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/LinearAlgebra/Prod.lean
|
LinearMap.coprod_zero_left
|
[] |
[
240,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
239,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.FinMeasAdditive.smul_measure_iff
|
[] |
[
144,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Computability/Halting.lean
|
Nat.Partrec'.tail
|
[
{
"state_after": "n : ℕ\nf : Vector ℕ n →. ℕ\nhf : Partrec' f\nv : Vector ℕ (succ n)\n⊢ f (ofFn fun i => Vector.get v (Fin.succ i)) = f (Vector.tail v)",
"state_before": "n : ℕ\nf : Vector ℕ n →. ℕ\nhf : Partrec' f\nv : Vector ℕ (succ n)\n⊢ (mOfFn fun i => (↑fun v => Vector.get v (Fin.succ i)) v) >>= f = f (Vector.tail v)",
"tactic": "simp"
},
{
"state_after": "n : ℕ\nf : Vector ℕ n →. ℕ\nhf : Partrec' f\nv : Vector ℕ (succ n)\n⊢ f (ofFn fun i => Vector.get v (Fin.succ i)) = f (ofFn (Vector.get (Vector.tail v)))",
"state_before": "n : ℕ\nf : Vector ℕ n →. ℕ\nhf : Partrec' f\nv : Vector ℕ (succ n)\n⊢ f (ofFn fun i => Vector.get v (Fin.succ i)) = f (Vector.tail v)",
"tactic": "rw [← ofFn_get v.tail]"
},
{
"state_after": "case e_a.e_a\nn : ℕ\nf : Vector ℕ n →. ℕ\nhf : Partrec' f\nv : Vector ℕ (succ n)\n⊢ (fun i => Vector.get v (Fin.succ i)) = Vector.get (Vector.tail v)",
"state_before": "n : ℕ\nf : Vector ℕ n →. ℕ\nhf : Partrec' f\nv : Vector ℕ (succ n)\n⊢ f (ofFn fun i => Vector.get v (Fin.succ i)) = f (ofFn (Vector.get (Vector.tail v)))",
"tactic": "congr"
},
{
"state_after": "case e_a.e_a.h\nn : ℕ\nf : Vector ℕ n →. ℕ\nhf : Partrec' f\nv : Vector ℕ (succ n)\ni : Fin n\n⊢ Vector.get v (Fin.succ i) = Vector.get (Vector.tail v) i",
"state_before": "case e_a.e_a\nn : ℕ\nf : Vector ℕ n →. ℕ\nhf : Partrec' f\nv : Vector ℕ (succ n)\n⊢ (fun i => Vector.get v (Fin.succ i)) = Vector.get (Vector.tail v)",
"tactic": "funext i"
},
{
"state_after": "no goals",
"state_before": "case e_a.e_a.h\nn : ℕ\nf : Vector ℕ n →. ℕ\nhf : Partrec' f\nv : Vector ℕ (succ n)\ni : Fin n\n⊢ Vector.get v (Fin.succ i) = Vector.get (Vector.tail v) i",
"tactic": "simp"
}
] |
[
323,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
321,
1
] |
Mathlib/Analysis/InnerProductSpace/Orthogonal.lean
|
Submodule.inf_orthogonal
|
[] |
[
176,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
HasFDerivWithinAt.cosh
|
[] |
[
1067,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1065,
1
] |
Mathlib/Data/Nat/Prime.lean
|
Nat.Prime.not_dvd_mul
|
[
{
"state_after": "no goals",
"state_before": "p m n : ℕ\npp : Prime p\nHm : ¬p ∣ m\nHn : ¬p ∣ n\n⊢ ¬(p ∣ m ∨ p ∣ n)",
"tactic": "simp [Hm, Hn]"
}
] |
[
570,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
569,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
edist_congr_right
|
[
{
"state_after": "case a\nα : Type u\nβ : Type v\nX : Type ?u.5599\ninst✝ : PseudoEMetricSpace α\nx y z : α\nh : edist x y = 0\n⊢ edist x z ≤ edist y z\n\ncase a\nα : Type u\nβ : Type v\nX : Type ?u.5599\ninst✝ : PseudoEMetricSpace α\nx y z : α\nh : edist x y = 0\n⊢ edist y z ≤ edist x z",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.5599\ninst✝ : PseudoEMetricSpace α\nx y z : α\nh : edist x y = 0\n⊢ edist x z = edist y z",
"tactic": "apply le_antisymm"
},
{
"state_after": "case a\nα : Type u\nβ : Type v\nX : Type ?u.5599\ninst✝ : PseudoEMetricSpace α\nx y z : α\nh : edist x y = 0\n⊢ edist x z ≤ edist x y + edist y z",
"state_before": "case a\nα : Type u\nβ : Type v\nX : Type ?u.5599\ninst✝ : PseudoEMetricSpace α\nx y z : α\nh : edist x y = 0\n⊢ edist x z ≤ edist y z",
"tactic": "rw [← zero_add (edist y z), ← h]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u\nβ : Type v\nX : Type ?u.5599\ninst✝ : PseudoEMetricSpace α\nx y z : α\nh : edist x y = 0\n⊢ edist x z ≤ edist x y + edist y z",
"tactic": "apply edist_triangle"
},
{
"state_after": "case a\nα : Type u\nβ : Type v\nX : Type ?u.5599\ninst✝ : PseudoEMetricSpace α\nx y z : α\nh : edist y x = 0\n⊢ edist y z ≤ edist x z",
"state_before": "case a\nα : Type u\nβ : Type v\nX : Type ?u.5599\ninst✝ : PseudoEMetricSpace α\nx y z : α\nh : edist x y = 0\n⊢ edist y z ≤ edist x z",
"tactic": "rw [edist_comm] at h"
},
{
"state_after": "case a\nα : Type u\nβ : Type v\nX : Type ?u.5599\ninst✝ : PseudoEMetricSpace α\nx y z : α\nh : edist y x = 0\n⊢ edist y z ≤ edist y x + edist x z",
"state_before": "case a\nα : Type u\nβ : Type v\nX : Type ?u.5599\ninst✝ : PseudoEMetricSpace α\nx y z : α\nh : edist y x = 0\n⊢ edist y z ≤ edist x z",
"tactic": "rw [← zero_add (edist x z), ← h]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u\nβ : Type v\nX : Type ?u.5599\ninst✝ : PseudoEMetricSpace α\nx y z : α\nh : edist y x = 0\n⊢ edist y z ≤ edist y x + edist x z",
"tactic": "apply edist_triangle"
}
] |
[
114,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/Data/PFunctor/Univariate/M.lean
|
PFunctor.M.ichildren_mk
|
[
{
"state_after": "F : PFunctor\nX : Type ?u.26833\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nx : Obj F (M F)\ni : IdxCat F\n⊢ (if H' : i.fst = head (M.mk x) then children (M.mk x) (cast (_ : B F i.fst = B F (head (M.mk x))) i.snd)\n else default) =\n if h : i.fst = x.fst then Sigma.snd x (cast (_ : B F i.fst = B F x.fst) i.snd) else default",
"state_before": "F : PFunctor\nX : Type ?u.26833\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nx : Obj F (M F)\ni : IdxCat F\n⊢ ichildren i (M.mk x) = Obj.iget x i",
"tactic": "dsimp only [ichildren, PFunctor.Obj.iget]"
},
{
"state_after": "no goals",
"state_before": "F : PFunctor\nX : Type ?u.26833\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nx : Obj F (M F)\ni : IdxCat F\n⊢ (if H' : i.fst = head (M.mk x) then children (M.mk x) (cast (_ : B F i.fst = B F (head (M.mk x))) i.snd)\n else default) =\n if h : i.fst = x.fst then Sigma.snd x (cast (_ : B F i.fst = B F x.fst) i.snd) else default",
"tactic": "congr with h"
}
] |
[
555,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
552,
1
] |
Mathlib/CategoryTheory/Simple.lean
|
CategoryTheory.epi_from_simple_zero_of_not_iso
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\ninst✝² : Abelian C\nX Y : C\ninst✝¹ : Simple X\nf : X ⟶ Y\ninst✝ : Epi f\nw : IsIso f → False\n⊢ f = 0",
"tactic": "classical\n by_contra h\n exact w (isIso_of_epi_of_nonzero h)"
},
{
"state_after": "C : Type u\ninst✝³ : Category C\ninst✝² : Abelian C\nX Y : C\ninst✝¹ : Simple X\nf : X ⟶ Y\ninst✝ : Epi f\nw : IsIso f → False\nh : ¬f = 0\n⊢ False",
"state_before": "C : Type u\ninst✝³ : Category C\ninst✝² : Abelian C\nX Y : C\ninst✝¹ : Simple X\nf : X ⟶ Y\ninst✝ : Epi f\nw : IsIso f → False\n⊢ f = 0",
"tactic": "by_contra h"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\ninst✝² : Abelian C\nX Y : C\ninst✝¹ : Simple X\nf : X ⟶ Y\ninst✝ : Epi f\nw : IsIso f → False\nh : ¬f = 0\n⊢ False",
"tactic": "exact w (isIso_of_epi_of_nonzero h)"
}
] |
[
186,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/Algebra/Ring/Idempotents.lean
|
IsIdempotentElem.one
|
[] |
[
65,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Topology/Connected.lean
|
Continuous.connectedComponentsMap_continuous
|
[] |
[
1562,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1560,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Icc_erase_left
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.72862\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrder α\na✝ b✝ c : α\ninst✝ : DecidableEq α\na b : α\n⊢ erase (Icc a b) a = Ioc a b",
"tactic": "simp [← coe_inj]"
}
] |
[
545,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
545,
1
] |
Mathlib/Init/Propext.lean
|
imp_congr_eq
|
[] |
[
20,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
19,
1
] |
Mathlib/Order/FixedPoints.lean
|
OrderHom.le_gfp
|
[] |
[
115,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Order/Monotone/Basic.lean
|
Monotone.imp
|
[] |
[
364,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
363,
1
] |
Mathlib/Algebra/Algebra/Basic.lean
|
algebraMap.coe_pow
|
[] |
[
169,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.cycleFactorsFinset_eq_empty_iff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.2748424\nα : Type u_1\nβ : Type ?u.2748430\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f : Perm α\n⊢ cycleFactorsFinset f = ∅ ↔ f = 1",
"tactic": "simpa [cycleFactorsFinset_eq_finset] using eq_comm"
}
] |
[
1443,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1442,
1
] |
Mathlib/Algebra/Order/Pointwise.lean
|
csSup_one
|
[] |
[
115,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
subset_interior_mul_right
|
[] |
[
1267,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1266,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Algebra.surjective_algebraMap_iff
|
[] |
[
952,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
946,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
LinearMap.isUnit_iff_range_eq_top
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nf : V →ₗ[K] V\n⊢ IsUnit f ↔ range f = ⊤",
"tactic": "rw [isUnit_iff_ker_eq_bot, ker_eq_bot_iff_range_eq_top]"
}
] |
[
1016,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1014,
1
] |
Mathlib/LinearAlgebra/SesquilinearForm.lean
|
LinearMap.ortho_smul_left
|
[
{
"state_after": "R : Type ?u.36929\nR₁ : Type ?u.36932\nR₂ : Type ?u.36935\nR₃ : Type ?u.36938\nM : Type ?u.36941\nM₁ : Type ?u.36944\nM₂ : Type ?u.36947\nMₗ₁ : Type ?u.36950\nMₗ₁' : Type ?u.36953\nMₗ₂ : Type ?u.36956\nMₗ₂' : Type ?u.36959\nK : Type u_2\nK₁ : Type u_1\nK₂ : Type u_5\nV : Type ?u.36971\nV₁ : Type u_3\nV₂ : Type u_4\nn : Type ?u.36980\ninst✝⁶ : Field K\ninst✝⁵ : Field K₁\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K₁ V₁\ninst✝² : Field K₂\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K₂ V₂\nI₁ : K₁ →+* K\nI₂ : K₂ →+* K\nI₁' : K₁ →+* K\nJ₁ J₂ : K →+* K\nB : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] K\nx : V₁\ny : V₂\na : K₁\nha : a ≠ 0\n⊢ ↑(↑B x) y = 0 ↔ ↑(↑B (a • x)) y = 0",
"state_before": "R : Type ?u.36929\nR₁ : Type ?u.36932\nR₂ : Type ?u.36935\nR₃ : Type ?u.36938\nM : Type ?u.36941\nM₁ : Type ?u.36944\nM₂ : Type ?u.36947\nMₗ₁ : Type ?u.36950\nMₗ₁' : Type ?u.36953\nMₗ₂ : Type ?u.36956\nMₗ₂' : Type ?u.36959\nK : Type u_2\nK₁ : Type u_1\nK₂ : Type u_5\nV : Type ?u.36971\nV₁ : Type u_3\nV₂ : Type u_4\nn : Type ?u.36980\ninst✝⁶ : Field K\ninst✝⁵ : Field K₁\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K₁ V₁\ninst✝² : Field K₂\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K₂ V₂\nI₁ : K₁ →+* K\nI₂ : K₂ →+* K\nI₁' : K₁ →+* K\nJ₁ J₂ : K →+* K\nB : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] K\nx : V₁\ny : V₂\na : K₁\nha : a ≠ 0\n⊢ IsOrtho B x y ↔ IsOrtho B (a • x) y",
"tactic": "dsimp only [IsOrtho]"
},
{
"state_after": "case mp\nR : Type ?u.36929\nR₁ : Type ?u.36932\nR₂ : Type ?u.36935\nR₃ : Type ?u.36938\nM : Type ?u.36941\nM₁ : Type ?u.36944\nM₂ : Type ?u.36947\nMₗ₁ : Type ?u.36950\nMₗ₁' : Type ?u.36953\nMₗ₂ : Type ?u.36956\nMₗ₂' : Type ?u.36959\nK : Type u_2\nK₁ : Type u_1\nK₂ : Type u_5\nV : Type ?u.36971\nV₁ : Type u_3\nV₂ : Type u_4\nn : Type ?u.36980\ninst✝⁶ : Field K\ninst✝⁵ : Field K₁\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K₁ V₁\ninst✝² : Field K₂\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K₂ V₂\nI₁ : K₁ →+* K\nI₂ : K₂ →+* K\nI₁' : K₁ →+* K\nJ₁ J₂ : K →+* K\nB : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] K\nx : V₁\ny : V₂\na : K₁\nha : a ≠ 0\nH : ↑(↑B x) y = 0\n⊢ ↑(↑B (a • x)) y = 0\n\ncase mpr\nR : Type ?u.36929\nR₁ : Type ?u.36932\nR₂ : Type ?u.36935\nR₃ : Type ?u.36938\nM : Type ?u.36941\nM₁ : Type ?u.36944\nM₂ : Type ?u.36947\nMₗ₁ : Type ?u.36950\nMₗ₁' : Type ?u.36953\nMₗ₂ : Type ?u.36956\nMₗ₂' : Type ?u.36959\nK : Type u_2\nK₁ : Type u_1\nK₂ : Type u_5\nV : Type ?u.36971\nV₁ : Type u_3\nV₂ : Type u_4\nn : Type ?u.36980\ninst✝⁶ : Field K\ninst✝⁵ : Field K₁\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K₁ V₁\ninst✝² : Field K₂\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K₂ V₂\nI₁ : K₁ →+* K\nI₂ : K₂ →+* K\nI₁' : K₁ →+* K\nJ₁ J₂ : K →+* K\nB : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] K\nx : V₁\ny : V₂\na : K₁\nha : a ≠ 0\nH : ↑(↑B (a • x)) y = 0\n⊢ ↑(↑B x) y = 0",
