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/Data/Set/Function.lean
|
Set.mapsTo_inter
|
[] |
[
495,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
491,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.conj_neg_I
|
[
{
"state_after": "no goals",
"state_before": "⊢ (↑(starRingEnd ℂ) (-I)).re = I.re ∧ (↑(starRingEnd ℂ) (-I)).im = I.im",
"tactic": "simp"
}
] |
[
542,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
541,
1
] |
Mathlib/Data/Multiset/Sort.lean
|
Multiset.sort_sorted
|
[] |
[
44,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
1
] |
Mathlib/Logic/Equiv/Fintype.lean
|
Equiv.extendSubtype_mem
|
[
{
"state_after": "case h.e'_1\nα : Type u_1\nβ : Type ?u.153621\ninst✝³ : Fintype α\ninst✝² : DecidableEq β\ne✝ : Perm α\nf : α ↪ β\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\ne : { x // p x } ≃ { x // q x }\nx : α\nhx : p x\n⊢ ↑(extendSubtype e) x = ↑(↑e { val := x, property := hx })",
"state_before": "α : Type u_1\nβ : Type ?u.153621\ninst✝³ : Fintype α\ninst✝² : DecidableEq β\ne✝ : Perm α\nf : α ↪ β\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\ne : { x // p x } ≃ { x // q x }\nx : α\nhx : p x\n⊢ q (↑(extendSubtype e) x)",
"tactic": "convert (e ⟨x, hx⟩).2"
},
{
"state_after": "no goals",
"state_before": "case h.e'_1\nα : Type u_1\nβ : Type ?u.153621\ninst✝³ : Fintype α\ninst✝² : DecidableEq β\ne✝ : Perm α\nf : α ↪ β\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\ne : { x // p x } ≃ { x // q x }\nx : α\nhx : p x\n⊢ ↑(extendSubtype e) x = ↑(↑e { val := x, property := hx })",
"tactic": "rw [e.extendSubtype_apply_of_mem _ hx]"
}
] |
[
130,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Algebra/Hom/Ring.lean
|
NonUnitalRingHom.coe_comp
|
[] |
[
258,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.leadingCoeff_one
|
[] |
[
837,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
836,
1
] |
Mathlib/Data/List/Zip.lean
|
List.zipWith_congr
|
[
{
"state_after": "case nil\nα : Type u\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.89176\nε : Type ?u.89179\nf g : α → β → γ\nla : List α\nlb : List β\n⊢ zipWith f [] [] = zipWith g [] []\n\ncase cons\nα : Type u\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.89176\nε : Type ?u.89179\nf g : α → β → γ\nla : List α\nlb : List β\na : α\nb : β\nas : List α\nbs : List β\nhfg : f a b = g a b\na✝ : Forall₂ (fun a b => f a b = g a b) as bs\nih : zipWith f as bs = zipWith g as bs\n⊢ zipWith f (a :: as) (b :: bs) = zipWith g (a :: as) (b :: bs)",
"state_before": "α : Type u\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.89176\nε : Type ?u.89179\nf g : α → β → γ\nla : List α\nlb : List β\nh : Forall₂ (fun a b => f a b = g a b) la lb\n⊢ zipWith f la lb = zipWith g la lb",
"tactic": "induction' h with a b as bs hfg _ ih"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.89176\nε : Type ?u.89179\nf g : α → β → γ\nla : List α\nlb : List β\n⊢ zipWith f [] [] = zipWith g [] []",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.89176\nε : Type ?u.89179\nf g : α → β → γ\nla : List α\nlb : List β\na : α\nb : β\nas : List α\nbs : List β\nhfg : f a b = g a b\na✝ : Forall₂ (fun a b => f a b = g a b) as bs\nih : zipWith f as bs = zipWith g as bs\n⊢ zipWith f (a :: as) (b :: bs) = zipWith g (a :: as) (b :: bs)",
"tactic": "exact congr_arg₂ _ hfg ih"
}
] |
[
282,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
278,
1
] |
Mathlib/Algebra/AddTorsor.lean
|
vsub_vadd_eq_vsub_sub
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np1 p2 : P\ng : G\n⊢ p1 -ᵥ (g +ᵥ p2) = p1 -ᵥ p2 - g",
"tactic": "rw [← add_right_inj (p2 -ᵥ p1 : G), vsub_add_vsub_cancel, ← neg_vsub_eq_vsub_rev, vadd_vsub, ←\n add_sub_assoc, ← neg_vsub_eq_vsub_rev, neg_add_self, zero_sub]"
}
] |
[
170,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/GroupTheory/Submonoid/Pointwise.lean
|
AddSubmonoid.one_eq_mrange
|
[] |
[
501,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
500,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean
|
CategoryTheory.Limits.hasTerminalChangeUniverse
|
[] |
[
257,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
255,
1
] |
Mathlib/Data/Polynomial/Degree/TrailingDegree.lean
|
Polynomial.trailingDegree_eq_iff_natTrailingDegree_eq
|
[
{
"state_after": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nhp : p ≠ 0\n⊢ ↑(natTrailingDegree p) = ↑n ↔ natTrailingDegree p = n",
"state_before": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nhp : p ≠ 0\n⊢ trailingDegree p = ↑n ↔ natTrailingDegree p = n",
"tactic": "rw [trailingDegree_eq_natTrailingDegree hp]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nhp : p ≠ 0\n⊢ ↑(natTrailingDegree p) = ↑n ↔ natTrailingDegree p = n",
"tactic": "exact WithTop.coe_eq_coe"
}
] |
[
117,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/LinearAlgebra/Dimension.lean
|
cardinal_lift_le_rank_of_linearIndependent'
|
[] |
[
261,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
259,
1
] |
Mathlib/Data/List/Basic.lean
|
List.getI_eq_default
|
[] |
[
4480,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4479,
1
] |
Mathlib/Logic/Relation.lean
|
Relation.reflTransGen_closed
|
[] |
[
569,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
567,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
Antitone.measurable
|
[] |
[
1183,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1181,
11
] |
Mathlib/Analysis/NormedSpace/lpSpace.lean
|
lp.norm_eq_zero_iff
|
[
{
"state_after": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E p }\nh : ‖f‖ = 0\n⊢ f = 0",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E p }\n⊢ ‖f‖ = 0 ↔ f = 0",
"tactic": "refine' ⟨fun h => _, by rintro rfl; exact norm_zero⟩"
},
{
"state_after": "case inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E 0 }\nh : ‖f‖ = 0\n⊢ f = 0\n\ncase inr.inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E ⊤ }\nh : ‖f‖ = 0\n⊢ f = 0\n\ncase inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E p }\nh : ‖f‖ = 0\nhp : 0 < ENNReal.toReal p\n⊢ f = 0",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E p }\nh : ‖f‖ = 0\n⊢ f = 0",
"tactic": "rcases p.trichotomy with (rfl | rfl | hp)"
},
{
"state_after": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\n⊢ ‖0‖ = 0",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E p }\n⊢ f = 0 → ‖f‖ = 0",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\n⊢ ‖0‖ = 0",
"tactic": "exact norm_zero"
},
{
"state_after": "case inl.h.h\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E 0 }\nh : ‖f‖ = 0\ni : α\n⊢ ↑f i = ↑0 i",
"state_before": "case inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E 0 }\nh : ‖f‖ = 0\n⊢ f = 0",
"tactic": "ext i"
},
{
"state_after": "case inl.h.h\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E 0 }\nh : ‖f‖ = 0\ni : α\nthis : {i | ¬↑f i = 0} = ∅\n⊢ ↑f i = ↑0 i",
"state_before": "case inl.h.h\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E 0 }\nh : ‖f‖ = 0\ni : α\n⊢ ↑f i = ↑0 i",
"tactic": "have : { i : α | ¬f i = 0 } = ∅ := by simpa [lp.norm_eq_card_dsupport f] using h"
},
{
"state_after": "case inl.h.h\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E 0 }\nh : ‖f‖ = 0\ni : α\nthis✝ : {i | ¬↑f i = 0} = ∅\nthis : (¬↑f i = 0) = False\n⊢ ↑f i = ↑0 i",
"state_before": "case inl.h.h\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E 0 }\nh : ‖f‖ = 0\ni : α\nthis : {i | ¬↑f i = 0} = ∅\n⊢ ↑f i = ↑0 i",
"tactic": "have : (¬f i = 0) = False := congr_fun this i"
},
{
"state_after": "no goals",
"state_before": "case inl.h.h\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E 0 }\nh : ‖f‖ = 0\ni : α\nthis✝ : {i | ¬↑f i = 0} = ∅\nthis : (¬↑f i = 0) = False\n⊢ ↑f i = ↑0 i",
"tactic": "tauto"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E 0 }\nh : ‖f‖ = 0\ni : α\n⊢ {i | ¬↑f i = 0} = ∅",
"tactic": "simpa [lp.norm_eq_card_dsupport f] using h"
},
{
"state_after": "case inr.inl.inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E ⊤ }\nh : ‖f‖ = 0\n_i : IsEmpty α\n⊢ f = 0\n\ncase inr.inl.inr\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E ⊤ }\nh : ‖f‖ = 0\n_i : Nonempty α\n⊢ f = 0",
"state_before": "case inr.inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E ⊤ }\nh : ‖f‖ = 0\n⊢ f = 0",
"tactic": "cases' isEmpty_or_nonempty α with _i _i"
},
{
"state_after": "case inr.inl.inr\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E ⊤ }\nh : ‖f‖ = 0\n_i : Nonempty α\nH : IsLUB (Set.range fun i => ‖↑f i‖) 0\n⊢ f = 0",
"state_before": "case inr.inl.inr\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E ⊤ }\nh : ‖f‖ = 0\n_i : Nonempty α\n⊢ f = 0",
"tactic": "have H : IsLUB (Set.range fun i => ‖f i‖) 0 := by simpa [h] using lp.isLUB_norm f"
},
{
"state_after": "case inr.inl.inr.h.h\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E ⊤ }\nh : ‖f‖ = 0\n_i : Nonempty α\nH : IsLUB (Set.range fun i => ‖↑f i‖) 0\ni : α\n⊢ ↑f i = ↑0 i",
"state_before": "case inr.inl.inr\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E ⊤ }\nh : ‖f‖ = 0\n_i : Nonempty α\nH : IsLUB (Set.range fun i => ‖↑f i‖) 0\n⊢ f = 0",
"tactic": "ext i"
},
{
"state_after": "case inr.inl.inr.h.h\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E ⊤ }\nh : ‖f‖ = 0\n_i : Nonempty α\nH : IsLUB (Set.range fun i => ‖↑f i‖) 0\ni : α\nthis : ‖↑f i‖ = 0\n⊢ ↑f i = ↑0 i",
"state_before": "case inr.inl.inr.h.h\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E ⊤ }\nh : ‖f‖ = 0\n_i : Nonempty α\nH : IsLUB (Set.range fun i => ‖↑f i‖) 0\ni : α\n⊢ ↑f i = ↑0 i",
"tactic": "have : ‖f i‖ = 0 := le_antisymm (H.1 ⟨i, rfl⟩) (norm_nonneg _)"
},
{
"state_after": "no goals",
"state_before": "case inr.inl.inr.h.h\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E ⊤ }\nh : ‖f‖ = 0\n_i : Nonempty α\nH : IsLUB (Set.range fun i => ‖↑f i‖) 0\ni : α\nthis : ‖↑f i‖ = 0\n⊢ ↑f i = ↑0 i",
"tactic": "simpa using this"
},
{
"state_after": "no goals",
"state_before": "case inr.inl.inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E ⊤ }\nh : ‖f‖ = 0\n_i : IsEmpty α\n⊢ f = 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E ⊤ }\nh : ‖f‖ = 0\n_i : Nonempty α\n⊢ IsLUB (Set.range fun i => ‖↑f i‖) 0",
"tactic": "simpa [h] using lp.isLUB_norm f"
},
{
"state_after": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E p }\nh : ‖f‖ = 0\nhp : 0 < ENNReal.toReal p\nhf : HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) 0\n⊢ f = 0",
"state_before": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E p }\nh : ‖f‖ = 0\nhp : 0 < ENNReal.toReal p\n⊢ f = 0",
"tactic": "have hf : HasSum (fun i : α => ‖f i‖ ^ p.toReal) 0 := by\n have := lp.hasSum_norm hp f\n rwa [h, Real.zero_rpow hp.ne'] at this"
},
{
"state_after": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E p }\nh : ‖f‖ = 0\nhp : 0 < ENNReal.toReal p\nhf : HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) 0\nthis : ∀ (i : α), 0 ≤ ‖↑f i‖ ^ ENNReal.toReal p\n⊢ f = 0",
"state_before": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E p }\nh : ‖f‖ = 0\nhp : 0 < ENNReal.toReal p\nhf : HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) 0\n⊢ f = 0",
"tactic": "have : ∀ i, 0 ≤ ‖f i‖ ^ p.toReal := fun i => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _"
},
{
"state_after": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E p }\nh : ‖f‖ = 0\nhp : 0 < ENNReal.toReal p\nhf : (fun i => ‖↑f i‖ ^ ENNReal.toReal p) = 0\nthis : ∀ (i : α), 0 ≤ ‖↑f i‖ ^ ENNReal.toReal p\n⊢ f = 0",
"state_before": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E p }\nh : ‖f‖ = 0\nhp : 0 < ENNReal.toReal p\nhf : HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) 0\nthis : ∀ (i : α), 0 ≤ ‖↑f i‖ ^ ENNReal.toReal p\n⊢ f = 0",
"tactic": "rw [hasSum_zero_iff_of_nonneg this] at hf"
},
{
"state_after": "case inr.inr.h.h\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E p }\nh : ‖f‖ = 0\nhp : 0 < ENNReal.toReal p\nhf : (fun i => ‖↑f i‖ ^ ENNReal.toReal p) = 0\nthis : ∀ (i : α), 0 ≤ ‖↑f i‖ ^ ENNReal.toReal p\ni : α\n⊢ ↑f i = ↑0 i",
"state_before": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E p }\nh : ‖f‖ = 0\nhp : 0 < ENNReal.toReal p\nhf : (fun i => ‖↑f i‖ ^ ENNReal.toReal p) = 0\nthis : ∀ (i : α), 0 ≤ ‖↑f i‖ ^ ENNReal.toReal p\n⊢ f = 0",
"tactic": "ext i"
},
{
"state_after": "case inr.inr.h.h\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E p }\nh : ‖f‖ = 0\nhp : 0 < ENNReal.toReal p\nhf : (fun i => ‖↑f i‖ ^ ENNReal.toReal p) = 0\nthis✝ : ∀ (i : α), 0 ≤ ‖↑f i‖ ^ ENNReal.toReal p\ni : α\nthis : ↑f i = 0 ∧ ENNReal.toReal p ≠ 0\n⊢ ↑f i = ↑0 i",
"state_before": "case inr.inr.h.h\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E p }\nh : ‖f‖ = 0\nhp : 0 < ENNReal.toReal p\nhf : (fun i => ‖↑f i‖ ^ ENNReal.toReal p) = 0\nthis : ∀ (i : α), 0 ≤ ‖↑f i‖ ^ ENNReal.toReal p\ni : α\n⊢ ↑f i = ↑0 i",
"tactic": "have : f i = 0 ∧ p.toReal ≠ 0 := by\n simpa [Real.rpow_eq_zero_iff_of_nonneg (norm_nonneg (f i))] using congr_fun hf i"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.h.h\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E p }\nh : ‖f‖ = 0\nhp : 0 < ENNReal.toReal p\nhf : (fun i => ‖↑f i‖ ^ ENNReal.toReal p) = 0\nthis✝ : ∀ (i : α), 0 ≤ ‖↑f i‖ ^ ENNReal.toReal p\ni : α\nthis : ↑f i = 0 ∧ ENNReal.toReal p ≠ 0\n⊢ ↑f i = ↑0 i",
"tactic": "exact this.1"
},
{
"state_after": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E p }\nh : ‖f‖ = 0\nhp : 0 < ENNReal.toReal p\nthis : HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) (‖f‖ ^ ENNReal.toReal p)\n⊢ HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) 0",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E p }\nh : ‖f‖ = 0\nhp : 0 < ENNReal.toReal p\n⊢ HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) 0",
