file_path
stringlengths 11
79
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stringlengths 2
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stringclasses 4
values | url
stringclasses 4
values | start
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|---|---|---|---|---|---|---|
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.verts_sSup
|
[] |
[
424,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
423,
1
] |
Mathlib/Topology/Algebra/Order/MonotoneConvergence.lean
|
tendsto_atTop_isLUB
|
[
{
"state_after": "α : Type u_2\nβ : Type ?u.5253\nι : Type u_1\ninst✝³ : Preorder ι\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : SupConvergenceClass α\nf : ι → α\na : α\nh_mono : Monotone f\nha : IsLUB (range f) a\nthis : Tendsto (rangeFactorization f) atTop atTop\n⊢ Tendsto f atTop (𝓝 a)\n\ncase this\nα : Type u_2\nβ : Type ?u.5253\nι : Type u_1\ninst✝³ : Preorder ι\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : SupConvergenceClass α\nf : ι → α\na : α\nh_mono : Monotone f\nha : IsLUB (range f) a\n⊢ Tendsto (rangeFactorization f) atTop atTop",
"state_before": "α : Type u_2\nβ : Type ?u.5253\nι : Type u_1\ninst✝³ : Preorder ι\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : SupConvergenceClass α\nf : ι → α\na : α\nh_mono : Monotone f\nha : IsLUB (range f) a\n⊢ Tendsto f atTop (𝓝 a)",
"tactic": "suffices : Tendsto (rangeFactorization f) atTop atTop"
},
{
"state_after": "case this\nα : Type u_2\nβ : Type ?u.5253\nι : Type u_1\ninst✝³ : Preorder ι\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : SupConvergenceClass α\nf : ι → α\na : α\nh_mono : Monotone f\nha : IsLUB (range f) a\n⊢ Tendsto (rangeFactorization f) atTop atTop",
"state_before": "α : Type u_2\nβ : Type ?u.5253\nι : Type u_1\ninst✝³ : Preorder ι\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : SupConvergenceClass α\nf : ι → α\na : α\nh_mono : Monotone f\nha : IsLUB (range f) a\nthis : Tendsto (rangeFactorization f) atTop atTop\n⊢ Tendsto f atTop (𝓝 a)\n\ncase this\nα : Type u_2\nβ : Type ?u.5253\nι : Type u_1\ninst✝³ : Preorder ι\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : SupConvergenceClass α\nf : ι → α\na : α\nh_mono : Monotone f\nha : IsLUB (range f) a\n⊢ Tendsto (rangeFactorization f) atTop atTop",
"tactic": "exact (SupConvergenceClass.tendsto_coe_atTop_isLUB _ _ ha).comp this"
},
{
"state_after": "no goals",
"state_before": "case this\nα : Type u_2\nβ : Type ?u.5253\nι : Type u_1\ninst✝³ : Preorder ι\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : SupConvergenceClass α\nf : ι → α\na : α\nh_mono : Monotone f\nha : IsLUB (range f) a\n⊢ Tendsto (rangeFactorization f) atTop atTop",
"tactic": "exact h_mono.rangeFactorization.tendsto_atTop_atTop fun b => b.2.imp fun a ha => ha.ge"
}
] |
[
102,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/RingTheory/Ideal/LocalRing.lean
|
isLocalRingHom_of_comp
|
[] |
[
258,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
1
] |
Mathlib/Analysis/SpecialFunctions/CompareExp.lean
|
Complex.IsExpCmpFilter.tendsto_abs
|
[] |
[
107,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.measure_union_lt_top
|
[] |
[
310,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
309,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.mem_bot
|
[] |
[
700,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
699,
1
] |
Mathlib/InformationTheory/Hamming.lean
|
hammingDist_comm
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.2207\nι : Type u_1\nβ : ι → Type u_2\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → DecidableEq (β i)\nγ : ι → Type ?u.2239\ninst✝ : (i : ι) → DecidableEq (γ i)\nx y : (i : ι) → β i\n⊢ hammingDist x y = hammingDist y x",
"tactic": "simp_rw [hammingDist, ne_comm]"
}
] |
[
61,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean
|
isBoundedLinearMap_prod_multilinear
|
[
{
"state_after": "case H\n𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.71705\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\ninst✝² : Fintype ι\nE : ι → Type u_1\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\np₁ p₂ : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\n⊢ ↑(ContinuousMultilinearMap.prod (p₁ + p₂).fst (p₁ + p₂).snd) m =\n ↑(ContinuousMultilinearMap.prod p₁.fst p₁.snd + ContinuousMultilinearMap.prod p₂.fst p₂.snd) m",
"state_before": "𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.71705\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\ninst✝² : Fintype ι\nE : ι → Type u_1\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\np₁ p₂ : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G\n⊢ ContinuousMultilinearMap.prod (p₁ + p₂).fst (p₁ + p₂).snd =\n ContinuousMultilinearMap.prod p₁.fst p₁.snd + ContinuousMultilinearMap.prod p₂.fst p₂.snd",
"tactic": "ext1 m"
},
{
"state_after": "no goals",
"state_before": "case H\n𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.71705\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\ninst✝² : Fintype ι\nE : ι → Type u_1\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\np₁ p₂ : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\n⊢ ↑(ContinuousMultilinearMap.prod (p₁ + p₂).fst (p₁ + p₂).snd) m =\n ↑(ContinuousMultilinearMap.prod p₁.fst p₁.snd + ContinuousMultilinearMap.prod p₂.fst p₂.snd) m",
"tactic": "rfl"
},
{
"state_after": "case H\n𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.71705\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\ninst✝² : Fintype ι\nE : ι → Type u_1\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\nc : 𝕜\np : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\n⊢ ↑(ContinuousMultilinearMap.prod (c • p).fst (c • p).snd) m = ↑(c • ContinuousMultilinearMap.prod p.fst p.snd) m",
"state_before": "𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.71705\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\ninst✝² : Fintype ι\nE : ι → Type u_1\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\nc : 𝕜\np : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G\n⊢ ContinuousMultilinearMap.prod (c • p).fst (c • p).snd = c • ContinuousMultilinearMap.prod p.fst p.snd",
"tactic": "ext1 m"
},
{
"state_after": "no goals",
"state_before": "case H\n𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.71705\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\ninst✝² : Fintype ι\nE : ι → Type u_1\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\nc : 𝕜\np : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\n⊢ ↑(ContinuousMultilinearMap.prod (c • p).fst (c • p).snd) m = ↑(c • ContinuousMultilinearMap.prod p.fst p.snd) m",
"tactic": "rfl"
},
{
"state_after": "𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.71705\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\ninst✝² : Fintype ι\nE : ι → Type u_1\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\np : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G\n⊢ ‖ContinuousMultilinearMap.prod p.fst p.snd‖ ≤ ‖p‖",
"state_before": "𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.71705\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\ninst✝² : Fintype ι\nE : ι → Type u_1\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\np : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G\n⊢ ‖ContinuousMultilinearMap.prod p.fst p.snd‖ ≤ 1 * ‖p‖",
"tactic": "rw [one_mul]"
},
{
"state_after": "𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.71705\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\ninst✝² : Fintype ι\nE : ι → Type u_1\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\np : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G\n⊢ ∀ (m : (i : ι) → E i), ‖↑(ContinuousMultilinearMap.prod p.fst p.snd) m‖ ≤ ‖p‖ * ∏ i : ι, ‖m i‖",
"state_before": "𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.71705\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\ninst✝² : Fintype ι\nE : ι → Type u_1\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\np : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G\n⊢ ‖ContinuousMultilinearMap.prod p.fst p.snd‖ ≤ ‖p‖",
"tactic": "apply ContinuousMultilinearMap.op_norm_le_bound _ (norm_nonneg _) _"
},
{
"state_after": "𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.71705\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\ninst✝² : Fintype ι\nE : ι → Type u_1\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\np : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\n⊢ ‖↑(ContinuousMultilinearMap.prod p.fst p.snd) m‖ ≤ ‖p‖ * ∏ i : ι, ‖m i‖",
"state_before": "𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.71705\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\ninst✝² : Fintype ι\nE : ι → Type u_1\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\np : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G\n⊢ ∀ (m : (i : ι) → E i), ‖↑(ContinuousMultilinearMap.prod p.fst p.snd) m‖ ≤ ‖p‖ * ∏ i : ι, ‖m i‖",
"tactic": "intro m"
},
{
"state_after": "𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.71705\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\ninst✝² : Fintype ι\nE : ι → Type u_1\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\np : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\n⊢ ‖(↑p.fst m, ↑p.snd m).fst‖ ≤ ‖p‖ * ∏ i : ι, ‖m i‖ ∧ ‖(↑p.fst m, ↑p.snd m).snd‖ ≤ ‖p‖ * ∏ i : ι, ‖m i‖",
"state_before": "𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.71705\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\ninst✝² : Fintype ι\nE : ι → Type u_1\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\np : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\n⊢ ‖↑(ContinuousMultilinearMap.prod p.fst p.snd) m‖ ≤ ‖p‖ * ∏ i : ι, ‖m i‖",
"tactic": "rw [ContinuousMultilinearMap.prod_apply, norm_prod_le_iff]"
},
{
"state_after": "case left\n𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.71705\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\ninst✝² : Fintype ι\nE : ι → Type u_1\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\np : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\n⊢ ‖(↑p.fst m, ↑p.snd m).fst‖ ≤ ‖p‖ * ∏ i : ι, ‖m i‖\n\ncase right\n𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.71705\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\ninst✝² : Fintype ι\nE : ι → Type u_1\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\np : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\n⊢ ‖(↑p.fst m, ↑p.snd m).snd‖ ≤ ‖p‖ * ∏ i : ι, ‖m i‖",
"state_before": "𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.71705\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\ninst✝² : Fintype ι\nE : ι → Type u_1\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\np : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\n⊢ ‖(↑p.fst m, ↑p.snd m).fst‖ ≤ ‖p‖ * ∏ i : ι, ‖m i‖ ∧ ‖(↑p.fst m, ↑p.snd m).snd‖ ≤ ‖p‖ * ∏ i : ι, ‖m i‖",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case left\n𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.71705\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\ninst✝² : Fintype ι\nE : ι → Type u_1\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\np : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\n⊢ ‖(↑p.fst m, ↑p.snd m).fst‖ ≤ ‖p‖ * ∏ i : ι, ‖m i‖",
"tactic": "exact (p.1.le_op_norm m).trans (mul_le_mul_of_nonneg_right (norm_fst_le p)\n (Finset.prod_nonneg fun i _ => norm_nonneg _))"
},
{
"state_after": "no goals",
"state_before": "case right\n𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.71705\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\ninst✝² : Fintype ι\nE : ι → Type u_1\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\np : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\n⊢ ‖(↑p.fst m, ↑p.snd m).snd‖ ≤ ‖p‖ * ∏ i : ι, ‖m i‖",
"tactic": "exact (p.2.le_op_norm m).trans (mul_le_mul_of_nonneg_right (norm_snd_le p)\n (Finset.prod_nonneg fun i _ => norm_nonneg _))"
}
] |
[
237,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/RingTheory/Localization/AtPrime.lean
|
Localization.AtPrime.comap_maximalIdeal
|
[] |
[
192,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Prod.lean
|
fderiv_fst
|
[] |
[
215,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/Data/Real/NNReal.lean
|
Real.toNNReal_le_toNNReal
|
[] |
[
666,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
665,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
RingHom.map_dotProduct
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.1289823\nm : Type ?u.1289826\nn : Type u_3\no : Type ?u.1289832\nm' : o → Type ?u.1289837\nn' : o → Type ?u.1289842\nR : Type u_1\nS : Type u_2\nα : Type v\nβ : Type w\nγ : Type ?u.1289855\ninst✝⁴ : Fintype n\ninst✝³ : NonAssocSemiring α\ninst✝² : NonAssocSemiring β\ninst✝¹ : NonAssocSemiring R\ninst✝ : NonAssocSemiring S\nf : R →+* S\nv w : n → R\n⊢ ↑f (v ⬝ᵥ w) = ↑f ∘ v ⬝ᵥ ↑f ∘ w",
"tactic": "simp only [Matrix.dotProduct, f.map_sum, f.map_mul, Function.comp]"
}
] |
[
2942,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2940,
1
] |
src/lean/Init/Data/List/Basic.lean
|
List.append_eq_appendTR
|
[
{
"state_after": "case h\nα : Type u\nβ : Type v\nγ : Type w\n⊢ ∀ (x : Type u_1), List.append = appendTR",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\n⊢ @List.append = @appendTR",
"tactic": "apply funext"
},
{
"state_after": "case h\nα✝ : Type u\nβ : Type v\nγ : Type w\nα : Type u_1\n⊢ List.append = appendTR",
"state_before": "case h\nα : Type u\nβ : Type v\nγ : Type w\n⊢ ∀ (x : Type u_1), List.append = appendTR",
"tactic": "intro α"
},
{
"state_after": "case h.h\nα✝ : Type u\nβ : Type v\nγ : Type w\nα : Type u_1\n⊢ ∀ (x : List α), List.append x = appendTR x",
"state_before": "case h\nα✝ : Type u\nβ : Type v\nγ : Type w\nα : Type u_1\n⊢ List.append = appendTR",
"tactic": "apply funext"
},
{
"state_after": "case h.h\nα✝ : Type u\nβ : Type v\nγ : Type w\nα : Type u_1\nas : List α\n⊢ List.append as = appendTR as",
"state_before": "case h.h\nα✝ : Type u\nβ : Type v\nγ : Type w\nα : Type u_1\n⊢ ∀ (x : List α), List.append x = appendTR x",
"tactic": "intro as"
},
{
"state_after": "case h.h.h\nα✝ : Type u\nβ : Type v\nγ : Type w\nα : Type u_1\nas : List α\n⊢ ∀ (x : List α), List.append as x = appendTR as x",
"state_before": "case h.h\nα✝ : Type u\nβ : Type v\nγ : Type w\nα : Type u_1\nas : List α\n⊢ List.append as = appendTR as",
"tactic": "apply funext"
},
{
"state_after": "case h.h.h\nα✝ : Type u\nβ : Type v\nγ : Type w\nα : Type u_1\nas bs : List α\n⊢ List.append as bs = appendTR as bs",
"state_before": "case h.h.h\nα✝ : Type u\nβ : Type v\nγ : Type w\nα : Type u_1\nas : List α\n⊢ ∀ (x : List α), List.append as x = appendTR as x",
"tactic": "intro bs"
},
{
"state_after": "case h.h.h\nα✝ : Type u\nβ : Type v\nγ : Type w\nα : Type u_1\nas bs : List α\n⊢ List.append as bs = reverseAux (reverseAux as nil) bs",
"state_before": "case h.h.h\nα✝ : Type u\nβ : Type v\nγ : Type w\nα : Type u_1\nas bs : List α\n⊢ List.append as bs = appendTR as bs",
"tactic": "simp [appendTR, reverse]"
},
{
"state_after": "no goals",
"state_before": "case h.h.h\nα✝ : Type u\nβ : Type v\nγ : Type w\nα : Type u_1\nas bs : List α\n⊢ List.append as bs = reverseAux (reverseAux as nil) bs",
"tactic": "induction as with\n| nil => rfl\n| cons a as ih =>\n simp [reverseAux, List.append, ih, reverseAux_reverseAux]"
},
{
"state_after": "no goals",
"state_before": "case h.h.h.nil\nα✝ : Type u\nβ : Type v\nγ : Type w\nα : Type u_1\nbs : List α\n⊢ List.append nil bs = reverseAux (reverseAux nil nil) bs",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case h.h.h.cons\nα✝ : Type u\nβ : Type v\nγ : Type w\nα : Type u_1\nbs : List α\na : α\nas : List α\nih : List.append as bs = reverseAux (reverseAux as nil) bs\n⊢ List.append (a :: as) bs = reverseAux (reverseAux (a :: as) nil) bs",
"tactic": "simp [reverseAux, List.append, ih, reverseAux_reverseAux]"
}