"state_before": "R : Type ?u.36929\nR₁ : Type ?u.36932\nR₂ : Type ?u.36935\nR₃ : Type ?u.36938\nM : Type ?u.36941\nM₁ : Type ?u.36944\nM₂ : Type ?u.36947\nMₗ₁ : Type ?u.36950\nMₗ₁' : Type ?u.36953\nMₗ₂ : Type ?u.36956\nMₗ₂' : Type ?u.36959\nK : Type u_2\nK₁ : Type u_1\nK₂ : Type u_5\nV : Type ?u.36971\nV₁ : Type u_3\nV₂ : Type u_4\nn : Type ?u.36980\ninst✝⁶ : Field K\ninst✝⁵ : Field K₁\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K₁ V₁\ninst✝² : Field K₂\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K₂ V₂\nI₁ : K₁ →+* K\nI₂ : K₂ →+* K\nI₁' : K₁ →+* K\nJ₁ J₂ : K →+* K\nB : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] K\nx : V₁\ny : V₂\na : K₁\nha : a ≠ 0\n⊢ ↑(↑B x) y = 0 ↔ ↑(↑B (a • x)) y = 0",
"tactic": "constructor <;> intro H"
},
{
"state_after": "no goals",
"state_before": "case mp\nR : Type ?u.36929\nR₁ : Type ?u.36932\nR₂ : Type ?u.36935\nR₃ : Type ?u.36938\nM : Type ?u.36941\nM₁ : Type ?u.36944\nM₂ : Type ?u.36947\nMₗ₁ : Type ?u.36950\nMₗ₁' : Type ?u.36953\nMₗ₂ : Type ?u.36956\nMₗ₂' : Type ?u.36959\nK : Type u_2\nK₁ : Type u_1\nK₂ : Type u_5\nV : Type ?u.36971\nV₁ : Type u_3\nV₂ : Type u_4\nn : Type ?u.36980\ninst✝⁶ : Field K\ninst✝⁵ : Field K₁\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K₁ V₁\ninst✝² : Field K₂\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K₂ V₂\nI₁ : K₁ →+* K\nI₂ : K₂ →+* K\nI₁' : K₁ →+* K\nJ₁ J₂ : K →+* K\nB : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] K\nx : V₁\ny : V₂\na : K₁\nha : a ≠ 0\nH : ↑(↑B x) y = 0\n⊢ ↑(↑B (a • x)) y = 0",
"tactic": "rw [map_smulₛₗ₂, H, smul_zero]"
},
{
"state_after": "case mpr\nR : Type ?u.36929\nR₁ : Type ?u.36932\nR₂ : Type ?u.36935\nR₃ : Type ?u.36938\nM : Type ?u.36941\nM₁ : Type ?u.36944\nM₂ : Type ?u.36947\nMₗ₁ : Type ?u.36950\nMₗ₁' : Type ?u.36953\nMₗ₂ : Type ?u.36956\nMₗ₂' : Type ?u.36959\nK : Type u_2\nK₁ : Type u_1\nK₂ : Type u_5\nV : Type ?u.36971\nV₁ : Type u_3\nV₂ : Type u_4\nn : Type ?u.36980\ninst✝⁶ : Field K\ninst✝⁵ : Field K₁\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K₁ V₁\ninst✝² : Field K₂\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K₂ V₂\nI₁ : K₁ →+* K\nI₂ : K₂ →+* K\nI₁' : K₁ →+* K\nJ₁ J₂ : K →+* K\nB : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] K\nx : V₁\ny : V₂\na : K₁\nha : a ≠ 0\nH : ↑I₁ a = 0 ∨ ↑(↑B x) y = 0\n⊢ ↑(↑B x) y = 0",
"state_before": "case mpr\nR : Type ?u.36929\nR₁ : Type ?u.36932\nR₂ : Type ?u.36935\nR₃ : Type ?u.36938\nM : Type ?u.36941\nM₁ : Type ?u.36944\nM₂ : Type ?u.36947\nMₗ₁ : Type ?u.36950\nMₗ₁' : Type ?u.36953\nMₗ₂ : Type ?u.36956\nMₗ₂' : Type ?u.36959\nK : Type u_2\nK₁ : Type u_1\nK₂ : Type u_5\nV : Type ?u.36971\nV₁ : Type u_3\nV₂ : Type u_4\nn : Type ?u.36980\ninst✝⁶ : Field K\ninst✝⁵ : Field K₁\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K₁ V₁\ninst✝² : Field K₂\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K₂ V₂\nI₁ : K₁ →+* K\nI₂ : K₂ →+* K\nI₁' : K₁ →+* K\nJ₁ J₂ : K →+* K\nB : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] K\nx : V₁\ny : V₂\na : K₁\nha : a ≠ 0\nH : ↑(↑B (a • x)) y = 0\n⊢ ↑(↑B x) y = 0",
"tactic": "rw [map_smulₛₗ₂, smul_eq_zero] at H"
},
{
"state_after": "case mpr.inl\nR : Type ?u.36929\nR₁ : Type ?u.36932\nR₂ : Type ?u.36935\nR₃ : Type ?u.36938\nM : Type ?u.36941\nM₁ : Type ?u.36944\nM₂ : Type ?u.36947\nMₗ₁ : Type ?u.36950\nMₗ₁' : Type ?u.36953\nMₗ₂ : Type ?u.36956\nMₗ₂' : Type ?u.36959\nK : Type u_2\nK₁ : Type u_1\nK₂ : Type u_5\nV : Type ?u.36971\nV₁ : Type u_3\nV₂ : Type u_4\nn : Type ?u.36980\ninst✝⁶ : Field K\ninst✝⁵ : Field K₁\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K₁ V₁\ninst✝² : Field K₂\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K₂ V₂\nI₁ : K₁ →+* K\nI₂ : K₂ →+* K\nI₁' : K₁ →+* K\nJ₁ J₂ : K →+* K\nB : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] K\nx : V₁\ny : V₂\na : K₁\nha : a ≠ 0\nH : ↑I₁ a = 0\n⊢ ↑(↑B x) y = 0\n\ncase mpr.inr\nR : Type ?u.36929\nR₁ : Type ?u.36932\nR₂ : Type ?u.36935\nR₃ : Type ?u.36938\nM : Type ?u.36941\nM₁ : Type ?u.36944\nM₂ : Type ?u.36947\nMₗ₁ : Type ?u.36950\nMₗ₁' : Type ?u.36953\nMₗ₂ : Type ?u.36956\nMₗ₂' : Type ?u.36959\nK : Type u_2\nK₁ : Type u_1\nK₂ : Type u_5\nV : Type ?u.36971\nV₁ : Type u_3\nV₂ : Type u_4\nn : Type ?u.36980\ninst✝⁶ : Field K\ninst✝⁵ : Field K₁\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K₁ V₁\ninst✝² : Field K₂\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K₂ V₂\nI₁ : K₁ →+* K\nI₂ : K₂ →+* K\nI₁' : K₁ →+* K\nJ₁ J₂ : K →+* K\nB : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] K\nx : V₁\ny : V₂\na : K₁\nha : a ≠ 0\nH : ↑(↑B x) y = 0\n⊢ ↑(↑B x) y = 0",
"state_before": "case mpr\nR : Type ?u.36929\nR₁ : Type ?u.36932\nR₂ : Type ?u.36935\nR₃ : Type ?u.36938\nM : Type ?u.36941\nM₁ : Type ?u.36944\nM₂ : Type ?u.36947\nMₗ₁ : Type ?u.36950\nMₗ₁' : Type ?u.36953\nMₗ₂ : Type ?u.36956\nMₗ₂' : Type ?u.36959\nK : Type u_2\nK₁ : Type u_1\nK₂ : Type u_5\nV : Type ?u.36971\nV₁ : Type u_3\nV₂ : Type u_4\nn : Type ?u.36980\ninst✝⁶ : Field K\ninst✝⁵ : Field K₁\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K₁ V₁\ninst✝² : Field K₂\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K₂ V₂\nI₁ : K₁ →+* K\nI₂ : K₂ →+* K\nI₁' : K₁ →+* K\nJ₁ J₂ : K →+* K\nB : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] K\nx : V₁\ny : V₂\na : K₁\nha : a ≠ 0\nH : ↑I₁ a = 0 ∨ ↑(↑B x) y = 0\n⊢ ↑(↑B x) y = 0",
"tactic": "cases' H with H H"
},
{
"state_after": "case mpr.inl\nR : Type ?u.36929\nR₁ : Type ?u.36932\nR₂ : Type ?u.36935\nR₃ : Type ?u.36938\nM : Type ?u.36941\nM₁ : Type ?u.36944\nM₂ : Type ?u.36947\nMₗ₁ : Type ?u.36950\nMₗ₁' : Type ?u.36953\nMₗ₂ : Type ?u.36956\nMₗ₂' : Type ?u.36959\nK : Type u_2\nK₁ : Type u_1\nK₂ : Type u_5\nV : Type ?u.36971\nV₁ : Type u_3\nV₂ : Type u_4\nn : Type ?u.36980\ninst✝⁶ : Field K\ninst✝⁵ : Field K₁\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K₁ V₁\ninst✝² : Field K₂\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K₂ V₂\nI₁ : K₁ →+* K\nI₂ : K₂ →+* K\nI₁' : K₁ →+* K\nJ₁ J₂ : K →+* K\nB : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] K\nx : V₁\ny : V₂\na : K₁\nha : a ≠ 0\nH : a = 0\n⊢ ↑(↑B x) y = 0",
"state_before": "case mpr.inl\nR : Type ?u.36929\nR₁ : Type ?u.36932\nR₂ : Type ?u.36935\nR₃ : Type ?u.36938\nM : Type ?u.36941\nM₁ : Type ?u.36944\nM₂ : Type ?u.36947\nMₗ₁ : Type ?u.36950\nMₗ₁' : Type ?u.36953\nMₗ₂ : Type ?u.36956\nMₗ₂' : Type ?u.36959\nK : Type u_2\nK₁ : Type u_1\nK₂ : Type u_5\nV : Type ?u.36971\nV₁ : Type u_3\nV₂ : Type u_4\nn : Type ?u.36980\ninst✝⁶ : Field K\ninst✝⁵ : Field K₁\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K₁ V₁\ninst✝² : Field K₂\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K₂ V₂\nI₁ : K₁ →+* K\nI₂ : K₂ →+* K\nI₁' : K₁ →+* K\nJ₁ J₂ : K →+* K\nB : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] K\nx : V₁\ny : V₂\na : K₁\nha : a ≠ 0\nH : ↑I₁ a = 0\n⊢ ↑(↑B x) y = 0",
"tactic": "rw [map_eq_zero I₁] at H"
},
{
"state_after": "no goals",
"state_before": "case mpr.inl\nR : Type ?u.36929\nR₁ : Type ?u.36932\nR₂ : Type ?u.36935\nR₃ : Type ?u.36938\nM : Type ?u.36941\nM₁ : Type ?u.36944\nM₂ : Type ?u.36947\nMₗ₁ : Type ?u.36950\nMₗ₁' : Type ?u.36953\nMₗ₂ : Type ?u.36956\nMₗ₂' : Type ?u.36959\nK : Type u_2\nK₁ : Type u_1\nK₂ : Type u_5\nV : Type ?u.36971\nV₁ : Type u_3\nV₂ : Type u_4\nn : Type ?u.36980\ninst✝⁶ : Field K\ninst✝⁵ : Field K₁\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K₁ V₁\ninst✝² : Field K₂\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K₂ V₂\nI₁ : K₁ →+* K\nI₂ : K₂ →+* K\nI₁' : K₁ →+* K\nJ₁ J₂ : K →+* K\nB : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] K\nx : V₁\ny : V₂\na : K₁\nha : a ≠ 0\nH : a = 0\n⊢ ↑(↑B x) y = 0",
"tactic": "trivial"
},
{
"state_after": "no goals",
"state_before": "case mpr.inr\nR : Type ?u.36929\nR₁ : Type ?u.36932\nR₂ : Type ?u.36935\nR₃ : Type ?u.36938\nM : Type ?u.36941\nM₁ : Type ?u.36944\nM₂ : Type ?u.36947\nMₗ₁ : Type ?u.36950\nMₗ₁' : Type ?u.36953\nMₗ₂ : Type ?u.36956\nMₗ₂' : Type ?u.36959\nK : Type u_2\nK₁ : Type u_1\nK₂ : Type u_5\nV : Type ?u.36971\nV₁ : Type u_3\nV₂ : Type u_4\nn : Type ?u.36980\ninst✝⁶ : Field K\ninst✝⁵ : Field K₁\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K₁ V₁\ninst✝² : Field K₂\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K₂ V₂\nI₁ : K₁ →+* K\nI₂ : K₂ →+* K\nI₁' : K₁ →+* K\nJ₁ J₂ : K →+* K\nB : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] K\nx : V₁\ny : V₂\na : K₁\nha : a ≠ 0\nH : ↑(↑B x) y = 0\n⊢ ↑(↑B x) y = 0",
"tactic": "exact H"
}
] |
[
124,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Analysis/NormedSpace/Pointwise.lean
|
exists_dist_lt_le
|
[
{
"state_after": "case intro.intro\n𝕜 : Type ?u.212520\nE : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nx y✝ z : E\nδ ε : ℝ\nhδ : 0 < δ\nhε : 0 ≤ ε\nh : dist x z < ε + δ\ny : E\nyz : dist z y ≤ ε\nxy : dist y x < δ\n⊢ ∃ y, dist x y < δ ∧ dist y z ≤ ε",
"state_before": "𝕜 : Type ?u.212520\nE : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nx y z : E\nδ ε : ℝ\nhδ : 0 < δ\nhε : 0 ≤ ε\nh : dist x z < ε + δ\n⊢ ∃ y, dist x y < δ ∧ dist y z ≤ ε",
"tactic": "obtain ⟨y, yz, xy⟩ :=\n exists_dist_le_lt hε hδ (show dist z x < δ + ε by simpa only [dist_comm, add_comm] using h)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\n𝕜 : Type ?u.212520\nE : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nx y✝ z : E\nδ ε : ℝ\nhδ : 0 < δ\nhε : 0 ≤ ε\nh : dist x z < ε + δ\ny : E\nyz : dist z y ≤ ε\nxy : dist y x < δ\n⊢ ∃ y, dist x y < δ ∧ dist y z ≤ ε",
"tactic": "exact ⟨y, by simp [dist_comm x y, dist_comm y z, *]⟩"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.212520\nE : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nx y z : E\nδ ε : ℝ\nhδ : 0 < δ\nhε : 0 ≤ ε\nh : dist x z < ε + δ\n⊢ dist z x < δ + ε",
"tactic": "simpa only [dist_comm, add_comm] using h"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.212520\nE : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nx y✝ z : E\nδ ε : ℝ\nhδ : 0 < δ\nhε : 0 ≤ ε\nh : dist x z < ε + δ\ny : E\nyz : dist z y ≤ ε\nxy : dist y x < δ\n⊢ dist x y < δ ∧ dist y z ≤ ε",
"tactic": "simp [dist_comm x y, dist_comm y z, *]"
}
] |
[
195,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.preimage_sub_const_Ici
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ (fun x => x - a) ⁻¹' Ici b = Ici (b + a)",
"tactic": "simp [sub_eq_add_neg]"
}
] |
[
177,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
176,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
|
CategoryTheory.Limits.comp_factorThruImage_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\ninst✝ : HasImage g\nh : f ≫ g = 0\n⊢ (f ≫ factorThruImage g) ≫ image.ι g = 0",
"tactic": "simp [h]"
}
] |
[
576,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
574,
1
] |
Mathlib/Topology/ContinuousFunction/Weierstrass.lean
|
polynomialFunctions_closure_eq_top
|
[
{
"state_after": "case pos\na b : ℝ\nh : a < b\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤\n\ncase neg\na b : ℝ\nh : ¬a < b\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤",
"state_before": "a b : ℝ\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤",
"tactic": "by_cases h : a < b"
},
{
"state_after": "case pos\na b : ℝ\nh : a < b\nW : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) :=\n compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤",
"state_before": "case pos\na b : ℝ\nh : a < b\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤",
"tactic": "let W : C(Set.Icc a b, ℝ) →ₐ[ℝ] C(I, ℝ) :=\n compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap"
},
{
