → Module R✝ (Nᵢ i)\nR : Type u_1\ninst✝² : OrderedRing R\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nh : ∀ (a : M₁) (b : M₂), ↑Q₁ a + ↑Q₂ b = 0 → a = 0 ∧ b = 0\nx : M₂\nhx : ↑Q₂ x = 0\n⊢ ↑Q₁ 0 + ↑Q₂ x = 0",
"state_before": "case right\nι : Type ?u.37629\nR✝ : Type ?u.37632\nM₁ : Type u_2\nM₂ : Type u_3\nN₁ : Type ?u.37641\nN₂ : Type ?u.37644\nMᵢ : ι → Type ?u.37649\nNᵢ : ι → Type ?u.37654\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Module R✝ M₁\ninst✝⁹ : Module R✝ M₂\ninst✝⁸ : Module R✝ N₁\ninst✝⁷ : Module R✝ N₂\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁴ : (i : ι) → Module R✝ (Mᵢ i)\ninst✝³ : (i : ι) → Module R✝ (Nᵢ i)\nR : Type u_1\ninst✝² : OrderedRing R\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nh : ∀ (a : M₁) (b : M₂), ↑Q₁ a + ↑Q₂ b = 0 → a = 0 ∧ b = 0\nx : M₂\nhx : ↑Q₂ x = 0\n⊢ x = 0",
"tactic": "refine' (h 0 x _).2"
},
{
"state_after": "no goals",
"state_before": "case right\nι : Type ?u.37629\nR✝ : Type ?u.37632\nM₁ : Type u_2\nM₂ : Type u_3\nN₁ : Type ?u.37641\nN₂ : Type ?u.37644\nMᵢ : ι → Type ?u.37649\nNᵢ : ι → Type ?u.37654\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Module R✝ M₁\ninst✝⁹ : Module R✝ M₂\ninst✝⁸ : Module R✝ N₁\ninst✝⁷ : Module R✝ N₂\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁴ : (i : ι) → Module R✝ (Mᵢ i)\ninst✝³ : (i : ι) → Module R✝ (Nᵢ i)\nR : Type u_1\ninst✝² : OrderedRing R\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nh : ∀ (a : M₁) (b : M₂), ↑Q₁ a + ↑Q₂ b = 0 → a = 0 ∧ b = 0\nx : M₂\nhx : ↑Q₂ x = 0\n⊢ ↑Q₁ 0 + ↑Q₂ x = 0",
"tactic": "rw [hx, add_zero, map_zero]"
}
] |
[
88,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
eq_mul_inv_iff_mul_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.54177\nβ : Type ?u.54180\nG : Type u_1\ninst✝ : Group G\na b c d : G\nh : a = b * c⁻¹\n⊢ a * c = b",
"tactic": "rw [h, inv_mul_cancel_right]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.54177\nβ : Type ?u.54180\nG : Type u_1\ninst✝ : Group G\na b c d : G\nh : a * c = b\n⊢ a = b * c⁻¹",
"tactic": "rw [← h, mul_inv_cancel_right]"
}
] |
[
693,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
692,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Ioc_subset_Ioo_right
|
[
{
"state_after": "ι : Type ?u.22707\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\nh : b₁ < b₂\n⊢ Set.Ioc a b₁ ⊆ Set.Ioo a b₂",
"state_before": "ι : Type ?u.22707\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\nh : b₁ < b₂\n⊢ Ioc a b₁ ⊆ Ioo a b₂",
"tactic": "rw [← coe_subset, coe_Ioc, coe_Ioo]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.22707\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\nh : b₁ < b₂\n⊢ Set.Ioc a b₁ ⊆ Set.Ioo a b₂",
"tactic": "exact Set.Ioc_subset_Ioo_right h"
}
] |
[
217,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.SimpleFunc.setToSimpleFunc_indicator
|
[
{
"state_after": "case inl\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\nx : F\nhs : MeasurableSet ∅\n⊢ setToSimpleFunc T (piecewise ∅ hs (const α x) (const α 0)) = ↑(T ∅) x\n\ncase inr\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\n⊢ setToSimpleFunc T (piecewise s hs (const α x) (const α 0)) = ↑(T s) x",
"state_before": "α : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\n⊢ setToSimpleFunc T (piecewise s hs (const α x) (const α 0)) = ↑(T s) x",
"tactic": "obtain rfl | hs_empty := s.eq_empty_or_nonempty"
},
{
"state_after": "case inr\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\n⊢ ∑ x_1 in SimpleFunc.range (piecewise s hs (const α x) (const α 0)),\n ↑(T (↑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 =\n ↑(T s) x",
"state_before": "case inr\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\n⊢ setToSimpleFunc T (piecewise s hs (const α x) (const α 0)) = ↑(T s) x",
"tactic": "simp_rw [setToSimpleFunc]"
},
{
"state_after": "case inr.inl\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\nx : F\nhs : MeasurableSet Set.univ\nhs_empty : Set.Nonempty Set.univ\n⊢ ∑ x_1 in SimpleFunc.range (piecewise Set.univ hs (const α x) (const α 0)),\n ↑(T (↑(piecewise Set.univ hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 =\n ↑(T Set.univ) x\n\ncase inr.inr\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\n⊢ ∑ x_1 in SimpleFunc.range (piecewise s hs (const α x) (const α 0)),\n ↑(T (↑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 =\n ↑(T s) x",
"state_before": "case inr\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\n⊢ ∑ x_1 in SimpleFunc.range (piecewise s hs (const α x) (const α 0)),\n ↑(T (↑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 =\n ↑(T s) x",
"tactic": "obtain rfl | hs_univ := eq_or_ne s univ"
},
{
"state_after": "case inr.inr\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\n⊢ ∑ x_1 in {x, 0}, ↑(T (↑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 = ↑(T s) x",
"state_before": "case inr.inr\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\n⊢ ∑ x_1 in SimpleFunc.range (piecewise s hs (const α x) (const α 0)),\n ↑(T (↑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 =\n ↑(T s) x",
"tactic": "rw [range_indicator hs hs_empty hs_univ]"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\nhx0 : x = 0\n⊢ ∑ x_1 in {x, 0}, ↑(T (↑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 = ↑(T s) x\n\ncase neg\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\nhx0 : ¬x = 0\n⊢ ∑ x_1 in {x, 0}, ↑(T (↑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 = ↑(T s) x",
"state_before": "case inr.inr\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\n⊢ ∑ x_1 in {x, 0}, ↑(T (↑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 = ↑(T s) x",
"tactic": "by_cases hx0 : x = 0"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\nhx0 : ¬x = 0\n⊢ ↑(T (↑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x})) x +\n ∑ x_1 in {0}, ↑(T (↑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 =\n ↑(T s) x\n\ncase neg\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\nhx0 : ¬x = 0\n⊢ ¬x ∈ {0}",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\nhx0 : ¬x = 0\n⊢ ∑ x_1 in {x, 0}, ↑(T (↑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 = ↑(T s) x",
"tactic": "rw [sum_insert]"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\nhx0 : ¬x = 0\n⊢ ¬x ∈ {0}\n\ncase neg\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\nhx0 : ¬x = 0\n⊢ ↑(T (↑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x})) x +\n ∑ x_1 in {0}, ↑(T (↑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 =\n ↑(T s) x",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\nhx0 : ¬x = 0\n⊢ ↑(T (↑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x})) x +\n ∑ x_1 in {0}, ↑(T (↑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 =\n ↑(T s) x\n\ncase neg\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\nhx0 : ¬x = 0\n⊢ ¬x ∈ {0}",
"tactic": "swap"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\nhx0 : ¬x = 0\n⊢ ↑(T (↑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x})) x = ↑(T s) x",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\nhx0 : ¬x = 0\n⊢ ↑(T (↑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x})) x +\n ∑ x_1 in {0}, ↑(T (↑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 =\n ↑(T s) x",
"tactic": "rw [sum_singleton, (T _).map_zero, add_zero]"
},
{
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{
"state_after": "case pos\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\nhx0 : x = 0\n⊢ ∑ x in {0, 0}, ↑(T (↑(piecewise s hs (const α 0) (const α 0)) ⁻¹' {x})) x = ↑(T s) 0",
"state_before": "case pos\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\nhx0 : x = 0\n⊢ ∑ x_1 in {x, 0}, ↑(T (↑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 = ↑(T s) x",
"tactic": "simp_rw [hx0]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\nhx0 : x = 0\n⊢ ∑ x in {0, 0}, ↑(T (↑(piecewise s hs (const α 0) (const α 0)) ⁻¹' {x})) x = ↑(T s) 0",
"tactic": "simp"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\nhx0 : ¬x = 0\n⊢ ¬x = 0",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\nhx0 : ¬x = 0\n⊢ ¬x ∈ {0}",
"tactic": "rw [Finset.mem_singleton]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\nhx0 : ¬x = 0\n⊢ ¬x = 0",
"tactic": "exact hx0"
},
{
"state_after": "no goals",
"state_before": "case neg.e_a.e_a\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\nhx0 : ¬x = 0\n⊢ x ∈ {x}",
"tactic": "exact Set.mem_singleton x"
},
{
"state_after": "case neg.e_a.e_a\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\nhx0 : ¬x = 0\n⊢ ¬0 = x",
"state_before": "case neg.e_a.e_a\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\nhx0 : ¬x = 0\n⊢ ¬0 ∈ {x}",
"tactic": "rw [Set.mem_singleton_iff]"
},
{
"state_after": "no goals",