] |
[
88,
62
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
82,
10
] |
Mathlib/CategoryTheory/Subobject/MonoOver.lean
|
CategoryTheory.MonoOver.lift_comm
|
[] |
[
182,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.dual_balanceR
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\n⊢ dual (balanceR l x r) = balanceL (dual r) x (dual l)",
"tactic": "rw [← dual_dual (balanceL _ _ _), dual_balanceL, dual_dual, dual_dual]"
}
] |
[
371,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
369,
1
] |
Mathlib/Data/Part.lean
|
Part.mem_coe
|
[] |
[
354,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
353,
1
] |
Mathlib/Topology/Order/Basic.lean
|
Monotone.map_ciSup_of_continuousAt
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderClosedTopology β\ninst✝ : Nonempty γ\nf : α → β\ng : γ → α\nCf : ContinuousAt f (⨆ (i : γ), g i)\nMf : Monotone f\nH : BddAbove (range g)\n⊢ sSup (range (f ∘ fun i => g i)) = sSup (range fun i => f (g i))",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderClosedTopology β\ninst✝ : Nonempty γ\nf : α → β\ng : γ → α\nCf : ContinuousAt f (⨆ (i : γ), g i)\nMf : Monotone f\nH : BddAbove (range g)\n⊢ f (⨆ (i : γ), g i) = ⨆ (i : γ), f (g i)",
"tactic": "rw [iSup, Mf.map_csSup_of_continuousAt Cf (range_nonempty _) H, ← range_comp, iSup]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderClosedTopology β\ninst✝ : Nonempty γ\nf : α → β\ng : γ → α\nCf : ContinuousAt f (⨆ (i : γ), g i)\nMf : Monotone f\nH : BddAbove (range g)\n⊢ sSup (range (f ∘ fun i => g i)) = sSup (range fun i => f (g i))",
"tactic": "rfl"
}
] |
[
2806,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2804,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
self_sub_toIcoMod
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ b - toIcoMod hp a b = toIcoDiv hp a b • p",
"tactic": "rw [toIcoMod, sub_sub_cancel]"
}
] |
[
146,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Data/Int/Interval.lean
|
Int.card_Ioo_of_lt
|
[
{
"state_after": "no goals",
"state_before": "a b : ℤ\nh : a < b\n⊢ ↑(card (Ioo a b)) = b - a - 1",
"tactic": "rw [card_Ioo, sub_sub, toNat_sub_of_le h]"
}
] |
[
143,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/CategoryTheory/Limits/Constructions/Equalizers.lean
|
CategoryTheory.Limits.HasEqualizersOfHasPullbacksAndBinaryProducts.pullbackFst_eq_pullback_snd
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\nG : C ⥤ D\ninst✝¹ : HasBinaryProducts C\ninst✝ : HasPullbacks C\nF : WalkingParallelPair ⥤ C\n⊢ pullbackFst F = pullback.snd",
"tactic": "convert (eq_whisker pullback.condition Limits.prod.fst :\n (_ : constructEqualizer F ⟶ F.obj WalkingParallelPair.zero) = _) <;> simp"
}
] |
[
58,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/Topology/Algebra/ContinuousMonoidHom.lean
|
ContinuousMonoidHom.ext
|
[] |
[
124,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/MeasureTheory/Lattice.lean
|
AEMeasurable.const_sup
|
[] |
[
122,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/Topology/NoetherianSpace.lean
|
TopologicalSpace.noetherianSpace_iff_isCompact
|
[] |
[
110,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Mathlib/RingTheory/RootsOfUnity/Complex.lean
|
Complex.card_primitiveRoots
|
[
{
"state_after": "case pos\nk : ℕ\nh : k = 0\n⊢ Finset.card (primitiveRoots k ℂ) = φ k\n\ncase neg\nk : ℕ\nh : ¬k = 0\n⊢ Finset.card (primitiveRoots k ℂ) = φ k",
"state_before": "k : ℕ\n⊢ Finset.card (primitiveRoots k ℂ) = φ k",
"tactic": "by_cases h : k = 0"
},
{
"state_after": "no goals",
"state_before": "case neg\nk : ℕ\nh : ¬k = 0\n⊢ Finset.card (primitiveRoots k ℂ) = φ k",
"tactic": "exact (isPrimitiveRoot_exp k h).card_primitiveRoots"
},
{
"state_after": "no goals",
"state_before": "case pos\nk : ℕ\nh : k = 0\n⊢ Finset.card (primitiveRoots k ℂ) = φ k",
"tactic": "simp [h]"
}
] |
[
101,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/NumberTheory/PythagoreanTriples.lean
|
pythagoreanTriple_comm
|
[
{
"state_after": "x y z : ℤ\n⊢ x * x + y * y = z * z ↔ y * y + x * x = z * z",
"state_before": "x y z : ℤ\n⊢ PythagoreanTriple x y z ↔ PythagoreanTriple y x z",
"tactic": "delta PythagoreanTriple"
},
{
"state_after": "no goals",
"state_before": "x y z : ℤ\n⊢ x * x + y * y = z * z ↔ y * y + x * x = z * z",
"tactic": "rw [add_comm]"
}
] |
[
57,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Algebra/Symmetrized.lean
|
SymAlg.sym_mul_self
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Semiring α\ninst✝ : Invertible 2\na : α\n⊢ ↑sym (a * a) = ↑sym a * ↑sym a",
"tactic": "rw [sym_mul_sym, ← two_mul, invOf_mul_self_assoc]"
}
] |
[
335,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
1
] |
Mathlib/Data/List/Perm.lean
|
List.DecEq_eq
|
[
{
"state_after": "case h.h\nα✝ : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α✝\nα : Type u_1\ninst✝ : DecidableEq α\nl₁ l₂ : List α\n⊢ List.beq l₁ l₂ = decide (l₁ = l₂)",
"state_before": "α✝ : Type uu\nβ : Type vv\nl₁ l₂ : List α✝\nα : Type u_1\ninst✝ : DecidableEq α\n⊢ List.beq = fun a b => decide (a = b)",
"tactic": "funext l₁ l₂"
},
{
"state_after": "case h.h\nα✝ : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α✝\nα : Type u_1\ninst✝ : DecidableEq α\nl₁ l₂ : List α\n⊢ (l₁ == l₂) = decide (l₁ = l₂)",
"state_before": "case h.h\nα✝ : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α✝\nα : Type u_1\ninst✝ : DecidableEq α\nl₁ l₂ : List α\n⊢ List.beq l₁ l₂ = decide (l₁ = l₂)",
"tactic": "show (l₁ == l₂) = _"
},
{
"state_after": "no goals",
"state_before": "case h.h\nα✝ : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α✝\nα : Type u_1\ninst✝ : DecidableEq α\nl₁ l₂ : List α\n⊢ (l₁ == l₂) = decide (l₁ = l₂)",
"tactic": "rw [Bool.eq_iff_eq_true_iff, @beq_iff_eq _ (_), decide_eq_true_iff]"
}
] |
[
1274,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1269,
9
] |
Mathlib/Algebra/Star/Subalgebra.lean
|
StarSubalgebra.star_subset_adjoin
|
[] |
[
448,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
447,
1
] |
Mathlib/Analysis/SpecificLimits/Normed.lean
|
NormedField.tendsto_zero_smul_of_tendsto_zero_of_bounded
|
[
{
"state_after": "α : Type ?u.57289\nβ : Type ?u.57292\nι✝ : Type ?u.57295\nι : Type u_1\n𝕜 : Type u_2\n𝔸 : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedAddCommGroup 𝔸\ninst✝ : NormedSpace 𝕜 𝔸\nl : Filter ι\nε : ι → 𝕜\nf : ι → 𝔸\nhε : ε =o[l] fun _x => 1\nhf : IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)\n⊢ (ε • f) =o[l] fun _x => 1",
"state_before": "α : Type ?u.57289\nβ : Type ?u.57292\nι✝ : Type ?u.57295\nι : Type u_1\n𝕜 : Type u_2\n𝔸 : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedAddCommGroup 𝔸\ninst✝ : NormedSpace 𝕜 𝔸\nl : Filter ι\nε : ι → 𝕜\nf : ι → 𝔸\nhε : Tendsto ε l (𝓝 0)\nhf : IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)\n⊢ Tendsto (ε • f) l (𝓝 0)",
"tactic": "rw [← isLittleO_one_iff 𝕜] at hε⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.57289\nβ : Type ?u.57292\nι✝ : Type ?u.57295\nι : Type u_1\n𝕜 : Type u_2\n𝔸 : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedAddCommGroup 𝔸\ninst✝ : NormedSpace 𝕜 𝔸\nl : Filter ι\nε : ι → 𝕜\nf : ι → 𝔸\nhε : ε =o[l] fun _x => 1\nhf : IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)\n⊢ (ε • f) =o[l] fun _x => 1",
"tactic": "simpa using IsLittleO.smul_isBigO hε (hf.isBigO_const (one_ne_zero : (1 : 𝕜) ≠ 0))"
}
] |
[
75,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/LinearAlgebra/Matrix/Charpoly/Eigs.lean
|
Matrix.det_eq_prod_roots_charpoly
|
[] |
[
86,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/ModelTheory/Syntax.lean
|
FirstOrder.Language.Formula.equivSentence_inf
|
[] |
[
1055,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1053,
1
] |
Mathlib/Data/Sign.lean
|
sign_pow
|
[] |
[
430,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
430,
1
] |
Mathlib/RingTheory/WittVector/IsPoly.lean
|
WittVector.IsPoly.comp₂
|
[
{
"state_after": "case mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.597461\nidx : Type ?u.597464\ninst✝¹ : CommRing R\ninst✝ : CommRing S\ng : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nf : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nhg : IsPoly p g\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\n⊢ IsPoly₂ p fun R _Rcr x y => g (f x y)",
"state_before": "p : ℕ\nR S : Type u\nσ : Type ?u.597461\nidx : Type ?u.597464\ninst✝¹ : CommRing R\ninst✝ : CommRing S\ng : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nf : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nhg : IsPoly p g\nhf : IsPoly₂ p f\n⊢ IsPoly₂ p fun R _Rcr x y => g (f x y)",
"tactic": "obtain ⟨φ, hf⟩ := hf"
},
{
"state_after": "case mk'.intro.mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.597461\nidx : Type ?u.597464\ninst✝¹ : CommRing R\ninst✝ : CommRing S\ng : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nf : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\n⊢ IsPoly₂ p fun R _Rcr x y => g (f x y)",
"state_before": "case mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.597461\nidx : Type ?u.597464\ninst✝¹ : CommRing R\ninst✝ : CommRing S\ng : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nf : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nhg : IsPoly p g\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\n⊢ IsPoly₂ p fun R _Rcr x y => g (f x y)",
"tactic": "obtain ⟨ψ, hg⟩ := hg"
},
{
"state_after": "case mk'.intro.mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.597461\nidx : Type ?u.597464\ninst✝¹ : CommRing R\ninst✝ : CommRing S\ng : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nf : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\n⊢ ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x y : 𝕎 R),\n (g (f x y)).coeff = fun n => peval ((fun n => ↑(bind₁ φ) (ψ n)) n) ![x.coeff, y.coeff]",
"state_before": "case mk'.intro.mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.597461\nidx : Type ?u.597464\ninst✝¹ : CommRing R\ninst✝ : CommRing S\ng : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nf : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\n⊢ IsPoly₂ p fun R _Rcr x y => g (f x y)",
"tactic": "use fun n => bind₁ φ (ψ n)"
},
{
"state_after": "case mk'.intro.mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.597461\nidx : Type ?u.597464\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ng : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nf : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u_1\ninst✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\n⊢ (g (f x✝ y✝)).coeff = fun n => peval ((fun n => ↑(bind₁ φ) (ψ n)) n) ![x✝.coeff, y✝.coeff]",
"state_before": "case mk'.intro.mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.597461\nidx : Type ?u.597464\ninst✝¹ : CommRing R\ninst✝ : CommRing S\ng : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nf : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\n⊢ ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x y : 𝕎 R),\n (g (f x y)).coeff = fun n => peval ((fun n => ↑(bind₁ φ) (ψ n)) n) ![x.coeff, y.coeff]",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "case mk'.intro.mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.597461\nidx : Type ?u.597464\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ng : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nf : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u_1\ninst✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\n⊢ (g (f x✝ y✝)).coeff = fun n => peval ((fun n => ↑(bind₁ φ) (ψ n)) n) ![x✝.coeff, y✝.coeff]",
"tactic": "simp only [peval, aeval_bind₁, Function.comp, hg, hf]"
}
] |
[
340,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.range_abs
|
[
{
"state_after": "no goals",
"state_before": "⊢ range ↑abs ⊆ Ici 0",
"tactic": "simp only [range_subset_iff, Ici, mem_setOf_eq, map_nonneg, forall_const]"
}
] |
[
1002,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
999,
1
] |
Mathlib/Data/PNat/Factors.lean
|
PrimeMultiset.card_ofPrime
|
[] |
[
55,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
Summable.map
|
[] |
[
268,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
11
] |
Mathlib/Algebra/GroupPower/Order.lean
|
Right.pow_le_one_of_le
|
[
{
"state_after": "β : Type ?u.61783\nA : Type ?u.61786\nG : Type ?u.61789\nM : Type u_1\nR : Type ?u.61795\ninst✝² : Monoid M\ninst✝¹ : Preorder M\ninst✝ : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nx : M\nhx : x ≤ 1\nn : ℕ\n⊢ x * x ^ n ≤ 1",
"state_before": "β : Type ?u.61783\nA : Type ?u.61786\nG : Type ?u.61789\nM : Type u_1\nR : Type ?u.61795\ninst✝² : Monoid M\ninst✝¹ : Preorder M\ninst✝ : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nx : M\nhx : x ≤ 1\nn : ℕ\n⊢ x ^ (n + 1) ≤ 1",
"tactic": "rw [pow_succ]"
},
{
"state_after": "no goals",
"state_before": "β : Type ?u.61783\nA : Type ?u.61786\nG : Type ?u.61789\nM : Type u_1\nR : Type ?u.61795\ninst✝² : Monoid M\ninst✝¹ : Preorder M\ninst✝ : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nx : M\nhx : x ≤ 1\nn : ℕ\n⊢ x * x ^ n ≤ 1",
"tactic": "exact Right.mul_le_one hx <| Right.pow_le_one_of_le hx"
}
] |
[
152,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.StronglyMeasurable.tendsto_approxBounded_ae
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ✝ : Type ?u.39374\nγ : Type ?u.39377\nι : Type ?u.39380\ninst✝³ : Countable ι\nf✝ g : α → β✝\ninst✝² : TopologicalSpace β✝\nβ : Type u_1\nf : α → β\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedSpace ℝ β\nm m0 : MeasurableSpace α\nμ : Measure α\nhf : StronglyMeasurable f\nc : ℝ\nhf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c\n⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n => ↑(approxBounded hf c n) x) atTop (𝓝 (f x))",
"tactic": "filter_upwards [hf_bound] with x hfx using tendsto_approxBounded_of_norm_le hf hfx"
}
] |
[
254,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
250,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
|
EuclideanGeometry.cos_angle_of_angle_eq_pi_div_two
|
[
{
"state_after": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\n⊢ Real.cos (∠ p₂ p₃ p₁) = dist p₃ p₂ / dist p₁ p₃",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ∠ p₁ p₂ p₃ = π / 2\n⊢ Real.cos (∠ p₂ p₃ p₁) = dist p₃ p₂ / dist p₁ p₃",
"tactic": "rw [angle, ← inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, ← neg_eq_zero, ←\n inner_neg_left, neg_vsub_eq_vsub_rev] at h"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\n⊢ Real.cos (∠ p₂ p₃ p₁) = dist p₃ p₂ / dist p₁ p₃",
"tactic": "rw [angle, dist_eq_norm_vsub' V p₃ p₂, dist_eq_norm_vsub V p₁ p₃, ← vsub_add_vsub_cancel p₁ p₂ p₃,\n add_comm, cos_angle_add_of_inner_eq_zero h]"
}
] |
[
439,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
434,
1
] |
Mathlib/Order/Disjoint.lean
|
IsCompl.symm
|
[] |
[
472,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
471,
11
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
tendsto_inv_nhdsWithin_Iio_inv
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalGroup G\ninst✝³ : TopologicalSpace α\nf : α → G\ns : Set α\nx : α\ninst✝² : TopologicalSpace H\ninst✝¹ : OrderedCommGroup H\ninst✝ : ContinuousInv H\na : H\n⊢ Tendsto Inv.inv (𝓝[Iio a⁻¹] a⁻¹) (𝓝[Ioi a] a)",
"tactic": "simpa only [inv_inv] using @tendsto_inv_nhdsWithin_Iio _ _ _ _ a⁻¹"
}
] |
[
579,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
578,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.op_norm_extend_le
|
[
{
"state_after": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\n⊢ ‖extend f e h_dense (_ : UniformInducing ↑e)‖ ≤ ↑N * ‖f‖",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\n⊢ ‖extend f e h_dense (_ : UniformInducing ↑e)‖ ≤ ↑N * ‖f‖",
"tactic": "have uni : UniformInducing e := (uniformEmbedding_of_bound _ h_e).toUniformInducing"
},
{
