file_path
stringlengths 11
79
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stringlengths 2
100
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list | end
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stringclasses 4
values | url
stringclasses 4
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---|---|---|---|---|---|---|
Mathlib/Topology/Covering.lean
|
IsEvenlyCovered.mem_toTrivialization_baseSet
|
[] |
[
52,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Cardinal.aleph_add_aleph
|
[
{
"state_after": "no goals",
"state_before": "o₁ o₂ : Ordinal\n⊢ aleph o₁ + aleph o₂ = aleph (max o₁ o₂)",
"tactic": "rw [Cardinal.add_eq_max (aleph0_le_aleph o₁), max_aleph_eq]"
}
] |
[
853,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
852,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
ENNReal.le_rpow_one_div_iff
|
[
{
"state_after": "x y : ℝ≥0∞\nz : ℝ\nhz : 0 < z\n⊢ x ^ 1 ≤ y ^ (1 / z) ↔ x ^ z ≤ y",
"state_before": "x y : ℝ≥0∞\nz : ℝ\nhz : 0 < z\n⊢ x ≤ y ^ (1 / z) ↔ x ^ z ≤ y",
"tactic": "nth_rw 1 [← rpow_one x]"
},
{
"state_after": "x y : ℝ≥0∞\nz : ℝ\nhz : 0 < z\n⊢ x ^ (z * z⁻¹) ≤ y ^ (1 / z) ↔ x ^ z ≤ y",
"state_before": "x y : ℝ≥0∞\nz : ℝ\nhz : 0 < z\n⊢ x ^ 1 ≤ y ^ (1 / z) ↔ x ^ z ≤ y",
"tactic": "nth_rw 1 [← @_root_.mul_inv_cancel _ _ z hz.ne']"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ≥0∞\nz : ℝ\nhz : 0 < z\n⊢ x ^ (z * z⁻¹) ≤ y ^ (1 / z) ↔ x ^ z ≤ y",
"tactic": "rw [rpow_mul, ← one_div, @rpow_le_rpow_iff _ _ (1 / z) (by simp [hz])]"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ≥0∞\nz : ℝ\nhz : 0 < z\n⊢ 0 < 1 / z",
"tactic": "simp [hz]"
}
] |
[
584,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
581,
1
] |
Mathlib/Topology/Instances/NNReal.lean
|
NNReal.tendsto_cofinite_zero_of_summable
|
[
{
"state_after": "α : Type u_1\nf : α → ℝ≥0\nhf : Summable fun a => ↑(f a)\n⊢ Tendsto (fun a => ↑(f a)) cofinite (𝓝 ↑0)",
"state_before": "α : Type u_1\nf : α → ℝ≥0\nhf : Summable f\n⊢ Tendsto f cofinite (𝓝 0)",
"tactic": "simp only [← summable_coe, ← tendsto_coe] at hf ⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nf : α → ℝ≥0\nhf : Summable fun a => ↑(f a)\n⊢ Tendsto (fun a => ↑(f a)) cofinite (𝓝 ↑0)",
"tactic": "exact hf.tendsto_cofinite_zero"
}
] |
[
238,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
235,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
fderiv_const
|
[
{
"state_after": "case h.h\n𝕜 : Type u_3\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.1385368\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.1385463\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nc : F\nm x✝ : E\n⊢ ↑(fderiv 𝕜 (fun x => c) m) x✝ = ↑(OfNat.ofNat 0 m) x✝",
"state_before": "𝕜 : Type u_3\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.1385368\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.1385463\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nc : F\n⊢ (fderiv 𝕜 fun x => c) = 0",
"tactic": "ext m"
},
{
"state_after": "case h.h\n𝕜 : Type u_3\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.1385368\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.1385463\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nc : F\nm x✝ : E\n⊢ ↑0 x✝ = ↑(OfNat.ofNat 0 m) x✝",
"state_before": "case h.h\n𝕜 : Type u_3\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.1385368\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.1385463\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nc : F\nm x✝ : E\n⊢ ↑(fderiv 𝕜 (fun x => c) m) x✝ = ↑(OfNat.ofNat 0 m) x✝",
"tactic": "rw [fderiv_const_apply]"
},
{
"state_after": "no goals",
"state_before": "case h.h\n𝕜 : Type u_3\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.1385368\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.1385463\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nc : F\nm x✝ : E\n⊢ ↑0 x✝ = ↑(OfNat.ofNat 0 m) x✝",
"tactic": "rfl"
}
] |
[
1094,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1091,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.Memℒp.neg
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.3183332\nG : Type ?u.3183335\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → E\nhf : Memℒp f p\n⊢ snorm (-f) p μ < ⊤",
"tactic": "simp [hf.right]"
}
] |
[
1025,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1024,
1
] |
Mathlib/GroupTheory/Submonoid/Operations.lean
|
Submonoid.prod_le_iff
|
[
{
"state_after": "case mp\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\n⊢ prod s t ≤ u → map (inl M N) s ≤ u ∧ map (inr M N) t ≤ u\n\ncase mpr\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\n⊢ map (inl M N) s ≤ u ∧ map (inr M N) t ≤ u → prod s t ≤ u",
"state_before": "M : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\n⊢ prod s t ≤ u ↔ map (inl M N) s ≤ u ∧ map (inr M N) t ≤ u",
"tactic": "constructor"
},
{
"state_after": "case mp\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nh : prod s t ≤ u\n⊢ map (inl M N) s ≤ u ∧ map (inr M N) t ≤ u",
"state_before": "case mp\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\n⊢ prod s t ≤ u → map (inl M N) s ≤ u ∧ map (inr M N) t ≤ u",
"tactic": "intro h"
},
{
"state_after": "case mp.left\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nh : prod s t ≤ u\n⊢ map (inl M N) s ≤ u\n\ncase mp.right\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nh : prod s t ≤ u\n⊢ map (inr M N) t ≤ u",
"state_before": "case mp\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nh : prod s t ≤ u\n⊢ map (inl M N) s ≤ u ∧ map (inr M N) t ≤ u",
"tactic": "constructor"
},
{
"state_after": "case mp.left.intro.intro\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nh : prod s t ≤ u\nx : M\nhx : x ∈ ↑s\n⊢ ↑(inl M N) x ∈ u",
"state_before": "case mp.left\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nh : prod s t ≤ u\n⊢ map (inl M N) s ≤ u",
"tactic": "rintro _ ⟨x, hx, rfl⟩"
},
{
"state_after": "case mp.left.intro.intro.a\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nh : prod s t ≤ u\nx : M\nhx : x ∈ ↑s\n⊢ ↑(inl M N) x ∈ prod s t",
"state_before": "case mp.left.intro.intro\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nh : prod s t ≤ u\nx : M\nhx : x ∈ ↑s\n⊢ ↑(inl M N) x ∈ u",
"tactic": "apply h"
},
{
"state_after": "no goals",
"state_before": "case mp.left.intro.intro.a\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nh : prod s t ≤ u\nx : M\nhx : x ∈ ↑s\n⊢ ↑(inl M N) x ∈ prod s t",
"tactic": "exact ⟨hx, Submonoid.one_mem _⟩"
},
{
"state_after": "case mp.right.intro.intro\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nh : prod s t ≤ u\nx : N\nhx : x ∈ ↑t\n⊢ ↑(inr M N) x ∈ u",
"state_before": "case mp.right\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nh : prod s t ≤ u\n⊢ map (inr M N) t ≤ u",
"tactic": "rintro _ ⟨x, hx, rfl⟩"
},
{
"state_after": "case mp.right.intro.intro.a\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nh : prod s t ≤ u\nx : N\nhx : x ∈ ↑t\n⊢ ↑(inr M N) x ∈ prod s t",
"state_before": "case mp.right.intro.intro\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nh : prod s t ≤ u\nx : N\nhx : x ∈ ↑t\n⊢ ↑(inr M N) x ∈ u",
"tactic": "apply h"
},
{
"state_after": "no goals",
"state_before": "case mp.right.intro.intro.a\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nh : prod s t ≤ u\nx : N\nhx : x ∈ ↑t\n⊢ ↑(inr M N) x ∈ prod s t",
"tactic": "exact ⟨Submonoid.one_mem _, hx⟩"
},
{
"state_after": "case mpr.intro.mk.intro\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nhH : map (inl M N) s ≤ u\nhK : map (inr M N) t ≤ u\nx1 : M\nx2 : N\nh1 : (x1, x2).fst ∈ ↑s\nh2 : (x1, x2).snd ∈ ↑t\n⊢ (x1, x2) ∈ u",
"state_before": "case mpr\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\n⊢ map (inl M N) s ≤ u ∧ map (inr M N) t ≤ u → prod s t ≤ u",
"tactic": "rintro ⟨hH, hK⟩ ⟨x1, x2⟩ ⟨h1, h2⟩"
},
{
"state_after": "case mpr.intro.mk.intro\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nhH : map (inl M N) s ≤ u\nhK : map (inr M N) t ≤ u\nx1 : M\nx2 : N\nh1 : (x1, x2).fst ∈ ↑s\nh2 : (x1, x2).snd ∈ ↑t\nh1' : ↑(inl M N) x1 ∈ u\n⊢ (x1, x2) ∈ u",
"state_before": "case mpr.intro.mk.intro\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nhH : map (inl M N) s ≤ u\nhK : map (inr M N) t ≤ u\nx1 : M\nx2 : N\nh1 : (x1, x2).fst ∈ ↑s\nh2 : (x1, x2).snd ∈ ↑t\n⊢ (x1, x2) ∈ u",
"tactic": "have h1' : inl M N x1 ∈ u := by\n apply hH\n simpa using h1"
},
{
"state_after": "case mpr.intro.mk.intro\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nhH : map (inl M N) s ≤ u\nhK : map (inr M N) t ≤ u\nx1 : M\nx2 : N\nh1 : (x1, x2).fst ∈ ↑s\nh2 : (x1, x2).snd ∈ ↑t\nh1' : ↑(inl M N) x1 ∈ u\nh2' : ↑(inr M N) x2 ∈ u\n⊢ (x1, x2) ∈ u",
"state_before": "case mpr.intro.mk.intro\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nhH : map (inl M N) s ≤ u\nhK : map (inr M N) t ≤ u\nx1 : M\nx2 : N\nh1 : (x1, x2).fst ∈ ↑s\nh2 : (x1, x2).snd ∈ ↑t\nh1' : ↑(inl M N) x1 ∈ u\n⊢ (x1, x2) ∈ u",
"tactic": "have h2' : inr M N x2 ∈ u := by\n apply hK\n simpa using h2"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.mk.intro\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nhH : map (inl M N) s ≤ u\nhK : map (inr M N) t ≤ u\nx1 : M\nx2 : N\nh1 : (x1, x2).fst ∈ ↑s\nh2 : (x1, x2).snd ∈ ↑t\nh1' : ↑(inl M N) x1 ∈ u\nh2' : ↑(inr M N) x2 ∈ u\n⊢ (x1, x2) ∈ u",
"tactic": "simpa using Submonoid.mul_mem _ h1' h2'"
},
{
"state_after": "case a\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nhH : map (inl M N) s ≤ u\nhK : map (inr M N) t ≤ u\nx1 : M\nx2 : N\nh1 : (x1, x2).fst ∈ ↑s\nh2 : (x1, x2).snd ∈ ↑t\n⊢ ↑(inl M N) x1 ∈ map (inl M N) s",
"state_before": "M : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nhH : map (inl M N) s ≤ u\nhK : map (inr M N) t ≤ u\nx1 : M\nx2 : N\nh1 : (x1, x2).fst ∈ ↑s\nh2 : (x1, x2).snd ∈ ↑t\n⊢ ↑(inl M N) x1 ∈ u",
"tactic": "apply hH"
},
{
"state_after": "no goals",
"state_before": "case a\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nhH : map (inl M N) s ≤ u\nhK : map (inr M N) t ≤ u\nx1 : M\nx2 : N\nh1 : (x1, x2).fst ∈ ↑s\nh2 : (x1, x2).snd ∈ ↑t\n⊢ ↑(inl M N) x1 ∈ map (inl M N) s",
"tactic": "simpa using h1"
},
{
"state_after": "case a\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nhH : map (inl M N) s ≤ u\nhK : map (inr M N) t ≤ u\nx1 : M\nx2 : N\nh1 : (x1, x2).fst ∈ ↑s\nh2 : (x1, x2).snd ∈ ↑t\nh1' : ↑(inl M N) x1 ∈ u\n⊢ ↑(inr M N) x2 ∈ map (inr M N) t",
"state_before": "M : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nhH : map (inl M N) s ≤ u\nhK : map (inr M N) t ≤ u\nx1 : M\nx2 : N\nh1 : (x1, x2).fst ∈ ↑s\nh2 : (x1, x2).snd ∈ ↑t\nh1' : ↑(inl M N) x1 ∈ u\n⊢ ↑(inr M N) x2 ∈ u",
"tactic": "apply hK"
},
{
"state_after": "no goals",
"state_before": "case a\nM : Type u_1\nN : Type u_2\nP : Type ?u.111775\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.111796\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nhH : map (inl M N) s ≤ u\nhK : map (inr M N) t ≤ u\nx1 : M\nx2 : N\nh1 : (x1, x2).fst ∈ ↑s\nh2 : (x1, x2).snd ∈ ↑t\nh1' : ↑(inl M N) x1 ∈ u\n⊢ ↑(inr M N) x2 ∈ map (inr M N) t",
"tactic": "simpa using h2"
}
] |
[
994,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
976,
1
] |
Mathlib/Combinatorics/Composition.lean
|
Composition.one_le_blocks
|
[] |
[
180,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Constructions.lean
|
Pmf.mem_support_seq_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.73172\nq : Pmf (α → β)\np : Pmf α\nb : β\n⊢ b ∈ support (seq q p) ↔ ∃ f, f ∈ support q ∧ b ∈ f '' support p",
"tactic": "simp"
}
] |
[
137,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Order/SuccPred/Basic.lean
|
Order.pred_lt
|
[] |
[
696,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
695,
1
] |
Mathlib/Analysis/Convex/Slope.lean
|
StrictConcaveOn.slope_anti_adjacent
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConcaveOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nthis : -(((-f) z - (-f) y) / (z - y)) < -(((-f) y - (-f) x) / (y - x))\n⊢ (f z - f y) / (z - y) < (f y - f x) / (y - x)",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConcaveOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\n⊢ (f z - f y) / (z - y) < (f y - f x) / (y - x)",
"tactic": "have := neg_lt_neg (StrictConvexOn.slope_strict_mono_adjacent hf.neg hx hz hxy hyz)"
},
{
"state_after": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConcaveOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nthis : (f z - f y) / (z - y) < (f y - f x) / (y - x)\n⊢ (f z - f y) / (z - y) < (f y - f x) / (y - x)",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConcaveOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nthis : -(((-f) z - (-f) y) / (z - y)) < -(((-f) y - (-f) x) / (y - x))\n⊢ (f z - f y) / (z - y) < (f y - f x) / (y - x)",
"tactic": "simp only [Pi.neg_apply, ← neg_div, neg_sub', neg_neg] at this"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConcaveOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nthis : (f z - f y) / (z - y) < (f y - f x) / (y - x)\n⊢ (f z - f y) / (z - y) < (f y - f x) / (y - x)",
"tactic": "exact this"
}
] |
[
101,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Computability/PartrecCode.lean
|
Nat.Partrec.Code.evaln_sound
|
[
{
"state_after": "no goals",
"state_before": "x✝ : Code\nn x : ℕ\nh : x ∈ evaln 0 x✝ n\n⊢ x ∈ eval x✝ n",
"tactic": "simp [evaln] at h"
},
{
"state_after": "case zero.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\nn x : ℕ\nleft✝ : n ≤ k\nh : pure 0 = some x\n⊢ x ∈ pure 0 n\n\ncase succ.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\nn x : ℕ\nleft✝ : n ≤ k\nh : pure (Nat.succ n) = some x\n⊢ x = Nat.succ n\n\ncase left.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\nn x : ℕ\nleft✝ : n ≤ k\nh : pure (unpair n).fst = some x\n⊢ x = (unpair n).fst\n\ncase right.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\nn x : ℕ\nleft✝ : n ≤ k\nh : pure (unpair n).snd = some x\n⊢ x = (unpair n).snd\n\ncase pair.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x : ℕ\nleft✝ : n ≤ k\nh : ∃ a, evaln (k + 1) cf n = some a ∧ ∃ a_1, evaln (k + 1) cg n = some a_1 ∧ Nat.pair a a_1 = x\n⊢ ∃ a, a ∈ eval cf n ∧ ∃ a_1, a_1 ∈ eval cg n ∧ Nat.pair a a_1 = x\n\ncase comp.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x : ℕ\nleft✝ : n ≤ k\nh : ∃ a, evaln (k + 1) cg n = some a ∧ evaln (k + 1) cf a = some x\n⊢ ∃ a, a ∈ eval cg n ∧ x ∈ eval cf a\n\ncase prec.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x : ℕ\nleft✝ : n ≤ k\nh :\n Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd =\n some x\n⊢ x ∈\n Nat.rec (eval cf (unpair n).fst)\n (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\n\ncase rfind'.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn x : ℕ\nleft✝ : n ≤ k\nh :\n ∃ a,\n evaln (k + 1) cf n = some a ∧\n (if a = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n some x\n⊢ ∃ a,\n (0 ∈ eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) ∧\n ∀ {m : ℕ}, m < a → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) ∧ ¬a = 0) ∧\n a + (unpair n).snd = x",
"state_before": "k : ℕ\nc : Code\nn x : ℕ\nh : x ∈ evaln (k + 1) c n\n⊢ x ∈ eval c n",
"tactic": "induction' c with cf cg hf hg cf cg hf hg cf cg hf hg cf hf generalizing x n <;>\n simp [eval, evaln, Bind.bind, Seq.seq] at h⊢ <;>\n cases' h with _ h"
},
{