"state_before": "case neg.e_a.e_a\nα : Type u_1\nE : Type ?u.425807\nF : Type u_2\nF' : Type u_3\nG : Type ?u.425816\n𝕜 : Type ?u.425819\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : Set.Nonempty s\nhs_univ : s ≠ Set.univ\nhx0 : ¬x = 0\n⊢ ¬0 = x",
"tactic": "exact Ne.symm hx0"
}
] |
[
632,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
607,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
AffineSubspace.nonempty_of_affineSpan_eq_top
|
[
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : Set P\nh : affineSpan k s = ⊤\n⊢ s ≠ ∅",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : Set P\nh : affineSpan k s = ⊤\n⊢ Set.Nonempty s",
"tactic": "rw [Set.nonempty_iff_ne_empty]"
},
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\nh : affineSpan k ∅ = ⊤\n⊢ False",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : Set P\nh : affineSpan k s = ⊤\n⊢ s ≠ ∅",
"tactic": "rintro rfl"
},
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\nh : ⊥ = ⊤\n⊢ False",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\nh : affineSpan k ∅ = ⊤\n⊢ False",
"tactic": "rw [AffineSubspace.span_empty] at h"
},
{
"state_after": "no goals",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\nh : ⊥ = ⊤\n⊢ False",
"tactic": "exact bot_ne_top k V P h"
}
] |
[
784,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
780,
1
] |
Mathlib/Data/Bool/Basic.lean
|
Bool.not_inj
|
[
{
"state_after": "no goals",
"state_before": "⊢ ∀ {a b : Bool}, (!decide (a = !b)) = true → a = b",
"tactic": "decide"
}
] |
[
314,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
314,
1
] |
Mathlib/Data/Finsupp/NeLocus.lean
|
Finsupp.zipWith_neLocus_eq_right
|
[
{
"state_after": "case a\nα : Type u_4\nM : Type u_1\nN : Type u_3\nP : Type u_2\ninst✝⁵ : DecidableEq α\ninst✝⁴ : DecidableEq M\ninst✝³ : Zero M\ninst✝² : DecidableEq P\ninst✝¹ : Zero P\ninst✝ : Zero N\nF : M → N → P\nF0 : F 0 0 = 0\nf₁ f₂ : α →₀ M\ng : α →₀ N\nhF : ∀ (g : N), Function.Injective fun f => F f g\na✝ : α\n⊢ a✝ ∈ neLocus (zipWith F F0 f₁ g) (zipWith F F0 f₂ g) ↔ a✝ ∈ neLocus f₁ f₂",
"state_before": "α : Type u_4\nM : Type u_1\nN : Type u_3\nP : Type u_2\ninst✝⁵ : DecidableEq α\ninst✝⁴ : DecidableEq M\ninst✝³ : Zero M\ninst✝² : DecidableEq P\ninst✝¹ : Zero P\ninst✝ : Zero N\nF : M → N → P\nF0 : F 0 0 = 0\nf₁ f₂ : α →₀ M\ng : α →₀ N\nhF : ∀ (g : N), Function.Injective fun f => F f g\n⊢ neLocus (zipWith F F0 f₁ g) (zipWith F F0 f₂ g) = neLocus f₁ f₂",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_4\nM : Type u_1\nN : Type u_3\nP : Type u_2\ninst✝⁵ : DecidableEq α\ninst✝⁴ : DecidableEq M\ninst✝³ : Zero M\ninst✝² : DecidableEq P\ninst✝¹ : Zero P\ninst✝ : Zero N\nF : M → N → P\nF0 : F 0 0 = 0\nf₁ f₂ : α →₀ M\ng : α →₀ N\nhF : ∀ (g : N), Function.Injective fun f => F f g\na✝ : α\n⊢ a✝ ∈ neLocus (zipWith F F0 f₁ g) (zipWith F F0 f₂ g) ↔ a✝ ∈ neLocus f₁ f₂",
"tactic": "simpa only [mem_neLocus] using (hF _).ne_iff"
}
] |
[
109,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
wbtw_const_vsub_iff
|
[] |
[
201,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
199,
1
] |
Mathlib/Topology/FiberBundle/Basic.lean
|
FiberPrebundle.continuous_proj
|
[
{
"state_after": "ι : Type ?u.65338\nB : Type u_1\nF : Type u_3\nX : Type ?u.65347\ninst✝³ : TopologicalSpace X\nE : B → Type u_2\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\ninst✝ : (x : B) → TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nthis : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\n⊢ Continuous TotalSpace.proj",
"state_before": "ι : Type ?u.65338\nB : Type u_1\nF : Type u_3\nX : Type ?u.65347\ninst✝³ : TopologicalSpace X\nE : B → Type u_2\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\ninst✝ : (x : B) → TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\n⊢ Continuous TotalSpace.proj",
"tactic": "letI := a.totalSpaceTopology"
},
{
"state_after": "ι : Type ?u.65338\nB : Type u_1\nF : Type u_3\nX : Type ?u.65347\ninst✝³ : TopologicalSpace X\nE : B → Type u_2\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\ninst✝ : (x : B) → TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nthis✝ : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\n⊢ Continuous TotalSpace.proj",
"state_before": "ι : Type ?u.65338\nB : Type u_1\nF : Type u_3\nX : Type ?u.65347\ninst✝³ : TopologicalSpace X\nE : B → Type u_2\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\ninst✝ : (x : B) → TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nthis : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\n⊢ Continuous TotalSpace.proj",
"tactic": "letI := a.toFiberBundle"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.65338\nB : Type u_1\nF : Type u_3\nX : Type ?u.65347\ninst✝³ : TopologicalSpace X\nE : B → Type u_2\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\ninst✝ : (x : B) → TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nthis✝ : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\n⊢ Continuous TotalSpace.proj",
"tactic": "exact FiberBundle.continuous_proj F E"
}
] |
[
911,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
908,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/SubboxInduction.lean
|
BoxIntegral.Prepartition.iUnion_toSubordinate
|
[] |
[
207,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Std/Classes/Order.lean
|
Std.TransCmp.lt_le_trans
|
[] |
[
68,
63
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
67,
1
] |
Mathlib/Topology/ContinuousFunction/Compact.lean
|
ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
|
[] |
[
319,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
317,
1
] |
Mathlib/MeasureTheory/Constructions/Pi.lean
|
MeasureTheory.Measure.pi_Iio_ae_eq_pi_Iic
|
[] |
[
491,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
489,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.ofPowerSeries_apply
|
[
{
"state_after": "Γ : Type ?u.2177048\nR : Type ?u.2177051\ninst✝¹ : Semiring R\ninst✝ : StrictOrderedSemiring Γ\nx : PowerSeries R\n⊢ ∀ {a b : ℕ}, ↑a ≤ ↑b ↔ a ≤ b",
"state_before": "Γ : Type ?u.2177048\nR : Type ?u.2177051\ninst✝¹ : Semiring R\ninst✝ : StrictOrderedSemiring Γ\nx : PowerSeries R\n⊢ ∀ {a b : ℕ},\n ↑{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } a ≤\n ↑{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } b ↔\n a ≤ b",
"tactic": "simp only [Function.Embedding.coeFn_mk]"
},
{
"state_after": "no goals",
"state_before": "Γ : Type ?u.2177048\nR : Type ?u.2177051\ninst✝¹ : Semiring R\ninst✝ : StrictOrderedSemiring Γ\nx : PowerSeries R\n⊢ ∀ {a b : ℕ}, ↑a ≤ ↑b ↔ a ≤ b",
"tactic": "exact Nat.cast_le"
}
] |
[
1186,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1179,
1
] |
Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean
|
CategoryTheory.Limits.biproduct.mapBiproduct_inv_map_desc
|
[
{
"state_after": "case w\nC : Type u₁\ninst✝⁶ : Category C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : HasZeroMorphisms C\ninst✝³ : HasZeroMorphisms D\nF : C ⥤ D\ninst✝² : PreservesZeroMorphisms F\nJ : Type w₁\nf : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : PreservesBiproduct f F\nW : C\ng : (j : J) → f j ⟶ W\n⊢ ∀ (j : J), ι (F.obj ∘ f) j ≫ (mapBiproduct F f).inv ≫ F.map (desc g) = ι (F.obj ∘ f) j ≫ desc fun j => F.map (g j)",
"state_before": "C : Type u₁\ninst✝⁶ : Category C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : HasZeroMorphisms C\ninst✝³ : HasZeroMorphisms D\nF : C ⥤ D\ninst✝² : PreservesZeroMorphisms F\nJ : Type w₁\nf : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : PreservesBiproduct f F\nW : C\ng : (j : J) → f j ⟶ W\n⊢ (mapBiproduct F f).inv ≫ F.map (desc g) = desc fun j => F.map (g j)",
"tactic": "apply biproduct.hom_ext'"
},
{
"state_after": "case w\nC : Type u₁\ninst✝⁶ : Category C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : HasZeroMorphisms C\ninst✝³ : HasZeroMorphisms D\nF : C ⥤ D\ninst✝² : PreservesZeroMorphisms F\nJ : Type w₁\nf : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : PreservesBiproduct f F\nW : C\ng : (j : J) → f j ⟶ W\nj : J\n⊢ ι (F.obj ∘ f) j ≫ (mapBiproduct F f).inv ≫ F.map (desc g) = ι (F.obj ∘ f) j ≫ desc fun j => F.map (g j)",
"state_before": "case w\nC : Type u₁\ninst✝⁶ : Category C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : HasZeroMorphisms C\ninst✝³ : HasZeroMorphisms D\nF : C ⥤ D\ninst✝² : PreservesZeroMorphisms F\nJ : Type w₁\nf : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : PreservesBiproduct f F\nW : C\ng : (j : J) → f j ⟶ W\n⊢ ∀ (j : J), ι (F.obj ∘ f) j ≫ (mapBiproduct F f).inv ≫ F.map (desc g) = ι (F.obj ∘ f) j ≫ desc fun j => F.map (g j)",
"tactic": "intro j"
},
{
"state_after": "case w\nC : Type u₁\ninst✝⁶ : Category C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : HasZeroMorphisms C\ninst✝³ : HasZeroMorphisms D\nF : C ⥤ D\ninst✝² : PreservesZeroMorphisms F\nJ : Type w₁\nf : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : PreservesBiproduct f F\nW : C\ng : (j : J) → f j ⟶ W\nj : J\n⊢ ι (fun x => F.obj (f x)) j ≫ (mapBiproduct F f).inv ≫ F.map (desc g) =\n ι (fun x => F.obj (f x)) j ≫ desc fun j => F.map (g j)",