"tactic": "have := lp.hasSum_norm hp f"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E p }\nh : ‖f‖ = 0\nhp : 0 < ENNReal.toReal p\nthis : HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) (‖f‖ ^ ENNReal.toReal p)\n⊢ HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) 0",
"tactic": "rwa [h, Real.zero_rpow hp.ne'] at this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E p }\nh : ‖f‖ = 0\nhp : 0 < ENNReal.toReal p\nhf : (fun i => ‖↑f i‖ ^ ENNReal.toReal p) = 0\nthis : ∀ (i : α), 0 ≤ ‖↑f i‖ ^ ENNReal.toReal p\ni : α\n⊢ ↑f i = 0 ∧ ENNReal.toReal p ≠ 0",
"tactic": "simpa [Real.rpow_eq_zero_iff_of_nonneg (norm_nonneg (f i))] using congr_fun hf i"
}
] |
[
477,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
456,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
ZNum.zneg_pred
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.658993\nn : ZNum\n⊢ -pred n = succ (-n)",
"tactic": "rw [← zneg_zneg (succ (-n)), zneg_succ, zneg_zneg]"
}
] |
[
1093,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1092,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.closure_eq
|
[] |
[
1037,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1036,
1
] |
Std/Data/Nat/Gcd.lean
|
Nat.coprime_mul_iff_left
|
[
{
"state_after": "no goals",
"state_before": "m n k : Nat\nx✝ : coprime m k ∧ coprime n k\nh : coprime m k\nright✝ : coprime n k\n⊢ coprime (m * n) k",
"tactic": "rwa [coprime_iff_gcd_eq_one, h.gcd_mul_left_cancel n]"
}
] |
[
335,
75
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
333,
1
] |
Mathlib/Topology/FiberBundle/IsHomeomorphicTrivialBundle.lean
|
isHomeomorphicTrivialFiberBundle_snd
|
[] |
[
79,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Topology/Algebra/Affine.lean
|
AffineMap.continuous_iff
|
[
{
"state_after": "case mp\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : TopologicalSpace F\ninst✝³ : TopologicalAddGroup F\ninst✝² : Ring R\ninst✝¹ : Module R E\ninst✝ : Module R F\nf : E →ᵃ[R] F\n⊢ Continuous ↑f → Continuous ↑f.linear\n\ncase mpr\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : TopologicalSpace F\ninst✝³ : TopologicalAddGroup F\ninst✝² : Ring R\ninst✝¹ : Module R E\ninst✝ : Module R F\nf : E →ᵃ[R] F\n⊢ Continuous ↑f.linear → Continuous ↑f",
"state_before": "R : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : TopologicalSpace F\ninst✝³ : TopologicalAddGroup F\ninst✝² : Ring R\ninst✝¹ : Module R E\ninst✝ : Module R F\nf : E →ᵃ[R] F\n⊢ Continuous ↑f ↔ Continuous ↑f.linear",
"tactic": "constructor"
},
{
"state_after": "case mp\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : TopologicalSpace F\ninst✝³ : TopologicalAddGroup F\ninst✝² : Ring R\ninst✝¹ : Module R E\ninst✝ : Module R F\nf : E →ᵃ[R] F\nhc : Continuous ↑f\n⊢ Continuous ↑f.linear",
"state_before": "case mp\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : TopologicalSpace F\ninst✝³ : TopologicalAddGroup F\ninst✝² : Ring R\ninst✝¹ : Module R E\ninst✝ : Module R F\nf : E →ᵃ[R] F\n⊢ Continuous ↑f → Continuous ↑f.linear",
"tactic": "intro hc"
},
{
"state_after": "case mp\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : TopologicalSpace F\ninst✝³ : TopologicalAddGroup F\ninst✝² : Ring R\ninst✝¹ : Module R E\ninst✝ : Module R F\nf : E →ᵃ[R] F\nhc : Continuous ↑f\n⊢ Continuous (↑f - fun x => ↑f 0)",
"state_before": "case mp\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : TopologicalSpace F\ninst✝³ : TopologicalAddGroup F\ninst✝² : Ring R\ninst✝¹ : Module R E\ninst✝ : Module R F\nf : E →ᵃ[R] F\nhc : Continuous ↑f\n⊢ Continuous ↑f.linear",
"tactic": "rw [decomp' f]"
},
{
"state_after": "no goals",
"state_before": "case mp\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : TopologicalSpace F\ninst✝³ : TopologicalAddGroup F\ninst✝² : Ring R\ninst✝¹ : Module R E\ninst✝ : Module R F\nf : E →ᵃ[R] F\nhc : Continuous ↑f\n⊢ Continuous (↑f - fun x => ↑f 0)",
"tactic": "exact hc.sub continuous_const"
},
{
"state_after": "case mpr\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : TopologicalSpace F\ninst✝³ : TopologicalAddGroup F\ninst✝² : Ring R\ninst✝¹ : Module R E\ninst✝ : Module R F\nf : E →ᵃ[R] F\nhc : Continuous ↑f.linear\n⊢ Continuous ↑f",
"state_before": "case mpr\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : TopologicalSpace F\ninst✝³ : TopologicalAddGroup F\ninst✝² : Ring R\ninst✝¹ : Module R E\ninst✝ : Module R F\nf : E →ᵃ[R] F\n⊢ Continuous ↑f.linear → Continuous ↑f",
"tactic": "intro hc"
},
{
"state_after": "case mpr\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : TopologicalSpace F\ninst✝³ : TopologicalAddGroup F\ninst✝² : Ring R\ninst✝¹ : Module R E\ninst✝ : Module R F\nf : E →ᵃ[R] F\nhc : Continuous ↑f.linear\n⊢ Continuous (↑f.linear + fun x => ↑f 0)",
"state_before": "case mpr\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : TopologicalSpace F\ninst✝³ : TopologicalAddGroup F\ninst✝² : Ring R\ninst✝¹ : Module R E\ninst✝ : Module R F\nf : E →ᵃ[R] F\nhc : Continuous ↑f.linear\n⊢ Continuous ↑f",
"tactic": "rw [decomp f]"
},
{
"state_after": "no goals",
"state_before": "case mpr\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : TopologicalSpace F\ninst✝³ : TopologicalAddGroup F\ninst✝² : Ring R\ninst✝¹ : Module R E\ninst✝ : Module R F\nf : E →ᵃ[R] F\nhc : Continuous ↑f.linear\n⊢ Continuous (↑f.linear + fun x => ↑f 0)",
"tactic": "exact hc.add continuous_const"
}
] |
[
48,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
41,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
neg_concaveOn_iff
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.650858\nα : Type ?u.650861\nβ : Type u_3\nι : Type ?u.650867\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid F\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf g : E → β\n⊢ ConcaveOn 𝕜 s (-f) ↔ ConvexOn 𝕜 s f",
"tactic": "rw [← neg_convexOn_iff, neg_neg f]"
}
] |
[
851,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
850,
1
] |
Mathlib/MeasureTheory/Covering/Differentiation.lean
|
VitaliFamily.ae_tendsto_measure_inter_div
|
[
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1)",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1)",
"tactic": "let t := toMeasurable μ s"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\nA : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1)",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1)",
"tactic": "have A :\n ∀ᵐ x ∂μ.restrict s,\n Tendsto (fun a => μ (t ∩ a) / μ a) (v.filterAt x) (𝓝 (t.indicator 1 x)) := by\n apply ae_mono restrict_le_self\n apply ae_tendsto_measure_inter_div_of_measurableSet\n exact measurableSet_toMeasurable _ _"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\nA : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))\nB : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1)",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\nA : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1)",
"tactic": "have B : ∀ᵐ x ∂μ.restrict s, t.indicator 1 x = (1 : ℝ≥0∞) := by\n refine' ae_restrict_of_ae_restrict_of_subset (subset_toMeasurable μ s) _\n filter_upwards [ae_restrict_mem (measurableSet_toMeasurable μ s)] with _ hx\n simp only [hx, Pi.one_apply, indicator_of_mem]"
},
{
"state_after": "case h\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\nA : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))\nB : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1\nx : α\nhx : Tendsto (fun a => ↑↑μ (toMeasurable μ s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator (toMeasurable μ s) 1 x))\nh'x : indicator (toMeasurable μ s) 1 x = 1\n⊢ Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1)",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\nA : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))\nB : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1)",
"tactic": "filter_upwards [A, B] with x hx h'x"
},
{
"state_after": "case h\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\nA : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))\nB : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1\nx : α\nhx : Tendsto (fun a => ↑↑μ (toMeasurable μ s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1)\nh'x : indicator (toMeasurable μ s) 1 x = 1\n⊢ Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1)",
"state_before": "case h\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\nA : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))\nB : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1\nx : α\nhx : Tendsto (fun a => ↑↑μ (toMeasurable μ s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator (toMeasurable μ s) 1 x))\nh'x : indicator (toMeasurable μ s) 1 x = 1\n⊢ Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1)",
"tactic": "rw [h'x] at hx"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\nA : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))\nB : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1\nx : α\nhx : Tendsto (fun a => ↑↑μ (toMeasurable μ s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1)\nh'x : indicator (toMeasurable μ s) 1 x = 1\n⊢ (fun a => ↑↑μ (toMeasurable μ s ∩ a) / ↑↑μ a) =ᶠ[filterAt v x] fun a => ↑↑μ (s ∩ a) / ↑↑μ a",
"state_before": "case h\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\nA : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))\nB : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1\nx : α\nhx : Tendsto (fun a => ↑↑μ (toMeasurable μ s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1)\nh'x : indicator (toMeasurable μ s) 1 x = 1\n⊢ Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1)",
"tactic": "apply hx.congr' _"
},
{
"state_after": "case h\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\nA : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))\nB : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1\nx : α\nhx : Tendsto (fun a => ↑↑μ (toMeasurable μ s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1)\nh'x : indicator (toMeasurable μ s) 1 x = 1\na✝ : Set α\nha : MeasurableSet a✝\n⊢ ↑↑μ (toMeasurable μ s ∩ a✝) / ↑↑μ a✝ = ↑↑μ (s ∩ a✝) / ↑↑μ a✝",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\nA : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))\nB : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1\nx : α\nhx : Tendsto (fun a => ↑↑μ (toMeasurable μ s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1)\nh'x : indicator (toMeasurable μ s) 1 x = 1\n⊢ (fun a => ↑↑μ (toMeasurable μ s ∩ a) / ↑↑μ a) =ᶠ[filterAt v x] fun a => ↑↑μ (s ∩ a) / ↑↑μ a",
"tactic": "filter_upwards [v.eventually_filterAt_measurableSet x] with _ ha"
},
{
"state_after": "case h.e_a\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\nA : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))\nB : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1\nx : α\nhx : Tendsto (fun a => ↑↑μ (toMeasurable μ s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1)\nh'x : indicator (toMeasurable μ s) 1 x = 1\na✝ : Set α\nha : MeasurableSet a✝\n⊢ ↑↑μ (toMeasurable μ s ∩ a✝) = ↑↑μ (s ∩ a✝)",
"state_before": "case h\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\nA : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))\nB : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1\nx : α\nhx : Tendsto (fun a => ↑↑μ (toMeasurable μ s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1)\nh'x : indicator (toMeasurable μ s) 1 x = 1\na✝ : Set α\nha : MeasurableSet a✝\n⊢ ↑↑μ (toMeasurable μ s ∩ a✝) / ↑↑μ a✝ = ↑↑μ (s ∩ a✝) / ↑↑μ a✝",
"tactic": "congr 1"
},
{
"state_after": "no goals",
"state_before": "case h.e_a\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\nA : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))\nB : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1\nx : α\nhx : Tendsto (fun a => ↑↑μ (toMeasurable μ s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1)\nh'x : indicator (toMeasurable μ s) 1 x = 1\na✝ : Set α\nha : MeasurableSet a✝\n⊢ ↑↑μ (toMeasurable μ s ∩ a✝) = ↑↑μ (s ∩ a✝)",
"tactic": "exact measure_toMeasurable_inter_of_sigmaFinite ha _"
},
{
"state_after": "case a\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\n⊢ {x | (fun x => Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))) x} ∈ ae μ",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))",
"tactic": "apply ae_mono restrict_le_self"
},
{
"state_after": "case a.hs\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\n⊢ MeasurableSet t",
"state_before": "case a\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\n⊢ {x | (fun x => Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))) x} ∈ ae μ",
"tactic": "apply ae_tendsto_measure_inter_div_of_measurableSet"
},
{
"state_after": "no goals",
"state_before": "case a.hs\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\n⊢ MeasurableSet t",
"tactic": "exact measurableSet_toMeasurable _ _"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\nA : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ (toMeasurable μ s), indicator t 1 x = 1",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\nA : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1",
"tactic": "refine' ae_restrict_of_ae_restrict_of_subset (subset_toMeasurable μ s) _"
},
{
"state_after": "case h\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\nA : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))\na✝ : α\nhx : a✝ ∈ toMeasurable μ s\n⊢ indicator (toMeasurable μ s) 1 a✝ = 1",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\nA : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ (toMeasurable μ s), indicator t 1 x = 1",