"state_after": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\n⊢ ‖extend f e h_dense (_ : UniformInducing ↑e)‖ ≤ ↑N * ‖f‖",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\n⊢ ‖extend f e h_dense (_ : UniformInducing ↑e)‖ ≤ ↑N * ‖f‖",
"tactic": "have eq : ∀ x, ψ (e x) = f x := uniformly_extend_of_ind uni h_dense f.uniformContinuous"
},
{
"state_after": "case pos\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : 0 ≤ N\n⊢ ‖extend f e h_dense (_ : UniformInducing ↑e)‖ ≤ ↑N * ‖f‖\n\ncase neg\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : ¬0 ≤ N\n⊢ ‖extend f e h_dense (_ : UniformInducing ↑e)‖ ≤ ↑N * ‖f‖",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\n⊢ ‖extend f e h_dense (_ : UniformInducing ↑e)‖ ≤ ↑N * ‖f‖",
"tactic": "by_cases N0 : 0 ≤ N"
},
{
"state_after": "case pos.refine'_1\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : 0 ≤ N\n⊢ 0 ≤ ↑N * ‖f‖\n\ncase pos.refine'_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : 0 ≤ N\n⊢ Continuous fun x => ‖↑(extend f e h_dense (_ : UniformInducing ↑e)) x‖\n\ncase pos.refine'_3\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : 0 ≤ N\n⊢ Continuous fun x => ↑N * ‖f‖ * ‖x‖\n\ncase pos.refine'_4\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : 0 ≤ N\n⊢ ∀ (a : E), ‖↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e a)‖ ≤ ↑N * ‖f‖ * ‖↑e a‖",
"state_before": "case pos\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : 0 ≤ N\n⊢ ‖extend f e h_dense (_ : UniformInducing ↑e)‖ ≤ ↑N * ‖f‖",
"tactic": "refine' op_norm_le_bound ψ _ (isClosed_property h_dense (isClosed_le _ _) _)"
},
{
"state_after": "no goals",
"state_before": "case pos.refine'_1\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : 0 ≤ N\n⊢ 0 ≤ ↑N * ‖f‖",
"tactic": "exact mul_nonneg N0 (norm_nonneg _)"
},
{
"state_after": "no goals",
"state_before": "case pos.refine'_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : 0 ≤ N\n⊢ Continuous fun x => ‖↑(extend f e h_dense (_ : UniformInducing ↑e)) x‖",
"tactic": "exact continuous_norm.comp (cont ψ)"
},
{
"state_after": "no goals",
"state_before": "case pos.refine'_3\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : 0 ≤ N\n⊢ Continuous fun x => ↑N * ‖f‖ * ‖x‖",
"tactic": "exact continuous_const.mul continuous_norm"
},
{
"state_after": "case pos.refine'_4\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx✝ y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : 0 ≤ N\nx : E\n⊢ ‖↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x)‖ ≤ ↑N * ‖f‖ * ‖↑e x‖",
"state_before": "case pos.refine'_4\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : 0 ≤ N\n⊢ ∀ (a : E), ‖↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e a)‖ ≤ ↑N * ‖f‖ * ‖↑e a‖",
"tactic": "intro x"
},
{
"state_after": "case pos.refine'_4\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx✝ y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : 0 ≤ N\nx : E\n⊢ ‖↑f x‖ ≤ ↑N * ‖f‖ * ‖↑e x‖",
"state_before": "case pos.refine'_4\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx✝ y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : 0 ≤ N\nx : E\n⊢ ‖↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x)‖ ≤ ↑N * ‖f‖ * ‖↑e x‖",
"tactic": "rw [eq]"
},
{
"state_after": "no goals",
"state_before": "case pos.refine'_4\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx✝ y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : 0 ≤ N\nx : E\n⊢ ‖↑f x‖ ≤ ↑N * ‖f‖ * ‖↑e x‖",
"tactic": "calc\n ‖f x‖ ≤ ‖f‖ * ‖x‖ := le_op_norm _ _\n _ ≤ ‖f‖ * (N * ‖e x‖) := (mul_le_mul_of_nonneg_left (h_e x) (norm_nonneg _))\n _ ≤ N * ‖f‖ * ‖e x‖ := by rw [mul_comm ↑N ‖f‖, mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx✝ y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : 0 ≤ N\nx : E\n⊢ ‖f‖ * (↑N * ‖↑e x‖) ≤ ↑N * ‖f‖ * ‖↑e x‖",
"tactic": "rw [mul_comm ↑N ‖f‖, mul_assoc]"
},
{
"state_after": "case neg\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : ¬0 ≤ N\nhe : ∀ (x : E), x = 0\n⊢ ‖extend f e h_dense (_ : UniformInducing ↑e)‖ ≤ ↑N * ‖f‖",
"state_before": "case neg\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : ¬0 ≤ N\n⊢ ‖extend f e h_dense (_ : UniformInducing ↑e)‖ ≤ ↑N * ‖f‖",
"tactic": "have he : ∀ x : E, x = 0 := by\n intro x\n have N0 : N ≤ 0 := le_of_lt (lt_of_not_ge N0)\n rw [← norm_le_zero_iff]\n exact le_trans (h_e x) (mul_nonpos_of_nonpos_of_nonneg N0 (norm_nonneg _))"
},
{
"state_after": "case neg\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : ¬0 ≤ N\nhe : ∀ (x : E), x = 0\nhf : f = 0\n⊢ ‖extend f e h_dense (_ : UniformInducing ↑e)‖ ≤ ↑N * ‖f‖",
"state_before": "case neg\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : ¬0 ≤ N\nhe : ∀ (x : E), x = 0\n⊢ ‖extend f e h_dense (_ : UniformInducing ↑e)‖ ≤ ↑N * ‖f‖",
"tactic": "have hf : f = 0 := by\n ext x\n simp only [he x, zero_apply, map_zero]"
},
{
"state_after": "case neg\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : ¬0 ≤ N\nhe : ∀ (x : E), x = 0\nhf : f = 0\nhψ : extend f e h_dense (_ : UniformInducing ↑e) = 0\n⊢ ‖extend f e h_dense (_ : UniformInducing ↑e)‖ ≤ ↑N * ‖f‖",
"state_before": "case neg\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : ¬0 ≤ N\nhe : ∀ (x : E), x = 0\nhf : f = 0\n⊢ ‖extend f e h_dense (_ : UniformInducing ↑e)‖ ≤ ↑N * ‖f‖",
"tactic": "have hψ : ψ = 0 := by\n rw [hf]\n apply extend_zero"
},
{
"state_after": "no goals",
"state_before": "case neg\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : ¬0 ≤ N\nhe : ∀ (x : E), x = 0\nhf : f = 0\nhψ : extend f e h_dense (_ : UniformInducing ↑e) = 0\n⊢ ‖extend f e h_dense (_ : UniformInducing ↑e)‖ ≤ ↑N * ‖f‖",
"tactic": "rw [hψ, hf, norm_zero, norm_zero, MulZeroClass.mul_zero]"
},
{
"state_after": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx✝ y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : ¬0 ≤ N\nx : E\n⊢ x = 0",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : ¬0 ≤ N\n⊢ ∀ (x : E), x = 0",
"tactic": "intro x"
},
{
"state_after": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx✝ y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0✝ : ¬0 ≤ N\nx : E\nN0 : N ≤ 0\n⊢ x = 0",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx✝ y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : ¬0 ≤ N\nx : E\n⊢ x = 0",
"tactic": "have N0 : N ≤ 0 := le_of_lt (lt_of_not_ge N0)"
},
{
"state_after": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx✝ y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0✝ : ¬0 ≤ N\nx : E\nN0 : N ≤ 0\n⊢ ‖x‖ ≤ 0",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx✝ y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0✝ : ¬0 ≤ N\nx : E\nN0 : N ≤ 0\n⊢ x = 0",
"tactic": "rw [← norm_le_zero_iff]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx✝ y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0✝ : ¬0 ≤ N\nx : E\nN0 : N ≤ 0\n⊢ ‖x‖ ≤ 0",
"tactic": "exact le_trans (h_e x) (mul_nonpos_of_nonpos_of_nonneg N0 (norm_nonneg _))"
},
{
"state_after": "case h\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx✝ y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : ¬0 ≤ N\nhe : ∀ (x : E), x = 0\nx : E\n⊢ ↑f x = ↑0 x",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : ¬0 ≤ N\nhe : ∀ (x : E), x = 0\n⊢ f = 0",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx✝ y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : ¬0 ≤ N\nhe : ∀ (x : E), x = 0\nx : E\n⊢ ↑f x = ↑0 x",
"tactic": "simp only [he x, zero_apply, map_zero]"
},
{
"state_after": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : ¬0 ≤ N\nhe : ∀ (x : E), x = 0\nhf : f = 0\n⊢ extend 0 e h_dense (_ : UniformInducing ↑e) = 0",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : ¬0 ≤ N\nhe : ∀ (x : E), x = 0\nhf : f = 0\n⊢ extend f e h_dense (_ : UniformInducing ↑e) = 0",
"tactic": "rw [hf]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.3539330\nE : Type u_5\nEₗ : Type ?u.3539336\nF : Type u_1\nFₗ : Type u_2\nG : Type ?u.3539345\nGₗ : Type ?u.3539348\n𝓕 : Type ?u.3539351\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝¹ : CompleteSpace F\ne : E →L[𝕜] Fₗ\nh_dense : DenseRange ↑e\nN : ℝ≥0\nh_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖↑e x‖\ninst✝ : RingHomIsometric σ₁₂\nuni : UniformInducing ↑e\neq : ∀ (x : E), ↑(extend f e h_dense (_ : UniformInducing ↑e)) (↑e x) = ↑f x\nN0 : ¬0 ≤ N\nhe : ∀ (x : E), x = 0\nhf : f = 0\n⊢ extend 0 e h_dense (_ : UniformInducing ↑e) = 0",
"tactic": "apply extend_zero"
}
] |
[
1765,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1740,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.one_mulVec
|
[
{
"state_after": "case h\nl : Type ?u.903237\nm : Type u_1\nn : Type ?u.903243\no : Type ?u.903246\nm' : o → Type ?u.903251\nn' : o → Type ?u.903256\nR : Type ?u.903259\nS : Type ?u.903262\nα : Type v\nβ : Type w\nγ : Type ?u.903269\ninst✝³ : NonAssocSemiring α\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : DecidableEq m\nv : m → α\nx✝ : m\n⊢ mulVec 1 v x✝ = v x✝",
"state_before": "l : Type ?u.903237\nm : Type u_1\nn : Type ?u.903243\no : Type ?u.903246\nm' : o → Type ?u.903251\nn' : o → Type ?u.903256\nR : Type ?u.903259\nS : Type ?u.903262\nα : Type v\nβ : Type w\nγ : Type ?u.903269\ninst✝³ : NonAssocSemiring α\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : DecidableEq m\nv : m → α\n⊢ mulVec 1 v = v",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nl : Type ?u.903237\nm : Type u_1\nn : Type ?u.903243\no : Type ?u.903246\nm' : o → Type ?u.903251\nn' : o → Type ?u.903256\nR : Type ?u.903259\nS : Type ?u.903262\nα : Type v\nβ : Type w\nγ : Type ?u.903269\ninst✝³ : NonAssocSemiring α\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : DecidableEq m\nv : m → α\nx✝ : m\n⊢ mulVec 1 v x✝ = v x✝",
"tactic": "rw [← diagonal_one, mulVec_diagonal, one_mul]"
}
] |
[
1881,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1879,
1
] |
Mathlib/Topology/Basic.lean
|
mem_closure_iff_comap_neBot
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\nA : Set α\nx : α\n⊢ x ∈ closure A ↔ NeBot (comap Subtype.val (𝓝 x))",
"tactic": "simp_rw [mem_closure_iff_nhds, comap_neBot_iff, Set.inter_nonempty_iff_exists_right,\n SetCoe.exists, exists_prop]"
}
] |
[
1343,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1340,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
tendsto_inv_nhdsWithin_Iic_inv
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalGroup G\ninst✝³ : TopologicalSpace α\nf : α → G\ns : Set α\nx : α\ninst✝² : TopologicalSpace H\ninst✝¹ : OrderedCommGroup H\ninst✝ : ContinuousInv H\na : H\n⊢ Tendsto Inv.inv (𝓝[Iic a⁻¹] a⁻¹) (𝓝[Ici a] a)",
"tactic": "simpa only [inv_inv] using @tendsto_inv_nhdsWithin_Iic _ _ _ _ a⁻¹"
}
] |
[
603,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
602,
1
] |
Mathlib/Topology/Separation.lean
|
t1Space_iff_specializes_imp_eq
|
[] |
[
520,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
519,
1
] |
Mathlib/Analysis/Convex/Integral.lean
|
StrictConvex.ae_eq_const_or_average_mem_interior
|
[
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.2067161\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : IsFiniteMeasure μ\nhs : StrictConvex ℝ s\nhsc : IsClosed s\nhfs : ∀ᵐ (x : α) ∂μ, f x ∈ s\nhfi : Integrable f\nthis : ∀ {t : Set α}, ↑↑μ t ≠ 0 → (⨍ (x : α) in t, f x ∂μ) ∈ s\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ (⨍ (x : α), f x ∂μ) ∈ interior s",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.2067161\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : IsFiniteMeasure μ\nhs : StrictConvex ℝ s\nhsc : IsClosed s\nhfs : ∀ᵐ (x : α) ∂μ, f x ∈ s\nhfi : Integrable f\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ (⨍ (x : α), f x ∂μ) ∈ interior s",
"tactic": "have : ∀ {t}, μ t ≠ 0 → (⨍ x in t, f x ∂μ) ∈ s := fun ht =>\n hs.convex.set_average_mem hsc ht (measure_ne_top _ _) (ae_restrict_of_ae hfs) hfi.integrableOn"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.2067161\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : IsFiniteMeasure μ\nhs : StrictConvex ℝ s\nhsc : IsClosed s\nhfs : ∀ᵐ (x : α) ∂μ, f x ∈ s\nhfi : Integrable f\nthis : ∀ {t : Set α}, ↑↑μ t ≠ 0 → (⨍ (x : α) in t, f x ∂μ) ∈ s\n⊢ (∃ t, MeasurableSet t ∧ ↑↑μ t ≠ 0 ∧ ↑↑μ (tᶜ) ≠ 0 ∧ (⨍ (x : α) in t, f x ∂μ) ≠ ⨍ (x : α) in tᶜ, f x ∂μ) →\n (⨍ (x : α), f x ∂μ) ∈ interior s",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.2067161\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : IsFiniteMeasure μ\nhs : StrictConvex ℝ s\nhsc : IsClosed s\nhfs : ∀ᵐ (x : α) ∂μ, f x ∈ s\nhfi : Integrable f\nthis : ∀ {t : Set α}, ↑↑μ t ≠ 0 → (⨍ (x : α) in t, f x ∂μ) ∈ s\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ (⨍ (x : α), f x ∂μ) ∈ interior s",
"tactic": "refine' (ae_eq_const_or_exists_average_ne_compl hfi).imp_right _"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nE : Type u_2\nF : Type ?u.2067161\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt✝ : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : IsFiniteMeasure μ\nhs : StrictConvex ℝ s\nhsc : IsClosed s\nhfs : ∀ᵐ (x : α) ∂μ, f x ∈ s\nhfi : Integrable f\nthis : ∀ {t : Set α}, ↑↑μ t ≠ 0 → (⨍ (x : α) in t, f x ∂μ) ∈ s\nt : Set α\nhm : MeasurableSet t\nh₀ : ↑↑μ t ≠ 0\nh₀' : ↑↑μ (tᶜ) ≠ 0\nhne : (⨍ (x : α) in t, f x ∂μ) ≠ ⨍ (x : α) in tᶜ, f x ∂μ\n⊢ (⨍ (x : α), f x ∂μ) ∈ interior s",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.2067161\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : IsFiniteMeasure μ\nhs : StrictConvex ℝ s\nhsc : IsClosed s\nhfs : ∀ᵐ (x : α) ∂μ, f x ∈ s\nhfi : Integrable f\nthis : ∀ {t : Set α}, ↑↑μ t ≠ 0 → (⨍ (x : α) in t, f x ∂μ) ∈ s\n⊢ (∃ t, MeasurableSet t ∧ ↑↑μ t ≠ 0 ∧ ↑↑μ (tᶜ) ≠ 0 ∧ (⨍ (x : α) in t, f x ∂μ) ≠ ⨍ (x : α) in tᶜ, f x ∂μ) →\n (⨍ (x : α), f x ∂μ) ∈ interior s",
"tactic": "rintro ⟨t, hm, h₀, h₀', hne⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nE : Type u_2\nF : Type ?u.2067161\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt✝ : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : IsFiniteMeasure μ\nhs : StrictConvex ℝ s\nhsc : IsClosed s\nhfs : ∀ᵐ (x : α) ∂μ, f x ∈ s\nhfi : Integrable f\nthis : ∀ {t : Set α}, ↑↑μ t ≠ 0 → (⨍ (x : α) in t, f x ∂μ) ∈ s\nt : Set α\nhm : MeasurableSet t\nh₀ : ↑↑μ t ≠ 0\nh₀' : ↑↑μ (tᶜ) ≠ 0\nhne : (⨍ (x : α) in t, f x ∂μ) ≠ ⨍ (x : α) in tᶜ, f x ∂μ\n⊢ (⨍ (x : α), f x ∂μ) ∈ interior s",
"tactic": "exact\n hs.openSegment_subset (this h₀) (this h₀') hne\n (average_mem_openSegment_compl_self hm.nullMeasurableSet h₀ h₀' hfi)"
}
] |
[
284,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
275,
1
] |
Mathlib/Order/Filter/CountableInter.lean
|
Filter.countableGenerate_isGreatest
|
[
{
"state_after": "ι : Sort ?u.24738\nα : Type u_1\nβ : Type ?u.24744\nl : Filter α\ninst✝ : CountableInterFilter l\ng : Set (Set α)\n⊢ countableGenerate g ∈ upperBounds {f | CountableInterFilter f ∧ g ⊆ f.sets}",
"state_before": "ι : Sort ?u.24738\nα : Type u_1\nβ : Type ?u.24744\nl : Filter α\ninst✝ : CountableInterFilter l\ng : Set (Set α)\n⊢ IsGreatest {f | CountableInterFilter f ∧ g ⊆ f.sets} (countableGenerate g)",
"tactic": "refine' ⟨⟨inferInstance, fun s => CountableGenerateSets.basic⟩, _⟩"
},
{
"state_after": "case intro\nι : Sort ?u.24738\nα : Type u_1\nβ : Type ?u.24744\nl : Filter α\ninst✝ : CountableInterFilter l\ng : Set (Set α)\nf : Filter α\nfct : CountableInterFilter f\nhf : g ⊆ f.sets\n⊢ f ≤ countableGenerate g",
"state_before": "ι : Sort ?u.24738\nα : Type u_1\nβ : Type ?u.24744\nl : Filter α\ninst✝ : CountableInterFilter l\ng : Set (Set α)\n⊢ countableGenerate g ∈ upperBounds {f | CountableInterFilter f ∧ g ⊆ f.sets}",
"tactic": "rintro f ⟨fct, hf⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nι : Sort ?u.24738\nα : Type u_1\nβ : Type ?u.24744\nl : Filter α\ninst✝ : CountableInterFilter l\ng : Set (Set α)\nf : Filter α\nfct : CountableInterFilter f\nhf : g ⊆ f.sets\n⊢ f ≤ countableGenerate g",
"tactic": "rwa [@le_countableGenerate_iff_of_countableInterFilter _ _ _ fct]"
}
] |
[