"state_after": "case pos\na b : ℝ\nh : a < b\nW : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) :=\n compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))\nW' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h))\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤",
"state_before": "case pos\na b : ℝ\nh : a < b\nW : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) :=\n compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤",
"tactic": "let W' : C(Set.Icc a b, ℝ) ≃ₜ C(I, ℝ) := compRightHomeomorph ℝ (iccHomeoI a b h).symm"
},
{
"state_after": "case pos\na b : ℝ\nh : a < b\nW : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) :=\n compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))\nW' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h))\nw : ↑W = ↑W'\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤",
"state_before": "case pos\na b : ℝ\nh : a < b\nW : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) :=\n compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))\nW' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h))\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤",
"tactic": "have w : (W : C(Set.Icc a b, ℝ) → C(I, ℝ)) = W' := rfl"
},
{
"state_after": "case pos\na b : ℝ\nh : a < b\nW : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) :=\n compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))\nW' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h))\nw : ↑W = ↑W'\np : Subalgebra.topologicalClosure (polynomialFunctions I) = ⊤\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤",
"state_before": "case pos\na b : ℝ\nh : a < b\nW : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) :=\n compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))\nW' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h))\nw : ↑W = ↑W'\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤",
"tactic": "have p := polynomialFunctions_closure_eq_top'"
},
{
"state_after": "case pos\na b : ℝ\nh : a < b\nW : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) :=\n compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))\nW' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h))\nw : ↑W = ↑W'\np : Subalgebra.comap W (Subalgebra.topologicalClosure (polynomialFunctions I)) = Subalgebra.comap W ⊤\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤",
"state_before": "case pos\na b : ℝ\nh : a < b\nW : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) :=\n compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))\nW' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h))\nw : ↑W = ↑W'\np : Subalgebra.topologicalClosure (polynomialFunctions I) = ⊤\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤",
"tactic": "apply_fun fun s => s.comap W at p"
},
{
"state_after": "case pos\na b : ℝ\nh : a < b\nW : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) :=\n compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))\nW' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h))\nw : ↑W = ↑W'\np :\n Subalgebra.comap (compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h))))\n (Subalgebra.topologicalClosure (polynomialFunctions I)) =\n ⊤\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤",
"state_before": "case pos\na b : ℝ\nh : a < b\nW : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) :=\n compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))\nW' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h))\nw : ↑W = ↑W'\np : Subalgebra.comap W (Subalgebra.topologicalClosure (polynomialFunctions I)) = Subalgebra.comap W ⊤\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤",
"tactic": "simp only [Algebra.comap_top] at p"
},
{
"state_after": "case pos\na b : ℝ\nh : a < b\nW : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) :=\n compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))\nW' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h))\nw : ↑W = ↑W'\np : Subalgebra.topologicalClosure (Subalgebra.comap W (polynomialFunctions I)) = ⊤\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤",
"state_before": "case pos\na b : ℝ\nh : a < b\nW : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) :=\n compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))\nW' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h))\nw : ↑W = ↑W'\np :\n Subalgebra.comap (compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h))))\n (Subalgebra.topologicalClosure (polynomialFunctions I)) =\n ⊤\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤",
"tactic": "rw [Subalgebra.topologicalClosure_comap_homeomorph _ W W' w] at p"
},
{
"state_after": "case pos\na b : ℝ\nh : a < b\nW : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) :=\n compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))\nW' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h))\nw : ↑W = ↑W'\np : Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤",
"state_before": "case pos\na b : ℝ\nh : a < b\nW : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) :=\n compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))\nW' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h))\nw : ↑W = ↑W'\np : Subalgebra.topologicalClosure (Subalgebra.comap W (polynomialFunctions I)) = ⊤\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤",
"tactic": "rw [polynomialFunctions.comap_compRightAlgHom_iccHomeoI] at p"
},
{
"state_after": "no goals",
"state_before": "case pos\na b : ℝ\nh : a < b\nW : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) :=\n compRightAlgHom ℝ ℝ (Homeomorph.toContinuousMap (Homeomorph.symm (iccHomeoI a b h)))\nW' : C(↑(Set.Icc a b), ℝ) ≃ₜ C(↑I, ℝ) := compRightHomeomorph ℝ (Homeomorph.symm (iccHomeoI a b h))\nw : ↑W = ↑W'\np : Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤",
"tactic": "exact p"
},
{
"state_after": "case neg\na b : ℝ\nh : ¬a < b\nthis : Subsingleton ↑(Set.Icc a b)\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤",
"state_before": "case neg\na b : ℝ\nh : ¬a < b\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤",
"tactic": "haveI : Subsingleton (Set.Icc a b) :=\n ⟨fun x y =>\n le_antisymm ((x.2.2.trans (not_lt.mp h)).trans y.2.1)\n ((y.2.2.trans (not_lt.mp h)).trans x.2.1)⟩"
},
{
"state_after": "no goals",
"state_before": "case neg\na b : ℝ\nh : ¬a < b\nthis : Subsingleton ↑(Set.Icc a b)\n⊢ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤",
"tactic": "apply Subsingleton.elim"
}
] |
[
86,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/Algebra/Homology/HomologicalComplex.lean
|
CochainComplex.of_d
|
[
{
"state_after": "ι : Type ?u.215700\nV : Type u\ninst✝⁴ : Category V\ninst✝³ : HasZeroMorphisms V\nα : Type u_1\ninst✝² : AddRightCancelSemigroup α\ninst✝¹ : One α\ninst✝ : DecidableEq α\nX : α → V\nd : (n : α) → X n ⟶ X (n + 1)\nsq : ∀ (n : α), d n ≫ d (n + 1) = 0\nj : α\n⊢ (if j + 1 = j + 1 then d j ≫ 𝟙 (X (j + 1)) else 0) = d j",
"state_before": "ι : Type ?u.215700\nV : Type u\ninst✝⁴ : Category V\ninst✝³ : HasZeroMorphisms V\nα : Type u_1\ninst✝² : AddRightCancelSemigroup α\ninst✝¹ : One α\ninst✝ : DecidableEq α\nX : α → V\nd : (n : α) → X n ⟶ X (n + 1)\nsq : ∀ (n : α), d n ≫ d (n + 1) = 0\nj : α\n⊢ HomologicalComplex.d (of X d sq) j (j + 1) = d j",
"tactic": "dsimp [of]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.215700\nV : Type u\ninst✝⁴ : Category V\ninst✝³ : HasZeroMorphisms V\nα : Type u_1\ninst✝² : AddRightCancelSemigroup α\ninst✝¹ : One α\ninst✝ : DecidableEq α\nX : α → V\nd : (n : α) → X n ⟶ X (n + 1)\nsq : ∀ (n : α), d n ≫ d (n + 1) = 0\nj : α\n⊢ (if j + 1 = j + 1 then d j ≫ 𝟙 (X (j + 1)) else 0) = d j",
"tactic": "rw [if_pos rfl, Category.comp_id]"
}
] |
[
907,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
905,
1
] |
Mathlib/Data/Nat/Parity.lean
|
Nat.bit1_div_two
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ\n⊢ bit1 n / 2 = n",
"tactic": "rw [← Nat.bit1_eq_bit1, bit1, bit0_eq_two_mul, Nat.two_mul_div_two_add_one_of_odd (odd_bit1 n)]"
}
] |
[
250,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
249,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_trim_ae
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1823472\nγ : Type ?u.1823475\nδ : Type ?u.1823478\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nf : α → ℝ≥0∞\nhf : AEMeasurable f\n⊢ (∫⁻ (a : α), f a ∂Measure.trim μ hm) = ∫⁻ (a : α), f a ∂μ",
"tactic": "rw [lintegral_congr_ae (ae_eq_of_ae_eq_trim hf.ae_eq_mk), lintegral_congr_ae hf.ae_eq_mk,\n lintegral_trim hm hf.measurable_mk]"
}
] |
[
1945,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1942,
1
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean
|
ContinuousLinearMap.strongTopology.topologicalAddGroup
|
[
{
"state_after": "𝕜₁ : Type u_3\n𝕜₂ : Type u_4\ninst✝¹³ : NormedField 𝕜₁\ninst✝¹² : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_2\nE' : Type ?u.40995\nF : Type u_1\nF' : Type ?u.41001\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : Module 𝕜₁ E\ninst✝⁹ : AddCommGroup E'\ninst✝⁸ : Module ℝ E'\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module ℝ F'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace E'\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\n𝔖 : Set (Set E)\nthis : UniformSpace F := TopologicalAddGroup.toUniformSpace F\n⊢ TopologicalAddGroup (E →SL[σ] F)",
"state_before": "𝕜₁ : Type u_3\n𝕜₂ : Type u_4\ninst✝¹³ : NormedField 𝕜₁\ninst✝¹² : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_2\nE' : Type ?u.40995\nF : Type u_1\nF' : Type ?u.41001\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : Module 𝕜₁ E\ninst✝⁹ : AddCommGroup E'\ninst✝⁸ : Module ℝ E'\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module ℝ F'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace E'\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\n𝔖 : Set (Set E)\n⊢ TopologicalAddGroup (E →SL[σ] F)",
"tactic": "letI : UniformSpace F := TopologicalAddGroup.toUniformSpace F"
},
{
"state_after": "𝕜₁ : Type u_3\n𝕜₂ : Type u_4\ninst✝¹³ : NormedField 𝕜₁\ninst✝¹² : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_2\nE' : Type ?u.40995\nF : Type u_1\nF' : Type ?u.41001\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : Module 𝕜₁ E\ninst✝⁹ : AddCommGroup E'\ninst✝⁸ : Module ℝ E'\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module ℝ F'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace E'\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\n𝔖 : Set (Set E)\nthis✝ : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis : UniformAddGroup F\n⊢ TopologicalAddGroup (E →SL[σ] F)",
"state_before": "𝕜₁ : Type u_3\n𝕜₂ : Type u_4\ninst✝¹³ : NormedField 𝕜₁\ninst✝¹² : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_2\nE' : Type ?u.40995\nF : Type u_1\nF' : Type ?u.41001\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : Module 𝕜₁ E\ninst✝⁹ : AddCommGroup E'\ninst✝⁸ : Module ℝ E'\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module ℝ F'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace E'\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\n𝔖 : Set (Set E)\nthis : UniformSpace F := TopologicalAddGroup.toUniformSpace F\n⊢ TopologicalAddGroup (E →SL[σ] F)",
"tactic": "haveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform"
},
{
"state_after": "𝕜₁ : Type u_3\n𝕜₂ : Type u_4\ninst✝¹³ : NormedField 𝕜₁\ninst✝¹² : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_2\nE' : Type ?u.40995\nF : Type u_1\nF' : Type ?u.41001\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : Module 𝕜₁ E\ninst✝⁹ : AddCommGroup E'\ninst✝⁸ : Module ℝ E'\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module ℝ F'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace E'\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\n𝔖 : Set (Set E)\nthis✝¹ : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis✝ : UniformAddGroup F\nthis : UniformSpace (E →SL[σ] F) := strongUniformity σ F 𝔖\n⊢ TopologicalAddGroup (E →SL[σ] F)",