"state_after": "case pair.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x : ℕ\nleft✝ : n ≤ k\nh : ∃ a, evaln (k + 1) cf n = some a ∧ ∃ a_1, evaln (k + 1) cg n = some a_1 ∧ Nat.pair a a_1 = x\n⊢ ∃ a, a ∈ eval cf n ∧ ∃ a_1, a_1 ∈ eval cg n ∧ Nat.pair a a_1 = x\n\ncase comp.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x : ℕ\nleft✝ : n ≤ k\nh : ∃ a, evaln (k + 1) cg n = some a ∧ evaln (k + 1) cf a = some x\n⊢ ∃ a, a ∈ eval cg n ∧ x ∈ eval cf a\n\ncase prec.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x : ℕ\nleft✝ : n ≤ k\nh :\n Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd =\n some x\n⊢ x ∈\n Nat.rec (eval cf (unpair n).fst)\n (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\n\ncase rfind'.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn x : ℕ\nleft✝ : n ≤ k\nh :\n ∃ a,\n evaln (k + 1) cf n = some a ∧\n (if a = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n some x\n⊢ ∃ a,\n (0 ∈ eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) ∧\n ∀ {m : ℕ}, m < a → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) ∧ ¬a = 0) ∧\n a + (unpair n).snd = x",
"state_before": "case zero.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\nn x : ℕ\nleft✝ : n ≤ k\nh : pure 0 = some x\n⊢ x ∈ pure 0 n\n\ncase succ.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\nn x : ℕ\nleft✝ : n ≤ k\nh : pure (Nat.succ n) = some x\n⊢ x = Nat.succ n\n\ncase left.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\nn x : ℕ\nleft✝ : n ≤ k\nh : pure (unpair n).fst = some x\n⊢ x = (unpair n).fst\n\ncase right.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\nn x : ℕ\nleft✝ : n ≤ k\nh : pure (unpair n).snd = some x\n⊢ x = (unpair n).snd\n\ncase pair.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x : ℕ\nleft✝ : n ≤ k\nh : ∃ a, evaln (k + 1) cf n = some a ∧ ∃ a_1, evaln (k + 1) cg n = some a_1 ∧ Nat.pair a a_1 = x\n⊢ ∃ a, a ∈ eval cf n ∧ ∃ a_1, a_1 ∈ eval cg n ∧ Nat.pair a a_1 = x\n\ncase comp.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x : ℕ\nleft✝ : n ≤ k\nh : ∃ a, evaln (k + 1) cg n = some a ∧ evaln (k + 1) cf a = some x\n⊢ ∃ a, a ∈ eval cg n ∧ x ∈ eval cf a\n\ncase prec.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x : ℕ\nleft✝ : n ≤ k\nh :\n Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd =\n some x\n⊢ x ∈\n Nat.rec (eval cf (unpair n).fst)\n (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\n\ncase rfind'.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn x : ℕ\nleft✝ : n ≤ k\nh :\n ∃ a,\n evaln (k + 1) cf n = some a ∧\n (if a = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n some x\n⊢ ∃ a,\n (0 ∈ eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) ∧\n ∀ {m : ℕ}, m < a → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) ∧ ¬a = 0) ∧\n a + (unpair n).snd = x",
"tactic": "iterate 4 simpa [pure, PFun.pure, eq_comm] using h"
},
{
"state_after": "case pair.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x : ℕ\nleft✝ : n ≤ k\nh : ∃ a, evaln (k + 1) cf n = some a ∧ ∃ a_1, evaln (k + 1) cg n = some a_1 ∧ Nat.pair a a_1 = x\n⊢ ∃ a, a ∈ eval cf n ∧ ∃ a_1, a_1 ∈ eval cg n ∧ Nat.pair a a_1 = x\n\ncase comp.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x : ℕ\nleft✝ : n ≤ k\nh : ∃ a, evaln (k + 1) cg n = some a ∧ evaln (k + 1) cf a = some x\n⊢ ∃ a, a ∈ eval cg n ∧ x ∈ eval cf a\n\ncase prec.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x : ℕ\nleft✝ : n ≤ k\nh :\n Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd =\n some x\n⊢ x ∈\n Nat.rec (eval cf (unpair n).fst)\n (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\n\ncase rfind'.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn x : ℕ\nleft✝ : n ≤ k\nh :\n ∃ a,\n evaln (k + 1) cf n = some a ∧\n (if a = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n some x\n⊢ ∃ a,\n (0 ∈ eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) ∧\n ∀ {m : ℕ}, m < a → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) ∧ ¬a = 0) ∧\n a + (unpair n).snd = x",
"state_before": "case right.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\nn x : ℕ\nleft✝ : n ≤ k\nh : pure (unpair n).snd = some x\n⊢ x = (unpair n).snd\n\ncase pair.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x : ℕ\nleft✝ : n ≤ k\nh : ∃ a, evaln (k + 1) cf n = some a ∧ ∃ a_1, evaln (k + 1) cg n = some a_1 ∧ Nat.pair a a_1 = x\n⊢ ∃ a, a ∈ eval cf n ∧ ∃ a_1, a_1 ∈ eval cg n ∧ Nat.pair a a_1 = x\n\ncase comp.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x : ℕ\nleft✝ : n ≤ k\nh : ∃ a, evaln (k + 1) cg n = some a ∧ evaln (k + 1) cf a = some x\n⊢ ∃ a, a ∈ eval cg n ∧ x ∈ eval cf a\n\ncase prec.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x : ℕ\nleft✝ : n ≤ k\nh :\n Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd =\n some x\n⊢ x ∈\n Nat.rec (eval cf (unpair n).fst)\n (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\n\ncase rfind'.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn x : ℕ\nleft✝ : n ≤ k\nh :\n ∃ a,\n evaln (k + 1) cf n = some a ∧\n (if a = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n some x\n⊢ ∃ a,\n (0 ∈ eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) ∧\n ∀ {m : ℕ}, m < a → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) ∧ ¬a = 0) ∧\n a + (unpair n).snd = x",
"tactic": "simpa [pure, PFun.pure, eq_comm] using h"
},
{
"state_after": "case pair.intro.intro.intro.intro.intro\nk : ℕ\nc : Code\nn✝ x : ℕ\nh : x ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn : ℕ\nleft✝ : n ≤ k\ny : ℕ\nef : evaln (k + 1) cf n = some y\nz : ℕ\neg : evaln (k + 1) cg n = some z\n⊢ ∃ a, a ∈ eval cf n ∧ ∃ a_1, a_1 ∈ eval cg n ∧ Nat.pair a a_1 = Nat.pair y z",
"state_before": "case pair.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x : ℕ\nleft✝ : n ≤ k\nh : ∃ a, evaln (k + 1) cf n = some a ∧ ∃ a_1, evaln (k + 1) cg n = some a_1 ∧ Nat.pair a a_1 = x\n⊢ ∃ a, a ∈ eval cf n ∧ ∃ a_1, a_1 ∈ eval cg n ∧ Nat.pair a a_1 = x",
"tactic": "rcases h with ⟨y, ef, z, eg, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case pair.intro.intro.intro.intro.intro\nk : ℕ\nc : Code\nn✝ x : ℕ\nh : x ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn : ℕ\nleft✝ : n ≤ k\ny : ℕ\nef : evaln (k + 1) cf n = some y\nz : ℕ\neg : evaln (k + 1) cg n = some z\n⊢ ∃ a, a ∈ eval cf n ∧ ∃ a_1, a_1 ∈ eval cg n ∧ Nat.pair a a_1 = Nat.pair y z",
"tactic": "exact ⟨_, hf _ _ ef, _, hg _ _ eg, rfl⟩"
},
{
"state_after": "case comp.intro.intro.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh : x✝ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x : ℕ\nleft✝ : n ≤ k\ny : ℕ\neg : evaln (k + 1) cg n = some y\nef : evaln (k + 1) cf y = some x\n⊢ ∃ a, a ∈ eval cg n ∧ x ∈ eval cf a",
"state_before": "case comp.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x : ℕ\nleft✝ : n ≤ k\nh : ∃ a, evaln (k + 1) cg n = some a ∧ evaln (k + 1) cf a = some x\n⊢ ∃ a, a ∈ eval cg n ∧ x ∈ eval cf a",
"tactic": "rcases h with ⟨y, eg, ef⟩"
},
{
"state_after": "no goals",
"state_before": "case comp.intro.intro.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh : x✝ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x : ℕ\nleft✝ : n ≤ k\ny : ℕ\neg : evaln (k + 1) cg n = some y\nef : evaln (k + 1) cf y = some x\n⊢ ∃ a, a ∈ eval cg n ∧ x ∈ eval cf a",
"tactic": "exact ⟨_, hg _ _ eg, hf _ _ ef⟩"
},
{
"state_after": "case prec.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh : x✝ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x : ℕ\nleft✝ : n ≤ k\n⊢ Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd =\n some x →\n x ∈\n Nat.rec (eval cf (unpair n).fst)\n (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd",
"state_before": "case prec.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x : ℕ\nleft✝ : n ≤ k\nh :\n Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd =\n some x\n⊢ x ∈\n Nat.rec (eval cf (unpair n).fst)\n (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd",
"tactic": "revert h"
},
{
"state_after": "case prec.intro.zero\nk : ℕ\nc : Code\nn✝ x✝¹ : ℕ\nh : x✝¹ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x✝ : ℕ\nleft✝ : n ≤ k\nx : ℕ\n⊢ evaln (k + 1) cf (unpair n).fst = some x → x ∈ eval cf (unpair n).fst\n\ncase prec.intro.succ\nk : ℕ\nc : Code\nn✝ x✝¹ : ℕ\nh : x✝¹ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x✝ : ℕ\nleft✝ : n ≤ k\nm : ℕ\nIH :\n ∀ (x : ℕ),\n Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n m =\n some x →\n x ∈\n Nat.rec (eval cf (unpair n).fst)\n (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) m\nx : ℕ\n⊢ ∀ (x_1 : ℕ),\n evaln k (prec cf cg) (Nat.pair (unpair n).fst m) = some x_1 →\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair m x_1)) = some x →\n ∃ a,\n a ∈\n Nat.rec (eval cf (unpair n).fst)\n (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) m ∧\n x ∈ eval cg (Nat.pair (unpair n).fst (Nat.pair m a))",
"state_before": "case prec.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh : x✝ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x : ℕ\nleft✝ : n ≤ k\n⊢ Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd =\n some x →\n x ∈\n Nat.rec (eval cf (unpair n).fst)\n (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd",
"tactic": "induction' n.unpair.2 with m IH generalizing x <;> simp"
},
{
"state_after": "no goals",
"state_before": "case prec.intro.zero\nk : ℕ\nc : Code\nn✝ x✝¹ : ℕ\nh : x✝¹ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x✝ : ℕ\nleft✝ : n ≤ k\nx : ℕ\n⊢ evaln (k + 1) cf (unpair n).fst = some x → x ∈ eval cf (unpair n).fst",
"tactic": "apply hf"
},
{
"state_after": "case prec.intro.succ.refine'_1\nk : ℕ\nc : Code\nn✝ x✝¹ : ℕ\nh : x✝¹ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x✝ : ℕ\nleft✝ : n ≤ k\nm : ℕ\nIH :\n ∀ (x : ℕ),\n Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n m =\n some x →\n x ∈\n Nat.rec (eval cf (unpair n).fst)\n (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) m\nx y : ℕ\nh₁ : evaln k (prec cf cg) (Nat.pair (unpair n).fst m) = some y\nh₂ : evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair m y)) = some x\n⊢ Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n m =\n some y\n\ncase prec.intro.succ.refine'_2\nk : ℕ\nc : Code\nn✝ x✝¹ : ℕ\nh : x✝¹ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x✝ : ℕ\nleft✝ : n ≤ k\nm : ℕ\nIH :\n ∀ (x : ℕ),\n Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n m =\n some x →\n x ∈\n Nat.rec (eval cf (unpair n).fst)\n (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) m\nx y : ℕ\nh₁ : evaln k (prec cf cg) (Nat.pair (unpair n).fst m) = some y\nh₂ : evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair m y)) = some x\n⊢ x ∈ eval cg (Nat.pair (unpair n).fst (Nat.pair m y))",
"state_before": "case prec.intro.succ\nk : ℕ\nc : Code\nn✝ x✝¹ : ℕ\nh : x✝¹ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x✝ : ℕ\nleft✝ : n ≤ k\nm : ℕ\nIH :\n ∀ (x : ℕ),\n Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n m =\n some x →\n x ∈\n Nat.rec (eval cf (unpair n).fst)\n (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) m\nx : ℕ\n⊢ ∀ (x_1 : ℕ),\n evaln k (prec cf cg) (Nat.pair (unpair n).fst m) = some x_1 →\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair m x_1)) = some x →\n ∃ a,\n a ∈\n Nat.rec (eval cf (unpair n).fst)\n (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) m ∧\n x ∈ eval cg (Nat.pair (unpair n).fst (Nat.pair m a))",
"tactic": "refine' fun y h₁ h₂ => ⟨y, IH _ _, _⟩"
},
{
"state_after": "case prec.intro.succ.refine'_1\nk : ℕ\nc : Code\nn✝ x✝¹ : ℕ\nh : x✝¹ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x✝ : ℕ\nleft✝ : n ≤ k\nm : ℕ\nIH :\n ∀ (x : ℕ),\n Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n m =\n some x →\n x ∈\n Nat.rec (eval cf (unpair n).fst)\n (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) m\nx y : ℕ\nh₁ : evaln k (prec cf cg) (Nat.pair (unpair n).fst m) = some y\nh₂ : evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair m y)) = some x\nthis : y ∈ evaln (Nat.succ k) (prec cf cg) (Nat.pair (unpair n).fst m)\n⊢ Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n m =\n some y",
"state_before": "case prec.intro.succ.refine'_1\nk : ℕ\nc : Code\nn✝ x✝¹ : ℕ\nh : x✝¹ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x✝ : ℕ\nleft✝ : n ≤ k\nm : ℕ\nIH :\n ∀ (x : ℕ),\n Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n m =\n some x →\n x ∈\n Nat.rec (eval cf (unpair n).fst)\n (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) m\nx y : ℕ\nh₁ : evaln k (prec cf cg) (Nat.pair (unpair n).fst m) = some y\nh₂ : evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair m y)) = some x\n⊢ Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n m =\n some y",
"tactic": "have := evaln_mono k.le_succ h₁"
},
{
"state_after": "case prec.intro.succ.refine'_1\nk : ℕ\nc : Code\nn✝ x✝¹ : ℕ\nh : x✝¹ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x✝ : ℕ\nleft✝ : n ≤ k\nm : ℕ\nIH :\n ∀ (x : ℕ),\n Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n m =\n some x →\n x ∈\n Nat.rec (eval cf (unpair n).fst)\n (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) m\nx y : ℕ\nh₁ : evaln k (prec cf cg) (Nat.pair (unpair n).fst m) = some y\nh₂ : evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair m y)) = some x\nthis :\n Nat.pair (unpair n).fst m ≤ k ∧\n Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n m =\n some y\n⊢ Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n m =\n some y",
"state_before": "case prec.intro.succ.refine'_1\nk : ℕ\nc : Code\nn✝ x✝¹ : ℕ\nh : x✝¹ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x✝ : ℕ\nleft✝ : n ≤ k\nm : ℕ\nIH :\n ∀ (x : ℕ),\n Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n m =\n some x →\n x ∈\n Nat.rec (eval cf (unpair n).fst)\n (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) m\nx y : ℕ\nh₁ : evaln k (prec cf cg) (Nat.pair (unpair n).fst m) = some y\nh₂ : evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair m y)) = some x\nthis : y ∈ evaln (Nat.succ k) (prec cf cg) (Nat.pair (unpair n).fst m)\n⊢ Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n m =\n some y",
"tactic": "simp [evaln, Bind.bind] at this"
},
{
"state_after": "no goals",
"state_before": "case prec.intro.succ.refine'_1\nk : ℕ\nc : Code\nn✝ x✝¹ : ℕ\nh : x✝¹ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x✝ : ℕ\nleft✝ : n ≤ k\nm : ℕ\nIH :\n ∀ (x : ℕ),\n Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n m =\n some x →\n x ∈\n Nat.rec (eval cf (unpair n).fst)\n (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) m\nx y : ℕ\nh₁ : evaln k (prec cf cg) (Nat.pair (unpair n).fst m) = some y\nh₂ : evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair m y)) = some x\nthis :\n Nat.pair (unpair n).fst m ≤ k ∧\n Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n m =\n some y\n⊢ Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n m =\n some y",
"tactic": "exact this.2"
},
{
"state_after": "no goals",