271,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
src/lean/Init/SimpLemmas.lean
|
dite_congr
|
[
{
"state_after": "no goals",
"state_before": "b c : Prop\nα : Sort u_1\nx✝ : Decidable b\ninst✝ : Decidable c\nx : b → α\nu : c → α\ny : ¬b → α\nv : ¬c → α\nh₁ : b = c\nh₂ : ∀ (h : c), x (_ : b) = u h\nh₃ : ∀ (h : ¬c), y (_ : ¬b) = v h\n⊢ dite b x y = dite c u v",
"tactic": "cases Decidable.em c with\n| inl h => rw [dif_pos h]; subst b; rw [dif_pos h]; exact h₂ h\n| inr h => rw [dif_neg h]; subst b; rw [dif_neg h]; exact h₃ h"
},
{
"state_after": "case inl\nb c : Prop\nα : Sort u_1\nx✝ : Decidable b\ninst✝ : Decidable c\nx : b → α\nu : c → α\ny : ¬b → α\nv : ¬c → α\nh₁ : b = c\nh₂ : ∀ (h : c), x (_ : b) = u h\nh₃ : ∀ (h : ¬c), y (_ : ¬b) = v h\nh : c\n⊢ dite b x y = u h",
"state_before": "case inl\nb c : Prop\nα : Sort u_1\nx✝ : Decidable b\ninst✝ : Decidable c\nx : b → α\nu : c → α\ny : ¬b → α\nv : ¬c → α\nh₁ : b = c\nh₂ : ∀ (h : c), x (_ : b) = u h\nh₃ : ∀ (h : ¬c), y (_ : ¬b) = v h\nh : c\n⊢ dite b x y = dite c u v",
"tactic": "rw [dif_pos h]"
},
{
"state_after": "case inl\nc : Prop\nα : Sort u_1\ninst✝ : Decidable c\nu : c → α\nv : ¬c → α\nh : c\nx✝ : Decidable c\nx : c → α\ny : ¬c → α\nh₂ : ∀ (h : c), x (_ : c) = u h\nh₃ : ∀ (h : ¬c), y (_ : ¬c) = v h\n⊢ dite c x y = u h",
"state_before": "case inl\nb c : Prop\nα : Sort u_1\nx✝ : Decidable b\ninst✝ : Decidable c\nx : b → α\nu : c → α\ny : ¬b → α\nv : ¬c → α\nh₁ : b = c\nh₂ : ∀ (h : c), x (_ : b) = u h\nh₃ : ∀ (h : ¬c), y (_ : ¬b) = v h\nh : c\n⊢ dite b x y = u h",
"tactic": "subst b"
},
{
"state_after": "case inl\nc : Prop\nα : Sort u_1\ninst✝ : Decidable c\nu : c → α\nv : ¬c → α\nh : c\nx✝ : Decidable c\nx : c → α\ny : ¬c → α\nh₂ : ∀ (h : c), x (_ : c) = u h\nh₃ : ∀ (h : ¬c), y (_ : ¬c) = v h\n⊢ x h = u h",
"state_before": "case inl\nc : Prop\nα : Sort u_1\ninst✝ : Decidable c\nu : c → α\nv : ¬c → α\nh : c\nx✝ : Decidable c\nx : c → α\ny : ¬c → α\nh₂ : ∀ (h : c), x (_ : c) = u h\nh₃ : ∀ (h : ¬c), y (_ : ¬c) = v h\n⊢ dite c x y = u h",
"tactic": "rw [dif_pos h]"
},
{
"state_after": "no goals",
"state_before": "case inl\nc : Prop\nα : Sort u_1\ninst✝ : Decidable c\nu : c → α\nv : ¬c → α\nh : c\nx✝ : Decidable c\nx : c → α\ny : ¬c → α\nh₂ : ∀ (h : c), x (_ : c) = u h\nh₃ : ∀ (h : ¬c), y (_ : ¬c) = v h\n⊢ x h = u h",
"tactic": "exact h₂ h"
},
{
"state_after": "case inr\nb c : Prop\nα : Sort u_1\nx✝ : Decidable b\ninst✝ : Decidable c\nx : b → α\nu : c → α\ny : ¬b → α\nv : ¬c → α\nh₁ : b = c\nh₂ : ∀ (h : c), x (_ : b) = u h\nh₃ : ∀ (h : ¬c), y (_ : ¬b) = v h\nh : ¬c\n⊢ dite b x y = v h",
"state_before": "case inr\nb c : Prop\nα : Sort u_1\nx✝ : Decidable b\ninst✝ : Decidable c\nx : b → α\nu : c → α\ny : ¬b → α\nv : ¬c → α\nh₁ : b = c\nh₂ : ∀ (h : c), x (_ : b) = u h\nh₃ : ∀ (h : ¬c), y (_ : ¬b) = v h\nh : ¬c\n⊢ dite b x y = dite c u v",
"tactic": "rw [dif_neg h]"
},
{
"state_after": "case inr\nc : Prop\nα : Sort u_1\ninst✝ : Decidable c\nu : c → α\nv : ¬c → α\nh : ¬c\nx✝ : Decidable c\nx : c → α\ny : ¬c → α\nh₂ : ∀ (h : c), x (_ : c) = u h\nh₃ : ∀ (h : ¬c), y (_ : ¬c) = v h\n⊢ dite c x y = v h",
"state_before": "case inr\nb c : Prop\nα : Sort u_1\nx✝ : Decidable b\ninst✝ : Decidable c\nx : b → α\nu : c → α\ny : ¬b → α\nv : ¬c → α\nh₁ : b = c\nh₂ : ∀ (h : c), x (_ : b) = u h\nh₃ : ∀ (h : ¬c), y (_ : ¬b) = v h\nh : ¬c\n⊢ dite b x y = v h",
"tactic": "subst b"
},
{
"state_after": "case inr\nc : Prop\nα : Sort u_1\ninst✝ : Decidable c\nu : c → α\nv : ¬c → α\nh : ¬c\nx✝ : Decidable c\nx : c → α\ny : ¬c → α\nh₂ : ∀ (h : c), x (_ : c) = u h\nh₃ : ∀ (h : ¬c), y (_ : ¬c) = v h\n⊢ y h = v h",
"state_before": "case inr\nc : Prop\nα : Sort u_1\ninst✝ : Decidable c\nu : c → α\nv : ¬c → α\nh : ¬c\nx✝ : Decidable c\nx : c → α\ny : ¬c → α\nh₂ : ∀ (h : c), x (_ : c) = u h\nh₃ : ∀ (h : ¬c), y (_ : ¬c) = v h\n⊢ dite c x y = v h",
"tactic": "rw [dif_neg h]"
},
{
"state_after": "no goals",
"state_before": "case inr\nc : Prop\nα : Sort u_1\ninst✝ : Decidable c\nu : c → α\nv : ¬c → α\nh : ¬c\nx✝ : Decidable c\nx : c → α\ny : ¬c → α\nh₂ : ∀ (h : c), x (_ : c) = u h\nh₃ : ∀ (h : ¬c), y (_ : ¬c) = v h\n⊢ y h = v h",
"tactic": "exact h₃ h"
}
] |
[
74,
65
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
66,
1
] |
Mathlib/RingTheory/Localization/Integer.lean
|
IsLocalization.map_integerMultiple
|
[] |
[
140,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/NumberTheory/Padics/Hensel.lean
|
newton_seq_is_cauchy
|
[
{
"state_after": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nε : ℝ\nhε : ε > 0\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → ‖newton_seq j - newton_seq i‖ < ε",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\n⊢ IsCauSeq norm newton_seq",
"tactic": "intro ε hε"
},
{
"state_after": "case intro\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nε : ℝ\nhε : ε > 0\nN : ℕ\nhN : ∀ {n : ℕ}, n ≥ N → ‖Polynomial.eval a (↑Polynomial.derivative F)‖ * T_gen p F a ^ 2 ^ n < ε\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → ‖newton_seq j - newton_seq i‖ < ε",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nε : ℝ\nhε : ε > 0\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → ‖newton_seq j - newton_seq i‖ < ε",
"tactic": "cases' bound hnorm hε with N hN"
},
{
"state_after": "case intro\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nε : ℝ\nhε : ε > 0\nN : ℕ\nhN : ∀ {n : ℕ}, n ≥ N → ‖Polynomial.eval a (↑Polynomial.derivative F)‖ * T_gen p F a ^ 2 ^ n < ε\n⊢ ∀ (j : ℕ), j ≥ N → ‖newton_seq j - newton_seq N‖ < ε",
"state_before": "case intro\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nε : ℝ\nhε : ε > 0\nN : ℕ\nhN : ∀ {n : ℕ}, n ≥ N → ‖Polynomial.eval a (↑Polynomial.derivative F)‖ * T_gen p F a ^ 2 ^ n < ε\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → ‖newton_seq j - newton_seq i‖ < ε",
"tactic": "exists N"
},
{
"state_after": "case intro\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nε : ℝ\nhε : ε > 0\nN : ℕ\nhN : ∀ {n : ℕ}, n ≥ N → ‖Polynomial.eval a (↑Polynomial.derivative F)‖ * T_gen p F a ^ 2 ^ n < ε\nj : ℕ\nhj : j ≥ N\n⊢ ‖newton_seq j - newton_seq N‖ < ε",
"state_before": "case intro\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nε : ℝ\nhε : ε > 0\nN : ℕ\nhN : ∀ {n : ℕ}, n ≥ N → ‖Polynomial.eval a (↑Polynomial.derivative F)‖ * T_gen p F a ^ 2 ^ n < ε\n⊢ ∀ (j : ℕ), j ≥ N → ‖newton_seq j - newton_seq N‖ < ε",
"tactic": "intro j hj"
},
{
"state_after": "case intro.a\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nε : ℝ\nhε : ε > 0\nN : ℕ\nhN : ∀ {n : ℕ}, n ≥ N → ‖Polynomial.eval a (↑Polynomial.derivative F)‖ * T_gen p F a ^ 2 ^ n < ε\nj : ℕ\nhj : j ≥ N\n⊢ ‖newton_seq j - newton_seq N‖ ≤ ?intro.b\n\ncase intro.a\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nε : ℝ\nhε : ε > 0\nN : ℕ\nhN : ∀ {n : ℕ}, n ≥ N → ‖Polynomial.eval a (↑Polynomial.derivative F)‖ * T_gen p F a ^ 2 ^ n < ε\nj : ℕ\nhj : j ≥ N\n⊢ ?intro.b < ε\n\ncase intro.b\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nε : ℝ\nhε : ε > 0\nN : ℕ\nhN : ∀ {n : ℕ}, n ≥ N → ‖Polynomial.eval a (↑Polynomial.derivative F)‖ * T_gen p F a ^ 2 ^ n < ε\nj : ℕ\nhj : j ≥ N\n⊢ ℝ",
"state_before": "case intro\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nε : ℝ\nhε : ε > 0\nN : ℕ\nhN : ∀ {n : ℕ}, n ≥ N → ‖Polynomial.eval a (↑Polynomial.derivative F)‖ * T_gen p F a ^ 2 ^ n < ε\nj : ℕ\nhj : j ≥ N\n⊢ ‖newton_seq j - newton_seq N‖ < ε",
"tactic": "apply lt_of_le_of_lt"
},
{
"state_after": "no goals",
"state_before": "case intro.a\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nε : ℝ\nhε : ε > 0\nN : ℕ\nhN : ∀ {n : ℕ}, n ≥ N → ‖Polynomial.eval a (↑Polynomial.derivative F)‖ * T_gen p F a ^ 2 ^ n < ε\nj : ℕ\nhj : j ≥ N\n⊢ ‖newton_seq j - newton_seq N‖ ≤ ?intro.b",
"tactic": "apply newton_seq_dist hnorm hj"
},
{
"state_after": "case intro.a.a\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nε : ℝ\nhε : ε > 0\nN : ℕ\nhN : ∀ {n : ℕ}, n ≥ N → ‖Polynomial.eval a (↑Polynomial.derivative F)‖ * T_gen p F a ^ 2 ^ n < ε\nj : ℕ\nhj : j ≥ N\n⊢ N ≥ N",
"state_before": "case intro.a\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nε : ℝ\nhε : ε > 0\nN : ℕ\nhN : ∀ {n : ℕ}, n ≥ N → ‖Polynomial.eval a (↑Polynomial.derivative F)‖ * T_gen p F a ^ 2 ^ n < ε\nj : ℕ\nhj : j ≥ N\n⊢ ‖Polynomial.eval a (↑Polynomial.derivative F)‖ * T_gen p F a ^ 2 ^ N < ε",
"tactic": "apply hN"
},
{
"state_after": "no goals",
"state_before": "case intro.a.a\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nε : ℝ\nhε : ε > 0\nN : ℕ\nhN : ∀ {n : ℕ}, n ≥ N → ‖Polynomial.eval a (↑Polynomial.derivative F)‖ * T_gen p F a ^ 2 ^ n < ε\nj : ℕ\nhj : j ≥ N\n⊢ N ≥ N",
"tactic": "exact le_rfl"
}
] |
[
382,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
374,
9
] |
Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean
|
List.Nat.antidiagonalTuple_zero_zero
|
[] |
[
76,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Algebra/Order/Sub/Defs.lean
|
tsub_nonpos
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.30124\ninst✝³ : Preorder α\ninst✝² : AddCommMonoid α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\n⊢ a - b ≤ 0 ↔ a ≤ b",
"tactic": "rw [tsub_le_iff_left, add_zero]"
}
] |
[
250,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
250,
1
] |
Mathlib/SetTheory/Lists.lean
|
Lists'.mem_of_subset
|
[] |
[
450,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
449,
1
] |
Mathlib/Algebra/MonoidAlgebra/Degree.lean
|
AddMonoidAlgebra.sup_support_list_prod_le
|
[
{
"state_after": "R : Type u_3\nA : Type u_2\nT : Type ?u.27671\nB : Type u_1\nι : Type ?u.27677\ninst✝¹¹ : SemilatticeSup B\ninst✝¹⁰ : OrderBot B\ninst✝⁹ : SemilatticeInf T\ninst✝⁸ : OrderTop T\ninst✝⁷ : Semiring R\ninst✝⁶ : AddMonoid A\ninst✝⁵ : AddMonoid B\ninst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝² : AddMonoid T\ninst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ndegb : A → B\ndegt : A → T\ndegb0 : degb 0 ≤ 0\ndegbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b\n⊢ ∀ (b : A), b ∈ 1.support → degb b ≤ 0",
"state_before": "R : Type u_3\nA : Type u_2\nT : Type ?u.27671\nB : Type u_1\nι : Type ?u.27677\ninst✝¹¹ : SemilatticeSup B\ninst✝¹⁰ : OrderBot B\ninst✝⁹ : SemilatticeInf T\ninst✝⁸ : OrderTop T\ninst✝⁷ : Semiring R\ninst✝⁶ : AddMonoid A\ninst✝⁵ : AddMonoid B\ninst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝² : AddMonoid T\ninst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ndegb : A → B\ndegt : A → T\ndegb0 : degb 0 ≤ 0\ndegbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b\n⊢ Finset.sup (List.prod []).support degb ≤ List.sum (List.map (fun f => Finset.sup f.support degb) [])",
"tactic": "rw [List.map_nil, Finset.sup_le_iff, List.prod_nil, List.sum_nil]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_3\nA : Type u_2\nT : Type ?u.27671\nB : Type u_1\nι : Type ?u.27677\ninst✝¹¹ : SemilatticeSup B\ninst✝¹⁰ : OrderBot B\ninst✝⁹ : SemilatticeInf T\ninst✝⁸ : OrderTop T\ninst✝⁷ : Semiring R\ninst✝⁶ : AddMonoid A\ninst✝⁵ : AddMonoid B\ninst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝² : AddMonoid T\ninst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ndegb : A → B\ndegt : A → T\ndegb0 : degb 0 ≤ 0\ndegbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b\n⊢ ∀ (b : A), b ∈ 1.support → degb b ≤ 0",
"tactic": "exact fun a ha => by rwa [Finset.mem_singleton.mp (Finsupp.support_single_subset ha)]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_3\nA : Type u_2\nT : Type ?u.27671\nB : Type u_1\nι : Type ?u.27677\ninst✝¹¹ : SemilatticeSup B\ninst✝¹⁰ : OrderBot B\ninst✝⁹ : SemilatticeInf T\ninst✝⁸ : OrderTop T\ninst✝⁷ : Semiring R\ninst✝⁶ : AddMonoid A\ninst✝⁵ : AddMonoid B\ninst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝² : AddMonoid T\ninst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ndegb : A → B\ndegt : A → T\ndegb0 : degb 0 ≤ 0\ndegbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b\na : A\nha : a ∈ 1.support\n⊢ degb a ≤ 0",
"tactic": "rwa [Finset.mem_singleton.mp (Finsupp.support_single_subset ha)]"
},
{
"state_after": "R : Type u_3\nA : Type u_2\nT : Type ?u.27671\nB : Type u_1\nι : Type ?u.27677\ninst✝¹¹ : SemilatticeSup B\ninst✝¹⁰ : OrderBot B\ninst✝⁹ : SemilatticeInf T\ninst✝⁸ : OrderTop T\ninst✝⁷ : Semiring R\ninst✝⁶ : AddMonoid A\ninst✝⁵ : AddMonoid B\ninst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝² : AddMonoid T\ninst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ndegb : A → B\ndegt : A → T\ndegb0 : degb 0 ≤ 0\ndegbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b\nf : AddMonoidAlgebra R A\nfs : List (AddMonoidAlgebra R A)\n⊢ Finset.sup (f * List.prod fs).support degb ≤\n Finset.sup f.support degb + List.sum (List.map (fun f => Finset.sup f.support degb) fs)",
"state_before": "R : Type u_3\nA : Type u_2\nT : Type ?u.27671\nB : Type u_1\nι : Type ?u.27677\ninst✝¹¹ : SemilatticeSup B\ninst✝¹⁰ : OrderBot B\ninst✝⁹ : SemilatticeInf T\ninst✝⁸ : OrderTop T\ninst✝⁷ : Semiring R\ninst✝⁶ : AddMonoid A\ninst✝⁵ : AddMonoid B\ninst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝² : AddMonoid T\ninst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ndegb : A → B\ndegt : A → T\ndegb0 : degb 0 ≤ 0\ndegbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b\nf : AddMonoidAlgebra R A\nfs : List (AddMonoidAlgebra R A)\n⊢ Finset.sup (List.prod (f :: fs)).support degb ≤ List.sum (List.map (fun f => Finset.sup f.support degb) (f :: fs))",
"tactic": "rw [List.prod_cons, List.map_cons, List.sum_cons]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_3\nA : Type u_2\nT : Type ?u.27671\nB : Type u_1\nι : Type ?u.27677\ninst✝¹¹ : SemilatticeSup B\ninst✝¹⁰ : OrderBot B\ninst✝⁹ : SemilatticeInf T\ninst✝⁸ : OrderTop T\ninst✝⁷ : Semiring R\ninst✝⁶ : AddMonoid A\ninst✝⁵ : AddMonoid B\ninst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝² : AddMonoid T\ninst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ndegb : A → B\ndegt : A → T\ndegb0 : degb 0 ≤ 0\ndegbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b\nf : AddMonoidAlgebra R A\nfs : List (AddMonoidAlgebra R A)\n⊢ Finset.sup (f * List.prod fs).support degb ≤\n Finset.sup f.support degb + List.sum (List.map (fun f => Finset.sup f.support degb) fs)",
"tactic": "exact (sup_support_mul_le (@fun a b => degbm a b) _ _).trans\n (add_le_add_left (sup_support_list_prod_le degb0 degbm fs) _)"
}
] |
[
104,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Algebra/Module/Submodule/Basic.lean
|
Submodule.eta
|
[] |
[
92,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
9
] |
Mathlib/Algebra/SMulWithZero.lean
|
smul_inv₀
|
[
{
"state_after": "case inl\nR : Type ?u.35696\nR' : Type ?u.35699\nM : Type ?u.35702\nM' : Type ?u.35705\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : GroupWithZero α\ninst✝³ : GroupWithZero β\ninst✝² : MulActionWithZero α β\ninst✝¹ : SMulCommClass α β β\ninst✝ : IsScalarTower α β β\nx : β\n⊢ (0 • x)⁻¹ = 0⁻¹ • x⁻¹\n\ncase inr\nR : Type ?u.35696\nR' : Type ?u.35699\nM : Type ?u.35702\nM' : Type ?u.35705\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : GroupWithZero α\ninst✝³ : GroupWithZero β\ninst✝² : MulActionWithZero α β\ninst✝¹ : SMulCommClass α β β\ninst✝ : IsScalarTower α β β\nc : α\nx : β\nhc : c ≠ 0\n⊢ (c • x)⁻¹ = c⁻¹ • x⁻¹",
"state_before": "R : Type ?u.35696\nR' : Type ?u.35699\nM : Type ?u.35702\nM' : Type ?u.35705\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : GroupWithZero α\ninst✝³ : GroupWithZero β\ninst✝² : MulActionWithZero α β\ninst✝¹ : SMulCommClass α β β\ninst✝ : IsScalarTower α β β\nc : α\nx : β\n⊢ (c • x)⁻¹ = c⁻¹ • x⁻¹",
"tactic": "obtain rfl | hc := eq_or_ne c 0"
},
{
