file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.Cofork.π_precompose
|
[] |
[
671,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
669,
1
] |
Mathlib/Analysis/Calculus/FDerivMeasurable.lean
|
aestronglyMeasurable_deriv
|
[] |
[
450,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
448,
1
] |
Mathlib/Algebra/Order/Hom/Monoid.lean
|
OrderMonoidWithZeroHom.toFun_eq_coe
|
[] |
[
609,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
608,
1
] |
Mathlib/Data/Dfinsupp/Interval.lean
|
Finset.card_dfinsupp
|
[] |
[
46,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.spanningSetsIndex_eq_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.728649\nγ : Type ?u.728652\nδ : Type ?u.728655\nι : Type ?u.728658\nR : Type ?u.728661\nR' : Type ?u.728664\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nx : α\nn : ℕ\n⊢ spanningSetsIndex μ x = n ↔ x ∈ disjointed (spanningSets μ) n",
"tactic": "convert Set.ext_iff.1 (preimage_spanningSetsIndex_singleton μ n) x"
}
] |
[
3507,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3505,
1
] |
Mathlib/MeasureTheory/Group/FundamentalDomain.lean
|
MeasureTheory.sdiff_fundamentalInterior
|
[] |
[
605,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
604,
1
] |
src/lean/Init/Core.lean
|
Decidable.byContradiction
|
[] |
[
739,
43
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
738,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isBigOWith_pi
|
[
{
"state_after": "α : Type u_3\nβ : Type ?u.690828\nE : Type ?u.690831\nF : Type ?u.690834\nG : Type ?u.690837\nE'✝ : Type ?u.690840\nF' : Type u_4\nG' : Type ?u.690846\nE'' : Type ?u.690849\nF'' : Type ?u.690852\nG'' : Type ?u.690855\nR : Type ?u.690858\nR' : Type ?u.690861\n𝕜 : Type ?u.690864\n𝕜' : Type ?u.690867\ninst✝¹⁴ : Norm E\ninst✝¹³ : Norm F\ninst✝¹² : Norm G\ninst✝¹¹ : SeminormedAddCommGroup E'✝\ninst✝¹⁰ : SeminormedAddCommGroup F'\ninst✝⁹ : SeminormedAddCommGroup G'\ninst✝⁸ : NormedAddCommGroup E''\ninst✝⁷ : NormedAddCommGroup F''\ninst✝⁶ : NormedAddCommGroup G''\ninst✝⁵ : SeminormedRing R\ninst✝⁴ : SeminormedRing R'\ninst✝³ : NormedField 𝕜\ninst✝² : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf✝ : α → E\ng : α → F\nk : α → G\nf' : α → E'✝\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nι : Type u_1\ninst✝¹ : Fintype ι\nE' : ι → Type u_2\ninst✝ : (i : ι) → NormedAddCommGroup (E' i)\nf : α → (i : ι) → E' i\nC : ℝ\nhC : 0 ≤ C\nthis : ∀ (x : α), 0 ≤ C * ‖g' x‖\n⊢ IsBigOWith C l f g' ↔ ∀ (i : ι), IsBigOWith C l (fun x => f x i) g'",
"state_before": "α : Type u_3\nβ : Type ?u.690828\nE : Type ?u.690831\nF : Type ?u.690834\nG : Type ?u.690837\nE'✝ : Type ?u.690840\nF' : Type u_4\nG' : Type ?u.690846\nE'' : Type ?u.690849\nF'' : Type ?u.690852\nG'' : Type ?u.690855\nR : Type ?u.690858\nR' : Type ?u.690861\n𝕜 : Type ?u.690864\n𝕜' : Type ?u.690867\ninst✝¹⁴ : Norm E\ninst✝¹³ : Norm F\ninst✝¹² : Norm G\ninst✝¹¹ : SeminormedAddCommGroup E'✝\ninst✝¹⁰ : SeminormedAddCommGroup F'\ninst✝⁹ : SeminormedAddCommGroup G'\ninst✝⁸ : NormedAddCommGroup E''\ninst✝⁷ : NormedAddCommGroup F''\ninst✝⁶ : NormedAddCommGroup G''\ninst✝⁵ : SeminormedRing R\ninst✝⁴ : SeminormedRing R'\ninst✝³ : NormedField 𝕜\ninst✝² : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf✝ : α → E\ng : α → F\nk : α → G\nf' : α → E'✝\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nι : Type u_1\ninst✝¹ : Fintype ι\nE' : ι → Type u_2\ninst✝ : (i : ι) → NormedAddCommGroup (E' i)\nf : α → (i : ι) → E' i\nC : ℝ\nhC : 0 ≤ C\n⊢ IsBigOWith C l f g' ↔ ∀ (i : ι), IsBigOWith C l (fun x => f x i) g'",
"tactic": "have : ∀ x, 0 ≤ C * ‖g' x‖ := fun x => mul_nonneg hC (norm_nonneg _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type ?u.690828\nE : Type ?u.690831\nF : Type ?u.690834\nG : Type ?u.690837\nE'✝ : Type ?u.690840\nF' : Type u_4\nG' : Type ?u.690846\nE'' : Type ?u.690849\nF'' : Type ?u.690852\nG'' : Type ?u.690855\nR : Type ?u.690858\nR' : Type ?u.690861\n𝕜 : Type ?u.690864\n𝕜' : Type ?u.690867\ninst✝¹⁴ : Norm E\ninst✝¹³ : Norm F\ninst✝¹² : Norm G\ninst✝¹¹ : SeminormedAddCommGroup E'✝\ninst✝¹⁰ : SeminormedAddCommGroup F'\ninst✝⁹ : SeminormedAddCommGroup G'\ninst✝⁸ : NormedAddCommGroup E''\ninst✝⁷ : NormedAddCommGroup F''\ninst✝⁶ : NormedAddCommGroup G''\ninst✝⁵ : SeminormedRing R\ninst✝⁴ : SeminormedRing R'\ninst✝³ : NormedField 𝕜\ninst✝² : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf✝ : α → E\ng : α → F\nk : α → G\nf' : α → E'✝\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nι : Type u_1\ninst✝¹ : Fintype ι\nE' : ι → Type u_2\ninst✝ : (i : ι) → NormedAddCommGroup (E' i)\nf : α → (i : ι) → E' i\nC : ℝ\nhC : 0 ≤ C\nthis : ∀ (x : α), 0 ≤ C * ‖g' x‖\n⊢ IsBigOWith C l f g' ↔ ∀ (i : ι), IsBigOWith C l (fun x => f x i) g'",
"tactic": "simp only [isBigOWith_iff, pi_norm_le_iff_of_nonneg (this _), eventually_all]"
}
] |
[
2138,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2134,
1
] |
Std/Data/String/Lemmas.lean
|
String.Iterator.Valid.setCurr
|
[
{
"state_after": "c : Char\nit : Iterator\nh✝ : Valid it\nl r : List Char\nh : ValidFor l r it\n⊢ Valid (Iterator.setCurr it c)",
"state_before": "c : Char\nit : Iterator\nh : Valid it\n⊢ Valid (Iterator.setCurr it c)",
"tactic": "let ⟨l, r, h⟩ := h.validFor"
},
{
"state_after": "no goals",
"state_before": "c : Char\nit : Iterator\nh✝ : Valid it\nl r : List Char\nh : ValidFor l r it\n⊢ Valid (Iterator.setCurr it c)",
"tactic": "exact h.setCurr'.valid"
}
] |
[
649,
27
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
646,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.mul_pure
|
[] |
[
360,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
359,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.mul_iSup
|
[
{
"state_after": "ι : Sort u_1\nf✝ f : ι → ℝ≥0\na : ℝ≥0\n⊢ (↑a * ⨆ (i : ι), ↑(f i)) = ⨆ (i : ι), ↑(a * f i)",
"state_before": "ι : Sort u_1\nf✝ f : ι → ℝ≥0\na : ℝ≥0\n⊢ (a * ⨆ (i : ι), f i) = ⨆ (i : ι), a * f i",
"tactic": "rw [← NNReal.coe_eq, NNReal.coe_mul, NNReal.coe_iSup, NNReal.coe_iSup]"
},
{
"state_after": "no goals",
"state_before": "ι : Sort u_1\nf✝ f : ι → ℝ≥0\na : ℝ≥0\n⊢ (↑a * ⨆ (i : ι), ↑(f i)) = ⨆ (i : ι), ↑(a * f i)",
"tactic": "exact Real.mul_iSup_of_nonneg (NNReal.coe_nonneg _) _"
}
] |
[
964,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
962,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Order.lean
|
hasSum_zero_iff
|
[] |
[
210,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
209,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.sum_eq_top_iff
|
[] |
[
1232,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1231,
1
] |
Mathlib/GroupTheory/Submonoid/Basic.lean
|
Submonoid.coe_top
|
[] |
[
277,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
276,
1
] |
Mathlib/Data/List/Basic.lean
|
List.enum_cons
|
[] |
[
3883,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3882,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.tendsto_nhds_top_iff_nat
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.40146\nγ : Type ?u.40149\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nm : α → ℝ≥0∞\nf : Filter α\nh : ∀ (x : ℝ≥0), ∀ᶠ (a : α) in f, ↑x < m a\nn : ℕ\n⊢ ∀ᶠ (a : α) in f, ↑n < m a",
"tactic": "simpa only [ENNReal.coe_nat] using h n"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.40146\nγ : Type ?u.40149\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx✝ y✝ z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nm : α → ℝ≥0∞\nf : Filter α\nh : ∀ (n : ℕ), ∀ᶠ (a : α) in f, ↑n < m a\nx : ℝ≥0\nn : ℕ\nhn : x < ↑n\ny : α\n⊢ ↑x < ↑n",
"tactic": "rwa [← ENNReal.coe_nat, coe_lt_coe]"
}
] |
[
173,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
UpperSet.mem_iInf₂_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.60515\nγ : Type ?u.60518\nι : Sort u_2\nκ : ι → Sort u_3\ninst✝ : LE α\nS : Set (UpperSet α)\ns t : UpperSet α\na : α\nf : (i : ι) → κ i → UpperSet α\n⊢ (a ∈ ⨅ (i : ι) (j : κ i), f i j) ↔ ∃ i j, a ∈ f i j",
"tactic": "simp_rw [mem_iInf_iff]"
}
] |
[
619,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
618,
1
] |
Mathlib/Topology/Algebra/Ring/Basic.lean
|
Subring.le_topologicalClosure
|
[] |
[
255,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/Analysis/Normed/Group/Quotient.lean
|
Ideal.Quotient.norm_mk_le
|
[] |
[
497,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
496,
1
] |
Mathlib/Order/Disjointed.lean
|
disjointed_le
|
[] |
[
74,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
LinearMap.toSpanSingleton_zero
|
[
{
"state_after": "case h\nR : Type u_1\nR₂ : Type ?u.332387\nK : Type ?u.332390\nM : Type u_2\nM₂ : Type ?u.332396\nV : Type ?u.332399\nS : Type ?u.332402\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\n⊢ ↑(toSpanSingleton R M 0) 1 = ↑0 1",
"state_before": "R : Type u_1\nR₂ : Type ?u.332387\nK : Type ?u.332390\nM : Type u_2\nM₂ : Type ?u.332396\nV : Type ?u.332399\nS : Type ?u.332402\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\n⊢ toSpanSingleton R M 0 = 0",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\nR₂ : Type ?u.332387\nK : Type ?u.332390\nM : Type u_2\nM₂ : Type ?u.332396\nV : Type ?u.332399\nS : Type ?u.332402\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\n⊢ ↑(toSpanSingleton R M 0) 1 = ↑0 1",
"tactic": "simp"
}
] |
[
933,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
931,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.vsub_empty
|
[] |
[
1478,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1477,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
iteratedFDerivWithin_const_smul_apply
|
[] |
[
1572,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1570,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
PosNum.one_add
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.30381\nn : PosNum\n⊢ 1 + n = succ n",
"tactic": "cases n <;> rfl"
}
] |
[
83,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Data/Finset/Interval.lean
|
Finset.card_Iic_finset
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\n⊢ card (Iic s) = 2 ^ card s",
"tactic": "rw [Iic_eq_powerset, card_powerset]"
}
] |
[
123,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.range_re
|
[] |
[
76,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
HasSum.sub
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.401752\nδ : Type ?u.401755\ninst✝² : AddCommGroup α\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalAddGroup α\nf g : β → α\na a₁ a₂ : α\nhf : HasSum f a₁\nhg : HasSum g a₂\n⊢ HasSum (fun b => f b + -g b) (a₁ + -a₂)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.401752\nδ : Type ?u.401755\ninst✝² : AddCommGroup α\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalAddGroup α\nf g : β → α\na a₁ a₂ : α\nhf : HasSum f a₁\nhg : HasSum g a₂\n⊢ HasSum (fun b => f b - g b) (a₁ - a₂)",
"tactic": "simp only [sub_eq_add_neg]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.401752\nδ : Type ?u.401755\ninst✝² : AddCommGroup α\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalAddGroup α\nf g : β → α\na a₁ a₂ : α\nhf : HasSum f a₁\nhg : HasSum g a₂\n⊢ HasSum (fun b => f b + -g b) (a₁ + -a₂)",
"tactic": "exact hf.add hg.neg"
}
] |
[
822,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
819,
1
] |
Mathlib/Data/List/Sections.lean
|
List.mem_sections_length
|
[] |
[
41,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Mathlib/Topology/UniformSpace/Completion.lean
|
UniformSpace.Completion.comap_coe_eq_uniformity
|
[
{
"state_after": "α : Type u_1\ninst✝² : UniformSpace α\nβ : Type ?u.37466\ninst✝¹ : UniformSpace β\nγ : Type ?u.37472\ninst✝ : UniformSpace γ\nthis :\n (fun x => (↑α x.fst, ↑α x.snd)) =\n (fun x =>\n (Quotient.mk (separationSetoid (CauchyFilter α)) x.fst,\n Quotient.mk (separationSetoid (CauchyFilter α)) x.snd)) ∘\n fun x => (pureCauchy x.fst, pureCauchy x.snd)\n⊢ Filter.comap (fun p => (↑α p.fst, ↑α p.snd)) (𝓤 (Completion α)) = 𝓤 α",
"state_before": "α : Type u_1\ninst✝² : UniformSpace α\nβ : Type ?u.37466\ninst✝¹ : UniformSpace β\nγ : Type ?u.37472\ninst✝ : UniformSpace γ\n⊢ Filter.comap (fun p => (↑α p.fst, ↑α p.snd)) (𝓤 (Completion α)) = 𝓤 α",
"tactic": "have :\n (fun x : α × α => ((x.1 : Completion α), (x.2 : Completion α))) =\n (fun x : CauchyFilter α × CauchyFilter α => (⟦x.1⟧, ⟦x.2⟧)) ∘ fun x : α × α =>\n (pureCauchy x.1, pureCauchy x.2) :=\n by ext ⟨a, b⟩ <;> simp <;> rfl"
},
{
"state_after": "α : Type u_1\ninst✝² : UniformSpace α\nβ : Type ?u.37466\ninst✝¹ : UniformSpace β\nγ : Type ?u.37472\ninst✝ : UniformSpace γ\nthis :\n (fun x => (↑α x.fst, ↑α x.snd)) =\n (fun x =>\n (Quotient.mk (separationSetoid (CauchyFilter α)) x.fst,\n Quotient.mk (separationSetoid (CauchyFilter α)) x.snd)) ∘\n fun x => (pureCauchy x.fst, pureCauchy x.snd)\n⊢ Filter.comap (fun x => (pureCauchy x.fst, pureCauchy x.snd))\n (Filter.comap\n (fun x =>\n (Quotient.mk (separationSetoid (CauchyFilter α)) x.fst,\n Quotient.mk (separationSetoid (CauchyFilter α)) x.snd))\n (𝓤 (Completion α))) =\n 𝓤 α",
"state_before": "α : Type u_1\ninst✝² : UniformSpace α\nβ : Type ?u.37466\ninst✝¹ : UniformSpace β\nγ : Type ?u.37472\ninst✝ : UniformSpace γ\nthis :\n (fun x => (↑α x.fst, ↑α x.snd)) =\n (fun x =>\n (Quotient.mk (separationSetoid (CauchyFilter α)) x.fst,\n Quotient.mk (separationSetoid (CauchyFilter α)) x.snd)) ∘\n fun x => (pureCauchy x.fst, pureCauchy x.snd)\n⊢ Filter.comap (fun p => (↑α p.fst, ↑α p.snd)) (𝓤 (Completion α)) = 𝓤 α",
"tactic": "rw [this, ← Filter.comap_comap]"
},
{
"state_after": "α : Type u_1\ninst✝² : UniformSpace α\nβ : Type ?u.37466\ninst✝¹ : UniformSpace β\nγ : Type ?u.37472\ninst✝ : UniformSpace γ\nthis :\n (fun x => (↑α x.fst, ↑α x.snd)) =\n (fun x =>\n (Quotient.mk (separationSetoid (CauchyFilter α)) x.fst,\n Quotient.mk (separationSetoid (CauchyFilter α)) x.snd)) ∘\n fun x => (pureCauchy x.fst, pureCauchy x.snd)\n⊢ Filter.comap (fun x => (pureCauchy x.fst, pureCauchy x.snd))\n (Filter.comap\n (fun x =>\n (Quotient.mk (separationSetoid (CauchyFilter α)) x.fst,\n Quotient.mk (separationSetoid (CauchyFilter α)) x.snd))\n (𝓤 (Quotient (separationSetoid (CauchyFilter α))))) =\n 𝓤 α",
"state_before": "α : Type u_1\ninst✝² : UniformSpace α\nβ : Type ?u.37466\ninst✝¹ : UniformSpace β\nγ : Type ?u.37472\ninst✝ : UniformSpace γ\nthis :\n (fun x => (↑α x.fst, ↑α x.snd)) =\n (fun x =>\n (Quotient.mk (separationSetoid (CauchyFilter α)) x.fst,\n Quotient.mk (separationSetoid (CauchyFilter α)) x.snd)) ∘\n fun x => (pureCauchy x.fst, pureCauchy x.snd)\n⊢ Filter.comap (fun x => (pureCauchy x.fst, pureCauchy x.snd))\n (Filter.comap\n (fun x =>\n (Quotient.mk (separationSetoid (CauchyFilter α)) x.fst,\n Quotient.mk (separationSetoid (CauchyFilter α)) x.snd))\n (𝓤 (Completion α))) =\n 𝓤 α",