"tactic": "filter_upwards [ae_restrict_mem (measurableSet_toMeasurable μ s)] with _ hx"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.6387832\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\ns : Set α\nt : Set α := toMeasurable μ s\nA : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))\na✝ : α\nhx : a✝ ∈ toMeasurable μ s\n⊢ indicator (toMeasurable μ s) 1 a✝ = 1",
"tactic": "simp only [hx, Pi.one_apply, indicator_of_mem]"
}
] |
[
773,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
755,
1
] |
Mathlib/Topology/FiberBundle/Trivialization.lean
|
Trivialization.preimageSingletonHomeomorph_symm_apply
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.38955\nB : Type ?u.38958\nF : Type ?u.38961\nE : B → Type ?u.38966\nZ : Type ?u.38969\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace F\nproj : Z → B\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace (TotalSpace E)\ne : Trivialization F proj\nx : Z\nb : B\nhb : b ∈ e.baseSet\np : F\n⊢ ↑(LocalHomeomorph.symm e.toLocalHomeomorph) (b, p) ∈ proj ⁻¹' {b}",
"tactic": "rw [mem_preimage, e.proj_symm_apply' hb, mem_singleton_iff]"
}
] |
[
512,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
509,
1
] |
Mathlib/Analysis/NormedSpace/lpSpace.lean
|
Memℓp.finite_dsupport
|
[] |
[
153,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Data/Polynomial/EraseLead.lean
|
Polynomial.eraseLead_C_mul_X_pow
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nr : R\nn : ℕ\n⊢ eraseLead (↑C r * X ^ n) = 0",
"tactic": "rw [C_mul_X_pow_eq_monomial, eraseLead_monomial]"
}
] |
[
157,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.restrictScalars_smul
|
[] |
[
1717,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1715,
1
] |
Mathlib/Topology/Connected.lean
|
connectedComponents_preimage_image
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.143197\nπ : ι → Type ?u.143202\ninst✝² : TopologicalSpace α\ns t u v : Set α\ninst✝¹ : TopologicalSpace β\ninst✝ : TotallyDisconnectedSpace β\nf : α → β\nU : Set α\n⊢ ConnectedComponents.mk ⁻¹' (ConnectedComponents.mk '' U) = ⋃ (x : α) (_ : x ∈ U), connectedComponent x",
"tactic": "simp only [connectedComponents_preimage_singleton, preimage_iUnion₂, image_eq_iUnion]"
}
] |
[
1540,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1538,
1
] |
Mathlib/Analysis/Calculus/Deriv/Inv.lean
|
hasStrictDerivAt_inv
|
[
{
"state_after": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\n⊢ (fun p => (p.fst - p.snd) * ((x * x)⁻¹ - (p.fst * p.snd)⁻¹)) =o[𝓝 (x, x)] fun p => (p.fst - p.snd) * 1",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\n⊢ HasStrictDerivAt Inv.inv (-(x ^ 2)⁻¹) x",
"tactic": "suffices\n (fun p : 𝕜 × 𝕜 => (p.1 - p.2) * ((x * x)⁻¹ - (p.1 * p.2)⁻¹)) =o[𝓝 (x, x)] fun p =>\n (p.1 - p.2) * 1 by\n refine' this.congr' _ (eventually_of_forall fun _ => mul_one _)\n refine' Eventually.mono ((isOpen_ne.prod isOpen_ne).mem_nhds ⟨hx, hx⟩) _\n rintro ⟨y, z⟩ ⟨hy, hz⟩\n simp only [mem_setOf_eq] at hy hz\n field_simp [hx, hy, hz]\n ring"
},
{
"state_after": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\n⊢ Tendsto (fun p => (x * x)⁻¹ - (p.fst * p.snd)⁻¹) (𝓝 (x, x)) (𝓝 0)",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\n⊢ (fun p => (p.fst - p.snd) * ((x * x)⁻¹ - (p.fst * p.snd)⁻¹)) =o[𝓝 (x, x)] fun p => (p.fst - p.snd) * 1",
"tactic": "refine' (isBigO_refl (fun p : 𝕜 × 𝕜 => p.1 - p.2) _).mul_isLittleO ((isLittleO_one_iff 𝕜).2 _)"
},
{
"state_after": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\n⊢ Tendsto (fun p => (x * x)⁻¹ - (p.fst * p.snd)⁻¹) (𝓝 (x, x)) (𝓝 ((x * x)⁻¹ - (x * x)⁻¹))",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\n⊢ Tendsto (fun p => (x * x)⁻¹ - (p.fst * p.snd)⁻¹) (𝓝 (x, x)) (𝓝 0)",
"tactic": "rw [← sub_self (x * x)⁻¹]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\n⊢ Tendsto (fun p => (x * x)⁻¹ - (p.fst * p.snd)⁻¹) (𝓝 (x, x)) (𝓝 ((x * x)⁻¹ - (x * x)⁻¹))",
"tactic": "exact tendsto_const_nhds.sub ((continuous_mul.tendsto (x, x)).inv₀ <| mul_ne_zero hx hx)"
},
{
"state_after": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\nthis : (fun p => (p.fst - p.snd) * ((x * x)⁻¹ - (p.fst * p.snd)⁻¹)) =o[𝓝 (x, x)] fun p => (p.fst - p.snd) * 1\n⊢ (fun p => (p.fst - p.snd) * ((x * x)⁻¹ - (p.fst * p.snd)⁻¹)) =ᶠ[𝓝 (x, x)] fun p =>\n p.fst⁻¹ - p.snd⁻¹ - ↑(smulRight 1 (-(x ^ 2)⁻¹)) (p.fst - p.snd)",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\nthis : (fun p => (p.fst - p.snd) * ((x * x)⁻¹ - (p.fst * p.snd)⁻¹)) =o[𝓝 (x, x)] fun p => (p.fst - p.snd) * 1\n⊢ HasStrictDerivAt Inv.inv (-(x ^ 2)⁻¹) x",
"tactic": "refine' this.congr' _ (eventually_of_forall fun _ => mul_one _)"
},
{
"state_after": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\nthis : (fun p => (p.fst - p.snd) * ((x * x)⁻¹ - (p.fst * p.snd)⁻¹)) =o[𝓝 (x, x)] fun p => (p.fst - p.snd) * 1\n⊢ ∀ (x_1 : 𝕜 × 𝕜),\n ({y | y ≠ 0} ×ˢ {y | y ≠ 0}) x_1 →\n (fun p => (p.fst - p.snd) * ((x * x)⁻¹ - (p.fst * p.snd)⁻¹)) x_1 =\n (fun p => p.fst⁻¹ - p.snd⁻¹ - ↑(smulRight 1 (-(x ^ 2)⁻¹)) (p.fst - p.snd)) x_1",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\nthis : (fun p => (p.fst - p.snd) * ((x * x)⁻¹ - (p.fst * p.snd)⁻¹)) =o[𝓝 (x, x)] fun p => (p.fst - p.snd) * 1\n⊢ (fun p => (p.fst - p.snd) * ((x * x)⁻¹ - (p.fst * p.snd)⁻¹)) =ᶠ[𝓝 (x, x)] fun p =>\n p.fst⁻¹ - p.snd⁻¹ - ↑(smulRight 1 (-(x ^ 2)⁻¹)) (p.fst - p.snd)",
"tactic": "refine' Eventually.mono ((isOpen_ne.prod isOpen_ne).mem_nhds ⟨hx, hx⟩) _"
},
{
"state_after": "case mk.intro\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\nthis : (fun p => (p.fst - p.snd) * ((x * x)⁻¹ - (p.fst * p.snd)⁻¹)) =o[𝓝 (x, x)] fun p => (p.fst - p.snd) * 1\ny z : 𝕜\nhy : (y, z).fst ∈ {y | y ≠ 0}\nhz : (y, z).snd ∈ {y | y ≠ 0}\n⊢ (fun p => (p.fst - p.snd) * ((x * x)⁻¹ - (p.fst * p.snd)⁻¹)) (y, z) =\n (fun p => p.fst⁻¹ - p.snd⁻¹ - ↑(smulRight 1 (-(x ^ 2)⁻¹)) (p.fst - p.snd)) (y, z)",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\nthis : (fun p => (p.fst - p.snd) * ((x * x)⁻¹ - (p.fst * p.snd)⁻¹)) =o[𝓝 (x, x)] fun p => (p.fst - p.snd) * 1\n⊢ ∀ (x_1 : 𝕜 × 𝕜),\n ({y | y ≠ 0} ×ˢ {y | y ≠ 0}) x_1 →\n (fun p => (p.fst - p.snd) * ((x * x)⁻¹ - (p.fst * p.snd)⁻¹)) x_1 =\n (fun p => p.fst⁻¹ - p.snd⁻¹ - ↑(smulRight 1 (-(x ^ 2)⁻¹)) (p.fst - p.snd)) x_1",
"tactic": "rintro ⟨y, z⟩ ⟨hy, hz⟩"
},
{
"state_after": "case mk.intro\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\nthis : (fun p => (p.fst - p.snd) * ((x * x)⁻¹ - (p.fst * p.snd)⁻¹)) =o[𝓝 (x, x)] fun p => (p.fst - p.snd) * 1\ny z : 𝕜\nhy : y ≠ 0\nhz : z ≠ 0\n⊢ (fun p => (p.fst - p.snd) * ((x * x)⁻¹ - (p.fst * p.snd)⁻¹)) (y, z) =\n (fun p => p.fst⁻¹ - p.snd⁻¹ - ↑(smulRight 1 (-(x ^ 2)⁻¹)) (p.fst - p.snd)) (y, z)",
"state_before": "case mk.intro\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\nthis : (fun p => (p.fst - p.snd) * ((x * x)⁻¹ - (p.fst * p.snd)⁻¹)) =o[𝓝 (x, x)] fun p => (p.fst - p.snd) * 1\ny z : 𝕜\nhy : (y, z).fst ∈ {y | y ≠ 0}\nhz : (y, z).snd ∈ {y | y ≠ 0}\n⊢ (fun p => (p.fst - p.snd) * ((x * x)⁻¹ - (p.fst * p.snd)⁻¹)) (y, z) =\n (fun p => p.fst⁻¹ - p.snd⁻¹ - ↑(smulRight 1 (-(x ^ 2)⁻¹)) (p.fst - p.snd)) (y, z)",
"tactic": "simp only [mem_setOf_eq] at hy hz"
},
{
"state_after": "case mk.intro\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\nthis : (fun p => (p.fst - p.snd) * ((x * x)⁻¹ - (p.fst * p.snd)⁻¹)) =o[𝓝 (x, x)] fun p => (p.fst - p.snd) * 1\ny z : 𝕜\nhy : y ≠ 0\nhz : z ≠ 0\n⊢ (y - z) * (y * z - x * x) * (y * z * (x ^ 2 * x ^ 2)) =\n ((z - y) * (x ^ 2 * x ^ 2) - y * z * (-(y * x ^ 2) + x ^ 2 * z)) * (x * x * (y * z))",
"state_before": "case mk.intro\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\nthis : (fun p => (p.fst - p.snd) * ((x * x)⁻¹ - (p.fst * p.snd)⁻¹)) =o[𝓝 (x, x)] fun p => (p.fst - p.snd) * 1\ny z : 𝕜\nhy : y ≠ 0\nhz : z ≠ 0\n⊢ (fun p => (p.fst - p.snd) * ((x * x)⁻¹ - (p.fst * p.snd)⁻¹)) (y, z) =\n (fun p => p.fst⁻¹ - p.snd⁻¹ - ↑(smulRight 1 (-(x ^ 2)⁻¹)) (p.fst - p.snd)) (y, z)",
"tactic": "field_simp [hx, hy, hz]"
},
{
"state_after": "no goals",
"state_before": "case mk.intro\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\nthis : (fun p => (p.fst - p.snd) * ((x * x)⁻¹ - (p.fst * p.snd)⁻¹)) =o[𝓝 (x, x)] fun p => (p.fst - p.snd) * 1\ny z : 𝕜\nhy : y ≠ 0\nhz : z ≠ 0\n⊢ (y - z) * (y * z - x * x) * (y * z * (x ^ 2 * x ^ 2)) =\n ((z - y) * (x ^ 2 * x ^ 2) - y * z * (-(y * x ^ 2) + x ^ 2 * z)) * (x * x * (y * z))",
"tactic": "ring"
}
] |
[
70,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/CategoryTheory/Sites/Plus.lean
|
CategoryTheory.GrothendieckTopology.toPlus_plusLift
|
[
{
"state_after": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nhQ : Presheaf.IsSheaf J Q\n⊢ toPlus J P ≫ plusMap J η ≫ (isoToPlus J Q hQ).inv = η",
"state_before": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nhQ : Presheaf.IsSheaf J Q\n⊢ toPlus J P ≫ plusLift J η hQ = η",
"tactic": "dsimp [plusLift]"
},
{
"state_after": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nhQ : Presheaf.IsSheaf J Q\n⊢ (toPlus J P ≫ plusMap J η) ≫ (isoToPlus J Q hQ).inv = η",
"state_before": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nhQ : Presheaf.IsSheaf J Q\n⊢ toPlus J P ≫ plusMap J η ≫ (isoToPlus J Q hQ).inv = η",
"tactic": "rw [← Category.assoc]"
},
{
"state_after": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nhQ : Presheaf.IsSheaf J Q\n⊢ toPlus J P ≫ plusMap J η = η ≫ (isoToPlus J Q hQ).hom",
"state_before": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nhQ : Presheaf.IsSheaf J Q\n⊢ (toPlus J P ≫ plusMap J η) ≫ (isoToPlus J Q hQ).inv = η",
"tactic": "rw [Iso.comp_inv_eq]"
},
{
"state_after": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nhQ : Presheaf.IsSheaf J Q\n⊢ toPlus J P ≫ plusMap J η = η ≫ toPlus J Q",
"state_before": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nhQ : Presheaf.IsSheaf J Q\n⊢ toPlus J P ≫ plusMap J η = η ≫ (isoToPlus J Q hQ).hom",
"tactic": "dsimp only [isoToPlus, asIso]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nhQ : Presheaf.IsSheaf J Q\n⊢ toPlus J P ≫ plusMap J η = η ≫ toPlus J Q",
"tactic": "rw [toPlus_naturality]"
}
] |
[
322,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
316,
1
] |
Mathlib/Order/Filter/Archimedean.lean
|
Filter.Tendsto.atTop_nsmul_const
|
[
{
"state_after": "α : Type u_2\nR : Type u_1\nl : Filter α\nf✝ : α → R\nr : R\ninst✝¹ : LinearOrderedCancelAddCommMonoid R\ninst✝ : Archimedean R\nf : α → ℕ\nhr : 0 < r\nhf : Tendsto f l atTop\ns : R\n⊢ ∀ᶠ (a : α) in l, s ≤ f a • r",
"state_before": "α : Type u_2\nR : Type u_1\nl : Filter α\nf✝ : α → R\nr : R\ninst✝¹ : LinearOrderedCancelAddCommMonoid R\ninst✝ : Archimedean R\nf : α → ℕ\nhr : 0 < r\nhf : Tendsto f l atTop\n⊢ Tendsto (fun x => f x • r) l atTop",
"tactic": "refine' tendsto_atTop.mpr fun s => _"
},
{
"state_after": "case intro\nα : Type u_2\nR : Type u_1\nl : Filter α\nf✝ : α → R\nr : R\ninst✝¹ : LinearOrderedCancelAddCommMonoid R\ninst✝ : Archimedean R\nf : α → ℕ\nhr : 0 < r\nhf : Tendsto f l atTop\ns : R\nn : ℕ\nhn : s ≤ n • r\n⊢ ∀ᶠ (a : α) in l, s ≤ f a • r",
"state_before": "α : Type u_2\nR : Type u_1\nl : Filter α\nf✝ : α → R\nr : R\ninst✝¹ : LinearOrderedCancelAddCommMonoid R\ninst✝ : Archimedean R\nf : α → ℕ\nhr : 0 < r\nhf : Tendsto f l atTop\ns : R\n⊢ ∀ᶠ (a : α) in l, s ≤ f a • r",
"tactic": "obtain ⟨n : ℕ, hn : s ≤ n • r⟩ := Archimedean.arch s hr"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_2\nR : Type u_1\nl : Filter α\nf✝ : α → R\nr : R\ninst✝¹ : LinearOrderedCancelAddCommMonoid R\ninst✝ : Archimedean R\nf : α → ℕ\nhr : 0 < r\nhf : Tendsto f l atTop\ns : R\nn : ℕ\nhn : s ≤ n • r\n⊢ ∀ᶠ (a : α) in l, s ≤ f a • r",
"tactic": "exact (tendsto_atTop.mp hf n).mono fun a ha => hn.trans (nsmul_le_nsmul hr.le ha)"
}
] |
[
218,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/Order/Hom/Bounded.lean
|
BoundedOrderHom.cancel_right
|
[] |
[
693,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
690,
1
] |
Mathlib/Analysis/SpecialFunctions/Exp.lean
|
Real.comap_exp_nhds_exp
|
[] |
[
358,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
357,
1
] |
Mathlib/MeasureTheory/Covering/Besicovitch.lean
|
Besicovitch.SatelliteConfig.hlast'
|
[
{
"state_after": "case inl\nα : Type u_1\ninst✝ : MetricSpace α\nN : ℕ\nτ : ℝ\na : SatelliteConfig α N τ\ni : Fin (Nat.succ N)\nh : 1 ≤ τ\nH : i < last N\n⊢ r a (last N) ≤ τ * r a i\n\ncase inr\nα : Type u_1\ninst✝ : MetricSpace α\nN : ℕ\nτ : ℝ\na : SatelliteConfig α N τ\ni : Fin (Nat.succ N)\nh : 1 ≤ τ\nH : last N ≤ i\n⊢ r a (last N) ≤ τ * r a i",
"state_before": "α : Type u_1\ninst✝ : MetricSpace α\nN : ℕ\nτ : ℝ\na : SatelliteConfig α N τ\ni : Fin (Nat.succ N)\nh : 1 ≤ τ\n⊢ r a (last N) ≤ τ * r a i",
"tactic": "rcases lt_or_le i (last N) with (H | H)"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\ninst✝ : MetricSpace α\nN : ℕ\nτ : ℝ\na : SatelliteConfig α N τ\ni : Fin (Nat.succ N)\nh : 1 ≤ τ\nH : i < last N\n⊢ r a (last N) ≤ τ * r a i",
"tactic": "exact (a.hlast i H).2"
},
{
"state_after": "case inr\nα : Type u_1\ninst✝ : MetricSpace α\nN : ℕ\nτ : ℝ\na : SatelliteConfig α N τ\ni : Fin (Nat.succ N)\nh : 1 ≤ τ\nH : last N ≤ i\nthis : i = last N\n⊢ r a (last N) ≤ τ * r a i",
"state_before": "case inr\nα : Type u_1\ninst✝ : MetricSpace α\nN : ℕ\nτ : ℝ\na : SatelliteConfig α N τ\ni : Fin (Nat.succ N)\nh : 1 ≤ τ\nH : last N ≤ i\n⊢ r a (last N) ≤ τ * r a i",
"tactic": "have : i = last N := top_le_iff.1 H"
},
{