"state_before": "case w\nC : Type u₁\ninst✝⁶ : Category C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : HasZeroMorphisms C\ninst✝³ : HasZeroMorphisms D\nF : C ⥤ D\ninst✝² : PreservesZeroMorphisms F\nJ : Type w₁\nf : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : PreservesBiproduct f F\nW : C\ng : (j : J) → f j ⟶ W\nj : J\n⊢ ι (F.obj ∘ f) j ≫ (mapBiproduct F f).inv ≫ F.map (desc g) = ι (F.obj ∘ f) j ≫ desc fun j => F.map (g j)",
"tactic": "dsimp [Function.comp]"
},
{
"state_after": "case w\nC : Type u₁\ninst✝⁶ : Category C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : HasZeroMorphisms C\ninst✝³ : HasZeroMorphisms D\nF : C ⥤ D\ninst✝² : PreservesZeroMorphisms F\nJ : Type w₁\nf : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : PreservesBiproduct f F\nW : C\ng : (j : J) → f j ⟶ W\nj : J\nthis : HasBiproduct fun j => F.obj (f j)\n⊢ ι (fun x => F.obj (f x)) j ≫ (mapBiproduct F f).inv ≫ F.map (desc g) =\n ι (fun x => F.obj (f x)) j ≫ desc fun j => F.map (g j)",
"state_before": "case w\nC : Type u₁\ninst✝⁶ : Category C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : HasZeroMorphisms C\ninst✝³ : HasZeroMorphisms D\nF : C ⥤ D\ninst✝² : PreservesZeroMorphisms F\nJ : Type w₁\nf : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : PreservesBiproduct f F\nW : C\ng : (j : J) → f j ⟶ W\nj : J\n⊢ ι (fun x => F.obj (f x)) j ≫ (mapBiproduct F f).inv ≫ F.map (desc g) =\n ι (fun x => F.obj (f x)) j ≫ desc fun j => F.map (g j)",
"tactic": "haveI : HasBiproduct fun j => F.obj (f j) := hasBiproduct_of_preserves F f"
},
{
"state_after": "no goals",
"state_before": "case w\nC : Type u₁\ninst✝⁶ : Category C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : HasZeroMorphisms C\ninst✝³ : HasZeroMorphisms D\nF : C ⥤ D\ninst✝² : PreservesZeroMorphisms F\nJ : Type w₁\nf : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : PreservesBiproduct f F\nW : C\ng : (j : J) → f j ⟶ W\nj : J\nthis : HasBiproduct fun j => F.obj (f j)\n⊢ ι (fun x => F.obj (f x)) j ≫ (mapBiproduct F f).inv ≫ F.map (desc g) =\n ι (fun x => F.obj (f x)) j ≫ desc fun j => F.map (g j)",
"tactic": "simp only [mapBiproduct_inv, ← Category.assoc, biproduct.ι_desc ,← F.map_comp]"
}
] |
[
437,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
430,
1
] |
Mathlib/Algebra/Homology/HomologicalComplex.lean
|
HomologicalComplex.eqToHom_comp_d
|
[
{
"state_after": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ni j : ι\nrij rij' : ComplexShape.Rel c i j\n⊢ eqToHom (_ : X C i = X C i) ≫ d C i j = d C i j",
"state_before": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ni i' j : ι\nrij : ComplexShape.Rel c i j\nrij' : ComplexShape.Rel c i' j\n⊢ eqToHom (_ : X C i = X C i') ≫ d C i' j = d C i j",
"tactic": "obtain rfl := c.prev_eq rij rij'"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ni j : ι\nrij rij' : ComplexShape.Rel c i j\n⊢ eqToHom (_ : X C i = X C i) ≫ d C i j = d C i j",
"tactic": "simp only [eqToHom_refl, id_comp]"
}
] |
[
332,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
329,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.top_mul_of_pos
|
[
{
"state_after": "x : EReal\nh : 0 < x\n⊢ x * ⊤ = ⊤",
"state_before": "x : EReal\nh : 0 < x\n⊢ ⊤ * x = ⊤",
"tactic": "rw [EReal.mul_comm]"
},
{
"state_after": "no goals",
"state_before": "x : EReal\nh : 0 < x\n⊢ x * ⊤ = ⊤",
"tactic": "exact mul_top_of_pos h"
}
] |
[
936,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
934,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
|
ContDiffOn.rpow
|
[] |
[
512,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
511,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.map_add_left_Ico
|
[
{
"state_after": "ι : Type ?u.215992\nα : Type u_1\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ ↑(addLeftEmbedding c) '' Set.Ico a b = Set.Ico (c + a) (c + b)",
"state_before": "ι : Type ?u.215992\nα : Type u_1\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ map (addLeftEmbedding c) (Ico a b) = Ico (c + a) (c + b)",
"tactic": "rw [← coe_inj, coe_map, coe_Ico, coe_Ico]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.215992\nα : Type u_1\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ ↑(addLeftEmbedding c) '' Set.Ico a b = Set.Ico (c + a) (c + b)",
"tactic": "exact Set.image_const_add_Ico _ _ _"
}
] |
[
1058,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1055,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Basic.lean
|
Pmf.toMeasure_apply_eq_toOuterMeasure_apply
|
[] |
[
256,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.coe_multiset_prod
|
[] |
[
325,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
324,
1
] |
Std/Data/Int/DivMod.lean
|
Int.dvd_trans
|
[
{
"state_after": "no goals",
"state_before": "a✝ d e : Int\n⊢ a✝ * d * e = a✝ * (d * e)",
"tactic": "rw [Int.mul_assoc]"
}
] |
[
598,
66
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
597,
11
] |
Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean
|
CategoryTheory.Groupoid.Free.freeGroupoidFunctor_comp
|
[
{
"state_after": "V : Type u\ninst✝² : Quiver V\nV' : Type u'\ninst✝¹ : Quiver V'\nV'' : Type u''\ninst✝ : Quiver V''\nφ : V ⥤q V'\nφ' : V' ⥤q V''\n⊢ lift (φ ⋙q φ' ⋙q of V'') = lift (φ ⋙q of V') ⋙ lift (φ' ⋙q of V'')",
"state_before": "V : Type u\ninst✝² : Quiver V\nV' : Type u'\ninst✝¹ : Quiver V'\nV'' : Type u''\ninst✝ : Quiver V''\nφ : V ⥤q V'\nφ' : V' ⥤q V''\n⊢ freeGroupoidFunctor (φ ⋙q φ') = freeGroupoidFunctor φ ⋙ freeGroupoidFunctor φ'",
"tactic": "dsimp only [freeGroupoidFunctor]"
},
{
"state_after": "V : Type u\ninst✝² : Quiver V\nV' : Type u'\ninst✝¹ : Quiver V'\nV'' : Type u''\ninst✝ : Quiver V''\nφ : V ⥤q V'\nφ' : V' ⥤q V''\n⊢ lift (φ ⋙q of V') ⋙ lift (φ' ⋙q of V'') = lift (φ ⋙q φ' ⋙q of V'')",
"state_before": "V : Type u\ninst✝² : Quiver V\nV' : Type u'\ninst✝¹ : Quiver V'\nV'' : Type u''\ninst✝ : Quiver V''\nφ : V ⥤q V'\nφ' : V' ⥤q V''\n⊢ lift (φ ⋙q φ' ⋙q of V'') = lift (φ ⋙q of V') ⋙ lift (φ' ⋙q of V'')",
"tactic": "symm"
},
{
"state_after": "case hΦ\nV : Type u\ninst✝² : Quiver V\nV' : Type u'\ninst✝¹ : Quiver V'\nV'' : Type u''\ninst✝ : Quiver V''\nφ : V ⥤q V'\nφ' : V' ⥤q V''\n⊢ of V ⋙q (lift (φ ⋙q of V') ⋙ lift (φ' ⋙q of V'')).toPrefunctor = φ ⋙q φ' ⋙q of V''",
"state_before": "V : Type u\ninst✝² : Quiver V\nV' : Type u'\ninst✝¹ : Quiver V'\nV'' : Type u''\ninst✝ : Quiver V''\nφ : V ⥤q V'\nφ' : V' ⥤q V''\n⊢ lift (φ ⋙q of V') ⋙ lift (φ' ⋙q of V'') = lift (φ ⋙q φ' ⋙q of V'')",
"tactic": "apply lift_unique"
},
{
"state_after": "no goals",
"state_before": "case hΦ\nV : Type u\ninst✝² : Quiver V\nV' : Type u'\ninst✝¹ : Quiver V'\nV'' : Type u''\ninst✝ : Quiver V''\nφ : V ⥤q V'\nφ' : V' ⥤q V''\n⊢ of V ⋙q (lift (φ ⋙q of V') ⋙ lift (φ' ⋙q of V'')).toPrefunctor = φ ⋙q φ' ⋙q of V''",
"tactic": "rfl"
}
] |
[
211,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isLittleO_const_iff_isLittleO_one
|
[] |
[
1319,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1316,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.empty_ssubset
|
[] |
[
642,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
641,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieModuleHom.coeSubmodule_range
|
[] |
[
1217,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1216,
1
] |
Mathlib/LinearAlgebra/Matrix/Reindex.lean
|
Matrix.reindexLinearEquiv_trans
|
[
{
"state_after": "case h.a.h\nl : Type ?u.20354\nm : Type u_1\nn : Type u_3\no : Type ?u.20363\nl' : Type ?u.20366\nm' : Type u_2\nn' : Type u_4\no' : Type ?u.20375\nm'' : Type u_5\nn'' : Type u_6\nR : Type u_8\nA : Type u_7\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid A\ninst✝ : Module R A\ne₁ : m ≃ m'\ne₂ : n ≃ n'\ne₁' : m' ≃ m''\ne₂' : n' ≃ n''\nx✝¹ : Matrix m n A\ni✝ : m''\nx✝ : n''\n⊢ ↑(LinearEquiv.trans (reindexLinearEquiv R A e₁ e₂) (reindexLinearEquiv R A e₁' e₂')) x✝¹ i✝ x✝ =\n ↑(reindexLinearEquiv R A (e₁.trans e₁') (e₂.trans e₂')) x✝¹ i✝ x✝",
"state_before": "l : Type ?u.20354\nm : Type u_1\nn : Type u_3\no : Type ?u.20363\nl' : Type ?u.20366\nm' : Type u_2\nn' : Type u_4\no' : Type ?u.20375\nm'' : Type u_5\nn'' : Type u_6\nR : Type u_8\nA : Type u_7\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid A\ninst✝ : Module R A\ne₁ : m ≃ m'\ne₂ : n ≃ n'\ne₁' : m' ≃ m''\ne₂' : n' ≃ n''\n⊢ LinearEquiv.trans (reindexLinearEquiv R A e₁ e₂) (reindexLinearEquiv R A e₁' e₂') =\n reindexLinearEquiv R A (e₁.trans e₁') (e₂.trans e₂')",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.a.h\nl : Type ?u.20354\nm : Type u_1\nn : Type u_3\no : Type ?u.20363\nl' : Type ?u.20366\nm' : Type u_2\nn' : Type u_4\no' : Type ?u.20375\nm'' : Type u_5\nn'' : Type u_6\nR : Type u_8\nA : Type u_7\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid A\ninst✝ : Module R A\ne₁ : m ≃ m'\ne₂ : n ≃ n'\ne₁' : m' ≃ m''\ne₂' : n' ≃ n''\nx✝¹ : Matrix m n A\ni✝ : m''\nx✝ : n''\n⊢ ↑(LinearEquiv.trans (reindexLinearEquiv R A e₁ e₂) (reindexLinearEquiv R A e₁' e₂')) x✝¹ i✝ x✝ =\n ↑(reindexLinearEquiv R A (e₁.trans e₁') (e₂.trans e₂')) x✝¹ i✝ x✝",