"tactic": "change Filter.comap _ (Filter.comap _ (𝓤 <| Quotient <| separationSetoid <| CauchyFilter α)) = 𝓤 α"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : UniformSpace α\nβ : Type ?u.37466\ninst✝¹ : UniformSpace β\nγ : Type ?u.37472\ninst✝ : UniformSpace γ\nthis :\n (fun x => (↑α x.fst, ↑α x.snd)) =\n (fun x =>\n (Quotient.mk (separationSetoid (CauchyFilter α)) x.fst,\n Quotient.mk (separationSetoid (CauchyFilter α)) x.snd)) ∘\n fun x => (pureCauchy x.fst, pureCauchy x.snd)\n⊢ Filter.comap (fun x => (pureCauchy x.fst, pureCauchy x.snd))\n (Filter.comap\n (fun x =>\n (Quotient.mk (separationSetoid (CauchyFilter α)) x.fst,\n Quotient.mk (separationSetoid (CauchyFilter α)) x.snd))\n (𝓤 (Quotient (separationSetoid (CauchyFilter α))))) =\n 𝓤 α",
"tactic": "rw [comap_quotient_eq_uniformity, uniformEmbedding_pureCauchy.comap_uniformity]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : UniformSpace α\nβ : Type ?u.37466\ninst✝¹ : UniformSpace β\nγ : Type ?u.37472\ninst✝ : UniformSpace γ\n⊢ (fun x => (↑α x.fst, ↑α x.snd)) =\n (fun x =>\n (Quotient.mk (separationSetoid (CauchyFilter α)) x.fst,\n Quotient.mk (separationSetoid (CauchyFilter α)) x.snd)) ∘\n fun x => (pureCauchy x.fst, pureCauchy x.snd)",
"tactic": "ext ⟨a, b⟩ <;> simp <;> rfl"
}
] |
[
408,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
399,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
zpow_le_zpow_iff'
|
[] |
[
409,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
408,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
MonoidAlgebra.smul_of
|
[
{
"state_after": "no goals",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type ?u.464052\ninst✝¹ : Semiring k\ninst✝ : MulOneClass G\ng : G\nr : k\n⊢ r • ↑(of k G) g = single g r",
"tactic": "rw [of_apply, smul_single', mul_one]"
}
] |
[
520,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
518,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
Metric.hausdorffDist_empty
|
[
{
"state_after": "case inl\nι : Sort ?u.74934\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\nh : s = ∅\n⊢ hausdorffDist s ∅ = 0\n\ncase inr\nι : Sort ?u.74934\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\nh : Set.Nonempty s\n⊢ hausdorffDist s ∅ = 0",
"state_before": "ι : Sort ?u.74934\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\n⊢ hausdorffDist s ∅ = 0",
"tactic": "cases' s.eq_empty_or_nonempty with h h"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Sort ?u.74934\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\nh : s = ∅\n⊢ hausdorffDist s ∅ = 0",
"tactic": "simp [h]"
},
{
"state_after": "no goals",
"state_before": "case inr\nι : Sort ?u.74934\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\nh : Set.Nonempty s\n⊢ hausdorffDist s ∅ = 0",
"tactic": "simp [hausdorffDist, hausdorffEdist_empty h]"
}
] |
[
734,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
731,
1
] |
Mathlib/GroupTheory/Exponent.lean
|
Nat.Prime.exists_orderOf_eq_pow_factorization_exponent
|
[
{
"state_after": "G : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\n⊢ ∃ g, orderOf g = p ^ ↑(Nat.factorization (exponent G)) p",
"state_before": "G : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\n⊢ ∃ g, orderOf g = p ^ ↑(Nat.factorization (exponent G)) p",
"tactic": "haveI := Fact.mk hp"
},
{
"state_after": "case inl\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : ↑(Nat.factorization (exponent G)) p = 0\n⊢ ∃ g, orderOf g = p ^ ↑(Nat.factorization (exponent G)) p\n\ncase inr\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : ↑(Nat.factorization (exponent G)) p ≠ 0\n⊢ ∃ g, orderOf g = p ^ ↑(Nat.factorization (exponent G)) p",
"state_before": "G : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\n⊢ ∃ g, orderOf g = p ^ ↑(Nat.factorization (exponent G)) p",
"tactic": "rcases eq_or_ne ((exponent G).factorization p) 0 with (h | h)"
},
{
"state_after": "case inr\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : ↑(Nat.factorization (exponent G)) p ≠ 0\nhe : 0 < exponent G\n⊢ ∃ g, orderOf g = p ^ ↑(Nat.factorization (exponent G)) p",
"state_before": "case inr\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : ↑(Nat.factorization (exponent G)) p ≠ 0\n⊢ ∃ g, orderOf g = p ^ ↑(Nat.factorization (exponent G)) p",
"tactic": "have he : 0 < exponent G :=\n Ne.bot_lt fun ht => by\n rw [ht] at h\n apply h\n rw [bot_eq_zero, Nat.factorization_zero, Finsupp.zero_apply]"
},
{
"state_after": "case inr\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : p ∈ (Nat.factorization (exponent G)).support\nhe : 0 < exponent G\n⊢ ∃ g, orderOf g = p ^ ↑(Nat.factorization (exponent G)) p",
"state_before": "case inr\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : ↑(Nat.factorization (exponent G)) p ≠ 0\nhe : 0 < exponent G\n⊢ ∃ g, orderOf g = p ^ ↑(Nat.factorization (exponent G)) p",
"tactic": "rw [← Finsupp.mem_support_iff] at h"
},
{
"state_after": "case inr.intro.intro\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : p ∈ (Nat.factorization (exponent G)).support\nhe : 0 < exponent G\ng : G\nhg : g ^ (exponent G / p) ≠ 1\nk : ℕ\nhk : exponent G = p ^ ↑(Nat.factorization (exponent G)) p * k\n⊢ ∃ g, orderOf g = p ^ ↑(Nat.factorization (exponent G)) p",
"state_before": "case inr.intro\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : p ∈ (Nat.factorization (exponent G)).support\nhe : 0 < exponent G\ng : G\nhg : g ^ (exponent G / p) ≠ 1\n⊢ ∃ g, orderOf g = p ^ ↑(Nat.factorization (exponent G)) p",
"tactic": "obtain ⟨k, hk : exponent G = p ^ _ * k⟩ := Nat.ord_proj_dvd _ _"
},
{
"state_after": "case inr.intro.intro.intro\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : p ∈ (Nat.factorization (exponent G)).support\nhe : 0 < exponent G\ng : G\nhg : g ^ (exponent G / p) ≠ 1\nk : ℕ\nhk : exponent G = p ^ ↑(Nat.factorization (exponent G)) p * k\nt : ℕ\nht : ↑(Nat.factorization (exponent G)) p = Nat.succ t\n⊢ ∃ g, orderOf g = p ^ ↑(Nat.factorization (exponent G)) p",
"state_before": "case inr.intro.intro\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : p ∈ (Nat.factorization (exponent G)).support\nhe : 0 < exponent G\ng : G\nhg : g ^ (exponent G / p) ≠ 1\nk : ℕ\nhk : exponent G = p ^ ↑(Nat.factorization (exponent G)) p * k\n⊢ ∃ g, orderOf g = p ^ ↑(Nat.factorization (exponent G)) p",
"tactic": "obtain ⟨t, ht⟩ := Nat.exists_eq_succ_of_ne_zero (Finsupp.mem_support_iff.mp h)"
},
{
"state_after": "case inr.intro.intro.intro\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : p ∈ (Nat.factorization (exponent G)).support\nhe : 0 < exponent G\ng : G\nhg : g ^ (exponent G / p) ≠ 1\nk : ℕ\nhk : exponent G = p ^ ↑(Nat.factorization (exponent G)) p * k\nt : ℕ\nht : ↑(Nat.factorization (exponent G)) p = Nat.succ t\n⊢ orderOf (g ^ k) = p ^ ↑(Nat.factorization (exponent G)) p",
"state_before": "case inr.intro.intro.intro\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : p ∈ (Nat.factorization (exponent G)).support\nhe : 0 < exponent G\ng : G\nhg : g ^ (exponent G / p) ≠ 1\nk : ℕ\nhk : exponent G = p ^ ↑(Nat.factorization (exponent G)) p * k\nt : ℕ\nht : ↑(Nat.factorization (exponent G)) p = Nat.succ t\n⊢ ∃ g, orderOf g = p ^ ↑(Nat.factorization (exponent G)) p",
"tactic": "refine' ⟨g ^ k, _⟩"
},
{
"state_after": "case inr.intro.intro.intro\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : p ∈ (Nat.factorization (exponent G)).support\nhe : 0 < exponent G\ng : G\nhg : g ^ (exponent G / p) ≠ 1\nk : ℕ\nhk : exponent G = p ^ ↑(Nat.factorization (exponent G)) p * k\nt : ℕ\nht : ↑(Nat.factorization (exponent G)) p = Nat.succ t\n⊢ orderOf (g ^ k) = p ^ Nat.succ t",
"state_before": "case inr.intro.intro.intro\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : p ∈ (Nat.factorization (exponent G)).support\nhe : 0 < exponent G\ng : G\nhg : g ^ (exponent G / p) ≠ 1\nk : ℕ\nhk : exponent G = p ^ ↑(Nat.factorization (exponent G)) p * k\nt : ℕ\nht : ↑(Nat.factorization (exponent G)) p = Nat.succ t\n⊢ orderOf (g ^ k) = p ^ ↑(Nat.factorization (exponent G)) p",
"tactic": "rw [ht]"
},
{
"state_after": "case inr.intro.intro.intro.hnot\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : p ∈ (Nat.factorization (exponent G)).support\nhe : 0 < exponent G\ng : G\nhg : g ^ (exponent G / p) ≠ 1\nk : ℕ\nhk : exponent G = p ^ ↑(Nat.factorization (exponent G)) p * k\nt : ℕ\nht : ↑(Nat.factorization (exponent G)) p = Nat.succ t\n⊢ ¬(g ^ k) ^ p ^ t = 1\n\ncase inr.intro.intro.intro.hfin\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : p ∈ (Nat.factorization (exponent G)).support\nhe : 0 < exponent G\ng : G\nhg : g ^ (exponent G / p) ≠ 1\nk : ℕ\nhk : exponent G = p ^ ↑(Nat.factorization (exponent G)) p * k\nt : ℕ\nht : ↑(Nat.factorization (exponent G)) p = Nat.succ t\n⊢ (g ^ k) ^ p ^ (t + 1) = 1",
"state_before": "case inr.intro.intro.intro\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : p ∈ (Nat.factorization (exponent G)).support\nhe : 0 < exponent G\ng : G\nhg : g ^ (exponent G / p) ≠ 1\nk : ℕ\nhk : exponent G = p ^ ↑(Nat.factorization (exponent G)) p * k\nt : ℕ\nht : ↑(Nat.factorization (exponent G)) p = Nat.succ t\n⊢ orderOf (g ^ k) = p ^ Nat.succ t",
"tactic": "apply orderOf_eq_prime_pow"
},
{
"state_after": "no goals",
"state_before": "case inl\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : ↑(Nat.factorization (exponent G)) p = 0\n⊢ ∃ g, orderOf g = p ^ ↑(Nat.factorization (exponent G)) p",
"tactic": "refine' ⟨1, by rw [h, pow_zero, orderOf_one]⟩"
},
{
"state_after": "no goals",
"state_before": "G : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : ↑(Nat.factorization (exponent G)) p = 0\n⊢ orderOf 1 = p ^ ↑(Nat.factorization (exponent G)) p",
"tactic": "rw [h, pow_zero, orderOf_one]"
},
{
"state_after": "G : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : ↑(Nat.factorization ⊥) p ≠ 0\nht : exponent G = ⊥\n⊢ False",
"state_before": "G : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : ↑(Nat.factorization (exponent G)) p ≠ 0\nht : exponent G = ⊥\n⊢ False",
"tactic": "rw [ht] at h"
},
{
"state_after": "G : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : ↑(Nat.factorization ⊥) p ≠ 0\nht : exponent G = ⊥\n⊢ ↑(Nat.factorization ⊥) p = 0",
"state_before": "G : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : ↑(Nat.factorization ⊥) p ≠ 0\nht : exponent G = ⊥\n⊢ False",
"tactic": "apply h"
},
{
"state_after": "no goals",
"state_before": "G : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : ↑(Nat.factorization ⊥) p ≠ 0\nht : exponent G = ⊥\n⊢ ↑(Nat.factorization ⊥) p = 0",
"tactic": "rw [bot_eq_zero, Nat.factorization_zero, Finsupp.zero_apply]"
},
{
"state_after": "G : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : p ∈ (Nat.factorization (exponent G)).support\nhe : 0 < exponent G\nkey : ¬exponent G ∣ exponent G / p\n⊢ ∃ g, g ^ (exponent G / p) ≠ 1\n\ncase key\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : p ∈ (Nat.factorization (exponent G)).support\nhe : 0 < exponent G\n⊢ ¬exponent G ∣ exponent G / p",
"state_before": "G : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : p ∈ (Nat.factorization (exponent G)).support\nhe : 0 < exponent G\n⊢ ∃ g, g ^ (exponent G / p) ≠ 1",
"tactic": "suffices key : ¬exponent G ∣ exponent G / p"
},
{
"state_after": "no goals",
"state_before": "case key\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : p ∈ (Nat.factorization (exponent G)).support\nhe : 0 < exponent G\n⊢ ¬exponent G ∣ exponent G / p",
"tactic": "exact fun hd =>\n hp.one_lt.not_le\n ((mul_le_iff_le_one_left he).mp <|\n Nat.le_of_dvd he <| Nat.mul_dvd_of_dvd_div (Nat.dvd_of_mem_factorization h) hd)"
},
{
"state_after": "no goals",
"state_before": "G : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : p ∈ (Nat.factorization (exponent G)).support\nhe : 0 < exponent G\nkey : ¬exponent G ∣ exponent G / p\n⊢ ∃ g, g ^ (exponent G / p) ≠ 1",
"tactic": "simpa using mt (exponent_dvd_of_forall_pow_eq_one G (exponent G / p)) key"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro.intro.hnot\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : p ∈ (Nat.factorization (exponent G)).support\nhe : 0 < exponent G\ng : G\nhg : g ^ (exponent G / p) ≠ 1\nk : ℕ\nhk : exponent G = p ^ ↑(Nat.factorization (exponent G)) p * k\nt : ℕ\nht : ↑(Nat.factorization (exponent G)) p = Nat.succ t\n⊢ ¬(g ^ k) ^ p ^ t = 1",
"tactic": "rwa [hk, mul_comm, ht, pow_succ', ← mul_assoc, Nat.mul_div_cancel _ hp.pos, pow_mul] at hg"
},
{
"state_after": "case inr.intro.intro.intro.hfin\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : p ∈ (Nat.factorization (exponent G)).support\nhe : 0 < exponent G\ng : G\nhg : g ^ (exponent G / p) ≠ 1\nk : ℕ\nhk : exponent G = p ^ ↑(Nat.factorization (exponent G)) p * k\nt : ℕ\nht : ↑(Nat.factorization (exponent G)) p = Nat.succ t\n⊢ g ^ exponent G = 1",
"state_before": "case inr.intro.intro.intro.hfin\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : p ∈ (Nat.factorization (exponent G)).support\nhe : 0 < exponent G\ng : G\nhg : g ^ (exponent G / p) ≠ 1\nk : ℕ\nhk : exponent G = p ^ ↑(Nat.factorization (exponent G)) p * k\nt : ℕ\nht : ↑(Nat.factorization (exponent G)) p = Nat.succ t\n⊢ (g ^ k) ^ p ^ (t + 1) = 1",
"tactic": "rw [← Nat.succ_eq_add_one, ← ht, ← pow_mul, mul_comm, ← hk]"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro.intro.hfin\nG : Type u\ninst✝ : Monoid G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nh : p ∈ (Nat.factorization (exponent G)).support\nhe : 0 < exponent G\ng : G\nhg : g ^ (exponent G / p) ≠ 1\nk : ℕ\nhk : exponent G = p ^ ↑(Nat.factorization (exponent G)) p * k\nt : ℕ\nht : ↑(Nat.factorization (exponent G)) p = Nat.succ t\n⊢ g ^ exponent G = 1",
"tactic": "exact pow_exponent_eq_one g"
}
] |
[
217,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Init/Logic.lean
|
false_or_iff
|
[] |
[
183,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.restr_coe_symm
|
[] |
[
579,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
578,
1
] |
Mathlib/ModelTheory/Substructures.lean
|
FirstOrder.Language.Substructure.map_map
|
[] |
[
466,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
465,
1
] |
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.AECover.inter
|
[] |
[
115,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
|
SeminormFamily.withSeminorms_iff_topologicalSpace_eq_iInf
|
[
{
"state_after": "𝕜 : Type u_1\n𝕜₂ : Type ?u.446340\n𝕝 : Type ?u.446343\n𝕝₂ : Type ?u.446346\nE : Type u_2\nF : Type ?u.446352\nG : Type ?u.446355\nι : Type u_3\nι' : Type ?u.446361\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\nt : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\n⊢ (𝓝 0 = ⨅ (i : ι), Filter.comap (↑(p i)) (𝓝 0)) ↔ 𝓝 0 = ⨅ (i : ι), 𝓝 0",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type ?u.446340\n𝕝 : Type ?u.446343\n𝕝₂ : Type ?u.446346\nE : Type u_2\nF : Type ?u.446352\nG : Type ?u.446355\nι : Type u_3\nι' : Type ?u.446361\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\nt : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\n⊢ WithSeminorms p ↔ t = ⨅ (i : ι), UniformSpace.toTopologicalSpace",
"tactic": "rw [p.withSeminorms_iff_nhds_eq_iInf,\n TopologicalAddGroup.ext_iff inferInstance (topologicalAddGroup_iInf fun i => inferInstance),\n nhds_iInf]"
},
{