"state_after": "case inr\nα : Type u_1\ninst✝ : MetricSpace α\nN : ℕ\nτ : ℝ\na : SatelliteConfig α N τ\ni : Fin (Nat.succ N)\nh : 1 ≤ τ\nH : last N ≤ i\nthis : i = last N\n⊢ r a (last N) ≤ τ * r a (last N)",
"state_before": "case inr\nα : Type u_1\ninst✝ : MetricSpace α\nN : ℕ\nτ : ℝ\na : SatelliteConfig α N τ\ni : Fin (Nat.succ N)\nh : 1 ≤ τ\nH : last N ≤ i\nthis : i = last N\n⊢ r a (last N) ≤ τ * r a i",
"tactic": "rw [this]"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\ninst✝ : MetricSpace α\nN : ℕ\nτ : ℝ\na : SatelliteConfig α N τ\ni : Fin (Nat.succ N)\nh : 1 ≤ τ\nH : last N ≤ i\nthis : i = last N\n⊢ r a (last N) ≤ τ * r a (last N)",
"tactic": "exact le_mul_of_one_le_left (a.rpos _).le h"
}
] |
[
186,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
1
] |
Mathlib/Analysis/Calculus/MeanValue.lean
|
Convex.lipschitzOnWith_of_nnnorm_fderiv_le
|
[] |
[
551,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
548,
1
] |
Mathlib/Algebra/Associated.lean
|
Associated.mul_left
|
[] |
[
525,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
524,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.dual_Iic
|
[] |
[
236,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
235,
1
] |
Std/Data/Int/DivMod.lean
|
Int.dvd_iff_dvd_of_dvd_sub
|
[] |
[
629,
54
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
627,
11
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.le_of_forall_lt_one_mul_le
|
[
{
"state_after": "α : Type ?u.113642\nβ : Type ?u.113645\nγ : Type ?u.113648\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx✝ y✝ z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nx y : ℝ≥0∞\nh : ∀ (a : ℝ≥0∞), a < 1 → a * x ≤ y\nthis : Tendsto (fun x_1 => x_1 * x) (𝓝[Iio 1] 1) (𝓝 x)\n⊢ x ≤ y",
"state_before": "α : Type ?u.113642\nβ : Type ?u.113645\nγ : Type ?u.113648\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx✝ y✝ z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nx y : ℝ≥0∞\nh : ∀ (a : ℝ≥0∞), a < 1 → a * x ≤ y\nthis : Tendsto (fun x_1 => x_1 * x) (𝓝[Iio 1] 1) (𝓝 (1 * x))\n⊢ x ≤ y",
"tactic": "rw [one_mul] at this"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.113642\nβ : Type ?u.113645\nγ : Type ?u.113648\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx✝ y✝ z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nx y : ℝ≥0∞\nh : ∀ (a : ℝ≥0∞), a < 1 → a * x ≤ y\nthis : Tendsto (fun x_1 => x_1 * x) (𝓝[Iio 1] 1) (𝓝 x)\n⊢ x ≤ y",
"tactic": "exact le_of_tendsto this (eventually_nhdsWithin_iff.2 <| eventually_of_forall h)"
}
] |
[
483,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
479,
1
] |
Mathlib/LinearAlgebra/PiTensorProduct.lean
|
PiTensorProduct.smul_tprodCoeff_aux
|
[] |
[
170,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/Geometry/Euclidean/Sphere/Power.lean
|
EuclideanGeometry.mul_dist_eq_mul_dist_of_cospherical
|
[
{
"state_after": "case intro.intro\nV : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_1\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : ∃ k₁, k₁ ≠ 1 ∧ b -ᵥ p = k₁ • (a -ᵥ p)\nhcpd : ∃ k₂, k₂ ≠ 1 ∧ d -ᵥ p = k₂ • (c -ᵥ p)\nq : P\nr : ℝ\nh' : ∀ (p : P), p ∈ {a, b, c, d} → dist p q = r\n⊢ dist a p * dist b p = dist c p * dist d p",
"state_before": "V : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_1\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : ∃ k₁, k₁ ≠ 1 ∧ b -ᵥ p = k₁ • (a -ᵥ p)\nhcpd : ∃ k₂, k₂ ≠ 1 ∧ d -ᵥ p = k₂ • (c -ᵥ p)\n⊢ dist a p * dist b p = dist c p * dist d p",
"tactic": "obtain ⟨q, r, h'⟩ := (cospherical_def {a, b, c, d}).mp h"
},
{
"state_after": "case intro.intro\nV : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_1\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : ∃ k₁, k₁ ≠ 1 ∧ b -ᵥ p = k₁ • (a -ᵥ p)\nhcpd : ∃ k₂, k₂ ≠ 1 ∧ d -ᵥ p = k₂ • (c -ᵥ p)\nq : P\nr : ℝ\nh' : ∀ (p : P), p ∈ {a, b, c, d} → dist p q = r\nha : dist a q = r\nhb : dist b q = r\nhc : dist c q = r\nhd : dist d q = r\n⊢ dist a p * dist b p = dist c p * dist d p",
"state_before": "case intro.intro\nV : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_1\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : ∃ k₁, k₁ ≠ 1 ∧ b -ᵥ p = k₁ • (a -ᵥ p)\nhcpd : ∃ k₂, k₂ ≠ 1 ∧ d -ᵥ p = k₂ • (c -ᵥ p)\nq : P\nr : ℝ\nh' : ∀ (p : P), p ∈ {a, b, c, d} → dist p q = r\n⊢ dist a p * dist b p = dist c p * dist d p",
"tactic": "obtain ⟨ha, hb, hc, hd⟩ := h' a (by simp), h' b (by simp), h' c (by simp), h' d (by simp)"
},
{
"state_after": "no goals",
"state_before": "V : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_1\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : ∃ k₁, k₁ ≠ 1 ∧ b -ᵥ p = k₁ • (a -ᵥ p)\nhcpd : ∃ k₂, k₂ ≠ 1 ∧ d -ᵥ p = k₂ • (c -ᵥ p)\nq : P\nr : ℝ\nh' : ∀ (p : P), p ∈ {a, b, c, d} → dist p q = r\n⊢ a ∈ {a, b, c, d}",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "V : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_1\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : ∃ k₁, k₁ ≠ 1 ∧ b -ᵥ p = k₁ • (a -ᵥ p)\nhcpd : ∃ k₂, k₂ ≠ 1 ∧ d -ᵥ p = k₂ • (c -ᵥ p)\nq : P\nr : ℝ\nh' : ∀ (p : P), p ∈ {a, b, c, d} → dist p q = r\n⊢ b ∈ {a, b, c, d}",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "V : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_1\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : ∃ k₁, k₁ ≠ 1 ∧ b -ᵥ p = k₁ • (a -ᵥ p)\nhcpd : ∃ k₂, k₂ ≠ 1 ∧ d -ᵥ p = k₂ • (c -ᵥ p)\nq : P\nr : ℝ\nh' : ∀ (p : P), p ∈ {a, b, c, d} → dist p q = r\n⊢ c ∈ {a, b, c, d}",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "V : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_1\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : ∃ k₁, k₁ ≠ 1 ∧ b -ᵥ p = k₁ • (a -ᵥ p)\nhcpd : ∃ k₂, k₂ ≠ 1 ∧ d -ᵥ p = k₂ • (c -ᵥ p)\nq : P\nr : ℝ\nh' : ∀ (p : P), p ∈ {a, b, c, d} → dist p q = r\n⊢ d ∈ {a, b, c, d}",
"tactic": "simp"
},
{
"state_after": "case intro.intro\nV : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_1\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : ∃ k₁, k₁ ≠ 1 ∧ b -ᵥ p = k₁ • (a -ᵥ p)\nhcpd : ∃ k₂, k₂ ≠ 1 ∧ d -ᵥ p = k₂ • (c -ᵥ p)\nq : P\nr : ℝ\nh' : ∀ (p : P), p ∈ {a, b, c, d} → dist p q = r\nha : dist a q = r\nhb : dist b q = r\nhc : dist c q = dist d q\nhd : dist d q = r\n⊢ dist a p * dist b p = dist c p * dist d p",
"state_before": "case intro.intro\nV : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_1\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : ∃ k₁, k₁ ≠ 1 ∧ b -ᵥ p = k₁ • (a -ᵥ p)\nhcpd : ∃ k₂, k₂ ≠ 1 ∧ d -ᵥ p = k₂ • (c -ᵥ p)\nq : P\nr : ℝ\nh' : ∀ (p : P), p ∈ {a, b, c, d} → dist p q = r\nha : dist a q = r\nhb : dist b q = r\nhc : dist c q = r\nhd : dist d q = r\n⊢ dist a p * dist b p = dist c p * dist d p",
"tactic": "rw [← hd] at hc"
},
{
"state_after": "case intro.intro\nV : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_1\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : ∃ k₁, k₁ ≠ 1 ∧ b -ᵥ p = k₁ • (a -ᵥ p)\nhcpd : ∃ k₂, k₂ ≠ 1 ∧ d -ᵥ p = k₂ • (c -ᵥ p)\nq : P\nr : ℝ\nh' : ∀ (p : P), p ∈ {a, b, c, d} → dist p q = r\nha : dist a q = dist b q\nhb : dist b q = r\nhc : dist c q = dist d q\nhd : dist d q = r\n⊢ dist a p * dist b p = dist c p * dist d p",
"state_before": "case intro.intro\nV : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_1\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : ∃ k₁, k₁ ≠ 1 ∧ b -ᵥ p = k₁ • (a -ᵥ p)\nhcpd : ∃ k₂, k₂ ≠ 1 ∧ d -ᵥ p = k₂ • (c -ᵥ p)\nq : P\nr : ℝ\nh' : ∀ (p : P), p ∈ {a, b, c, d} → dist p q = r\nha : dist a q = r\nhb : dist b q = r\nhc : dist c q = dist d q\nhd : dist d q = r\n⊢ dist a p * dist b p = dist c p * dist d p",
"tactic": "rw [← hb] at ha"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nV : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_1\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : ∃ k₁, k₁ ≠ 1 ∧ b -ᵥ p = k₁ • (a -ᵥ p)\nhcpd : ∃ k₂, k₂ ≠ 1 ∧ d -ᵥ p = k₂ • (c -ᵥ p)\nq : P\nr : ℝ\nh' : ∀ (p : P), p ∈ {a, b, c, d} → dist p q = r\nha : dist a q = dist b q\nhb : dist b q = r\nhc : dist c q = dist d q\nhd : dist d q = r\n⊢ dist a p * dist b p = dist c p * dist d p",
"tactic": "rw [mul_dist_eq_abs_sub_sq_dist hapb ha, hb, mul_dist_eq_abs_sub_sq_dist hcpd hc, hd]"
}
] |
[
112,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.update_eq_piecewise
|
[] |
[
2534,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2532,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.preimage_mul_const_Ioi_of_neg
|
[] |
[
563,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
561,
1
] |
Mathlib/Analysis/Calculus/Conformal/InnerProduct.lean
|
conformalAt_iff
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : InnerProductSpace ℝ F\nf : E → F\nx : E\nf' : E →L[ℝ] F\nh : HasFDerivAt f f' x\n⊢ ConformalAt f x ↔ ∃ c, 0 < c ∧ ∀ (u v : E), inner (↑f' u) (↑f' v) = c * inner u v",
"tactic": "simp only [conformalAt_iff', h.fderiv]"
}
] |
[
43,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
41,
1
] |
Mathlib/Algebra/Hom/Group.lean
|
MulHom.coe_coe
|
[] |
[
335,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
335,
1
] |
Mathlib/Data/Nat/Digits.lean
|
Nat.ofDigits_lt_base_pow_length'
|
[
{
"state_after": "case nil\nn b : ℕ\nl : List ℕ\nhl✝ : ∀ (x : ℕ), x ∈ l → x < b + 2\nhl : ∀ (x : ℕ), x ∈ [] → x < b + 2\n⊢ ofDigits (b + 2) [] < (b + 2) ^ List.length []\n\ncase cons\nn b : ℕ\nl : List ℕ\nhl✝ : ∀ (x : ℕ), x ∈ l → x < b + 2\nhd : ℕ\ntl : List ℕ\nIH : (∀ (x : ℕ), x ∈ tl → x < b + 2) → ofDigits (b + 2) tl < (b + 2) ^ List.length tl\nhl : ∀ (x : ℕ), x ∈ hd :: tl → x < b + 2\n⊢ ofDigits (b + 2) (hd :: tl) < (b + 2) ^ List.length (hd :: tl)",
"state_before": "n b : ℕ\nl : List ℕ\nhl : ∀ (x : ℕ), x ∈ l → x < b + 2\n⊢ ofDigits (b + 2) l < (b + 2) ^ List.length l",
"tactic": "induction' l with hd tl IH"
},
{
"state_after": "no goals",
"state_before": "case nil\nn b : ℕ\nl : List ℕ\nhl✝ : ∀ (x : ℕ), x ∈ l → x < b + 2\nhl : ∀ (x : ℕ), x ∈ [] → x < b + 2\n⊢ ofDigits (b + 2) [] < (b + 2) ^ List.length []",
"tactic": "simp [ofDigits]"
},
{
"state_after": "case cons\nn b : ℕ\nl : List ℕ\nhl✝ : ∀ (x : ℕ), x ∈ l → x < b + 2\nhd : ℕ\ntl : List ℕ\nIH : (∀ (x : ℕ), x ∈ tl → x < b + 2) → ofDigits (b + 2) tl < (b + 2) ^ List.length tl\nhl : ∀ (x : ℕ), x ∈ hd :: tl → x < b + 2\n⊢ ↑hd + (b + 2) * ofDigits (b + 2) tl < (b + 2) ^ List.length tl * (b + 2)",
"state_before": "case cons\nn b : ℕ\nl : List ℕ\nhl✝ : ∀ (x : ℕ), x ∈ l → x < b + 2\nhd : ℕ\ntl : List ℕ\nIH : (∀ (x : ℕ), x ∈ tl → x < b + 2) → ofDigits (b + 2) tl < (b + 2) ^ List.length tl\nhl : ∀ (x : ℕ), x ∈ hd :: tl → x < b + 2\n⊢ ofDigits (b + 2) (hd :: tl) < (b + 2) ^ List.length (hd :: tl)",
"tactic": "rw [ofDigits, List.length_cons, pow_succ]"
},
{
"state_after": "case cons\nn b : ℕ\nl : List ℕ\nhl✝ : ∀ (x : ℕ), x ∈ l → x < b + 2\nhd : ℕ\ntl : List ℕ\nIH : (∀ (x : ℕ), x ∈ tl → x < b + 2) → ofDigits (b + 2) tl < (b + 2) ^ List.length tl\nhl : ∀ (x : ℕ), x ∈ hd :: tl → x < b + 2\nthis : (ofDigits (b + 2) tl + 1) * (b + 2) ≤ (b + 2) ^ List.length tl * (b + 2)\n⊢ ↑hd + (b + 2) * ofDigits (b + 2) tl < (b + 2) ^ List.length tl * (b + 2)",
"state_before": "case cons\nn b : ℕ\nl : List ℕ\nhl✝ : ∀ (x : ℕ), x ∈ l → x < b + 2\nhd : ℕ\ntl : List ℕ\nIH : (∀ (x : ℕ), x ∈ tl → x < b + 2) → ofDigits (b + 2) tl < (b + 2) ^ List.length tl\nhl : ∀ (x : ℕ), x ∈ hd :: tl → x < b + 2\n⊢ ↑hd + (b + 2) * ofDigits (b + 2) tl < (b + 2) ^ List.length tl * (b + 2)",
"tactic": "have : (ofDigits (b + 2) tl + 1) * (b + 2) ≤ (b + 2) ^ tl.length * (b + 2) :=\n mul_le_mul (IH fun x hx => hl _ (List.mem_cons_of_mem _ hx)) (by rfl) (by simp only [zero_le])\n (Nat.zero_le _)"
},
{
"state_after": "case cons\nn b : ℕ\nl : List ℕ\nhl✝ : ∀ (x : ℕ), x ∈ l → x < b + 2\nhd : ℕ\ntl : List ℕ\nIH : (∀ (x : ℕ), x ∈ tl → x < b + 2) → ofDigits (b + 2) tl < (b + 2) ^ List.length tl\nhl : ∀ (x : ℕ), x ∈ hd :: tl → x < b + 2\nthis : (ofDigits (b + 2) tl + 1) * (b + 2) ≤ (b + 2) ^ List.length tl * (b + 2)\n⊢ hd < b + 2",
"state_before": "case cons\nn b : ℕ\nl : List ℕ\nhl✝ : ∀ (x : ℕ), x ∈ l → x < b + 2\nhd : ℕ\ntl : List ℕ\nIH : (∀ (x : ℕ), x ∈ tl → x < b + 2) → ofDigits (b + 2) tl < (b + 2) ^ List.length tl\nhl : ∀ (x : ℕ), x ∈ hd :: tl → x < b + 2\nthis : (ofDigits (b + 2) tl + 1) * (b + 2) ≤ (b + 2) ^ List.length tl * (b + 2)\n⊢ ↑hd + (b + 2) * ofDigits (b + 2) tl < (b + 2) ^ List.length tl * (b + 2)",
"tactic": "suffices ↑hd < b + 2 by linarith"
},
{
"state_after": "case cons\nn b : ℕ\nl : List ℕ\nhl✝ : ∀ (x : ℕ), x ∈ l → x < b + 2\nhd : ℕ\ntl : List ℕ\nIH : (∀ (x : ℕ), x ∈ tl → x < b + 2) → ofDigits (b + 2) tl < (b + 2) ^ List.length tl\nhl : ∀ (x : ℕ), x ∈ hd :: tl → x < b + 2\nthis : (ofDigits (b + 2) tl + 1) * (b + 2) ≤ (b + 2) ^ List.length tl * (b + 2)\n⊢ hd < b + 2",
"state_before": "case cons\nn b : ℕ\nl : List ℕ\nhl✝ : ∀ (x : ℕ), x ∈ l → x < b + 2\nhd : ℕ\ntl : List ℕ\nIH : (∀ (x : ℕ), x ∈ tl → x < b + 2) → ofDigits (b + 2) tl < (b + 2) ^ List.length tl\nhl : ∀ (x : ℕ), x ∈ hd :: tl → x < b + 2\nthis : (ofDigits (b + 2) tl + 1) * (b + 2) ≤ (b + 2) ^ List.length tl * (b + 2)\n⊢ hd < b + 2",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "case cons\nn b : ℕ\nl : List ℕ\nhl✝ : ∀ (x : ℕ), x ∈ l → x < b + 2\nhd : ℕ\ntl : List ℕ\nIH : (∀ (x : ℕ), x ∈ tl → x < b + 2) → ofDigits (b + 2) tl < (b + 2) ^ List.length tl\nhl : ∀ (x : ℕ), x ∈ hd :: tl → x < b + 2\nthis : (ofDigits (b + 2) tl + 1) * (b + 2) ≤ (b + 2) ^ List.length tl * (b + 2)\n⊢ hd < b + 2",
"tactic": "exact hl hd (List.mem_cons_self _ _)"
},
{
"state_after": "no goals",