"state_before": "case prec.intro.succ.refine'_2\nk : ℕ\nc : Code\nn✝ x✝¹ : ℕ\nh : x✝¹ ∈ evaln (k + 1) c n✝\ncf cg : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nhg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ eval cg n\nn x✝ : ℕ\nleft✝ : n ≤ k\nm : ℕ\nIH :\n ∀ (x : ℕ),\n Nat.rec (evaln (k + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n m =\n some x →\n x ∈\n Nat.rec (eval cf (unpair n).fst)\n (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) m\nx y : ℕ\nh₁ : evaln k (prec cf cg) (Nat.pair (unpair n).fst m) = some y\nh₂ : evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair m y)) = some x\n⊢ x ∈ eval cg (Nat.pair (unpair n).fst (Nat.pair m y))",
"tactic": "exact hg _ _ h₂"
},
{
"state_after": "case rfind'.intro.intro.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh : x✝ ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn x : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nh₂ :\n (if m = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) = some x\n⊢ ∃ a,\n (0 ∈ eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) ∧\n ∀ {m : ℕ}, m < a → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) ∧ ¬a = 0) ∧\n a + (unpair n).snd = x",
"state_before": "case rfind'.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh✝ : x✝ ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn x : ℕ\nleft✝ : n ≤ k\nh :\n ∃ a,\n evaln (k + 1) cf n = some a ∧\n (if a = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n some x\n⊢ ∃ a,\n (0 ∈ eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) ∧\n ∀ {m : ℕ}, m < a → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) ∧ ¬a = 0) ∧\n a + (unpair n).snd = x",
"tactic": "rcases h with ⟨m, h₁, h₂⟩"
},
{
"state_after": "case pos\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh : x✝ ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn x : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : m = 0\nh₂ : pure (unpair n).snd = some x\n⊢ ∃ a,\n (0 ∈ eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) ∧\n ∀ {m : ℕ}, m < a → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) ∧ ¬a = 0) ∧\n a + (unpair n).snd = x\n\ncase neg\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh : x✝ ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn x : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : ¬m = 0\nh₂ : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some x\n⊢ ∃ a,\n (0 ∈ eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) ∧\n ∀ {m : ℕ}, m < a → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) ∧ ¬a = 0) ∧\n a + (unpair n).snd = x",
"state_before": "case rfind'.intro.intro.intro\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh : x✝ ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn x : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nh₂ :\n (if m = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) = some x\n⊢ ∃ a,\n (0 ∈ eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) ∧\n ∀ {m : ℕ}, m < a → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) ∧ ¬a = 0) ∧\n a + (unpair n).snd = x",
"tactic": "by_cases m0 : m = 0 <;> simp [m0] at h₂"
},
{
"state_after": "no goals",
"state_before": "case pos\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh : x✝ ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn x : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : m = 0\nh₂ : pure (unpair n).snd = some x\n⊢ ∃ a,\n (0 ∈ eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) ∧\n ∀ {m : ℕ}, m < a → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) ∧ ¬a = 0) ∧\n a + (unpair n).snd = x",
"tactic": "exact\n ⟨0, ⟨by simpa [m0] using hf _ _ h₁, fun {m} => (Nat.not_lt_zero _).elim⟩, by\n injection h₂ with h₂; simp [h₂]⟩"
},
{
"state_after": "no goals",
"state_before": "k : ℕ\nc : Code\nn✝ x✝ : ℕ\nh : x✝ ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn x : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : m = 0\nh₂ : pure (unpair n).snd = some x\n⊢ 0 ∈ eval cf (Nat.pair (unpair n).fst (0 + (unpair n).snd))",
"tactic": "simpa [m0] using hf _ _ h₁"
},
{
"state_after": "k : ℕ\nc : Code\nn✝ x✝ : ℕ\nh : x✝ ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn x : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : m = 0\nh₂ : (unpair n).snd = x\n⊢ 0 + (unpair n).snd = x",
"state_before": "k : ℕ\nc : Code\nn✝ x✝ : ℕ\nh : x✝ ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn x : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : m = 0\nh₂ : pure (unpair n).snd = some x\n⊢ 0 + (unpair n).snd = x",
"tactic": "injection h₂ with h₂"
},
{
"state_after": "no goals",
"state_before": "k : ℕ\nc : Code\nn✝ x✝ : ℕ\nh : x✝ ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn x : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : m = 0\nh₂ : (unpair n).snd = x\n⊢ 0 + (unpair n).snd = x",
"tactic": "simp [h₂]"
},
{
"state_after": "case neg\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh : x✝ ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn x : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : ¬m = 0\nh₂ : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some x\nthis : x ∈ eval (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))\n⊢ ∃ a,\n (0 ∈ eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) ∧\n ∀ {m : ℕ}, m < a → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) ∧ ¬a = 0) ∧\n a + (unpair n).snd = x",
"state_before": "case neg\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh : x✝ ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn x : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : ¬m = 0\nh₂ : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some x\n⊢ ∃ a,\n (0 ∈ eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) ∧\n ∀ {m : ℕ}, m < a → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) ∧ ¬a = 0) ∧\n a + (unpair n).snd = x",
"tactic": "have := evaln_sound h₂"
},
{
"state_after": "case neg\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh : x✝ ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn x : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : ¬m = 0\nh₂ : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some x\nthis :\n ∃ a,\n (0 ∈ eval cf (Nat.pair (unpair n).fst (a + ((unpair n).snd + 1))) ∧\n ∀ {m : ℕ}, m < a → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) ∧ ¬a = 0) ∧\n a + ((unpair n).snd + 1) = x\n⊢ ∃ a,\n (0 ∈ eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) ∧\n ∀ {m : ℕ}, m < a → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) ∧ ¬a = 0) ∧\n a + (unpair n).snd = x",
"state_before": "case neg\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh : x✝ ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn x : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : ¬m = 0\nh₂ : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some x\nthis : x ∈ eval (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))\n⊢ ∃ a,\n (0 ∈ eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) ∧\n ∀ {m : ℕ}, m < a → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) ∧ ¬a = 0) ∧\n a + (unpair n).snd = x",
"tactic": "simp [eval] at this"
},
{
"state_after": "case neg.intro.intro.intro\nk : ℕ\nc : Code\nn✝ x : ℕ\nh : x ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : ¬m = 0\ny : ℕ\nhy₁ : 0 ∈ eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy₂ : ∀ {m : ℕ}, m < y → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) ∧ ¬a = 0\nh₂ : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\n⊢ ∃ a,\n (0 ∈ eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) ∧\n ∀ {m : ℕ}, m < a → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) ∧ ¬a = 0) ∧\n a + (unpair n).snd = y + ((unpair n).snd + 1)",
"state_before": "case neg\nk : ℕ\nc : Code\nn✝ x✝ : ℕ\nh : x✝ ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn x : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : ¬m = 0\nh₂ : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some x\nthis :\n ∃ a,\n (0 ∈ eval cf (Nat.pair (unpair n).fst (a + ((unpair n).snd + 1))) ∧\n ∀ {m : ℕ}, m < a → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) ∧ ¬a = 0) ∧\n a + ((unpair n).snd + 1) = x\n⊢ ∃ a,\n (0 ∈ eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) ∧\n ∀ {m : ℕ}, m < a → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) ∧ ¬a = 0) ∧\n a + (unpair n).snd = x",
"tactic": "rcases this with ⟨y, ⟨hy₁, hy₂⟩, rfl⟩"
},
{
"state_after": "case neg.intro.intro.intro\nk : ℕ\nc : Code\nn✝ x : ℕ\nh : x ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : ¬m = 0\ny : ℕ\nhy₁ : 0 ∈ eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy₂ : ∀ {m : ℕ}, m < y → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) ∧ ¬a = 0\nh₂ : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\ni : ℕ\nim : i < y + 1\n⊢ ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (i + (unpair n).snd)) ∧ ¬a = 0",
"state_before": "case neg.intro.intro.intro\nk : ℕ\nc : Code\nn✝ x : ℕ\nh : x ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : ¬m = 0\ny : ℕ\nhy₁ : 0 ∈ eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy₂ : ∀ {m : ℕ}, m < y → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) ∧ ¬a = 0\nh₂ : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\n⊢ ∃ a,\n (0 ∈ eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) ∧\n ∀ {m : ℕ}, m < a → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) ∧ ¬a = 0) ∧\n a + (unpair n).snd = y + ((unpair n).snd + 1)",
"tactic": "refine'\n ⟨y + 1, ⟨by simpa [add_comm, add_left_comm] using hy₁, fun {i} im => _⟩, by\n simp [add_comm, add_left_comm]⟩"
},
{
"state_after": "case neg.intro.intro.intro.zero\nk : ℕ\nc : Code\nn✝ x : ℕ\nh : x ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : ¬m = 0\ny : ℕ\nhy₁ : 0 ∈ eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy₂ : ∀ {m : ℕ}, m < y → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) ∧ ¬a = 0\nh₂ : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\nim : Nat.zero < y + 1\n⊢ ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (Nat.zero + (unpair n).snd)) ∧ ¬a = 0\n\ncase neg.intro.intro.intro.succ\nk : ℕ\nc : Code\nn✝ x : ℕ\nh : x ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : ¬m = 0\ny : ℕ\nhy₁ : 0 ∈ eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy₂ : ∀ {m : ℕ}, m < y → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) ∧ ¬a = 0\nh₂ : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\ni : ℕ\nim : Nat.succ i < y + 1\n⊢ ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (Nat.succ i + (unpair n).snd)) ∧ ¬a = 0",
"state_before": "case neg.intro.intro.intro\nk : ℕ\nc : Code\nn✝ x : ℕ\nh : x ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : ¬m = 0\ny : ℕ\nhy₁ : 0 ∈ eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy₂ : ∀ {m : ℕ}, m < y → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) ∧ ¬a = 0\nh₂ : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\ni : ℕ\nim : i < y + 1\n⊢ ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (i + (unpair n).snd)) ∧ ¬a = 0",
"tactic": "cases' i with i"
},
{
"state_after": "no goals",
"state_before": "k : ℕ\nc : Code\nn✝ x : ℕ\nh : x ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : ¬m = 0\ny : ℕ\nhy₁ : 0 ∈ eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy₂ : ∀ {m : ℕ}, m < y → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) ∧ ¬a = 0\nh₂ : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\n⊢ 0 ∈ eval cf (Nat.pair (unpair n).fst (y + 1 + (unpair n).snd))",
"tactic": "simpa [add_comm, add_left_comm] using hy₁"
},
{
"state_after": "no goals",
"state_before": "k : ℕ\nc : Code\nn✝ x : ℕ\nh : x ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : ¬m = 0\ny : ℕ\nhy₁ : 0 ∈ eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy₂ : ∀ {m : ℕ}, m < y → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) ∧ ¬a = 0\nh₂ : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\n⊢ y + 1 + (unpair n).snd = y + ((unpair n).snd + 1)",
"tactic": "simp [add_comm, add_left_comm]"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.intro.zero\nk : ℕ\nc : Code\nn✝ x : ℕ\nh : x ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : ¬m = 0\ny : ℕ\nhy₁ : 0 ∈ eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy₂ : ∀ {m : ℕ}, m < y → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) ∧ ¬a = 0\nh₂ : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\nim : Nat.zero < y + 1\n⊢ ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (Nat.zero + (unpair n).snd)) ∧ ¬a = 0",
"tactic": "exact ⟨m, by simpa using hf _ _ h₁, m0⟩"
},
{
"state_after": "no goals",
"state_before": "k : ℕ\nc : Code\nn✝ x : ℕ\nh : x ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : ¬m = 0\ny : ℕ\nhy₁ : 0 ∈ eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy₂ : ∀ {m : ℕ}, m < y → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) ∧ ¬a = 0\nh₂ : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\nim : Nat.zero < y + 1\n⊢ m ∈ eval cf (Nat.pair (unpair n).fst (Nat.zero + (unpair n).snd))",
"tactic": "simpa using hf _ _ h₁"
},
{
"state_after": "case neg.intro.intro.intro.succ.intro.intro\nk : ℕ\nc : Code\nn✝ x : ℕ\nh : x ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : ¬m = 0\ny : ℕ\nhy₁ : 0 ∈ eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy₂ : ∀ {m : ℕ}, m < y → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) ∧ ¬a = 0\nh₂ : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\ni : ℕ\nim : Nat.succ i < y + 1\nz : ℕ\nhz : z ∈ eval cf (Nat.pair (unpair n).fst (i + ((unpair n).snd + 1)))\nz0 : ¬z = 0\n⊢ ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (Nat.succ i + (unpair n).snd)) ∧ ¬a = 0",
"state_before": "case neg.intro.intro.intro.succ\nk : ℕ\nc : Code\nn✝ x : ℕ\nh : x ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : ¬m = 0\ny : ℕ\nhy₁ : 0 ∈ eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy₂ : ∀ {m : ℕ}, m < y → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) ∧ ¬a = 0\nh₂ : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\ni : ℕ\nim : Nat.succ i < y + 1\n⊢ ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (Nat.succ i + (unpair n).snd)) ∧ ¬a = 0",
"tactic": "rcases hy₂ (Nat.lt_of_succ_lt_succ im) with ⟨z, hz, z0⟩"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.intro.succ.intro.intro\nk : ℕ\nc : Code\nn✝ x : ℕ\nh : x ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : ¬m = 0\ny : ℕ\nhy₁ : 0 ∈ eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy₂ : ∀ {m : ℕ}, m < y → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) ∧ ¬a = 0\nh₂ : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\ni : ℕ\nim : Nat.succ i < y + 1\nz : ℕ\nhz : z ∈ eval cf (Nat.pair (unpair n).fst (i + ((unpair n).snd + 1)))\nz0 : ¬z = 0\n⊢ ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (Nat.succ i + (unpair n).snd)) ∧ ¬a = 0",
"tactic": "exact ⟨z, by simpa [Nat.succ_eq_add_one, add_comm, add_left_comm] using hz, z0⟩"
},
{
"state_after": "no goals",
"state_before": "k : ℕ\nc : Code\nn✝ x : ℕ\nh : x ∈ evaln (k + 1) c n✝\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ eval cf n\nn : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : ¬m = 0\ny : ℕ\nhy₁ : 0 ∈ eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy₂ : ∀ {m : ℕ}, m < y → ∃ a, a ∈ eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) ∧ ¬a = 0\nh₂ : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\ni : ℕ\nim : Nat.succ i < y + 1\nz : ℕ\nhz : z ∈ eval cf (Nat.pair (unpair n).fst (i + ((unpair n).snd + 1)))\nz0 : ¬z = 0\n⊢ z ∈ eval cf (Nat.pair (unpair n).fst (Nat.succ i + (unpair n).snd))",