"state_after": "𝕜 : Type u_1\n𝕜₂ : Type ?u.446340\n𝕝 : Type ?u.446343\n𝕝₂ : Type ?u.446346\nE : Type u_2\nF : Type ?u.446352\nG : Type ?u.446355\nι : Type u_3\nι' : Type ?u.446361\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\nt : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\n⊢ (𝓝 0 = ⨅ (i : ι), Filter.comap (↑(p i)) (𝓝 0)) = (𝓝 0 = ⨅ (i : ι), 𝓝 0)",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type ?u.446340\n𝕝 : Type ?u.446343\n𝕝₂ : Type ?u.446346\nE : Type u_2\nF : Type ?u.446352\nG : Type ?u.446355\nι : Type u_3\nι' : Type ?u.446361\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\nt : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\n⊢ (𝓝 0 = ⨅ (i : ι), Filter.comap (↑(p i)) (𝓝 0)) ↔ 𝓝 0 = ⨅ (i : ι), 𝓝 0",
"tactic": "refine Eq.to_iff ?_"
},
{
"state_after": "case e_a.e_s\n𝕜 : Type u_1\n𝕜₂ : Type ?u.446340\n𝕝 : Type ?u.446343\n𝕝₂ : Type ?u.446346\nE : Type u_2\nF : Type ?u.446352\nG : Type ?u.446355\nι : Type u_3\nι' : Type ?u.446361\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\nt : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\n⊢ (fun i => Filter.comap (↑(p i)) (𝓝 0)) = fun i => 𝓝 0",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type ?u.446340\n𝕝 : Type ?u.446343\n𝕝₂ : Type ?u.446346\nE : Type u_2\nF : Type ?u.446352\nG : Type ?u.446355\nι : Type u_3\nι' : Type ?u.446361\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\nt : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\n⊢ (𝓝 0 = ⨅ (i : ι), Filter.comap (↑(p i)) (𝓝 0)) = (𝓝 0 = ⨅ (i : ι), 𝓝 0)",
"tactic": "congr"
},
{
"state_after": "case e_a.e_s.h\n𝕜 : Type u_1\n𝕜₂ : Type ?u.446340\n𝕝 : Type ?u.446343\n𝕝₂ : Type ?u.446346\nE : Type u_2\nF : Type ?u.446352\nG : Type ?u.446355\nι : Type u_3\nι' : Type ?u.446361\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\nt : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\ni : ι\n⊢ Filter.comap (↑(p i)) (𝓝 0) = 𝓝 0",
"state_before": "case e_a.e_s\n𝕜 : Type u_1\n𝕜₂ : Type ?u.446340\n𝕝 : Type ?u.446343\n𝕝₂ : Type ?u.446346\nE : Type u_2\nF : Type ?u.446352\nG : Type ?u.446355\nι : Type u_3\nι' : Type ?u.446361\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\nt : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\n⊢ (fun i => Filter.comap (↑(p i)) (𝓝 0)) = fun i => 𝓝 0",
"tactic": "funext i"
},
{
"state_after": "no goals",
"state_before": "case e_a.e_s.h\n𝕜 : Type u_1\n𝕜₂ : Type ?u.446340\n𝕝 : Type ?u.446343\n𝕝₂ : Type ?u.446346\nE : Type u_2\nF : Type ?u.446352\nG : Type ?u.446355\nι : Type u_3\nι' : Type ?u.446361\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\nt : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\ni : ι\n⊢ Filter.comap (↑(p i)) (𝓝 0) = 𝓝 0",
"tactic": "exact @comap_norm_nhds_zero _ (p i).toAddGroupSeminorm.toSeminormedAddGroup"
}
] |
[
463,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
452,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Independent.lean
|
affineIndependent_iff_linearIndependent_vsub
|
[
{
"state_after": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\n⊢ AffineIndependent k p → LinearIndependent k fun i => p ↑i -ᵥ p i1\n\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\n⊢ (LinearIndependent k fun i => p ↑i -ᵥ p i1) → AffineIndependent k p",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\n⊢ AffineIndependent k p ↔ LinearIndependent k fun i => p ↑i -ᵥ p i1",
"tactic": "constructor"
},
{
"state_after": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\n⊢ LinearIndependent k fun i => p ↑i -ᵥ p i1",
"state_before": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\n⊢ AffineIndependent k p → LinearIndependent k fun i => p ↑i -ᵥ p i1",
"tactic": "intro h"
},
{
"state_after": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\n⊢ ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k),\n ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ (i : { x // x ≠ i1 }), i ∈ s → g i = 0",
"state_before": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\n⊢ LinearIndependent k fun i => p ↑i -ᵥ p i1",
"tactic": "rw [linearIndependent_iff']"
},
{
"state_after": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\n⊢ g i = 0",
"state_before": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\n⊢ ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k),\n ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ (i : { x // x ≠ i1 }), i ∈ s → g i = 0",
"tactic": "intro s g hg i hi"
},
{
"state_after": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\n⊢ g i = 0",
"state_before": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\n⊢ g i = 0",
"tactic": "set f : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g ⟨x, hx⟩ with hfdef"
},
{
"state_after": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\n⊢ g i = 0",
"state_before": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\n⊢ g i = 0",
"tactic": "let s2 : Finset ι := insert i1 (s.map (Embedding.subtype _))"
},
{
"state_after": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\n⊢ g i = 0",
"state_before": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\n⊢ g i = 0",
"tactic": "have hfg : ∀ x : { x // x ≠ i1 }, g x = f x := by\n intro x\n rw [hfdef]\n dsimp only\n erw [dif_neg x.property, Subtype.coe_eta]"
},
{
"state_after": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\n⊢ f ↑i = 0",
"state_before": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\n⊢ g i = 0",
"tactic": "rw [hfg]"
},
{
"state_after": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\nhf : ∑ ι in s2, f ι = 0\n⊢ f ↑i = 0",
"state_before": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\n⊢ f ↑i = 0",
"tactic": "have hf : (∑ ι in s2, f ι) = 0 := by\n rw [Finset.sum_insert\n (Finset.not_mem_map_subtype_of_not_property s (Classical.not_not.2 rfl)),\n Finset.sum_subtype_map_embedding fun x _ => (hfg x).symm]\n rw [hfdef]\n dsimp only\n rw [dif_pos rfl]\n exact neg_add_self _"
},
{
"state_after": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\nhf : ∑ ι in s2, f ι = 0\nhs2 : ↑(Finset.weightedVSub s2 p) f = 0\n⊢ f ↑i = 0",
"state_before": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\nhf : ∑ ι in s2, f ι = 0\n⊢ f ↑i = 0",
"tactic": "have hs2 : s2.weightedVSub p f = (0 : V) := by\n set f2 : ι → V := fun x => f x • (p x -ᵥ p i1) with hf2def\n set g2 : { x // x ≠ i1 } → V := fun x => g x • (p x -ᵥ p i1)\n have hf2g2 : ∀ x : { x // x ≠ i1 }, f2 x = g2 x := by\n simp only [hf2def]\n refine' fun x => _\n rw [hfg]\n rw [Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero s2 f p hf (p i1),\n Finset.weightedVSubOfPoint_insert, Finset.weightedVSubOfPoint_apply,\n Finset.sum_subtype_map_embedding fun x _ => hf2g2 x]\n exact hg"
},
{
"state_after": "no goals",
"state_before": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\nhf : ∑ ι in s2, f ι = 0\nhs2 : ↑(Finset.weightedVSub s2 p) f = 0\n⊢ f ↑i = 0",
"tactic": "exact h s2 f hf hs2 i (Finset.mem_insert_of_mem (Finset.mem_map.2 ⟨i, hi, rfl⟩))"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nx : { x // x ≠ i1 }\n⊢ g x = f ↑x",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\n⊢ ∀ (x : { x // x ≠ i1 }), g x = f ↑x",
"tactic": "intro x"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nx : { x // x ≠ i1 }\n⊢ g x = (fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }) ↑x",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nx : { x // x ≠ i1 }\n⊢ g x = f ↑x",
"tactic": "rw [hfdef]"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nx : { x // x ≠ i1 }\n⊢ g x = if hx : ↑x = i1 then -∑ y in s, g y else g { val := ↑x, property := hx }",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nx : { x // x ≠ i1 }\n⊢ g x = (fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }) ↑x",
"tactic": "dsimp only"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nx : { x // x ≠ i1 }\n⊢ g x = if hx : ↑x = i1 then -∑ y in s, g y else g { val := ↑x, property := hx }",
"tactic": "erw [dif_neg x.property, Subtype.coe_eta]"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\n⊢ f i1 + ∑ x in s, g x = 0",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\n⊢ ∑ ι in s2, f ι = 0",
"tactic": "rw [Finset.sum_insert\n (Finset.not_mem_map_subtype_of_not_property s (Classical.not_not.2 rfl)),\n Finset.sum_subtype_map_embedding fun x _ => (hfg x).symm]"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\n⊢ (fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }) i1 + ∑ x in s, g x = 0",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\n⊢ f i1 + ∑ x in s, g x = 0",
"tactic": "rw [hfdef]"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\n⊢ (if hx : i1 = i1 then -∑ y in s, g y else g { val := i1, property := hx }) + ∑ x in s, g x = 0",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\n⊢ (fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }) i1 + ∑ x in s, g x = 0",
"tactic": "dsimp only"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\n⊢ -∑ y in s, g y + ∑ x in s, g x = 0",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\n⊢ (if hx : i1 = i1 then -∑ y in s, g y else g { val := i1, property := hx }) + ∑ x in s, g x = 0",
"tactic": "rw [dif_pos rfl]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\n⊢ -∑ y in s, g y + ∑ x in s, g x = 0",
"tactic": "exact neg_add_self _"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\nhf : ∑ ι in s2, f ι = 0\nf2 : ι → V := fun x => f x • (p x -ᵥ p i1)\nhf2def : f2 = fun x => f x • (p x -ᵥ p i1)\n⊢ ↑(Finset.weightedVSub s2 p) f = 0",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\nhf : ∑ ι in s2, f ι = 0\n⊢ ↑(Finset.weightedVSub s2 p) f = 0",
"tactic": "set f2 : ι → V := fun x => f x • (p x -ᵥ p i1) with hf2def"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\nhf : ∑ ι in s2, f ι = 0\nf2 : ι → V := fun x => f x • (p x -ᵥ p i1)\nhf2def : f2 = fun x => f x • (p x -ᵥ p i1)\ng2 : { x // x ≠ i1 } → V := fun x => g x • (p ↑x -ᵥ p i1)\nhg : Finset.sum s g2 = 0\n⊢ ↑(Finset.weightedVSub s2 p) f = 0",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\nhf : ∑ ι in s2, f ι = 0\nf2 : ι → V := fun x => f x • (p x -ᵥ p i1)\nhf2def : f2 = fun x => f x • (p x -ᵥ p i1)\n⊢ ↑(Finset.weightedVSub s2 p) f = 0",
"tactic": "set g2 : { x // x ≠ i1 } → V := fun x => g x • (p x -ᵥ p i1)"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\nhf : ∑ ι in s2, f ι = 0\nf2 : ι → V := fun x => f x • (p x -ᵥ p i1)\nhf2def : f2 = fun x => f x • (p x -ᵥ p i1)\ng2 : { x // x ≠ i1 } → V := fun x => g x • (p ↑x -ᵥ p i1)\nhg : Finset.sum s g2 = 0\nhf2g2 : ∀ (x : { x // x ≠ i1 }), f2 ↑x = g2 x\n⊢ ↑(Finset.weightedVSub s2 p) f = 0",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\nhf : ∑ ι in s2, f ι = 0\nf2 : ι → V := fun x => f x • (p x -ᵥ p i1)\nhf2def : f2 = fun x => f x • (p x -ᵥ p i1)\ng2 : { x // x ≠ i1 } → V := fun x => g x • (p ↑x -ᵥ p i1)\nhg : Finset.sum s g2 = 0\n⊢ ↑(Finset.weightedVSub s2 p) f = 0",
"tactic": "have hf2g2 : ∀ x : { x // x ≠ i1 }, f2 x = g2 x := by\n simp only [hf2def]\n refine' fun x => _\n rw [hfg]"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\nhf : ∑ ι in s2, f ι = 0\nf2 : ι → V := fun x => f x • (p x -ᵥ p i1)\nhf2def : f2 = fun x => f x • (p x -ᵥ p i1)\ng2 : { x // x ≠ i1 } → V := fun x => g x • (p ↑x -ᵥ p i1)\nhg : Finset.sum s g2 = 0\nhf2g2 : ∀ (x : { x // x ≠ i1 }), f2 ↑x = g2 x\n⊢ ∑ x in s, g2 x = 0",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\nhf : ∑ ι in s2, f ι = 0\nf2 : ι → V := fun x => f x • (p x -ᵥ p i1)\nhf2def : f2 = fun x => f x • (p x -ᵥ p i1)\ng2 : { x // x ≠ i1 } → V := fun x => g x • (p ↑x -ᵥ p i1)\nhg : Finset.sum s g2 = 0\nhf2g2 : ∀ (x : { x // x ≠ i1 }), f2 ↑x = g2 x\n⊢ ↑(Finset.weightedVSub s2 p) f = 0",
"tactic": "rw [Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero s2 f p hf (p i1),\n Finset.weightedVSubOfPoint_insert, Finset.weightedVSubOfPoint_apply,\n Finset.sum_subtype_map_embedding fun x _ => hf2g2 x]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\nhf : ∑ ι in s2, f ι = 0\nf2 : ι → V := fun x => f x • (p x -ᵥ p i1)\nhf2def : f2 = fun x => f x • (p x -ᵥ p i1)\ng2 : { x // x ≠ i1 } → V := fun x => g x • (p ↑x -ᵥ p i1)\nhg : Finset.sum s g2 = 0\nhf2g2 : ∀ (x : { x // x ≠ i1 }), f2 ↑x = g2 x\n⊢ ∑ x in s, g2 x = 0",
"tactic": "exact hg"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\nhf : ∑ ι in s2, f ι = 0\nf2 : ι → V := fun x => f x • (p x -ᵥ p i1)\nhf2def : f2 = fun x => f x • (p x -ᵥ p i1)\ng2 : { x // x ≠ i1 } → V := fun x => g x • (p ↑x -ᵥ p i1)\nhg : Finset.sum s g2 = 0\n⊢ ∀ (x : { x // x ≠ i1 }),\n (if hx : ↑x = i1 then -∑ y in s, g y else g { val := ↑x, property := hx }) • (p ↑x -ᵥ p i1) = g x • (p ↑x -ᵥ p i1)",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\nhf : ∑ ι in s2, f ι = 0\nf2 : ι → V := fun x => f x • (p x -ᵥ p i1)\nhf2def : f2 = fun x => f x • (p x -ᵥ p i1)\ng2 : { x // x ≠ i1 } → V := fun x => g x • (p ↑x -ᵥ p i1)\nhg : Finset.sum s g2 = 0\n⊢ ∀ (x : { x // x ≠ i1 }), f2 ↑x = g2 x",
"tactic": "simp only [hf2def]"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\nhf : ∑ ι in s2, f ι = 0\nf2 : ι → V := fun x => f x • (p x -ᵥ p i1)\nhf2def : f2 = fun x => f x • (p x -ᵥ p i1)\ng2 : { x // x ≠ i1 } → V := fun x => g x • (p ↑x -ᵥ p i1)\nhg : Finset.sum s g2 = 0\nx : { x // x ≠ i1 }\n⊢ (if hx : ↑x = i1 then -∑ y in s, g y else g { val := ↑x, property := hx }) • (p ↑x -ᵥ p i1) = g x • (p ↑x -ᵥ p i1)",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\nhf : ∑ ι in s2, f ι = 0\nf2 : ι → V := fun x => f x • (p x -ᵥ p i1)\nhf2def : f2 = fun x => f x • (p x -ᵥ p i1)\ng2 : { x // x ≠ i1 } → V := fun x => g x • (p ↑x -ᵥ p i1)\nhg : Finset.sum s g2 = 0\n⊢ ∀ (x : { x // x ≠ i1 }),\n (if hx : ↑x = i1 then -∑ y in s, g y else g { val := ↑x, property := hx }) • (p ↑x -ᵥ p i1) = g x • (p ↑x -ᵥ p i1)",
"tactic": "refine' fun x => _"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\ni : { x // x ≠ i1 }\nhi : i ∈ s\nf : ι → k := fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -∑ y in s, g y else g { val := x, property := hx }\ns2 : Finset ι := insert i1 (Finset.map (Embedding.subtype fun x => x ≠ i1) s)\nhfg : ∀ (x : { x // x ≠ i1 }), g x = f ↑x\nhf : ∑ ι in s2, f ι = 0\nf2 : ι → V := fun x => f x • (p x -ᵥ p i1)\nhf2def : f2 = fun x => f x • (p x -ᵥ p i1)\ng2 : { x // x ≠ i1 } → V := fun x => g x • (p ↑x -ᵥ p i1)\nhg : Finset.sum s g2 = 0\nx : { x // x ≠ i1 }\n⊢ (if hx : ↑x = i1 then -∑ y in s, g y else g { val := ↑x, property := hx }) • (p ↑x -ᵥ p i1) = g x • (p ↑x -ᵥ p i1)",
"tactic": "rw [hfg]"
},
{
"state_after": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : LinearIndependent k fun i => p ↑i -ᵥ p i1\n⊢ AffineIndependent k p",
"state_before": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\n⊢ (LinearIndependent k fun i => p ↑i -ᵥ p i1) → AffineIndependent k p",
"tactic": "intro h"
},
{
"state_after": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh :\n ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k),\n ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ (i : { x // x ≠ i1 }), i ∈ s → g i = 0\n⊢ AffineIndependent k p",
"state_before": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : LinearIndependent k fun i => p ↑i -ᵥ p i1\n⊢ AffineIndependent k p",