"tactic": "rfl"
}
] |
[
74,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/GroupTheory/Coset.lean
|
eq_cosets_of_normal
|
[
{
"state_after": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\nN : Normal s\ng a : α\n⊢ g⁻¹ * a ∈ s ↔ a * g⁻¹ ∈ s",
"state_before": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\nN : Normal s\ng a : α\n⊢ a ∈ g *l ↑s ↔ a ∈ ↑s *r g",
"tactic": "simp [mem_leftCoset_iff, mem_rightCoset_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\nN : Normal s\ng a : α\n⊢ g⁻¹ * a ∈ s ↔ a * g⁻¹ ∈ s",
"tactic": "rw [N.mem_comm_iff]"
}
] |
[
254,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
|
Real.volume_Ici
|
[
{
"state_after": "ι : Type ?u.247709\ninst✝ : Fintype ι\na : ℝ\n⊢ ↑↑volume (Ioi a) = ⊤",
"state_before": "ι : Type ?u.247709\ninst✝ : Fintype ι\na : ℝ\n⊢ ↑↑volume (Ici a) = ⊤",
"tactic": "rw [← measure_congr Ioi_ae_eq_Ici]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.247709\ninst✝ : Fintype ι\na : ℝ\n⊢ ↑↑volume (Ioi a) = ⊤",
"tactic": "simp"
}
] |
[
157,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
1
] |
Mathlib/Analysis/NormedSpace/CompactOperator.lean
|
IsCompactOperator.codRestrict
|
[] |
[
291,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
287,
1
] |
Mathlib/Probability/Kernel/Basic.lean
|
ProbabilityTheory.kernel.set_lintegral_deterministic'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type ?u.398685\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\nf : β → ℝ≥0∞\ng : α → β\na : α\nhg : Measurable g\nhf : Measurable f\ns : Set β\nhs : MeasurableSet s\ninst✝ : Decidable (g a ∈ s)\n⊢ (∫⁻ (x : β) in s, f x ∂↑(deterministic g hg) a) = if g a ∈ s then f (g a) else 0",
"tactic": "rw [kernel.deterministic_apply, set_lintegral_dirac' hf hs]"
}
] |
[
385,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
382,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
upperBounds_mono_set
|
[] |
[
192,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/Data/Nat/Bitwise.lean
|
Nat.lxor'_zero
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ lxor' n 0 = n",
"tactic": "simp [lxor']"
}
] |
[
194,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.join_append
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nS T : WSeq (WSeq α)\n⊢ ∀ {s t : WSeq α},\n (fun s1 s2 => ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))) s t →\n Computation.LiftRel\n (LiftRelO (fun x x_1 => x = x_1) fun s1 s2 =>\n ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T)))\n (destruct s) (destruct t)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nS T : WSeq (WSeq α)\n⊢ join (append S T) ~ʷ append (join S) (join T)",
"tactic": "refine'\n ⟨fun s1 s2 =>\n ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T)),\n ⟨nil, S, T, by simp, by simp⟩, _⟩"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nS T : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))\n⊢ Computation.LiftRel\n (LiftRelO (fun x x_1 => x = x_1) fun s1 s2 =>\n ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T)))\n (destruct s1) (destruct s2)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nS T : WSeq (WSeq α)\n⊢ ∀ {s t : WSeq α},\n (fun s1 s2 => ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))) s t →\n Computation.LiftRel\n (LiftRelO (fun x x_1 => x = x_1) fun s1 s2 =>\n ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T)))\n (destruct s) (destruct t)",
"tactic": "intro s1 s2 h"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nS T : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))\n⊢ ∀ {ca cb : Computation (Option (α × WSeq α))},\n (fun c1 c2 =>\n ∃ s S T, c1 = destruct (append s (join (append S T))) ∧ c2 = destruct (append s (append (join S) (join T))))\n ca cb →\n LiftRelAux\n (LiftRelO (fun x x_1 => x = x_1) fun s1 s2 =>\n ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T)))\n (fun c1 c2 =>\n ∃ s S T, c1 = destruct (append s (join (append S T))) ∧ c2 = destruct (append s (append (join S) (join T))))\n (Computation.destruct ca) (Computation.destruct cb)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nS T : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))\n⊢ Computation.LiftRel\n (LiftRelO (fun x x_1 => x = x_1) fun s1 s2 =>\n ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T)))\n (destruct s1) (destruct s2)",
"tactic": "apply\n lift_rel_rec\n (fun c1 c2 =>\n ∃ (s : WSeq α) (S T : _),\n c1 = destruct (append s (join (append S T))) ∧\n c2 = destruct (append s (append (join S) (join T))))\n _ _ _\n (let ⟨s, S, T, h1, h2⟩ := h\n ⟨s, S, T, congr_arg destruct h1, congr_arg destruct h2⟩)"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nS✝ T✝ : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))\ns : WSeq α\nS T : WSeq (WSeq α)\n⊢ LiftRelAux\n (LiftRelO (fun x x_1 => x = x_1) fun s1 s2 =>\n ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T)))\n (fun c1 c2 =>\n ∃ s S T, c1 = destruct (append s (join (append S T))) ∧ c2 = destruct (append s (append (join S) (join T))))\n (Computation.destruct (destruct (append s (join (append S T)))))\n (Computation.destruct (destruct (append s (append (join S) (join T)))))",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nS T : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))\n⊢ ∀ {ca cb : Computation (Option (α × WSeq α))},\n (fun c1 c2 =>\n ∃ s S T, c1 = destruct (append s (join (append S T))) ∧ c2 = destruct (append s (append (join S) (join T))))\n ca cb →\n LiftRelAux\n (LiftRelO (fun x x_1 => x = x_1) fun s1 s2 =>\n ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T)))\n (fun c1 c2 =>\n ∃ s S T, c1 = destruct (append s (join (append S T))) ∧ c2 = destruct (append s (append (join S) (join T))))\n (Computation.destruct ca) (Computation.destruct cb)",
"tactic": "rintro c1 c2 ⟨s, S, T, rfl, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro.h1\nα : Type u\nβ : Type v\nγ : Type w\nS✝ T✝ : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))\nS T : WSeq (WSeq α)\n⊢ LiftRelAux\n (fun x x_1 =>\n match x, x_1 with\n | none, none => True\n | some (a, s), some (b, t) =>\n a = b ∧ ∃ s_1 S T, s = append s_1 (join (append S T)) ∧ t = append s_1 (append (join S) (join T))\n | x, x_2 => False)\n (fun c1 c2 =>\n ∃ s S T, c1 = destruct (append s (join (append S T))) ∧ c2 = destruct (append s (append (join S) (join T))))\n (Computation.destruct (destruct (join (append S T)))) (Computation.destruct (destruct (append (join S) (join T))))\n\ncase intro.intro.intro.intro.h2\nα : Type u\nβ : Type v\nγ : Type w\nS✝ T✝ : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))\nS T : WSeq (WSeq α)\na : α\ns : WSeq α\n⊢ ∃ s_1 S_1 T_1,\n append s (join (append S T)) = append s_1 (join (append S_1 T_1)) ∧\n append s (append (join S) (join T)) = append s_1 (append (join S_1) (join T_1))\n\ncase intro.intro.intro.intro.h3\nα : Type u\nβ : Type v\nγ : Type w\nS✝ T✝ : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))\nS T : WSeq (WSeq α)\ns : WSeq α\n⊢ ∃ s_1 S_1 T_1,\n destruct (append s (join (append S T))) = destruct (append s_1 (join (append S_1 T_1))) ∧\n destruct (append s (append (join S) (join T))) = destruct (append s_1 (append (join S_1) (join T_1)))",
"state_before": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nS✝ T✝ : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))\ns : WSeq α\nS T : WSeq (WSeq α)\n⊢ LiftRelAux\n (LiftRelO (fun x x_1 => x = x_1) fun s1 s2 =>\n ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T)))\n (fun c1 c2 =>\n ∃ s S T, c1 = destruct (append s (join (append S T))) ∧ c2 = destruct (append s (append (join S) (join T))))\n (Computation.destruct (destruct (append s (join (append S T)))))\n (Computation.destruct (destruct (append s (append (join S) (join T)))))",
"tactic": "induction' s using WSeq.recOn with a s s <;> simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nS T : WSeq (WSeq α)\n⊢ join (append S T) = append nil (join (append S T))",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nS T : WSeq (WSeq α)\n⊢ append (join S) (join T) = append nil (append (join S) (join T))",
"tactic": "simp"
},
{