"tactic": "simpa [Nat.succ_eq_add_one, add_comm, add_left_comm] using hz"
}
] |
[
857,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
820,
1
] |
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
|
intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_ge
|
[] |
[
390,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
383,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.coequalizerComparison_map_desc
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nX Y : C\nf g : X ⟶ Y\nD : Type u₂\ninst✝² : Category D\nG : C ⥤ D\ninst✝¹ : HasCoequalizer f g\ninst✝ : HasCoequalizer (G.map f) (G.map g)\nZ : C\nh : Y ⟶ Z\nw : f ≫ h = g ≫ h\n⊢ G.map f ≫ G.map h = G.map g ≫ G.map h",
"tactic": "simp only [← G.map_comp, w]"
},
{
"state_after": "case h\nC : Type u\ninst✝³ : Category C\nX Y : C\nf g : X ⟶ Y\nD : Type u₂\ninst✝² : Category D\nG : C ⥤ D\ninst✝¹ : HasCoequalizer f g\ninst✝ : HasCoequalizer (G.map f) (G.map g)\nZ : C\nh : Y ⟶ Z\nw : f ≫ h = g ≫ h\n⊢ coequalizer.π (G.map f) (G.map g) ≫ coequalizerComparison f g G ≫ G.map (coequalizer.desc h w) =\n coequalizer.π (G.map f) (G.map g) ≫ coequalizer.desc (G.map h) (_ : G.map f ≫ G.map h = G.map g ≫ G.map h)",
"state_before": "C : Type u\ninst✝³ : Category C\nX Y : C\nf g : X ⟶ Y\nD : Type u₂\ninst✝² : Category D\nG : C ⥤ D\ninst✝¹ : HasCoequalizer f g\ninst✝ : HasCoequalizer (G.map f) (G.map g)\nZ : C\nh : Y ⟶ Z\nw : f ≫ h = g ≫ h\n⊢ coequalizerComparison f g G ≫ G.map (coequalizer.desc h w) =\n coequalizer.desc (G.map h) (_ : G.map f ≫ G.map h = G.map g ≫ G.map h)",
"tactic": "apply coequalizer.hom_ext"
},
{
"state_after": "no goals",
"state_before": "case h\nC : Type u\ninst✝³ : Category C\nX Y : C\nf g : X ⟶ Y\nD : Type u₂\ninst✝² : Category D\nG : C ⥤ D\ninst✝¹ : HasCoequalizer f g\ninst✝ : HasCoequalizer (G.map f) (G.map g)\nZ : C\nh : Y ⟶ Z\nw : f ≫ h = g ≫ h\n⊢ coequalizer.π (G.map f) (G.map g) ≫ coequalizerComparison f g G ≫ G.map (coequalizer.desc h w) =\n coequalizer.π (G.map f) (G.map g) ≫ coequalizer.desc (G.map h) (_ : G.map f ≫ G.map h = G.map g ≫ G.map h)",
"tactic": "simp [← G.map_comp]"
}
] |
[
1158,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1153,
1
] |
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
|
Seminorm.continuous_of_continuous_comp
|
[
{
"state_after": "𝕜 : Type ?u.514785\n𝕜₂ : Type ?u.514788\n𝕝 : Type u_5\n𝕝₂ : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type ?u.514803\nι : Type ?u.514806\nι' : Type u_3\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : AddCommGroup E\ninst✝¹⁵ : Module 𝕜 E\ninst✝¹⁴ : NormedField 𝕝\ninst✝¹³ : Module 𝕝 E\ninst✝¹² : NontriviallyNormedField 𝕜₂\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : Module 𝕜₂ F\ninst✝⁹ : NormedField 𝕝₂\ninst✝⁸ : Module 𝕝₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁷ : RingHomIsometric σ₁₂\nτ₁₂ : 𝕝 →+* 𝕝₂\ninst✝⁶ : RingHomIsometric τ₁₂\ninst✝⁵ : Nonempty ι\ninst✝⁴ : Nonempty ι'\nq : SeminormFamily 𝕝₂ F ι'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalAddGroup E\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\nhq : WithSeminorms q\nf : E →ₛₗ[τ₁₂] F\nhf : ∀ (i : ι'), Continuous ↑(comp (q i) f)\n⊢ ContinuousAt (↑f) 0",
"state_before": "𝕜 : Type ?u.514785\n𝕜₂ : Type ?u.514788\n𝕝 : Type u_5\n𝕝₂ : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type ?u.514803\nι : Type ?u.514806\nι' : Type u_3\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : AddCommGroup E\ninst✝¹⁵ : Module 𝕜 E\ninst✝¹⁴ : NormedField 𝕝\ninst✝¹³ : Module 𝕝 E\ninst✝¹² : NontriviallyNormedField 𝕜₂\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : Module 𝕜₂ F\ninst✝⁹ : NormedField 𝕝₂\ninst✝⁸ : Module 𝕝₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁷ : RingHomIsometric σ₁₂\nτ₁₂ : 𝕝 →+* 𝕝₂\ninst✝⁶ : RingHomIsometric τ₁₂\ninst✝⁵ : Nonempty ι\ninst✝⁴ : Nonempty ι'\nq : SeminormFamily 𝕝₂ F ι'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalAddGroup E\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\nhq : WithSeminorms q\nf : E →ₛₗ[τ₁₂] F\nhf : ∀ (i : ι'), Continuous ↑(comp (q i) f)\n⊢ Continuous ↑f",
"tactic": "refine' continuous_of_continuousAt_zero f _"
},
{
"state_after": "𝕜 : Type ?u.514785\n𝕜₂ : Type ?u.514788\n𝕝 : Type u_5\n𝕝₂ : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type ?u.514803\nι : Type ?u.514806\nι' : Type u_3\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : AddCommGroup E\ninst✝¹⁵ : Module 𝕜 E\ninst✝¹⁴ : NormedField 𝕝\ninst✝¹³ : Module 𝕝 E\ninst✝¹² : NontriviallyNormedField 𝕜₂\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : Module 𝕜₂ F\ninst✝⁹ : NormedField 𝕝₂\ninst✝⁸ : Module 𝕝₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁷ : RingHomIsometric σ₁₂\nτ₁₂ : 𝕝 →+* 𝕝₂\ninst✝⁶ : RingHomIsometric τ₁₂\ninst✝⁵ : Nonempty ι\ninst✝⁴ : Nonempty ι'\nq : SeminormFamily 𝕝₂ F ι'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalAddGroup E\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\nhq : WithSeminorms q\nf : E →ₛₗ[τ₁₂] F\nhf : ∀ (i : ι'), Continuous ↑(comp (q i) f)\n⊢ ∀ (i : ι'), Filter.Tendsto (↑(q i) ∘ ↑f) (𝓝 0) (𝓝 0)",
"state_before": "𝕜 : Type ?u.514785\n𝕜₂ : Type ?u.514788\n𝕝 : Type u_5\n𝕝₂ : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type ?u.514803\nι : Type ?u.514806\nι' : Type u_3\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : AddCommGroup E\ninst✝¹⁵ : Module 𝕜 E\ninst✝¹⁴ : NormedField 𝕝\ninst✝¹³ : Module 𝕝 E\ninst✝¹² : NontriviallyNormedField 𝕜₂\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : Module 𝕜₂ F\ninst✝⁹ : NormedField 𝕝₂\ninst✝⁸ : Module 𝕝₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁷ : RingHomIsometric σ₁₂\nτ₁₂ : 𝕝 →+* 𝕝₂\ninst✝⁶ : RingHomIsometric τ₁₂\ninst✝⁵ : Nonempty ι\ninst✝⁴ : Nonempty ι'\nq : SeminormFamily 𝕝₂ F ι'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalAddGroup E\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\nhq : WithSeminorms q\nf : E →ₛₗ[τ₁₂] F\nhf : ∀ (i : ι'), Continuous ↑(comp (q i) f)\n⊢ ContinuousAt (↑f) 0",
"tactic": "simp_rw [ContinuousAt, f.map_zero, q.withSeminorms_iff_nhds_eq_iInf.mp hq, Filter.tendsto_iInf,\n Filter.tendsto_comap_iff]"
},
{
"state_after": "𝕜 : Type ?u.514785\n𝕜₂ : Type ?u.514788\n𝕝 : Type u_5\n𝕝₂ : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type ?u.514803\nι : Type ?u.514806\nι' : Type u_3\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : AddCommGroup E\ninst✝¹⁵ : Module 𝕜 E\ninst✝¹⁴ : NormedField 𝕝\ninst✝¹³ : Module 𝕝 E\ninst✝¹² : NontriviallyNormedField 𝕜₂\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : Module 𝕜₂ F\ninst✝⁹ : NormedField 𝕝₂\ninst✝⁸ : Module 𝕝₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁷ : RingHomIsometric σ₁₂\nτ₁₂ : 𝕝 →+* 𝕝₂\ninst✝⁶ : RingHomIsometric τ₁₂\ninst✝⁵ : Nonempty ι\ninst✝⁴ : Nonempty ι'\nq : SeminormFamily 𝕝₂ F ι'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalAddGroup E\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\nhq : WithSeminorms q\nf : E →ₛₗ[τ₁₂] F\nhf : ∀ (i : ι'), Continuous ↑(comp (q i) f)\ni : ι'\n⊢ Filter.Tendsto (↑(q i) ∘ ↑f) (𝓝 0) (𝓝 0)",
"state_before": "𝕜 : Type ?u.514785\n𝕜₂ : Type ?u.514788\n𝕝 : Type u_5\n𝕝₂ : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type ?u.514803\nι : Type ?u.514806\nι' : Type u_3\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : AddCommGroup E\ninst✝¹⁵ : Module 𝕜 E\ninst✝¹⁴ : NormedField 𝕝\ninst✝¹³ : Module 𝕝 E\ninst✝¹² : NontriviallyNormedField 𝕜₂\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : Module 𝕜₂ F\ninst✝⁹ : NormedField 𝕝₂\ninst✝⁸ : Module 𝕝₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁷ : RingHomIsometric σ₁₂\nτ₁₂ : 𝕝 →+* 𝕝₂\ninst✝⁶ : RingHomIsometric τ₁₂\ninst✝⁵ : Nonempty ι\ninst✝⁴ : Nonempty ι'\nq : SeminormFamily 𝕝₂ F ι'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalAddGroup E\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\nhq : WithSeminorms q\nf : E →ₛₗ[τ₁₂] F\nhf : ∀ (i : ι'), Continuous ↑(comp (q i) f)\n⊢ ∀ (i : ι'), Filter.Tendsto (↑(q i) ∘ ↑f) (𝓝 0) (𝓝 0)",
"tactic": "intro i"
},
{
"state_after": "case h.e'_5.h.e'_3\n𝕜 : Type ?u.514785\n𝕜₂ : Type ?u.514788\n𝕝 : Type u_5\n𝕝₂ : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type ?u.514803\nι : Type ?u.514806\nι' : Type u_3\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : AddCommGroup E\ninst✝¹⁵ : Module 𝕜 E\ninst✝¹⁴ : NormedField 𝕝\ninst✝¹³ : Module 𝕝 E\ninst✝¹² : NontriviallyNormedField 𝕜₂\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : Module 𝕜₂ F\ninst✝⁹ : NormedField 𝕝₂\ninst✝⁸ : Module 𝕝₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁷ : RingHomIsometric σ₁₂\nτ₁₂ : 𝕝 →+* 𝕝₂\ninst✝⁶ : RingHomIsometric τ₁₂\ninst✝⁵ : Nonempty ι\ninst✝⁴ : Nonempty ι'\nq : SeminormFamily 𝕝₂ F ι'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalAddGroup E\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\nhq : WithSeminorms q\nf : E →ₛₗ[τ₁₂] F\nhf : ∀ (i : ι'), Continuous ↑(comp (q i) f)\ni : ι'\n⊢ 0 = ↑(comp (q i) f) 0",
"state_before": "𝕜 : Type ?u.514785\n𝕜₂ : Type ?u.514788\n𝕝 : Type u_5\n𝕝₂ : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type ?u.514803\nι : Type ?u.514806\nι' : Type u_3\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : AddCommGroup E\ninst✝¹⁵ : Module 𝕜 E\ninst✝¹⁴ : NormedField 𝕝\ninst✝¹³ : Module 𝕝 E\ninst✝¹² : NontriviallyNormedField 𝕜₂\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : Module 𝕜₂ F\ninst✝⁹ : NormedField 𝕝₂\ninst✝⁸ : Module 𝕝₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁷ : RingHomIsometric σ₁₂\nτ₁₂ : 𝕝 →+* 𝕝₂\ninst✝⁶ : RingHomIsometric τ₁₂\ninst✝⁵ : Nonempty ι\ninst✝⁴ : Nonempty ι'\nq : SeminormFamily 𝕝₂ F ι'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalAddGroup E\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\nhq : WithSeminorms q\nf : E →ₛₗ[τ₁₂] F\nhf : ∀ (i : ι'), Continuous ↑(comp (q i) f)\ni : ι'\n⊢ Filter.Tendsto (↑(q i) ∘ ↑f) (𝓝 0) (𝓝 0)",
"tactic": "convert (hf i).continuousAt.tendsto"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h.e'_3\n𝕜 : Type ?u.514785\n𝕜₂ : Type ?u.514788\n𝕝 : Type u_5\n𝕝₂ : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type ?u.514803\nι : Type ?u.514806\nι' : Type u_3\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : AddCommGroup E\ninst✝¹⁵ : Module 𝕜 E\ninst✝¹⁴ : NormedField 𝕝\ninst✝¹³ : Module 𝕝 E\ninst✝¹² : NontriviallyNormedField 𝕜₂\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : Module 𝕜₂ F\ninst✝⁹ : NormedField 𝕝₂\ninst✝⁸ : Module 𝕝₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁷ : RingHomIsometric σ₁₂\nτ₁₂ : 𝕝 →+* 𝕝₂\ninst✝⁶ : RingHomIsometric τ₁₂\ninst✝⁵ : Nonempty ι\ninst✝⁴ : Nonempty ι'\nq : SeminormFamily 𝕝₂ F ι'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalAddGroup E\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\nhq : WithSeminorms q\nf : E →ₛₗ[τ₁₂] F\nhf : ∀ (i : ι'), Continuous ↑(comp (q i) f)\ni : ι'\n⊢ 0 = ↑(comp (q i) f) 0",
"tactic": "exact (map_zero _).symm"
}
] |
[
616,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
608,
1
] |
Mathlib/GroupTheory/OrderOfElement.lean
|
orderOf_submonoid
|
[] |
[
341,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
340,
1
] |
Mathlib/RingTheory/Subring/Pointwise.lean
|
Subring.smul_mem_pointwise_smul
|
[] |
[
78,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.iSup_restrict_spanningSets
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.731158\nγ : Type ?u.731161\nδ : Type ?u.731164\nι : Type ?u.731167\nR : Type ?u.731170\nR' : Type ?u.731173\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\ninst✝ : SigmaFinite μ\nhs : MeasurableSet s\n⊢ ↑↑(restrict μ (⋃ (i : ℕ), spanningSets μ i)) s = ↑↑μ s",
"tactic": "rw [iUnion_spanningSets, restrict_univ]"
}
] |
[
3537,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3532,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.zero_smul_set_subset
|
[] |
[
815,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
814,
1
] |
Mathlib/Data/Multiset/Functor.lean
|
Multiset.map_comp_coe
|
[
{
"state_after": "case h\nF : Type u → Type u\ninst✝¹ : Applicative F\ninst✝ : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nα β : Type u_1\nh : α → β\nx✝ : List α\n⊢ (Functor.map h ∘ Coe.coe) x✝ = (Coe.coe ∘ Functor.map h) x✝",
"state_before": "F : Type u → Type u\ninst✝¹ : Applicative F\ninst✝ : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nα β : Type u_1\nh : α → β\n⊢ Functor.map h ∘ Coe.coe = Coe.coe ∘ Functor.map h",
"tactic": "funext"
},
{
"state_after": "no goals",
"state_before": "case h\nF : Type u → Type u\ninst✝¹ : Applicative F\ninst✝ : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nα β : Type u_1\nh : α → β\nx✝ : List α\n⊢ (Functor.map h ∘ Coe.coe) x✝ = (Coe.coe ∘ Functor.map h) x✝",
"tactic": "simp only [Function.comp_apply, Coe.coe, fmap_def, coe_map, List.map_eq_map]"
}
] |
[
100,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.flatten_equiv
|
[
{
"state_after": "case h1\nα : Type u\nβ : Type v\nγ : Type w\nc : Computation (WSeq α)\ns : WSeq α\nh : s ∈ c\n⊢ flatten (Computation.pure s) ~ʷ s\n\ncase h2\nα : Type u\nβ : Type v\nγ : Type w\nc : Computation (WSeq α)\ns : WSeq α\nh : s ∈ c\n⊢ ∀ (s_1 : Computation (WSeq α)), flatten s_1 ~ʷ s → flatten (Computation.think s_1) ~ʷ s",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nc : Computation (WSeq α)\ns : WSeq α\nh : s ∈ c\n⊢ flatten c ~ʷ s",
"tactic": "apply Computation.memRecOn h"
},
{
"state_after": "no goals",
"state_before": "case h1\nα : Type u\nβ : Type v\nγ : Type w\nc : Computation (WSeq α)\ns : WSeq α\nh : s ∈ c\n⊢ flatten (Computation.pure s) ~ʷ s",
"tactic": "simp [Equiv.refl]"
},
{
"state_after": "case h2\nα : Type u\nβ : Type v\nγ : Type w\nc : Computation (WSeq α)\ns : WSeq α\nh : s ∈ c\ns' : Computation (WSeq α)\n⊢ flatten s' ~ʷ s → flatten (Computation.think s') ~ʷ s",
"state_before": "case h2\nα : Type u\nβ : Type v\nγ : Type w\nc : Computation (WSeq α)\ns : WSeq α\nh : s ∈ c\n⊢ ∀ (s_1 : Computation (WSeq α)), flatten s_1 ~ʷ s → flatten (Computation.think s_1) ~ʷ s",
"tactic": "intro s'"
},
{
"state_after": "case h2.a\nα : Type u\nβ : Type v\nγ : Type w\nc : Computation (WSeq α)\ns : WSeq α\nh : s ∈ c\ns' : Computation (WSeq α)\n⊢ flatten (Computation.think s') ~ʷ flatten s'",
"state_before": "case h2\nα : Type u\nβ : Type v\nγ : Type w\nc : Computation (WSeq α)\ns : WSeq α\nh : s ∈ c\ns' : Computation (WSeq α)\n⊢ flatten s' ~ʷ s → flatten (Computation.think s') ~ʷ s",
"tactic": "apply Equiv.trans"
},
{
"state_after": "no goals",
"state_before": "case h2.a\nα : Type u\nβ : Type v\nγ : Type w\nc : Computation (WSeq α)\ns : WSeq α\nh : s ∈ c\ns' : Computation (WSeq α)\n⊢ flatten (Computation.think s') ~ʷ flatten s'",
"tactic": "simp [think_equiv]"
}
] |
[
1153,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1148,
1
] |
Mathlib/AlgebraicTopology/ExtraDegeneracy.lean
|
CategoryTheory.Arrow.AugmentedCechNerve.ExtraDegeneracy.s_comp_base
|
[
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝¹ : Category C\nf : Arrow C\ninst✝ : ∀ (n : ℕ), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nS : SplitEpi f.hom\nn : ℕ\n⊢ (s f S n ≫ WidePullback.base fun x => f.hom) = WidePullback.base fun x => f.hom",
"tactic": "apply WidePullback.lift_base"
}
] |
[