"state_after": "case inr.inl\nR : Type ?u.35696\nR' : Type ?u.35699\nM : Type ?u.35702\nM' : Type ?u.35705\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : GroupWithZero α\ninst✝³ : GroupWithZero β\ninst✝² : MulActionWithZero α β\ninst✝¹ : SMulCommClass α β β\ninst✝ : IsScalarTower α β β\nc : α\nhc : c ≠ 0\n⊢ (c • 0)⁻¹ = c⁻¹ • 0⁻¹\n\ncase inr.inr\nR : Type ?u.35696\nR' : Type ?u.35699\nM : Type ?u.35702\nM' : Type ?u.35705\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : GroupWithZero α\ninst✝³ : GroupWithZero β\ninst✝² : MulActionWithZero α β\ninst✝¹ : SMulCommClass α β β\ninst✝ : IsScalarTower α β β\nc : α\nx : β\nhc : c ≠ 0\nhx : x ≠ 0\n⊢ (c • x)⁻¹ = c⁻¹ • x⁻¹",
"state_before": "case inr\nR : Type ?u.35696\nR' : Type ?u.35699\nM : Type ?u.35702\nM' : Type ?u.35705\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : GroupWithZero α\ninst✝³ : GroupWithZero β\ninst✝² : MulActionWithZero α β\ninst✝¹ : SMulCommClass α β β\ninst✝ : IsScalarTower α β β\nc : α\nx : β\nhc : c ≠ 0\n⊢ (c • x)⁻¹ = c⁻¹ • x⁻¹",
"tactic": "obtain rfl | hx := eq_or_ne x 0"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type ?u.35696\nR' : Type ?u.35699\nM : Type ?u.35702\nM' : Type ?u.35705\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : GroupWithZero α\ninst✝³ : GroupWithZero β\ninst✝² : MulActionWithZero α β\ninst✝¹ : SMulCommClass α β β\ninst✝ : IsScalarTower α β β\nx : β\n⊢ (0 • x)⁻¹ = 0⁻¹ • x⁻¹",
"tactic": "simp only [inv_zero, zero_smul]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nR : Type ?u.35696\nR' : Type ?u.35699\nM : Type ?u.35702\nM' : Type ?u.35705\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : GroupWithZero α\ninst✝³ : GroupWithZero β\ninst✝² : MulActionWithZero α β\ninst✝¹ : SMulCommClass α β β\ninst✝ : IsScalarTower α β β\nc : α\nhc : c ≠ 0\n⊢ (c • 0)⁻¹ = c⁻¹ • 0⁻¹",
"tactic": "simp only [inv_zero, smul_zero]"
},
{
"state_after": "case inr.inr\nR : Type ?u.35696\nR' : Type ?u.35699\nM : Type ?u.35702\nM' : Type ?u.35705\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : GroupWithZero α\ninst✝³ : GroupWithZero β\ninst✝² : MulActionWithZero α β\ninst✝¹ : SMulCommClass α β β\ninst✝ : IsScalarTower α β β\nc : α\nx : β\nhc : c ≠ 0\nhx : x ≠ 0\n⊢ c⁻¹ • x⁻¹ * c • x = 1",
"state_before": "case inr.inr\nR : Type ?u.35696\nR' : Type ?u.35699\nM : Type ?u.35702\nM' : Type ?u.35705\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : GroupWithZero α\ninst✝³ : GroupWithZero β\ninst✝² : MulActionWithZero α β\ninst✝¹ : SMulCommClass α β β\ninst✝ : IsScalarTower α β β\nc : α\nx : β\nhc : c ≠ 0\nhx : x ≠ 0\n⊢ (c • x)⁻¹ = c⁻¹ • x⁻¹",
"tactic": "refine' inv_eq_of_mul_eq_one_left _"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nR : Type ?u.35696\nR' : Type ?u.35699\nM : Type ?u.35702\nM' : Type ?u.35705\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : GroupWithZero α\ninst✝³ : GroupWithZero β\ninst✝² : MulActionWithZero α β\ninst✝¹ : SMulCommClass α β β\ninst✝ : IsScalarTower α β β\nc : α\nx : β\nhc : c ≠ 0\nhx : x ≠ 0\n⊢ c⁻¹ • x⁻¹ * c • x = 1",
"tactic": "rw [smul_mul_smul, inv_mul_cancel hc, inv_mul_cancel hx, one_smul]"
}
] |
[
222,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/MeasureTheory/Constructions/Pi.lean
|
MeasureTheory.Measure.univ_pi_Ioc_ae_eq_Icc
|
[
{
"state_after": "ι : Type u_2\nι' : Type ?u.4569072\nα : ι → Type u_1\ninst✝⁴ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝³ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝² : ∀ (i : ι), SigmaFinite (μ i)\ninst✝¹ : (i : ι) → PartialOrder (α i)\ninst✝ : ∀ (i : ι), NoAtoms (μ i)\nf g : (i : ι) → α i\n⊢ (Set.pi univ fun i => Ioc (f i) (g i)) =ᶠ[ae (Measure.pi μ)] Set.pi univ fun i => Icc (f i) (g i)",
"state_before": "ι : Type u_2\nι' : Type ?u.4569072\nα : ι → Type u_1\ninst✝⁴ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝³ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝² : ∀ (i : ι), SigmaFinite (μ i)\ninst✝¹ : (i : ι) → PartialOrder (α i)\ninst✝ : ∀ (i : ι), NoAtoms (μ i)\nf g : (i : ι) → α i\n⊢ (Set.pi univ fun i => Ioc (f i) (g i)) =ᶠ[ae (Measure.pi μ)] Icc f g",
"tactic": "rw [← pi_univ_Icc]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nι' : Type ?u.4569072\nα : ι → Type u_1\ninst✝⁴ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝³ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝² : ∀ (i : ι), SigmaFinite (μ i)\ninst✝¹ : (i : ι) → PartialOrder (α i)\ninst✝ : ∀ (i : ι), NoAtoms (μ i)\nf g : (i : ι) → α i\n⊢ (Set.pi univ fun i => Ioc (f i) (g i)) =ᶠ[ae (Measure.pi μ)] Set.pi univ fun i => Icc (f i) (g i)",
"tactic": "exact pi_Ioc_ae_eq_pi_Icc"
}
] |
[
531,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
529,
1
] |
Mathlib/Data/Int/GCD.lean
|
Int.gcd_dvd_right
|
[] |
[
251,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
250,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.vsub_subset_vsub
|
[] |
[
1519,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1518,
1
] |
Mathlib/Data/Nat/Factorial/Basic.lean
|
Nat.descFactorial_self
|
[
{
"state_after": "no goals",
"state_before": "⊢ descFactorial 0 0 = 0!",
"tactic": "rw [descFactorial_zero, factorial_zero]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ descFactorial (succ n) (succ n) = (succ n)!",
"tactic": "rw [succ_descFactorial_succ, descFactorial_self n, factorial_succ]"
}
] |
[
393,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
391,
1
] |
Mathlib/Order/Basic.lean
|
min_rec
|
[] |
[
941,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
939,
1
] |
Mathlib/Data/Nat/Order/Basic.lean
|
Nat.pred_le_iff
|
[
{
"state_after": "case zero\nn k l : ℕ\n⊢ zero ≤ succ n → pred zero ≤ n\n\ncase succ\nn k l n✝ : ℕ\n⊢ succ n✝ ≤ succ n → pred (succ n✝) ≤ n",
"state_before": "m n k l : ℕ\n⊢ m ≤ succ n → pred m ≤ n",
"tactic": "cases m"
},
{
"state_after": "no goals",
"state_before": "case succ\nn k l n✝ : ℕ\n⊢ succ n✝ ≤ succ n → pred (succ n✝) ≤ n",
"tactic": "exact le_of_succ_le_succ"
},
{
"state_after": "no goals",
"state_before": "case zero\nn k l : ℕ\n⊢ zero ≤ succ n → pred zero ≤ n",
"tactic": "exact fun _ => zero_le n"
}
] |
[
238,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
1
] |
Mathlib/Dynamics/FixedPoints/Basic.lean
|
Function.IsFixedPt.perm_inv
|
[] |
[
109,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
11
] |
Mathlib/Topology/Covering.lean
|
IsCoveringMapOn.isLocallyHomeomorphOn
|
[
{
"state_after": "E : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\nf : E → X\ns : Set X\nhf : IsCoveringMapOn f s\nx : E\nhx : x ∈ f ⁻¹' s\n⊢ ∃ e, x ∈ e.source ∧ ∀ (y : E), y ∈ e.source → f y = ↑e y",
"state_before": "E : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\nf : E → X\ns : Set X\nhf : IsCoveringMapOn f s\n⊢ IsLocallyHomeomorphOn f (f ⁻¹' s)",
"tactic": "refine' IsLocallyHomeomorphOn.mk f (f ⁻¹' s) fun x hx => _"
},
{
"state_after": "E : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\nf : E → X\ns : Set X\nhf : IsCoveringMapOn f s\nx : E\nhx : x ∈ f ⁻¹' s\ne : Trivialization (↑(f ⁻¹' {f x})) f := IsEvenlyCovered.toTrivialization (_ : IsEvenlyCovered f (f x) ↑(f ⁻¹' {f x}))\n⊢ ∃ e, x ∈ e.source ∧ ∀ (y : E), y ∈ e.source → f y = ↑e y",
"state_before": "E : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\nf : E → X\ns : Set X\nhf : IsCoveringMapOn f s\nx : E\nhx : x ∈ f ⁻¹' s\n⊢ ∃ e, x ∈ e.source ∧ ∀ (y : E), y ∈ e.source → f y = ↑e y",
"tactic": "let e := (hf (f x) hx).toTrivialization"
},
{
"state_after": "E : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\nf : E → X\ns : Set X\nhf : IsCoveringMapOn f s\nx : E\nhx : x ∈ f ⁻¹' s\ne : Trivialization (↑(f ⁻¹' {f x})) f := IsEvenlyCovered.toTrivialization (_ : IsEvenlyCovered f (f x) ↑(f ⁻¹' {f x}))\nh : f x ∈ (IsEvenlyCovered.toTrivialization (_ : IsEvenlyCovered f (f x) ↑(f ⁻¹' {f x}))).baseSet\n⊢ ∃ e, x ∈ e.source ∧ ∀ (y : E), y ∈ e.source → f y = ↑e y",
"state_before": "E : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\nf : E → X\ns : Set X\nhf : IsCoveringMapOn f s\nx : E\nhx : x ∈ f ⁻¹' s\ne : Trivialization (↑(f ⁻¹' {f x})) f := IsEvenlyCovered.toTrivialization (_ : IsEvenlyCovered f (f x) ↑(f ⁻¹' {f x}))\n⊢ ∃ e, x ∈ e.source ∧ ∀ (y : E), y ∈ e.source → f y = ↑e y",
"tactic": "have h := (hf (f x) hx).mem_toTrivialization_baseSet"
},
{
"state_after": "E : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\nf : E → X\ns : Set X\nhf : IsCoveringMapOn f s\nx : E\nhx : x ∈ f ⁻¹' s\ne : Trivialization (↑(f ⁻¹' {f x})) f := IsEvenlyCovered.toTrivialization (_ : IsEvenlyCovered f (f x) ↑(f ⁻¹' {f x}))\nh : f x ∈ (IsEvenlyCovered.toTrivialization (_ : IsEvenlyCovered f (f x) ↑(f ⁻¹' {f x}))).baseSet\nhe : x ∈ e.source := Iff.mpr (Trivialization.mem_source e) h\n⊢ ∃ e, x ∈ e.source ∧ ∀ (y : E), y ∈ e.source → f y = ↑e y",
"state_before": "E : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\nf : E → X\ns : Set X\nhf : IsCoveringMapOn f s\nx : E\nhx : x ∈ f ⁻¹' s\ne : Trivialization (↑(f ⁻¹' {f x})) f := IsEvenlyCovered.toTrivialization (_ : IsEvenlyCovered f (f x) ↑(f ⁻¹' {f x}))\nh : f x ∈ (IsEvenlyCovered.toTrivialization (_ : IsEvenlyCovered f (f x) ↑(f ⁻¹' {f x}))).baseSet\n⊢ ∃ e, x ∈ e.source ∧ ∀ (y : E), y ∈ e.source → f y = ↑e y",
"tactic": "let he := e.mem_source.2 h"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\nf : E → X\ns : Set X\nhf : IsCoveringMapOn f s\nx : E\nhx : x ∈ f ⁻¹' s\ne : Trivialization (↑(f ⁻¹' {f x})) f := IsEvenlyCovered.toTrivialization (_ : IsEvenlyCovered f (f x) ↑(f ⁻¹' {f x}))\nh : f x ∈ (IsEvenlyCovered.toTrivialization (_ : IsEvenlyCovered f (f x) ↑(f ⁻¹' {f x}))).baseSet\nhe : x ∈ e.source := Iff.mpr (Trivialization.mem_source e) h\n⊢ ∃ e, x ∈ e.source ∧ ∀ (y : E), y ∈ e.source → f y = ↑e y",
"tactic": "refine'\n ⟨e.toLocalHomeomorph.trans\n { toFun := fun p => p.1\n invFun := fun p => ⟨p, x, rfl⟩\n source := e.baseSet ×ˢ ({⟨x, rfl⟩} : Set (f ⁻¹' {f x}))\n target := e.baseSet\n open_source :=\n e.open_baseSet.prod (singletons_open_iff_discrete.2 (hf (f x) hx).1 ⟨x, rfl⟩)\n open_target := e.open_baseSet\n map_source' := fun p => And.left\n map_target' := fun p hp => ⟨hp, rfl⟩\n left_inv' := fun p hp => Prod.ext rfl hp.2.symm\n right_inv' := fun p _ => rfl\n continuous_toFun := continuous_fst.continuousOn\n continuous_invFun := (continuous_id'.prod_mk continuous_const).continuousOn },\n ⟨he, by rwa [e.toLocalHomeomorph.symm_symm, e.proj_toFun x he],\n (hf (f x) hx).toTrivialization_apply⟩,\n fun p h => (e.proj_toFun p h.1).symm⟩"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\nf : E → X\ns : Set X\nhf : IsCoveringMapOn f s\nx : E\nhx : x ∈ f ⁻¹' s\ne : Trivialization (↑(f ⁻¹' {f x})) f := IsEvenlyCovered.toTrivialization (_ : IsEvenlyCovered f (f x) ↑(f ⁻¹' {f x}))\nh : f x ∈ (IsEvenlyCovered.toTrivialization (_ : IsEvenlyCovered f (f x) ↑(f ⁻¹' {f x}))).baseSet\nhe : x ∈ e.source := Iff.mpr (Trivialization.mem_source e) h\n⊢ (↑(LocalHomeomorph.symm (LocalHomeomorph.symm e.toLocalHomeomorph)) x).fst ∈ e.baseSet",
"tactic": "rwa [e.toLocalHomeomorph.symm_symm, e.proj_toFun x he]"
}
] |
[
130,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
11
] |
Mathlib/Topology/Bornology/Constructions.lean
|
Bornology.isBounded_induced
|
[] |
[
136,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
src/lean/Init/Data/Nat/Basic.lean
|
Nat.succ_ne_zero
|
[] |
[
433,
29
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
432,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/NormalMono/Equalizers.lean
|
CategoryTheory.NormalMonoCategory.epi_of_zero_cokernel
|
[
{
"state_after": "no goals",
"state_before": "C : Type ?u.76906\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteProducts C\ninst✝¹ : HasKernels C\ninst✝ : NormalMonoCategory C\nX Y : C\nf : X ⟶ Y\nZ : C\n⊢ f ≫ 0 = 0",
"tactic": "simp"
},
{
"state_after": "case mk\nC : Type u_2\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteProducts C\ninst✝¹ : HasKernels C\ninst✝ : NormalMonoCategory C\nX Y : C\nf : X ⟶ Y\nZ : C\nl : IsColimit (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0))\nZ✝ : C\nu v : Y ⟶ Z✝\nhuv : f ≫ u = f ≫ v\nW : C\nw : Y ⟶ W\nhw : equalizer.ι u v ≫ w = 0\nhl : IsLimit (KernelFork.ofι (equalizer.ι u v) hw)\n⊢ u = v",
"state_before": "C : Type u_2\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteProducts C\ninst✝¹ : HasKernels C\ninst✝ : NormalMonoCategory C\nX Y : C\nf : X ⟶ Y\nZ : C\nl : IsColimit (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0))\nZ✝ : C\nu v : Y ⟶ Z✝\nhuv : f ≫ u = f ≫ v\n⊢ u = v",
"tactic": "obtain ⟨W, w, hw, hl⟩ := normalMonoOfMono (equalizer.ι u v)"
},
{
"state_after": "case mk.mk\nC : Type u_2\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteProducts C\ninst✝¹ : HasKernels C\ninst✝ : NormalMonoCategory C\nX Y : C\nf : X ⟶ Y\nZ : C\nl : IsColimit (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0))\nZ✝ : C\nu v : Y ⟶ Z✝\nhuv : f ≫ u = f ≫ v\nW : C\nw : Y ⟶ W\nhw : equalizer.ι u v ≫ w = 0\nhl : IsLimit (KernelFork.ofι (equalizer.ι u v) hw)\nm : X ⟶ equalizer u v\nhm : m ≫ equalizer.ι u v = f\n⊢ u = v",
"state_before": "case mk\nC : Type u_2\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteProducts C\ninst✝¹ : HasKernels C\ninst✝ : NormalMonoCategory C\nX Y : C\nf : X ⟶ Y\nZ : C\nl : IsColimit (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0))\nZ✝ : C\nu v : Y ⟶ Z✝\nhuv : f ≫ u = f ≫ v\nW : C\nw : Y ⟶ W\nhw : equalizer.ι u v ≫ w = 0\nhl : IsLimit (KernelFork.ofι (equalizer.ι u v) hw)\n⊢ u = v",
"tactic": "obtain ⟨m, hm⟩ := equalizer.lift' f huv"
},
{
"state_after": "case mk.mk\nC : Type u_2\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteProducts C\ninst✝¹ : HasKernels C\ninst✝ : NormalMonoCategory C\nX Y : C\nf : X ⟶ Y\nZ : C\nl : IsColimit (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0))\nZ✝ : C\nu v : Y ⟶ Z✝\nhuv : f ≫ u = f ≫ v\nW : C\nw : Y ⟶ W\nhw : equalizer.ι u v ≫ w = 0\nhl : IsLimit (KernelFork.ofι (equalizer.ι u v) hw)\nm : X ⟶ equalizer u v\nhm : m ≫ equalizer.ι u v = f\nhwf : f ≫ w = 0\n⊢ u = v",
"state_before": "case mk.mk\nC : Type u_2\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteProducts C\ninst✝¹ : HasKernels C\ninst✝ : NormalMonoCategory C\nX Y : C\nf : X ⟶ Y\nZ : C\nl : IsColimit (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0))\nZ✝ : C\nu v : Y ⟶ Z✝\nhuv : f ≫ u = f ≫ v\nW : C\nw : Y ⟶ W\nhw : equalizer.ι u v ≫ w = 0\nhl : IsLimit (KernelFork.ofι (equalizer.ι u v) hw)\nm : X ⟶ equalizer u v\nhm : m ≫ equalizer.ι u v = f\n⊢ u = v",
"tactic": "have hwf : f ≫ w = 0 := by rw [← hm, Category.assoc, hw, comp_zero]"
},
{
"state_after": "case mk.mk.mk\nC : Type u_2\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteProducts C\ninst✝¹ : HasKernels C\ninst✝ : NormalMonoCategory C\nX Y : C\nf : X ⟶ Y\nZ : C\nl : IsColimit (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0))\nZ✝ : C\nu v : Y ⟶ Z✝\nhuv : f ≫ u = f ≫ v\nW : C\nw : Y ⟶ W\nhw : equalizer.ι u v ≫ w = 0\nhl : IsLimit (KernelFork.ofι (equalizer.ι u v) hw)\nm : X ⟶ equalizer u v\nhm : m ≫ equalizer.ι u v = f\nhwf : f ≫ w = 0\nn : (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0)).pt ⟶ W\nhn : Cofork.π (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0)) ≫ n = w\n⊢ u = v",
"state_before": "case mk.mk\nC : Type u_2\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteProducts C\ninst✝¹ : HasKernels C\ninst✝ : NormalMonoCategory C\nX Y : C\nf : X ⟶ Y\nZ : C\nl : IsColimit (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0))\nZ✝ : C\nu v : Y ⟶ Z✝\nhuv : f ≫ u = f ≫ v\nW : C\nw : Y ⟶ W\nhw : equalizer.ι u v ≫ w = 0\nhl : IsLimit (KernelFork.ofι (equalizer.ι u v) hw)\nm : X ⟶ equalizer u v\nhm : m ≫ equalizer.ι u v = f\nhwf : f ≫ w = 0\n⊢ u = v",
"tactic": "obtain ⟨n, hn⟩ := CokernelCofork.IsColimit.desc' l _ hwf"
},
{