"state_before": "n b : ℕ\nl : List ℕ\nhl✝ : ∀ (x : ℕ), x ∈ l → x < b + 2\nhd : ℕ\ntl : List ℕ\nIH : (∀ (x : ℕ), x ∈ tl → x < b + 2) → ofDigits (b + 2) tl < (b + 2) ^ List.length tl\nhl : ∀ (x : ℕ), x ∈ hd :: tl → x < b + 2\n⊢ b + 2 ≤ b + 2",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "n b : ℕ\nl : List ℕ\nhl✝ : ∀ (x : ℕ), x ∈ l → x < b + 2\nhd : ℕ\ntl : List ℕ\nIH : (∀ (x : ℕ), x ∈ tl → x < b + 2) → ofDigits (b + 2) tl < (b + 2) ^ List.length tl\nhl : ∀ (x : ℕ), x ∈ hd :: tl → x < b + 2\n⊢ 0 ≤ b + 2",
"tactic": "simp only [zero_le]"
},
{
"state_after": "no goals",
"state_before": "n b : ℕ\nl : List ℕ\nhl✝ : ∀ (x : ℕ), x ∈ l → x < b + 2\nhd : ℕ\ntl : List ℕ\nIH : (∀ (x : ℕ), x ∈ tl → x < b + 2) → ofDigits (b + 2) tl < (b + 2) ^ List.length tl\nhl : ∀ (x : ℕ), x ∈ hd :: tl → x < b + 2\nthis✝ : (ofDigits (b + 2) tl + 1) * (b + 2) ≤ (b + 2) ^ List.length tl * (b + 2)\nthis : hd < b + 2\n⊢ ↑hd + (b + 2) * ofDigits (b + 2) tl < (b + 2) ^ List.length tl * (b + 2)",
"tactic": "linarith"
}
] |
[
405,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
395,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
|
CategoryTheory.Limits.biprod.conePointUniqueUpToIso_hom
|
[] |
[
1548,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1545,
1
] |
Mathlib/Combinatorics/SimpleGraph/Partition.lean
|
SimpleGraph.Partition.to_colorable
|
[] |
[
122,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Real.lean
|
dist_le_tsum_of_dist_le_of_tendsto
|
[
{
"state_after": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na✝ : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f (Nat.succ n)) ≤ d n\nhd : Summable d\na : α\nha : Tendsto f atTop (𝓝 a)\nn m : ℕ\nhnm : m ≥ n\n⊢ dist (f n) (f m) ≤ ∑' (m : ℕ), d (n + m)",
"state_before": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na✝ : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f (Nat.succ n)) ≤ d n\nhd : Summable d\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\n⊢ dist (f n) a ≤ ∑' (m : ℕ), d (n + m)",
"tactic": "refine' le_of_tendsto (tendsto_const_nhds.dist ha) (eventually_atTop.2 ⟨n, fun m hnm => _⟩)"
},
{
"state_after": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na✝ : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f (Nat.succ n)) ≤ d n\nhd : Summable d\na : α\nha : Tendsto f atTop (𝓝 a)\nn m : ℕ\nhnm : m ≥ n\n⊢ ∑ i in Ico n m, d i ≤ ∑' (m : ℕ), d (n + m)",
"state_before": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na✝ : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f (Nat.succ n)) ≤ d n\nhd : Summable d\na : α\nha : Tendsto f atTop (𝓝 a)\nn m : ℕ\nhnm : m ≥ n\n⊢ dist (f n) (f m) ≤ ∑' (m : ℕ), d (n + m)",
"tactic": "refine' le_trans (dist_le_Ico_sum_of_dist_le hnm fun _ _ => hf _) _"
},
{
"state_after": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na✝ : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f (Nat.succ n)) ≤ d n\nhd : Summable d\na : α\nha : Tendsto f atTop (𝓝 a)\nn m : ℕ\nhnm : m ≥ n\n⊢ ∑ k in range (m - n), d (n + k) ≤ ∑' (m : ℕ), d (n + m)",
"state_before": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na✝ : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f (Nat.succ n)) ≤ d n\nhd : Summable d\na : α\nha : Tendsto f atTop (𝓝 a)\nn m : ℕ\nhnm : m ≥ n\n⊢ ∑ i in Ico n m, d i ≤ ∑' (m : ℕ), d (n + m)",
"tactic": "rw [sum_Ico_eq_sum_range]"
},
{
"state_after": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na✝ : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f (Nat.succ n)) ≤ d n\nhd : Summable d\na : α\nha : Tendsto f atTop (𝓝 a)\nn m : ℕ\nhnm : m ≥ n\n⊢ Summable fun k => d (n + k)",
"state_before": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na✝ : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f (Nat.succ n)) ≤ d n\nhd : Summable d\na : α\nha : Tendsto f atTop (𝓝 a)\nn m : ℕ\nhnm : m ≥ n\n⊢ ∑ k in range (m - n), d (n + k) ≤ ∑' (m : ℕ), d (n + m)",
"tactic": "refine' sum_le_tsum (range _) (fun _ _ => le_trans dist_nonneg (hf _)) _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na✝ : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f (Nat.succ n)) ≤ d n\nhd : Summable d\na : α\nha : Tendsto f atTop (𝓝 a)\nn m : ℕ\nhnm : m ≥ n\n⊢ Summable fun k => d (n + k)",
"tactic": "exact hd.comp_injective (add_right_injective n)"
}
] |
[
82,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.map_principal
|
[] |
[
1813,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1812,
1
] |
Mathlib/Topology/Algebra/Monoid.lean
|
Filter.tendsto_cocompact_mul_right
|
[
{
"state_after": "ι : Type ?u.374432\nα : Type ?u.374435\nX : Type ?u.374438\nM : Type u_1\nN : Type ?u.374444\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\na b : M\nha : a * b = 1\n⊢ Tendsto ((fun b_1 => b_1 * b) ∘ fun x => x * a) (cocompact M) (cocompact M)",
"state_before": "ι : Type ?u.374432\nα : Type ?u.374435\nX : Type ?u.374438\nM : Type u_1\nN : Type ?u.374444\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\na b : M\nha : a * b = 1\n⊢ Tendsto (fun x => x * a) (cocompact M) (cocompact M)",
"tactic": "refine Filter.Tendsto.of_tendsto_comp ?_ (Filter.comap_cocompact_le (continuous_mul_right b))"
},
{
"state_after": "ι : Type ?u.374432\nα : Type ?u.374435\nX : Type ?u.374438\nM : Type u_1\nN : Type ?u.374444\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\na b : M\nha : a * b = 1\n⊢ Tendsto (fun x => x) (cocompact M) (cocompact M)",
"state_before": "ι : Type ?u.374432\nα : Type ?u.374435\nX : Type ?u.374438\nM : Type u_1\nN : Type ?u.374444\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\na b : M\nha : a * b = 1\n⊢ Tendsto ((fun b_1 => b_1 * b) ∘ fun x => x * a) (cocompact M) (cocompact M)",
"tactic": "simp only [comp_mul_right, ha, mul_one]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.374432\nα : Type ?u.374435\nX : Type ?u.374438\nM : Type u_1\nN : Type ?u.374444\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\na b : M\nha : a * b = 1\n⊢ Tendsto (fun x => x) (cocompact M) (cocompact M)",
"tactic": "exact Filter.tendsto_id"
}
] |
[
643,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
639,
1
] |
Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean
|
AlgebraicTopology.karoubi_alternatingFaceMapComplex_d
|
[
{
"state_after": "C : Type u_1\ninst✝¹ : Category C\ninst✝ : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : ℕ\n⊢ (HomologicalComplex.d (AlternatingFaceMapComplex.obj (KaroubiFunctorCategoryEmbedding.obj P)) (n + 1) n).f =\n P.p.app [n + 1].op ≫ HomologicalComplex.d (AlternatingFaceMapComplex.obj P.X) (n + 1) n",
"state_before": "C : Type u_1\ninst✝¹ : Category C\ninst✝ : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : ℕ\n⊢ (HomologicalComplex.d (AlternatingFaceMapComplex.obj (KaroubiFunctorCategoryEmbedding.obj P)) (n + 1) n).f =\n P.p.app [n + 1].op ≫ HomologicalComplex.d (AlternatingFaceMapComplex.obj P.X) (n + 1) n",
"tactic": "dsimp"
},
{
"state_after": "C : Type u_1\ninst✝¹ : Category C\ninst✝ : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : ℕ\n⊢ ∑ x : Fin (n + 2), (-1) ^ ↑x • (SimplicialObject.δ (KaroubiFunctorCategoryEmbedding.obj P) x).f =\n ∑ x : Fin (n + 2), (-1) ^ ↑x • P.p.app [n + 1].op ≫ SimplicialObject.δ P.X x",
"state_before": "C : Type u_1\ninst✝¹ : Category C\ninst✝ : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : ℕ\n⊢ (HomologicalComplex.d (AlternatingFaceMapComplex.obj (KaroubiFunctorCategoryEmbedding.obj P)) (n + 1) n).f =\n P.p.app [n + 1].op ≫ HomologicalComplex.d (AlternatingFaceMapComplex.obj P.X) (n + 1) n",
"tactic": "simp only [AlternatingFaceMapComplex.obj_d_eq, Karoubi.sum_hom, Preadditive.comp_sum,\n Karoubi.zsmul_hom, Preadditive.comp_zsmul]"
},
{
"state_after": "no goals",
"state_before": "C : Type u_1\ninst✝¹ : Category C\ninst✝ : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : ℕ\n⊢ ∑ x : Fin (n + 2), (-1) ^ ↑x • (SimplicialObject.δ (KaroubiFunctorCategoryEmbedding.obj P) x).f =\n ∑ x : Fin (n + 2), (-1) ^ ↑x • P.p.app [n + 1].op ≫ SimplicialObject.δ P.X x",
"tactic": "rfl"
}
] |
[
223,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
LinearEquiv.isometryOfOrthonormal_toLinearEquiv
|
[] |
[
1384,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1381,
1
] |
Mathlib/LinearAlgebra/Matrix/Determinant.lean
|
Matrix.det_succ_column
|
[
{
"state_after": "m : Type ?u.2570289\nn✝ : Type ?u.2570292\ninst✝⁴ : DecidableEq n✝\ninst✝³ : Fintype n✝\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nn : ℕ\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nj : Fin (Nat.succ n)\n⊢ ∑ j_1 : Fin (Nat.succ n), (-1) ^ (↑j + ↑j_1) * Aᵀ j j_1 * det (submatrix Aᵀ ↑(Fin.succAbove j) ↑(Fin.succAbove j_1)) =\n ∑ i : Fin (Nat.succ n), (-1) ^ (↑i + ↑j) * A i j * det (submatrix A ↑(Fin.succAbove i) ↑(Fin.succAbove j))",
"state_before": "m : Type ?u.2570289\nn✝ : Type ?u.2570292\ninst✝⁴ : DecidableEq n✝\ninst✝³ : Fintype n✝\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nn : ℕ\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nj : Fin (Nat.succ n)\n⊢ det A = ∑ i : Fin (Nat.succ n), (-1) ^ (↑i + ↑j) * A i j * det (submatrix A ↑(Fin.succAbove i) ↑(Fin.succAbove j))",
"tactic": "rw [← det_transpose, det_succ_row _ j]"
},
{
"state_after": "m : Type ?u.2570289\nn✝ : Type ?u.2570292\ninst✝⁴ : DecidableEq n✝\ninst✝³ : Fintype n✝\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nn : ℕ\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nj i : Fin (Nat.succ n)\nx✝ : i ∈ univ\n⊢ (-1) ^ (↑j + ↑i) * Aᵀ j i * det (submatrix Aᵀ ↑(Fin.succAbove j) ↑(Fin.succAbove i)) =\n (-1) ^ (↑i + ↑j) * A i j * det (submatrix A ↑(Fin.succAbove i) ↑(Fin.succAbove j))",
"state_before": "m : Type ?u.2570289\nn✝ : Type ?u.2570292\ninst✝⁴ : DecidableEq n✝\ninst✝³ : Fintype n✝\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nn : ℕ\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nj : Fin (Nat.succ n)\n⊢ ∑ j_1 : Fin (Nat.succ n), (-1) ^ (↑j + ↑j_1) * Aᵀ j j_1 * det (submatrix Aᵀ ↑(Fin.succAbove j) ↑(Fin.succAbove j_1)) =\n ∑ i : Fin (Nat.succ n), (-1) ^ (↑i + ↑j) * A i j * det (submatrix A ↑(Fin.succAbove i) ↑(Fin.succAbove j))",
"tactic": "refine' Finset.sum_congr rfl fun i _ => _"
},
{
"state_after": "no goals",
"state_before": "m : Type ?u.2570289\nn✝ : Type ?u.2570292\ninst✝⁴ : DecidableEq n✝\ninst✝³ : Fintype n✝\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nn : ℕ\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nj i : Fin (Nat.succ n)\nx✝ : i ∈ univ\n⊢ (-1) ^ (↑j + ↑i) * Aᵀ j i * det (submatrix Aᵀ ↑(Fin.succAbove j) ↑(Fin.succAbove i)) =\n (-1) ^ (↑i + ↑j) * A i j * det (submatrix A ↑(Fin.succAbove i) ↑(Fin.succAbove j))",
"tactic": "rw [add_comm, ← det_transpose, transpose_apply, transpose_submatrix, transpose_transpose]"
}
] |
[
781,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
776,
1
] |
Mathlib/Data/Int/ModEq.lean
|
Int.coe_nat_modEq_iff
|
[
{
"state_after": "m n✝ a✝ b✝ c d : ℤ\na b n : ℕ\n⊢ ↑a % ↑n = ↑b % ↑n ↔ a % n = b % n",
"state_before": "m n✝ a✝ b✝ c d : ℤ\na b n : ℕ\n⊢ ↑a ≡ ↑b [ZMOD ↑n] ↔ a ≡ b [MOD n]",
"tactic": "unfold ModEq Nat.ModEq"
},
{
"state_after": "m n✝ a✝ b✝ c d : ℤ\na b n : ℕ\n⊢ ↑a % ↑n = ↑b % ↑n ↔ ↑(a % n) = ↑(b % n)",
"state_before": "m n✝ a✝ b✝ c d : ℤ\na b n : ℕ\n⊢ ↑a % ↑n = ↑b % ↑n ↔ a % n = b % n",
"tactic": "rw [← Int.ofNat_inj]"
},
{
"state_after": "no goals",
"state_before": "m n✝ a✝ b✝ c d : ℤ\na b n : ℕ\n⊢ ↑a % ↑n = ↑b % ↑n ↔ ↑(a % n) = ↑(b % n)",
"tactic": "simp [coe_nat_mod]"
}
] |
[
82,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Analysis/Normed/Order/Lattice.lean
|
LatticeOrderedAddCommGroup.isSolid_ball
|
[] |
[
64,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
LinearEquiv.finiteDimensional
|
[] |
[
817,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
815,
11
] |
Mathlib/MeasureTheory/Measure/Sub.lean
|
MeasureTheory.Measure.sub_def
|
[] |
[
42,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
42,
1
] |
Mathlib/Algebra/Module/Equiv.lean
|
LinearEquiv.ext
|
[] |
[
238,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.union_ae_eq_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.105395\nγ : Type ?u.105398\nδ : Type ?u.105401\nι : Type ?u.105404\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\n⊢ s ∪ t =ᵐ[μ] t ↔ ↑↑μ (s \\ t) = 0",
"tactic": "simp [eventuallyLE_antisymm_iff, ae_le_set, union_diff_right,\n diff_eq_empty.2 (Set.subset_union_right _ _)]"
}
] |
[
471,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
469,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.isIrreducible_zeroLocus_iff
|
[] |
[
555,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
553,
1
] |
Mathlib/MeasureTheory/Function/SimpleFuncDense.lean
|
MeasureTheory.SimpleFunc.tendsto_nearestPt
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.27019\nι : Type ?u.27022\nE : Type ?u.27025\nF : Type ?u.27028\n𝕜 : Type ?u.27031\ninst✝² : MeasurableSpace α\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : OpensMeasurableSpace α\ne : ℕ → α\nx : α\nhx : x ∈ closure (Set.range e)\nε : ℝ≥0∞\nhε : 0 < ε\n⊢ ∃ ia, True ∧ ∀ (x_1 : ℕ), x_1 ∈ Set.Ici ia → ↑(nearestPt e x_1) x ∈ ball x ε",
"state_before": "α : Type u_1\nβ : Type ?u.27019\nι : Type ?u.27022\nE : Type ?u.27025\nF : Type ?u.27028\n𝕜 : Type ?u.27031\ninst✝² : MeasurableSpace α\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : OpensMeasurableSpace α\ne : ℕ → α\nx : α\nhx : x ∈ closure (Set.range e)\n⊢ Tendsto (fun N => ↑(nearestPt e N) x) atTop (𝓝 x)",
"tactic": "refine' (atTop_basis.tendsto_iff nhds_basis_eball).2 fun ε hε => _"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.27019\nι : Type ?u.27022\nE : Type ?u.27025\nF : Type ?u.27028\n𝕜 : Type ?u.27031\ninst✝² : MeasurableSpace α\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : OpensMeasurableSpace α\ne : ℕ → α\nx : α\nhx : x ∈ closure (Set.range e)\nε : ℝ≥0∞\nhε : 0 < ε\nN : ℕ\nhN : edist x (e N) < ε\n⊢ ∃ ia, True ∧ ∀ (x_1 : ℕ), x_1 ∈ Set.Ici ia → ↑(nearestPt e x_1) x ∈ ball x ε",