302,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
300,
1
] |
Mathlib/Computability/RegularExpressions.lean
|
RegularExpression.one_rmatch_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.33505\nγ : Type ?u.33508\ndec : DecidableEq α\na b : α\nx : List α\n⊢ rmatch 1 x = true ↔ x = []",
"tactic": "induction x <;> simp [rmatch, matchEpsilon, *]"
}
] |
[
224,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/Data/Real/Sqrt.lean
|
Real.sqrt_div
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : 0 ≤ x\ny : ℝ\n⊢ sqrt (x / y) = sqrt x / sqrt y",
"tactic": "rw [division_def, sqrt_mul hx, sqrt_inv, division_def]"
}
] |
[
411,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
410,
1
] |
Mathlib/Topology/Basic.lean
|
topologicalSpace_eq
|
[] |
[
115,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Analysis/Normed/Group/Quotient.lean
|
quotient_norm_neg
|
[
{
"state_after": "M : Type u_1\nN : Type ?u.81421\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M ⧸ S\n⊢ sInf (norm '' {m | ↑m = -x}) = sInf (norm '' {m | ↑m = x})",
"state_before": "M : Type u_1\nN : Type ?u.81421\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M ⧸ S\n⊢ ‖-x‖ = ‖x‖",
"tactic": "simp only [AddSubgroup.quotient_norm_eq]"
},
{
"state_after": "case e_a.h\nM : Type u_1\nN : Type ?u.81421\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M ⧸ S\nr : ℝ\n⊢ r ∈ norm '' {m | ↑m = -x} ↔ r ∈ norm '' {m | ↑m = x}",
"state_before": "M : Type u_1\nN : Type ?u.81421\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M ⧸ S\n⊢ sInf (norm '' {m | ↑m = -x}) = sInf (norm '' {m | ↑m = x})",
"tactic": "congr 1 with r"
},
{
"state_after": "no goals",
"state_before": "case e_a.h\nM : Type u_1\nN : Type ?u.81421\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M ⧸ S\nr : ℝ\n⊢ r ∈ norm '' {m | ↑m = -x} ↔ r ∈ norm '' {m | ↑m = x}",
"tactic": "constructor <;> { rintro ⟨m, hm, rfl⟩; use -m; simpa [neg_eq_iff_eq_neg] using hm }"
}
] |
[
146,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Algebra/Order/Field/Power.lean
|
Odd.zpow_pos_iff
|
[
{
"state_after": "case intro\nα : Type u_1\ninst✝ : LinearOrderedField α\na b c d : α\nn k : ℤ\nhk : n = 2 * k + 1\n⊢ 0 < a ^ n ↔ 0 < a",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\nhn : Odd n\n⊢ 0 < a ^ n ↔ 0 < a",
"tactic": "cases' hn with k hk"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\ninst✝ : LinearOrderedField α\na b c d : α\nn k : ℤ\nhk : n = 2 * k + 1\n⊢ 0 < a ^ n ↔ 0 < a",
"tactic": "simpa only [hk, two_mul] using zpow_bit1_pos_iff"
}
] |
[
202,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
201,
1
] |
Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
|
MeasureTheory.Measure.singularPart_add
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.53774\nm : MeasurableSpace α\nμ ν✝ μ₁ μ₂ ν : Measure α\ninst✝¹ : HaveLebesgueDecomposition μ₁ ν\ninst✝ : HaveLebesgueDecomposition μ₂ ν\n⊢ μ₁ + μ₂ = singularPart μ₁ ν + singularPart μ₂ ν + withDensity ν fun a => rnDeriv μ₁ ν a + rnDeriv μ₂ ν a",
"state_before": "α : Type u_1\nβ : Type ?u.53774\nm : MeasurableSpace α\nμ ν✝ μ₁ μ₂ ν : Measure α\ninst✝¹ : HaveLebesgueDecomposition μ₁ ν\ninst✝ : HaveLebesgueDecomposition μ₂ ν\n⊢ singularPart (μ₁ + μ₂) ν = singularPart μ₁ ν + singularPart μ₂ ν",
"tactic": "refine'\n (eq_singularPart ((measurable_rnDeriv μ₁ ν).add (measurable_rnDeriv μ₂ ν))\n ((haveLebesgueDecomposition_spec _ _).2.1.add_left\n (haveLebesgueDecomposition_spec _ _).2.1)\n _).symm"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.53774\nm : MeasurableSpace α\nμ ν✝ μ₁ μ₂ ν : Measure α\ninst✝¹ : HaveLebesgueDecomposition μ₁ ν\ninst✝ : HaveLebesgueDecomposition μ₂ ν\n⊢ μ₁ + μ₂ =\n singularPart μ₁ ν + singularPart μ₂ ν + (withDensity ν (rnDeriv μ₁ ν) + withDensity ν fun a => rnDeriv μ₂ ν a)",
"state_before": "α : Type u_1\nβ : Type ?u.53774\nm : MeasurableSpace α\nμ ν✝ μ₁ μ₂ ν : Measure α\ninst✝¹ : HaveLebesgueDecomposition μ₁ ν\ninst✝ : HaveLebesgueDecomposition μ₂ ν\n⊢ μ₁ + μ₂ = singularPart μ₁ ν + singularPart μ₂ ν + withDensity ν fun a => rnDeriv μ₁ ν a + rnDeriv μ₂ ν a",
"tactic": "erw [withDensity_add_left (measurable_rnDeriv μ₁ ν)]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.53774\nm : MeasurableSpace α\nμ ν✝ μ₁ μ₂ ν : Measure α\ninst✝¹ : HaveLebesgueDecomposition μ₁ ν\ninst✝ : HaveLebesgueDecomposition μ₂ ν\n⊢ μ₁ + μ₂ =\n (singularPart μ₁ ν + withDensity ν (rnDeriv μ₁ ν) + withDensity ν fun a => rnDeriv μ₂ ν a) + singularPart μ₂ ν",
"state_before": "α : Type u_1\nβ : Type ?u.53774\nm : MeasurableSpace α\nμ ν✝ μ₁ μ₂ ν : Measure α\ninst✝¹ : HaveLebesgueDecomposition μ₁ ν\ninst✝ : HaveLebesgueDecomposition μ₂ ν\n⊢ μ₁ + μ₂ =\n singularPart μ₁ ν + singularPart μ₂ ν + (withDensity ν (rnDeriv μ₁ ν) + withDensity ν fun a => rnDeriv μ₂ ν a)",
"tactic": "conv_rhs => rw [add_assoc, add_comm (μ₂.singularPart ν), ← add_assoc, ← add_assoc]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.53774\nm : MeasurableSpace α\nμ ν✝ μ₁ μ₂ ν : Measure α\ninst✝¹ : HaveLebesgueDecomposition μ₁ ν\ninst✝ : HaveLebesgueDecomposition μ₂ ν\n⊢ μ₁ + μ₂ =\n (singularPart μ₁ ν + withDensity ν (rnDeriv μ₁ ν) + withDensity ν fun a => rnDeriv μ₂ ν a) + singularPart μ₂ ν",
"tactic": "rw [← haveLebesgueDecomposition_add μ₁ ν, add_assoc, add_comm (ν.withDensity (μ₂.rnDeriv ν)),\n ← haveLebesgueDecomposition_add μ₂ ν]"
}
] |
[
299,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
288,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
LocalHomeomorph.extend_source_mem_nhds
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_1\nH : Type u_2\nE' : Type ?u.134059\nM' : Type ?u.134062\nH' : Type ?u.134065\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx✝ : M\ns t : Set M\nx : M\nh : x ∈ f.source\n⊢ x ∈ (extend f I).source",
"tactic": "rwa [f.extend_source I]"
}
] |
[
819,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
818,
1
] |
Mathlib/Analysis/SpecialFunctions/Integrals.lean
|
integral_one_div_of_neg
|
[
{
"state_after": "no goals",
"state_before": "a b : ℝ\nn : ℕ\nha : a < 0\nhb : b < 0\n⊢ (∫ (x : ℝ) in a..b, 1 / x) = Real.log (b / a)",
"tactic": "simp only [one_div, integral_inv_of_neg ha hb]"
}
] |
[
452,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
451,
1
] |
Mathlib/Topology/Compactification/OnePoint.lean
|
OnePoint.continuousAt_coe
|
[
{
"state_after": "X : Type u_2\ninst✝¹ : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nY : Type u_1\ninst✝ : TopologicalSpace Y\nf : OnePoint X → Y\nx : X\n⊢ Tendsto (f ∘ some) (𝓝 x) (𝓝 (f ↑x)) ↔ Tendsto (f ∘ some) (𝓝 x) (𝓝 ((f ∘ some) x))",
"state_before": "X : Type u_2\ninst✝¹ : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nY : Type u_1\ninst✝ : TopologicalSpace Y\nf : OnePoint X → Y\nx : X\n⊢ ContinuousAt f ↑x ↔ ContinuousAt (f ∘ some) x",
"tactic": "rw [ContinuousAt, nhds_coe_eq, tendsto_map'_iff, ContinuousAt]"
},
{
"state_after": "no goals",
"state_before": "X : Type u_2\ninst✝¹ : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nY : Type u_1\ninst✝ : TopologicalSpace Y\nf : OnePoint X → Y\nx : X\n⊢ Tendsto (f ∘ some) (𝓝 x) (𝓝 (f ↑x)) ↔ Tendsto (f ∘ some) (𝓝 x) (𝓝 ((f ∘ some) x))",
"tactic": "rfl"
}
] |
[
386,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
384,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
le_compl_compl
|
[] |
[
899,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
898,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.fromBlocks_conjTranspose
|
[
{
"state_after": "no goals",
"state_before": "l : Type u_3\nm : Type u_4\nn : Type u_2\no : Type u_5\np : Type ?u.17851\nq : Type ?u.17854\nm' : o → Type ?u.17859\nn' : o → Type ?u.17864\np' : o → Type ?u.17869\nR : Type ?u.17872\nS : Type ?u.17875\nα : Type u_1\nβ : Type ?u.17881\ninst✝ : Star α\nA : Matrix n l α\nB : Matrix n m α\nC : Matrix o l α\nD : Matrix o m α\n⊢ (fromBlocks A B C D)ᴴ = fromBlocks Aᴴ Cᴴ Bᴴ Dᴴ",
"tactic": "simp only [conjTranspose, fromBlocks_transpose, fromBlocks_map]"
}
] |
[
160,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Equiv.sumArrowEquivProdArrow_apply_snd
|
[] |
[
886,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
884,
1
] |
Mathlib/Topology/Order/Basic.lean
|
Ioo_mem_nhds
|
[] |
[
345,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
344,
1
] |
Mathlib/Order/Hom/Basic.lean
|
OrderIso.symm_injective
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.81630\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.81639\nδ : Type ?u.81642\ninst✝² : LE α\ninst✝¹ : LE β\ninst✝ : LE γ\ne e' : α ≃o β\nh : symm e = symm e'\n⊢ e = e'",
"tactic": "rw [← e.symm_symm, h, e'.symm_symm]"
}
] |
[
885,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
884,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.snorm_sub_le
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.3047771\nG : Type ?u.3047774\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf g : α → E\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nhp : 1 ≤ p\n⊢ snorm (f - g) p μ ≤ snorm f p μ + snorm g p μ",
"tactic": "simpa [LpAddConst_of_one_le hp] using snorm_sub_le' hf hg p"
}
] |
[
877,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
875,
1
] |
Mathlib/Algebra/RingQuot.lean
|
RingQuot.neg_quot
|
[
{
"state_after": "R✝ : Type u₁\ninst✝⁴ : Semiring R✝\nS : Type u₂\ninst✝³ : CommSemiring S\nA : Type u₃\ninst✝² : Semiring A\ninst✝¹ : Algebra S A\nr✝ : R✝ → R✝ → Prop\nR : Type u₁\ninst✝ : Ring R\nr : R → R → Prop\na : R\n⊢ RingQuot.neg r { toQuot := Quot.mk (Rel r) a } = { toQuot := Quot.mk (Rel r) (-a) }",
"state_before": "R✝ : Type u₁\ninst✝⁴ : Semiring R✝\nS : Type u₂\ninst✝³ : CommSemiring S\nA : Type u₃\ninst✝² : Semiring A\ninst✝¹ : Algebra S A\nr✝ : R✝ → R✝ → Prop\nR : Type u₁\ninst✝ : Ring R\nr : R → R → Prop\na : R\n⊢ -{ toQuot := Quot.mk (Rel r) a } = { toQuot := Quot.mk (Rel r) (-a) }",
"tactic": "show neg r _ = _"
},
{
"state_after": "R✝ : Type u₁\ninst✝⁴ : Semiring R✝\nS : Type u₂\ninst✝³ : CommSemiring S\nA : Type u₃\ninst✝² : Semiring A\ninst✝¹ : Algebra S A\nr✝ : R✝ → R✝ → Prop\nR : Type u₁\ninst✝ : Ring R\nr : R → R → Prop\na : R\n⊢ (match { toQuot := Quot.mk (Rel r) a } with\n | { toQuot := a } => { toQuot := Quot.map (fun a => -a) (_ : ∀ ⦃a b : R⦄, Rel r a b → Rel r (-a) (-b)) a }) =\n { toQuot := Quot.mk (Rel r) (-a) }",
"state_before": "R✝ : Type u₁\ninst✝⁴ : Semiring R✝\nS : Type u₂\ninst✝³ : CommSemiring S\nA : Type u₃\ninst✝² : Semiring A\ninst✝¹ : Algebra S A\nr✝ : R✝ → R✝ → Prop\nR : Type u₁\ninst✝ : Ring R\nr : R → R → Prop\na : R\n⊢ RingQuot.neg r { toQuot := Quot.mk (Rel r) a } = { toQuot := Quot.mk (Rel r) (-a) }",
"tactic": "rw [neg_def]"
},
{
"state_after": "no goals",
"state_before": "R✝ : Type u₁\ninst✝⁴ : Semiring R✝\nS : Type u₂\ninst✝³ : CommSemiring S\nA : Type u₃\ninst✝² : Semiring A\ninst✝¹ : Algebra S A\nr✝ : R✝ → R✝ → Prop\nR : Type u₁\ninst✝ : Ring R\nr : R → R → Prop\na : R\n⊢ (match { toQuot := Quot.mk (Rel r) a } with\n | { toQuot := a } => { toQuot := Quot.map (fun a => -a) (_ : ∀ ⦃a b : R⦄, Rel r a b → Rel r (-a) (-b)) a }) =\n { toQuot := Quot.mk (Rel r) (-a) }",
"tactic": "rfl"
}
] |
[
257,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/Data/Finsupp/Fin.lean
|
Finsupp.cons_zero
|
[] |
[
48,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
1
] |
Mathlib/Analysis/Convex/Cone/Basic.lean
|
Convex.toCone_isLeast
|
[
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.176423\nG : Type ?u.176426\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nhs : Convex 𝕜 s\nx✝ : E\nt : ConvexCone 𝕜 E\nht : t ∈ {t | s ⊆ ↑t}\nx : E\nhx : x ∈ toCone s hs\n⊢ x ∈ t",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.176423\nG : Type ?u.176426\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nhs : Convex 𝕜 s\nx : E\n⊢ IsLeast {t | s ⊆ ↑t} (toCone s hs)",
"tactic": "refine' ⟨hs.subset_toCone, fun t ht x hx => _⟩"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.176423\nG : Type ?u.176426\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nhs : Convex 𝕜 s\nx : E\nt : ConvexCone 𝕜 E\nht : t ∈ {t | s ⊆ ↑t}\nc : 𝕜\nhc : 0 < c\ny : E\nhy : y ∈ s\nhx : c • y ∈ toCone s hs\n⊢ c • y ∈ t",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.176423\nG : Type ?u.176426\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nhs : Convex 𝕜 s\nx✝ : E\nt : ConvexCone 𝕜 E\nht : t ∈ {t | s ⊆ ↑t}\nx : E\nhx : x ∈ toCone s hs\n⊢ x ∈ t",
"tactic": "rcases hs.mem_toCone.1 hx with ⟨c, hc, y, hy, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.176423\nG : Type ?u.176426\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nhs : Convex 𝕜 s\nx : E\nt : ConvexCone 𝕜 E\nht : t ∈ {t | s ⊆ ↑t}\nc : 𝕜\nhc : 0 < c\ny : E\nhy : y ∈ s\nhx : c • y ∈ toCone s hs\n⊢ c • y ∈ t",
"tactic": "exact t.smul_mem hc (ht hy)"
}
] |
[
680,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
677,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.dotProduct_one
|
[] |
[
736,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
736,
1
] |
Mathlib/Order/LiminfLimsup.lean
|
Filter.sdiff_liminf
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.176896\nι : Type ?u.176899\ninst✝ : CompleteBooleanAlgebra α\nf : Filter β\nu : β → α\na : α\n⊢ (a \\ liminf u f)ᶜ = limsup (fun b => a \\ u b) fᶜ",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.176896\nι : Type ?u.176899\ninst✝ : CompleteBooleanAlgebra α\nf : Filter β\nu : β → α\na : α\n⊢ a \\ liminf u f = limsup (fun b => a \\ u b) f",
"tactic": "rw [← compl_inj_iff]"
}
] |
[
1044,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1042,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
Finset.affineCombination_sdiff_sub
|
[
{
"state_after": "k : Type u_4\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.352836\ns₂✝ : Finset ι₂\ninst✝ : DecidableEq ι\ns₂ : Finset ι\nh : s₂ ⊆ s\nw : ι → k\np : ι → P\n⊢ ↑(weightedVSubOfPoint (s \\ s₂) p (Classical.choice (_ : Nonempty P))) w -\n ↑(weightedVSubOfPoint s₂ p (Classical.choice (_ : Nonempty P))) (-w) =\n ↑(weightedVSub s p) w",
"state_before": "k : Type u_4\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.352836\ns₂✝ : Finset ι₂\ninst✝ : DecidableEq ι\ns₂ : Finset ι\nh : s₂ ⊆ s\nw : ι → k\np : ι → P\n⊢ ↑(affineCombination k (s \\ s₂) p) w -ᵥ ↑(affineCombination k s₂ p) (-w) = ↑(weightedVSub s p) w",
"tactic": "simp_rw [affineCombination_apply, vadd_vsub_vadd_cancel_right]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_4\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.352836\ns₂✝ : Finset ι₂\ninst✝ : DecidableEq ι\ns₂ : Finset ι\nh : s₂ ⊆ s\nw : ι → k\np : ι → P\n⊢ ↑(weightedVSubOfPoint (s \\ s₂) p (Classical.choice (_ : Nonempty P))) w -\n ↑(weightedVSubOfPoint s₂ p (Classical.choice (_ : Nonempty P))) (-w) =\n ↑(weightedVSub s p) w",
"tactic": "exact s.weightedVSub_sdiff_sub h _ _"
}
] |
[
540,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
536,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Nat.one_le_floor_iff
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.29827\nα : Type u_1\nβ : Type ?u.29833\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nx : α\n⊢ 1 ≤ ⌊x⌋₊ ↔ 1 ≤ x",
"tactic": "exact_mod_cast @le_floor_iff' α _ _ x 1 one_ne_zero"
}
] |
[
206,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean
|