"state_after": "case mk.mk.mk\nC : Type u_2\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteProducts C\ninst✝¹ : HasKernels C\ninst✝ : NormalMonoCategory C\nX Y : C\nf : X ⟶ Y\nZ : C\nl : IsColimit (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0))\nZ✝ : C\nu v : Y ⟶ Z✝\nhuv : f ≫ u = f ≫ v\nW : C\nw : Y ⟶ W\nhw : equalizer.ι u v ≫ w = 0\nhl : IsLimit (KernelFork.ofι (equalizer.ι u v) hw)\nm : X ⟶ equalizer u v\nhm : m ≫ equalizer.ι u v = f\nhwf : f ≫ w = 0\nn : (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0)).pt ⟶ W\nhn : 0 = w\n⊢ u = v",
"state_before": "case mk.mk.mk\nC : Type u_2\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteProducts C\ninst✝¹ : HasKernels C\ninst✝ : NormalMonoCategory C\nX Y : C\nf : X ⟶ Y\nZ : C\nl : IsColimit (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0))\nZ✝ : C\nu v : Y ⟶ Z✝\nhuv : f ≫ u = f ≫ v\nW : C\nw : Y ⟶ W\nhw : equalizer.ι u v ≫ w = 0\nhl : IsLimit (KernelFork.ofι (equalizer.ι u v) hw)\nm : X ⟶ equalizer u v\nhm : m ≫ equalizer.ι u v = f\nhwf : f ≫ w = 0\nn : (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0)).pt ⟶ W\nhn : Cofork.π (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0)) ≫ n = w\n⊢ u = v",
"tactic": "rw [Cofork.π_ofπ, zero_comp] at hn"
},
{
"state_after": "case mk.mk.mk\nC : Type u_2\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteProducts C\ninst✝¹ : HasKernels C\ninst✝ : NormalMonoCategory C\nX Y : C\nf : X ⟶ Y\nZ : C\nl : IsColimit (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0))\nZ✝ : C\nu v : Y ⟶ Z✝\nhuv : f ≫ u = f ≫ v\nW : C\nw : Y ⟶ W\nhw : equalizer.ι u v ≫ w = 0\nhl : IsLimit (KernelFork.ofι (equalizer.ι u v) hw)\nm : X ⟶ equalizer u v\nhm : m ≫ equalizer.ι u v = f\nhwf : f ≫ w = 0\nn : (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0)).pt ⟶ W\nhn : 0 = w\nthis : IsIso (equalizer.ι u v)\n⊢ u = v",
"state_before": "case mk.mk.mk\nC : Type u_2\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteProducts C\ninst✝¹ : HasKernels C\ninst✝ : NormalMonoCategory C\nX Y : C\nf : X ⟶ Y\nZ : C\nl : IsColimit (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0))\nZ✝ : C\nu v : Y ⟶ Z✝\nhuv : f ≫ u = f ≫ v\nW : C\nw : Y ⟶ W\nhw : equalizer.ι u v ≫ w = 0\nhl : IsLimit (KernelFork.ofι (equalizer.ι u v) hw)\nm : X ⟶ equalizer u v\nhm : m ≫ equalizer.ι u v = f\nhwf : f ≫ w = 0\nn : (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0)).pt ⟶ W\nhn : 0 = w\n⊢ u = v",
"tactic": "have : IsIso (equalizer.ι u v) := by apply isIso_limit_cone_parallelPair_of_eq hn.symm hl"
},
{
"state_after": "case mk.mk.mk\nC : Type u_2\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteProducts C\ninst✝¹ : HasKernels C\ninst✝ : NormalMonoCategory C\nX Y : C\nf : X ⟶ Y\nZ : C\nl : IsColimit (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0))\nZ✝ : C\nu v : Y ⟶ Z✝\nhuv : f ≫ u = f ≫ v\nW : C\nw : Y ⟶ W\nhw : equalizer.ι u v ≫ w = 0\nhl : IsLimit (KernelFork.ofι (equalizer.ι u v) hw)\nm : X ⟶ equalizer u v\nhm : m ≫ equalizer.ι u v = f\nhwf : f ≫ w = 0\nn : (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0)).pt ⟶ W\nhn : 0 = w\nthis : IsIso (equalizer.ι u v)\n⊢ equalizer.ι u v ≫ u = equalizer.ι u v ≫ v",
"state_before": "case mk.mk.mk\nC : Type u_2\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteProducts C\ninst✝¹ : HasKernels C\ninst✝ : NormalMonoCategory C\nX Y : C\nf : X ⟶ Y\nZ : C\nl : IsColimit (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0))\nZ✝ : C\nu v : Y ⟶ Z✝\nhuv : f ≫ u = f ≫ v\nW : C\nw : Y ⟶ W\nhw : equalizer.ι u v ≫ w = 0\nhl : IsLimit (KernelFork.ofι (equalizer.ι u v) hw)\nm : X ⟶ equalizer u v\nhm : m ≫ equalizer.ι u v = f\nhwf : f ≫ w = 0\nn : (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0)).pt ⟶ W\nhn : 0 = w\nthis : IsIso (equalizer.ι u v)\n⊢ u = v",
"tactic": "apply (cancel_epi (equalizer.ι u v)).1"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.mk\nC : Type u_2\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteProducts C\ninst✝¹ : HasKernels C\ninst✝ : NormalMonoCategory C\nX Y : C\nf : X ⟶ Y\nZ : C\nl : IsColimit (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0))\nZ✝ : C\nu v : Y ⟶ Z✝\nhuv : f ≫ u = f ≫ v\nW : C\nw : Y ⟶ W\nhw : equalizer.ι u v ≫ w = 0\nhl : IsLimit (KernelFork.ofι (equalizer.ι u v) hw)\nm : X ⟶ equalizer u v\nhm : m ≫ equalizer.ι u v = f\nhwf : f ≫ w = 0\nn : (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0)).pt ⟶ W\nhn : 0 = w\nthis : IsIso (equalizer.ι u v)\n⊢ equalizer.ι u v ≫ u = equalizer.ι u v ≫ v",
"tactic": "exact equalizer.condition _ _"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteProducts C\ninst✝¹ : HasKernels C\ninst✝ : NormalMonoCategory C\nX Y : C\nf : X ⟶ Y\nZ : C\nl : IsColimit (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0))\nZ✝ : C\nu v : Y ⟶ Z✝\nhuv : f ≫ u = f ≫ v\nW : C\nw : Y ⟶ W\nhw : equalizer.ι u v ≫ w = 0\nhl : IsLimit (KernelFork.ofι (equalizer.ι u v) hw)\nm : X ⟶ equalizer u v\nhm : m ≫ equalizer.ι u v = f\n⊢ f ≫ w = 0",
"tactic": "rw [← hm, Category.assoc, hw, comp_zero]"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteProducts C\ninst✝¹ : HasKernels C\ninst✝ : NormalMonoCategory C\nX Y : C\nf : X ⟶ Y\nZ : C\nl : IsColimit (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0))\nZ✝ : C\nu v : Y ⟶ Z✝\nhuv : f ≫ u = f ≫ v\nW : C\nw : Y ⟶ W\nhw : equalizer.ι u v ≫ w = 0\nhl : IsLimit (KernelFork.ofι (equalizer.ι u v) hw)\nm : X ⟶ equalizer u v\nhm : m ≫ equalizer.ι u v = f\nhwf : f ≫ w = 0\nn : (CokernelCofork.ofπ 0 (_ : f ≫ 0 = 0)).pt ⟶ W\nhn : 0 = w\n⊢ IsIso (equalizer.ι u v)",
"tactic": "apply isIso_limit_cone_parallelPair_of_eq hn.symm hl"
}
] |
[
168,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Topology/Algebra/UniformConvergence.lean
|
UniformOnFun.hasBasis_nhds_one_of_basis
|
[
{
"state_after": "α : Type u_1\nG : Type u_2\nι : Type u_3\ninst✝² : Group G\n𝔖✝ : Set (Set α)\ninst✝¹ : UniformSpace G\ninst✝ : UniformGroup G\n𝔖 : Set (Set α)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\np : ι → Prop\nb : ι → Set G\nh : HasBasis (𝓝 1) p b\nthis : HasBasis (comap (fun p => p.fst / p.snd) (𝓝 1)) p fun i => (fun p => p.fst / p.snd) ⁻¹' b i\n⊢ HasBasis (𝓝 1) (fun Si => Si.fst ∈ 𝔖 ∧ p Si.snd) fun Si => {f | ∀ (x : α), x ∈ Si.fst → f x ∈ b Si.snd}",
"state_before": "α : Type u_1\nG : Type u_2\nι : Type u_3\ninst✝² : Group G\n𝔖✝ : Set (Set α)\ninst✝¹ : UniformSpace G\ninst✝ : UniformGroup G\n𝔖 : Set (Set α)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\np : ι → Prop\nb : ι → Set G\nh : HasBasis (𝓝 1) p b\n⊢ HasBasis (𝓝 1) (fun Si => Si.fst ∈ 𝔖 ∧ p Si.snd) fun Si => {f | ∀ (x : α), x ∈ Si.fst → f x ∈ b Si.snd}",
"tactic": "have := h.comap fun p : G × G => p.1 / p.2"
},
{
"state_after": "α : Type u_1\nG : Type u_2\nι : Type u_3\ninst✝² : Group G\n𝔖✝ : Set (Set α)\ninst✝¹ : UniformSpace G\ninst✝ : UniformGroup G\n𝔖 : Set (Set α)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\np : ι → Prop\nb : ι → Set G\nh : HasBasis (𝓝 1) p b\nthis : HasBasis (uniformity G) p fun i => (fun p => p.fst / p.snd) ⁻¹' b i\n⊢ HasBasis (𝓝 1) (fun Si => Si.fst ∈ 𝔖 ∧ p Si.snd) fun Si => {f | ∀ (x : α), x ∈ Si.fst → f x ∈ b Si.snd}",
"state_before": "α : Type u_1\nG : Type u_2\nι : Type u_3\ninst✝² : Group G\n𝔖✝ : Set (Set α)\ninst✝¹ : UniformSpace G\ninst✝ : UniformGroup G\n𝔖 : Set (Set α)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\np : ι → Prop\nb : ι → Set G\nh : HasBasis (𝓝 1) p b\nthis : HasBasis (comap (fun p => p.fst / p.snd) (𝓝 1)) p fun i => (fun p => p.fst / p.snd) ⁻¹' b i\n⊢ HasBasis (𝓝 1) (fun Si => Si.fst ∈ 𝔖 ∧ p Si.snd) fun Si => {f | ∀ (x : α), x ∈ Si.fst → f x ∈ b Si.snd}",
"tactic": "rw [← uniformity_eq_comap_nhds_one_swapped] at this"
},
{
"state_after": "case h.e'_5.h.h.e'_2.h.a\nα : Type u_1\nG : Type u_2\nι : Type u_3\ninst✝² : Group G\n𝔖✝ : Set (Set α)\ninst✝¹ : UniformSpace G\ninst✝ : UniformGroup G\n𝔖 : Set (Set α)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\np : ι → Prop\nb : ι → Set G\nh : HasBasis (𝓝 1) p b\nthis : HasBasis (uniformity G) p fun i => (fun p => p.fst / p.snd) ⁻¹' b i\nx✝¹ : Set α × ι\nx✝ : α →ᵤ[𝔖] G\n⊢ (∀ (x : α), x ∈ x✝¹.fst → x✝ x ∈ b x✝¹.snd) ↔\n (x✝, 1) ∈ UniformOnFun.gen 𝔖 x✝¹.fst ((fun p => p.fst / p.snd) ⁻¹' b x✝¹.snd)",
"state_before": "α : Type u_1\nG : Type u_2\nι : Type u_3\ninst✝² : Group G\n𝔖✝ : Set (Set α)\ninst✝¹ : UniformSpace G\ninst✝ : UniformGroup G\n𝔖 : Set (Set α)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\np : ι → Prop\nb : ι → Set G\nh : HasBasis (𝓝 1) p b\nthis : HasBasis (uniformity G) p fun i => (fun p => p.fst / p.snd) ⁻¹' b i\n⊢ HasBasis (𝓝 1) (fun Si => Si.fst ∈ 𝔖 ∧ p Si.snd) fun Si => {f | ∀ (x : α), x ∈ Si.fst → f x ∈ b Si.snd}",
"tactic": "convert UniformOnFun.hasBasis_nhds_of_basis α _ 𝔖 (1 : α →ᵤ[𝔖] G) h𝔖₁ h𝔖₂ this"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h.h.e'_2.h.a\nα : Type u_1\nG : Type u_2\nι : Type u_3\ninst✝² : Group G\n𝔖✝ : Set (Set α)\ninst✝¹ : UniformSpace G\ninst✝ : UniformGroup G\n𝔖 : Set (Set α)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\np : ι → Prop\nb : ι → Set G\nh : HasBasis (𝓝 1) p b\nthis : HasBasis (uniformity G) p fun i => (fun p => p.fst / p.snd) ⁻¹' b i\nx✝¹ : Set α × ι\nx✝ : α →ᵤ[𝔖] G\n⊢ (∀ (x : α), x ∈ x✝¹.fst → x✝ x ∈ b x✝¹.snd) ↔\n (x✝, 1) ∈ UniformOnFun.gen 𝔖 x✝¹.fst ((fun p => p.fst / p.snd) ⁻¹' b x✝¹.snd)",
"tactic": "simp [UniformOnFun.gen]"
}
] |
[
190,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
11
] |
Mathlib/GroupTheory/Perm/Fin.lean
|
Equiv.Perm.decomposeFin_symm_of_refl
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\np : Fin (n + 1)\n⊢ ↑decomposeFin.symm (p, Equiv.refl (Fin n)) = swap 0 p",
"tactic": "simp [Equiv.Perm.decomposeFin, Equiv.permCongr_def]"
}
] |
[
34,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
32,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.brange_const
|
[
{
"state_after": "α : Type u_2\nβ : Type ?u.276922\nγ : Type ?u.276925\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nho : o ≠ 0\nc : α\n⊢ range (familyOfBFamily o fun x x => c) = {c}",
"state_before": "α : Type u_2\nβ : Type ?u.276922\nγ : Type ?u.276925\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nho : o ≠ 0\nc : α\n⊢ (brange o fun x x => c) = {c}",
"tactic": "rw [← range_familyOfBFamily]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.276922\nγ : Type ?u.276925\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nho : o ≠ 0\nc : α\n⊢ range (familyOfBFamily o fun x x => c) = {c}",
"tactic": "exact @Set.range_const _ o.out.α (out_nonempty_iff_ne_zero.2 ho) c"
}
] |
[
1217,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1215,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.union_mul_inter_subset_union
|
[] |
[
472,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
471,
1
] |
src/lean/Init/Data/Nat/Basic.lean
|
Nat.le.intro
|
[] |
[
358,
23
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
357,
1
] |
Mathlib/ModelTheory/Basic.lean
|
FirstOrder.Language.Embedding.ext
|
[] |
[
657,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
656,
1
] |
Mathlib/Order/CountableDenseLinearOrder.lean
|
Order.embedding_from_countable_to_dense
|
[
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : LinearOrder β\ninst✝² : Encodable α\ninst✝¹ : DenselyOrdered β\ninst✝ : Nontrivial β\nx y : β\nhxy : x < y\n⊢ Nonempty (α ↪o β)",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : LinearOrder β\ninst✝² : Encodable α\ninst✝¹ : DenselyOrdered β\ninst✝ : Nontrivial β\n⊢ Nonempty (α ↪o β)",
"tactic": "rcases exists_pair_lt β with ⟨x, y, hxy⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : LinearOrder β\ninst✝² : Encodable α\ninst✝¹ : DenselyOrdered β\ninst✝ : Nontrivial β\nx y : β\nhxy : x < y\na : β\nha : x < a ∧ a < y\n⊢ Nonempty (α ↪o β)",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : LinearOrder β\ninst✝² : Encodable α\ninst✝¹ : DenselyOrdered β\ninst✝ : Nontrivial β\nx y : β\nhxy : x < y\n⊢ Nonempty (α ↪o β)",
"tactic": "cases' exists_between hxy with a ha"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : LinearOrder β\ninst✝² : Encodable α\ninst✝¹ : DenselyOrdered β\ninst✝ : Nontrivial β\nx y : β\nhxy : x < y\na : β\nha : x < a ∧ a < y\nthis : Nonempty ↑(Set.Ioo x y)\n⊢ Nonempty (α ↪o β)",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : LinearOrder β\ninst✝² : Encodable α\ninst✝¹ : DenselyOrdered β\ninst✝ : Nontrivial β\nx y : β\nhxy : x < y\na : β\nha : x < a ∧ a < y\n⊢ Nonempty (α ↪o β)",
"tactic": "haveI : Nonempty (Set.Ioo x y) := ⟨⟨a, ha⟩⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : LinearOrder β\ninst✝² : Encodable α\ninst✝¹ : DenselyOrdered β\ninst✝ : Nontrivial β\nx y : β\nhxy : x < y\na : β\nha : x < a ∧ a < y\nthis : Nonempty ↑(Set.Ioo x y)\nour_ideal : Ideal (PartialIso α ↑(Set.Ioo x y)) := idealOfCofinals default (definedAtLeft ↑(Set.Ioo x y))\n⊢ Nonempty (α ↪o β)",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : LinearOrder β\ninst✝² : Encodable α\ninst✝¹ : DenselyOrdered β\ninst✝ : Nontrivial β\nx y : β\nhxy : x < y\na : β\nha : x < a ∧ a < y\nthis : Nonempty ↑(Set.Ioo x y)\n⊢ Nonempty (α ↪o β)",
"tactic": "let our_ideal : Ideal (PartialIso α _) :=\n idealOfCofinals default (definedAtLeft (Set.Ioo x y))"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : LinearOrder β\ninst✝² : Encodable α\ninst✝¹ : DenselyOrdered β\ninst✝ : Nontrivial β\nx y : β\nhxy : x < y\na : β\nha : x < a ∧ a < y\nthis : Nonempty ↑(Set.Ioo x y)\nour_ideal : Ideal (PartialIso α ↑(Set.Ioo x y)) := idealOfCofinals default (definedAtLeft ↑(Set.Ioo x y))\nF : (a : α) → { b // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun a =>\n funOfIdeal a our_ideal\n (_ : ∃ x_1, x_1 ∈ definedAtLeft (↑(Set.Ioo x y)) a ∧ x_1 ∈ idealOfCofinals default (definedAtLeft ↑(Set.Ioo x y)))\n⊢ Nonempty (α ↪o β)",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : LinearOrder β\ninst✝² : Encodable α\ninst✝¹ : DenselyOrdered β\ninst✝ : Nontrivial β\nx y : β\nhxy : x < y\na : β\nha : x < a ∧ a < y\nthis : Nonempty ↑(Set.Ioo x y)\nour_ideal : Ideal (PartialIso α ↑(Set.Ioo x y)) := idealOfCofinals default (definedAtLeft ↑(Set.Ioo x y))\n⊢ Nonempty (α ↪o β)",
"tactic": "let F a := funOfIdeal a our_ideal (cofinal_meets_idealOfCofinals _ _ a)"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : LinearOrder β\ninst✝² : Encodable α\ninst✝¹ : DenselyOrdered β\ninst✝ : Nontrivial β\nx y : β\nhxy : x < y\na : β\nha : x < a ∧ a < y\nthis : Nonempty ↑(Set.Ioo x y)\nour_ideal : Ideal (PartialIso α ↑(Set.Ioo x y)) := idealOfCofinals default (definedAtLeft ↑(Set.Ioo x y))\nF : (a : α) → { b // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun a =>\n funOfIdeal a our_ideal\n (_ : ∃ x_1, x_1 ∈ definedAtLeft (↑(Set.Ioo x y)) a ∧ x_1 ∈ idealOfCofinals default (definedAtLeft ↑(Set.Ioo x y)))\na₁ a₂ : α\n⊢ a₁ < a₂ → (fun a => ↑(F a)) a₁ < (fun a => ↑(F a)) a₂",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : LinearOrder β\ninst✝² : Encodable α\ninst✝¹ : DenselyOrdered β\ninst✝ : Nontrivial β\nx y : β\nhxy : x < y\na : β\nha : x < a ∧ a < y\nthis : Nonempty ↑(Set.Ioo x y)\nour_ideal : Ideal (PartialIso α ↑(Set.Ioo x y)) := idealOfCofinals default (definedAtLeft ↑(Set.Ioo x y))\nF : (a : α) → { b // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun a =>\n funOfIdeal a our_ideal\n (_ : ∃ x_1, x_1 ∈ definedAtLeft (↑(Set.Ioo x y)) a ∧ x_1 ∈ idealOfCofinals default (definedAtLeft ↑(Set.Ioo x y)))\n⊢ Nonempty (α ↪o β)",