"state_before": "𝕜₁ : Type u_3\n𝕜₂ : Type u_4\ninst✝¹³ : NormedField 𝕜₁\ninst✝¹² : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_2\nE' : Type ?u.40995\nF : Type u_1\nF' : Type ?u.41001\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : Module 𝕜₁ E\ninst✝⁹ : AddCommGroup E'\ninst✝⁸ : Module ℝ E'\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module ℝ F'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace E'\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\n𝔖 : Set (Set E)\nthis✝ : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis : UniformAddGroup F\n⊢ TopologicalAddGroup (E →SL[σ] F)",
"tactic": "letI : UniformSpace (E →SL[σ] F) := strongUniformity σ F 𝔖"
},
{
"state_after": "𝕜₁ : Type u_3\n𝕜₂ : Type u_4\ninst✝¹³ : NormedField 𝕜₁\ninst✝¹² : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_2\nE' : Type ?u.40995\nF : Type u_1\nF' : Type ?u.41001\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : Module 𝕜₁ E\ninst✝⁹ : AddCommGroup E'\ninst✝⁸ : Module ℝ E'\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module ℝ F'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace E'\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\n𝔖 : Set (Set E)\nthis✝² : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis✝¹ : UniformAddGroup F\nthis✝ : UniformSpace (E →SL[σ] F) := strongUniformity σ F 𝔖\nthis : UniformAddGroup (E →SL[σ] F)\n⊢ TopologicalAddGroup (E →SL[σ] F)",
"state_before": "𝕜₁ : Type u_3\n𝕜₂ : Type u_4\ninst✝¹³ : NormedField 𝕜₁\ninst✝¹² : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_2\nE' : Type ?u.40995\nF : Type u_1\nF' : Type ?u.41001\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : Module 𝕜₁ E\ninst✝⁹ : AddCommGroup E'\ninst✝⁸ : Module ℝ E'\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module ℝ F'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace E'\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\n𝔖 : Set (Set E)\nthis✝¹ : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis✝ : UniformAddGroup F\nthis : UniformSpace (E →SL[σ] F) := strongUniformity σ F 𝔖\n⊢ TopologicalAddGroup (E →SL[σ] F)",
"tactic": "haveI : UniformAddGroup (E →SL[σ] F) := strongUniformity.uniformAddGroup σ F 𝔖"
},
{
"state_after": "no goals",
"state_before": "𝕜₁ : Type u_3\n𝕜₂ : Type u_4\ninst✝¹³ : NormedField 𝕜₁\ninst✝¹² : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_2\nE' : Type ?u.40995\nF : Type u_1\nF' : Type ?u.41001\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : Module 𝕜₁ E\ninst✝⁹ : AddCommGroup E'\ninst✝⁸ : Module ℝ E'\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module ℝ F'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace E'\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\n𝔖 : Set (Set E)\nthis✝² : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis✝¹ : UniformAddGroup F\nthis✝ : UniformSpace (E →SL[σ] F) := strongUniformity σ F 𝔖\nthis : UniformAddGroup (E →SL[σ] F)\n⊢ TopologicalAddGroup (E →SL[σ] F)",
"tactic": "infer_instance"
}
] |
[
128,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/CategoryTheory/Bicategory/Free.lean
|
CategoryTheory.FreeBicategory.mk_left_unitor_hom
|
[] |
[
290,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
RingHom.rangeS_top_of_surjective
|
[] |
[
1188,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1186,
1
] |
Mathlib/Analysis/Convex/Join.lean
|
convexJoin_assoc
|
[
{
"state_after": "ι : Sort ?u.66107\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx y : E\ns t u : Set E\n⊢ convexJoin 𝕜 s (convexJoin 𝕜 t u) ⊆ convexJoin 𝕜 (convexJoin 𝕜 s t) u",
"state_before": "ι : Sort ?u.66107\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx y : E\ns t u : Set E\n⊢ convexJoin 𝕜 (convexJoin 𝕜 s t) u = convexJoin 𝕜 s (convexJoin 𝕜 t u)",
"tactic": "refine' (convexJoin_assoc_aux _ _ _).antisymm _"
},
{
"state_after": "ι : Sort ?u.66107\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx y : E\ns t u : Set E\n⊢ convexJoin 𝕜 (convexJoin 𝕜 u t) s ⊆ convexJoin 𝕜 u (convexJoin 𝕜 t s)",
"state_before": "ι : Sort ?u.66107\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx y : E\ns t u : Set E\n⊢ convexJoin 𝕜 s (convexJoin 𝕜 t u) ⊆ convexJoin 𝕜 (convexJoin 𝕜 s t) u",
"tactic": "simp_rw [convexJoin_comm s, convexJoin_comm _ u]"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.66107\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx y : E\ns t u : Set E\n⊢ convexJoin 𝕜 (convexJoin 𝕜 u t) s ⊆ convexJoin 𝕜 u (convexJoin 𝕜 t s)",
"tactic": "exact convexJoin_assoc_aux _ _ _"
}
] |
[
161,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
1
] |
Mathlib/Data/Nat/Bitwise.lean
|
Nat.lor'_assoc
|
[
{
"state_after": "no goals",
"state_before": "n m k : ℕ\n⊢ lor' (lor' n m) k = lor' n (lor' m k)",
"tactic": "bitwise_assoc_tac"
}
] |
[
234,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
1
] |
Mathlib/ModelTheory/Substructures.lean
|
FirstOrder.Language.Substructure.comap_iInf_map_of_injective
|
[] |
[
585,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
583,
1
] |
Mathlib/Algebra/Group/Opposite.lean
|
MulOpposite.op_div
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : DivInvMonoid α\nx y : α\n⊢ op (x / y) = (op y)⁻¹ * op x",
"tactic": "simp [div_eq_mul_inv]"
}
] |
[
212,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Topology/FiberBundle/Basic.lean
|
FiberBundleCore.localTrivAsLocalEquiv_symm
|
[] |
[
676,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
674,
1
] |
Mathlib/Topology/Separation.lean
|
Bornology.relativelyCompact.isBounded_iff
|
[
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns : Set α\n⊢ sᶜ ∈ coclosedCompact α ↔ IsCompact (closure s)",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns : Set α\n⊢ IsBounded s ↔ IsCompact (closure s)",
"tactic": "change sᶜ ∈ Filter.coclosedCompact α ↔ _"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns : Set α\n⊢ (∃ t, IsClosed t ∧ IsCompact t ∧ tᶜ ⊆ sᶜ) ↔ IsCompact (closure s)",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns : Set α\n⊢ sᶜ ∈ coclosedCompact α ↔ IsCompact (closure s)",
"tactic": "rw [Filter.mem_coclosedCompact]"
},
{
"state_after": "case mp\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns : Set α\n⊢ (∃ t, IsClosed t ∧ IsCompact t ∧ tᶜ ⊆ sᶜ) → IsCompact (closure s)\n\ncase mpr\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns : Set α\n⊢ IsCompact (closure s) → ∃ t, IsClosed t ∧ IsCompact t ∧ tᶜ ⊆ sᶜ",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns : Set α\n⊢ (∃ t, IsClosed t ∧ IsCompact t ∧ tᶜ ⊆ sᶜ) ↔ IsCompact (closure s)",
"tactic": "constructor"
},
{
"state_after": "case mp.intro.intro.intro\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns t : Set α\nht₁ : IsClosed t\nht₂ : IsCompact t\nhst : tᶜ ⊆ sᶜ\n⊢ IsCompact (closure s)",
"state_before": "case mp\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns : Set α\n⊢ (∃ t, IsClosed t ∧ IsCompact t ∧ tᶜ ⊆ sᶜ) → IsCompact (closure s)",
"tactic": "rintro ⟨t, ht₁, ht₂, hst⟩"
},
{
"state_after": "case mp.intro.intro.intro\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns t : Set α\nht₁ : IsClosed t\nht₂ : IsCompact t\nhst : s ⊆ t\n⊢ IsCompact (closure s)",
"state_before": "case mp.intro.intro.intro\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns t : Set α\nht₁ : IsClosed t\nht₂ : IsCompact t\nhst : tᶜ ⊆ sᶜ\n⊢ IsCompact (closure s)",
"tactic": "rw [compl_subset_compl] at hst"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns t : Set α\nht₁ : IsClosed t\nht₂ : IsCompact t\nhst : s ⊆ t\n⊢ IsCompact (closure s)",
"tactic": "exact isCompact_of_isClosed_subset ht₂ isClosed_closure (closure_minimal hst ht₁)"
},
{
"state_after": "case mpr\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns : Set α\nh : IsCompact (closure s)\n⊢ ∃ t, IsClosed t ∧ IsCompact t ∧ tᶜ ⊆ sᶜ",
"state_before": "case mpr\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns : Set α\n⊢ IsCompact (closure s) → ∃ t, IsClosed t ∧ IsCompact t ∧ tᶜ ⊆ sᶜ",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mpr\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns : Set α\nh : IsCompact (closure s)\n⊢ ∃ t, IsClosed t ∧ IsCompact t ∧ tᶜ ⊆ sᶜ",
"tactic": "exact ⟨closure s, isClosed_closure, h, compl_subset_compl.mpr subset_closure⟩"
}
] |
[
451,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
442,
1
] |
Mathlib/Analysis/BoxIntegral/Basic.lean
|
BoxIntegral.integral_smul
|
[
{
"state_after": "case inl\nι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\n⊢ integral I l (fun x => 0 • f x) vol = 0 • integral I l f vol\n\ncase inr\nι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nc : ℝ\nhc : c ≠ 0\n⊢ integral I l (fun x => c • f x) vol = c • integral I l f vol",
"state_before": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nc : ℝ\n⊢ integral I l (fun x => c • f x) vol = c • integral I l f vol",
"tactic": "rcases eq_or_ne c 0 with (rfl | hc)"
},
{
"state_after": "case pos\nι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nc : ℝ\nhc : c ≠ 0\nhf : Integrable I l f vol\n⊢ integral I l (fun x => c • f x) vol = c • integral I l f vol\n\ncase neg\nι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nc : ℝ\nhc : c ≠ 0\nhf : ¬Integrable I l f vol\n⊢ integral I l (fun x => c • f x) vol = c • integral I l f vol",
"state_before": "case inr\nι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nc : ℝ\nhc : c ≠ 0\n⊢ integral I l (fun x => c • f x) vol = c • integral I l f vol",
"tactic": "by_cases hf : Integrable I l f vol"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\n⊢ integral I l (fun x => 0 • f x) vol = 0 • integral I l f vol",
"tactic": "simp only [zero_smul, integral_zero]"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nc : ℝ\nhc : c ≠ 0\nhf : Integrable I l f vol\n⊢ integral I l (fun x => c • f x) vol = c • integral I l f vol",
"tactic": "exact (hf.hasIntegral.smul c).integral_eq"
},
{
"state_after": "case neg\nι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nc : ℝ\nhc : c ≠ 0\nhf : ¬Integrable I l f vol\nthis : ¬Integrable I l (fun x => c • f x) vol\n⊢ integral I l (fun x => c • f x) vol = c • integral I l f vol",
"state_before": "case neg\nι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nc : ℝ\nhc : c ≠ 0\nhf : ¬Integrable I l f vol\n⊢ integral I l (fun x => c • f x) vol = c • integral I l f vol",
"tactic": "have : ¬Integrable I l (fun x => c • f x) vol := mt (fun h => h.of_smul hc) hf"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nc : ℝ\nhc : c ≠ 0\nhf : ¬Integrable I l f vol\nthis : ¬Integrable I l (fun x => c • f x) vol\n⊢ integral I l (fun x => c • f x) vol = c • integral I l f vol",
"tactic": "rw [integral, integral, dif_neg hf, dif_neg this, smul_zero]"
}
] |
[
371,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
366,
1
] |
Mathlib/Data/Multiset/Fintype.lean
|
Multiset.toEmbedding_coeEquiv_trans
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nm✝ m : Multiset α\n⊢ Function.Embedding.trans (Equiv.toEmbedding (coeEquiv m)) (Function.Embedding.subtype fun x => x ∈ toEnumFinset m) =\n coeEmbedding m",
"tactic": "ext <;> rfl"
}
] |
[
192,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_insert_one
|
[] |
[
341,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
340,
1
] |
Mathlib/CategoryTheory/Adjunction/Opposites.lean
|
CategoryTheory.Adjunction.leftAdjointUniq_hom_app_counit
|
[
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nx : D\n⊢ (leftAdjointUniq adj1 adj2).hom.app (G.obj x) ≫ adj2.counit.app x =\n (whiskerLeft G (leftAdjointUniq adj1 adj2).hom ≫ adj2.counit).app x",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nx : D\n⊢ (leftAdjointUniq adj1 adj2).hom.app (G.obj x) ≫ adj2.counit.app x = adj1.counit.app x",
"tactic": "rw [← leftAdjointUniq_hom_counit adj1 adj2]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nx : D\n⊢ (leftAdjointUniq adj1 adj2).hom.app (G.obj x) ≫ adj2.counit.app x =\n (whiskerLeft G (leftAdjointUniq adj1 adj2).hom ≫ adj2.counit).app x",