CategoryTheory.MonoidalCategory.leftUnitor_tensor_inv
|
[
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : MonoidalCategory C\nX Y : C\n⊢ (λ_ (X ⊗ Y)).inv = ((λ_ X).inv ⊗ 𝟙 Y) ≫ (α_ (𝟙_ C) X Y).hom",
"tactic": "coherence"
}
] |
[
45,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearEquiv.subsingleton_or_norm_symm_pos
|
[
{
"state_after": "case inl\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.3812785\nE : Type u_3\nEₗ : Type ?u.3812791\nF : Type u_4\nFₗ : Type ?u.3812797\nG : Type ?u.3812800\nGₗ : Type ?u.3812803\n𝓕 : Type ?u.3812806\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₁ : 𝕜₂ →+* 𝕜\ninst✝³ : RingHomInvPair σ₁₂ σ₂₁\ninst✝² : RingHomInvPair σ₂₁ σ₁₂\ninst✝¹ : RingHomIsometric σ₂₁\ninst✝ : RingHomIsometric σ₁₂\ne : E ≃SL[σ₁₂] F\n_i : Subsingleton E\n⊢ Subsingleton E ∨ 0 < ‖↑(ContinuousLinearEquiv.symm e)‖\n\ncase inr\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.3812785\nE : Type u_3\nEₗ : Type ?u.3812791\nF : Type u_4\nFₗ : Type ?u.3812797\nG : Type ?u.3812800\nGₗ : Type ?u.3812803\n𝓕 : Type ?u.3812806\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₁ : 𝕜₂ →+* 𝕜\ninst✝³ : RingHomInvPair σ₁₂ σ₂₁\ninst✝² : RingHomInvPair σ₂₁ σ₁₂\ninst✝¹ : RingHomIsometric σ₂₁\ninst✝ : RingHomIsometric σ₁₂\ne : E ≃SL[σ₁₂] F\n_i : Nontrivial E\n⊢ Subsingleton E ∨ 0 < ‖↑(ContinuousLinearEquiv.symm e)‖",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.3812785\nE : Type u_3\nEₗ : Type ?u.3812791\nF : Type u_4\nFₗ : Type ?u.3812797\nG : Type ?u.3812800\nGₗ : Type ?u.3812803\n𝓕 : Type ?u.3812806\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₁ : 𝕜₂ →+* 𝕜\ninst✝³ : RingHomInvPair σ₁₂ σ₂₁\ninst✝² : RingHomInvPair σ₂₁ σ₁₂\ninst✝¹ : RingHomIsometric σ₂₁\ninst✝ : RingHomIsometric σ₁₂\ne : E ≃SL[σ₁₂] F\n⊢ Subsingleton E ∨ 0 < ‖↑(ContinuousLinearEquiv.symm e)‖",
"tactic": "rcases subsingleton_or_nontrivial E with (_i | _i) <;> skip"
},
{
"state_after": "case inl.h\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.3812785\nE : Type u_3\nEₗ : Type ?u.3812791\nF : Type u_4\nFₗ : Type ?u.3812797\nG : Type ?u.3812800\nGₗ : Type ?u.3812803\n𝓕 : Type ?u.3812806\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₁ : 𝕜₂ →+* 𝕜\ninst✝³ : RingHomInvPair σ₁₂ σ₂₁\ninst✝² : RingHomInvPair σ₂₁ σ₁₂\ninst✝¹ : RingHomIsometric σ₂₁\ninst✝ : RingHomIsometric σ₁₂\ne : E ≃SL[σ₁₂] F\n_i : Subsingleton E\n⊢ Subsingleton E",
"state_before": "case inl\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.3812785\nE : Type u_3\nEₗ : Type ?u.3812791\nF : Type u_4\nFₗ : Type ?u.3812797\nG : Type ?u.3812800\nGₗ : Type ?u.3812803\n𝓕 : Type ?u.3812806\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₁ : 𝕜₂ →+* 𝕜\ninst✝³ : RingHomInvPair σ₁₂ σ₂₁\ninst✝² : RingHomInvPair σ₂₁ σ₁₂\ninst✝¹ : RingHomIsometric σ₂₁\ninst✝ : RingHomIsometric σ₁₂\ne : E ≃SL[σ₁₂] F\n_i : Subsingleton E\n⊢ Subsingleton E ∨ 0 < ‖↑(ContinuousLinearEquiv.symm e)‖",
"tactic": "left"
},
{
"state_after": "no goals",
"state_before": "case inl.h\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.3812785\nE : Type u_3\nEₗ : Type ?u.3812791\nF : Type u_4\nFₗ : Type ?u.3812797\nG : Type ?u.3812800\nGₗ : Type ?u.3812803\n𝓕 : Type ?u.3812806\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₁ : 𝕜₂ →+* 𝕜\ninst✝³ : RingHomInvPair σ₁₂ σ₂₁\ninst✝² : RingHomInvPair σ₂₁ σ₁₂\ninst✝¹ : RingHomIsometric σ₂₁\ninst✝ : RingHomIsometric σ₁₂\ne : E ≃SL[σ₁₂] F\n_i : Subsingleton E\n⊢ Subsingleton E",
"tactic": "infer_instance"
},
{
"state_after": "case inr.h\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.3812785\nE : Type u_3\nEₗ : Type ?u.3812791\nF : Type u_4\nFₗ : Type ?u.3812797\nG : Type ?u.3812800\nGₗ : Type ?u.3812803\n𝓕 : Type ?u.3812806\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₁ : 𝕜₂ →+* 𝕜\ninst✝³ : RingHomInvPair σ₁₂ σ₂₁\ninst✝² : RingHomInvPair σ₂₁ σ₁₂\ninst✝¹ : RingHomIsometric σ₂₁\ninst✝ : RingHomIsometric σ₁₂\ne : E ≃SL[σ₁₂] F\n_i : Nontrivial E\n⊢ 0 < ‖↑(ContinuousLinearEquiv.symm e)‖",
"state_before": "case inr\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.3812785\nE : Type u_3\nEₗ : Type ?u.3812791\nF : Type u_4\nFₗ : Type ?u.3812797\nG : Type ?u.3812800\nGₗ : Type ?u.3812803\n𝓕 : Type ?u.3812806\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₁ : 𝕜₂ →+* 𝕜\ninst✝³ : RingHomInvPair σ₁₂ σ₂₁\ninst✝² : RingHomInvPair σ₂₁ σ₁₂\ninst✝¹ : RingHomIsometric σ₂₁\ninst✝ : RingHomIsometric σ₁₂\ne : E ≃SL[σ₁₂] F\n_i : Nontrivial E\n⊢ Subsingleton E ∨ 0 < ‖↑(ContinuousLinearEquiv.symm e)‖",
"tactic": "right"
},
{
"state_after": "no goals",
"state_before": "case inr.h\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.3812785\nE : Type u_3\nEₗ : Type ?u.3812791\nF : Type u_4\nFₗ : Type ?u.3812797\nG : Type ?u.3812800\nGₗ : Type ?u.3812803\n𝓕 : Type ?u.3812806\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup G\ninst✝⁹ : NormedAddCommGroup Fₗ\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField 𝕜₃\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₁ : 𝕜₂ →+* 𝕜\ninst✝³ : RingHomInvPair σ₁₂ σ₂₁\ninst✝² : RingHomInvPair σ₂₁ σ₁₂\ninst✝¹ : RingHomIsometric σ₂₁\ninst✝ : RingHomIsometric σ₁₂\ne : E ≃SL[σ₁₂] F\n_i : Nontrivial E\n⊢ 0 < ‖↑(ContinuousLinearEquiv.symm e)‖",
"tactic": "exact e.norm_symm_pos"
}
] |
[
1976,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1970,
1
] |
Mathlib/Topology/ContinuousFunction/Algebra.lean
|
ContinuousMap.coe_one
|
[] |
[
93,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Analysis/BoxIntegral/Box/Basic.lean
|
BoxIntegral.Box.le_TFAE
|
[
{
"state_after": "case tfae_1_iff_2\nι : Type u_1\nI J : Box ι\nx y : ι → ℝ\n⊢ I ≤ J ↔ ↑I ⊆ ↑J\n\nι : Type u_1\nI J : Box ι\nx y : ι → ℝ\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\n⊢ List.TFAE [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper]",
"state_before": "ι : Type u_1\nI J : Box ι\nx y : ι → ℝ\n⊢ List.TFAE [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper]",
"tactic": "tfae_have 1 ↔ 2"
},
{
"state_after": "ι : Type u_1\nI J : Box ι\nx y : ι → ℝ\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\n⊢ List.TFAE [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper]",
"state_before": "case tfae_1_iff_2\nι : Type u_1\nI J : Box ι\nx y : ι → ℝ\n⊢ I ≤ J ↔ ↑I ⊆ ↑J\n\nι : Type u_1\nI J : Box ι\nx y : ι → ℝ\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\n⊢ List.TFAE [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper]",
"tactic": "exact Iff.rfl"
},
{
"state_after": "case tfae_2_to_3\nι : Type u_1\nI J : Box ι\nx y : ι → ℝ\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\n⊢ ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper\n\nι : Type u_1\nI J : Box ι\nx y : ι → ℝ\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\ntfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper\n⊢ List.TFAE [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper]",
"state_before": "ι : Type u_1\nI J : Box ι\nx y : ι → ℝ\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\n⊢ List.TFAE [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper]",
"tactic": "tfae_have 2 → 3"
},
{
"state_after": "case tfae_3_iff_4\nι : Type u_1\nI J : Box ι\nx y : ι → ℝ\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\ntfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper\n⊢ Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper\n\nι : Type u_1\nI J : Box ι\nx y : ι → ℝ\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\ntfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper\ntfae_3_iff_4 : Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper\n⊢ List.TFAE [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper]",
"state_before": "ι : Type u_1\nI J : Box ι\nx y : ι → ℝ\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\ntfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper\n⊢ List.TFAE [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper]",
"tactic": "tfae_have 3 ↔ 4"
},
{
"state_after": "ι : Type u_1\nI J : Box ι\nx y : ι → ℝ\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\ntfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper\ntfae_3_iff_4 : Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper\n⊢ List.TFAE [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper]",
"state_before": "case tfae_3_iff_4\nι : Type u_1\nI J : Box ι\nx y : ι → ℝ\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\ntfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper\n⊢ Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper\n\nι : Type u_1\nI J : Box ι\nx y : ι → ℝ\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\ntfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper\ntfae_3_iff_4 : Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper\n⊢ List.TFAE [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper]",
"tactic": "exact Icc_subset_Icc_iff I.lower_le_upper"
},
{
"state_after": "case tfae_4_to_2\nι : Type u_1\nI J : Box ι\nx y : ι → ℝ\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\ntfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper\ntfae_3_iff_4 : Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper\n⊢ J.lower ≤ I.lower ∧ I.upper ≤ J.upper → ↑I ⊆ ↑J\n\nι : Type u_1\nI J : Box ι\nx y : ι → ℝ\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\ntfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper\ntfae_3_iff_4 : Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper\ntfae_4_to_2 : J.lower ≤ I.lower ∧ I.upper ≤ J.upper → ↑I ⊆ ↑J\n⊢ List.TFAE [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper]",
"state_before": "ι : Type u_1\nI J : Box ι\nx y : ι → ℝ\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\ntfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper\ntfae_3_iff_4 : Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper\n⊢ List.TFAE [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper]",
"tactic": "tfae_have 4 → 2"
},
{
"state_after": "ι : Type u_1\nI J : Box ι\nx y : ι → ℝ\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\ntfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper\ntfae_3_iff_4 : Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper\ntfae_4_to_2 : J.lower ≤ I.lower ∧ I.upper ≤ J.upper → ↑I ⊆ ↑J\n⊢ List.TFAE [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper]",
"state_before": "case tfae_4_to_2\nι : Type u_1\nI J : Box ι\nx y : ι → ℝ\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\ntfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper\ntfae_3_iff_4 : Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper\n⊢ J.lower ≤ I.lower ∧ I.upper ≤ J.upper → ↑I ⊆ ↑J\n\nι : Type u_1\nI J : Box ι\nx y : ι → ℝ\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\ntfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper\ntfae_3_iff_4 : Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper\ntfae_4_to_2 : J.lower ≤ I.lower ∧ I.upper ≤ J.upper → ↑I ⊆ ↑J\n⊢ List.TFAE [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper]",
"tactic": "exact fun h x hx i ↦ Ioc_subset_Ioc (h.1 i) (h.2 i) (hx i)"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nI J : Box ι\nx y : ι → ℝ\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\ntfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper\ntfae_3_iff_4 : Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper\ntfae_4_to_2 : J.lower ≤ I.lower ∧ I.upper ≤ J.upper → ↑I ⊆ ↑J\n⊢ List.TFAE [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper]",
"tactic": "tfae_finish"
},
{
"state_after": "case tfae_2_to_3\nι : Type u_1\nI J : Box ι\nx y : ι → ℝ\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\nh : ↑I ⊆ ↑J\n⊢ Icc I.lower I.upper ⊆ Icc J.lower J.upper",
"state_before": "case tfae_2_to_3\nι : Type u_1\nI J : Box ι\nx y : ι → ℝ\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\n⊢ ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case tfae_2_to_3\nι : Type u_1\nI J : Box ι\nx y : ι → ℝ\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\nh : ↑I ⊆ ↑J\n⊢ Icc I.lower I.upper ⊆ Icc J.lower J.upper",
"tactic": "simpa [coe_eq_pi, closure_pi_set, lower_ne_upper] using closure_mono h"
}
] |
[
164,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.equalizer.fork_π_app_zero
|
[] |
[
776,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
775,
1
] |
Mathlib/Topology/Algebra/Order/LeftRight.lean
|
continuousWithinAt_Iio_iff_Iic
|
[] |
[
44,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
42,
1
] |
Mathlib/Order/MinMax.lean
|
min_lt_min
|
[] |
[
223,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
222,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Cardinal.blsub_lt_ord_lift_of_isRegular
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.158839\nr : α → α → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\nc : Cardinal\nhc : IsRegular c\nho : lift (card o) < c\n⊢ lift (card o) < Ordinal.cof (ord c)",
"tactic": "rwa [hc.cof_eq]"
}
] |
[
1084,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1081,
1
] |
Mathlib/Deprecated/Group.lean
|
IsGroupHom.to_isMonoidHom
|
[] |
[
288,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
287,
1
] |
Mathlib/Topology/Sequences.lean
|
UniformSpace.compactSpace_iff_seqCompactSpace
|
[
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type ?u.22819\ninst✝¹ : UniformSpace X\ns : Set X\ninst✝ : IsCountablyGenerated (𝓤 X)\n⊢ CompactSpace X ↔ SeqCompactSpace X",
"tactic": "simp only [← isCompact_univ_iff, seqCompactSpace_iff, UniformSpace.isCompact_iff_isSeqCompact]"
}
] |
[
399,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
398,
1
] |
Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean
|
CompHaus.effectiveEpiFamily_tfae
|
[
{
"state_after": "case tfae_1_to_2\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\n⊢ EffectiveEpiFamily X π → Epi (Sigma.desc π)\n\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\n⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b]",
"state_before": "α : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\n⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b]",
"tactic": "tfae_have 1 → 2"
},
{
"state_after": "case tfae_2_to_3\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\n⊢ Epi (Sigma.desc π) → ∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b\n\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ntfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b\n⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b]",
"state_before": "α : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\n⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b]",
"tactic": "tfae_have 2 → 3"
},
{
"state_after": "case tfae_3_to_1\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ntfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b\n⊢ (∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b) → EffectiveEpiFamily X π\n\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ntfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b\ntfae_3_to_1 : (∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b) → EffectiveEpiFamily X π\n⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b]",
"state_before": "α : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ntfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b\n⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b]",
"tactic": "tfae_have 3 → 1"