"tactic": "refine\n ⟨RelEmbedding.trans (OrderEmbedding.ofStrictMono (fun a ↦ (F a).val) fun a₁ a₂ ↦ ?_)\n (OrderEmbedding.subtype _)⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : LinearOrder β\ninst✝² : Encodable α\ninst✝¹ : DenselyOrdered β\ninst✝ : Nontrivial β\nx y : β\nhxy : x < y\na : β\nha : x < a ∧ a < y\nthis : Nonempty ↑(Set.Ioo x y)\nour_ideal : Ideal (PartialIso α ↑(Set.Ioo x y)) := idealOfCofinals default (definedAtLeft ↑(Set.Ioo x y))\nF : (a : α) → { b // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun a =>\n funOfIdeal a our_ideal\n (_ : ∃ x_1, x_1 ∈ definedAtLeft (↑(Set.Ioo x y)) a ∧ x_1 ∈ idealOfCofinals default (definedAtLeft ↑(Set.Ioo x y)))\na₁ a₂ : α\nf : PartialIso α ↑(Set.Ioo x y)\nhf : f ∈ our_ideal\nha₁ : (a₁, ↑(F a₁)) ∈ ↑f\n⊢ a₁ < a₂ → (fun a => ↑(F a)) a₁ < (fun a => ↑(F a)) a₂",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : LinearOrder β\ninst✝² : Encodable α\ninst✝¹ : DenselyOrdered β\ninst✝ : Nontrivial β\nx y : β\nhxy : x < y\na : β\nha : x < a ∧ a < y\nthis : Nonempty ↑(Set.Ioo x y)\nour_ideal : Ideal (PartialIso α ↑(Set.Ioo x y)) := idealOfCofinals default (definedAtLeft ↑(Set.Ioo x y))\nF : (a : α) → { b // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun a =>\n funOfIdeal a our_ideal\n (_ : ∃ x_1, x_1 ∈ definedAtLeft (↑(Set.Ioo x y)) a ∧ x_1 ∈ idealOfCofinals default (definedAtLeft ↑(Set.Ioo x y)))\na₁ a₂ : α\n⊢ a₁ < a₂ → (fun a => ↑(F a)) a₁ < (fun a => ↑(F a)) a₂",
"tactic": "rcases(F a₁).prop with ⟨f, hf, ha₁⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : LinearOrder β\ninst✝² : Encodable α\ninst✝¹ : DenselyOrdered β\ninst✝ : Nontrivial β\nx y : β\nhxy : x < y\na : β\nha : x < a ∧ a < y\nthis : Nonempty ↑(Set.Ioo x y)\nour_ideal : Ideal (PartialIso α ↑(Set.Ioo x y)) := idealOfCofinals default (definedAtLeft ↑(Set.Ioo x y))\nF : (a : α) → { b // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun a =>\n funOfIdeal a our_ideal\n (_ : ∃ x_1, x_1 ∈ definedAtLeft (↑(Set.Ioo x y)) a ∧ x_1 ∈ idealOfCofinals default (definedAtLeft ↑(Set.Ioo x y)))\na₁ a₂ : α\nf : PartialIso α ↑(Set.Ioo x y)\nhf : f ∈ our_ideal\nha₁ : (a₁, ↑(F a₁)) ∈ ↑f\ng : PartialIso α ↑(Set.Ioo x y)\nhg : g ∈ our_ideal\nha₂ : (a₂, ↑(F a₂)) ∈ ↑g\n⊢ a₁ < a₂ → (fun a => ↑(F a)) a₁ < (fun a => ↑(F a)) a₂",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : LinearOrder β\ninst✝² : Encodable α\ninst✝¹ : DenselyOrdered β\ninst✝ : Nontrivial β\nx y : β\nhxy : x < y\na : β\nha : x < a ∧ a < y\nthis : Nonempty ↑(Set.Ioo x y)\nour_ideal : Ideal (PartialIso α ↑(Set.Ioo x y)) := idealOfCofinals default (definedAtLeft ↑(Set.Ioo x y))\nF : (a : α) → { b // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun a =>\n funOfIdeal a our_ideal\n (_ : ∃ x_1, x_1 ∈ definedAtLeft (↑(Set.Ioo x y)) a ∧ x_1 ∈ idealOfCofinals default (definedAtLeft ↑(Set.Ioo x y)))\na₁ a₂ : α\nf : PartialIso α ↑(Set.Ioo x y)\nhf : f ∈ our_ideal\nha₁ : (a₁, ↑(F a₁)) ∈ ↑f\n⊢ a₁ < a₂ → (fun a => ↑(F a)) a₁ < (fun a => ↑(F a)) a₂",
"tactic": "rcases(F a₂).prop with ⟨g, hg, ha₂⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : LinearOrder β\ninst✝² : Encodable α\ninst✝¹ : DenselyOrdered β\ninst✝ : Nontrivial β\nx y : β\nhxy : x < y\na : β\nha : x < a ∧ a < y\nthis : Nonempty ↑(Set.Ioo x y)\nour_ideal : Ideal (PartialIso α ↑(Set.Ioo x y)) := idealOfCofinals default (definedAtLeft ↑(Set.Ioo x y))\nF : (a : α) → { b // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun a =>\n funOfIdeal a our_ideal\n (_ : ∃ x_1, x_1 ∈ definedAtLeft (↑(Set.Ioo x y)) a ∧ x_1 ∈ idealOfCofinals default (definedAtLeft ↑(Set.Ioo x y)))\na₁ a₂ : α\nf : PartialIso α ↑(Set.Ioo x y)\nhf : f ∈ our_ideal\nha₁ : (a₁, ↑(F a₁)) ∈ ↑f\ng : PartialIso α ↑(Set.Ioo x y)\nhg : g ∈ our_ideal\nha₂ : (a₂, ↑(F a₂)) ∈ ↑g\nm : PartialIso α ↑(Set.Ioo x y)\n_hm : m ∈ ↑our_ideal\nfm : f ≤ m\ngm : g ≤ m\n⊢ a₁ < a₂ → (fun a => ↑(F a)) a₁ < (fun a => ↑(F a)) a₂",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : LinearOrder β\ninst✝² : Encodable α\ninst✝¹ : DenselyOrdered β\ninst✝ : Nontrivial β\nx y : β\nhxy : x < y\na : β\nha : x < a ∧ a < y\nthis : Nonempty ↑(Set.Ioo x y)\nour_ideal : Ideal (PartialIso α ↑(Set.Ioo x y)) := idealOfCofinals default (definedAtLeft ↑(Set.Ioo x y))\nF : (a : α) → { b // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun a =>\n funOfIdeal a our_ideal\n (_ : ∃ x_1, x_1 ∈ definedAtLeft (↑(Set.Ioo x y)) a ∧ x_1 ∈ idealOfCofinals default (definedAtLeft ↑(Set.Ioo x y)))\na₁ a₂ : α\nf : PartialIso α ↑(Set.Ioo x y)\nhf : f ∈ our_ideal\nha₁ : (a₁, ↑(F a₁)) ∈ ↑f\ng : PartialIso α ↑(Set.Ioo x y)\nhg : g ∈ our_ideal\nha₂ : (a₂, ↑(F a₂)) ∈ ↑g\n⊢ a₁ < a₂ → (fun a => ↑(F a)) a₁ < (fun a => ↑(F a)) a₂",
"tactic": "rcases our_ideal.directed _ hf _ hg with ⟨m, _hm, fm, gm⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : LinearOrder β\ninst✝² : Encodable α\ninst✝¹ : DenselyOrdered β\ninst✝ : Nontrivial β\nx y : β\nhxy : x < y\na : β\nha : x < a ∧ a < y\nthis : Nonempty ↑(Set.Ioo x y)\nour_ideal : Ideal (PartialIso α ↑(Set.Ioo x y)) := idealOfCofinals default (definedAtLeft ↑(Set.Ioo x y))\nF : (a : α) → { b // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun a =>\n funOfIdeal a our_ideal\n (_ : ∃ x_1, x_1 ∈ definedAtLeft (↑(Set.Ioo x y)) a ∧ x_1 ∈ idealOfCofinals default (definedAtLeft ↑(Set.Ioo x y)))\na₁ a₂ : α\nf : PartialIso α ↑(Set.Ioo x y)\nhf : f ∈ our_ideal\nha₁ : (a₁, ↑(F a₁)) ∈ ↑f\ng : PartialIso α ↑(Set.Ioo x y)\nhg : g ∈ our_ideal\nha₂ : (a₂, ↑(F a₂)) ∈ ↑g\nm : PartialIso α ↑(Set.Ioo x y)\n_hm : m ∈ ↑our_ideal\nfm : f ≤ m\ngm : g ≤ m\n⊢ a₁ < a₂ → (fun a => ↑(F a)) a₁ < (fun a => ↑(F a)) a₂",
"tactic": "exact (lt_iff_lt_of_cmp_eq_cmp <| m.prop (a₁, _) (fm ha₁) (a₂, _) (gm ha₂)).mp"
}
] |
[
219,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/Algebra/DirectSum/Module.lean
|
DirectSum.IsInternal.isCompl
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : Semiring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nA✝ A : ι → Submodule R M\ni j : ι\nhij : i ≠ j\nh : Set.univ = {i, j}\nhi : IsInternal A\n⊢ iSup A = A i ⊔ A j",
"tactic": "rw [← sSup_pair, iSup, ← Set.image_univ, h, Set.image_insert_eq, Set.image_singleton]"
}
] |
[
385,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
381,
1
] |
Mathlib/Order/SuccPred/Basic.lean
|
Order.pred_wcovby
|
[] |
[
629,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
628,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
measurable_nndist
|
[] |
[
1571,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1570,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.walkingParallelPairOpEquiv_counitIso_zero
|
[] |
[
196,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/GroupTheory/GroupAction/Units.lean
|
IsUnit.inv_smul
|
[] |
[
50,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/Algebra/Hom/GroupAction.lean
|
DistribMulActionHom.map_neg
|
[] |
[
313,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
312,
11
] |
Mathlib/Algebra/Lie/Basic.lean
|
LieHom.one_apply
|
[] |
[
372,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
371,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.vsub_self_mono
|
[] |
[
681,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
680,
1
] |
Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
|
MeasureTheory.FiniteMeasure.tendsto_iff_forall_testAgainstNN_tendsto
|
[
{
"state_after": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.142439\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nμ : FiniteMeasure Ω\n⊢ (∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => ↑(toWeakDualBCNN (μs i)) f) F (𝓝 (↑(toWeakDualBCNN μ) f))) ↔\n ∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => testAgainstNN (μs i) f) F (𝓝 (testAgainstNN μ f))",
"state_before": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.142439\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nμ : FiniteMeasure Ω\n⊢ Tendsto μs F (𝓝 μ) ↔ ∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => testAgainstNN (μs i) f) F (𝓝 (testAgainstNN μ f))",
"tactic": "rw [FiniteMeasure.tendsto_iff_forall_toWeakDualBCNN_tendsto]"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.142439\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nμ : FiniteMeasure Ω\n⊢ (∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => ↑(toWeakDualBCNN (μs i)) f) F (𝓝 (↑(toWeakDualBCNN μ) f))) ↔\n ∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => testAgainstNN (μs i) f) F (𝓝 (testAgainstNN μ f))",
"tactic": "rfl"
}
] |
[
510,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
506,
1
] |
Mathlib/ModelTheory/Syntax.lean
|
FirstOrder.Language.BoundedFormula.mapTermRel_mapTermRel
|
[
{
"state_after": "case falsum\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.61221\nP : Type ?u.61224\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type u_5\nn✝¹ : ℕ\nL'' : Language\nft : (n : ℕ) → Term L (α ⊕ Fin n) → Term L' (β ⊕ Fin n)\nfr : (n : ℕ) → Relations L n → Relations L' n\nft' : (n : ℕ) → Term L' (β ⊕ Fin n) → Term L'' (γ ⊕ Fin n)\nfr' : (n : ℕ) → Relations L' n → Relations L'' n\nn n✝ : ℕ\n⊢ mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) falsum) =\n mapTermRel (fun x => ft' x ∘ ft x) (fun x => fr' x ∘ fr x) (fun x => id) falsum\n\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.61221\nP : Type ?u.61224\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type u_5\nn✝¹ : ℕ\nL'' : Language\nft : (n : ℕ) → Term L (α ⊕ Fin n) → Term L' (β ⊕ Fin n)\nfr : (n : ℕ) → Relations L n → Relations L' n\nft' : (n : ℕ) → Term L' (β ⊕ Fin n) → Term L'' (γ ⊕ Fin n)\nfr' : (n : ℕ) → Relations L' n → Relations L'' n\nn n✝ : ℕ\nt₁✝ t₂✝ : Term L (α ⊕ Fin n✝)\n⊢ mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) (equal t₁✝ t₂✝)) =\n mapTermRel (fun x => ft' x ∘ ft x) (fun x => fr' x ∘ fr x) (fun x => id) (equal t₁✝ t₂✝)\n\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.61221\nP : Type ?u.61224\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type u_5\nn✝¹ : ℕ\nL'' : Language\nft : (n : ℕ) → Term L (α ⊕ Fin n) → Term L' (β ⊕ Fin n)\nfr : (n : ℕ) → Relations L n → Relations L' n\nft' : (n : ℕ) → Term L' (β ⊕ Fin n) → Term L'' (γ ⊕ Fin n)\nfr' : (n : ℕ) → Relations L' n → Relations L'' n\nn n✝ l✝ : ℕ\nR✝ : Relations L l✝\nts✝ : Fin l✝ → Term L (α ⊕ Fin n✝)\n⊢ mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) (rel R✝ ts✝)) =\n mapTermRel (fun x => ft' x ∘ ft x) (fun x => fr' x ∘ fr x) (fun x => id) (rel R✝ ts✝)\n\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.61221\nP : Type ?u.61224\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type u_5\nn✝¹ : ℕ\nL'' : Language\nft : (n : ℕ) → Term L (α ⊕ Fin n) → Term L' (β ⊕ Fin n)\nfr : (n : ℕ) → Relations L n → Relations L' n\nft' : (n : ℕ) → Term L' (β ⊕ Fin n) → Term L'' (γ ⊕ Fin n)\nfr' : (n : ℕ) → Relations L' n → Relations L'' n\nn n✝ : ℕ\nf₁✝ f₂✝ : BoundedFormula L α n✝\nih1 :\n mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) f₁✝) =\n mapTermRel (fun x => ft' x ∘ ft x) (fun x => fr' x ∘ fr x) (fun x => id) f₁✝\nih2 :\n mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) f₂✝) =\n mapTermRel (fun x => ft' x ∘ ft x) (fun x => fr' x ∘ fr x) (fun x => id) f₂✝\n⊢ mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) (imp f₁✝ f₂✝)) =\n mapTermRel (fun x => ft' x ∘ ft x) (fun x => fr' x ∘ fr x) (fun x => id) (imp f₁✝ f₂✝)\n\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.61221\nP : Type ?u.61224\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type u_5\nn✝¹ : ℕ\nL'' : Language\nft : (n : ℕ) → Term L (α ⊕ Fin n) → Term L' (β ⊕ Fin n)\nfr : (n : ℕ) → Relations L n → Relations L' n\nft' : (n : ℕ) → Term L' (β ⊕ Fin n) → Term L'' (γ ⊕ Fin n)\nfr' : (n : ℕ) → Relations L' n → Relations L'' n\nn n✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 :\n mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) f✝) =\n mapTermRel (fun x => ft' x ∘ ft x) (fun x => fr' x ∘ fr x) (fun x => id) f✝\n⊢ mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) (all f✝)) =\n mapTermRel (fun x => ft' x ∘ ft x) (fun x => fr' x ∘ fr x) (fun x => id) (all f✝)",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.61221\nP : Type ?u.61224\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type u_5\nn✝ : ℕ\nL'' : Language\nft : (n : ℕ) → Term L (α ⊕ Fin n) → Term L' (β ⊕ Fin n)\nfr : (n : ℕ) → Relations L n → Relations L' n\nft' : (n : ℕ) → Term L' (β ⊕ Fin n) → Term L'' (γ ⊕ Fin n)\nfr' : (n : ℕ) → Relations L' n → Relations L'' n\nn : ℕ\nφ : BoundedFormula L α n\n⊢ mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) φ) =\n mapTermRel (fun x => ft' x ∘ ft x) (fun x => fr' x ∘ fr x) (fun x => id) φ",
"tactic": "induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3"
},
{
"state_after": "no goals",
"state_before": "case falsum\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.61221\nP : Type ?u.61224\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type u_5\nn✝¹ : ℕ\nL'' : Language\nft : (n : ℕ) → Term L (α ⊕ Fin n) → Term L' (β ⊕ Fin n)\nfr : (n : ℕ) → Relations L n → Relations L' n\nft' : (n : ℕ) → Term L' (β ⊕ Fin n) → Term L'' (γ ⊕ Fin n)\nfr' : (n : ℕ) → Relations L' n → Relations L'' n\nn n✝ : ℕ\n⊢ mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) falsum) =\n mapTermRel (fun x => ft' x ∘ ft x) (fun x => fr' x ∘ fr x) (fun x => id) falsum",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case equal\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.61221\nP : Type ?u.61224\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type u_5\nn✝¹ : ℕ\nL'' : Language\nft : (n : ℕ) → Term L (α ⊕ Fin n) → Term L' (β ⊕ Fin n)\nfr : (n : ℕ) → Relations L n → Relations L' n\nft' : (n : ℕ) → Term L' (β ⊕ Fin n) → Term L'' (γ ⊕ Fin n)\nfr' : (n : ℕ) → Relations L' n → Relations L'' n\nn n✝ : ℕ\nt₁✝ t₂✝ : Term L (α ⊕ Fin n✝)\n⊢ mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) (equal t₁✝ t₂✝)) =\n mapTermRel (fun x => ft' x ∘ ft x) (fun x => fr' x ∘ fr x) (fun x => id) (equal t₁✝ t₂✝)",
"tactic": "simp [mapTermRel]"
},
{
"state_after": "no goals",
"state_before": "case rel\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.61221\nP : Type ?u.61224\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type u_5\nn✝¹ : ℕ\nL'' : Language\nft : (n : ℕ) → Term L (α ⊕ Fin n) → Term L' (β ⊕ Fin n)\nfr : (n : ℕ) → Relations L n → Relations L' n\nft' : (n : ℕ) → Term L' (β ⊕ Fin n) → Term L'' (γ ⊕ Fin n)\nfr' : (n : ℕ) → Relations L' n → Relations L'' n\nn n✝ l✝ : ℕ\nR✝ : Relations L l✝\nts✝ : Fin l✝ → Term L (α ⊕ Fin n✝)\n⊢ mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) (rel R✝ ts✝)) =\n mapTermRel (fun x => ft' x ∘ ft x) (fun x => fr' x ∘ fr x) (fun x => id) (rel R✝ ts✝)",
"tactic": "simp [mapTermRel]"
},
{
"state_after": "no goals",
"state_before": "case imp\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.61221\nP : Type ?u.61224\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type u_5\nn✝¹ : ℕ\nL'' : Language\nft : (n : ℕ) → Term L (α ⊕ Fin n) → Term L' (β ⊕ Fin n)\nfr : (n : ℕ) → Relations L n → Relations L' n\nft' : (n : ℕ) → Term L' (β ⊕ Fin n) → Term L'' (γ ⊕ Fin n)\nfr' : (n : ℕ) → Relations L' n → Relations L'' n\nn n✝ : ℕ\nf₁✝ f₂✝ : BoundedFormula L α n✝\nih1 :\n mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) f₁✝) =\n mapTermRel (fun x => ft' x ∘ ft x) (fun x => fr' x ∘ fr x) (fun x => id) f₁✝\nih2 :\n mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) f₂✝) =\n mapTermRel (fun x => ft' x ∘ ft x) (fun x => fr' x ∘ fr x) (fun x => id) f₂✝\n⊢ mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) (imp f₁✝ f₂✝)) =\n mapTermRel (fun x => ft' x ∘ ft x) (fun x => fr' x ∘ fr x) (fun x => id) (imp f₁✝ f₂✝)",
"tactic": "simp [mapTermRel, ih1, ih2]"
},
{
"state_after": "no goals",
"state_before": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.61221\nP : Type ?u.61224\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type u_5\nn✝¹ : ℕ\nL'' : Language\nft : (n : ℕ) → Term L (α ⊕ Fin n) → Term L' (β ⊕ Fin n)\nfr : (n : ℕ) → Relations L n → Relations L' n\nft' : (n : ℕ) → Term L' (β ⊕ Fin n) → Term L'' (γ ⊕ Fin n)\nfr' : (n : ℕ) → Relations L' n → Relations L'' n\nn n✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 :\n mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) f✝) =\n mapTermRel (fun x => ft' x ∘ ft x) (fun x => fr' x ∘ fr x) (fun x => id) f✝\n⊢ mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) (all f✝)) =\n mapTermRel (fun x => ft' x ∘ ft x) (fun x => fr' x ∘ fr x) (fun x => id) (all f✝)",