"state_after": "case intro.intro.intro.intro.h1.h1\nα : Type u\nβ : Type v\nγ : Type w\nS T✝ : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))\nT : WSeq (WSeq α)\n⊢ LiftRelAux\n (fun x x_1 =>\n match x, x_1 with\n | none, none => True\n | some (a, s), some (b, t) =>\n a = b ∧ ∃ s_1 S T, s = append s_1 (join (append S T)) ∧ t = append s_1 (append (join S) (join T))\n | x, x_2 => False)\n (fun c1 c2 =>\n ∃ s S T, c1 = destruct (append s (join (append S T))) ∧ c2 = destruct (append s (append (join S) (join T))))\n (Computation.destruct (destruct (join T))) (Computation.destruct (destruct (join T)))\n\ncase intro.intro.intro.intro.h1.h2\nα : Type u\nβ : Type v\nγ : Type w\nS✝ T✝ : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))\nT : WSeq (WSeq α)\ns : WSeq α\nS : WSeq (WSeq α)\n⊢ ∃ s_1 S_1 T_1,\n destruct (append s (join (append S T))) = destruct (append s_1 (join (append S_1 T_1))) ∧\n destruct (append s (append (join S) (join T))) = destruct (append s_1 (append (join S_1) (join T_1)))\n\ncase intro.intro.intro.intro.h1.h3\nα : Type u\nβ : Type v\nγ : Type w\nS✝ T✝ : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))\nT S : WSeq (WSeq α)\n⊢ ∃ s S_1 T_1,\n destruct (join (append S T)) = destruct (append s (join (append S_1 T_1))) ∧\n destruct (append (join S) (join T)) = destruct (append s (append (join S_1) (join T_1)))",
"state_before": "case intro.intro.intro.intro.h1\nα : Type u\nβ : Type v\nγ : Type w\nS✝ T✝ : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))\nS T : WSeq (WSeq α)\n⊢ LiftRelAux\n (fun x x_1 =>\n match x, x_1 with\n | none, none => True\n | some (a, s), some (b, t) =>\n a = b ∧ ∃ s_1 S T, s = append s_1 (join (append S T)) ∧ t = append s_1 (append (join S) (join T))\n | x, x_2 => False)\n (fun c1 c2 =>\n ∃ s S T, c1 = destruct (append s (join (append S T))) ∧ c2 = destruct (append s (append (join S) (join T))))\n (Computation.destruct (destruct (join (append S T)))) (Computation.destruct (destruct (append (join S) (join T))))",
"tactic": "induction' S using WSeq.recOn with s S S <;> simp"
},
{
"state_after": "case intro.intro.intro.intro.h1.h1.h2\nα : Type u\nβ : Type v\nγ : Type w\nS T✝ : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))\ns : WSeq α\nT : WSeq (WSeq α)\n⊢ ∃ s_1 S T_1,\n destruct (append s (join T)) = destruct (append s_1 (join (append S T_1))) ∧\n destruct (append s (join T)) = destruct (append s_1 (append (join S) (join T_1)))\n\ncase intro.intro.intro.intro.h1.h1.h3\nα : Type u\nβ : Type v\nγ : Type w\nS T✝ : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))\nT : WSeq (WSeq α)\n⊢ ∃ s S T_1,\n destruct (join T) = destruct (append s (join (append S T_1))) ∧\n destruct (join T) = destruct (append s (append (join S) (join T_1)))",
"state_before": "case intro.intro.intro.intro.h1.h1\nα : Type u\nβ : Type v\nγ : Type w\nS T✝ : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))\nT : WSeq (WSeq α)\n⊢ LiftRelAux\n (fun x x_1 =>\n match x, x_1 with\n | none, none => True\n | some (a, s), some (b, t) =>\n a = b ∧ ∃ s_1 S T, s = append s_1 (join (append S T)) ∧ t = append s_1 (append (join S) (join T))\n | x, x_2 => False)\n (fun c1 c2 =>\n ∃ s S T, c1 = destruct (append s (join (append S T))) ∧ c2 = destruct (append s (append (join S) (join T))))\n (Computation.destruct (destruct (join T))) (Computation.destruct (destruct (join T)))",
"tactic": "induction' T using WSeq.recOn with s T T <;> simp"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.h1.h1.h2\nα : Type u\nβ : Type v\nγ : Type w\nS T✝ : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))\ns : WSeq α\nT : WSeq (WSeq α)\n⊢ ∃ s_1 S T_1,\n destruct (append s (join T)) = destruct (append s_1 (join (append S T_1))) ∧\n destruct (append s (join T)) = destruct (append s_1 (append (join S) (join T_1)))",
"tactic": "refine' ⟨s, nil, T, _, _⟩ <;> simp"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.h1.h1.h3\nα : Type u\nβ : Type v\nγ : Type w\nS T✝ : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))\nT : WSeq (WSeq α)\n⊢ ∃ s S T_1,\n destruct (join T) = destruct (append s (join (append S T_1))) ∧\n destruct (join T) = destruct (append s (append (join S) (join T_1)))",
"tactic": "refine' ⟨nil, nil, T, _, _⟩ <;> simp"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.h1.h2\nα : Type u\nβ : Type v\nγ : Type w\nS✝ T✝ : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))\nT : WSeq (WSeq α)\ns : WSeq α\nS : WSeq (WSeq α)\n⊢ ∃ s_1 S_1 T_1,\n destruct (append s (join (append S T))) = destruct (append s_1 (join (append S_1 T_1))) ∧\n destruct (append s (append (join S) (join T))) = destruct (append s_1 (append (join S_1) (join T_1)))",
"tactic": "exact ⟨s, S, T, rfl, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.h1.h3\nα : Type u\nβ : Type v\nγ : Type w\nS✝ T✝ : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))\nT S : WSeq (WSeq α)\n⊢ ∃ s S_1 T_1,\n destruct (join (append S T)) = destruct (append s (join (append S_1 T_1))) ∧\n destruct (append (join S) (join T)) = destruct (append s (append (join S_1) (join T_1)))",
"tactic": "refine' ⟨nil, S, T, _, _⟩ <;> simp"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.h2\nα : Type u\nβ : Type v\nγ : Type w\nS✝ T✝ : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))\nS T : WSeq (WSeq α)\na : α\ns : WSeq α\n⊢ ∃ s_1 S_1 T_1,\n append s (join (append S T)) = append s_1 (join (append S_1 T_1)) ∧\n append s (append (join S) (join T)) = append s_1 (append (join S_1) (join T_1))",
"tactic": "exact ⟨s, S, T, rfl, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.h3\nα : Type u\nβ : Type v\nγ : Type w\nS✝ T✝ : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))\nS T : WSeq (WSeq α)\ns : WSeq α\n⊢ ∃ s_1 S_1 T_1,\n destruct (append s (join (append S T))) = destruct (append s_1 (join (append S_1 T_1))) ∧\n destruct (append s (append (join S) (join T))) = destruct (append s_1 (append (join S_1) (join T_1)))",
"tactic": "exact ⟨s, S, T, rfl, rfl⟩"
}
] |
[
1732,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1708,
1
] |
Mathlib/Data/Vector/Mem.lean
|
Vector.mem_of_mem_tail
|
[
{
"state_after": "case zero\nα : Type u_1\nβ : Type ?u.2767\nn : ℕ\na a' : α\nv✝ : Vector α n\nha✝ : a ∈ toList (tail v✝)\nv : Vector α Nat.zero\nha : a ∈ toList (tail v)\n⊢ a ∈ toList v\n\ncase succ\nα : Type u_1\nβ : Type ?u.2767\nn✝ : ℕ\na a' : α\nv✝ : Vector α n✝\nha✝ : a ∈ toList (tail v✝)\nn : ℕ\nn_ih✝ : ∀ (v : Vector α n), a ∈ toList (tail v) → a ∈ toList v\nv : Vector α (Nat.succ n)\nha : a ∈ toList (tail v)\n⊢ a ∈ toList v",
"state_before": "α : Type u_1\nβ : Type ?u.2767\nn : ℕ\na a' : α\nv : Vector α n\nha : a ∈ toList (tail v)\n⊢ a ∈ toList v",
"tactic": "induction' n with n _"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u_1\nβ : Type ?u.2767\nn : ℕ\na a' : α\nv✝ : Vector α n\nha✝ : a ∈ toList (tail v✝)\nv : Vector α Nat.zero\nha : a ∈ toList (tail v)\n⊢ a ∈ toList v",
"tactic": "exact False.elim (Vector.not_mem_zero a v.tail ha)"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type u_1\nβ : Type ?u.2767\nn✝ : ℕ\na a' : α\nv✝ : Vector α n✝\nha✝ : a ∈ toList (tail v✝)\nn : ℕ\nn_ih✝ : ∀ (v : Vector α n), a ∈ toList (tail v) → a ∈ toList v\nv : Vector α (Nat.succ n)\nha : a ∈ toList (tail v)\n⊢ a ∈ toList v",
"tactic": "exact (mem_succ_iff a v).2 (Or.inr ha)"
}
] |
[
76,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Std/Data/Option/Lemmas.lean
|
Option.ne_none_iff_exists'
|
[] |
[
78,
60
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
77,
1
] |
Mathlib/Data/Int/Basic.lean
|
Int.sign_negSucc
|
[] |
[
136,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Cardinal.eq_of_add_eq_of_aleph0_le
|
[
{
"state_after": "case a\na b c : Cardinal\nh : a + b = c\nha : a < c\nhc : ℵ₀ ≤ c\n⊢ b ≤ c\n\ncase a\na b c : Cardinal\nh : a + b = c\nha : a < c\nhc : ℵ₀ ≤ c\n⊢ c ≤ b",
"state_before": "a b c : Cardinal\nh : a + b = c\nha : a < c\nhc : ℵ₀ ≤ c\n⊢ b = c",
"tactic": "apply le_antisymm"
},
{
"state_after": "case a\na b c : Cardinal\nh : a + b = c\nha : a < c\nhc : ℵ₀ ≤ c\n⊢ ¬b < c",
"state_before": "case a\na b c : Cardinal\nh : a + b = c\nha : a < c\nhc : ℵ₀ ≤ c\n⊢ c ≤ b",
"tactic": "rw [← not_lt]"
},
{
"state_after": "case a\na b c : Cardinal\nh : a + b = c\nha : a < c\nhc : ℵ₀ ≤ c\nhb : b < c\n⊢ False",
"state_before": "case a\na b c : Cardinal\nh : a + b = c\nha : a < c\nhc : ℵ₀ ≤ c\n⊢ ¬b < c",
"tactic": "intro hb"
},
{
"state_after": "case a\na b c : Cardinal\nh : a + b = c\nha : a < c\nhc : ℵ₀ ≤ c\nhb : b < c\nthis : a + b < c\n⊢ False",
"state_before": "case a\na b c : Cardinal\nh : a + b = c\nha : a < c\nhc : ℵ₀ ≤ c\nhb : b < c\n⊢ False",
"tactic": "have : a + b < c := add_lt_of_lt hc ha hb"
},
{
"state_after": "no goals",
"state_before": "case a\na b c : Cardinal\nh : a + b = c\nha : a < c\nhc : ℵ₀ ≤ c\nhb : b < c\nthis : a + b < c\n⊢ False",
"tactic": "simp [h, lt_irrefl] at this"
},
{
"state_after": "case a\na b c : Cardinal\nh : a + b = c\nha : a < c\nhc : ℵ₀ ≤ c\n⊢ b ≤ a + b",
"state_before": "case a\na b c : Cardinal\nh : a + b = c\nha : a < c\nhc : ℵ₀ ≤ c\n⊢ b ≤ c",
"tactic": "rw [← h]"
},
{
"state_after": "no goals",
"state_before": "case a\na b c : Cardinal\nh : a + b = c\nha : a < c\nhc : ℵ₀ ≤ c\n⊢ b ≤ a + b",