"tactic": "rw [linearIndependent_iff'] at h"
},
{
"state_after": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh :\n ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k),\n ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ (i : { x // x ≠ i1 }), i ∈ s → g i = 0\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni : ι\nhi : i ∈ s\n⊢ w i = 0",
"state_before": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh :\n ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k),\n ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ (i : { x // x ≠ i1 }), i ∈ s → g i = 0\n⊢ AffineIndependent k p",
"tactic": "intro s w hw hs i hi"
},
{
"state_after": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh :\n ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k),\n ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ (i : { x // x ≠ i1 }), i ∈ s → g i = 0\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ∑ i in Finset.erase s i1, w i • (p i -ᵥ p i1) = 0\ni : ι\nhi : i ∈ s\n⊢ w i = 0",
"state_before": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh :\n ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k),\n ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ (i : { x // x ≠ i1 }), i ∈ s → g i = 0\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni : ι\nhi : i ∈ s\n⊢ w i = 0",
"tactic": "rw [Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero s w p hw (p i1), ←\n s.weightedVSubOfPoint_erase w p i1, Finset.weightedVSubOfPoint_apply] at hs"
},
{
"state_after": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh :\n ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k),\n ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ (i : { x // x ≠ i1 }), i ∈ s → g i = 0\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ∑ i in Finset.erase s i1, w i • (p i -ᵥ p i1) = 0\ni : ι\nhi : i ∈ s\nf : ι → V := fun i => w i • (p i -ᵥ p i1)\n⊢ w i = 0",
"state_before": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh :\n ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k),\n ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ (i : { x // x ≠ i1 }), i ∈ s → g i = 0\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ∑ i in Finset.erase s i1, w i • (p i -ᵥ p i1) = 0\ni : ι\nhi : i ∈ s\n⊢ w i = 0",
"tactic": "let f : ι → V := fun i => w i • (p i -ᵥ p i1)"
},
{
"state_after": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh :\n ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k),\n ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ (i : { x // x ≠ i1 }), i ∈ s → g i = 0\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ∑ i in Finset.erase s i1, w i • (p i -ᵥ p i1) = 0\ni : ι\nhi : i ∈ s\nf : ι → V := fun i => w i • (p i -ᵥ p i1)\nhs2 : ∑ i in Finset.subtype (fun i => i ≠ i1) (Finset.erase s i1), f ↑i = 0\n⊢ w i = 0",
"state_before": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh :\n ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k),\n ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ (i : { x // x ≠ i1 }), i ∈ s → g i = 0\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ∑ i in Finset.erase s i1, w i • (p i -ᵥ p i1) = 0\ni : ι\nhi : i ∈ s\nf : ι → V := fun i => w i • (p i -ᵥ p i1)\n⊢ w i = 0",
"tactic": "have hs2 : (∑ i in (s.erase i1).subtype fun i => i ≠ i1, f i) = 0 := by\n rw [← hs]\n convert Finset.sum_subtype_of_mem f fun x => Finset.ne_of_mem_erase"
},
{
"state_after": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh :\n ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k),\n ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ (i : { x // x ≠ i1 }), i ∈ s → g i = 0\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ∑ i in Finset.erase s i1, w i • (p i -ᵥ p i1) = 0\ni : ι\nhi : i ∈ s\nf : ι → V := fun i => w i • (p i -ᵥ p i1)\nhs2 : ∑ i in Finset.subtype (fun i => i ≠ i1) (Finset.erase s i1), f ↑i = 0\nh2 : ∀ (i : { x // x ≠ i1 }), i ∈ Finset.subtype (fun i => i ≠ i1) (Finset.erase s i1) → (fun x => w ↑x) i = 0\n⊢ w i = 0",
"state_before": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh :\n ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k),\n ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ (i : { x // x ≠ i1 }), i ∈ s → g i = 0\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ∑ i in Finset.erase s i1, w i • (p i -ᵥ p i1) = 0\ni : ι\nhi : i ∈ s\nf : ι → V := fun i => w i • (p i -ᵥ p i1)\nhs2 : ∑ i in Finset.subtype (fun i => i ≠ i1) (Finset.erase s i1), f ↑i = 0\n⊢ w i = 0",
"tactic": "have h2 := h ((s.erase i1).subtype fun i => i ≠ i1) (fun x => w x) hs2"
},
{
"state_after": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh :\n ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k),\n ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ (i : { x // x ≠ i1 }), i ∈ s → g i = 0\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ∑ i in Finset.erase s i1, w i • (p i -ᵥ p i1) = 0\ni : ι\nhi : i ∈ s\nf : ι → V := fun i => w i • (p i -ᵥ p i1)\nhs2 : ∑ i in Finset.subtype (fun i => i ≠ i1) (Finset.erase s i1), f ↑i = 0\nh2 : ∀ (i : { x // x ≠ i1 }), ↑i ∈ Finset.erase s i1 → w ↑i = 0\n⊢ w i = 0",
"state_before": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh :\n ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k),\n ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ (i : { x // x ≠ i1 }), i ∈ s → g i = 0\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ∑ i in Finset.erase s i1, w i • (p i -ᵥ p i1) = 0\ni : ι\nhi : i ∈ s\nf : ι → V := fun i => w i • (p i -ᵥ p i1)\nhs2 : ∑ i in Finset.subtype (fun i => i ≠ i1) (Finset.erase s i1), f ↑i = 0\nh2 : ∀ (i : { x // x ≠ i1 }), i ∈ Finset.subtype (fun i => i ≠ i1) (Finset.erase s i1) → (fun x => w ↑x) i = 0\n⊢ w i = 0",
"tactic": "simp_rw [Finset.mem_subtype] at h2"
},
{
"state_after": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh :\n ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k),\n ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ (i : { x // x ≠ i1 }), i ∈ s → g i = 0\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ∑ i in Finset.erase s i1, w i • (p i -ᵥ p i1) = 0\ni : ι\nhi : i ∈ s\nf : ι → V := fun i => w i • (p i -ᵥ p i1)\nhs2 : ∑ i in Finset.subtype (fun i => i ≠ i1) (Finset.erase s i1), f ↑i = 0\nh2 : ∀ (i : { x // x ≠ i1 }), ↑i ∈ Finset.erase s i1 → w ↑i = 0\nh2b : ∀ (i : ι), i ∈ s → i ≠ i1 → w i = 0\n⊢ w i = 0",
"state_before": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh :\n ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k),\n ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ (i : { x // x ≠ i1 }), i ∈ s → g i = 0\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ∑ i in Finset.erase s i1, w i • (p i -ᵥ p i1) = 0\ni : ι\nhi : i ∈ s\nf : ι → V := fun i => w i • (p i -ᵥ p i1)\nhs2 : ∑ i in Finset.subtype (fun i => i ≠ i1) (Finset.erase s i1), f ↑i = 0\nh2 : ∀ (i : { x // x ≠ i1 }), ↑i ∈ Finset.erase s i1 → w ↑i = 0\n⊢ w i = 0",
"tactic": "have h2b : ∀ i ∈ s, i ≠ i1 → w i = 0 := fun i his hi =>\n h2 ⟨i, hi⟩ (Finset.mem_erase_of_ne_of_mem hi his)"
},
{
"state_after": "no goals",
"state_before": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh :\n ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k),\n ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ (i : { x // x ≠ i1 }), i ∈ s → g i = 0\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ∑ i in Finset.erase s i1, w i • (p i -ᵥ p i1) = 0\ni : ι\nhi : i ∈ s\nf : ι → V := fun i => w i • (p i -ᵥ p i1)\nhs2 : ∑ i in Finset.subtype (fun i => i ≠ i1) (Finset.erase s i1), f ↑i = 0\nh2 : ∀ (i : { x // x ≠ i1 }), ↑i ∈ Finset.erase s i1 → w ↑i = 0\nh2b : ∀ (i : ι), i ∈ s → i ≠ i1 → w i = 0\n⊢ w i = 0",
"tactic": "exact Finset.eq_zero_of_sum_eq_zero hw h2b i hi"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh :\n ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k),\n ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ (i : { x // x ≠ i1 }), i ∈ s → g i = 0\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ∑ i in Finset.erase s i1, w i • (p i -ᵥ p i1) = 0\ni : ι\nhi : i ∈ s\nf : ι → V := fun i => w i • (p i -ᵥ p i1)\n⊢ ∑ i in Finset.subtype (fun i => i ≠ i1) (Finset.erase s i1), f ↑i = ∑ i in Finset.erase s i1, w i • (p i -ᵥ p i1)",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh :\n ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k),\n ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ (i : { x // x ≠ i1 }), i ∈ s → g i = 0\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ∑ i in Finset.erase s i1, w i • (p i -ᵥ p i1) = 0\ni : ι\nhi : i ∈ s\nf : ι → V := fun i => w i • (p i -ᵥ p i1)\n⊢ ∑ i in Finset.subtype (fun i => i ≠ i1) (Finset.erase s i1), f ↑i = 0",
"tactic": "rw [← hs]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh :\n ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k),\n ∑ i in s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ (i : { x // x ≠ i1 }), i ∈ s → g i = 0\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ∑ i in Finset.erase s i1, w i • (p i -ᵥ p i1) = 0\ni : ι\nhi : i ∈ s\nf : ι → V := fun i => w i • (p i -ᵥ p i1)\n⊢ ∑ i in Finset.subtype (fun i => i ≠ i1) (Finset.erase s i1), f ↑i = ∑ i in Finset.erase s i1, w i • (p i -ᵥ p i1)",
"tactic": "convert Finset.sum_subtype_of_mem f fun x => Finset.ne_of_mem_erase"
}
] |
[
139,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Analysis/BoxIntegral/Box/Basic.lean
|
BoxIntegral.Box.ne_of_disjoint_coe
|
[] |
[
195,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean
|
NormedAddGroupHom.SurjectiveOnWith.surjOn
|
[] |
[
205,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/Analysis/Calculus/Deriv/ZPow.lean
|
iter_deriv_pow'
|
[] |
[
141,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.abs_sin_eq_of_two_nsmul_eq
|
[
{
"state_after": "θ ψ : Angle\nh : θ = ψ ∨ θ = ψ + ↑π\n⊢ abs (sin θ) = abs (sin ψ)",
"state_before": "θ ψ : Angle\nh : 2 • θ = 2 • ψ\n⊢ abs (sin θ) = abs (sin ψ)",
"tactic": "rw [two_nsmul_eq_iff] at h"
},
{
"state_after": "case inl\nθ : Angle\n⊢ abs (sin θ) = abs (sin θ)\n\ncase inr\nψ : Angle\n⊢ abs (sin (ψ + ↑π)) = abs (sin ψ)",
"state_before": "θ ψ : Angle\nh : θ = ψ ∨ θ = ψ + ↑π\n⊢ abs (sin θ) = abs (sin ψ)",
"tactic": "rcases h with (rfl | rfl)"
},
{
"state_after": "no goals",
"state_before": "case inl\nθ : Angle\n⊢ abs (sin θ) = abs (sin θ)",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case inr\nψ : Angle\n⊢ abs (sin (ψ + ↑π)) = abs (sin ψ)",
"tactic": "rw [sin_add_pi, abs_neg]"
}
] |
[
484,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
479,
1
] |
Mathlib/Topology/Connected.lean
|
IsPreirreducible.isPreconnected
|
[] |
[
77,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/Deprecated/Group.lean
|
IsMonoidHom.id
|
[] |
[
209,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Data/Nat/Factorial/DoubleFactorial.lean
|
Nat.factorial_eq_mul_doubleFactorial
|
[
{
"state_after": "no goals",
"state_before": "k : ℕ\n⊢ (k + 1 + 1)! = (k + 1 + 1)‼ * (k + 1)‼",
"tactic": "rw [doubleFactorial_add_two, factorial, factorial_eq_mul_doubleFactorial _, mul_comm _ k‼,\n mul_assoc]"
}
] |
[
53,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/Analysis/Convex/Slope.lean
|
ConvexOn.secant_mono
|
[
{
"state_after": "case inl\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhxa : x ≠ a\nhy : x ∈ s\nhya : x ≠ a\nhxy : x ≤ x\n⊢ (f x - f a) / (x - a) ≤ (f x - f a) / (x - a)\n\ncase inr\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x y : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhy : y ∈ s\nhxa : x ≠ a\nhya : y ≠ a\nhxy✝ : x ≤ y\nhxy : x < y\n⊢ (f x - f a) / (x - a) ≤ (f y - f a) / (y - a)",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x y : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhy : y ∈ s\nhxa : x ≠ a\nhya : y ≠ a\nhxy : x ≤ y\n⊢ (f x - f a) / (x - a) ≤ (f y - f a) / (y - a)",
"tactic": "rcases eq_or_lt_of_le hxy with (rfl | hxy)"
},
{
"state_after": "case inr.inl\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x y : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhy : y ∈ s\nhxa✝ : x ≠ a\nhya : y ≠ a\nhxy✝ : x ≤ y\nhxy : x < y\nhxa : x < a\n⊢ (f x - f a) / (x - a) ≤ (f y - f a) / (y - a)\n\ncase inr.inr\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x y : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhy : y ∈ s\nhxa✝ : x ≠ a\nhya : y ≠ a\nhxy✝ : x ≤ y\nhxy : x < y\nhxa : x > a\n⊢ (f x - f a) / (x - a) ≤ (f y - f a) / (y - a)",
"state_before": "case inr\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x y : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhy : y ∈ s\nhxa : x ≠ a\nhya : y ≠ a\nhxy✝ : x ≤ y\nhxy : x < y\n⊢ (f x - f a) / (x - a) ≤ (f y - f a) / (y - a)",
"tactic": "cases' lt_or_gt_of_ne hxa with hxa hxa"
},
{
"state_after": "no goals",
"state_before": "case inl\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhxa : x ≠ a\nhy : x ∈ s\nhya : x ≠ a\nhxy : x ≤ x\n⊢ (f x - f a) / (x - a) ≤ (f x - f a) / (x - a)",
"tactic": "simp"
},
{
"state_after": "case inr.inl.inl\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x y : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhy : y ∈ s\nhxa✝ : x ≠ a\nhya✝ : y ≠ a\nhxy✝ : x ≤ y\nhxy : x < y\nhxa : x < a\nhya : y < a\n⊢ (f x - f a) / (x - a) ≤ (f y - f a) / (y - a)\n\ncase inr.inl.inr\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x y : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhy : y ∈ s\nhxa✝ : x ≠ a\nhya✝ : y ≠ a\nhxy✝ : x ≤ y\nhxy : x < y\nhxa : x < a\nhya : y > a\n⊢ (f x - f a) / (x - a) ≤ (f y - f a) / (y - a)",
"state_before": "case inr.inl\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x y : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhy : y ∈ s\nhxa✝ : x ≠ a\nhya : y ≠ a\nhxy✝ : x ≤ y\nhxy : x < y\nhxa : x < a\n⊢ (f x - f a) / (x - a) ≤ (f y - f a) / (y - a)",
"tactic": "cases' lt_or_gt_of_ne hya with hya hya"
},
{
"state_after": "no goals",
"state_before": "case inr.inl.inl\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x y : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhy : y ∈ s\nhxa✝ : x ≠ a\nhya✝ : y ≠ a\nhxy✝ : x ≤ y\nhxy : x < y\nhxa : x < a\nhya : y < a\n⊢ (f x - f a) / (x - a) ≤ (f y - f a) / (y - a)",
"tactic": "convert hf.secant_mono_aux3 hx ha hxy hya using 1 <;> rw [← neg_div_neg_eq] <;> field_simp"
},
{
"state_after": "case h.e'_3\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x y : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhy : y ∈ s\nhxa✝ : x ≠ a\nhya✝ : y ≠ a\nhxy✝ : x ≤ y\nhxy : x < y\nhxa : x < a\nhya : y > a\n⊢ (f x - f a) / (x - a) = (f a - f x) / (a - x)",
"state_before": "case inr.inl.inr\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x y : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhy : y ∈ s\nhxa✝ : x ≠ a\nhya✝ : y ≠ a\nhxy✝ : x ≤ y\nhxy : x < y\nhxa : x < a\nhya : y > a\n⊢ (f x - f a) / (x - a) ≤ (f y - f a) / (y - a)",
"tactic": "convert hf.slope_mono_adjacent hx hy hxa hya using 1"
},
{