"tactic": "rfl"
}
] |
[
174,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
gcd_eq_right_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\na b : α\nh : ↑normalize b = b\n⊢ gcd a b = b ↔ b ∣ a",
"tactic": "simpa only [gcd_comm a b] using gcd_eq_left_iff b a h"
}
] |
[
480,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
479,
1
] |
Mathlib/Data/Vector/Basic.lean
|
Vector.toList_map
|
[
{
"state_after": "case mk\nn : ℕ\nα : Type u_2\nβ : Type u_1\nf : α → β\nval✝ : List α\nproperty✝ : List.length val✝ = n\n⊢ toList (map f { val := val✝, property := property✝ }) = List.map f (toList { val := val✝, property := property✝ })",
"state_before": "n : ℕ\nα : Type u_2\nβ : Type u_1\nv : Vector α n\nf : α → β\n⊢ toList (map f v) = List.map f (toList v)",
"tactic": "cases v"
},
{
"state_after": "no goals",
"state_before": "case mk\nn : ℕ\nα : Type u_2\nβ : Type u_1\nf : α → β\nval✝ : List α\nproperty✝ : List.length val✝ = n\n⊢ toList (map f { val := val✝, property := property✝ }) = List.map f (toList { val := val✝, property := property✝ })",
"tactic": "rfl"
}
] |
[
101,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Topology/FiberBundle/Trivialization.lean
|
Trivialization.apply_mk_symm
|
[] |
[
627,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
625,
1
] |
Mathlib/LinearAlgebra/Determinant.lean
|
LinearMap.det_zero'
|
[
{
"state_after": "R : Type ?u.871535\ninst✝¹¹ : CommRing R\nM : Type u_3\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\nM' : Type ?u.872126\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : Module R M'\nι✝ : Type ?u.872668\ninst✝⁶ : DecidableEq ι✝\ninst✝⁵ : Fintype ι✝\ne : Basis ι✝ R M\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Module A M\nκ : Type ?u.873644\ninst✝² : Fintype κ\nι : Type u_1\ninst✝¹ : Finite ι\ninst✝ : Nonempty ι\nb : Basis ι A M\nthis : DecidableEq ι\n⊢ ↑LinearMap.det 0 = 0",
"state_before": "R : Type ?u.871535\ninst✝¹¹ : CommRing R\nM : Type u_3\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\nM' : Type ?u.872126\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : Module R M'\nι✝ : Type ?u.872668\ninst✝⁶ : DecidableEq ι✝\ninst✝⁵ : Fintype ι✝\ne : Basis ι✝ R M\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Module A M\nκ : Type ?u.873644\ninst✝² : Fintype κ\nι : Type u_1\ninst✝¹ : Finite ι\ninst✝ : Nonempty ι\nb : Basis ι A M\n⊢ ↑LinearMap.det 0 = 0",
"tactic": "haveI := Classical.decEq ι"
},
{
"state_after": "case intro\nR : Type ?u.871535\ninst✝¹¹ : CommRing R\nM : Type u_3\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\nM' : Type ?u.872126\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : Module R M'\nι✝ : Type ?u.872668\ninst✝⁶ : DecidableEq ι✝\ninst✝⁵ : Fintype ι✝\ne : Basis ι✝ R M\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Module A M\nκ : Type ?u.873644\ninst✝² : Fintype κ\nι : Type u_1\ninst✝¹ : Finite ι\ninst✝ : Nonempty ι\nb : Basis ι A M\nthis : DecidableEq ι\nval✝ : Fintype ι\n⊢ ↑LinearMap.det 0 = 0",
"state_before": "R : Type ?u.871535\ninst✝¹¹ : CommRing R\nM : Type u_3\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\nM' : Type ?u.872126\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : Module R M'\nι✝ : Type ?u.872668\ninst✝⁶ : DecidableEq ι✝\ninst✝⁵ : Fintype ι✝\ne : Basis ι✝ R M\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Module A M\nκ : Type ?u.873644\ninst✝² : Fintype κ\nι : Type u_1\ninst✝¹ : Finite ι\ninst✝ : Nonempty ι\nb : Basis ι A M\nthis : DecidableEq ι\n⊢ ↑LinearMap.det 0 = 0",
"tactic": "cases nonempty_fintype ι"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type ?u.871535\ninst✝¹¹ : CommRing R\nM : Type u_3\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\nM' : Type ?u.872126\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : Module R M'\nι✝ : Type ?u.872668\ninst✝⁶ : DecidableEq ι✝\ninst✝⁵ : Fintype ι✝\ne : Basis ι✝ R M\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Module A M\nκ : Type ?u.873644\ninst✝² : Fintype κ\nι : Type u_1\ninst✝¹ : Finite ι\ninst✝ : Nonempty ι\nb : Basis ι A M\nthis : DecidableEq ι\nval✝ : Fintype ι\n⊢ ↑LinearMap.det 0 = 0",
"tactic": "rwa [← det_toMatrix b, LinearEquiv.map_zero, det_zero]"
}
] |
[
280,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
276,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
real_inner_self_eq_norm_sq
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.2330841\nE : Type ?u.2330844\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx : F\n⊢ inner x x = ‖x‖ ^ 2",
"tactic": "rw [pow_two, real_inner_self_eq_norm_mul_norm]"
}
] |
[
1023,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1022,
1
] |
Mathlib/Data/Option/Basic.lean
|
Option.map_coe
|
[] |
[
108,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.comp_equiv_dotProduct_comp_equiv
|
[
{
"state_after": "l : Type ?u.148579\nm : Type u_1\nn : Type u_2\no : Type ?u.148588\nm' : o → Type ?u.148593\nn' : o → Type ?u.148598\nR : Type ?u.148601\nS : Type ?u.148604\nα : Type v\nβ : Type w\nγ : Type ?u.148611\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : NonUnitalNonAssocSemiring α\nu v w : m → α\nx y : n → α\ne : m ≃ n\n⊢ x ⬝ᵥ (y ∘ ↑e) ∘ ↑e.symm = x ⬝ᵥ y",
"state_before": "l : Type ?u.148579\nm : Type u_1\nn : Type u_2\no : Type ?u.148588\nm' : o → Type ?u.148593\nn' : o → Type ?u.148598\nR : Type ?u.148601\nS : Type ?u.148604\nα : Type v\nβ : Type w\nγ : Type ?u.148611\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : NonUnitalNonAssocSemiring α\nu v w : m → α\nx y : n → α\ne : m ≃ n\n⊢ x ∘ ↑e ⬝ᵥ y ∘ ↑e = x ⬝ᵥ y",
"tactic": "rw [← dotProduct_comp_equiv_symm]"
},
{
"state_after": "no goals",
"state_before": "l : Type ?u.148579\nm : Type u_1\nn : Type u_2\no : Type ?u.148588\nm' : o → Type ?u.148593\nn' : o → Type ?u.148598\nR : Type ?u.148601\nS : Type ?u.148604\nα : Type v\nβ : Type w\nγ : Type ?u.148611\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : NonUnitalNonAssocSemiring α\nu v w : m → α\nx y : n → α\ne : m ≃ n\n⊢ x ⬝ᵥ (y ∘ ↑e) ∘ ↑e.symm = x ⬝ᵥ y",
"tactic": "simp only [Function.comp, Equiv.apply_symm_apply]"
}
] |
[
798,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
796,
1
] |
Mathlib/Analysis/Complex/Basic.lean
|
IsROrC.hasSum_conj'
|
[] |
[
490,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
489,
1
] |
Mathlib/Algebra/CovariantAndContravariant.lean
|
Contravariant.flip
|
[] |
[
144,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean
|
AffineSubspace.isometryEquivMap.toAffineMap_eq
|
[] |
[
919,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
917,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.blockDiag_apply
|
[] |
[
509,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
507,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.eventually_nhdsWithin'
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.34648\nδ : Type ?u.34651\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne✝ : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\ne : LocalHomeomorph α β\nx : α\np : α → Prop\ns : Set α\nhx : x ∈ e.source\n⊢ (∀ᶠ (x : α) in 𝓝[s] x, p (↑(LocalHomeomorph.symm e) (↑e x))) ↔ ∀ᶠ (x : α) in 𝓝[s] x, p x",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.34648\nδ : Type ?u.34651\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne✝ : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\ne : LocalHomeomorph α β\nx : α\np : α → Prop\ns : Set α\nhx : x ∈ e.source\n⊢ (∀ᶠ (y : β) in 𝓝[↑(LocalHomeomorph.symm e) ⁻¹' s] ↑e x, p (↑(LocalHomeomorph.symm e) y)) ↔ ∀ᶠ (x : α) in 𝓝[s] x, p x",
"tactic": "rw [e.eventually_nhdsWithin _ hx]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.34648\nδ : Type ?u.34651\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne✝ : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\ne : LocalHomeomorph α β\nx : α\np : α → Prop\ns : Set α\nhx : x ∈ e.source\ny : α\nhy : ↑(LocalHomeomorph.symm e) (↑e y) = y\n⊢ p (↑(LocalHomeomorph.symm e) (↑e y)) ↔ p y",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.34648\nδ : Type ?u.34651\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne✝ : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\ne : LocalHomeomorph α β\nx : α\np : α → Prop\ns : Set α\nhx : x ∈ e.source\n⊢ (∀ᶠ (x : α) in 𝓝[s] x, p (↑(LocalHomeomorph.symm e) (↑e x))) ↔ ∀ᶠ (x : α) in 𝓝[s] x, p x",
"tactic": "refine'\n eventually_congr\n ((eventually_nhdsWithin_of_eventually_nhds <| e.eventually_left_inverse hx).mono fun y hy =>\n _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.34648\nδ : Type ?u.34651\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne✝ : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\ne : LocalHomeomorph α β\nx : α\np : α → Prop\ns : Set α\nhx : x ∈ e.source\ny : α\nhy : ↑(LocalHomeomorph.symm e) (↑e y) = y\n⊢ p (↑(LocalHomeomorph.symm e) (↑e y)) ↔ p y",
"tactic": "rw [hy]"
}
] |
[
421,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
414,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
|
CategoryTheory.Limits.coprod.map_comp_id
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nX✝ Y✝ X Y Z W : C\nf : X ⟶ Y\ng : Y ⟶ Z\ninst✝² : HasBinaryCoproduct Z W\ninst✝¹ : HasBinaryCoproduct Y W\ninst✝ : HasBinaryCoproduct X W\n⊢ map (f ≫ g) (𝟙 W) = map f (𝟙 W) ≫ map g (𝟙 W)",
"tactic": "simp"
}
] |
[
898,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
896,
1
] |
Mathlib/RingTheory/IntegralClosure.lean
|
RingHom.is_integral_one
|
[] |
[
485,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
484,
1
] |
Mathlib/Logic/Equiv/Defs.lean
|
Equiv.eq_symm_comp
|
[] |
[
598,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
597,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearEquiv.ker_comp
|
[] |
[
2166,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2163,
1
] |
Mathlib/Algebra/Category/Ring/Instances.lean
|
IsLocalization.epi
|
[] |
[
35,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
33,
1
] |
Mathlib/Topology/Sets/Compacts.lean
|
TopologicalSpace.Compacts.ext
|
[] |
[
66,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
11
] |
Mathlib/Topology/UniformSpace/CompactConvergence.lean
|
ContinuousMap.compactConvNhd_compact_entourage_nonempty
|
[] |
[
139,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Data/Multiset/Range.lean
|
Multiset.range_subset
|
[] |
[
48,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
eventually_nhdsWithin_iff
|
[] |
[
54,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/GroupTheory/Perm/Basic.lean
|
Equiv.Perm.inv_aux
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\np : α → Prop\nf : Perm α\n⊢ (∀ (x : α), p (↑f⁻¹ x) ↔ p (↑f (↑f⁻¹ x))) ↔ ∀ (x : α), p x ↔ p (↑f⁻¹ x)",
"tactic": "simp_rw [f.apply_inv_self, Iff.comm]"
}
] |
[
386,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
385,
9
] |
Mathlib/MeasureTheory/Constructions/Prod/Integral.lean
|
MeasureTheory.integral_fn_integral_sub
|
[
{
"state_after": "α : Type u_1\nα' : Type ?u.2419590\nβ : Type u_2\nβ' : Type ?u.2419596\nγ : Type ?u.2419599\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type u_4\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nF : E → E'\nhf : Integrable f\nhg : Integrable g\n⊢ (fun x => F (∫ (y : β), f (x, y) - g (x, y) ∂ν)) =ᶠ[ae μ] fun x =>\n F ((∫ (y : β), f (x, y) ∂ν) - ∫ (y : β), g (x, y) ∂ν)",
"state_before": "α : Type u_1\nα' : Type ?u.2419590\nβ : Type u_2\nβ' : Type ?u.2419596\nγ : Type ?u.2419599\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type u_4\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nF : E → E'\nhf : Integrable f\nhg : Integrable g\n⊢ (∫ (x : α), F (∫ (y : β), f (x, y) - g (x, y) ∂ν) ∂μ) =\n ∫ (x : α), F ((∫ (y : β), f (x, y) ∂ν) - ∫ (y : β), g (x, y) ∂ν) ∂μ",
"tactic": "refine' integral_congr_ae _"
},
{