"tactic": "simp [mapTermRel, ih3]"
}
] |
[
545,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
533,
1
] |
Mathlib/Topology/Semicontinuous.lean
|
IsOpen.upperSemicontinuousOn_indicator
|
[] |
[
745,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
743,
1
] |
Mathlib/MeasureTheory/Measure/NullMeasurable.lean
|
MeasureTheory.nullMeasurableSet_eq
|
[] |
[
362,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
EMetric.infEdist_empty
|
[] |
[
61,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Algebra/FreeMonoid/Basic.lean
|
FreeMonoid.map_id
|
[] |
[
352,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
352,
1
] |
Mathlib/Data/Prod/Basic.lean
|
Prod.Lex.refl_left
|
[] |
[
245,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
244,
1
] |
Mathlib/Control/LawfulFix.lean
|
Part.fix_eq
|
[
{
"state_after": "α✝ : Type u_1\nβ : α✝ → Type u_2\nα : Type ?u.14701\nf : ((a : α✝) → Part (β a)) →o (a : α✝) → Part (β a)\nhc : Continuous f\n⊢ ωSup (approxChain f) = ωSup (Chain.map (approxChain f) f)",
"state_before": "α✝ : Type u_1\nβ : α✝ → Type u_2\nα : Type ?u.14701\nf : ((a : α✝) → Part (β a)) →o (a : α✝) → Part (β a)\nhc : Continuous f\n⊢ Part.fix ↑f = ↑f (Part.fix ↑f)",
"tactic": "rw [fix_eq_ωSup f, hc]"
},
{
"state_after": "case a\nα✝ : Type u_1\nβ : α✝ → Type u_2\nα : Type ?u.14701\nf : ((a : α✝) → Part (β a)) →o (a : α✝) → Part (β a)\nhc : Continuous f\n⊢ ωSup (approxChain f) ≤ ωSup (Chain.map (approxChain f) f)\n\ncase a\nα✝ : Type u_1\nβ : α✝ → Type u_2\nα : Type ?u.14701\nf : ((a : α✝) → Part (β a)) →o (a : α✝) → Part (β a)\nhc : Continuous f\n⊢ ωSup (Chain.map (approxChain f) f) ≤ ωSup (approxChain f)",
"state_before": "α✝ : Type u_1\nβ : α✝ → Type u_2\nα : Type ?u.14701\nf : ((a : α✝) → Part (β a)) →o (a : α✝) → Part (β a)\nhc : Continuous f\n⊢ ωSup (approxChain f) = ωSup (Chain.map (approxChain f) f)",
"tactic": "apply le_antisymm"
},
{
"state_after": "α✝ : Type u_1\nβ : α✝ → Type u_2\nα : Type ?u.14701\nf : ((a : α✝) → Part (β a)) →o (a : α✝) → Part (β a)\nhc : Continuous f\n⊢ approxChain f ≤ Chain.map (approxChain f) f",
"state_before": "case a\nα✝ : Type u_1\nβ : α✝ → Type u_2\nα : Type ?u.14701\nf : ((a : α✝) → Part (β a)) →o (a : α✝) → Part (β a)\nhc : Continuous f\n⊢ ωSup (approxChain f) ≤ ωSup (Chain.map (approxChain f) f)",
"tactic": "apply ωSup_le_ωSup_of_le _"
},
{
"state_after": "α✝ : Type u_1\nβ : α✝ → Type u_2\nα : Type ?u.14701\nf : ((a : α✝) → Part (β a)) →o (a : α✝) → Part (β a)\nhc : Continuous f\ni : ℕ\n⊢ ∃ j, ↑(approxChain f) i ≤ ↑(Chain.map (approxChain f) f) j",
"state_before": "α✝ : Type u_1\nβ : α✝ → Type u_2\nα : Type ?u.14701\nf : ((a : α✝) → Part (β a)) →o (a : α✝) → Part (β a)\nhc : Continuous f\n⊢ approxChain f ≤ Chain.map (approxChain f) f",
"tactic": "intro i"
},
{
"state_after": "α✝ : Type u_1\nβ : α✝ → Type u_2\nα : Type ?u.14701\nf : ((a : α✝) → Part (β a)) →o (a : α✝) → Part (β a)\nhc : Continuous f\ni : ℕ\n⊢ ↑(approxChain f) i ≤ ↑(Chain.map (approxChain f) f) i",
"state_before": "α✝ : Type u_1\nβ : α✝ → Type u_2\nα : Type ?u.14701\nf : ((a : α✝) → Part (β a)) →o (a : α✝) → Part (β a)\nhc : Continuous f\ni : ℕ\n⊢ ∃ j, ↑(approxChain f) i ≤ ↑(Chain.map (approxChain f) f) j",
"tactic": "exists i"
},
{
"state_after": "α✝ : Type u_1\nβ : α✝ → Type u_2\nα : Type ?u.14701\nf : ((a : α✝) → Part (β a)) →o (a : α✝) → Part (β a)\nhc : Continuous f\ni : ℕ\nx : α✝\n⊢ ↑(approxChain f) i x ≤ ↑(Chain.map (approxChain f) f) i x",
"state_before": "α✝ : Type u_1\nβ : α✝ → Type u_2\nα : Type ?u.14701\nf : ((a : α✝) → Part (β a)) →o (a : α✝) → Part (β a)\nhc : Continuous f\ni : ℕ\n⊢ ↑(approxChain f) i ≤ ↑(Chain.map (approxChain f) f) i",
"tactic": "intro x"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\nβ : α✝ → Type u_2\nα : Type ?u.14701\nf : ((a : α✝) → Part (β a)) →o (a : α✝) → Part (β a)\nhc : Continuous f\ni : ℕ\nx : α✝\n⊢ ↑(approxChain f) i x ≤ ↑(Chain.map (approxChain f) f) i x",
"tactic": "apply le_f_of_mem_approx _ ⟨i, rfl⟩"
},
{
"state_after": "α✝ : Type u_1\nβ : α✝ → Type u_2\nα : Type ?u.14701\nf : ((a : α✝) → Part (β a)) →o (a : α✝) → Part (β a)\nhc : Continuous f\n⊢ Chain.map (approxChain f) f ≤ approxChain f",
"state_before": "case a\nα✝ : Type u_1\nβ : α✝ → Type u_2\nα : Type ?u.14701\nf : ((a : α✝) → Part (β a)) →o (a : α✝) → Part (β a)\nhc : Continuous f\n⊢ ωSup (Chain.map (approxChain f) f) ≤ ωSup (approxChain f)",
"tactic": "apply ωSup_le_ωSup_of_le _"
},
{
"state_after": "α✝ : Type u_1\nβ : α✝ → Type u_2\nα : Type ?u.14701\nf : ((a : α✝) → Part (β a)) →o (a : α✝) → Part (β a)\nhc : Continuous f\ni : ℕ\n⊢ ∃ j, ↑(Chain.map (approxChain f) f) i ≤ ↑(approxChain f) j",
"state_before": "α✝ : Type u_1\nβ : α✝ → Type u_2\nα : Type ?u.14701\nf : ((a : α✝) → Part (β a)) →o (a : α✝) → Part (β a)\nhc : Continuous f\n⊢ Chain.map (approxChain f) f ≤ approxChain f",
"tactic": "intro i"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\nβ : α✝ → Type u_2\nα : Type ?u.14701\nf : ((a : α✝) → Part (β a)) →o (a : α✝) → Part (β a)\nhc : Continuous f\ni : ℕ\n⊢ ∃ j, ↑(Chain.map (approxChain f) f) i ≤ ↑(approxChain f) j",
"tactic": "exists i.succ"
}
] |
[
189,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/Order/Monotone/Basic.lean
|
monotone_toDual_comp_iff
|
[] |
[
152,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/CategoryTheory/LiftingProperties/Basic.lean
|
CategoryTheory.HasLiftingProperty.of_arrow_iso_right
|
[
{
"state_after": "C : Type u_2\ninst✝ : Category C\nA✝ B✝ B' X✝ Y✝ Y'✝ : C\ni✝ : A✝ ⟶ B✝\ni' : B✝ ⟶ B'\np✝ : X✝ ⟶ Y✝\np'✝ : Y✝ ⟶ Y'✝\nA B X Y X' Y' : C\ni : A ⟶ B\np : X ⟶ Y\np' : X' ⟶ Y'\ne : Arrow.mk p ≅ Arrow.mk p'\nhip : HasLiftingProperty i p\n⊢ HasLiftingProperty i (e.inv.left ≫ p ≫ e.hom.right)",
"state_before": "C : Type u_2\ninst✝ : Category C\nA✝ B✝ B' X✝ Y✝ Y'✝ : C\ni✝ : A✝ ⟶ B✝\ni' : B✝ ⟶ B'\np✝ : X✝ ⟶ Y✝\np'✝ : Y✝ ⟶ Y'✝\nA B X Y X' Y' : C\ni : A ⟶ B\np : X ⟶ Y\np' : X' ⟶ Y'\ne : Arrow.mk p ≅ Arrow.mk p'\nhip : HasLiftingProperty i p\n⊢ HasLiftingProperty i p'",
"tactic": "rw [Arrow.iso_w' e]"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝ : Category C\nA✝ B✝ B' X✝ Y✝ Y'✝ : C\ni✝ : A✝ ⟶ B✝\ni' : B✝ ⟶ B'\np✝ : X✝ ⟶ Y✝\np'✝ : Y✝ ⟶ Y'✝\nA B X Y X' Y' : C\ni : A ⟶ B\np : X ⟶ Y\np' : X' ⟶ Y'\ne : Arrow.mk p ≅ Arrow.mk p'\nhip : HasLiftingProperty i p\n⊢ HasLiftingProperty i (e.inv.left ≫ p ≫ e.hom.right)",
"tactic": "infer_instance"
}
] |
[
134,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Data/Nat/PartENat.lean
|
PartENat.coe_lt_coe
|
[
{
"state_after": "no goals",
"state_before": "x y : ℕ\n⊢ ↑x < ↑y ↔ x < y",
"tactic": "rw [lt_iff_le_not_le, lt_iff_le_not_le, coe_le_coe, coe_le_coe]"
}
] |
[
281,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
1
] |
Mathlib/Geometry/Manifold/LocalInvariantProperties.lean
|
StructureGroupoid.LocalInvariantProp.liftPropWithinAt_mono
|
[
{
"state_after": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type ?u.49347\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : LocalInvariantProp G G' P\nmono : ∀ ⦃s : Set H⦄ ⦃x : H⦄ ⦃t : Set H⦄ ⦃f : H → H'⦄, t ⊆ s → P f s x → P f t x\nh : LiftPropWithinAt P g s x\nhts : t ⊆ s\ny : H\nhy : y ∈ ↑(LocalHomeomorph.symm (chartAt H x)) ⁻¹' t\n⊢ y ∈ ↑(LocalHomeomorph.symm (chartAt H x)) ⁻¹' s",
"state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type ?u.49347\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : LocalInvariantProp G G' P\nmono : ∀ ⦃s : Set H⦄ ⦃x : H⦄ ⦃t : Set H⦄ ⦃f : H → H'⦄, t ⊆ s → P f s x → P f t x\nh : LiftPropWithinAt P g s x\nhts : t ⊆ s\n⊢ LiftPropWithinAt P g t x",
"tactic": "refine' ⟨h.1.mono hts, mono (fun y hy ↦ _) h.2⟩"
},
{
"state_after": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type ?u.49347\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : LocalInvariantProp G G' P\nmono : ∀ ⦃s : Set H⦄ ⦃x : H⦄ ⦃t : Set H⦄ ⦃f : H → H'⦄, t ⊆ s → P f s x → P f t x\nh : LiftPropWithinAt P g s x\nhts : t ⊆ s\ny : H\nhy : ↑(LocalHomeomorph.symm (chartAt H x)) y ∈ t\n⊢ y ∈ ↑(LocalHomeomorph.symm (chartAt H x)) ⁻¹' s",
"state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type ?u.49347\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : LocalInvariantProp G G' P\nmono : ∀ ⦃s : Set H⦄ ⦃x : H⦄ ⦃t : Set H⦄ ⦃f : H → H'⦄, t ⊆ s → P f s x → P f t x\nh : LiftPropWithinAt P g s x\nhts : t ⊆ s\ny : H\nhy : y ∈ ↑(LocalHomeomorph.symm (chartAt H x)) ⁻¹' t\n⊢ y ∈ ↑(LocalHomeomorph.symm (chartAt H x)) ⁻¹' s",
"tactic": "simp only [mfld_simps] at hy"
},
{
"state_after": "no goals",
"state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type ?u.49347\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : LocalInvariantProp G G' P\nmono : ∀ ⦃s : Set H⦄ ⦃x : H⦄ ⦃t : Set H⦄ ⦃f : H → H'⦄, t ⊆ s → P f s x → P f t x\nh : LiftPropWithinAt P g s x\nhts : t ⊆ s\ny : H\nhy : ↑(LocalHomeomorph.symm (chartAt H x)) y ∈ t\n⊢ y ∈ ↑(LocalHomeomorph.symm (chartAt H x)) ⁻¹' s",
"tactic": "simp only [hy, hts _, mfld_simps]"
}
] |
[
459,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
455,
1
] |
Mathlib/Algebra/Module/LocalizedModule.lean
|
LocalizedModule.mk_neg
|
[] |
[
237,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.head_nil
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\n⊢ head nil = Computation.pure none",
"tactic": "simp [head]"
}
] |
[
658,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
658,
1
] |
Mathlib/Analysis/InnerProductSpace/Orthogonal.lean
|
Submodule.inf_orthogonal_eq_bot
|
[
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.37226\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nK : Submodule 𝕜 E\n⊢ K ⊓ Kᗮ ≤ ⊥",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.37226\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nK : Submodule 𝕜 E\n⊢ K ⊓ Kᗮ = ⊥",
"tactic": "rw [eq_bot_iff]"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.37226\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nK : Submodule 𝕜 E\nx : E\n⊢ x ∈ K ⊓ Kᗮ → x ∈ ⊥",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.37226\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nK : Submodule 𝕜 E\n⊢ K ⊓ Kᗮ ≤ ⊥",
"tactic": "intro x"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.37226\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nK : Submodule 𝕜 E\nx : E\n⊢ x ∈ K ∧ x ∈ Kᗮ → x ∈ ⊥",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.37226\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nK : Submodule 𝕜 E\nx : E\n⊢ x ∈ K ⊓ Kᗮ → x ∈ ⊥",
"tactic": "rw [mem_inf]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.37226\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nK : Submodule 𝕜 E\nx : E\n⊢ x ∈ K ∧ x ∈ Kᗮ → x ∈ ⊥",
"tactic": "exact fun ⟨hx, ho⟩ => inner_self_eq_zero.1 (ho x hx)"
}
] |
[
113,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.map_injective
|
[
{
"state_after": "case ofFractionRing.ofFractionRing\nK : Type u\ninst✝⁵ : CommRing K\nG₀ : Type ?u.685775\nL : Type ?u.685778\nR : Type u_2\nS : Type u_3\nF : Type u_1\ninst✝⁴ : CommGroupWithZero G₀\ninst✝³ : Field L\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : MonoidHomClass F R[X] S[X]\nφ : F\nhφ : R[X]⁰ ≤ Submonoid.comap φ S[X]⁰\nhf : Function.Injective ↑φ\nx y : FractionRing R[X]\nh : ↑(map φ hφ) { toFractionRing := x } = ↑(map φ hφ) { toFractionRing := y }\n⊢ { toFractionRing := x } = { toFractionRing := y }",
"state_before": "K : Type u\ninst✝⁵ : CommRing K\nG₀ : Type ?u.685775\nL : Type ?u.685778\nR : Type u_2\nS : Type u_3\nF : Type u_1\ninst✝⁴ : CommGroupWithZero G₀\ninst✝³ : Field L\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : MonoidHomClass F R[X] S[X]\nφ : F\nhφ : R[X]⁰ ≤ Submonoid.comap φ S[X]⁰\nhf : Function.Injective ↑φ\n⊢ Function.Injective ↑(map φ hφ)",
"tactic": "rintro ⟨x⟩ ⟨y⟩ h"
},
{
"state_after": "case ofFractionRing.ofFractionRing.H\nK : Type u\ninst✝⁵ : CommRing K\nG₀ : Type ?u.685775\nL : Type ?u.685778\nR : Type u_2\nS : Type u_3\nF : Type u_1\ninst✝⁴ : CommGroupWithZero G₀\ninst✝³ : Field L\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : MonoidHomClass F R[X] S[X]\nφ : F\nhφ : R[X]⁰ ≤ Submonoid.comap φ S[X]⁰\nhf : Function.Injective ↑φ\ny : FractionRing R[X]\ny✝ : R[X] × { x // x ∈ R[X]⁰ }\nh : ↑(map φ hφ) { toFractionRing := Localization.mk y✝.fst y✝.snd } = ↑(map φ hφ) { toFractionRing := y }\n⊢ { toFractionRing := Localization.mk y✝.fst y✝.snd } = { toFractionRing := y }",
"state_before": "case ofFractionRing.ofFractionRing\nK : Type u\ninst✝⁵ : CommRing K\nG₀ : Type ?u.685775\nL : Type ?u.685778\nR : Type u_2\nS : Type u_3\nF : Type u_1\ninst✝⁴ : CommGroupWithZero G₀\ninst✝³ : Field L\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : MonoidHomClass F R[X] S[X]\nφ : F\nhφ : R[X]⁰ ≤ Submonoid.comap φ S[X]⁰\nhf : Function.Injective ↑φ\nx y : FractionRing R[X]\nh : ↑(map φ hφ) { toFractionRing := x } = ↑(map φ hφ) { toFractionRing := y }\n⊢ { toFractionRing := x } = { toFractionRing := y }",
"tactic": "induction x using Localization.induction_on"
},
{
"state_after": "case ofFractionRing.ofFractionRing.H.H\nK : Type u\ninst✝⁵ : CommRing K\nG₀ : Type ?u.685775\nL : Type ?u.685778\nR : Type u_2\nS : Type u_3\nF : Type u_1\ninst✝⁴ : CommGroupWithZero G₀\ninst✝³ : Field L\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : MonoidHomClass F R[X] S[X]\nφ : F\nhφ : R[X]⁰ ≤ Submonoid.comap φ S[X]⁰\nhf : Function.Injective ↑φ\ny✝¹ y✝ : R[X] × { x // x ∈ R[X]⁰ }\nh :\n ↑(map φ hφ) { toFractionRing := Localization.mk y✝¹.fst y✝¹.snd } =\n ↑(map φ hφ) { toFractionRing := Localization.mk y✝.fst y✝.snd }\n⊢ { toFractionRing := Localization.mk y✝¹.fst y✝¹.snd } = { toFractionRing := Localization.mk y✝.fst y✝.snd }",
"state_before": "case ofFractionRing.ofFractionRing.H\nK : Type u\ninst✝⁵ : CommRing K\nG₀ : Type ?u.685775\nL : Type ?u.685778\nR : Type u_2\nS : Type u_3\nF : Type u_1\ninst✝⁴ : CommGroupWithZero G₀\ninst✝³ : Field L\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : MonoidHomClass F R[X] S[X]\nφ : F\nhφ : R[X]⁰ ≤ Submonoid.comap φ S[X]⁰\nhf : Function.Injective ↑φ\ny : FractionRing R[X]\ny✝ : R[X] × { x // x ∈ R[X]⁰ }\nh : ↑(map φ hφ) { toFractionRing := Localization.mk y✝.fst y✝.snd } = ↑(map φ hφ) { toFractionRing := y }\n⊢ { toFractionRing := Localization.mk y✝.fst y✝.snd } = { toFractionRing := y }",
"tactic": "induction y using Localization.induction_on"
},
{
"state_after": "no goals",
"state_before": "case ofFractionRing.ofFractionRing.H.H\nK : Type u\ninst✝⁵ : CommRing K\nG₀ : Type ?u.685775\nL : Type ?u.685778\nR : Type u_2\nS : Type u_3\nF : Type u_1\ninst✝⁴ : CommGroupWithZero G₀\ninst✝³ : Field L\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : MonoidHomClass F R[X] S[X]\nφ : F\nhφ : R[X]⁰ ≤ Submonoid.comap φ S[X]⁰\nhf : Function.Injective ↑φ\ny✝¹ y✝ : R[X] × { x // x ∈ R[X]⁰ }\nh :\n ↑(map φ hφ) { toFractionRing := Localization.mk y✝¹.fst y✝¹.snd } =\n ↑(map φ hφ) { toFractionRing := Localization.mk y✝.fst y✝.snd }\n⊢ { toFractionRing := Localization.mk y✝¹.fst y✝¹.snd } = { toFractionRing := Localization.mk y✝.fst y✝.snd }",
"tactic": "simpa only [map_apply_ofFractionRing_mk, ofFractionRing_injective.eq_iff,\n Localization.mk_eq_mk_iff, Localization.r_iff_exists, mul_cancel_left_coe_nonZeroDivisors,\n exists_const, ← map_mul, hf.eq_iff] using h"
}
] |
[
668,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
660,
1
] |
Mathlib/Topology/MetricSpace/Infsep.lean
|
Set.einfsep_iUnion_mem_option
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.18395\ninst✝ : EDist α\nx y : α\ns✝ t : Set α\nι : Type u_1\no : Option ι\ns : ι → Set α\n⊢ einfsep (⋃ (i : ι) (_ : i ∈ o), s i) = ⨅ (i : ι) (_ : i ∈ o), einfsep (s i)",
"tactic": "cases o <;> simp"
}
] |
[
141,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
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