"state_after": "case h.e'_3\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x y : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhy : y ∈ s\nhxa✝ : x ≠ a\nhya✝ : y ≠ a\nhxy✝ : x ≤ y\nhxy : x < y\nhxa : x < a\nhya : y > a\n⊢ -(f x - f a) / -(x - a) = (f a - f x) / (a - x)",
"state_before": "case h.e'_3\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x y : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhy : y ∈ s\nhxa✝ : x ≠ a\nhya✝ : y ≠ a\nhxy✝ : x ≤ y\nhxy : x < y\nhxa : x < a\nhya : y > a\n⊢ (f x - f a) / (x - a) = (f a - f x) / (a - x)",
"tactic": "rw [← neg_div_neg_eq]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x y : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhy : y ∈ s\nhxa✝ : x ≠ a\nhya✝ : y ≠ a\nhxy✝ : x ≤ y\nhxy : x < y\nhxa : x < a\nhya : y > a\n⊢ -(f x - f a) / -(x - a) = (f a - f x) / (a - x)",
"tactic": "field_simp"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x y : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhy : y ∈ s\nhxa✝ : x ≠ a\nhya : y ≠ a\nhxy✝ : x ≤ y\nhxy : x < y\nhxa : x > a\n⊢ (f x - f a) / (x - a) ≤ (f y - f a) / (y - a)",
"tactic": "exact hf.secant_mono_aux2 ha hy hxa hxy"
}
] |
[
287,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.dist_eq_iSup
|
[
{
"state_after": "case inl\nF : Type ?u.306605\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\nh✝ : IsEmpty α\n⊢ dist f g = ⨆ (x : α), dist (↑f x) (↑g x)\n\ncase inr\nF : Type ?u.306605\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\nh✝ : Nonempty α\n⊢ dist f g = ⨆ (x : α), dist (↑f x) (↑g x)",
"state_before": "F : Type ?u.306605\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\n⊢ dist f g = ⨆ (x : α), dist (↑f x) (↑g x)",
"tactic": "cases isEmpty_or_nonempty α"
},
{
"state_after": "case inr\nF : Type ?u.306605\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\nh✝ : Nonempty α\n⊢ BddAbove (range fun x => dist (↑f x) (↑g x))",
"state_before": "case inr\nF : Type ?u.306605\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\nh✝ : Nonempty α\n⊢ dist f g = ⨆ (x : α), dist (↑f x) (↑g x)",
"tactic": "refine' (dist_le_iff_of_nonempty.mpr <| le_ciSup _).antisymm (ciSup_le dist_coe_le_dist)"
},
{
"state_after": "no goals",
"state_before": "case inr\nF : Type ?u.306605\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\nh✝ : Nonempty α\n⊢ BddAbove (range fun x => dist (↑f x) (↑g x))",
"tactic": "exact dist_set_exists.imp fun C hC => forall_range_iff.2 hC.2"
},
{
"state_after": "no goals",
"state_before": "case inl\nF : Type ?u.306605\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\nh✝ : IsEmpty α\n⊢ dist f g = ⨆ (x : α), dist (↑f x) (↑g x)",
"tactic": "rw [iSup_of_empty', Real.sSup_empty, dist_zero_of_empty]"
}
] |
[
256,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/Algebra/Module/Equiv.lean
|
LinearEquiv.refl_symm
|
[] |
[
462,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
461,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.eq_one_of_prod_eq_one
|
[
{
"state_after": "ι : Type ?u.804263\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\na : α\nhp : ∏ x in s, f x = 1\nh1 : ∀ (x : α), x ∈ s → x ≠ a → f x = 1\nx : α\nhx : x ∈ s\n⊢ f x = 1",
"state_before": "ι : Type ?u.804263\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\na : α\nhp : ∏ x in s, f x = 1\nh1 : ∀ (x : α), x ∈ s → x ≠ a → f x = 1\n⊢ ∀ (x : α), x ∈ s → f x = 1",
"tactic": "intro x hx"
},
{
"state_after": "case pos\nι : Type ?u.804263\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\na : α\nhp : ∏ x in s, f x = 1\nh1 : ∀ (x : α), x ∈ s → x ≠ a → f x = 1\nx : α\nhx : x ∈ s\nh : x = a\n⊢ f x = 1\n\ncase neg\nι : Type ?u.804263\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\na : α\nhp : ∏ x in s, f x = 1\nh1 : ∀ (x : α), x ∈ s → x ≠ a → f x = 1\nx : α\nhx : x ∈ s\nh : ¬x = a\n⊢ f x = 1",
"state_before": "ι : Type ?u.804263\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\na : α\nhp : ∏ x in s, f x = 1\nh1 : ∀ (x : α), x ∈ s → x ≠ a → f x = 1\nx : α\nhx : x ∈ s\n⊢ f x = 1",
"tactic": "by_cases h : x = a"
},
{
"state_after": "case pos\nι : Type ?u.804263\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\na : α\nhp : ∏ x in s, f x = 1\nh1 : ∀ (x : α), x ∈ s → x ≠ a → f x = 1\nx : α\nhx : x ∈ s\nh : x = a\n⊢ f a = 1",
"state_before": "case pos\nι : Type ?u.804263\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\na : α\nhp : ∏ x in s, f x = 1\nh1 : ∀ (x : α), x ∈ s → x ≠ a → f x = 1\nx : α\nhx : x ∈ s\nh : x = a\n⊢ f x = 1",
"tactic": "rw [h]"
},
{
"state_after": "case pos\nι : Type ?u.804263\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\na : α\nhp : ∏ x in s, f x = 1\nh1 : ∀ (x : α), x ∈ s → x ≠ a → f x = 1\nx : α\nhx : a ∈ s\nh : x = a\n⊢ f a = 1",
"state_before": "case pos\nι : Type ?u.804263\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\na : α\nhp : ∏ x in s, f x = 1\nh1 : ∀ (x : α), x ∈ s → x ≠ a → f x = 1\nx : α\nhx : x ∈ s\nh : x = a\n⊢ f a = 1",
"tactic": "rw [h] at hx"
},
{
"state_after": "case pos\nι : Type ?u.804263\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\na : α\nhp : f a = 1\nh1 : ∀ (x : α), x ∈ s → x ≠ a → f x = 1\nx : α\nhx : a ∈ s\nh : x = a\n⊢ f a = 1",
"state_before": "case pos\nι : Type ?u.804263\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\na : α\nhp : ∏ x in s, f x = 1\nh1 : ∀ (x : α), x ∈ s → x ≠ a → f x = 1\nx : α\nhx : a ∈ s\nh : x = a\n⊢ f a = 1",
"tactic": "rw [← prod_subset (singleton_subset_iff.2 hx) fun t ht ha => h1 t ht (not_mem_singleton.1 ha),\n prod_singleton] at hp"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type ?u.804263\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\na : α\nhp : f a = 1\nh1 : ∀ (x : α), x ∈ s → x ≠ a → f x = 1\nx : α\nhx : a ∈ s\nh : x = a\n⊢ f a = 1",
"tactic": "exact hp"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type ?u.804263\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\na : α\nhp : ∏ x in s, f x = 1\nh1 : ∀ (x : α), x ∈ s → x ≠ a → f x = 1\nx : α\nhx : x ∈ s\nh : ¬x = a\n⊢ f x = 1",
"tactic": "exact h1 x hx h"
}
] |
[
1719,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1709,
1
] |
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
|
intervalIntegral.integral_le_sub_of_hasDeriv_right_of_le
|
[
{
"state_after": "ι : Type ?u.1832713\n𝕜 : Type ?u.1832716\nE : Type ?u.1832719\nF : Type ?u.1832722\nA : Type ?u.1832725\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\ng' g φ : ℝ → ℝ\na b : ℝ\nhab : a ≤ b\nhcont : ContinuousOn g (Icc a b)\nhderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivWithinAt g (g' x) (Ioi x) x\nφint : IntegrableOn φ (Icc a b)\nhφg : ∀ (x : ℝ), x ∈ Ioo a b → φ x ≤ g' x\n⊢ -(g b - g a) ≤ -∫ (y : ℝ) in a..b, φ y",
"state_before": "ι : Type ?u.1832713\n𝕜 : Type ?u.1832716\nE : Type ?u.1832719\nF : Type ?u.1832722\nA : Type ?u.1832725\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\ng' g φ : ℝ → ℝ\na b : ℝ\nhab : a ≤ b\nhcont : ContinuousOn g (Icc a b)\nhderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivWithinAt g (g' x) (Ioi x) x\nφint : IntegrableOn φ (Icc a b)\nhφg : ∀ (x : ℝ), x ∈ Ioo a b → φ x ≤ g' x\n⊢ (∫ (y : ℝ) in a..b, φ y) ≤ g b - g a",
"tactic": "rw [← neg_le_neg_iff]"
},
{
"state_after": "case h.e'_3\nι : Type ?u.1832713\n𝕜 : Type ?u.1832716\nE : Type ?u.1832719\nF : Type ?u.1832722\nA : Type ?u.1832725\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\ng' g φ : ℝ → ℝ\na b : ℝ\nhab : a ≤ b\nhcont : ContinuousOn g (Icc a b)\nhderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivWithinAt g (g' x) (Ioi x) x\nφint : IntegrableOn φ (Icc a b)\nhφg : ∀ (x : ℝ), x ∈ Ioo a b → φ x ≤ g' x\n⊢ -(g b - g a) = -g b - -g a\n\ncase h.e'_4\nι : Type ?u.1832713\n𝕜 : Type ?u.1832716\nE : Type ?u.1832719\nF : Type ?u.1832722\nA : Type ?u.1832725\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\ng' g φ : ℝ → ℝ\na b : ℝ\nhab : a ≤ b\nhcont : ContinuousOn g (Icc a b)\nhderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivWithinAt g (g' x) (Ioi x) x\nφint : IntegrableOn φ (Icc a b)\nhφg : ∀ (x : ℝ), x ∈ Ioo a b → φ x ≤ g' x\n⊢ (-∫ (y : ℝ) in a..b, φ y) = ∫ (y : ℝ) in a..b, (-φ) y",
"state_before": "ι : Type ?u.1832713\n𝕜 : Type ?u.1832716\nE : Type ?u.1832719\nF : Type ?u.1832722\nA : Type ?u.1832725\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\ng' g φ : ℝ → ℝ\na b : ℝ\nhab : a ≤ b\nhcont : ContinuousOn g (Icc a b)\nhderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivWithinAt g (g' x) (Ioi x) x\nφint : IntegrableOn φ (Icc a b)\nhφg : ∀ (x : ℝ), x ∈ Ioo a b → φ x ≤ g' x\n⊢ -(g b - g a) ≤ -∫ (y : ℝ) in a..b, φ y",
"tactic": "convert sub_le_integral_of_hasDeriv_right_of_le hab hcont.neg (fun x hx => (hderiv x hx).neg)\n φint.neg fun x hx => neg_le_neg (hφg x hx) using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nι : Type ?u.1832713\n𝕜 : Type ?u.1832716\nE : Type ?u.1832719\nF : Type ?u.1832722\nA : Type ?u.1832725\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\ng' g φ : ℝ → ℝ\na b : ℝ\nhab : a ≤ b\nhcont : ContinuousOn g (Icc a b)\nhderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivWithinAt g (g' x) (Ioi x) x\nφint : IntegrableOn φ (Icc a b)\nhφg : ∀ (x : ℝ), x ∈ Ioo a b → φ x ≤ g' x\n⊢ -(g b - g a) = -g b - -g a",
"tactic": "abel"
},
{
"state_after": "case h.e'_4\nι : Type ?u.1832713\n𝕜 : Type ?u.1832716\nE : Type ?u.1832719\nF : Type ?u.1832722\nA : Type ?u.1832725\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\ng' g φ : ℝ → ℝ\na b : ℝ\nhab : a ≤ b\nhcont : ContinuousOn g (Icc a b)\nhderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivWithinAt g (g' x) (Ioi x) x\nφint : IntegrableOn φ (Icc a b)\nhφg : ∀ (x : ℝ), x ∈ Ioo a b → φ x ≤ g' x\n⊢ (∫ (x : ℝ) in a..b, -φ x) = ∫ (y : ℝ) in a..b, (-φ) y",
"state_before": "case h.e'_4\nι : Type ?u.1832713\n𝕜 : Type ?u.1832716\nE : Type ?u.1832719\nF : Type ?u.1832722\nA : Type ?u.1832725\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\ng' g φ : ℝ → ℝ\na b : ℝ\nhab : a ≤ b\nhcont : ContinuousOn g (Icc a b)\nhderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivWithinAt g (g' x) (Ioi x) x\nφint : IntegrableOn φ (Icc a b)\nhφg : ∀ (x : ℝ), x ∈ Ioo a b → φ x ≤ g' x\n⊢ (-∫ (y : ℝ) in a..b, φ y) = ∫ (y : ℝ) in a..b, (-φ) y",
"tactic": "simp only [← integral_neg]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nι : Type ?u.1832713\n𝕜 : Type ?u.1832716\nE : Type ?u.1832719\nF : Type ?u.1832722\nA : Type ?u.1832725\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\ng' g φ : ℝ → ℝ\na b : ℝ\nhab : a ≤ b\nhcont : ContinuousOn g (Icc a b)\nhderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivWithinAt g (g' x) (Ioi x) x\nφint : IntegrableOn φ (Icc a b)\nhφg : ∀ (x : ℝ), x ∈ Ioo a b → φ x ≤ g' x\n⊢ (∫ (x : ℝ) in a..b, -φ x) = ∫ (y : ℝ) in a..b, (-φ) y",
"tactic": "rfl"
}
] |
[
1156,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1149,
1
] |
Mathlib/Data/Set/Intervals/Pi.lean
|
Set.pi_univ_Ioc_update_right
|
[
{
"state_after": "ι : Type u_2\nα : ι → Type u_1\ninst✝¹ : (i : ι) → Preorder (α i)\nx✝ y✝ : (i : ι) → α i\ninst✝ : DecidableEq ι\nx y : (i : ι) → α i\ni₀ : ι\nm : α i₀\nhm : m ≤ y i₀\nthis : Ioc (x i₀) m = Iic m ∩ Ioc (x i₀) (y i₀)\n⊢ (pi univ fun i => Ioc (x i) (update y i₀ m i)) = {z | z i₀ ≤ m} ∩ pi univ fun i => Ioc (x i) (y i)",
"state_before": "ι : Type u_2\nα : ι → Type u_1\ninst✝¹ : (i : ι) → Preorder (α i)\nx✝ y✝ : (i : ι) → α i\ninst✝ : DecidableEq ι\nx y : (i : ι) → α i\ni₀ : ι\nm : α i₀\nhm : m ≤ y i₀\n⊢ (pi univ fun i => Ioc (x i) (update y i₀ m i)) = {z | z i₀ ≤ m} ∩ pi univ fun i => Ioc (x i) (y i)",
"tactic": "have : Ioc (x i₀) m = Iic m ∩ Ioc (x i₀) (y i₀) := by\n rw [← Ioi_inter_Iic, ← Ioi_inter_Iic, inter_left_comm,\n inter_eq_self_of_subset_left (Iic_subset_Iic.2 hm)]"
},
{
"state_after": "ι : Type u_2\nα : ι → Type u_1\ninst✝¹ : (i : ι) → Preorder (α i)\nx✝ y✝ : (i : ι) → α i\ninst✝ : DecidableEq ι\nx y : (i : ι) → α i\ni₀ : ι\nm : α i₀\nhm : m ≤ y i₀\nthis : Ioc (x i₀) m = Iic m ∩ Ioc (x i₀) (y i₀)\n⊢ ({x_1 | x_1 i₀ ∈ Iic m ∩ Ioc (x i₀) (y i₀)} ∩ pi ({i₀}ᶜ) fun j => Ioc (x j) (y j)) =\n {z | z i₀ ≤ m} ∩ {x_1 | x_1 i₀ ∈ Ioc (x i₀) (y i₀)} ∩ pi ({i₀}ᶜ) fun j => Ioc (x j) (y j)",
"state_before": "ι : Type u_2\nα : ι → Type u_1\ninst✝¹ : (i : ι) → Preorder (α i)\nx✝ y✝ : (i : ι) → α i\ninst✝ : DecidableEq ι\nx y : (i : ι) → α i\ni₀ : ι\nm : α i₀\nhm : m ≤ y i₀\nthis : Ioc (x i₀) m = Iic m ∩ Ioc (x i₀) (y i₀)\n⊢ (pi univ fun i => Ioc (x i) (update y i₀ m i)) = {z | z i₀ ≤ m} ∩ pi univ fun i => Ioc (x i) (y i)",
"tactic": "simp_rw [univ_pi_update i₀ y m fun i z ↦ Ioc (x i) z, ← pi_inter_compl ({i₀} : Set ι),\n singleton_pi', ← inter_assoc, this]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nα : ι → Type u_1\ninst✝¹ : (i : ι) → Preorder (α i)\nx✝ y✝ : (i : ι) → α i\ninst✝ : DecidableEq ι\nx y : (i : ι) → α i\ni₀ : ι\nm : α i₀\nhm : m ≤ y i₀\nthis : Ioc (x i₀) m = Iic m ∩ Ioc (x i₀) (y i₀)\n⊢ ({x_1 | x_1 i₀ ∈ Iic m ∩ Ioc (x i₀) (y i₀)} ∩ pi ({i₀}ᶜ) fun j => Ioc (x j) (y j)) =\n {z | z i₀ ≤ m} ∩ {x_1 | x_1 i₀ ∈ Ioc (x i₀) (y i₀)} ∩ pi ({i₀}ᶜ) fun j => Ioc (x j) (y j)",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nα : ι → Type u_1\ninst✝¹ : (i : ι) → Preorder (α i)\nx✝ y✝ : (i : ι) → α i\ninst✝ : DecidableEq ι\nx y : (i : ι) → α i\ni₀ : ι\nm : α i₀\nhm : m ≤ y i₀\n⊢ Ioc (x i₀) m = Iic m ∩ Ioc (x i₀) (y i₀)",
"tactic": "rw [← Ioi_inter_Iic, ← Ioi_inter_Iic, inter_left_comm,\n inter_eq_self_of_subset_left (Iic_subset_Iic.2 hm)]"
}
] |
[
112,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/Algebra/ModEq.lean
|
AddCommGroup.ModEq.trans
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddCommGroup α\np a a₁ a₂ b b₁ b₂ c : α\nn✝ : ℕ\nz : ℤ\nx✝¹ : a ≡ b [PMOD p]\nx✝ : b ≡ c [PMOD p]\nm : ℤ\nhm : b - a = m • p\nn : ℤ\nhn : c - b = n • p\n⊢ c - a = (m + n) • p",
"tactic": "simp [add_smul, ← hm, ← hn]"
}
] |
[
75,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Set.toFinset_ssubset_toFinset
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.85081\nγ : Type ?u.85084\ns t : Set α\ninst✝¹ : Fintype ↑s\ninst✝ : Fintype ↑t\n⊢ toFinset s ⊂ toFinset t ↔ s ⊂ t",
"tactic": "simp only [Finset.ssubset_def, toFinset_subset_toFinset, ssubset_def]"
}
] |
[
673,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
672,
1
] |
Mathlib/Analysis/Convex/Segment.lean
|
Pi.segment_subset
|
[
{
"state_after": "case intro.intro.intro.intro.intro\n𝕜 : Type u_3\nE : Type ?u.376048\nF : Type ?u.376051\nG : Type ?u.376054\nι : Type u_1\nπ : ι → Type u_2\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : (i : ι) → AddCommMonoid (π i)\ninst✝ : (i : ι) → Module 𝕜 (π i)\ns : Set ι\nx y z : (i : ι) → π i\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhz : a • x + b • y = z\ni : ι\n⊢ z i ∈ (fun i => [x i-[𝕜]y i]) i",
"state_before": "𝕜 : Type u_3\nE : Type ?u.376048\nF : Type ?u.376051\nG : Type ?u.376054\nι : Type u_1\nπ : ι → Type u_2\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : (i : ι) → AddCommMonoid (π i)\ninst✝ : (i : ι) → Module 𝕜 (π i)\ns : Set ι\nx y : (i : ι) → π i\n⊢ [x-[𝕜]y] ⊆ pi s fun i => [x i-[𝕜]y i]",