"state_before": "α : Type u_1\nβ : Type ?u.27019\nι : Type ?u.27022\nE : Type ?u.27025\nF : Type ?u.27028\n𝕜 : Type ?u.27031\ninst✝² : MeasurableSpace α\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : OpensMeasurableSpace α\ne : ℕ → α\nx : α\nhx : x ∈ closure (Set.range e)\nε : ℝ≥0∞\nhε : 0 < ε\n⊢ ∃ ia, True ∧ ∀ (x_1 : ℕ), x_1 ∈ Set.Ici ia → ↑(nearestPt e x_1) x ∈ ball x ε",
"tactic": "rcases EMetric.mem_closure_iff.1 hx ε hε with ⟨_, ⟨N, rfl⟩, hN⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.27019\nι : Type ?u.27022\nE : Type ?u.27025\nF : Type ?u.27028\n𝕜 : Type ?u.27031\ninst✝² : MeasurableSpace α\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : OpensMeasurableSpace α\ne : ℕ → α\nx : α\nhx : x ∈ closure (Set.range e)\nε : ℝ≥0∞\nhε : 0 < ε\nN : ℕ\nhN : edist (e N) x < ε\n⊢ ∃ ia, True ∧ ∀ (x_1 : ℕ), x_1 ∈ Set.Ici ia → ↑(nearestPt e x_1) x ∈ ball x ε",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.27019\nι : Type ?u.27022\nE : Type ?u.27025\nF : Type ?u.27028\n𝕜 : Type ?u.27031\ninst✝² : MeasurableSpace α\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : OpensMeasurableSpace α\ne : ℕ → α\nx : α\nhx : x ∈ closure (Set.range e)\nε : ℝ≥0∞\nhε : 0 < ε\nN : ℕ\nhN : edist x (e N) < ε\n⊢ ∃ ia, True ∧ ∀ (x_1 : ℕ), x_1 ∈ Set.Ici ia → ↑(nearestPt e x_1) x ∈ ball x ε",
"tactic": "rw [edist_comm] at hN"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.27019\nι : Type ?u.27022\nE : Type ?u.27025\nF : Type ?u.27028\n𝕜 : Type ?u.27031\ninst✝² : MeasurableSpace α\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : OpensMeasurableSpace α\ne : ℕ → α\nx : α\nhx : x ∈ closure (Set.range e)\nε : ℝ≥0∞\nhε : 0 < ε\nN : ℕ\nhN : edist (e N) x < ε\n⊢ ∃ ia, True ∧ ∀ (x_1 : ℕ), x_1 ∈ Set.Ici ia → ↑(nearestPt e x_1) x ∈ ball x ε",
"tactic": "exact ⟨N, trivial, fun n hn => (edist_nearestPt_le e x hn).trans_lt hN⟩"
}
] |
[
123,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/Data/List/Count.lean
|
List.filter_beq'
|
[
{
"state_after": "α : Type u_1\nl✝ : List α\ninst✝ : DecidableEq α\nl : List α\na : α\n⊢ True ∧ ∀ (b : α), b ∈ l ∧ b = a → b = a",
"state_before": "α : Type u_1\nl✝ : List α\ninst✝ : DecidableEq α\nl : List α\na : α\n⊢ filter (fun x => x == a) l = replicate (count a l) a",
"tactic": "simp only [count, countp_eq_length_filter, eq_replicate, mem_filter, beq_iff_eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl✝ : List α\ninst✝ : DecidableEq α\nl : List α\na : α\n⊢ True ∧ ∀ (b : α), b ∈ l ∧ b = a → b = a",
"tactic": "exact ⟨trivial, fun _ h => h.2⟩"
}
] |
[
285,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.le_or_gf
|
[
{
"state_after": "x y : PGame\n⊢ x ≤ y ∨ ¬x ≤ y",
"state_before": "x y : PGame\n⊢ x ≤ y ∨ y ⧏ x",
"tactic": "rw [← PGame.not_le]"
},
{
"state_after": "no goals",
"state_before": "x y : PGame\n⊢ x ≤ y ∨ ¬x ≤ y",
"tactic": "apply em"
}
] |
[
464,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
462,
1
] |
Mathlib/LinearAlgebra/Alternating.lean
|
MultilinearMap.domCoprod_alternization
|
[
{
"state_after": "case a\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\n⊢ ↑(↑alternatization (domCoprod a b)) = ↑(AlternatingMap.domCoprod (↑alternatization a) (↑alternatization b))",
"state_before": "R : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\n⊢ ↑alternatization (domCoprod a b) = AlternatingMap.domCoprod (↑alternatization a) (↑alternatization b)",
"tactic": "apply coe_multilinearMap_injective"
},
{
"state_after": "case a\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\n⊢ ∑ xbar in Finset.image Quotient.mk'' Finset.univ,\n ∑ y in\n Finset.filter (fun x => Quotient.mk (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb))) x = xbar)\n Finset.univ,\n ↑Perm.sign y • domDomCongr y (domCoprod a b) =\n ∑ σ : Perm.ModSumCongr ιa ιb, domCoprod.summand (↑alternatization a) (↑alternatization b) σ",
"state_before": "case a\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\n⊢ ↑(↑alternatization (domCoprod a b)) = ↑(AlternatingMap.domCoprod (↑alternatization a) (↑alternatization b))",
"tactic": "rw [domCoprod_coe, MultilinearMap.alternatization_coe,\n Finset.sum_partition (QuotientGroup.leftRel (Perm.sumCongrHom ιa ιb).range)]"
},
{
"state_after": "case a.e_f\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\n⊢ (fun xbar =>\n ∑ y in\n Finset.filter (fun x => Quotient.mk (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb))) x = xbar)\n Finset.univ,\n ↑Perm.sign y • domDomCongr y (domCoprod a b)) =\n fun σ => domCoprod.summand (↑alternatization a) (↑alternatization b) σ",
"state_before": "case a\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\n⊢ ∑ xbar in Finset.image Quotient.mk'' Finset.univ,\n ∑ y in\n Finset.filter (fun x => Quotient.mk (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb))) x = xbar)\n Finset.univ,\n ↑Perm.sign y • domDomCongr y (domCoprod a b) =\n ∑ σ : Perm.ModSumCongr ιa ιb, domCoprod.summand (↑alternatization a) (↑alternatization b) σ",
"tactic": "congr 1"
},
{
"state_after": "case a.e_f.h\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\nσ : Quotient (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb)))\n⊢ ∑ y in\n Finset.filter (fun x => Quotient.mk (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb))) x = σ)\n Finset.univ,\n ↑Perm.sign y • domDomCongr y (domCoprod a b) =\n domCoprod.summand (↑alternatization a) (↑alternatization b) σ",
"state_before": "case a.e_f\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\n⊢ (fun xbar =>\n ∑ y in\n Finset.filter (fun x => Quotient.mk (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb))) x = xbar)\n Finset.univ,\n ↑Perm.sign y • domDomCongr y (domCoprod a b)) =\n fun σ => domCoprod.summand (↑alternatization a) (↑alternatization b) σ",
"tactic": "ext1 σ"
},
{
"state_after": "case a.e_f.h\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\nσ✝ : Quotient (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb)))\nσ : Perm (ιa ⊕ ιb)\n⊢ ∑ y in\n Finset.filter\n (fun x => Quotient.mk (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb))) x = Quotient.mk'' σ)\n Finset.univ,\n ↑Perm.sign y • domDomCongr y (domCoprod a b) =\n domCoprod.summand (↑alternatization a) (↑alternatization b) (Quotient.mk'' σ)",
"state_before": "case a.e_f.h\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\nσ : Quotient (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb)))\n⊢ ∑ y in\n Finset.filter (fun x => Quotient.mk (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb))) x = σ)\n Finset.univ,\n ↑Perm.sign y • domDomCongr y (domCoprod a b) =\n domCoprod.summand (↑alternatization a) (↑alternatization b) σ",
"tactic": "refine Quotient.inductionOn' σ fun σ => ?_"
},
{
"state_after": "case a.e_f.h\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\nσ✝ : Quotient (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb)))\nσ : Perm (ιa ⊕ ιb)\n⊢ ∑ y in Finset.filter (fun x => x⁻¹ * σ ∈ MonoidHom.range (Perm.sumCongrHom ιa ιb)) Finset.univ,\n ↑Perm.sign y • domDomCongr y (domCoprod a b) =\n domCoprod.summand (↑alternatization a) (↑alternatization b) (Quotient.mk'' σ)",
"state_before": "case a.e_f.h\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\nσ✝ : Quotient (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb)))\nσ : Perm (ιa ⊕ ιb)\n⊢ ∑ y in\n Finset.filter\n (fun x => Quotient.mk (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb))) x = Quotient.mk'' σ)\n Finset.univ,\n ↑Perm.sign y • domDomCongr y (domCoprod a b) =\n domCoprod.summand (↑alternatization a) (↑alternatization b) (Quotient.mk'' σ)",
"tactic": "erw\n [@Finset.filter_congr _ _ (fun a => @Quotient.decidableEq _ _\n (QuotientGroup.leftRelDecidable (MonoidHom.range (Perm.sumCongrHom ιa ιb)))\n (Quotient.mk (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb))) a)\n (Quotient.mk'' σ)) _ (s := Finset.univ)\n fun x _ => QuotientGroup.eq' (s := MonoidHom.range (Perm.sumCongrHom ιa ιb)) (a := x) (b := σ)]"
},
{
"state_after": "case a.e_f.h\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\nσ✝ : Quotient (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb)))\nσ : Perm (ιa ⊕ ιb)\n⊢ ∑ x in\n Finset.filter\n ((fun x => x⁻¹ * σ ∈ MonoidHom.range (Perm.sumCongrHom ιa ιb)) ∘ ↑(Equiv.toEmbedding (Equiv.mulLeft σ)))\n Finset.univ,\n ↑Perm.sign (↑(Equiv.toEmbedding (Equiv.mulLeft σ)) x) •\n domDomCongr (↑(Equiv.toEmbedding (Equiv.mulLeft σ)) x) (domCoprod a b) =\n domCoprod.summand (↑alternatization a) (↑alternatization b) (Quotient.mk'' σ)",
"state_before": "case a.e_f.h\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\nσ✝ : Quotient (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb)))\nσ : Perm (ιa ⊕ ιb)\n⊢ ∑ y in Finset.filter (fun x => x⁻¹ * σ ∈ MonoidHom.range (Perm.sumCongrHom ιa ιb)) Finset.univ,\n ↑Perm.sign y • domDomCongr y (domCoprod a b) =\n domCoprod.summand (↑alternatization a) (↑alternatization b) (Quotient.mk'' σ)",
"tactic": "rw [← Finset.map_univ_equiv (Equiv.mulLeft σ), Finset.filter_map, Finset.sum_map]"
},
{
"state_after": "case a.e_f.h\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\nσ✝ : Quotient (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb)))\nσ : Perm (ιa ⊕ ιb)\n⊢ ∑ x : Perm ιa × Perm ιb,\n ↑Perm.sign (σ * ↑(Perm.sumCongrHom ιa ιb) x) • domDomCongr (σ * ↑(Perm.sumCongrHom ιa ιb) x) (domCoprod a b) =\n ∑ x : Perm ιa × Perm ιb,\n ↑Perm.sign σ •\n ↑(domDomCongrEquiv σ)\n (↑Perm.sign x.fst • ↑Perm.sign x.snd • domCoprod (domDomCongr x.fst a) (domDomCongr x.snd b))",
"state_before": "case a.e_f.h\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\nσ✝ : Quotient (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb)))\nσ : Perm (ιa ⊕ ιb)\n⊢ ∑ x : Perm ιa × Perm ιb,\n ↑Perm.sign (σ * ↑(Perm.sumCongrHom ιa ιb) x) • domDomCongr (σ * ↑(Perm.sumCongrHom ιa ιb) x) (domCoprod a b) =\n domCoprod.summand (↑alternatization a) (↑alternatization b) (Quotient.mk'' σ)",
"tactic": "rw [domCoprod.summand_mk'', MultilinearMap.domCoprod_alternization_coe, ← Finset.sum_product',\n Finset.univ_product_univ, ← MultilinearMap.domDomCongrEquiv_apply, _root_.map_sum,\n Finset.smul_sum]"
},
{
"state_after": "case a.e_f.h.e_f\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\nσ✝ : Quotient (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb)))\nσ : Perm (ιa ⊕ ιb)\n⊢ (fun x =>\n ↑Perm.sign (σ * ↑(Perm.sumCongrHom ιa ιb) x) • domDomCongr (σ * ↑(Perm.sumCongrHom ιa ιb) x) (domCoprod a b)) =\n fun x =>\n ↑Perm.sign σ •\n ↑(domDomCongrEquiv σ)\n (↑Perm.sign x.fst • ↑Perm.sign x.snd • domCoprod (domDomCongr x.fst a) (domDomCongr x.snd b))",
"state_before": "case a.e_f.h\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\nσ✝ : Quotient (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb)))\nσ : Perm (ιa ⊕ ιb)\n⊢ ∑ x : Perm ιa × Perm ιb,\n ↑Perm.sign (σ * ↑(Perm.sumCongrHom ιa ιb) x) • domDomCongr (σ * ↑(Perm.sumCongrHom ιa ιb) x) (domCoprod a b) =\n ∑ x : Perm ιa × Perm ιb,\n ↑Perm.sign σ •\n ↑(domDomCongrEquiv σ)\n (↑Perm.sign x.fst • ↑Perm.sign x.snd • domCoprod (domDomCongr x.fst a) (domDomCongr x.snd b))",
"tactic": "congr 1"
},
{
"state_after": "case a.e_f.h.e_f.h.mk\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\nσ✝ : Quotient (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb)))\nσ : Perm (ιa ⊕ ιb)\nal : Perm ιa\nar : Perm ιb\n⊢ ↑Perm.sign (σ * ↑(Perm.sumCongrHom ιa ιb) (al, ar)) •\n domDomCongr (σ * ↑(Perm.sumCongrHom ιa ιb) (al, ar)) (domCoprod a b) =\n ↑Perm.sign σ •\n ↑(domDomCongrEquiv σ)\n (↑Perm.sign (al, ar).fst •\n ↑Perm.sign (al, ar).snd • domCoprod (domDomCongr (al, ar).fst a) (domDomCongr (al, ar).snd b))",
"state_before": "case a.e_f.h.e_f\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\nσ✝ : Quotient (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb)))\nσ : Perm (ιa ⊕ ιb)\n⊢ (fun x =>\n ↑Perm.sign (σ * ↑(Perm.sumCongrHom ιa ιb) x) • domDomCongr (σ * ↑(Perm.sumCongrHom ιa ιb) x) (domCoprod a b)) =\n fun x =>\n ↑Perm.sign σ •\n ↑(domDomCongrEquiv σ)\n (↑Perm.sign x.fst • ↑Perm.sign x.snd • domCoprod (domDomCongr x.fst a) (domDomCongr x.snd b))",
"tactic": "ext1 ⟨al, ar⟩"
},
{
"state_after": "case a.e_f.h.e_f.h.mk\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\nσ✝ : Quotient (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb)))\nσ : Perm (ιa ⊕ ιb)\nal : Perm ιa\nar : Perm ιb\n⊢ ↑Perm.sign (σ * ↑(Perm.sumCongrHom ιa ιb) (al, ar)) •\n domDomCongr (σ * ↑(Perm.sumCongrHom ιa ιb) (al, ar)) (domCoprod a b) =\n ↑Perm.sign σ •\n ↑(domDomCongrEquiv σ) (↑Perm.sign al • ↑Perm.sign ar • domCoprod (domDomCongr al a) (domDomCongr ar b))",
"state_before": "case a.e_f.h.e_f.h.mk\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\nσ✝ : Quotient (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb)))\nσ : Perm (ιa ⊕ ιb)\nal : Perm ιa\nar : Perm ιb\n⊢ ↑Perm.sign (σ * ↑(Perm.sumCongrHom ιa ιb) (al, ar)) •\n domDomCongr (σ * ↑(Perm.sumCongrHom ιa ιb) (al, ar)) (domCoprod a b) =\n ↑Perm.sign σ •\n ↑(domDomCongrEquiv σ)\n (↑Perm.sign (al, ar).fst •\n ↑Perm.sign (al, ar).snd • domCoprod (domDomCongr (al, ar).fst a) (domDomCongr (al, ar).snd b))",
"tactic": "dsimp only"
},
{
"state_after": "case a.e_f.h.e_f.h.mk\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\nσ✝ : Quotient (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb)))\nσ : Perm (ιa ⊕ ιb)\nal : Perm ιa\nar : Perm ιb\n⊢ ↑Perm.sign (σ * ↑(Perm.sumCongrHom ιa ιb) (al, ar)) •\n domDomCongr (σ * ↑(Perm.sumCongrHom ιa ιb) (al, ar)) (domCoprod a b) =\n ↑Perm.sign σ • ↑Perm.sign al • ↑Perm.sign ar • domDomCongr σ (domCoprod (domDomCongr al a) (domDomCongr ar b))",