},
{
"state_after": "no goals",
"state_before": "α : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ntfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b\ntfae_3_to_1 : (∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b) → EffectiveEpiFamily X π\n⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b]",
"tactic": "tfae_finish"
},
{
"state_after": "case tfae_1_to_2\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\n✝ : EffectiveEpiFamily X π\n⊢ Epi (Sigma.desc π)",
"state_before": "case tfae_1_to_2\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\n⊢ EffectiveEpiFamily X π → Epi (Sigma.desc π)",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "case tfae_1_to_2\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\n✝ : EffectiveEpiFamily X π\n⊢ Epi (Sigma.desc π)",
"tactic": "infer_instance"
},
{
"state_after": "case tfae_2_to_3\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Epi (Sigma.desc π)\n⊢ ∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b",
"state_before": "case tfae_2_to_3\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\n⊢ Epi (Sigma.desc π) → ∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b",
"tactic": "intro e"
},
{
"state_after": "case tfae_2_to_3\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\n⊢ ∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b",
"state_before": "case tfae_2_to_3\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Epi (Sigma.desc π)\n⊢ ∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b",
"tactic": "rw [epi_iff_surjective] at e"
},
{
"state_after": "case tfae_2_to_3\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\n⊢ ∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b",
"state_before": "case tfae_2_to_3\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\n⊢ ∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b",
"tactic": "let i : ∐ X ≅ finiteCoproduct X :=\n (colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _)"
},
{
"state_after": "case tfae_2_to_3\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nb : ↑B.toTop\n⊢ ∃ a x, (forget CompHaus).map (π a) x = b",
"state_before": "case tfae_2_to_3\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\n⊢ ∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b",
"tactic": "intro b"
},
{
"state_after": "case tfae_2_to_3.intro\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\n⊢ ∃ a x, (forget CompHaus).map (π a) x = (forget CompHaus).map (Sigma.desc π) t",
"state_before": "case tfae_2_to_3\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nb : ↑B.toTop\n⊢ ∃ a x, (forget CompHaus).map (π a) x = b",
"tactic": "obtain ⟨t,rfl⟩ := e b"
},
{
"state_after": "case tfae_2_to_3.intro\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\n⊢ ∃ a x, (forget CompHaus).map (π a) x = (forget CompHaus).map (Sigma.desc π) t",
"state_before": "case tfae_2_to_3.intro\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\n⊢ ∃ a x, (forget CompHaus).map (π a) x = (forget CompHaus).map (Sigma.desc π) t",
"tactic": "let q := i.hom t"
},
{
"state_after": "case tfae_2_to_3.intro\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\n⊢ (forget CompHaus).map (π q.fst) q.snd = (forget CompHaus).map (Sigma.desc π) t",
"state_before": "case tfae_2_to_3.intro\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\n⊢ ∃ a x, (forget CompHaus).map (π a) x = (forget CompHaus).map (Sigma.desc π) t",
"tactic": "refine ⟨q.1,q.2,?_⟩"
},
{
"state_after": "case tfae_2_to_3.intro\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\nthis : t = (forget CompHaus).map i.inv ((forget CompHaus).map i.hom t)\n⊢ (forget CompHaus).map (π q.fst) q.snd = (forget CompHaus).map (Sigma.desc π) t",
"state_before": "case tfae_2_to_3.intro\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\n⊢ (forget CompHaus).map (π q.fst) q.snd = (forget CompHaus).map (Sigma.desc π) t",
"tactic": "have : t = i.inv (i.hom t) := show t = (i.hom ≫ i.inv) t by simp only [i.hom_inv_id] ; rfl"
},
{
"state_after": "case tfae_2_to_3.intro\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\nthis : t = (forget CompHaus).map i.inv ((forget CompHaus).map i.hom t)\n⊢ (forget CompHaus).map (π q.fst) q.snd =\n (forget CompHaus).map (Sigma.desc π) ((forget CompHaus).map i.inv ((forget CompHaus).map i.hom t))",
"state_before": "case tfae_2_to_3.intro\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\nthis : t = (forget CompHaus).map i.inv ((forget CompHaus).map i.hom t)\n⊢ (forget CompHaus).map (π q.fst) q.snd = (forget CompHaus).map (Sigma.desc π) t",
"tactic": "rw [this]"
},
{
"state_after": "case tfae_2_to_3.intro\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\nthis : t = (forget CompHaus).map i.inv ((forget CompHaus).map i.hom t)\n⊢ (forget CompHaus).map (π q.fst) q.snd = (forget CompHaus).map (i.inv ≫ Sigma.desc π) ((forget CompHaus).map i.hom t)",
"state_before": "case tfae_2_to_3.intro\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\nthis : t = (forget CompHaus).map i.inv ((forget CompHaus).map i.hom t)\n⊢ (forget CompHaus).map (π q.fst) q.snd =\n (forget CompHaus).map (Sigma.desc π) ((forget CompHaus).map i.inv ((forget CompHaus).map i.hom t))",
"tactic": "show _ = (i.inv ≫ Sigma.desc π) (i.hom t)"
},
{
"state_after": "case tfae_2_to_3.intro\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\nthis : t = (forget CompHaus).map i.inv ((forget CompHaus).map i.hom t)\n⊢ i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π",
"state_before": "case tfae_2_to_3.intro\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\nthis : t = (forget CompHaus).map i.inv ((forget CompHaus).map i.hom t)\n⊢ (forget CompHaus).map (π q.fst) q.snd = (forget CompHaus).map (i.inv ≫ Sigma.desc π) ((forget CompHaus).map i.hom t)",
"tactic": "suffices i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π by\n rw [this] ; rfl"
},
{
"state_after": "case tfae_2_to_3.intro\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\nthis : t = (forget CompHaus).map i.inv ((forget CompHaus).map i.hom t)\n⊢ Sigma.desc π = i.hom ≫ finiteCoproduct.desc X π",
"state_before": "case tfae_2_to_3.intro\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\nthis : t = (forget CompHaus).map i.inv ((forget CompHaus).map i.hom t)\n⊢ i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π",
"tactic": "rw [Iso.inv_comp_eq]"
},
{
"state_after": "case tfae_2_to_3.intro.w\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\nthis : t = (forget CompHaus).map i.inv ((forget CompHaus).map i.hom t)\n⊢ ∀ (j : Discrete α),\n colimit.ι (Discrete.functor X) j ≫ Sigma.desc π =\n colimit.ι (Discrete.functor X) j ≫ i.hom ≫ finiteCoproduct.desc X π",
"state_before": "case tfae_2_to_3.intro\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\nthis : t = (forget CompHaus).map i.inv ((forget CompHaus).map i.hom t)\n⊢ Sigma.desc π = i.hom ≫ finiteCoproduct.desc X π",
"tactic": "apply colimit.hom_ext"
},
{
"state_after": "case tfae_2_to_3.intro.w.mk\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\nthis : t = (forget CompHaus).map i.inv ((forget CompHaus).map i.hom t)\na : α\n⊢ colimit.ι (Discrete.functor X) { as := a } ≫ Sigma.desc π =\n colimit.ι (Discrete.functor X) { as := a } ≫ i.hom ≫ finiteCoproduct.desc X π",
"state_before": "case tfae_2_to_3.intro.w\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\nthis : t = (forget CompHaus).map i.inv ((forget CompHaus).map i.hom t)\n⊢ ∀ (j : Discrete α),\n colimit.ι (Discrete.functor X) j ≫ Sigma.desc π =\n colimit.ι (Discrete.functor X) j ≫ i.hom ≫ finiteCoproduct.desc X π",
"tactic": "rintro ⟨a⟩"
},
{
"state_after": "case tfae_2_to_3.intro.w.mk\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\nthis : t = (forget CompHaus).map i.inv ((forget CompHaus).map i.hom t)\na : α\n⊢ π a = (finiteCoproduct.cocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π",
"state_before": "case tfae_2_to_3.intro.w.mk\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\nthis : t = (forget CompHaus).map i.inv ((forget CompHaus).map i.hom t)\na : α\n⊢ colimit.ι (Discrete.functor X) { as := a } ≫ Sigma.desc π =\n colimit.ι (Discrete.functor X) { as := a } ≫ i.hom ≫ finiteCoproduct.desc X π",
"tactic": "simp only [Discrete.functor_obj, colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app,\n colimit.comp_coconePointUniqueUpToIso_hom_assoc]"
},
{
"state_after": "case tfae_2_to_3.intro.w.mk.w\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\nthis : t = (forget CompHaus).map i.inv ((forget CompHaus).map i.hom t)\na : α\nx✝ : (forget CompHaus).obj (X a)\n⊢ (forget CompHaus).map (π a) x✝ =\n (forget CompHaus).map ((finiteCoproduct.cocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π) x✝",
"state_before": "case tfae_2_to_3.intro.w.mk\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\nthis : t = (forget CompHaus).map i.inv ((forget CompHaus).map i.hom t)\na : α\n⊢ π a = (finiteCoproduct.cocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case tfae_2_to_3.intro.w.mk.w\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\nthis : t = (forget CompHaus).map i.inv ((forget CompHaus).map i.hom t)\na : α\nx✝ : (forget CompHaus).obj (X a)\n⊢ (forget CompHaus).map (π a) x✝ =\n (forget CompHaus).map ((finiteCoproduct.cocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π) x✝",
"tactic": "rfl"
},
{
"state_after": "α : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\n⊢ t = (forget CompHaus).map (𝟙 (∐ X)) t",
"state_before": "α : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\n⊢ t = (forget CompHaus).map (i.hom ≫ i.inv) t",
"tactic": "simp only [i.hom_inv_id]"
},
{
"state_after": "no goals",
"state_before": "α : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\n⊢ t = (forget CompHaus).map (𝟙 (∐ X)) t",
"tactic": "rfl"
},
{
"state_after": "α : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\nthis✝ : t = (forget CompHaus).map i.inv ((forget CompHaus).map i.hom t)\nthis : i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π\n⊢ (forget CompHaus).map (π q.fst) q.snd =\n (forget CompHaus).map (finiteCoproduct.desc X π) ((forget CompHaus).map i.hom t)",
"state_before": "α : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\nthis✝ : t = (forget CompHaus).map i.inv ((forget CompHaus).map i.hom t)\nthis : i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π\n⊢ (forget CompHaus).map (π q.fst) q.snd = (forget CompHaus).map (i.inv ≫ Sigma.desc π) ((forget CompHaus).map i.hom t)",
"tactic": "rw [this]"
},
{
"state_after": "no goals",
"state_before": "α : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ne : Function.Surjective ((forget CompHaus).map (Sigma.desc π))\ni : ∐ X ≅ finiteCoproduct X :=\n IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)\nt : (forget CompHaus).obj (∐ fun b => X b)\nq : (forget CompHaus).obj (finiteCoproduct X) := (forget CompHaus).map i.hom t\nthis✝ : t = (forget CompHaus).map i.inv ((forget CompHaus).map i.hom t)\nthis : i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π\n⊢ (forget CompHaus).map (π q.fst) q.snd =\n (forget CompHaus).map (finiteCoproduct.desc X π) ((forget CompHaus).map i.hom t)",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case tfae_3_to_1\nα : Type\ninst✝ : Fintype α\nB : CompHaus\nX : α → CompHaus\nπ : (a : α) → X a ⟶ B\ntfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)\ntfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b\n⊢ (∀ (b : ↑B.toTop), ∃ a x, (forget CompHaus).map (π a) x = b) → EffectiveEpiFamily X π",
"tactic": "apply effectiveEpiFamily_of_jointly_surjective"
}
] |
[
225,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Logic/Basic.lean
|
forall_true_iff'
|
[] |
[
719,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
718,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.degree_smul_le
|
[
{
"state_after": "R : Type u\nS : Type v\na✝ b c d : R\nn m✝ : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.705453\na : R\np : R[X]\nm : ℕ\nhm : degree p < ↑m\n⊢ coeff (a • p) m = 0",
"state_before": "R : Type u\nS : Type v\na✝ b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.705453\na : R\np : R[X]\n⊢ degree (a • p) ≤ degree p",
"tactic": "refine (degree_le_iff_coeff_zero _ _).2 fun m hm => ?_"
},
{
"state_after": "R : Type u\nS : Type v\na✝ b c d : R\nn m✝ : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.705453\na : R\np : R[X]\nm : ℕ\nhm : ∀ (m_1 : ℕ), m ≤ m_1 → coeff p m_1 = 0\n⊢ coeff (a • p) m = 0",
"state_before": "R : Type u\nS : Type v\na✝ b c d : R\nn m✝ : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.705453\na : R\np : R[X]\nm : ℕ\nhm : degree p < ↑m\n⊢ coeff (a • p) m = 0",
"tactic": "rw [degree_lt_iff_coeff_zero] at hm"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na✝ b c d : R\nn m✝ : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.705453\na : R\np : R[X]\nm : ℕ\nhm : ∀ (m_1 : ℕ), m ≤ m_1 → coeff p m_1 = 0\n⊢ coeff (a • p) m = 0",
"tactic": "simp [hm m le_rfl]"
}
] |
[
1098,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1095,
1
] |
Mathlib/Analysis/NormedSpace/Star/Multiplier.lean
|
DoubleCentralizer.nat_cast_fst
|
[] |
[
307,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
306,
1
] |
Mathlib/Topology/Order/Basic.lean
|
interior_Ioo
|
[] |
[
321,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
320,
1
] |
Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
|
Finpartition.nonUniforms_mono
|
[] |
[
217,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/Algebra/Order/Monoid/WithTop.lean
|
WithTop.one_lt_coe
|
[] |
[
77,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/GroupTheory/GroupAction/Defs.lean
|
smul_mul_assoc
|
[] |
[
420,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
418,
1
] |
Mathlib/CategoryTheory/Endofunctor/Algebra.lean
|
CategoryTheory.Endofunctor.Coalgebra.id_eq_id
|
[] |
[
329,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Mathlib/CategoryTheory/Abelian/Homology.lean
|
homology.hom_to_ext
|
[
{
"state_after": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : W ⟶ homology f g w\nh :\n a ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w) =\n b ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w)\n⊢ a = b",
"state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : W ⟶ homology f g w\nh : a ≫ ι f g w = b ≫ ι f g w\n⊢ a = b",
"tactic": "dsimp [ι] at h"
},
{
"state_after": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : W ⟶ homology f g w\nh :\n a ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w) =\n b ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w)\n⊢ (fun e => e ≫ (homologyIsoKernelDesc f g w).hom) a = (fun e => e ≫ (homologyIsoKernelDesc f g w).hom) b\n\ncase inj\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : W ⟶ homology f g w\nh :\n a ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w) =\n b ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w)\n⊢ Function.Injective fun e => e ≫ (homologyIsoKernelDesc f g w).hom",
"state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : W ⟶ homology f g w\nh :\n a ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w) =\n b ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w)\n⊢ a = b",
"tactic": "apply_fun fun e => e ≫ (homologyIsoKernelDesc f g w).hom"