"tactic": "rintro z ⟨a, b, ha, hb, hab, hz⟩ i -"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\n𝕜 : Type u_3\nE : Type ?u.376048\nF : Type ?u.376051\nG : Type ?u.376054\nι : Type u_1\nπ : ι → Type u_2\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : (i : ι) → AddCommMonoid (π i)\ninst✝ : (i : ι) → Module 𝕜 (π i)\ns : Set ι\nx y z : (i : ι) → π i\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhz : a • x + b • y = z\ni : ι\n⊢ z i ∈ (fun i => [x i-[𝕜]y i]) i",
"tactic": "exact ⟨a, b, ha, hb, hab, congr_fun hz i⟩"
}
] |
[
648,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
646,
1
] |
Mathlib/Analysis/Normed/Group/AddCircle.lean
|
AddCircle.norm_eq
|
[
{
"state_after": "⊢ ∀ (x : ℝ), ‖↑x‖ = abs (x - ↑(round x))",
"state_before": "p x : ℝ\n⊢ ∀ (x : ℝ), ‖↑x‖ = abs (x - ↑(round x))",
"tactic": "clear! x p"
},
{
"state_after": "x : ℝ\n⊢ ‖↑x‖ = abs (x - ↑(round x))",
"state_before": "⊢ ∀ (x : ℝ), ‖↑x‖ = abs (x - ↑(round x))",
"tactic": "intros x"
},
{
"state_after": "x : ℝ\n⊢ sInf (norm '' {m | ↑m = ↑x}) = min (fract x) (1 - fract x)",
"state_before": "x : ℝ\n⊢ ‖↑x‖ = abs (x - ↑(round x))",
"tactic": "rw [quotient_norm_eq, abs_sub_round_eq_min]"
},
{
"state_after": "x : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\n⊢ sInf (norm '' {m | ↑m = ↑x}) = min (fract x) (1 - fract x)",
"state_before": "x : ℝ\n⊢ sInf (norm '' {m | ↑m = ↑x}) = min (fract x) (1 - fract x)",
"tactic": "have h₁ : BddBelow (abs '' { m : ℝ | (m : AddCircle (1 : ℝ)) = x }) :=\n ⟨0, by simp [mem_lowerBounds]⟩"
},
{
"state_after": "x : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\n⊢ sInf (norm '' {m | ↑m = ↑x}) = min (fract x) (1 - fract x)",
"state_before": "x : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\n⊢ sInf (norm '' {m | ↑m = ↑x}) = min (fract x) (1 - fract x)",
"tactic": "have h₂ : (abs '' { m : ℝ | (m : AddCircle (1 : ℝ)) = x }).Nonempty := ⟨|x|, ⟨x, rfl, rfl⟩⟩"
},
{
"state_after": "case a\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\n⊢ sInf (norm '' {m | ↑m = ↑x}) ≤ min (fract x) (1 - fract x)\n\ncase a\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\n⊢ min (fract x) (1 - fract x) ≤ sInf (norm '' {m | ↑m = ↑x})",
"state_before": "x : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\n⊢ sInf (norm '' {m | ↑m = ↑x}) = min (fract x) (1 - fract x)",
"tactic": "apply le_antisymm"
},
{
"state_after": "case inl\nx : ℝ\nthis : ∀ (x : ℝ), ‖↑x‖ = abs (x - ↑(round x))\n⊢ ‖↑x‖ = abs (x - ↑(round (0⁻¹ * x)) * 0)\n\ncase inr\np x : ℝ\nthis : ∀ (x : ℝ), ‖↑x‖ = abs (x - ↑(round x))\nhp : p ≠ 0\n⊢ ‖↑x‖ = abs (x - ↑(round (p⁻¹ * x)) * p)",
"state_before": "p x : ℝ\nthis : ∀ (x : ℝ), ‖↑x‖ = abs (x - ↑(round x))\n⊢ ‖↑x‖ = abs (x - ↑(round (p⁻¹ * x)) * p)",
"tactic": "rcases eq_or_ne p 0 with (rfl | hp)"
},
{
"state_after": "case inr\np x : ℝ\nthis : ∀ (x : ℝ), ‖↑x‖ = abs (x - ↑(round x))\nhp : p ≠ 0\n⊢ ‖↑x‖ = abs (x - ↑(round (p⁻¹ * x)) * p)",
"state_before": "case inr\np x : ℝ\nthis : ∀ (x : ℝ), ‖↑x‖ = abs (x - ↑(round x))\nhp : p ≠ 0\n⊢ ‖↑x‖ = abs (x - ↑(round (p⁻¹ * x)) * p)",
"tactic": "intros"
},
{
"state_after": "case inr\np x : ℝ\nthis : ∀ (x : ℝ), ‖↑x‖ = abs (x - ↑(round x))\nhp : p ≠ 0\nhx : ‖↑(p⁻¹ * x)‖ = abs p⁻¹ * ‖↑x‖\n⊢ ‖↑x‖ = abs (x - ↑(round (p⁻¹ * x)) * p)",
"state_before": "case inr\np x : ℝ\nthis : ∀ (x : ℝ), ‖↑x‖ = abs (x - ↑(round x))\nhp : p ≠ 0\n⊢ ‖↑x‖ = abs (x - ↑(round (p⁻¹ * x)) * p)",
"tactic": "have hx := norm_coe_mul p x p⁻¹"
},
{
"state_after": "case inr\np x : ℝ\nthis : ∀ (x : ℝ), ‖↑x‖ = abs (x - ↑(round x))\nhp : p ≠ 0\nhx : abs p * ‖↑(p⁻¹ * x)‖ = ‖↑x‖\n⊢ ‖↑x‖ = abs (x - ↑(round (p⁻¹ * x)) * p)",
"state_before": "case inr\np x : ℝ\nthis : ∀ (x : ℝ), ‖↑x‖ = abs (x - ↑(round x))\nhp : p ≠ 0\nhx : ‖↑(p⁻¹ * x)‖ = abs p⁻¹ * ‖↑x‖\n⊢ ‖↑x‖ = abs (x - ↑(round (p⁻¹ * x)) * p)",
"tactic": "rw [abs_inv, eq_inv_mul_iff_mul_eq₀ ((not_congr abs_eq_zero).mpr hp)] at hx"
},
{
"state_after": "no goals",
"state_before": "case inr\np x : ℝ\nthis : ∀ (x : ℝ), ‖↑x‖ = abs (x - ↑(round x))\nhp : p ≠ 0\nhx : abs p * ‖↑(p⁻¹ * x)‖ = ‖↑x‖\n⊢ ‖↑x‖ = abs (x - ↑(round (p⁻¹ * x)) * p)",
"tactic": "rw [← hx, inv_mul_cancel hp, this, ← abs_mul, mul_sub, mul_inv_cancel_left₀ hp, mul_comm p]"
},
{
"state_after": "no goals",
"state_before": "case inl\nx : ℝ\nthis : ∀ (x : ℝ), ‖↑x‖ = abs (x - ↑(round x))\n⊢ ‖↑x‖ = abs (x - ↑(round (0⁻¹ * x)) * 0)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\n⊢ 0 ∈ lowerBounds (abs '' {m | ↑m = ↑x})",
"tactic": "simp [mem_lowerBounds]"
},
{
"state_after": "case a\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\n⊢ ∀ (b : ℝ), b ∈ lowerBounds (abs '' {m | ↑m = ↑x}) → b ≤ fract x ∧ b ≤ 1 - fract x",
"state_before": "case a\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\n⊢ sInf (norm '' {m | ↑m = ↑x}) ≤ min (fract x) (1 - fract x)",
"tactic": "simp_rw [Real.norm_eq_abs, csInf_le_iff h₁ h₂, le_min_iff]"
},
{
"state_after": "case a\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\nb : ℝ\nh : b ∈ lowerBounds (abs '' {m | ↑m = ↑x})\n⊢ b ≤ fract x ∧ b ≤ 1 - fract x",
"state_before": "case a\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\n⊢ ∀ (b : ℝ), b ∈ lowerBounds (abs '' {m | ↑m = ↑x}) → b ≤ fract x ∧ b ≤ 1 - fract x",
"tactic": "intro b h"
},
{
"state_after": "case a.refine'_1\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\nb : ℝ\nh : b ∈ lowerBounds (abs '' {m | ↑m = ↑x})\n⊢ fract x ∈ {m | ↑m = ↑x}\n\ncase a.refine'_2\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\nb : ℝ\nh : b ∈ lowerBounds (abs '' {m | ↑m = ↑x})\n⊢ fract x - 1 ∈ {m | ↑m = ↑x}",
"state_before": "case a\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\nb : ℝ\nh : b ∈ lowerBounds (abs '' {m | ↑m = ↑x})\n⊢ b ≤ fract x ∧ b ≤ 1 - fract x",
"tactic": "refine'\n ⟨mem_lowerBounds.1 h _ ⟨fract x, _, abs_fract⟩,\n mem_lowerBounds.1 h _ ⟨fract x - 1, _, by rw [abs_sub_comm, abs_one_sub_fract]⟩⟩"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\nb : ℝ\nh : b ∈ lowerBounds (abs '' {m | ↑m = ↑x})\n⊢ abs (fract x - 1) = 1 - fract x",
"tactic": "rw [abs_sub_comm, abs_one_sub_fract]"
},
{
"state_after": "no goals",
"state_before": "case a.refine'_1\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\nb : ℝ\nh : b ∈ lowerBounds (abs '' {m | ↑m = ↑x})\n⊢ fract x ∈ {m | ↑m = ↑x}",
"tactic": "simp only [mem_setOf, fract, sub_eq_self, QuotientAddGroup.mk_sub,\n QuotientAddGroup.eq_zero_iff, int_cast_mem_zmultiples_one]"
},
{
"state_after": "no goals",
"state_before": "case a.refine'_2\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\nb : ℝ\nh : b ∈ lowerBounds (abs '' {m | ↑m = ↑x})\n⊢ fract x - 1 ∈ {m | ↑m = ↑x}",
"tactic": "simp only [mem_setOf, fract, sub_eq_self, QuotientAddGroup.mk_sub,\n QuotientAddGroup.eq_zero_iff, int_cast_mem_zmultiples_one, sub_sub,\n (by norm_cast : (⌊x⌋ : ℝ) + 1 = (↑(⌊x⌋ + 1) : ℝ))]"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\nb : ℝ\nh : b ∈ lowerBounds (abs '' {m | ↑m = ↑x})\n⊢ ↑⌊x⌋ + 1 = ↑(⌊x⌋ + 1)",
"tactic": "norm_cast"
},
{
"state_after": "case a\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\n⊢ ∀ (b : ℝ), b ∈ abs '' {m | ↑m = ↑x} → min (fract x) (1 - fract x) ≤ b",
"state_before": "case a\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\n⊢ min (fract x) (1 - fract x) ≤ sInf (norm '' {m | ↑m = ↑x})",
"tactic": "simp only [QuotientAddGroup.mk'_apply, Real.norm_eq_abs, le_csInf_iff h₁ h₂]"
},
{
"state_after": "case a.intro.intro\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\nb : ℝ\nhb : b ∈ {m | ↑m = ↑x}\n⊢ min (fract x) (1 - fract x) ≤ abs b",
"state_before": "case a\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\n⊢ ∀ (b : ℝ), b ∈ abs '' {m | ↑m = ↑x} → min (fract x) (1 - fract x) ≤ b",
"tactic": "rintro b' ⟨b, hb, rfl⟩"
},
{
"state_after": "case a.intro.intro\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\nb : ℝ\nhb : ∃ k, ↑k = b - x\n⊢ min (fract x) (1 - fract x) ≤ abs b",
"state_before": "case a.intro.intro\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\nb : ℝ\nhb : b ∈ {m | ↑m = ↑x}\n⊢ min (fract x) (1 - fract x) ≤ abs b",
"tactic": "simp only [mem_setOf, QuotientAddGroup.eq_iff_sub_mem, mem_zmultiples_iff,\n smul_one_eq_coe] at hb"
},
{
"state_after": "case a.intro.intro.intro\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\nb : ℝ\nz : ℤ\nhz : ↑z = b - x\n⊢ min (fract x) (1 - fract x) ≤ abs b",
"state_before": "case a.intro.intro\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\nb : ℝ\nhb : ∃ k, ↑k = b - x\n⊢ min (fract x) (1 - fract x) ≤ abs b",
"tactic": "obtain ⟨z, hz⟩ := hb"
},
{
"state_after": "case a.intro.intro.intro\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\nb : ℝ\nz : ℤ\nhz : ↑z = b - x\n⊢ abs (b - ↑(round b)) ≤ abs b",
"state_before": "case a.intro.intro.intro\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\nb : ℝ\nz : ℤ\nhz : ↑z = b - x\n⊢ min (fract x) (1 - fract x) ≤ abs b",
"tactic": "rw [(by rw [hz]; abel : x = b - z), fract_sub_int, ← abs_sub_round_eq_min]"
},
{
"state_after": "case h.e'_4.h.e'_3\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\nb : ℝ\nz : ℤ\nhz : ↑z = b - x\n⊢ b = b - ↑0",
"state_before": "case a.intro.intro.intro\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\nb : ℝ\nz : ℤ\nhz : ↑z = b - x\n⊢ abs (b - ↑(round b)) ≤ abs b",
"tactic": "convert round_le b 0"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4.h.e'_3\nx : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\nb : ℝ\nz : ℤ\nhz : ↑z = b - x\n⊢ b = b - ↑0",
"tactic": "simp"
},
{
"state_after": "x : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\nb : ℝ\nz : ℤ\nhz : ↑z = b - x\n⊢ x = b - (b - x)",
"state_before": "x : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\nb : ℝ\nz : ℤ\nhz : ↑z = b - x\n⊢ x = b - ↑z",
"tactic": "rw [hz]"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nh₁ : BddBelow (abs '' {m | ↑m = ↑x})\nh₂ : Set.Nonempty (abs '' {m | ↑m = ↑x})\nb : ℝ\nz : ℤ\nhz : ↑z = b - x\n⊢ x = b - (b - x)",
"tactic": "abel"
}
] |
[
121,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/Order/Bounded.lean
|
Set.unbounded_lt_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nr : α → α → Prop\ns t : Set α\ninst✝ : LinearOrder α\n⊢ Unbounded (fun x x_1 => x < x_1) s ↔ ∀ (a : α), ∃ b, b ∈ s ∧ a ≤ b",
"tactic": "simp only [Unbounded, not_lt]"
}
] |
[
57,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.prod_bot_sup_bot_prod
|
[] |
[
1261,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1255,
1
] |
Mathlib/RingTheory/IntegralClosure.lean
|
IsIntegralClosure.mk'_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nA : Type u_3\nB : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra R B\ninst✝¹ : Algebra A B\ninst✝ : IsIntegralClosure A R B\nh : optParam (IsIntegral R 0) (_ : IsIntegral R 0)\n⊢ ↑(algebraMap A B) (mk' A 0 h) = ↑(algebraMap A B) 0",
"tactic": "rw [algebraMap_mk', RingHom.map_zero]"
}
] |
[
878,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
877,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.HasAntitoneBasis.eventually_subset
|
[] |
[
1766,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1763,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometry.diam_range
|
[] |
[
337,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
336,
1
] |
Mathlib/Analysis/NormedSpace/AddTorsorBases.lean
|
IsOpen.exists_between_affineIndependent_span_eq_top
|
[
{
"state_after": "case intro\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\n⊢ ∃ t, s ⊆ t ∧ t ⊆ u ∧ AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ t = ⊤",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nhne : Set.Nonempty s\nh : AffineIndependent ℝ Subtype.val\n⊢ ∃ t, s ⊆ t ∧ t ⊆ u ∧ AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ t = ⊤",
"tactic": "obtain ⟨q, hq⟩ := hne"
},
{
"state_after": "case intro.intro.intro\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\n⊢ ∃ t, s ⊆ t ∧ t ⊆ u ∧ AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ t = ⊤",
"state_before": "case intro\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\n⊢ ∃ t, s ⊆ t ∧ t ⊆ u ∧ AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ t = ⊤",
"tactic": "obtain ⟨ε, ε0, hεu⟩ := Metric.nhds_basis_closedBall.mem_iff.1 (hu.mem_nhds <| hsu hq)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\n⊢ ∃ t, s ⊆ t ∧ t ⊆ u ∧ AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ t = ⊤",
"state_before": "case intro.intro.intro\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\n⊢ ∃ t, s ⊆ t ∧ t ⊆ u ∧ AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ t = ⊤",
"tactic": "obtain ⟨t, ht₁, ht₂, ht₃⟩ := exists_subset_affineIndependent_affineSpan_eq_top h"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\n⊢ ∃ t, s ⊆ t ∧ t ⊆ u ∧ AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ t = ⊤",
"state_before": "case intro.intro.intro.intro.intro.intro\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\n⊢ ∃ t, s ⊆ t ∧ t ⊆ u ∧ AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ t = ⊤",
"tactic": "let f : P → P := fun y => lineMap q y (ε / dist y q)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\nhf : ∀ (y : P), f y ∈ u\n⊢ ∃ t, s ⊆ t ∧ t ⊆ u ∧ AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ t = ⊤",
"state_before": "case intro.intro.intro.intro.intro.intro\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\n⊢ ∃ t, s ⊆ t ∧ t ⊆ u ∧ AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ t = ⊤",
"tactic": "have hf : ∀ y, f y ∈ u := by\n refine' fun y => hεu _\n simp only\n rw [Metric.mem_closedBall, lineMap_apply, dist_vadd_left, norm_smul, Real.norm_eq_abs,\n dist_eq_norm_vsub V y q, abs_div, abs_of_pos ε0, abs_of_nonneg (norm_nonneg _), div_mul_comm]\n exact mul_le_of_le_one_left ε0.le (div_self_le_one _)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\nhf : ∀ (y : P), f y ∈ u\nhεyq : ∀ (y : P), ¬y ∈ s → ε / dist y q ≠ 0\n⊢ ∃ t, s ⊆ t ∧ t ⊆ u ∧ AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ t = ⊤",
"state_before": "case intro.intro.intro.intro.intro.intro\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\nhf : ∀ (y : P), f y ∈ u\n⊢ ∃ t, s ⊆ t ∧ t ⊆ u ∧ AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ t = ⊤",
"tactic": "have hεyq : ∀ (y) (_ : y ∉ s), ε / dist y q ≠ 0 := fun y hy =>\n div_ne_zero ε0.ne' (dist_ne_zero.2 (ne_of_mem_of_not_mem hq hy).symm)"
},
{
"state_after": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\ny : P\n⊢ f y ∈ Metric.closedBall q ε",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\n⊢ ∀ (y : P), f y ∈ u",
"tactic": "refine' fun y => hεu _"
},
{
"state_after": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\ny : P\n⊢ ↑(lineMap q y) (ε / dist y q) ∈ Metric.closedBall q ε",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\ny : P\n⊢ f y ∈ Metric.closedBall q ε",
"tactic": "simp only"
},
{
"state_after": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\ny : P\n⊢ ‖y -ᵥ q‖ / ‖y -ᵥ q‖ * ε ≤ ε",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\ny : P\n⊢ ↑(lineMap q y) (ε / dist y q) ∈ Metric.closedBall q ε",