"state_before": "case a.e_f.h.e_f.h.mk\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\nσ✝ : Quotient (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb)))\nσ : Perm (ιa ⊕ ιb)\nal : Perm ιa\nar : Perm ιb\n⊢ ↑Perm.sign (σ * ↑(Perm.sumCongrHom ιa ιb) (al, ar)) •\n domDomCongr (σ * ↑(Perm.sumCongrHom ιa ιb) (al, ar)) (domCoprod a b) =\n ↑Perm.sign σ •\n ↑(domDomCongrEquiv σ) (↑Perm.sign al • ↑Perm.sign ar • domCoprod (domDomCongr al a) (domDomCongr ar b))",
"tactic": "rw [← AddEquiv.coe_toAddMonoidHom, ← AddMonoidHom.coe_toIntLinearMap, LinearMap.map_smul_of_tower,\n LinearMap.map_smul_of_tower, AddMonoidHom.coe_toIntLinearMap, AddEquiv.coe_toAddMonoidHom,\n MultilinearMap.domDomCongrEquiv_apply]"
},
{
"state_after": "no goals",
"state_before": "case a.e_f.h.e_f.h.mk\nR : Type ?u.1140652\ninst✝²¹ : Semiring R\nM : Type ?u.1140658\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1140690\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1140720\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1140750\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1141138\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1141526\nι' : Type ?u.1141529\nι'' : Type ?u.1141532\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : MultilinearMap R' (fun x => Mᵢ) N₁\nb : MultilinearMap R' (fun x => Mᵢ) N₂\nσ✝ : Quotient (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb)))\nσ : Perm (ιa ⊕ ιb)\nal : Perm ιa\nar : Perm ιb\n⊢ ↑Perm.sign (σ * ↑(Perm.sumCongrHom ιa ιb) (al, ar)) •\n domDomCongr (σ * ↑(Perm.sumCongrHom ιa ιb) (al, ar)) (domCoprod a b) =\n ↑Perm.sign σ • ↑Perm.sign al • ↑Perm.sign ar • domDomCongr σ (domCoprod (domDomCongr al a) (domDomCongr ar b))",
"tactic": "rw [MultilinearMap.domDomCongr_mul, Perm.sign_mul, Perm.sumCongrHom_apply,\n MultilinearMap.domCoprod_domDomCongr_sumCongr, Perm.sign_sumCongr, mul_smul, mul_smul]"
}
] |
[
1181,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1145,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
hasFTaylorSeriesUpToOn_zero_iff
|
[
{
"state_after": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nH : ContinuousOn f s ∧ ∀ (x : E), x ∈ s → ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nm : ℕ\nhm : ↑m ≤ 0\n⊢ ContinuousOn (fun x => p x m) s",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\n⊢ HasFTaylorSeriesUpToOn 0 f p s ↔ ContinuousOn f s ∧ ∀ (x : E), x ∈ s → ContinuousMultilinearMap.uncurry0 (p x 0) = f x",
"tactic": "refine ⟨fun H => ⟨H.continuousOn, H.zero_eq⟩, fun H =>\n ⟨H.2, fun m hm => False.elim (not_le.2 hm bot_le), fun m hm ↦ ?_⟩⟩"
},
{
"state_after": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nH : ContinuousOn f s ∧ ∀ (x : E), x ∈ s → ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nhm : ↑0 ≤ 0\n⊢ ContinuousOn (fun x => p x 0) s",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nH : ContinuousOn f s ∧ ∀ (x : E), x ∈ s → ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nm : ℕ\nhm : ↑m ≤ 0\n⊢ ContinuousOn (fun x => p x m) s",
"tactic": "obtain rfl : m = 0 := by exact_mod_cast hm.antisymm (zero_le _)"
},
{
"state_after": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nH : ContinuousOn f s ∧ ∀ (x : E), x ∈ s → ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nhm : ↑0 ≤ 0\nthis : EqOn (fun x => p x 0) (↑(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 𝕜 E F)) ∘ f) s\n⊢ ContinuousOn f s",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nH : ContinuousOn f s ∧ ∀ (x : E), x ∈ s → ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nhm : ↑0 ≤ 0\nthis : EqOn (fun x => p x 0) (↑(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 𝕜 E F)) ∘ f) s\n⊢ ContinuousOn (fun x => p x 0) s",
"tactic": "rw [continuousOn_congr this, LinearIsometryEquiv.comp_continuousOn_iff]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nH : ContinuousOn f s ∧ ∀ (x : E), x ∈ s → ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nhm : ↑0 ≤ 0\nthis : EqOn (fun x => p x 0) (↑(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 𝕜 E F)) ∘ f) s\n⊢ ContinuousOn f s",
"tactic": "exact H.1"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nH : ContinuousOn f s ∧ ∀ (x : E), x ∈ s → ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nm : ℕ\nhm : ↑m ≤ 0\n⊢ m = 0",
"tactic": "exact_mod_cast hm.antisymm (zero_le _)"
}
] |
[
236,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
1
] |
Mathlib/CategoryTheory/Limits/Constructions/EpiMono.lean
|
CategoryTheory.preserves_mono_of_preservesLimit
|
[
{
"state_after": "C : Type u₁\nD : Type u₂\ninst✝³ : Category C\ninst✝² : Category D\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\ninst✝¹ : PreservesLimit (cospan f f) F\ninst✝ : Mono f\nthis :\n let_fun this := (_ : F.map (𝟙 X) ≫ F.map f = F.map (𝟙 X) ≫ F.map f);\n IsLimit (PullbackCone.mk (F.map (𝟙 X)) (F.map (𝟙 X)) this)\n⊢ Mono (F.map f)",
"state_before": "C : Type u₁\nD : Type u₂\ninst✝³ : Category C\ninst✝² : Category D\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\ninst✝¹ : PreservesLimit (cospan f f) F\ninst✝ : Mono f\n⊢ Mono (F.map f)",
"tactic": "have := isLimitPullbackConeMapOfIsLimit F _ (PullbackCone.isLimitMkIdId f)"
},
{
"state_after": "C : Type u₁\nD : Type u₂\ninst✝³ : Category C\ninst✝² : Category D\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\ninst✝¹ : PreservesLimit (cospan f f) F\ninst✝ : Mono f\nthis : IsLimit (PullbackCone.mk (𝟙 (F.obj X)) (𝟙 (F.obj X)) (_ : 𝟙 (F.obj X) ≫ F.map f = 𝟙 (F.obj X) ≫ F.map f))\n⊢ Mono (F.map f)",
"state_before": "C : Type u₁\nD : Type u₂\ninst✝³ : Category C\ninst✝² : Category D\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\ninst✝¹ : PreservesLimit (cospan f f) F\ninst✝ : Mono f\nthis :\n let_fun this := (_ : F.map (𝟙 X) ≫ F.map f = F.map (𝟙 X) ≫ F.map f);\n IsLimit (PullbackCone.mk (F.map (𝟙 X)) (F.map (𝟙 X)) this)\n⊢ Mono (F.map f)",
"tactic": "simp_rw [F.map_id] at this"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\nD : Type u₂\ninst✝³ : Category C\ninst✝² : Category D\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\ninst✝¹ : PreservesLimit (cospan f f) F\ninst✝ : Mono f\nthis : IsLimit (PullbackCone.mk (𝟙 (F.obj X)) (𝟙 (F.obj X)) (_ : 𝟙 (F.obj X) ≫ F.map f = 𝟙 (F.obj X) ≫ F.map f))\n⊢ Mono (F.map f)",
"tactic": "apply PullbackCone.mono_of_isLimitMkIdId _ this"
}
] |
[
40,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
36,
1
] |
Mathlib/Order/OmegaCompletePartialOrder.lean
|
OmegaCompletePartialOrder.id_continuous'
|
[] |
[
293,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
292,
1
] |
Mathlib/Order/JordanHolder.lean
|
CompositionSeries.append_succ_castAdd
|
[
{
"state_after": "no goals",
"state_before": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nh : top s₁ = bot s₂\ni : Fin s₁.length\n⊢ series (append s₁ s₂ h) (Fin.succ (↑(Fin.castAdd s₂.length) i)) = series s₁ (Fin.succ i)",
"tactic": "rw [coe_append, append_succ_castAdd_aux _ _ _ h]"
}
] |
[
529,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
527,
1
] |
Mathlib/Topology/Algebra/Order/LiminfLimsup.lean
|
iInf_eq_of_forall_le_of_tendsto
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Type u_2\nR : Type u_1\ninst✝³ : CompleteLinearOrder R\ninst✝² : TopologicalSpace R\ninst✝¹ : OrderTopology R\nx : R\nas : ι → R\nx_le : ∀ (i : ι), x ≤ as i\nF : Filter ι\ninst✝ : NeBot F\nas_lim : Tendsto as F (𝓝 x)\n⊢ ∀ (w : R), x < w → ∃ i, as i < w",
"state_before": "α : Type u\nβ : Type v\nι : Type u_2\nR : Type u_1\ninst✝³ : CompleteLinearOrder R\ninst✝² : TopologicalSpace R\ninst✝¹ : OrderTopology R\nx : R\nas : ι → R\nx_le : ∀ (i : ι), x ≤ as i\nF : Filter ι\ninst✝ : NeBot F\nas_lim : Tendsto as F (𝓝 x)\n⊢ (⨅ (i : ι), as i) = x",
"tactic": "refine' iInf_eq_of_forall_ge_of_forall_gt_exists_lt (fun i ↦ x_le i) _"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Type u_2\nR : Type u_1\ninst✝³ : CompleteLinearOrder R\ninst✝² : TopologicalSpace R\ninst✝¹ : OrderTopology R\nx : R\nas : ι → R\nx_le : ∀ (i : ι), x ≤ as i\nF : Filter ι\ninst✝ : NeBot F\nas_lim : Tendsto as F (𝓝 x)\n⊢ ∀ (w : R), x < w → ∃ i, as i < w",
"tactic": "apply fun w x_lt_w ↦ ‹Filter.NeBot F›.nonempty_of_mem (eventually_lt_of_tendsto_lt x_lt_w as_lim)"
}
] |
[
425,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
422,
1
] |
Mathlib/LinearAlgebra/Quotient.lean
|
Submodule.quotEquivOfEq_mk
|
[] |
[
656,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
653,
1
] |
Mathlib/Tactic/IntervalCases.lean
|
Mathlib.Tactic.IntervalCases.of_not_lt_left
|
[] |
[
132,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Independent.lean
|
Affine.Simplex.mkOfPoint_points
|
[] |
[
804,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
803,
1
] |
Mathlib/RingTheory/Ideal/QuotientOperations.lean
|
DoubleQuot.coe_quotQuotEquivCommₐ
|
[] |
[
796,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
795,
1
] |
Mathlib/Algebra/DirectLimit.lean
|
Field.DirectLimit.mul_inv_cancel
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁴ : Ring R\nι : Type v\ndec_ι : DecidableEq ι\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : Nonempty ι\ninst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝ : (i : ι) → Field (G i)\nf : (i j : ι) → i ≤ j → G i → G j\nf' : (i j : ι) → i ≤ j → G i →+* G j\np : Ring.DirectLimit G f\nhp : p ≠ 0\n⊢ p * inv G f p = 1",
"tactic": "rw [inv, dif_neg hp, Classical.choose_spec (DirectLimit.exists_inv G f hp)]"
}
] |
[
732,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
731,
11
] |
Mathlib/Logic/Encodable/Basic.lean
|
Directed.le_sequence
|
[] |
[
668,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
667,
1
] |
Mathlib/Data/Fin/VecNotation.lean
|
Matrix.cons_val_one
|
[
{
"state_after": "α : Type u\nm n o : ℕ\nm' : Type ?u.22403\nn' : Type ?u.22406\no' : Type ?u.22409\nx : α\nu : Fin (Nat.succ m) → α\n⊢ u 0 = vecHead u",
"state_before": "α : Type u\nm n o : ℕ\nm' : Type ?u.22403\nn' : Type ?u.22406\no' : Type ?u.22409\nx : α\nu : Fin (Nat.succ m) → α\n⊢ vecCons x u 1 = vecHead u",
"tactic": "rw [← Fin.succ_zero_eq_one, cons_val_succ]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nm n o : ℕ\nm' : Type ?u.22403\nn' : Type ?u.22406\no' : Type ?u.22409\nx : α\nu : Fin (Nat.succ m) → α\n⊢ u 0 = vecHead u",
"tactic": "rfl"
}
] |
[
207,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/RingTheory/PrincipalIdealDomain.lean
|
PrincipalIdealRing.ne_zero_of_mem_factors
|
[] |
[
291,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
1
] |
Mathlib/Algebra/Homology/Homotopy.lean
|
Homotopy.prevD_zero_cochainComplex
|
[
{
"state_after": "ι : Type ?u.383391\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\n⊢ f 0 (ComplexShape.prev (ComplexShape.up ℕ) 0) ≫ d Q (ComplexShape.prev (ComplexShape.up ℕ) 0) 0 = 0",
"state_before": "ι : Type ?u.383391\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\n⊢ ↑(prevD 0) f = 0",
"tactic": "dsimp [prevD]"
},
{
"state_after": "case a\nι : Type ?u.383391\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\n⊢ ¬ComplexShape.Rel (ComplexShape.up ℕ) (ComplexShape.prev (ComplexShape.up ℕ) 0) 0",
"state_before": "ι : Type ?u.383391\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\n⊢ f 0 (ComplexShape.prev (ComplexShape.up ℕ) 0) ≫ d Q (ComplexShape.prev (ComplexShape.up ℕ) 0) 0 = 0",
"tactic": "rw [Q.shape, comp_zero]"
},
{
"state_after": "case a\nι : Type ?u.383391\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\n⊢ ¬ComplexShape.Rel (ComplexShape.up ℕ) 0 0",
"state_before": "case a\nι : Type ?u.383391\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\n⊢ ¬ComplexShape.Rel (ComplexShape.up ℕ) (ComplexShape.prev (ComplexShape.up ℕ) 0) 0",
"tactic": "rw [CochainComplex.prev_nat_zero]"
},
{
"state_after": "case a\nι : Type ?u.383391\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\n⊢ ¬0 + 1 = 0",
"state_before": "case a\nι : Type ?u.383391\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\n⊢ ¬ComplexShape.Rel (ComplexShape.up ℕ) 0 0",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case a\nι : Type ?u.383391\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\n⊢ ¬0 + 1 = 0",
"tactic": "decide"
}
] |
[
606,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
603,
1
] |
Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.PreservesPushout.iso_hom
|
[] |
[
312,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
311,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
|
Complex.arg_nonneg_iff
|
[
{
"state_after": "case inl\n\n⊢ 0 ≤ arg 0 ↔ 0 ≤ 0.im\n\ncase inr\nz : ℂ\nh₀ : z ≠ 0\n⊢ 0 ≤ arg z ↔ 0 ≤ z.im",
"state_before": "z : ℂ\n⊢ 0 ≤ arg z ↔ 0 ≤ z.im",
"tactic": "rcases eq_or_ne z 0 with (rfl | h₀)"
},
{
"state_after": "no goals",
"state_before": "case inr\nz : ℂ\nh₀ : z ≠ 0\n⊢ 0 ≤ arg z ↔ 0 ≤ z.im",
"tactic": "calc\n 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) :=\n ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by\n contrapose!\n intro h\n exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩\n _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), MulZeroClass.zero_mul]"
},
{
"state_after": "no goals",
"state_before": "case inl\n\n⊢ 0 ≤ arg 0 ↔ 0 ≤ 0.im",
"tactic": "simp"
},
{
"state_after": "z : ℂ\nh₀ : z ≠ 0\n⊢ arg z < 0 → Real.sin (arg z) < 0",
"state_before": "z : ℂ\nh₀ : z ≠ 0\n⊢ 0 ≤ Real.sin (arg z) → 0 ≤ arg z",
"tactic": "contrapose!"