"tactic": "apply self_le_add_left"
}
] |
[
776,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
769,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
ciSup_mono
|
[
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.55772\nγ : Type ?u.55775\nι : Sort u_2\ninst✝ : ConditionallyCompleteLattice α\ns t : Set α\na b : α\nf g : ι → α\nB : BddAbove (range g)\nH : ∀ (x : ι), f x ≤ g x\nh✝ : IsEmpty ι\n⊢ iSup f ≤ iSup g\n\ncase inr\nα : Type u_1\nβ : Type ?u.55772\nγ : Type ?u.55775\nι : Sort u_2\ninst✝ : ConditionallyCompleteLattice α\ns t : Set α\na b : α\nf g : ι → α\nB : BddAbove (range g)\nH : ∀ (x : ι), f x ≤ g x\nh✝ : Nonempty ι\n⊢ iSup f ≤ iSup g",
"state_before": "α : Type u_1\nβ : Type ?u.55772\nγ : Type ?u.55775\nι : Sort u_2\ninst✝ : ConditionallyCompleteLattice α\ns t : Set α\na b : α\nf g : ι → α\nB : BddAbove (range g)\nH : ∀ (x : ι), f x ≤ g x\n⊢ iSup f ≤ iSup g",
"tactic": "cases isEmpty_or_nonempty ι"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.55772\nγ : Type ?u.55775\nι : Sort u_2\ninst✝ : ConditionallyCompleteLattice α\ns t : Set α\na b : α\nf g : ι → α\nB : BddAbove (range g)\nH : ∀ (x : ι), f x ≤ g x\nh✝ : IsEmpty ι\n⊢ iSup f ≤ iSup g",
"tactic": "rw [iSup_of_empty', iSup_of_empty']"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.55772\nγ : Type ?u.55775\nι : Sort u_2\ninst✝ : ConditionallyCompleteLattice α\ns t : Set α\na b : α\nf g : ι → α\nB : BddAbove (range g)\nH : ∀ (x : ι), f x ≤ g x\nh✝ : Nonempty ι\n⊢ iSup f ≤ iSup g",
"tactic": "exact ciSup_le fun x => le_ciSup_of_le B x (H x)"
}
] |
[
798,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
794,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.mem_map_of_injective
|
[] |
[
1275,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1273,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.card_le_of_finset
|
[] |
[
1337,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1336,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
Submodule.sup_eq_range
|
[
{
"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\nM₅ : Type ?u.269908\nM₆ : Type ?u.269911\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np q : Submodule R M\nx : M\n⊢ x ∈ p ⊔ q ↔ x ∈ range (coprod (Submodule.subtype p) (Submodule.subtype q))",
"tactic": "simp [Submodule.mem_sup, SetLike.exists]"
}
] |
[
545,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
544,
1
] |
Mathlib/Algebra/IsPrimePow.lean
|
isPrimePow_iff_pow_succ
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommMonoidWithZero R\nn p✝ : R\nk✝ : ℕ\nx✝ : ∃ p k, Prime p ∧ 0 < k ∧ p ^ k = n\np : R\nk : ℕ\nhp : Prime p\nhk : 0 < k\nhn : p ^ k = n\n⊢ p ^ (?m.56323 x✝ p k hp hk hn + 1) = n",
"tactic": "rwa [Nat.sub_add_cancel hk]"
}
] |
[
38,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
35,
1
] |
Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean
|
ProjectiveSpectrum.zeroLocus_iSup_ideal
|
[] |
[
255,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/GroupTheory/Perm/Support.lean
|
Equiv.Perm.disjoint_one_left
|
[] |
[
72,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/SetTheory/Ordinal/NaturalOps.lean
|
Ordinal.nadd_le_nadd_iff_right
|
[] |
[
431,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
430,
1
] |
Mathlib/Algebra/Hom/Group.lean
|
OneHom.ext
|
[] |
[
644,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
643,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.iUnion_single
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nh : J ≤ I\n⊢ Prepartition.iUnion (single I J h) = ↑J",
"tactic": "simp [iUnion_def]"
}
] |
[
230,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
230,
1
] |
src/lean/Init/Data/Fin/Basic.lean
|
Fin.val_ne_of_ne
|
[] |
[
110,
39
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
109,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
closedBall_pi
|
[
{
"state_after": "case h\nα : Type u\nβ : Type v\nX : Type ?u.403409\nι : Type ?u.403412\ninst✝² : PseudoMetricSpace α\nπ : β → Type u_1\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (π b)\nx : (b : β) → π b\nr : ℝ\nhr : 0 ≤ r\np : (b : β) → π b\n⊢ p ∈ closedBall x r ↔ p ∈ Set.pi Set.univ fun b => closedBall (x b) r",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.403409\nι : Type ?u.403412\ninst✝² : PseudoMetricSpace α\nπ : β → Type u_1\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (π b)\nx : (b : β) → π b\nr : ℝ\nhr : 0 ≤ r\n⊢ closedBall x r = Set.pi Set.univ fun b => closedBall (x b) r",
"tactic": "ext p"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u\nβ : Type v\nX : Type ?u.403409\nι : Type ?u.403412\ninst✝² : PseudoMetricSpace α\nπ : β → Type u_1\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (π b)\nx : (b : β) → π b\nr : ℝ\nhr : 0 ≤ r\np : (b : β) → π b\n⊢ p ∈ closedBall x r ↔ p ∈ Set.pi Set.univ fun b => closedBall (x b) r",
"tactic": "simp [dist_pi_le_iff hr]"
}
] |
[
2067,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2064,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean
|
Zsqrtd.le_total
|
[
{
"state_after": "d : ℕ\na b : ℤ√↑d\nt : Nonneg (b - a) ∨ Nonneg (-(b - a))\n⊢ a ≤ b ∨ b ≤ a",
"state_before": "d : ℕ\na b : ℤ√↑d\n⊢ a ≤ b ∨ b ≤ a",
"tactic": "have t := (b - a).nonneg_total"
},
{
"state_after": "no goals",
"state_before": "d : ℕ\na b : ℤ√↑d\nt : Nonneg (b - a) ∨ Nonneg (-(b - a))\n⊢ a ≤ b ∨ b ≤ a",
"tactic": "rwa [neg_sub] at t"
}
] |
[
756,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
754,
11
] |
Mathlib/NumberTheory/LucasLehmer.lean
|
LucasLehmer.X.int_coe_snd
|
[] |
[
329,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Mathlib/Topology/MetricSpace/PiNat.lean
|
PiNat.mem_cylinder_iff_dist_le
|
[
{
"state_after": "case inl\nE : ℕ → Type u_1\ny : (n : ℕ) → E n\nn : ℕ\n⊢ y ∈ cylinder y n ↔ dist y y ≤ (1 / 2) ^ n\n\ncase inr\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhne : y ≠ x\n⊢ y ∈ cylinder x n ↔ dist y x ≤ (1 / 2) ^ n",
"state_before": "E : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\n⊢ y ∈ cylinder x n ↔ dist y x ≤ (1 / 2) ^ n",
"tactic": "rcases eq_or_ne y x with (rfl | hne)"
},
{
"state_after": "case inr\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhne : y ≠ x\n⊢ (∀ (i : ℕ), i < n → y i = x i) ↔ n ≤ firstDiff y x",
"state_before": "case inr\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhne : y ≠ x\n⊢ y ∈ cylinder x n ↔ dist y x ≤ (1 / 2) ^ n",
"tactic": "suffices (∀ i : ℕ, i < n → y i = x i) ↔ n ≤ firstDiff y x by simpa [dist_eq_of_ne hne]"
},
{
"state_after": "case inr.mp\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhne : y ≠ x\n⊢ (∀ (i : ℕ), i < n → y i = x i) → n ≤ firstDiff y x\n\ncase inr.mpr\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhne : y ≠ x\n⊢ n ≤ firstDiff y x → ∀ (i : ℕ), i < n → y i = x i",
"state_before": "case inr\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhne : y ≠ x\n⊢ (∀ (i : ℕ), i < n → y i = x i) ↔ n ≤ firstDiff y x",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case inl\nE : ℕ → Type u_1\ny : (n : ℕ) → E n\nn : ℕ\n⊢ y ∈ cylinder y n ↔ dist y y ≤ (1 / 2) ^ n",
"tactic": "simp [PiNat.dist_self]"
},
{
"state_after": "no goals",
"state_before": "E : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhne : y ≠ x\nthis : (∀ (i : ℕ), i < n → y i = x i) ↔ n ≤ firstDiff y x\n⊢ y ∈ cylinder x n ↔ dist y x ≤ (1 / 2) ^ n",
"tactic": "simpa [dist_eq_of_ne hne]"
},
{
"state_after": "case inr.mp\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhne : y ≠ x\nhy : ∀ (i : ℕ), i < n → y i = x i\n⊢ n ≤ firstDiff y x",
"state_before": "case inr.mp\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhne : y ≠ x\n⊢ (∀ (i : ℕ), i < n → y i = x i) → n ≤ firstDiff y x",
"tactic": "intro hy"
},
{
"state_after": "case inr.mp\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhne : y ≠ x\nhy : ∀ (i : ℕ), i < n → y i = x i\nH : firstDiff y x < n\n⊢ False",
"state_before": "case inr.mp\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhne : y ≠ x\nhy : ∀ (i : ℕ), i < n → y i = x i\n⊢ n ≤ firstDiff y x",
"tactic": "by_contra' H"
},
{
"state_after": "no goals",
"state_before": "case inr.mp\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhne : y ≠ x\nhy : ∀ (i : ℕ), i < n → y i = x i\nH : firstDiff y x < n\n⊢ False",
"tactic": "exact apply_firstDiff_ne hne (hy _ H)"
},
{
"state_after": "case inr.mpr\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhne : y ≠ x\nh : n ≤ firstDiff y x\ni : ℕ\nhi : i < n\n⊢ y i = x i",
"state_before": "case inr.mpr\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhne : y ≠ x\n⊢ n ≤ firstDiff y x → ∀ (i : ℕ), i < n → y i = x i",
"tactic": "intro h i hi"
},
{
"state_after": "no goals",
"state_before": "case inr.mpr\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhne : y ≠ x\nh : n ≤ firstDiff y x\ni : ℕ\nhi : i < n\n⊢ y i = x i",