"tactic": "rw [Metric.mem_closedBall, lineMap_apply, dist_vadd_left, norm_smul, Real.norm_eq_abs,\n dist_eq_norm_vsub V y q, abs_div, abs_of_pos ε0, abs_of_nonneg (norm_nonneg _), div_mul_comm]"
},
{
"state_after": "no goals",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\ny : P\n⊢ ‖y -ᵥ q‖ / ‖y -ᵥ q‖ * ε ≤ ε",
"tactic": "exact mul_le_of_le_one_left ε0.le (div_self_le_one _)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\nhf : ∀ (y : P), f y ∈ u\nhεyq : ∀ (y : P), ¬y ∈ s → ε / dist y q ≠ 0\nw : ↑t → ℝˣ := fun p => if hp : ↑p ∈ s then 1 else Units.mk0 (ε / dist (↑p) q) (_ : ε / dist (↑p) q ≠ 0)\n⊢ ∃ t, s ⊆ t ∧ t ⊆ u ∧ AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ t = ⊤",
"state_before": "case intro.intro.intro.intro.intro.intro\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\nhf : ∀ (y : P), f y ∈ u\nhεyq : ∀ (y : P), ¬y ∈ s → ε / dist y q ≠ 0\n⊢ ∃ t, s ⊆ t ∧ t ⊆ u ∧ AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ t = ⊤",
"tactic": "let w : t → ℝˣ := fun p => if hp : (p : P) ∈ s then 1 else Units.mk0 _ (hεyq (↑p) hp)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.refine'_1\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\nhf : ∀ (y : P), f y ∈ u\nhεyq : ∀ (y : P), ¬y ∈ s → ε / dist y q ≠ 0\nw : ↑t → ℝˣ := fun p => if hp : ↑p ∈ s then 1 else Units.mk0 (ε / dist (↑p) q) (_ : ε / dist (↑p) q ≠ 0)\n⊢ s ⊆ range fun p => ↑(lineMap q ↑p) ↑(w p)\n\ncase intro.intro.intro.intro.intro.intro.refine'_2\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\nhf : ∀ (y : P), f y ∈ u\nhεyq : ∀ (y : P), ¬y ∈ s → ε / dist y q ≠ 0\nw : ↑t → ℝˣ := fun p => if hp : ↑p ∈ s then 1 else Units.mk0 (ε / dist (↑p) q) (_ : ε / dist (↑p) q ≠ 0)\n⊢ (range fun p => ↑(lineMap q ↑p) ↑(w p)) ⊆ u\n\ncase intro.intro.intro.intro.intro.intro.refine'_3\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\nhf : ∀ (y : P), f y ∈ u\nhεyq : ∀ (y : P), ¬y ∈ s → ε / dist y q ≠ 0\nw : ↑t → ℝˣ := fun p => if hp : ↑p ∈ s then 1 else Units.mk0 (ε / dist (↑p) q) (_ : ε / dist (↑p) q ≠ 0)\n⊢ AffineIndependent ℝ Subtype.val\n\ncase intro.intro.intro.intro.intro.intro.refine'_4\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\nhf : ∀ (y : P), f y ∈ u\nhεyq : ∀ (y : P), ¬y ∈ s → ε / dist y q ≠ 0\nw : ↑t → ℝˣ := fun p => if hp : ↑p ∈ s then 1 else Units.mk0 (ε / dist (↑p) q) (_ : ε / dist (↑p) q ≠ 0)\n⊢ affineSpan ℝ (range fun p => ↑(lineMap q ↑p) ↑(w p)) = ⊤",
"state_before": "case intro.intro.intro.intro.intro.intro\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\nhf : ∀ (y : P), f y ∈ u\nhεyq : ∀ (y : P), ¬y ∈ s → ε / dist y q ≠ 0\nw : ↑t → ℝˣ := fun p => if hp : ↑p ∈ s then 1 else Units.mk0 (ε / dist (↑p) q) (_ : ε / dist (↑p) q ≠ 0)\n⊢ ∃ t, s ⊆ t ∧ t ⊆ u ∧ AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ t = ⊤",
"tactic": "refine' ⟨Set.range fun p : t => lineMap q p (w p : ℝ), _, _, _, _⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.refine'_1\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\nhf : ∀ (y : P), f y ∈ u\nhεyq : ∀ (y : P), ¬y ∈ s → ε / dist y q ≠ 0\nw : ↑t → ℝˣ := fun p => if hp : ↑p ∈ s then 1 else Units.mk0 (ε / dist (↑p) q) (_ : ε / dist (↑p) q ≠ 0)\np : P\nhp : p ∈ s\n⊢ p ∈ range fun p => ↑(lineMap q ↑p) ↑(w p)",
"state_before": "case intro.intro.intro.intro.intro.intro.refine'_1\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\nhf : ∀ (y : P), f y ∈ u\nhεyq : ∀ (y : P), ¬y ∈ s → ε / dist y q ≠ 0\nw : ↑t → ℝˣ := fun p => if hp : ↑p ∈ s then 1 else Units.mk0 (ε / dist (↑p) q) (_ : ε / dist (↑p) q ≠ 0)\n⊢ s ⊆ range fun p => ↑(lineMap q ↑p) ↑(w p)",
"tactic": "intro p hp"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.refine'_1\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\nhf : ∀ (y : P), f y ∈ u\nhεyq : ∀ (y : P), ¬y ∈ s → ε / dist y q ≠ 0\nw : ↑t → ℝˣ := fun p => if hp : ↑p ∈ s then 1 else Units.mk0 (ε / dist (↑p) q) (_ : ε / dist (↑p) q ≠ 0)\np : P\nhp : p ∈ s\n⊢ (fun p => ↑(lineMap q ↑p) ↑(w p)) { val := p, property := (_ : p ∈ t) } = p",
"state_before": "case intro.intro.intro.intro.intro.intro.refine'_1\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\nhf : ∀ (y : P), f y ∈ u\nhεyq : ∀ (y : P), ¬y ∈ s → ε / dist y q ≠ 0\nw : ↑t → ℝˣ := fun p => if hp : ↑p ∈ s then 1 else Units.mk0 (ε / dist (↑p) q) (_ : ε / dist (↑p) q ≠ 0)\np : P\nhp : p ∈ s\n⊢ p ∈ range fun p => ↑(lineMap q ↑p) ↑(w p)",
"tactic": "use ⟨p, ht₁ hp⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.refine'_1\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\nhf : ∀ (y : P), f y ∈ u\nhεyq : ∀ (y : P), ¬y ∈ s → ε / dist y q ≠ 0\nw : ↑t → ℝˣ := fun p => if hp : ↑p ∈ s then 1 else Units.mk0 (ε / dist (↑p) q) (_ : ε / dist (↑p) q ≠ 0)\np : P\nhp : p ∈ s\n⊢ (fun p => ↑(lineMap q ↑p) ↑(w p)) { val := p, property := (_ : p ∈ t) } = p",
"tactic": "simp [hp]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.refine'_2.intro.mk\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\nhf : ∀ (y : P), f y ∈ u\nhεyq : ∀ (y : P), ¬y ∈ s → ε / dist y q ≠ 0\nw : ↑t → ℝˣ := fun p => if hp : ↑p ∈ s then 1 else Units.mk0 (ε / dist (↑p) q) (_ : ε / dist (↑p) q ≠ 0)\np : P\nhp : p ∈ t\n⊢ (fun p => ↑(lineMap q ↑p) ↑(w p)) { val := p, property := hp } ∈ u",
"state_before": "case intro.intro.intro.intro.intro.intro.refine'_2\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\nhf : ∀ (y : P), f y ∈ u\nhεyq : ∀ (y : P), ¬y ∈ s → ε / dist y q ≠ 0\nw : ↑t → ℝˣ := fun p => if hp : ↑p ∈ s then 1 else Units.mk0 (ε / dist (↑p) q) (_ : ε / dist (↑p) q ≠ 0)\n⊢ (range fun p => ↑(lineMap q ↑p) ↑(w p)) ⊆ u",
"tactic": "rintro y ⟨⟨p, hp⟩, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.refine'_2.intro.mk\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\nhf : ∀ (y : P), f y ∈ u\nhεyq : ∀ (y : P), ¬y ∈ s → ε / dist y q ≠ 0\nw : ↑t → ℝˣ := fun p => if hp : ↑p ∈ s then 1 else Units.mk0 (ε / dist (↑p) q) (_ : ε / dist (↑p) q ≠ 0)\np : P\nhp : p ∈ t\n⊢ (fun p => ↑(lineMap q ↑p) ↑(w p)) { val := p, property := hp } ∈ u",
"tactic": "by_cases hps : p ∈ s <;>\nsimp only [hps, lineMap_apply_one, Units.val_mk0, dif_neg, dif_pos, not_false_iff,\n Units.val_one, Subtype.coe_mk] <;>\n[exact hsu hps; exact hf p]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.refine'_3\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\nhf : ∀ (y : P), f y ∈ u\nhεyq : ∀ (y : P), ¬y ∈ s → ε / dist y q ≠ 0\nw : ↑t → ℝˣ := fun p => if hp : ↑p ∈ s then 1 else Units.mk0 (ε / dist (↑p) q) (_ : ε / dist (↑p) q ≠ 0)\n⊢ AffineIndependent ℝ Subtype.val",
"tactic": "exact (ht₂.units_lineMap ⟨q, ht₁ hq⟩ w).range"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.refine'_4\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ : AffineIndependent ℝ fun p => ↑p\nht₃ : affineSpan ℝ t = ⊤\nf : P → P := fun y => ↑(lineMap q y) (ε / dist y q)\nhf : ∀ (y : P), f y ∈ u\nhεyq : ∀ (y : P), ¬y ∈ s → ε / dist y q ≠ 0\nw : ↑t → ℝˣ := fun p => if hp : ↑p ∈ s then 1 else Units.mk0 (ε / dist (↑p) q) (_ : ε / dist (↑p) q ≠ 0)\n⊢ affineSpan ℝ (range fun p => ↑(lineMap q ↑p) ↑(w p)) = ⊤",
"tactic": "rw [affineSpan_eq_affineSpan_lineMap_units (ht₁ hq) w, ht₃]"
}
] |
[
118,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/Dynamics/Ergodic/MeasurePreserving.lean
|
MeasureTheory.MeasurePreserving.comp_right_iff
|
[
{
"state_after": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.163029\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nμa : Measure α\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\ng : α → β\ne : γ ≃ᵐ α\nh : MeasurePreserving ↑e\nhg : MeasurePreserving (g ∘ ↑e)\n⊢ MeasurePreserving g",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.163029\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nμa : Measure α\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\ng : α → β\ne : γ ≃ᵐ α\nh : MeasurePreserving ↑e\n⊢ MeasurePreserving (g ∘ ↑e) ↔ MeasurePreserving g",
"tactic": "refine' ⟨fun hg => _, fun hg => hg.comp h⟩"
},
{
"state_after": "case h.e'_5\nα : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.163029\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nμa : Measure α\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\ng : α → β\ne : γ ≃ᵐ α\nh : MeasurePreserving ↑e\nhg : MeasurePreserving (g ∘ ↑e)\n⊢ g = (g ∘ ↑e) ∘ ↑(MeasurableEquiv.symm e)",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.163029\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nμa : Measure α\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\ng : α → β\ne : γ ≃ᵐ α\nh : MeasurePreserving ↑e\nhg : MeasurePreserving (g ∘ ↑e)\n⊢ MeasurePreserving g",
"tactic": "convert hg.comp (MeasurePreserving.symm e h)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5\nα : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.163029\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nμa : Measure α\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\ng : α → β\ne : γ ≃ᵐ α\nh : MeasurePreserving ↑e\nhg : MeasurePreserving (g ∘ ↑e)\n⊢ g = (g ∘ ↑e) ∘ ↑(MeasurableEquiv.symm e)",
"tactic": "simp [Function.comp.assoc g e e.symm]"
}
] |
[
116,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
11
] |
Mathlib/Topology/Separation.lean
|
t1Space_iff_exists_open
|
[] |
[
508,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
506,
1
] |
Mathlib/Data/Finset/Sups.lean
|
Finset.biUnion_image_sup_left
|
[] |
[
187,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Std/Data/String/Lemmas.lean
|
String.Iterator.Valid.remainingToString
|
[
{
"state_after": "case refl\nl r : List Char\nh : ValidFor l r { s := { data := List.reverseAux l r }, i := { byteIdx := utf8Len l } }\n⊢ Iterator.remainingToString { s := { data := List.reverseAux l r }, i := { byteIdx := utf8Len l } } = { data := r }",
"state_before": "l r : List Char\nit : Iterator\nh : ValidFor l r it\n⊢ Iterator.remainingToString it = { data := r }",
"tactic": "cases h.out"
},
{
"state_after": "no goals",
"state_before": "case refl\nl r : List Char\nh : ValidFor l r { s := { data := List.reverseAux l r }, i := { byteIdx := utf8Len l } }\n⊢ Iterator.remainingToString { s := { data := List.reverseAux l r }, i := { byteIdx := utf8Len l } } = { data := r }",
"tactic": "simpa [Iterator.remainingToString, List.reverseAux_eq] using extract_of_valid l.reverse r []"
}
] |
[
655,
95
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
653,
1
] |
Mathlib/Algebra/Order/Ring/WithTop.lean
|
WithBot.mul_bot
|
[] |
[
240,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
239,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.cos_add_nat_mul_two_pi
|
[] |
[
356,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
355,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
exp_ℝ_ℂ_eq_exp_ℂ_ℂ
|
[] |
[
668,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
667,
1
] |
Mathlib/FieldTheory/PerfectClosure.lean
|
RingHom.map_pthRoot
|
[] |
[
115,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Combinatorics/Young/YoungDiagram.lean
|
YoungDiagram.coe_sup
|
[] |
[
128,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
Metric.mem_thickening_iff_infDist_lt
|
[] |
[
973,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
971,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.union_diff_cancel_left
|
[] |
[
1834,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1833,
1
] |
src/lean/Init/Control/ExceptCps.lean
|
ExceptCpsT.run_bind_lift
|
[] |
[
58,
162
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
58,
9
] |
Mathlib/GroupTheory/Complement.lean
|
Subgroup.isComplement'_bot_right
|
[] |
[
194,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.IsImage.apply_mem_iff
|
[] |
[
478,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
477,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
padicNormE.image
|
[
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nq : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk equiv f ≠ 0\nthis : ¬f ≈ 0\nn : ℤ\nhn : PadicSeq.norm f = ↑p ^ (-n)\n⊢ ‖Quotient.mk equiv f‖ = ↑(PadicSeq.norm f)",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nq : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk equiv f ≠ 0\nthis : ¬f ≈ 0\nn : ℤ\nhn : PadicSeq.norm f = ↑p ^ (-n)\n⊢ ‖Quotient.mk equiv f‖ = ↑(↑p ^ (-n))",
"tactic": "rw [← hn]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nq : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk equiv f ≠ 0\nthis : ¬f ≈ 0\nn : ℤ\nhn : PadicSeq.norm f = ↑p ^ (-n)\n⊢ ‖Quotient.mk equiv f‖ = ↑(PadicSeq.norm f)",
"tactic": "rfl"
}
] |
[
876,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
872,
11
] |
Mathlib/LinearAlgebra/TensorPower.lean
|
TensorPower.algebraMap₀_one
|
[] |
[
241,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.lintegral_sum
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.884606\nγ : Type ?u.884609\nδ : Type ?u.884612\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nι : Type u_2\nf : α →ₛ ℝ≥0∞\nμ : ι → Measure α\n⊢ (∑' (x : { x // x ∈ SimpleFunc.range f }) (i : ι), ↑x * ↑↑(μ i) (↑f ⁻¹' {↑x})) =\n ∑' (i : ι) (x : { x // x ∈ SimpleFunc.range f }), ↑x * ↑↑(μ i) (↑f ⁻¹' {↑x})",
"state_before": "α : Type u_1\nβ : Type ?u.884606\nγ : Type ?u.884609\nδ : Type ?u.884612\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nι : Type u_2\nf : α →ₛ ℝ≥0∞\nμ : ι → Measure α\n⊢ lintegral f (Measure.sum μ) = ∑' (i : ι), lintegral f (μ i)",
"tactic": "simp only [lintegral, Measure.sum_apply, f.measurableSet_preimage, ← Finset.tsum_subtype, ←\n ENNReal.tsum_mul_left]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.884606\nγ : Type ?u.884609\nδ : Type ?u.884612\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nι : Type u_2\nf : α →ₛ ℝ≥0∞\nμ : ι → Measure α\n⊢ (∑' (x : { x // x ∈ SimpleFunc.range f }) (i : ι), ↑x * ↑↑(μ i) (↑f ⁻¹' {↑x})) =\n ∑' (i : ι) (x : { x // x ∈ SimpleFunc.range f }), ↑x * ↑↑(μ i) (↑f ⁻¹' {↑x})",
"tactic": "apply ENNReal.tsum_comm"
}
] |
[
1062,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1058,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.embedding_coe
|
[
{
"state_after": "α : Type ?u.1479\nβ : Type ?u.1482\nγ : Type ?u.1485\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\n⊢ OrdConnected (Iio ⊤)",
"state_before": "α : Type ?u.1479\nβ : Type ?u.1482\nγ : Type ?u.1485\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\n⊢ OrdConnected (range some)",
"tactic": "rw [range_coe']"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.1479\nβ : Type ?u.1482\nγ : Type ?u.1485\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\n⊢ OrdConnected (Iio ⊤)",
"tactic": "exact ordConnected_Iio"
}
] |
[
54,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Algebra/GroupPower/Ring.lean
|
eq_or_eq_neg_of_sq_eq_sq
|
[] |
[
301,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
300,
1
] |
Mathlib/Data/Set/Prod.lean
|
Set.snd_image_prod
|
[] |
[
373,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
370,
1
] |
Mathlib/Data/Finset/Card.lean
|
Finset.strongInductionOn_eq