},
{
"state_after": "z : ℂ\nh₀ : z ≠ 0\nh : arg z < 0\n⊢ Real.sin (arg z) < 0",
"state_before": "z : ℂ\nh₀ : z ≠ 0\n⊢ arg z < 0 → Real.sin (arg z) < 0",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "z : ℂ\nh₀ : z ≠ 0\nh : arg z < 0\n⊢ Real.sin (arg z) < 0",
"tactic": "exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)"
},
{
"state_after": "no goals",
"state_before": "z : ℂ\nh₀ : z ≠ 0\n⊢ 0 ≤ Real.sin (arg z) ↔ 0 ≤ z.im",
"tactic": "rw [sin_arg, le_div_iff (abs.pos h₀), MulZeroClass.zero_mul]"
}
] |
[
172,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.mul_assoc
|
[
{
"state_after": "case a.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o → Type ?u.375873\nn' : o → Type ?u.375878\nR : Type ?u.375881\nS : Type ?u.375884\nα : Type v\nβ : Type w\nγ : Type ?u.375891\ninst✝² : NonUnitalSemiring α\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nL : Matrix l m α\nM : Matrix m n α\nN : Matrix n o α\ni✝ : l\nx✝ : o\n⊢ (L ⬝ M ⬝ N) i✝ x✝ = (L ⬝ (M ⬝ N)) i✝ x✝",
"state_before": "l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o → Type ?u.375873\nn' : o → Type ?u.375878\nR : Type ?u.375881\nS : Type ?u.375884\nα : Type v\nβ : Type w\nγ : Type ?u.375891\ninst✝² : NonUnitalSemiring α\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nL : Matrix l m α\nM : Matrix m n α\nN : Matrix n o α\n⊢ L ⬝ M ⬝ N = L ⬝ (M ⬝ N)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o → Type ?u.375873\nn' : o → Type ?u.375878\nR : Type ?u.375881\nS : Type ?u.375884\nα : Type v\nβ : Type w\nγ : Type ?u.375891\ninst✝² : NonUnitalSemiring α\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nL : Matrix l m α\nM : Matrix m n α\nN : Matrix n o α\ni✝ : l\nx✝ : o\n⊢ (L ⬝ M ⬝ N) i✝ x✝ = (L ⬝ (M ⬝ N)) i✝ x✝",
"tactic": "apply dotProduct_assoc"
}
] |
[
1159,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1156,
11
] |
Mathlib/FieldTheory/IntermediateField.lean
|
IntermediateField.prod_mem
|
[] |
[
229,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
11
] |
Mathlib/Topology/Separation.lean
|
t1Space_iff_disjoint_nhds_pure
|
[] |
[
516,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
515,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
MonoidHom.ker_toHomUnits
|
[
{
"state_after": "case h\nG : Type u_2\nG' : Type ?u.511685\ninst✝⁶ : Group G\ninst✝⁵ : Group G'\nA : Type ?u.511694\ninst✝⁴ : AddGroup A\nN : Type ?u.511700\nP : Type ?u.511703\ninst✝³ : Group N\ninst✝² : Group P\nK : Subgroup G\nM✝ : Type ?u.511720\ninst✝¹ : MulOneClass M✝\nM : Type u_1\ninst✝ : Monoid M\nf : G →* M\nx : G\n⊢ x ∈ ker (toHomUnits f) ↔ x ∈ ker f",
"state_before": "G : Type u_2\nG' : Type ?u.511685\ninst✝⁶ : Group G\ninst✝⁵ : Group G'\nA : Type ?u.511694\ninst✝⁴ : AddGroup A\nN : Type ?u.511700\nP : Type ?u.511703\ninst✝³ : Group N\ninst✝² : Group P\nK : Subgroup G\nM✝ : Type ?u.511720\ninst✝¹ : MulOneClass M✝\nM : Type u_1\ninst✝ : Monoid M\nf : G →* M\n⊢ ker (toHomUnits f) = ker f",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nG : Type u_2\nG' : Type ?u.511685\ninst✝⁶ : Group G\ninst✝⁵ : Group G'\nA : Type ?u.511694\ninst✝⁴ : AddGroup A\nN : Type ?u.511700\nP : Type ?u.511703\ninst✝³ : Group N\ninst✝² : Group P\nK : Subgroup G\nM✝ : Type ?u.511720\ninst✝¹ : MulOneClass M✝\nM : Type u_1\ninst✝ : Monoid M\nf : G →* M\nx : G\n⊢ x ∈ ker (toHomUnits f) ↔ x ∈ ker f",
"tactic": "simp [mem_ker, Units.ext_iff]"
}
] |
[
2780,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2778,
1
] |
Mathlib/Topology/Order.lean
|
tendsto_nhds_true
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : TopologicalSpace α\nl : Filter α\np : α → Prop\n⊢ Tendsto p l (𝓝 True) ↔ ∀ᶠ (x : α) in l, p x",
"tactic": "simp"
}
] |
[
916,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
915,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.inv_im
|
[
{
"state_after": "no goals",
"state_before": "K : Type u_1\nE : Type ?u.5463776\ninst✝ : IsROrC K\nz : K\n⊢ ↑im z⁻¹ = -↑im z / ↑normSq z",
"tactic": "rw [inv_def, normSq_eq_def', mul_comm, ofReal_mul_im, conj_im, div_eq_inv_mul]"
}
] |
[
561,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
560,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.map_prod
|
[] |
[
1196,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1194,
11
] |
Std/Data/String/Lemmas.lean
|
String.Iterator.ValidFor.prev_nil
|
[
{
"state_after": "r : List Char\nit : Iterator\nh : ValidFor [] r it\n⊢ ValidFor [] r { s := { data := r }, i := 0 }",
"state_before": "r : List Char\nit : Iterator\nh : ValidFor [] r it\n⊢ ValidFor [] r (Iterator.prev it)",
"tactic": "simp [Iterator.prev, h.toString, h.pos]"
},
{
"state_after": "no goals",
"state_before": "r : List Char\nit : Iterator\nh : ValidFor [] r it\n⊢ ValidFor [] r { s := { data := r }, i := 0 }",
"tactic": "constructor"
}
] |
[
562,
69
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
561,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Types.lean
|
CategoryTheory.Limits.Types.coequalizerIso_π_comp_hom
|
[] |
[
514,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
512,
1
] |
Mathlib/Topology/Sets/Opens.lean
|
TopologicalSpace.Opens.coe_top
|
[] |
[
197,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/Data/Dfinsupp/WellFounded.lean
|
Pi.Lex.wellFounded
|
[
{
"state_after": "case inl\nι : Type u_1\nα : ι → Type u_2\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝¹ : IsStrictTotalOrder ι r\ninst✝ : Finite ι\nhs : ∀ (i : ι), WellFounded (s i)\nh : IsEmpty ((i : ι) → α i)\n⊢ WellFounded (Pi.Lex r fun {i} => s i)\n\ncase inr.intro\nι : Type u_1\nα : ι → Type u_2\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝¹ : IsStrictTotalOrder ι r\ninst✝ : Finite ι\nhs : ∀ (i : ι), WellFounded (s i)\nx : (i : ι) → α i\n⊢ WellFounded (Pi.Lex r fun {i} => s i)",
"state_before": "ι : Type u_1\nα : ι → Type u_2\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝¹ : IsStrictTotalOrder ι r\ninst✝ : Finite ι\nhs : ∀ (i : ι), WellFounded (s i)\n⊢ WellFounded (Pi.Lex r fun {i} => s i)",
"tactic": "obtain h | ⟨⟨x⟩⟩ := isEmpty_or_nonempty (∀ i, α i)"
},
{
"state_after": "case inr.intro\nι : Type u_1\nα : ι → Type u_2\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝¹ : IsStrictTotalOrder ι r\ninst✝ : Finite ι\nhs : ∀ (i : ι), WellFounded (s i)\nx : (i : ι) → α i\nthis : (i : ι) → Zero (α i) := fun i => { zero := WellFounded.min (_ : WellFounded (s i)) ⊤ (_ : ∃ x, x ∈ ⊤) }\n⊢ WellFounded (Pi.Lex r fun {i} => s i)",
"state_before": "case inr.intro\nι : Type u_1\nα : ι → Type u_2\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝¹ : IsStrictTotalOrder ι r\ninst✝ : Finite ι\nhs : ∀ (i : ι), WellFounded (s i)\nx : (i : ι) → α i\n⊢ WellFounded (Pi.Lex r fun {i} => s i)",
"tactic": "letI : ∀ i, Zero (α i) := fun i => ⟨(hs i).min ⊤ ⟨x i, trivial⟩⟩"
},
{
"state_after": "case inr.intro\nι : Type u_1\nα : ι → Type u_2\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝¹ : IsStrictTotalOrder ι r\ninst✝ : Finite ι\nhs : ∀ (i : ι), WellFounded (s i)\nx : (i : ι) → α i\nthis✝ : (i : ι) → Zero (α i) := fun i => { zero := WellFounded.min (_ : WellFounded (s i)) ⊤ (_ : ∃ x, x ∈ ⊤) }\nthis : IsTrans ι (Function.swap r)\n⊢ WellFounded (Pi.Lex r fun {i} => s i)",
"state_before": "case inr.intro\nι : Type u_1\nα : ι → Type u_2\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝¹ : IsStrictTotalOrder ι r\ninst✝ : Finite ι\nhs : ∀ (i : ι), WellFounded (s i)\nx : (i : ι) → α i\nthis : (i : ι) → Zero (α i) := fun i => { zero := WellFounded.min (_ : WellFounded (s i)) ⊤ (_ : ∃ x, x ∈ ⊤) }\n⊢ WellFounded (Pi.Lex r fun {i} => s i)",
"tactic": "haveI := IsTrans.swap r"
},
{
"state_after": "case inr.intro\nι : Type u_1\nα : ι → Type u_2\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝¹ : IsStrictTotalOrder ι r\ninst✝ : Finite ι\nhs : ∀ (i : ι), WellFounded (s i)\nx : (i : ι) → α i\nthis✝¹ : (i : ι) → Zero (α i) := fun i => { zero := WellFounded.min (_ : WellFounded (s i)) ⊤ (_ : ∃ x, x ∈ ⊤) }\nthis✝ : IsTrans ι (Function.swap r)\nthis : IsIrrefl ι (Function.swap r)\n⊢ WellFounded (Pi.Lex r fun {i} => s i)",
"state_before": "case inr.intro\nι : Type u_1\nα : ι → Type u_2\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝¹ : IsStrictTotalOrder ι r\ninst✝ : Finite ι\nhs : ∀ (i : ι), WellFounded (s i)\nx : (i : ι) → α i\nthis✝ : (i : ι) → Zero (α i) := fun i => { zero := WellFounded.min (_ : WellFounded (s i)) ⊤ (_ : ∃ x, x ∈ ⊤) }\nthis : IsTrans ι (Function.swap r)\n⊢ WellFounded (Pi.Lex r fun {i} => s i)",
"tactic": "haveI := IsIrrefl.swap r"
},
{
"state_after": "case inr.intro\nι : Type u_1\nα : ι → Type u_2\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝¹ : IsStrictTotalOrder ι r\ninst✝ : Finite ι\nhs : ∀ (i : ι), WellFounded (s i)\nx : (i : ι) → α i\nthis✝² : (i : ι) → Zero (α i) := fun i => { zero := WellFounded.min (_ : WellFounded (s i)) ⊤ (_ : ∃ x, x ∈ ⊤) }\nthis✝¹ : IsTrans ι (Function.swap r)\nthis✝ : IsIrrefl ι (Function.swap r)\nthis : Fintype ι\n⊢ WellFounded (Pi.Lex r fun {i} => s i)",
"state_before": "case inr.intro\nι : Type u_1\nα : ι → Type u_2\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝¹ : IsStrictTotalOrder ι r\ninst✝ : Finite ι\nhs : ∀ (i : ι), WellFounded (s i)\nx : (i : ι) → α i\nthis✝¹ : (i : ι) → Zero (α i) := fun i => { zero := WellFounded.min (_ : WellFounded (s i)) ⊤ (_ : ∃ x, x ∈ ⊤) }\nthis✝ : IsTrans ι (Function.swap r)\nthis : IsIrrefl ι (Function.swap r)\n⊢ WellFounded (Pi.Lex r fun {i} => s i)",
"tactic": "haveI := Fintype.ofFinite ι"
},
{
"state_after": "case inr.intro.refine'_1\nι : Type u_1\nα : ι → Type u_2\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝¹ : IsStrictTotalOrder ι r\ninst✝ : Finite ι\nhs : ∀ (i : ι), WellFounded (s i)\nx : (i : ι) → α i\nthis✝² : (i : ι) → Zero (α i) := fun i => { zero := WellFounded.min (_ : WellFounded (s i)) ⊤ (_ : ∃ x, x ∈ ⊤) }\nthis✝¹ : IsTrans ι (Function.swap r)\nthis✝ : IsIrrefl ι (Function.swap r)\nthis : Fintype ι\ni : ι\na : α i\n⊢ ¬s i a 0\n\ncase inr.intro.refine'_2\nι : Type u_1\nα : ι → Type u_2\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝¹ : IsStrictTotalOrder ι r\ninst✝ : Finite ι\nhs : ∀ (i : ι), WellFounded (s i)\nx : (i : ι) → α i\nthis✝² : (i : ι) → Zero (α i) := fun i => { zero := WellFounded.min (_ : WellFounded (s i)) ⊤ (_ : ∃ x, x ∈ ⊤) }\nthis✝¹ : IsTrans ι (Function.swap r)\nthis✝ : IsIrrefl ι (Function.swap r)\nthis : Fintype ι\n⊢ WellFounded (Function.swap fun j i => r j i)",
"state_before": "case inr.intro\nι : Type u_1\nα : ι → Type u_2\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝¹ : IsStrictTotalOrder ι r\ninst✝ : Finite ι\nhs : ∀ (i : ι), WellFounded (s i)\nx : (i : ι) → α i\nthis✝² : (i : ι) → Zero (α i) := fun i => { zero := WellFounded.min (_ : WellFounded (s i)) ⊤ (_ : ∃ x, x ∈ ⊤) }\nthis✝¹ : IsTrans ι (Function.swap r)\nthis✝ : IsIrrefl ι (Function.swap r)\nthis : Fintype ι\n⊢ WellFounded (Pi.Lex r fun {i} => s i)",
"tactic": "refine' InvImage.wf equivFunOnFintype.symm (Lex.wellFounded' (fun i a => _) hs _)"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.refine'_1\nι : Type u_1\nα : ι → Type u_2\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝¹ : IsStrictTotalOrder ι r\ninst✝ : Finite ι\nhs : ∀ (i : ι), WellFounded (s i)\nx : (i : ι) → α i\nthis✝² : (i : ι) → Zero (α i) := fun i => { zero := WellFounded.min (_ : WellFounded (s i)) ⊤ (_ : ∃ x, x ∈ ⊤) }\nthis✝¹ : IsTrans ι (Function.swap r)\nthis✝ : IsIrrefl ι (Function.swap r)\nthis : Fintype ι\ni : ι\na : α i\n⊢ ¬s i a 0\n\ncase inr.intro.refine'_2\nι : Type u_1\nα : ι → Type u_2\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝¹ : IsStrictTotalOrder ι r\ninst✝ : Finite ι\nhs : ∀ (i : ι), WellFounded (s i)\nx : (i : ι) → α i\nthis✝² : (i : ι) → Zero (α i) := fun i => { zero := WellFounded.min (_ : WellFounded (s i)) ⊤ (_ : ∃ x, x ∈ ⊤) }\nthis✝¹ : IsTrans ι (Function.swap r)\nthis✝ : IsIrrefl ι (Function.swap r)\nthis : Fintype ι\n⊢ WellFounded (Function.swap fun j i => r j i)",
"tactic": "exacts [(hs i).not_lt_min ⊤ _ trivial, Finite.wellFounded_of_trans_of_irrefl (Function.swap r)]"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Type u_1\nα : ι → Type u_2\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝¹ : IsStrictTotalOrder ι r\ninst✝ : Finite ι\nhs : ∀ (i : ι), WellFounded (s i)\nh : IsEmpty ((i : ι) → α i)\n⊢ WellFounded (Pi.Lex r fun {i} => s i)",
"tactic": "convert emptyWf.wf"
}
] |
[
194,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.sSup_eq_sup
|
[] |
[
1252,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1251,
1
] |
Mathlib/MeasureTheory/Integral/CircleIntegral.lean
|
measurable_circleMap
|
[] |
[
194,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Combinatorics/Composition.lean
|
List.splitWrtComposition_join
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\nα : Type u_1\nL : List (List α)\nc : Composition (length (join L))\nh : map length L = c.blocks\n⊢ splitWrtComposition (join L) c = L",
"tactic": "simp only [eq_self_iff_true, and_self_iff, eq_iff_join_eq, join_splitWrtComposition,\n map_length_splitWrtComposition, h]"
}
] |
[
777,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
774,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometryEquiv.toIsometryEquiv_symm
|
[] |
[
768,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
767,
1
] |
Mathlib/Order/SuccPred/LinearLocallyFinite.lean
|
toZ_of_eq
|
[
{
"state_after": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\n⊢ ↑(Nat.find (_ : ∃ n, (succ^[n]) i0 = i0)) = 0",
"state_before": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\n⊢ toZ i0 i0 = 0",
"tactic": "rw [toZ_of_ge le_rfl]"
},
{
"state_after": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\n⊢ Nat.find (_ : ∃ n, (succ^[n]) i0 = i0) = 0",
"state_before": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\n⊢ ↑(Nat.find (_ : ∃ n, (succ^[n]) i0 = i0)) = 0",
"tactic": "norm_cast"
},
{
"state_after": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\n⊢ (succ^[0]) i0 = i0",
"state_before": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\n⊢ Nat.find (_ : ∃ n, (succ^[n]) i0 = i0) = 0",
"tactic": "refine' le_antisymm (Nat.find_le _) (zero_le _)"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\n⊢ (succ^[0]) i0 = i0",
"tactic": "rw [Function.iterate_zero, id.def]"
}
] |
[
213,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
209,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.sameCycle_subtypePerm
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.206150\nα : Type u_1\nβ : Type ?u.206156\nf g : Perm α\np : α → Prop\nx✝ y✝ z : α\nh : ∀ (x : α), p x ↔ p (↑f x)\nx y : { x // p x }\nn : ℤ\n⊢ ↑(subtypePerm f h ^ n) x = y ↔ ↑(f ^ n) ↑x = ↑y",
"tactic": "simp [Subtype.ext_iff]"
}
] |
[
226,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
224,
1
] |
Mathlib/Order/Filter/Germ.lean
|
Filter.Germ.coe_one
|
[] |
[
357,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
356,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.node3L_size
|
[
{
"state_after": "α : Type u_1\nl : Ordnode α\nx : α\nm : Ordnode α\ny : α\nr : Ordnode α\n⊢ (((match l with\n | nil => 0\n | node sz l x r => sz) +\n match m with\n | nil => 0\n | node sz l x r => sz) +\n 1 +\n match r with\n | nil => 0\n | node sz l x r => sz) +\n 1 =\n (((match l with\n | nil => 0\n | node sz l x r => sz) +\n match m with\n | nil => 0\n | node sz l x r => sz) +\n match r with\n | nil => 0\n | node sz l x r => sz) +\n 2",
"state_before": "α : Type u_1\nl : Ordnode α\nx : α\nm : Ordnode α\ny : α\nr : Ordnode α\n⊢ size (node3L l x m y r) = size l + size m + size r + 2",
"tactic": "dsimp [node3L, node', size]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : Ordnode α\nx : α\nm : Ordnode α\ny : α\nr : Ordnode α\n⊢ (((match l with\n | nil => 0\n | node sz l x r => sz) +\n match m with\n | nil => 0\n | node sz l x r => sz) +\n 1 +\n match r with\n | nil => 0\n | node sz l x r => sz) +\n 1 =\n (((match l with\n | nil => 0\n | node sz l x r => sz) +\n match m with\n | nil => 0\n | node sz l x r => sz) +\n match r with\n | nil => 0\n | node sz l x r => sz) +\n 2",
"tactic": "rw [add_right_comm _ 1]"
}
] |
[
390,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
389,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.