"tactic": "exact apply_eq_of_lt_firstDiff (hi.trans_le h)"
}
] |
[
327,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
317,
1
] |
Mathlib/CategoryTheory/Products/Bifunctor.lean
|
CategoryTheory.Bifunctor.map_id
|
[] |
[
31,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
29,
1
] |
Mathlib/Computability/TMToPartrec.lean
|
Turing.PartrecToTM2.tr_ret_halt
|
[] |
[
1147,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1147,
1
] |
Mathlib/Init/Algebra/Order.lean
|
le_of_not_lt
|
[] |
[
329,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
325,
1
] |
Mathlib/Topology/Instances/Matrix.lean
|
Continuous.matrix_updateRow
|
[] |
[
216,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.measure_eq_inducedOuterMeasure
|
[] |
[
172,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Std/Data/String/Lemmas.lean
|
String.valid_toSubstring
|
[] |
[
973,
31
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
972,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
ZNum.dvd_to_int
|
[
{
"state_after": "α : Type ?u.1071772\nm n : ZNum\nx✝ : ↑m ∣ ↑n\nk : ℤ\ne : ↑n = ↑m * k\n⊢ ↑(↑m * k) = m * ↑k",
"state_before": "α : Type ?u.1071772\nm n : ZNum\nx✝ : ↑m ∣ ↑n\nk : ℤ\ne : ↑n = ↑m * k\n⊢ n = m * ↑k",
"tactic": "rw [← of_to_int n, e]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.1071772\nm n : ZNum\nx✝ : ↑m ∣ ↑n\nk : ℤ\ne : ↑n = ↑m * k\n⊢ ↑(↑m * k) = m * ↑k",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.1071772\nm n : ZNum\nx✝ : m ∣ n\nk : ZNum\ne : n = m * k\n⊢ ↑n = ↑m * ↑k",
"tactic": "simp [e]"
}
] |
[
1575,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1574,
1
] |
Mathlib/Algebra/Homology/QuasiIso.lean
|
HomologicalComplex.Hom.from_single₀_exact_f_d_at_zero
|
[
{
"state_after": "ι : Type ?u.92227\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\n⊢ ∃ w, Nonempty (_root_.homology (HomologicalComplex.Hom.f f 0) (d X 0 1) w ≅ 0)",
"state_before": "ι : Type ?u.92227\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\n⊢ Exact (HomologicalComplex.Hom.f f 0) (d X 0 1)",
"tactic": "rw [Preadditive.exact_iff_homology_zero]"
},
{
"state_after": "ι : Type ?u.92227\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\nh : HomologicalComplex.Hom.f f 0 ≫ d X 0 1 = 0\n⊢ ∃ w, Nonempty (_root_.homology (HomologicalComplex.Hom.f f 0) (d X 0 1) w ≅ 0)",
"state_before": "ι : Type ?u.92227\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\n⊢ ∃ w, Nonempty (_root_.homology (HomologicalComplex.Hom.f f 0) (d X 0 1) w ≅ 0)",
"tactic": "have h : f.f 0 ≫ X.d 0 1 = 0 := by\n simp only [HomologicalComplex.Hom.comm, CochainComplex.single₀_obj_X_d, zero_comp]"
},
{
"state_after": "ι : Type ?u.92227\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\nh : HomologicalComplex.Hom.f f 0 ≫ d X 0 1 = 0\n⊢ cokernel (kernel.lift (d X 0 1) (HomologicalComplex.Hom.f f 0) h) ≅ 0",
"state_before": "ι : Type ?u.92227\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\nh : HomologicalComplex.Hom.f f 0 ≫ d X 0 1 = 0\n⊢ ∃ w, Nonempty (_root_.homology (HomologicalComplex.Hom.f f 0) (d X 0 1) w ≅ 0)",
"tactic": "refine' ⟨h, Nonempty.intro (homologyIsoCokernelLift _ _ _ ≪≫ _)⟩"
},
{
"state_after": "ι : Type ?u.92227\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\nh : HomologicalComplex.Hom.f f 0 ≫ d X 0 1 = 0\n⊢ IsIso (kernel.lift (d X 0 1) (HomologicalComplex.Hom.f f 0) h)",
"state_before": "ι : Type ?u.92227\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\nh : HomologicalComplex.Hom.f f 0 ≫ d X 0 1 = 0\n⊢ cokernel (kernel.lift (d X 0 1) (HomologicalComplex.Hom.f f 0) h) ≅ 0",
"tactic": "suffices IsIso (kernel.lift (X.d 0 1) (f.f 0) h) by apply cokernel.ofEpi"
},
{
"state_after": "ι : Type ?u.92227\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\nh : HomologicalComplex.Hom.f f 0 ≫ d X 0 1 = 0\n⊢ IsIso (fromSingle₀KernelAtZeroIso f).inv",
"state_before": "ι : Type ?u.92227\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\nh : HomologicalComplex.Hom.f f 0 ≫ d X 0 1 = 0\n⊢ IsIso (kernel.lift (d X 0 1) (HomologicalComplex.Hom.f f 0) h)",
"tactic": "rw [← fromSingle₀KernelAtZeroIso_inv_eq f]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.92227\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\nh : HomologicalComplex.Hom.f f 0 ≫ d X 0 1 = 0\n⊢ IsIso (fromSingle₀KernelAtZeroIso f).inv",
"tactic": "infer_instance"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.92227\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\n⊢ HomologicalComplex.Hom.f f 0 ≫ d X 0 1 = 0",
"tactic": "simp only [HomologicalComplex.Hom.comm, CochainComplex.single₀_obj_X_d, zero_comp]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.92227\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\nh : HomologicalComplex.Hom.f f 0 ≫ d X 0 1 = 0\nthis : IsIso (kernel.lift (d X 0 1) (HomologicalComplex.Hom.f f 0) h)\n⊢ cokernel (kernel.lift (d X 0 1) (HomologicalComplex.Hom.f f 0) h) ≅ 0",
"tactic": "apply cokernel.ofEpi"
}
] |
[
193,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/Analysis/Calculus/Dslope.lean
|
dslope_same
|
[] |
[
40,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
Submodule.topologicalClosure_map
|
[] |
[
555,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
550,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.coe_eq_zero_iff_isEmpty
|
[] |
[
115,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Algebra/Lie/OfAssociative.lean
|
LinearEquiv.lieConj_symm
|
[] |
[
339,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Fin.image_succAbove_univ
|
[
{
"state_after": "case a\nα : Type ?u.100123\nβ : Type ?u.100126\nγ : Type ?u.100129\nn : ℕ\ni m : Fin (n + 1)\n⊢ m ∈ image (↑(succAbove i)) univ ↔ m ∈ {i}ᶜ",
"state_before": "α : Type ?u.100123\nβ : Type ?u.100126\nγ : Type ?u.100129\nn : ℕ\ni : Fin (n + 1)\n⊢ image (↑(succAbove i)) univ = {i}ᶜ",
"tactic": "ext m"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type ?u.100123\nβ : Type ?u.100126\nγ : Type ?u.100129\nn : ℕ\ni m : Fin (n + 1)\n⊢ m ∈ image (↑(succAbove i)) univ ↔ m ∈ {i}ᶜ",
"tactic": "simp"
}
] |
[
817,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
815,
1
] |
Mathlib/Data/Int/Cast/Basic.lean
|
Int.cast_negOfNat
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : AddGroupWithOne R\nn : ℕ\n⊢ ↑(negOfNat n) = -↑n",
"tactic": "simp [Int.cast_neg, negOfNat_eq]"
}
] |
[
106,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.bounded_range_of_cauchy_map_cofinite
|
[] |
[
2435,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2433,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.lt_inf'_iff
|
[] |
[
1113,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1112,
1
] |
Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean
|
gramSchmidtNormed_unit_length_coe
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nn : ι\nh₀ : LinearIndependent 𝕜 (f ∘ Subtype.val)\n⊢ ‖gramSchmidtNormed 𝕜 f n‖ = 1",
"tactic": "simp only [gramSchmidt_ne_zero_coe n h₀, gramSchmidtNormed, norm_smul_inv_norm, Ne.def,\n not_false_iff]"
}
] |
[
278,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
275,
1
] |
Mathlib/Algebra/Lie/Solvable.lean
|
LieAlgebra.abelian_iff_derived_succ_eq_bot
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI✝ J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nI : LieIdeal R L\nk : ℕ\n⊢ IsLieAbelian { x // x ∈ ↑(D k I) } ↔ D (k + 1) I = ⊥",
"tactic": "rw [add_comm, derivedSeriesOfIdeal_add I 1 k, abelian_iff_derived_one_eq_bot]"
}
] |
[
143,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
mem_lowerBounds_image2
|
[] |
[
1363,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1361,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.mem_centralizer_iff_commutator_eq_one
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\nG' : Type ?u.412647\ninst✝² : Group G\ninst✝¹ : Group G'\nA : Type ?u.412656\ninst✝ : AddGroup A\nH K : Subgroup G\ng : G\n⊢ g ∈ centralizer H ↔ ∀ (h : G), h ∈ H → h * g * h⁻¹ * g⁻¹ = 1",
"tactic": "simp only [mem_centralizer_iff, mul_inv_eq_iff_eq_mul, one_mul]"
}
] |
[
2288,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2286,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.sUnion_eq_iUnion
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.167312\nγ : Type ?u.167315\nι : Sort ?u.167318\nι' : Sort ?u.167321\nι₂ : Sort ?u.167324\nκ : ι → Sort ?u.167329\nκ₁ : ι → Sort ?u.167334\nκ₂ : ι → Sort ?u.167339\nκ' : ι' → Sort ?u.167344\ns : Set (Set α)\n⊢ ⋃₀ s = ⋃ (i : ↑s), ↑i",
"tactic": "simp only [← sUnion_range, Subtype.range_coe]"
}
] |
[
1322,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1321,
1
] |
Mathlib/Topology/Order/Basic.lean
|
nhdsWithin_Iio_self_neBot
|
[] |
[
2421,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2420,
1
] |
Mathlib/Data/List/Basic.lean
|
List.mem_of_mem_head?