"state_after": "case h\nα : Type u_1\nα' : Type ?u.2419590\nβ : Type u_2\nβ' : Type ?u.2419596\nγ : Type ?u.2419599\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type u_4\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nF : E → E'\nhf : Integrable f\nhg : Integrable g\na✝ : α\nh2f : Integrable fun y => f (a✝, y)\nh2g : Integrable fun y => g (a✝, y)\n⊢ F (∫ (y : β), f (a✝, y) - g (a✝, y) ∂ν) = F ((∫ (y : β), f (a✝, y) ∂ν) - ∫ (y : β), g (a✝, y) ∂ν)",
"state_before": "α : Type u_1\nα' : Type ?u.2419590\nβ : Type u_2\nβ' : Type ?u.2419596\nγ : Type ?u.2419599\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type u_4\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nF : E → E'\nhf : Integrable f\nhg : Integrable g\n⊢ (fun x => F (∫ (y : β), f (x, y) - g (x, y) ∂ν)) =ᶠ[ae μ] fun x =>\n F ((∫ (y : β), f (x, y) ∂ν) - ∫ (y : β), g (x, y) ∂ν)",
"tactic": "filter_upwards [hf.prod_right_ae, hg.prod_right_ae] with _ h2f h2g"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nα' : Type ?u.2419590\nβ : Type u_2\nβ' : Type ?u.2419596\nγ : Type ?u.2419599\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type u_4\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nF : E → E'\nhf : Integrable f\nhg : Integrable g\na✝ : α\nh2f : Integrable fun y => f (a✝, y)\nh2g : Integrable fun y => g (a✝, y)\n⊢ F (∫ (y : β), f (a✝, y) - g (a✝, y) ∂ν) = F ((∫ (y : β), f (a✝, y) ∂ν) - ∫ (y : β), g (a✝, y) ∂ν)",
"tactic": "simp [integral_sub h2f h2g]"
}
] |
[
375,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
369,
1
] |
Mathlib/Data/List/Rotate.lean
|
List.Nodup.cyclicPermutations
|
[
{
"state_after": "case nil\nα : Type u\nl l' : List α\nhn : Nodup []\n⊢ Nodup (List.cyclicPermutations [])\n\ncase cons\nα : Type u\nl✝ l' : List α\nx : α\nl : List α\nhn : Nodup (x :: l)\n⊢ Nodup (List.cyclicPermutations (x :: l))",
"state_before": "α : Type u\nl✝ l' l : List α\nhn : Nodup l\n⊢ Nodup (List.cyclicPermutations l)",
"tactic": "cases' l with x l"
},
{
"state_after": "case cons\nα : Type u\nl✝ l' : List α\nx : α\nl : List α\nhn : Nodup (x :: l)\n⊢ ∀ (i j : ℕ) (h₁ : i < length (List.cyclicPermutations (x :: l))) (h₂ : j < length (List.cyclicPermutations (x :: l))),\n nthLe (List.cyclicPermutations (x :: l)) i h₁ = nthLe (List.cyclicPermutations (x :: l)) j h₂ → i = j",
"state_before": "case cons\nα : Type u\nl✝ l' : List α\nx : α\nl : List α\nhn : Nodup (x :: l)\n⊢ Nodup (List.cyclicPermutations (x :: l))",
"tactic": "rw [nodup_iff_nthLe_inj]"
},
{
"state_after": "case cons\nα : Type u\nl✝ l' : List α\nx : α\nl : List α\nhn : Nodup (x :: l)\ni j : ℕ\nhi : i < length (List.cyclicPermutations (x :: l))\nhj : j < length (List.cyclicPermutations (x :: l))\nh : nthLe (List.cyclicPermutations (x :: l)) i hi = nthLe (List.cyclicPermutations (x :: l)) j hj\n⊢ i = j",
"state_before": "case cons\nα : Type u\nl✝ l' : List α\nx : α\nl : List α\nhn : Nodup (x :: l)\n⊢ ∀ (i j : ℕ) (h₁ : i < length (List.cyclicPermutations (x :: l))) (h₂ : j < length (List.cyclicPermutations (x :: l))),\n nthLe (List.cyclicPermutations (x :: l)) i h₁ = nthLe (List.cyclicPermutations (x :: l)) j h₂ → i = j",
"tactic": "intro i j hi hj h"
},
{
"state_after": "case cons\nα : Type u\nl✝ l' : List α\nx : α\nl : List α\nhn : Nodup (x :: l)\ni j : ℕ\nhi✝ : i < length (List.cyclicPermutations (x :: l))\nhj✝ : j < length (List.cyclicPermutations (x :: l))\nh : nthLe (List.cyclicPermutations (x :: l)) i hi✝ = nthLe (List.cyclicPermutations (x :: l)) j hj✝\nhi : i < length l + 1\nhj : j < length l + 1\n⊢ i = j",
"state_before": "case cons\nα : Type u\nl✝ l' : List α\nx : α\nl : List α\nhn : Nodup (x :: l)\ni j : ℕ\nhi : i < length (List.cyclicPermutations (x :: l))\nhj : j < length (List.cyclicPermutations (x :: l))\nh : nthLe (List.cyclicPermutations (x :: l)) i hi = nthLe (List.cyclicPermutations (x :: l)) j hj\n⊢ i = j",
"tactic": "simp only [length_cyclicPermutations_cons] at hi hj"
},
{
"state_after": "case cons\nα : Type u\nl✝ l' : List α\nx : α\nl : List α\nhn : Nodup (x :: l)\ni j : ℕ\nhi✝ : i < length (List.cyclicPermutations (x :: l))\nhj✝ : j < length (List.cyclicPermutations (x :: l))\nh : nthLe (List.cyclicPermutations (x :: l)) i hi✝ = nthLe (List.cyclicPermutations (x :: l)) j hj✝\nhi : i < length l + 1\nhj : j < length l + 1\n⊢ i % (length l + 1) = j % (length l + 1)",
"state_before": "case cons\nα : Type u\nl✝ l' : List α\nx : α\nl : List α\nhn : Nodup (x :: l)\ni j : ℕ\nhi✝ : i < length (List.cyclicPermutations (x :: l))\nhj✝ : j < length (List.cyclicPermutations (x :: l))\nh : nthLe (List.cyclicPermutations (x :: l)) i hi✝ = nthLe (List.cyclicPermutations (x :: l)) j hj✝\nhi : i < length l + 1\nhj : j < length l + 1\n⊢ i = j",
"tactic": "rw [← mod_eq_of_lt hi, ← mod_eq_of_lt hj]"
},
{
"state_after": "case cons.hn\nα : Type u\nl✝ l' : List α\nx : α\nl : List α\nhn : Nodup (x :: l)\ni j : ℕ\nhi✝ : i < length (List.cyclicPermutations (x :: l))\nhj✝ : j < length (List.cyclicPermutations (x :: l))\nh : nthLe (List.cyclicPermutations (x :: l)) i hi✝ = nthLe (List.cyclicPermutations (x :: l)) j hj✝\nhi : i < length l + 1\nhj : j < length l + 1\n⊢ x :: l ≠ []\n\ncase cons.h\nα : Type u\nl✝ l' : List α\nx : α\nl : List α\nhn : Nodup (x :: l)\ni j : ℕ\nhi✝ : i < length (List.cyclicPermutations (x :: l))\nhj✝ : j < length (List.cyclicPermutations (x :: l))\nh : nthLe (List.cyclicPermutations (x :: l)) i hi✝ = nthLe (List.cyclicPermutations (x :: l)) j hj✝\nhi : i < length l + 1\nhj : j < length l + 1\n⊢ rotate (x :: l) i = rotate (x :: l) j",
"state_before": "case cons\nα : Type u\nl✝ l' : List α\nx : α\nl : List α\nhn : Nodup (x :: l)\ni j : ℕ\nhi✝ : i < length (List.cyclicPermutations (x :: l))\nhj✝ : j < length (List.cyclicPermutations (x :: l))\nh : nthLe (List.cyclicPermutations (x :: l)) i hi✝ = nthLe (List.cyclicPermutations (x :: l)) j hj✝\nhi : i < length l + 1\nhj : j < length l + 1\n⊢ i % (length l + 1) = j % (length l + 1)",
"tactic": "apply hn.rotate_congr"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u\nl l' : List α\nhn : Nodup []\n⊢ Nodup (List.cyclicPermutations [])",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons.hn\nα : Type u\nl✝ l' : List α\nx : α\nl : List α\nhn : Nodup (x :: l)\ni j : ℕ\nhi✝ : i < length (List.cyclicPermutations (x :: l))\nhj✝ : j < length (List.cyclicPermutations (x :: l))\nh : nthLe (List.cyclicPermutations (x :: l)) i hi✝ = nthLe (List.cyclicPermutations (x :: l)) j hj✝\nhi : i < length l + 1\nhj : j < length l + 1\n⊢ x :: l ≠ []",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons.h\nα : Type u\nl✝ l' : List α\nx : α\nl : List α\nhn : Nodup (x :: l)\ni j : ℕ\nhi✝ : i < length (List.cyclicPermutations (x :: l))\nhj✝ : j < length (List.cyclicPermutations (x :: l))\nh : nthLe (List.cyclicPermutations (x :: l)) i hi✝ = nthLe (List.cyclicPermutations (x :: l)) j hj✝\nhi : i < length l + 1\nhj : j < length l + 1\n⊢ rotate (x :: l) i = rotate (x :: l) j",
"tactic": "simpa using h"
}
] |
[
667,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
658,
1
] |
Mathlib/Data/List/Basic.lean
|
List.set_of_mem_cons
|
[] |
[
79,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
coe_setBasisOfLinearIndependentOfCardEqFinrank
|
[
{
"state_after": "K : Type u\nV : Type v\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ns : Set V\ninst✝¹ : Nonempty ↑s\ninst✝ : Fintype ↑s\nlin_ind : LinearIndependent K Subtype.val\ncard_eq : Finset.card (Set.toFinset s) = finrank K V\n⊢ ↑(basisOfLinearIndependentOfCardEqFinrank lin_ind (_ : Fintype.card ↑s = finrank K V)) = Subtype.val",
"state_before": "K : Type u\nV : Type v\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ns : Set V\ninst✝¹ : Nonempty ↑s\ninst✝ : Fintype ↑s\nlin_ind : LinearIndependent K Subtype.val\ncard_eq : Finset.card (Set.toFinset s) = finrank K V\n⊢ ↑(setBasisOfLinearIndependentOfCardEqFinrank lin_ind card_eq) = Subtype.val",
"tactic": "rw [setBasisOfLinearIndependentOfCardEqFinrank]"
},
{
"state_after": "no goals",
"state_before": "K : Type u\nV : Type v\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ns : Set V\ninst✝¹ : Nonempty ↑s\ninst✝ : Fintype ↑s\nlin_ind : LinearIndependent K Subtype.val\ncard_eq : Finset.card (Set.toFinset s) = finrank K V\n⊢ ↑(basisOfLinearIndependentOfCardEqFinrank lin_ind (_ : Fintype.card ↑s = finrank K V)) = Subtype.val",
"tactic": "exact Basis.coe_mk _ _"
}
] |
[
1234,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1229,
1
] |
Mathlib/LinearAlgebra/Alternating.lean
|
AlternatingMap.coe_zero
|
[] |
[
358,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
357,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
|
InnerProductGeometry.norm_sub_sq_eq_norm_sq_add_norm_sq'
|
[] |
[
72,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.univ_finite_iff_nonempty_fintype
|
[] |
[
1049,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1048,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.boundedBy_apply
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nm : Set α → ℝ≥0∞\ns : Set α\n⊢ ↑(boundedBy m) s = ⨅ (t : ℕ → Set α) (_ : s ⊆ iUnion t), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), m (t n)",
"tactic": "simp [boundedBy, ofFunction_apply]"
}
] |
[
860,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
857,
1
] |
Mathlib/Algebra/Module/LocalizedModule.lean
|
LocalizedModule.smul'_mk
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\ns : { x // x ∈ S }\nm : M\n⊢ r • mk m s = mk (r • m) s",
"tactic": "erw [mk_smul_mk r m 1 s, one_mul]"
}
] |
[
420,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
419,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Submonoid.LocalizationMap.ofMulEquivOfDom_comp_symm
|
[] |
[
1528,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1526,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.singleton_smul
|
[] |
[
1645,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1644,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.dist_le
|
[] |
[
184,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/CategoryTheory/MorphismProperty.lean
|
CategoryTheory.MorphismProperty.StableUnderComposition.epimorphisms
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\nD : Type ?u.64328\ninst✝ : Category D\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : Epi f\nhg : Epi g\n⊢ Epi (f ≫ g)",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type ?u.64328\ninst✝ : Category D\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : MorphismProperty.epimorphisms C f\nhg : MorphismProperty.epimorphisms C g\n⊢ MorphismProperty.epimorphisms C (f ≫ g)",
"tactic": "rw [epimorphisms.iff] at hf hg⊢"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\nD : Type ?u.64328\ninst✝ : Category D\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : Epi f\nhg : Epi g\nthis : Epi f\n⊢ Epi (f ≫ g)",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type ?u.64328\ninst✝ : Category D\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : Epi f\nhg : Epi g\n⊢ Epi (f ≫ g)",
"tactic": "haveI := hf"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\nD : Type ?u.64328\ninst✝ : Category D\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : Epi f\nhg : Epi g\nthis✝ : Epi f\nthis : Epi g\n⊢ Epi (f ≫ g)",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type ?u.64328\ninst✝ : Category D\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : Epi f\nhg : Epi g\nthis : Epi f\n⊢ Epi (f ≫ g)",
"tactic": "haveI := hg"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type ?u.64328\ninst✝ : Category D\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : Epi f\nhg : Epi g\nthis✝ : Epi f\nthis : Epi g\n⊢ Epi (f ≫ g)",
"tactic": "apply epi_comp"
}
] |
[
474,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
469,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
LinearMap.toMatrix_basis_equiv
|
[
{