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.74655\ns✝ t : Finset α\nf : α → β\nn : ℕ\np : Finset α → Sort u_2\ns : Finset α\nH : (s : Finset α) → ((t : Finset α) → t ⊂ s → p t) → p s\n⊢ strongInduction H s = H s fun t x => strongInduction H t",
"state_before": "α : Type u_1\nβ : Type ?u.74655\ns✝ t : Finset α\nf : α → β\nn : ℕ\np : Finset α → Sort u_2\ns : Finset α\nH : (s : Finset α) → ((t : Finset α) → t ⊂ s → p t) → p s\n⊢ strongInductionOn s H = H s fun t x => strongInductionOn t H",
"tactic": "dsimp only [strongInductionOn]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.74655\ns✝ t : Finset α\nf : α → β\nn : ℕ\np : Finset α → Sort u_2\ns : Finset α\nH : (s : Finset α) → ((t : Finset α) → t ⊂ s → p t) → p s\n⊢ strongInduction H s = H s fun t x => strongInduction H t",
"tactic": "rw [strongInduction]"
}
] |
[
660,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
656,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.node4L_size
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : Ordnode α\nx : α\nm : Ordnode α\ny : α\nr : Ordnode α\nhm : Sized m\n⊢ size (node4L l x m y r) = size l + size m + size r + 2",
"tactic": "cases m <;> simp [node4L, node3L, node'] <;> [skip; simp [size, hm.1]] <;> abel"
}
] |
[
399,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
397,
1
] |
Mathlib/Topology/LocallyConstant/Basic.lean
|
LocallyConstant.congr_arg
|
[] |
[
278,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean
|
NormedAddGroupHom.zsmul_apply
|
[] |
[
581,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
580,
1
] |
Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean
|
MeasureTheory.ProbabilityMeasure.coeFn_univ
|
[] |
[
144,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean
|
pow_dvd_pow
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.103818\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : Monoid R\na : R\nm n : ℕ\nh : m ≤ n\n⊢ a ^ n = a ^ m * a ^ (n - m)",
"tactic": "rw [← pow_add, Nat.add_comm, Nat.sub_add_cancel h]"
}
] |
[
444,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
443,
1
] |
Mathlib/Data/QPF/Multivariate/Basic.lean
|
MvQPF.comp_map
|
[
{
"state_after": "n : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα β γ : TypeVec n\nf : α ⟹ β\ng : β ⟹ γ\nx : F α\n⊢ (g ⊚ f) <$$> abs (repr x) = g <$$> f <$$> abs (repr x)",
"state_before": "n : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα β γ : TypeVec n\nf : α ⟹ β\ng : β ⟹ γ\nx : F α\n⊢ (g ⊚ f) <$$> x = g <$$> f <$$> x",
"tactic": "rw [← abs_repr x]"
},
{
"state_after": "case mk\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα β γ : TypeVec n\nf✝ : α ⟹ β\ng : β ⟹ γ\nx : F α\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\n⊢ (g ⊚ f✝) <$$> abs { fst := a, snd := f } = g <$$> f✝ <$$> abs { fst := a, snd := f }",
"state_before": "n : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα β γ : TypeVec n\nf : α ⟹ β\ng : β ⟹ γ\nx : F α\n⊢ (g ⊚ f) <$$> abs (repr x) = g <$$> f <$$> abs (repr x)",
"tactic": "cases' repr x with a f"
},
{
"state_after": "case mk\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα β γ : TypeVec n\nf✝ : α ⟹ β\ng : β ⟹ γ\nx : F α\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\n⊢ abs ((g ⊚ f✝) <$$> { fst := a, snd := f }) = abs (g <$$> f✝ <$$> { fst := a, snd := f })",
"state_before": "case mk\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα β γ : TypeVec n\nf✝ : α ⟹ β\ng : β ⟹ γ\nx : F α\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\n⊢ (g ⊚ f✝) <$$> abs { fst := a, snd := f } = g <$$> f✝ <$$> abs { fst := a, snd := f }",
"tactic": "rw [← abs_map, ← abs_map, ← abs_map]"
},
{
"state_after": "no goals",
"state_before": "case mk\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα β γ : TypeVec n\nf✝ : α ⟹ β\ng : β ⟹ γ\nx : F α\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\n⊢ abs ((g ⊚ f✝) <$$> { fst := a, snd := f }) = abs (g <$$> f✝ <$$> { fst := a, snd := f })",
"tactic": "rfl"
}
] |
[
120,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Analysis/SpecialFunctions/Exp.lean
|
Real.continuousOn_exp
|
[] |
[
128,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
ContinuousMap.hasSum_of_hasSum_Lp
|
[
{
"state_after": "case h.e'_6\nα : Type u_2\nE : Type u_4\nF : Type ?u.15733246\nG : Type ?u.15733249\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedAddCommGroup G\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : BorelSpace α\ninst✝⁶ : SecondCountableTopologyEither α E\ninst✝⁵ : CompactSpace α\ninst✝⁴ : IsFiniteMeasure μ\n𝕜 : Type u_3\ninst✝³ : Fact (1 ≤ p)\nβ : Type u_1\ninst✝² : Measure.IsOpenPosMeasure μ\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\ng : β → C(α, E)\nf : C(α, E)\nhg : Summable g\nhg2 : HasSum (↑(toLp p μ 𝕜) ∘ g) (↑(toLp p μ 𝕜) f)\n⊢ f = ∑' (b : β), g b",
"state_before": "α : Type u_2\nE : Type u_4\nF : Type ?u.15733246\nG : Type ?u.15733249\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedAddCommGroup G\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : BorelSpace α\ninst✝⁶ : SecondCountableTopologyEither α E\ninst✝⁵ : CompactSpace α\ninst✝⁴ : IsFiniteMeasure μ\n𝕜 : Type u_3\ninst✝³ : Fact (1 ≤ p)\nβ : Type u_1\ninst✝² : Measure.IsOpenPosMeasure μ\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\ng : β → C(α, E)\nf : C(α, E)\nhg : Summable g\nhg2 : HasSum (↑(toLp p μ 𝕜) ∘ g) (↑(toLp p μ 𝕜) f)\n⊢ HasSum g f",
"tactic": "convert Summable.hasSum hg"
},
{
"state_after": "no goals",
"state_before": "case h.e'_6\nα : Type u_2\nE : Type u_4\nF : Type ?u.15733246\nG : Type ?u.15733249\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedAddCommGroup G\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : BorelSpace α\ninst✝⁶ : SecondCountableTopologyEither α E\ninst✝⁵ : CompactSpace α\ninst✝⁴ : IsFiniteMeasure μ\n𝕜 : Type u_3\ninst✝³ : Fact (1 ≤ p)\nβ : Type u_1\ninst✝² : Measure.IsOpenPosMeasure μ\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\ng : β → C(α, E)\nf : C(α, E)\nhg : Summable g\nhg2 : HasSum (↑(toLp p μ 𝕜) ∘ g) (↑(toLp p μ 𝕜) f)\n⊢ f = ∑' (b : β), g b",
"tactic": "exact toLp_injective μ (hg2.unique ((toLp p μ 𝕜).hasSum <| Summable.hasSum hg))"
}
] |
[
1759,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1755,
1
] |
Mathlib/CategoryTheory/Bicategory/Basic.lean
|
CategoryTheory.Bicategory.pentagon_inv_hom_hom_hom_inv
|
[
{
"state_after": "no goals",
"state_before": "B : Type u\ninst✝ : Bicategory B\na b c d e : B\nf : a ⟶ b\ng : b ⟶ c\nh : c ⟶ d\ni : d ⟶ e\n⊢ inv ((α_ (f ≫ g) h i).inv ≫ (α_ f g h).hom ▷ i ≫ (α_ f (g ≫ h) i).hom) =\n inv ((α_ f g (h ≫ i)).hom ≫ f ◁ (α_ g h i).inv)",
"tactic": "simp"
}
] |
[
275,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
272,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
real_inner_comm
|
[] |
[
448,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
447,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean
|
LinearMap.linearIndependent_iff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.109994\nR : Type u_1\nK : Type ?u.110000\nM : Type u_2\nM' : Type u_3\nM'' : Type ?u.110009\nV : Type u\nV' : Type ?u.110014\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nf : M →ₗ[R] M'\nhf_inj : ker f = ⊥\nh : LinearIndependent R v\n⊢ Disjoint (span R (Set.range v)) (ker f)",
"tactic": "simp only [hf_inj, disjoint_bot_right]"
}
] |
[
246,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
244,
11
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Cardinal.eq_aleph_of_eq_card_ord
|
[
{
"state_after": "case intro\no : Ordinal\nho : ord (card o) = o\nho' : ω ≤ o\na : Ordinal\nha : ord (aleph' a) = o\n⊢ ∃ a, ord (aleph a) = o",
"state_before": "o : Ordinal\nho : ord (card o) = o\nho' : ω ≤ o\n⊢ ∃ a, ord (aleph a) = o",
"tactic": "cases' eq_aleph'_of_eq_card_ord ho with a ha"
},
{
"state_after": "case intro\no : Ordinal\nho : ord (card o) = o\nho' : ω ≤ o\na : Ordinal\nha : ord (aleph' a) = o\n⊢ ord (aleph (a - ω)) = o",
"state_before": "case intro\no : Ordinal\nho : ord (card o) = o\nho' : ω ≤ o\na : Ordinal\nha : ord (aleph' a) = o\n⊢ ∃ a, ord (aleph a) = o",
"tactic": "use a - ω"
},
{
"state_after": "case intro\no : Ordinal\nho : ord (card o) = o\nho' : ω ≤ o\na : Ordinal\nha : ord (aleph' a) = o\n⊢ ord (aleph' (ω + (a - ω))) = o",
"state_before": "case intro\no : Ordinal\nho : ord (card o) = o\nho' : ω ≤ o\na : Ordinal\nha : ord (aleph' a) = o\n⊢ ord (aleph (a - ω)) = o",
"tactic": "unfold aleph"
},
{
"state_after": "case intro\no : Ordinal\nho : ord (card o) = o\nho' : ω ≤ o\na : Ordinal\nha : ord (aleph' a) = o\n⊢ ω ≤ a",
"state_before": "case intro\no : Ordinal\nho : ord (card o) = o\nho' : ω ≤ o\na : Ordinal\nha : ord (aleph' a) = o\n⊢ ord (aleph' (ω + (a - ω))) = o",
"tactic": "rwa [Ordinal.add_sub_cancel_of_le]"
},
{
"state_after": "no goals",
"state_before": "case intro\no : Ordinal\nho : ord (card o) = o\nho' : ω ≤ o\na : Ordinal\nha : ord (aleph' a) = o\n⊢ ω ≤ a",
"tactic": "rwa [← aleph0_le_aleph', ← ord_le_ord, ha, ord_aleph0]"
}
] |
[
389,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
383,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.one_lt_iff_nontrivial
|
[
{
"state_after": "no goals",
"state_before": "α✝ β α : Type u\n⊢ 1 < (#α) ↔ Nontrivial α",
"tactic": "rw [← not_le, le_one_iff_subsingleton, ← not_nontrivial_iff_subsingleton, Classical.not_not]"
}
] |
[
746,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
745,
1
] |
Mathlib/Order/Disjoint.lean
|
Disjoint.of_disjoint_inf_of_le'
|
[] |
[
177,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
176,
1
] |
Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean
|
Polynomial.IsWeaklyEisensteinAt.exists_mem_adjoin_mul_eq_pow_natDegree_le
|
[
{
"state_after": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\ni : ℕ\nhi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i\n⊢ ∃ y, y ∈ adjoin R {x} ∧ ↑(algebraMap R S) p * y = x ^ i",
"state_before": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\n⊢ ∀ (i : ℕ), natDegree (Polynomial.map (algebraMap R S) f) ≤ i → ∃ y, y ∈ adjoin R {x} ∧ ↑(algebraMap R S) p * y = x ^ i",
"tactic": "intro i hi"
},
{
"state_after": "case intro\nR : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\ni : ℕ\nhi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i\nk : ℕ\nhk : i = natDegree (Polynomial.map (algebraMap R S) f) + k\n⊢ ∃ y, y ∈ adjoin R {x} ∧ ↑(algebraMap R S) p * y = x ^ i",
"state_before": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\ni : ℕ\nhi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i\n⊢ ∃ y, y ∈ adjoin R {x} ∧ ↑(algebraMap R S) p * y = x ^ i",
"tactic": "obtain ⟨k, hk⟩ := exists_add_of_le hi"
},
{
"state_after": "case intro\nR : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\ni : ℕ\nhi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i\nk : ℕ\nhk : i = natDegree (Polynomial.map (algebraMap R S) f) + k\n⊢ ∃ y, y ∈ adjoin R {x} ∧ ↑(algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f) * x ^ k",
"state_before": "case intro\nR : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\ni : ℕ\nhi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i\nk : ℕ\nhk : i = natDegree (Polynomial.map (algebraMap R S) f) + k\n⊢ ∃ y, y ∈ adjoin R {x} ∧ ↑(algebraMap R S) p * y = x ^ i",
"tactic": "rw [hk, pow_add]"
},
{
"state_after": "case intro.intro.intro\nR : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\ni : ℕ\nhi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i\nk : ℕ\nhk : i = natDegree (Polynomial.map (algebraMap R S) f) + k\ny : S\nhy : y ∈ adjoin R {x}\nH : ↑(algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f)\n⊢ ∃ y, y ∈ adjoin R {x} ∧ ↑(algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f) * x ^ k",
"state_before": "case intro\nR : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\ni : ℕ\nhi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i\nk : ℕ\nhk : i = natDegree (Polynomial.map (algebraMap R S) f) + k\n⊢ ∃ y, y ∈ adjoin R {x} ∧ ↑(algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f) * x ^ k",
"tactic": "obtain ⟨y, hy, H⟩ := exists_mem_adjoin_mul_eq_pow_natDegree hx hmo hf"
},
{
"state_after": "case intro.intro.intro.refine'_1\nR : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\ni : ℕ\nhi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i\nk : ℕ\nhk : i = natDegree (Polynomial.map (algebraMap R S) f) + k\ny : S\nhy : y ∈ adjoin R {x}\nH : ↑(algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f)\n⊢ y * x ^ k ∈ adjoin R {x}\n\ncase intro.intro.intro.refine'_2\nR : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\ni : ℕ\nhi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i\nk : ℕ\nhk : i = natDegree (Polynomial.map (algebraMap R S) f) + k\ny : S\nhy : y ∈ adjoin R {x}\nH : ↑(algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f)\n⊢ ↑(algebraMap R S) p * (y * x ^ k) = x ^ natDegree (Polynomial.map (algebraMap R S) f) * x ^ k",
"state_before": "case intro.intro.intro\nR : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\ni : ℕ\nhi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i\nk : ℕ\nhk : i = natDegree (Polynomial.map (algebraMap R S) f) + k\ny : S\nhy : y ∈ adjoin R {x}\nH : ↑(algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f)\n⊢ ∃ y, y ∈ adjoin R {x} ∧ ↑(algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f) * x ^ k",
"tactic": "refine' ⟨y * x ^ k, _, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.refine'_1\nR : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\ni : ℕ\nhi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i\nk : ℕ\nhk : i = natDegree (Polynomial.map (algebraMap R S) f) + k\ny : S\nhy : y ∈ adjoin R {x}\nH : ↑(algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f)\n⊢ y * x ^ k ∈ adjoin R {x}",
"tactic": "exact Subalgebra.mul_mem _ hy (Subalgebra.pow_mem _ (subset_adjoin (Set.mem_singleton x)) _)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.refine'_2\nR : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\ni : ℕ\nhi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i\nk : ℕ\nhk : i = natDegree (Polynomial.map (algebraMap R S) f) + k\ny : S\nhy : y ∈ adjoin R {x}\nH : ↑(algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f)\n⊢ ↑(algebraMap R S) p * (y * x ^ k) = x ^ natDegree (Polynomial.map (algebraMap R S) f) * x ^ k",
"tactic": "rw [← mul_assoc _ y, H]"
}
] |
[
125,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Algebra/Order/SMul.lean
|
Tactic.smul_ne_zero_of_pos_of_ne_zero
|
[] |
[
321,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
320,
9
] |
Mathlib/Data/Rat/Defs.lean
|
Rat.num_div_den
|
[
{
"state_after": "a b c r : ℚ\n⊢ ↑r.num / ↑↑r.den = r",
"state_before": "a b c r : ℚ\n⊢ ↑r.num / ↑r.den = r",
"tactic": "rw [← Int.cast_ofNat]"
},
{
"state_after": "no goals",
"state_before": "a b c r : ℚ\n⊢ ↑r.num / ↑↑r.den = r",
"tactic": "erw [← divInt_eq_div, num_den]"
}
] |
[
511,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
509,
1
] |
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