|
[] |
[
883,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
882,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.monomial_zero_eq_C_apply
|
[] |
[
378,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
377,
1
] |
Mathlib/RingTheory/OreLocalization/Basic.lean
|
OreLocalization.universalHom_unique
|
[] |
[
824,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
822,
1
] |
Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
|
DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.of_ufd_of_unique_irreducible
|
[
{
"state_after": "case intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\n⊢ HasUnitMulPowIrreducibleFactorization R",
"state_before": "R : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\n⊢ HasUnitMulPowIrreducibleFactorization R",
"tactic": "obtain ⟨p, hp⟩ := h₁"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\n⊢ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x",
"state_before": "case intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\n⊢ HasUnitMulPowIrreducibleFactorization R",
"tactic": "refine' ⟨p, hp, _⟩"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\n⊢ ∃ n, Associated (p ^ n) x",
"state_before": "case intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\n⊢ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x",
"tactic": "intro x hx"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ (b : R), b ∈ fx → Irreducible b) ∧ Associated (Multiset.prod fx) x\n⊢ ∃ n, Associated (p ^ n) x",
"state_before": "case intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\n⊢ ∃ n, Associated (p ^ n) x",
"tactic": "cases' WfDvdMonoid.exists_factors x hx with fx hfx"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ (b : R), b ∈ fx → Irreducible b) ∧ Associated (Multiset.prod fx) x\n⊢ Associated (p ^ ↑Multiset.card fx) x",
"state_before": "case intro.intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ (b : R), b ∈ fx → Irreducible b) ∧ Associated (Multiset.prod fx) x\n⊢ ∃ n, Associated (p ^ n) x",
"tactic": "refine' ⟨Multiset.card fx, _⟩"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ (b : R), b ∈ fx → Irreducible b) ∧ Associated (Multiset.prod fx) x\nH : Associated (Multiset.prod fx) x\n⊢ Associated (p ^ ↑Multiset.card fx) x",
"state_before": "case intro.intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ (b : R), b ∈ fx → Irreducible b) ∧ Associated (Multiset.prod fx) x\n⊢ Associated (p ^ ↑Multiset.card fx) x",
"tactic": "have H := hfx.2"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ (b : R), b ∈ fx → Irreducible b) ∧ Associated (Multiset.prod fx) x\nH : Associates.mk (Multiset.prod fx) = Associates.mk x\n⊢ Associates.mk (p ^ ↑Multiset.card fx) = Associates.mk x",
"state_before": "case intro.intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ (b : R), b ∈ fx → Irreducible b) ∧ Associated (Multiset.prod fx) x\nH : Associated (Multiset.prod fx) x\n⊢ Associated (p ^ ↑Multiset.card fx) x",
"tactic": "rw [← Associates.mk_eq_mk_iff_associated] at H⊢"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ (b : R), b ∈ fx → Irreducible b) ∧ Associated (Multiset.prod fx) x\nH : Associates.mk (Multiset.prod fx) = Associates.mk x\n⊢ Multiset.prod (Multiset.replicate (↑Multiset.card fx) (Associates.mk p)) =\n Multiset.prod (Multiset.map Associates.mk fx)",
"state_before": "case intro.intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ (b : R), b ∈ fx → Irreducible b) ∧ Associated (Multiset.prod fx) x\nH : Associates.mk (Multiset.prod fx) = Associates.mk x\n⊢ Associates.mk (p ^ ↑Multiset.card fx) = Associates.mk x",
"tactic": "rw [← H, ← Associates.prod_mk, Associates.mk_pow, ← Multiset.prod_replicate]"
},
{
"state_after": "case intro.intro.e_a\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ (b : R), b ∈ fx → Irreducible b) ∧ Associated (Multiset.prod fx) x\nH : Associates.mk (Multiset.prod fx) = Associates.mk x\n⊢ Multiset.replicate (↑Multiset.card fx) (Associates.mk p) = Multiset.map Associates.mk fx",
"state_before": "case intro.intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ (b : R), b ∈ fx → Irreducible b) ∧ Associated (Multiset.prod fx) x\nH : Associates.mk (Multiset.prod fx) = Associates.mk x\n⊢ Multiset.prod (Multiset.replicate (↑Multiset.card fx) (Associates.mk p)) =\n Multiset.prod (Multiset.map Associates.mk fx)",
"tactic": "congr 1"
},
{
"state_after": "case intro.intro.e_a\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ (b : R), b ∈ fx → Irreducible b) ∧ Associated (Multiset.prod fx) x\nH : Associates.mk (Multiset.prod fx) = Associates.mk x\n⊢ Multiset.map Associates.mk fx = Multiset.replicate (↑Multiset.card fx) (Associates.mk p)",
"state_before": "case intro.intro.e_a\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ (b : R), b ∈ fx → Irreducible b) ∧ Associated (Multiset.prod fx) x\nH : Associates.mk (Multiset.prod fx) = Associates.mk x\n⊢ Multiset.replicate (↑Multiset.card fx) (Associates.mk p) = Multiset.map Associates.mk fx",
"tactic": "symm"
},
{
"state_after": "case intro.intro.e_a\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ (b : R), b ∈ fx → Irreducible b) ∧ Associated (Multiset.prod fx) x\nH : Associates.mk (Multiset.prod fx) = Associates.mk x\n⊢ ↑Multiset.card (Multiset.map Associates.mk fx) = ↑Multiset.card fx ∧\n ∀ (b : Associates R), b ∈ Multiset.map Associates.mk fx → b = Associates.mk p",
"state_before": "case intro.intro.e_a\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ (b : R), b ∈ fx → Irreducible b) ∧ Associated (Multiset.prod fx) x\nH : Associates.mk (Multiset.prod fx) = Associates.mk x\n⊢ Multiset.map Associates.mk fx = Multiset.replicate (↑Multiset.card fx) (Associates.mk p)",
"tactic": "rw [Multiset.eq_replicate]"
},
{
"state_after": "case intro.intro.e_a\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ (b : R), b ∈ fx → Irreducible b) ∧ Associated (Multiset.prod fx) x\nH : Associates.mk (Multiset.prod fx) = Associates.mk x\n⊢ ∀ (b : Associates R) (x : R), x ∈ fx → Associates.mk x = b → b = Associates.mk p",
"state_before": "case intro.intro.e_a\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ (b : R), b ∈ fx → Irreducible b) ∧ Associated (Multiset.prod fx) x\nH : Associates.mk (Multiset.prod fx) = Associates.mk x\n⊢ ↑Multiset.card (Multiset.map Associates.mk fx) = ↑Multiset.card fx ∧\n ∀ (b : Associates R), b ∈ Multiset.map Associates.mk fx → b = Associates.mk p",
"tactic": "simp only [true_and_iff, and_imp, Multiset.card_map, eq_self_iff_true, Multiset.mem_map,\n exists_imp]"
},
{
"state_after": "case intro.intro.e_a\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ (b : R), b ∈ fx → Irreducible b) ∧ Associated (Multiset.prod fx) x\nH : Associates.mk (Multiset.prod fx) = Associates.mk x\nq : R\nhq : q ∈ fx\n⊢ Associates.mk q = Associates.mk p",
"state_before": "case intro.intro.e_a\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ (b : R), b ∈ fx → Irreducible b) ∧ Associated (Multiset.prod fx) x\nH : Associates.mk (Multiset.prod fx) = Associates.mk x\n⊢ ∀ (b : Associates R) (x : R), x ∈ fx → Associates.mk x = b → b = Associates.mk p",
"tactic": "rintro _ q hq rfl"
},
{
"state_after": "case intro.intro.e_a\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ (b : R), b ∈ fx → Irreducible b) ∧ Associated (Multiset.prod fx) x\nH : Associates.mk (Multiset.prod fx) = Associates.mk x\nq : R\nhq : q ∈ fx\n⊢ Associated q p",
"state_before": "case intro.intro.e_a\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ (b : R), b ∈ fx → Irreducible b) ∧ Associated (Multiset.prod fx) x\nH : Associates.mk (Multiset.prod fx) = Associates.mk x\nq : R\nhq : q ∈ fx\n⊢ Associates.mk q = Associates.mk p",
"tactic": "rw [Associates.mk_eq_mk_iff_associated]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.e_a\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ (b : R), b ∈ fx → Irreducible b) ∧ Associated (Multiset.prod fx) x\nH : Associates.mk (Multiset.prod fx) = Associates.mk x\nq : R\nhq : q ∈ fx\n⊢ Associated q p",
"tactic": "apply h₂ (hfx.1 _ hq) hp"
}
] |
[
249,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/Data/MvPolynomial/PDeriv.lean
|
MvPolynomial.pderiv_monomial
|
[
{
"state_after": "R : Type u\nσ : Type v\na a' a₁ a₂ : R\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\ni : σ\n⊢ (sum s fun a_1 b => ↑(monomial (s - single a_1 1)) (a * ↑b) * Pi.single i 1 a_1) =\n ↑(monomial (s - single i 1)) (a * ↑(↑s i))",
"state_before": "R : Type u\nσ : Type v\na a' a₁ a₂ : R\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\ni : σ\n⊢ ↑(pderiv i) (↑(monomial s) a) = ↑(monomial (s - single i 1)) (a * ↑(↑s i))",
"tactic": "simp only [pderiv_def, mkDerivation_monomial, Finsupp.smul_sum, smul_eq_mul, ← smul_mul_assoc,\n ← (monomial _).map_smul]"
},
{
"state_after": "case refine'_1\nR : Type u\nσ : Type v\na a' a₁ a₂ : R\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\ni j : σ\nx✝ : j ∈ s.support\nhne : j ≠ i\n⊢ (fun a_1 b => ↑(monomial (s - single a_1 1)) (a * ↑b) * Pi.single i 1 a_1) j (↑s j) = 0\n\ncase refine'_2\nR : Type u\nσ : Type v\na a' a₁ a₂ : R\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\ni : σ\nhi : ¬i ∈ s.support\n⊢ (fun a_1 b => ↑(monomial (s - single a_1 1)) (a * ↑b) * Pi.single i 1 a_1) i (↑s i) = 0\n\ncase refine'_3\nR : Type u\nσ : Type v\na a' a₁ a₂ : R\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\ni : σ\n⊢ (fun a_1 b => ↑(monomial (s - single a_1 1)) (a * ↑b) * Pi.single i 1 a_1) i (↑s i) =\n ↑(monomial (s - single i 1)) (a * ↑(↑s i))",
"state_before": "R : Type u\nσ : Type v\na a' a₁ a₂ : R\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\ni : σ\n⊢ (sum s fun a_1 b => ↑(monomial (s - single a_1 1)) (a * ↑b) * Pi.single i 1 a_1) =\n ↑(monomial (s - single i 1)) (a * ↑(↑s i))",
"tactic": "refine' (Finset.sum_eq_single i (fun j _ hne => _) fun hi => _).trans _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nR : Type u\nσ : Type v\na a' a₁ a₂ : R\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\ni j : σ\nx✝ : j ∈ s.support\nhne : j ≠ i\n⊢ (fun a_1 b => ↑(monomial (s - single a_1 1)) (a * ↑b) * Pi.single i 1 a_1) j (↑s j) = 0",
"tactic": "simp [Pi.single_eq_of_ne hne]"
},
{
"state_after": "case refine'_2\nR : Type u\nσ : Type v\na a' a₁ a₂ : R\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\ni : σ\nhi : ↑s i = 0\n⊢ (fun a_1 b => ↑(monomial (s - single a_1 1)) (a * ↑b) * Pi.single i 1 a_1) i (↑s i) = 0",
"state_before": "case refine'_2\nR : Type u\nσ : Type v\na a' a₁ a₂ : R\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\ni : σ\nhi : ¬i ∈ s.support\n⊢ (fun a_1 b => ↑(monomial (s - single a_1 1)) (a * ↑b) * Pi.single i 1 a_1) i (↑s i) = 0",
"tactic": "rw [Finsupp.not_mem_support_iff] at hi"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nR : Type u\nσ : Type v\na a' a₁ a₂ : R\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\ni : σ\nhi : ↑s i = 0\n⊢ (fun a_1 b => ↑(monomial (s - single a_1 1)) (a * ↑b) * Pi.single i 1 a_1) i (↑s i) = 0",
"tactic": "simp [hi]"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nR : Type u\nσ : Type v\na a' a₁ a₂ : R\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\ni : σ\n⊢ (fun a_1 b => ↑(monomial (s - single a_1 1)) (a * ↑b) * Pi.single i 1 a_1) i (↑s i) =\n ↑(monomial (s - single i 1)) (a * ↑(↑s i))",
"tactic": "simp"
}
] |
[
82,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/CategoryTheory/Bicategory/Basic.lean
|
CategoryTheory.Bicategory.inv_hom_whiskerLeft
|
[
{
"state_after": "no goals",
"state_before": "B : Type u\ninst✝ : Bicategory B\na b c d e : B\nf : a ⟶ b\ng h : b ⟶ c\nη : g ≅ h\n⊢ f ◁ η.inv ≫ f ◁ η.hom = 𝟙 (f ≫ h)",
"tactic": "rw [← whiskerLeft_comp, inv_hom_id, whiskerLeft_id]"
}
] |
[
210,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
209,
1
] |
Mathlib/Logic/Relation.lean
|
Reflexive.ne_imp_iff
|
[] |
[
71,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.range_some_inter_none
|
[] |
[
1197,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1196,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.succ_mul_succ_eq
|
[
{
"state_after": "a b : Nat\n⊢ a * b + b + succ a = a * b + b + a + 1",
"state_before": "a b : Nat\n⊢ succ a * succ b = a * b + a + b + 1",
"tactic": "rw [mul_succ, succ_mul, Nat.add_right_comm _ a]"
},
{
"state_after": "no goals",
"state_before": "a b : Nat\n⊢ a * b + b + succ a = a * b + b + a + 1",
"tactic": "rfl"
}
] |
[
422,
55
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
421,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
differentiableAt_of_deriv_ne_zero
|
[] |
[
248,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
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