"state_after": "case a.h\nR : Type u_2\ninst✝¹¹ : CommSemiring R\nl : Type u_1\nm : Type ?u.2073940\nn : Type ?u.2073943\ninst✝¹⁰ : Fintype n\ninst✝⁹ : Fintype m\ninst✝⁸ : DecidableEq n\nM₁ : Type u_3\nM₂ : Type u_4\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : AddCommMonoid M₂\ninst✝⁵ : Module R M₁\ninst✝⁴ : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\nM₃ : Type ?u.2074466\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M₃\nv₃ : Basis l R M₃\ninst✝¹ : Fintype l\ninst✝ : DecidableEq l\nb : Basis l R M₁\nb' : Basis l R M₂\ni j : l\n⊢ ↑(toMatrix b' b) (↑(Basis.equiv b' b (Equiv.refl l))) i j = OfNat.ofNat 1 i j",
"state_before": "R : Type u_2\ninst✝¹¹ : CommSemiring R\nl : Type u_1\nm : Type ?u.2073940\nn : Type ?u.2073943\ninst✝¹⁰ : Fintype n\ninst✝⁹ : Fintype m\ninst✝⁸ : DecidableEq n\nM₁ : Type u_3\nM₂ : Type u_4\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : AddCommMonoid M₂\ninst✝⁵ : Module R M₁\ninst✝⁴ : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\nM₃ : Type ?u.2074466\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M₃\nv₃ : Basis l R M₃\ninst✝¹ : Fintype l\ninst✝ : DecidableEq l\nb : Basis l R M₁\nb' : Basis l R M₂\n⊢ ↑(toMatrix b' b) ↑(Basis.equiv b' b (Equiv.refl l)) = 1",
"tactic": "ext (i j)"
},
{
"state_after": "no goals",
"state_before": "case a.h\nR : Type u_2\ninst✝¹¹ : CommSemiring R\nl : Type u_1\nm : Type ?u.2073940\nn : Type ?u.2073943\ninst✝¹⁰ : Fintype n\ninst✝⁹ : Fintype m\ninst✝⁸ : DecidableEq n\nM₁ : Type u_3\nM₂ : Type u_4\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : AddCommMonoid M₂\ninst✝⁵ : Module R M₁\ninst✝⁴ : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\nM₃ : Type ?u.2074466\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M₃\nv₃ : Basis l R M₃\ninst✝¹ : Fintype l\ninst✝ : DecidableEq l\nb : Basis l R M₁\nb' : Basis l R M₂\ni j : l\n⊢ ↑(toMatrix b' b) (↑(Basis.equiv b' b (Equiv.refl l))) i j = OfNat.ofNat 1 i j",
"tactic": "simp [LinearMap.toMatrix_apply, Matrix.one_apply, Finsupp.single_apply, eq_comm]"
}
] |
[
678,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
674,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.sUnion_pair
|
[] |
[
1159,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1158,
1
] |
Mathlib/Data/Finmap.lean
|
Finmap.mem_replace
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\na a' : α\nb : β a\ns✝ : Finmap β\ns : AList β\n⊢ a' ∈ replace a b ⟦s⟧ ↔ a' ∈ ⟦s⟧",
"tactic": "simp"
}
] |
[
385,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
384,
1
] |
Mathlib/Data/List/Basic.lean
|
List.zipWith_nil
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.144456\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → β → γ\nl : List α\n⊢ zipWith f l [] = []",
"tactic": "cases l <;> rfl"
}
] |
[
1900,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1900,
1
] |
Mathlib/MeasureTheory/Function/UniformIntegrable.lean
|
MeasureTheory.UnifIntegrable.ae_eq
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhfg : ∀ (n : ι), f n =ᵐ[μ] g n\nε : ℝ\nhε : 0 < ε\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal ε",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhfg : ∀ (n : ι), f n =ᵐ[μ] g n\n⊢ UnifIntegrable g p μ",
"tactic": "intro ε hε"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhfg : ∀ (n : ι), f n =ᵐ[μ] g n\nε : ℝ\nhε : 0 < ε\nδ : ℝ\nhδ_pos : 0 < δ\nhfδ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal ε\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal ε",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhfg : ∀ (n : ι), f n =ᵐ[μ] g n\nε : ℝ\nhε : 0 < ε\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal ε",
"tactic": "obtain ⟨δ, hδ_pos, hfδ⟩ := hf hε"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhfg : ∀ (n : ι), f n =ᵐ[μ] g n\nε : ℝ\nhε : 0 < ε\nδ : ℝ\nhδ_pos : 0 < δ\nhfδ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal ε\nn : ι\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≤ ENNReal.ofReal δ\n⊢ Set.indicator s (g n) =ᵐ[μ] Set.indicator s (f n)",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhfg : ∀ (n : ι), f n =ᵐ[μ] g n\nε : ℝ\nhε : 0 < ε\nδ : ℝ\nhδ_pos : 0 < δ\nhfδ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal ε\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal ε",
"tactic": "refine' ⟨δ, hδ_pos, fun n s hs hμs => (le_of_eq <| snorm_congr_ae _).trans (hfδ n s hs hμs)⟩"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhfg : ∀ (n : ι), f n =ᵐ[μ] g n\nε : ℝ\nhε : 0 < ε\nδ : ℝ\nhδ_pos : 0 < δ\nhfδ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal ε\nn : ι\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≤ ENNReal.ofReal δ\nx : α\nhx : f n x = g n x\n⊢ Set.indicator s (g n) x = Set.indicator s (f n) x",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhfg : ∀ (n : ι), f n =ᵐ[μ] g n\nε : ℝ\nhε : 0 < ε\nδ : ℝ\nhδ_pos : 0 < δ\nhfδ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal ε\nn : ι\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≤ ENNReal.ofReal δ\n⊢ Set.indicator s (g n) =ᵐ[μ] Set.indicator s (f n)",
"tactic": "filter_upwards [hfg n] with x hx"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhfg : ∀ (n : ι), f n =ᵐ[μ] g n\nε : ℝ\nhε : 0 < ε\nδ : ℝ\nhδ_pos : 0 < δ\nhfδ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal ε\nn : ι\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≤ ENNReal.ofReal δ\nx : α\nhx : f n x = g n x\n⊢ Set.indicator s (g n) x = Set.indicator s (f n) x",
"tactic": "simp_rw [Set.indicator_apply, hx]"
}
] |
[
146,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
11
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
HasStrictFDerivAt.sub
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.441700\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.441795\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nhf : HasStrictFDerivAt f f' x\nhg : HasStrictFDerivAt g g' x\n⊢ HasStrictFDerivAt (fun x => f x - g x) (f' - g') x",
"tactic": "simpa only [sub_eq_add_neg] using hf.add hg.neg"
}
] |
[
478,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
476,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.add_nonpos
|
[] |
[
832,
40
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
831,
11
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.vsub_singleton
|
[] |
[
1505,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1504,
1
] |
Mathlib/Topology/MetricSpace/Infsep.lean
|
Set.infsep_le_dist_of_mem
|
[] |
[
426,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
425,
1
] |
Mathlib/ModelTheory/Substructures.lean
|
FirstOrder.Language.Embedding.domRestrict_apply
|
[] |
[
912,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
911,
1
] |
Mathlib/Order/LocallyFinite.lean
|
Multiset.mem_Ioo
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.33022\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\na b x : α\n⊢ x ∈ Ioo a b ↔ a < x ∧ x < b",
"tactic": "rw [Ioo, ← Finset.mem_def, Finset.mem_Ioo]"
}
] |
[
568,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
567,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.le_iff_nonempty_imp_le_and_iUnion_subset
|
[
{
"state_after": "case mp\nι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\n⊢ π₁ ≤ π₂ →\n (∀ (J : Box ι), J ∈ π₁ → ∀ (J' : Box ι), J' ∈ π₂ → Set.Nonempty (↑J ∩ ↑J') → J ≤ J') ∧\n Prepartition.iUnion π₁ ⊆ Prepartition.iUnion π₂\n\ncase mpr\nι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\n⊢ (∀ (J : Box ι), J ∈ π₁ → ∀ (J' : Box ι), J' ∈ π₂ → Set.Nonempty (↑J ∩ ↑J') → J ≤ J') ∧\n Prepartition.iUnion π₁ ⊆ Prepartition.iUnion π₂ →\n π₁ ≤ π₂",
"state_before": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\n⊢ π₁ ≤ π₂ ↔\n (∀ (J : Box ι), J ∈ π₁ → ∀ (J' : Box ι), J' ∈ π₂ → Set.Nonempty (↑J ∩ ↑J') → J ≤ J') ∧\n Prepartition.iUnion π₁ ⊆ Prepartition.iUnion π₂",
"tactic": "constructor"
},
{
"state_after": "case mp\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nH : π₁ ≤ π₂\nJ : Box ι\nhJ : J ∈ π₁\nJ' : Box ι\nhJ' : J' ∈ π₂\nHne : Set.Nonempty (↑J ∩ ↑J')\n⊢ J ≤ J'",
"state_before": "case mp\nι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\n⊢ π₁ ≤ π₂ →\n (∀ (J : Box ι), J ∈ π₁ → ∀ (J' : Box ι), J' ∈ π₂ → Set.Nonempty (↑J ∩ ↑J') → J ≤ J') ∧\n Prepartition.iUnion π₁ ⊆ Prepartition.iUnion π₂",
"tactic": "refine' fun H => ⟨fun J hJ J' hJ' Hne => _, iUnion_mono H⟩"
},
{
"state_after": "case mp.intro.intro\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nH : π₁ ≤ π₂\nJ : Box ι\nhJ : J ∈ π₁\nJ' : Box ι\nhJ' : J' ∈ π₂\nHne : Set.Nonempty (↑J ∩ ↑J')\nJ'' : Box ι\nhJ'' : J'' ∈ π₂\nHle : J ≤ J''\n⊢ J ≤ J'",
"state_before": "case mp\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nH : π₁ ≤ π₂\nJ : Box ι\nhJ : J ∈ π₁\nJ' : Box ι\nhJ' : J' ∈ π₂\nHne : Set.Nonempty (↑J ∩ ↑J')\n⊢ J ≤ J'",
"tactic": "rcases H hJ with ⟨J'', hJ'', Hle⟩"
},
{
"state_after": "case mp.intro.intro.intro.intro\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx✝ : ι → ℝ\nH : π₁ ≤ π₂\nJ : Box ι\nhJ : J ∈ π₁\nJ' : Box ι\nhJ' : J' ∈ π₂\nJ'' : Box ι\nhJ'' : J'' ∈ π₂\nHle : J ≤ J''\nx : ι → ℝ\nhx : x ∈ ↑J\nhx' : x ∈ ↑J'\n⊢ J ≤ J'",
"state_before": "case mp.intro.intro\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nH : π₁ ≤ π₂\nJ : Box ι\nhJ : J ∈ π₁\nJ' : Box ι\nhJ' : J' ∈ π₂\nHne : Set.Nonempty (↑J ∩ ↑J')\nJ'' : Box ι\nhJ'' : J'' ∈ π₂\nHle : J ≤ J''\n⊢ J ≤ J'",
"tactic": "rcases Hne with ⟨x, hx, hx'⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx✝ : ι → ℝ\nH : π₁ ≤ π₂\nJ : Box ι\nhJ : J ∈ π₁\nJ' : Box ι\nhJ' : J' ∈ π₂\nJ'' : Box ι\nhJ'' : J'' ∈ π₂\nHle : J ≤ J''\nx : ι → ℝ\nhx : x ∈ ↑J\nhx' : x ∈ ↑J'\n⊢ J ≤ J'",
"tactic": "rwa [π₂.eq_of_mem_of_mem hJ' hJ'' hx' (Hle hx)]"
},
{
"state_after": "case mpr.intro\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nH : ∀ (J : Box ι), J ∈ π₁ → ∀ (J' : Box ι), J' ∈ π₂ → Set.Nonempty (↑J ∩ ↑J') → J ≤ J'\nHU : Prepartition.iUnion π₁ ⊆ Prepartition.iUnion π₂\nJ : Box ι\nhJ : J ∈ π₁\n⊢ ∃ I', I' ∈ π₂ ∧ J ≤ I'",
"state_before": "case mpr\nι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\n⊢ (∀ (J : Box ι), J ∈ π₁ → ∀ (J' : Box ι), J' ∈ π₂ → Set.Nonempty (↑J ∩ ↑J') → J ≤ J') ∧\n Prepartition.iUnion π₁ ⊆ Prepartition.iUnion π₂ →\n π₁ ≤ π₂",
"tactic": "rintro ⟨H, HU⟩ J hJ"
},
{
"state_after": "case mpr.intro\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nH : ∀ (J : Box ι), J ∈ π₁ → ∀ (J' : Box ι), J' ∈ π₂ → Set.Nonempty (↑J ∩ ↑J') → J ≤ J'\nJ : Box ι\nhJ : J ∈ π₁\nHU : ∀ (x : ι → ℝ), (∃ J, J ∈ π₁ ∧ x ∈ J) → ∃ J, J ∈ π₂ ∧ x ∈ J\n⊢ ∃ I', I' ∈ π₂ ∧ J ≤ I'",
"state_before": "case mpr.intro\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nH : ∀ (J : Box ι), J ∈ π₁ → ∀ (J' : Box ι), J' ∈ π₂ → Set.Nonempty (↑J ∩ ↑J') → J ≤ J'\nHU : Prepartition.iUnion π₁ ⊆ Prepartition.iUnion π₂\nJ : Box ι\nhJ : J ∈ π₁\n⊢ ∃ I', I' ∈ π₂ ∧ J ≤ I'",
"tactic": "simp only [Set.subset_def, mem_iUnion] at HU"
},
{
"state_after": "case mpr.intro.intro.intro\nι : Type u_1\nI J✝ J₁ J₂✝ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nH : ∀ (J : Box ι), J ∈ π₁ → ∀ (J' : Box ι), J' ∈ π₂ → Set.Nonempty (↑J ∩ ↑J') → J ≤ J'\nJ : Box ι\nhJ : J ∈ π₁\nHU : ∀ (x : ι → ℝ), (∃ J, J ∈ π₁ ∧ x ∈ J) → ∃ J, J ∈ π₂ ∧ x ∈ J\nJ₂ : Box ι\nhJ₂ : J₂ ∈ π₂\nhx : J.upper ∈ J₂\n⊢ ∃ I', I' ∈ π₂ ∧ J ≤ I'",
"state_before": "case mpr.intro\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nH : ∀ (J : Box ι), J ∈ π₁ → ∀ (J' : Box ι), J' ∈ π₂ → Set.Nonempty (↑J ∩ ↑J') → J ≤ J'\nJ : Box ι\nhJ : J ∈ π₁\nHU : ∀ (x : ι → ℝ), (∃ J, J ∈ π₁ ∧ x ∈ J) → ∃ J, J ∈ π₂ ∧ x ∈ J\n⊢ ∃ I', I' ∈ π₂ ∧ J ≤ I'",
"tactic": "rcases HU J.upper ⟨J, hJ, J.upper_mem⟩ with ⟨J₂, hJ₂, hx⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro\nι : Type u_1\nI J✝ J₁ J₂✝ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nH : ∀ (J : Box ι), J ∈ π₁ → ∀ (J' : Box ι), J' ∈ π₂ → Set.Nonempty (↑J ∩ ↑J') → J ≤ J'\nJ : Box ι\nhJ : J ∈ π₁\nHU : ∀ (x : ι → ℝ), (∃ J, J ∈ π₁ ∧ x ∈ J) → ∃ J, J ∈ π₂ ∧ x ∈ J\nJ₂ : Box ι\nhJ₂ : J₂ ∈ π₂\nhx : J.upper ∈ J₂\n⊢ ∃ I', I' ∈ π₂ ∧ J ≤ I'",
"tactic": "exact ⟨J₂, hJ₂, H _ hJ _ hJ₂ ⟨_, J.upper_mem, hx⟩⟩"
}
] |
[
279,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
268,
1
] |
Mathlib/Algebra/Hom/Equiv/Basic.lean
|
MulEquiv.bijective
|
[] |
[
238,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
11
] |
Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean
|
TopCat.inducing_pullback_to_prod
|
[
{
"state_after": "no goals",
"state_before": "J : Type v\ninst✝ : SmallCategory J\nX✝ Y✝ Z✝ X Y Z : TopCat\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ topologicalSpace_forget (pullback f g) =\n induced ((forget TopCat).map (prod.lift pullback.fst pullback.snd)) (topologicalSpace_forget (X ⨯ Y))",
"tactic": "simp [prod_topology, pullback_topology, induced_compose, ← coe_comp]"
}
] |
[
180,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.toReal_eq_one_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.24050\nβ : Type ?u.24053\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx : ℝ≥0∞\n⊢ ENNReal.toReal x = 1 ↔ x = 1",
"tactic": "rw [ENNReal.toReal, NNReal.coe_eq_one, ENNReal.toNNReal_eq_one_iff]"
}
] |
[
280,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
279,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
|
Real.strictAntiOn_log
|
[
{
"state_after": "x✝ y✝ x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x < y\n⊢ log y < log x",
"state_before": "x y : ℝ\n⊢ StrictAntiOn log (Iio 0)",
"tactic": "rintro x (hx : x < 0) y (hy : y < 0) hxy"
},
{
"state_after": "x✝ y✝ x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x < y\n⊢ log (abs y) < log (abs x)",
"state_before": "x✝ y✝ x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x < y\n⊢ log y < log x",
"tactic": "rw [← log_abs y, ← log_abs x]"
},
{
"state_after": "x✝ y✝ x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x < y\n⊢ abs y < abs x",
"state_before": "x✝ y✝ x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x < y\n⊢ log (abs y) < log (abs x)",
"tactic": "refine' log_lt_log (abs_pos.2 hy.ne) _"
},
{
"state_after": "no goals",
"state_before": "x✝ y✝ x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x < y\n⊢ abs y < abs x",
"tactic": "rwa [abs_of_neg hy, abs_of_neg hx, neg_lt_neg_iff]"
}
] |
[
215,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
1
] |
Mathlib/Data/List/Rdrop.lean
|
List.rtakeWhile_idempotent
|
[] |
[
249,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
248,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.inr_comp_pushoutComparison
|
[] |
[
1487,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1485,
1
] |
Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean
|
IsBoundedLinearMap.fst
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.27451\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nx : E × F\n⊢ ‖↑(LinearMap.fst 𝕜 E F) x‖ ≤ 1 * ‖x‖",
"state_before": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.27451\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\n⊢ IsBoundedLinearMap 𝕜 fun x => x.fst",
"tactic": "refine' (LinearMap.fst 𝕜 E F).isLinear.with_bound 1 fun x => _"
},
{
"state_after": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.27451\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nx : E × F\n⊢ ‖↑(LinearMap.fst 𝕜 E F) x‖ ≤ ‖x‖",
"state_before": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.27451\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nx : E × F\n⊢ ‖↑(LinearMap.fst 𝕜 E F) x‖ ≤ 1 * ‖x‖",
"tactic": "rw [one_mul]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.27451\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nx : E × F\n⊢ ‖↑(LinearMap.fst 𝕜 E F) x‖ ≤ ‖x‖",
"tactic": "exact le_max_left _ _"
}
] |
[
118,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/MeasureTheory/Function/EssSup.lean
|
essSup_congr_ae
|
[] |
[
58,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/GroupTheory/Submonoid/Membership.lean
|
IsScalarTower.of_mclosure_eq_top
|
[
{
"state_after": "case refine'_1\nM : Type u_3\nA : Type ?u.235426\nB : Type ?u.235429\nN : Type u_1\nα : Type u_2\ninst✝³ : Monoid M\ninst✝² : MulAction M N\ninst✝¹ : SMul N α\ninst✝ : MulAction M α\ns : Set M\nhtop : Submonoid.closure s = ⊤\nhs : ∀ (x : M), x ∈ s → ∀ (y : N) (z : α), (x • y) • z = x • y • z\nx : M\n⊢ ∀ (y : N) (z : α), (1 • y) • z = 1 • y • z\n\ncase refine'_2\nM : Type u_3\nA : Type ?u.235426\nB : Type ?u.235429\nN : Type u_1\nα : Type u_2\ninst✝³ : Monoid M\ninst✝² : MulAction M N\ninst✝¹ : SMul N α\ninst✝ : MulAction M α\ns : Set M\nhtop : Submonoid.closure s = ⊤\nhs : ∀ (x : M), x ∈ s → ∀ (y : N) (z : α), (x • y) • z = x • y • z\nx : M\n⊢ ∀ (x : M),\n x ∈ s →\n ∀ (y : M),\n (∀ (y_1 : N) (z : α), (y • y_1) • z = y • y_1 • z) →\n ∀ (y_1 : N) (z : α), ((x * y) • y_1) • z = (x * y) • y_1 • z",
"state_before": "M : Type u_3\nA : Type ?u.235426\nB : Type ?u.235429\nN : Type u_1\nα : Type u_2\ninst✝³ : Monoid M\ninst✝² : MulAction M N\ninst✝¹ : SMul N α\ninst✝ : MulAction M α\ns : Set M\nhtop : Submonoid.closure s = ⊤\nhs : ∀ (x : M), x ∈ s → ∀ (y : N) (z : α), (x • y) • z = x • y • z\n⊢ IsScalarTower M N α",
"tactic": "refine' ⟨fun x => Submonoid.induction_of_closure_eq_top_left htop x _ _⟩"
},
{
"state_after": "case refine'_1\nM : Type u_3\nA : Type ?u.235426\nB : Type ?u.235429\nN : Type u_1\nα : Type u_2\ninst✝³ : Monoid M\ninst✝² : MulAction M N\ninst✝¹ : SMul N α\ninst✝ : MulAction M α\ns : Set M\nhtop : Submonoid.closure s = ⊤\nhs : ∀ (x : M), x ∈ s → ∀ (y : N) (z : α), (x • y) • z = x • y • z\nx : M\ny : N\nz : α\n⊢ (1 • y) • z = 1 • y • z",
"state_before": "case refine'_1\nM : Type u_3\nA : Type ?u.235426\nB : Type ?u.235429\nN : Type u_1\nα : Type u_2\ninst✝³ : Monoid M\ninst✝² : MulAction M N\ninst✝¹ : SMul N α\ninst✝ : MulAction M α\ns : Set M\nhtop : Submonoid.closure s = ⊤\nhs : ∀ (x : M), x ∈ s → ∀ (y : N) (z : α), (x • y) • z = x • y • z\nx : M\n⊢ ∀ (y : N) (z : α), (1 • y) • z = 1 • y • z",
"tactic": "intro y z"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nM : Type u_3\nA : Type ?u.235426\nB : Type ?u.235429\nN : Type u_1\nα : Type u_2\ninst✝³ : Monoid M\ninst✝² : MulAction M N\ninst✝¹ : SMul N α\ninst✝ : MulAction M α\ns : Set M\nhtop : Submonoid.closure s = ⊤\nhs : ∀ (x : M), x ∈ s → ∀ (y : N) (z : α), (x • y) • z = x • y • z\nx : M\ny : N\nz : α\n⊢ (1 • y) • z = 1 • y • z",
"tactic": "rw [one_smul, one_smul]"
},
{
"state_after": "case refine'_2\nM : Type u_3\nA : Type ?u.235426\nB : Type ?u.235429\nN : Type u_1\nα : Type u_2\ninst✝³ : Monoid M\ninst✝² : MulAction M N\ninst✝¹ : SMul N α\ninst✝ : MulAction M α\ns : Set M\nhtop : Submonoid.closure s = ⊤\nhs : ∀ (x : M), x ∈ s → ∀ (y : N) (z : α), (x • y) • z = x • y • z\n⊢ ∀ (x : M),\n x ∈ s →\n ∀ (y : M),\n (∀ (y_1 : N) (z : α), (y • y_1) • z = y • y_1 • z) →\n ∀ (y_1 : N) (z : α), ((x * y) • y_1) • z = (x * y) • y_1 • z",
"state_before": "case refine'_2\nM : Type u_3\nA : Type ?u.235426\nB : Type ?u.235429\nN : Type u_1\nα : Type u_2\ninst✝³ : Monoid M\ninst✝² : MulAction M N\ninst✝¹ : SMul N α\ninst✝ : MulAction M α\ns : Set M\nhtop : Submonoid.closure s = ⊤\nhs : ∀ (x : M), x ∈ s → ∀ (y : N) (z : α), (x • y) • z = x • y • z\nx : M\n⊢ ∀ (x : M),\n x ∈ s →\n ∀ (y : M),\n (∀ (y_1 : N) (z : α), (y • y_1) • z = y • y_1 • z) →\n ∀ (y_1 : N) (z : α), ((x * y) • y_1) • z = (x * y) • y_1 • z",
"tactic": "clear x"
},
{
"state_after": "case refine'_2\nM : Type u_3\nA : Type ?u.235426\nB : Type ?u.235429\nN : Type u_1\nα : Type u_2\ninst✝³ : Monoid M\ninst✝² : MulAction M N\ninst✝¹ : SMul N α\ninst✝ : MulAction M α\ns : Set M\nhtop : Submonoid.closure s = ⊤\nhs : ∀ (x : M), x ∈ s → ∀ (y : N) (z : α), (x • y) • z = x • y • z\nx : M\nhx : x ∈ s\nx' : M\nhx' : ∀ (y : N) (z : α), (x' • y) • z = x' • y • z\ny : N\nz : α\n⊢ ((x * x') • y) • z = (x * x') • y • z",
"state_before": "case refine'_2\nM : Type u_3\nA : Type ?u.235426\nB : Type ?u.235429\nN : Type u_1\nα : Type u_2\ninst✝³ : Monoid M\ninst✝² : MulAction M N\ninst✝¹ : SMul N α\ninst✝ : MulAction M α\ns : Set M\nhtop : Submonoid.closure s = ⊤\nhs : ∀ (x : M), x ∈ s → ∀ (y : N) (z : α), (x • y) • z = x • y • z\n⊢ ∀ (x : M),\n x ∈ s →\n ∀ (y : M),\n (∀ (y_1 : N) (z : α), (y • y_1) • z = y • y_1 • z) →\n ∀ (y_1 : N) (z : α), ((x * y) • y_1) • z = (x * y) • y_1 • z",
"tactic": "intro x hx x' hx' y z"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nM : Type u_3\nA : Type ?u.235426\nB : Type ?u.235429\nN : Type u_1\nα : Type u_2\ninst✝³ : Monoid M\ninst✝² : MulAction M N\ninst✝¹ : SMul N α\ninst✝ : MulAction M α\ns : Set M\nhtop : Submonoid.closure s = ⊤\nhs : ∀ (x : M), x ∈ s → ∀ (y : N) (z : α), (x • y) • z = x • y • z\nx : M\nhx : x ∈ s\nx' : M\nhx' : ∀ (y : N) (z : α), (x' • y) • z = x' • y • z\ny : N\nz : α\n⊢ ((x * x') • y) • z = (x * x') • y • z",
"tactic": "rw [mul_smul, mul_smul, hs x hx, hx']"
}
] |
[
546,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
538,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
|
CategoryTheory.Limits.coprod.triangle
|
[
{
"state_after": "C : Type u\ninst✝³ : Category C\nX✝ Y✝ : C\ninst✝² : Category C\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : HasInitial C\nX Y : C\n⊢ desc (desc inl (inl ≫ inr)) (inr ≫ inr) ≫ map (𝟙 X) (desc (initial.to Y) (𝟙 Y)) =\n map (desc (𝟙 X) (initial.to X)) (𝟙 Y)",
"state_before": "C : Type u\ninst✝³ : Category C\nX✝ Y✝ : C\ninst✝² : Category C\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : HasInitial C\nX Y : C\n⊢ (associator X (⊥_ C) Y).hom ≫ map (𝟙 X) (leftUnitor Y).hom = map (rightUnitor X).hom (𝟙 Y)",
"tactic": "dsimp"
},
{
"state_after": "case h₁\nC : Type u\ninst✝³ : Category C\nX✝ Y✝ : C\ninst✝² : Category C\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : HasInitial C\nX Y : C\n⊢ inl ≫ desc (desc inl (inl ≫ inr)) (inr ≫ inr) ≫ map (𝟙 X) (desc (initial.to Y) (𝟙 Y)) =\n inl ≫ map (desc (𝟙 X) (initial.to X)) (𝟙 Y)\n\ncase h₂\nC : Type u\ninst✝³ : Category C\nX✝ Y✝ : C\ninst✝² : Category C\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : HasInitial C\nX Y : C\n⊢ inr ≫ desc (desc inl (inl ≫ inr)) (inr ≫ inr) ≫ map (𝟙 X) (desc (initial.to Y) (𝟙 Y)) =\n inr ≫ map (desc (𝟙 X) (initial.to X)) (𝟙 Y)",
"state_before": "C : Type u\ninst✝³ : Category C\nX✝ Y✝ : C\ninst✝² : Category C\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : HasInitial C\nX Y : C\n⊢ desc (desc inl (inl ≫ inr)) (inr ≫ inr) ≫ map (𝟙 X) (desc (initial.to Y) (𝟙 Y)) =\n map (desc (𝟙 X) (initial.to X)) (𝟙 Y)",
"tactic": "apply coprod.hom_ext"
},
{
"state_after": "case h₂\nC : Type u\ninst✝³ : Category C\nX✝ Y✝ : C\ninst✝² : Category C\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : HasInitial C\nX Y : C\n⊢ inr ≫ desc (desc inl (inl ≫ inr)) (inr ≫ inr) ≫ map (𝟙 X) (desc (initial.to Y) (𝟙 Y)) =\n inr ≫ map (desc (𝟙 X) (initial.to X)) (𝟙 Y)",
"state_before": "case h₁\nC : Type u\ninst✝³ : Category C\nX✝ Y✝ : C\ninst✝² : Category C\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : HasInitial C\nX Y : C\n⊢ inl ≫ desc (desc inl (inl ≫ inr)) (inr ≫ inr) ≫ map (𝟙 X) (desc (initial.to Y) (𝟙 Y)) =\n inl ≫ map (desc (𝟙 X) (initial.to X)) (𝟙 Y)\n\ncase h₂\nC : Type u\ninst✝³ : Category C\nX✝ Y✝ : C\ninst✝² : Category C\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : HasInitial C\nX Y : C\n⊢ inr ≫ desc (desc inl (inl ≫ inr)) (inr ≫ inr) ≫ map (𝟙 X) (desc (initial.to Y) (𝟙 Y)) =\n inr ≫ map (desc (𝟙 X) (initial.to X)) (𝟙 Y)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case h₂\nC : Type u\ninst✝³ : Category C\nX✝ Y✝ : C\ninst✝² : Category C\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : HasInitial C\nX Y : C\n⊢ inr ≫ desc (desc inl (inl ≫ inr)) (inr ≫ inr) ≫ map (𝟙 X) (desc (initial.to Y) (𝟙 Y)) =\n inr ≫ map (desc (𝟙 X) (initial.to X)) (𝟙 Y)",
"tactic": "simp"
}
] |
[
1187,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1184,
1
] |
Mathlib/LinearAlgebra/Projection.lean
|
LinearMap.ofIsComplProd_apply
|
[] |
[
290,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
288,
1
] |
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