},
{
"state_after": "case inj\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : W ⟶ homology f g w\nh :\n a ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w) =\n b ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w)\n⊢ Function.Injective fun e => e ≫ (homologyIsoKernelDesc f g w).hom\n\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : W ⟶ homology f g w\nh :\n a ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w) =\n b ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w)\n⊢ (fun e => e ≫ (homologyIsoKernelDesc f g w).hom) a = (fun e => e ≫ (homologyIsoKernelDesc f g w).hom) b",
"state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : W ⟶ homology f g w\nh :\n a ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w) =\n b ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w)\n⊢ (fun e => e ≫ (homologyIsoKernelDesc f g w).hom) a = (fun e => e ≫ (homologyIsoKernelDesc f g w).hom) b\n\ncase inj\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : W ⟶ homology f g w\nh :\n a ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w) =\n b ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w)\n⊢ Function.Injective fun e => e ≫ (homologyIsoKernelDesc f g w).hom",
"tactic": "swap"
},
{
"state_after": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : W ⟶ homology f g w\nh :\n (a ≫ (homologyIsoKernelDesc f g w).hom) ≫ kernel.ι (cokernel.desc f g w) =\n (b ≫ (homologyIsoKernelDesc f g w).hom) ≫ kernel.ι (cokernel.desc f g w)\n⊢ (fun e => e ≫ (homologyIsoKernelDesc f g w).hom) a = (fun e => e ≫ (homologyIsoKernelDesc f g w).hom) b",
"state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : W ⟶ homology f g w\nh :\n a ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w) =\n b ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w)\n⊢ (fun e => e ≫ (homologyIsoKernelDesc f g w).hom) a = (fun e => e ≫ (homologyIsoKernelDesc f g w).hom) b",
"tactic": "simp only [← Category.assoc] at h"
},
{
"state_after": "no goals",
"state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : W ⟶ homology f g w\nh :\n (a ≫ (homologyIsoKernelDesc f g w).hom) ≫ kernel.ι (cokernel.desc f g w) =\n (b ≫ (homologyIsoKernelDesc f g w).hom) ≫ kernel.ι (cokernel.desc f g w)\n⊢ (fun e => e ≫ (homologyIsoKernelDesc f g w).hom) a = (fun e => e ≫ (homologyIsoKernelDesc f g w).hom) b",
"tactic": "exact equalizer.hom_ext h"
},
{
"state_after": "case inj\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : W ⟶ homology f g w\nh :\n a ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w) =\n b ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w)\ni j : W ⟶ homology f g w\nhh : (fun e => e ≫ (homologyIsoKernelDesc f g w).hom) i = (fun e => e ≫ (homologyIsoKernelDesc f g w).hom) j\n⊢ i = j",
"state_before": "case inj\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : W ⟶ homology f g w\nh :\n a ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w) =\n b ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w)\n⊢ Function.Injective fun e => e ≫ (homologyIsoKernelDesc f g w).hom",
"tactic": "intro i j hh"
},
{
"state_after": "case inj\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : W ⟶ homology f g w\nh :\n a ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w) =\n b ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w)\ni j : W ⟶ homology f g w\nhh :\n (fun e => e ≫ (homologyIsoKernelDesc f g w).hom) i ≫ (homologyIsoKernelDesc f g w).inv =\n (fun e => e ≫ (homologyIsoKernelDesc f g w).hom) j ≫ (homologyIsoKernelDesc f g w).inv\n⊢ i = j",
"state_before": "case inj\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : W ⟶ homology f g w\nh :\n a ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w) =\n b ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w)\ni j : W ⟶ homology f g w\nhh : (fun e => e ≫ (homologyIsoKernelDesc f g w).hom) i = (fun e => e ≫ (homologyIsoKernelDesc f g w).hom) j\n⊢ i = j",
"tactic": "apply_fun fun e => e ≫ (homologyIsoKernelDesc f g w).inv at hh"
},
{
"state_after": "no goals",
"state_before": "case inj\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : W ⟶ homology f g w\nh :\n a ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w) =\n b ≫ (homologyIsoKernelDesc f g w).hom ≫ kernel.ι (cokernel.desc f g w)\ni j : W ⟶ homology f g w\nhh :\n (fun e => e ≫ (homologyIsoKernelDesc f g w).hom) i ≫ (homologyIsoKernelDesc f g w).inv =\n (fun e => e ≫ (homologyIsoKernelDesc f g w).hom) j ≫ (homologyIsoKernelDesc f g w).inv\n⊢ i = j",
"tactic": "simpa using hh"
}
] |
[
187,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.le_iInf_add_iInf
|
[
{
"state_after": "ι : Sort u_1\nι' : Sort u_2\ninst✝¹ : Nonempty ι\ninst✝ : Nonempty ι'\nf : ι → ℝ≥0\ng : ι' → ℝ≥0\na : ℝ≥0\nh : ∀ (i : ι) (j : ι'), a ≤ f i + g j\n⊢ ↑a ≤ (⨅ (i : ι), ↑(f i)) + ⨅ (i : ι'), ↑(g i)",
"state_before": "ι : Sort u_1\nι' : Sort u_2\ninst✝¹ : Nonempty ι\ninst✝ : Nonempty ι'\nf : ι → ℝ≥0\ng : ι' → ℝ≥0\na : ℝ≥0\nh : ∀ (i : ι) (j : ι'), a ≤ f i + g j\n⊢ a ≤ (⨅ (i : ι), f i) + ⨅ (j : ι'), g j",
"tactic": "rw [← NNReal.coe_le_coe, NNReal.coe_add, coe_iInf, coe_iInf]"
},
{
"state_after": "no goals",
"state_before": "ι : Sort u_1\nι' : Sort u_2\ninst✝¹ : Nonempty ι\ninst✝ : Nonempty ι'\nf : ι → ℝ≥0\ng : ι' → ℝ≥0\na : ℝ≥0\nh : ∀ (i : ι) (j : ι'), a ≤ f i + g j\n⊢ ↑a ≤ (⨅ (i : ι), ↑(f i)) + ⨅ (i : ι'), ↑(g i)",
"tactic": "exact le_ciInf_add_ciInf h"
}
] |
[
523,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
520,
1
] |
Mathlib/LinearAlgebra/TensorPower.lean
|
TensorPower.gOne_def
|
[] |
[
78,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Topology/Instances/Rat.lean
|
Nat.closedEmbedding_coe_rat
|
[
{
"state_after": "no goals",
"state_before": "⊢ Pairwise fun x y => 1 ≤ dist ↑x ↑y",
"tactic": "simpa using Nat.pairwise_one_le_dist"
}
] |
[
71,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearEquiv.symm_image_image
|
[] |
[
2053,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2052,
1
] |
Mathlib/Analysis/InnerProductSpace/Orthogonal.lean
|
Submodule.isOrtho_iSup_right
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.143983\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nι : Sort u_1\nU : Submodule 𝕜 E\nV : ι → Submodule 𝕜 E\n⊢ (∀ (i : ι), V i ⟂ U) ↔ ∀ (i : ι), U ⟂ V i",
"tactic": "simp_rw [isOrtho_comm]"
}
] |
[
374,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
372,
1
] |
Mathlib/Algebra/Order/Pointwise.lean
|
csSup_div
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : ConditionallyCompleteLattice α\ninst✝² : Group α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ns t : Set α\nhs₀ : Set.Nonempty s\nhs₁ : BddAbove s\nht₀ : Set.Nonempty t\nht₁ : BddBelow t\n⊢ sSup (s / t) = sSup s / sInf t",
"tactic": "rw [div_eq_mul_inv, csSup_mul hs₀ hs₁ ht₀.inv ht₁.inv, csSup_inv ht₀ ht₁, div_eq_mul_inv]"
}
] |
[
165,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/Topology/Algebra/UniformGroup.lean
|
CauchySeq.const_mul
|
[] |
[
456,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
454,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
ContinuousLinearMap.coeFn_compLp'
|
[] |
[
955,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
954,
1
] |
Mathlib/Order/Filter/Germ.lean
|
Filter.Germ.le_def
|
[] |
[
657,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
656,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.blsub_eq_zero_iff
|
[
{
"state_after": "α : Type ?u.357763\nβ : Type ?u.357766\nγ : Type ?u.357769\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\n⊢ IsEmpty (Quotient.out o).α ↔ o = 0",
"state_before": "α : Type ?u.357763\nβ : Type ?u.357766\nγ : Type ?u.357769\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\n⊢ blsub o f = 0 ↔ o = 0",
"tactic": "rw [← lsub_eq_blsub, lsub_eq_zero_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.357763\nβ : Type ?u.357766\nγ : Type ?u.357769\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\n⊢ IsEmpty (Quotient.out o).α ↔ o = 0",
"tactic": "exact out_empty_iff_eq_zero"
}
] |
[
1885,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1883,
1
] |
Mathlib/Combinatorics/SimpleGraph/Coloring.lean
|
SimpleGraph.Colorable.cliqueFree
|
[
{
"state_after": "V : Type u\nG : SimpleGraph V\nα : Type v\nC : Coloring G α\nn m : ℕ\nhc : Colorable G n\nhm : n < m\nh : ¬CliqueFree G m\n⊢ False",
"state_before": "V : Type u\nG : SimpleGraph V\nα : Type v\nC : Coloring G α\nn m : ℕ\nhc : Colorable G n\nhm : n < m\n⊢ CliqueFree G m",
"tactic": "by_contra h"
},
{
"state_after": "V : Type u\nG : SimpleGraph V\nα : Type v\nC : Coloring G α\nn m : ℕ\nhc : Colorable G n\nhm : n < m\nh : ∃ x, IsClique G ↑x ∧ Finset.card x = m\n⊢ False",
"state_before": "V : Type u\nG : SimpleGraph V\nα : Type v\nC : Coloring G α\nn m : ℕ\nhc : Colorable G n\nhm : n < m\nh : ¬CliqueFree G m\n⊢ False",
"tactic": "simp only [CliqueFree, isNClique_iff, not_forall, Classical.not_not] at h"
},
{
"state_after": "case intro.intro\nV : Type u\nG : SimpleGraph V\nα : Type v\nC : Coloring G α\nn : ℕ\nhc : Colorable G n\ns : Finset V\nh : IsClique G ↑s\nhm : n < Finset.card s\n⊢ False",
"state_before": "V : Type u\nG : SimpleGraph V\nα : Type v\nC : Coloring G α\nn m : ℕ\nhc : Colorable G n\nhm : n < m\nh : ∃ x, IsClique G ↑x ∧ Finset.card x = m\n⊢ False",
"tactic": "obtain ⟨s, h, rfl⟩ := h"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nV : Type u\nG : SimpleGraph V\nα : Type v\nC : Coloring G α\nn : ℕ\nhc : Colorable G n\ns : Finset V\nh : IsClique G ↑s\nhm : n < Finset.card s\n⊢ False",
"tactic": "exact Nat.lt_le_antisymm hm (h.card_le_of_colorable hc)"
}
] |
[
452,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
447,
11
] |
Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean
|
CategoryTheory.tensorLeftHomEquiv_symm_coevaluation_comp_tensor_id
|
[
{
"state_after": "C : Type u₁\ninst✝³ : Category C\ninst✝² : MonoidalCategory C\nX Y : C\ninst✝¹ : HasRightDual X\ninst✝ : HasRightDual Y\nf : X ⟶ Y\n⊢ (𝟙 Yᘁ ⊗ η_ X Xᘁ ≫ (f ⊗ 𝟙 Xᘁ)) ≫ (α_ Yᘁ Y Xᘁ).inv ≫ (ε_ Y Yᘁ ⊗ 𝟙 Xᘁ) ≫ (λ_ Xᘁ).hom =\n (ρ_ Yᘁ).hom ≫ (ρ_ Yᘁ).inv ≫ (𝟙 Yᘁ ⊗ η_ X Xᘁ) ≫ (𝟙 Yᘁ ⊗ f ⊗ 𝟙 Xᘁ) ≫ (α_ Yᘁ Y Xᘁ).inv ≫ (ε_ Y Yᘁ ⊗ 𝟙 Xᘁ) ≫ (λ_ Xᘁ).hom",
"state_before": "C : Type u₁\ninst✝³ : Category C\ninst✝² : MonoidalCategory C\nX Y : C\ninst✝¹ : HasRightDual X\ninst✝ : HasRightDual Y\nf : X ⟶ Y\n⊢ ↑(tensorLeftHomEquiv (𝟙_ C) Y Yᘁ Xᘁ).symm (η_ X Xᘁ ≫ (f ⊗ 𝟙 Xᘁ)) = (ρ_ Yᘁ).hom ≫ fᘁ",
"tactic": "dsimp [tensorLeftHomEquiv, rightAdjointMate]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝³ : Category C\ninst✝² : MonoidalCategory C\nX Y : C\ninst✝¹ : HasRightDual X\ninst✝ : HasRightDual Y\nf : X ⟶ Y\n⊢ (𝟙 Yᘁ ⊗ η_ X Xᘁ ≫ (f ⊗ 𝟙 Xᘁ)) ≫ (α_ Yᘁ Y Xᘁ).inv ≫ (ε_ Y Yᘁ ⊗ 𝟙 Xᘁ) ≫ (λ_ Xᘁ).hom =\n (ρ_ Yᘁ).hom ≫ (ρ_ Yᘁ).inv ≫ (𝟙 Yᘁ ⊗ η_ X Xᘁ) ≫ (𝟙 Yᘁ ⊗ f ⊗ 𝟙 Xᘁ) ≫ (α_ Yᘁ Y Xᘁ).inv ≫ (ε_ Y Yᘁ ⊗ 𝟙 Xᘁ) ≫ (λ_ Xᘁ).hom",
"tactic": "simp"
}
] |
[
486,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
482,
1
] |
Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
|
MeasureTheory.FiniteMeasure.continuous_mass
|
[
{
"state_after": "Ω : Type u_1\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.126329\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\n⊢ Continuous fun μ => testAgainstNN μ 1",
"state_before": "Ω : Type u_1\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.126329\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\n⊢ Continuous fun μ => mass μ",
"tactic": "simp_rw [← testAgainstNN_one]"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.126329\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\n⊢ Continuous fun μ => testAgainstNN μ 1",
"tactic": "exact continuous_testAgainstNN_eval 1"
}
] |
[
484,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
483,
1
] |
Mathlib/Algebra/BigOperators/Order.lean
|
Finset.le_prod_nonempty_of_submultiplicative_on_pred
|
[
{
"state_after": "case refine'_1\nι : Type u_3\nα : Type ?u.59\nβ : Type ?u.62\nM : Type u_2\nN : Type u_1\nG : Type ?u.71\nk : Type ?u.74\nR : Type ?u.77\ninst✝¹ : CommMonoid M\ninst✝ : OrderedCommMonoid N\nf : M → N\np : M → Prop\nh_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y\nhp_mul : ∀ (x y : M), p x → p y → p (x * y)\ng : ι → M\ns : Finset ι\nhs_nonempty : Finset.Nonempty s\nhs : ∀ (i : ι), i ∈ s → p (g i)\n⊢ Multiset.map (fun i => g i) s.val ≠ ∅\n\ncase refine'_2\nι : Type u_3\nα : Type ?u.59\nβ : Type ?u.62\nM : Type u_2\nN : Type u_1\nG : Type ?u.71\nk : Type ?u.74\nR : Type ?u.77\ninst✝¹ : CommMonoid M\ninst✝ : OrderedCommMonoid N\nf : M → N\np : M → Prop\nh_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y\nhp_mul : ∀ (x y : M), p x → p y → p (x * y)\ng : ι → M\ns : Finset ι\nhs_nonempty : Finset.Nonempty s\nhs : ∀ (i : ι), i ∈ s → p (g i)\n⊢ ∀ (a : M), a ∈ Multiset.map (fun i => g i) s.val → p a\n\ncase refine'_3\nι : Type u_3\nα : Type ?u.59\nβ : Type ?u.62\nM : Type u_2\nN : Type u_1\nG : Type ?u.71\nk : Type ?u.74\nR : Type ?u.77\ninst✝¹ : CommMonoid M\ninst✝ : OrderedCommMonoid N\nf : M → N\np : M → Prop\nh_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y\nhp_mul : ∀ (x y : M), p x → p y → p (x * y)\ng : ι → M\ns : Finset ι\nhs_nonempty : Finset.Nonempty s\nhs : ∀ (i : ι), i ∈ s → p (g i)\n⊢ Multiset.prod (Multiset.map f (Multiset.map (fun i => g i) s.val)) ≤ ∏ i in s, f (g i)",
"state_before": "ι : Type u_3\nα : Type ?u.59\nβ : Type ?u.62\nM : Type u_2\nN : Type u_1\nG : Type ?u.71\nk : Type ?u.74\nR : Type ?u.77\ninst✝¹ : CommMonoid M\ninst✝ : OrderedCommMonoid N\nf : M → N\np : M → Prop\nh_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y\nhp_mul : ∀ (x y : M), p x → p y → p (x * y)\ng : ι → M\ns : Finset ι\nhs_nonempty : Finset.Nonempty s\nhs : ∀ (i : ι), i ∈ s → p (g i)\n⊢ f (∏ i in s, g i) ≤ ∏ i in s, f (g i)",
"tactic": "refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _"
},
{
"state_after": "case refine'_3\nι : Type u_3\nα : Type ?u.59\nβ : Type ?u.62\nM : Type u_2\nN : Type u_1\nG : Type ?u.71\nk : Type ?u.74\nR : Type ?u.77\ninst✝¹ : CommMonoid M\ninst✝ : OrderedCommMonoid N\nf : M → N\np : M → Prop\nh_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y\nhp_mul : ∀ (x y : M), p x → p y → p (x * y)\ng : ι → M\ns : Finset ι\nhs_nonempty : Finset.Nonempty s\nhs : ∀ (i : ι), i ∈ s → p (g i)\n⊢ Multiset.prod (Multiset.map (f ∘ fun i => g i) s.val) ≤ ∏ i in s, f (g i)",
"state_before": "case refine'_3\nι : Type u_3\nα : Type ?u.59\nβ : Type ?u.62\nM : Type u_2\nN : Type u_1\nG : Type ?u.71\nk : Type ?u.74\nR : Type ?u.77\ninst✝¹ : CommMonoid M\ninst✝ : OrderedCommMonoid N\nf : M → N\np : M → Prop\nh_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y\nhp_mul : ∀ (x y : M), p x → p y → p (x * y)\ng : ι → M\ns : Finset ι\nhs_nonempty : Finset.Nonempty s\nhs : ∀ (i : ι), i ∈ s → p (g i)\n⊢ Multiset.prod (Multiset.map f (Multiset.map (fun i => g i) s.val)) ≤ ∏ i in s, f (g i)",
"tactic": "rw [Multiset.map_map]"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nι : Type u_3\nα : Type ?u.59\nβ : Type ?u.62\nM : Type u_2\nN : Type u_1\nG : Type ?u.71\nk : Type ?u.74\nR : Type ?u.77\ninst✝¹ : CommMonoid M\ninst✝ : OrderedCommMonoid N\nf : M → N\np : M → Prop\nh_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y\nhp_mul : ∀ (x y : M), p x → p y → p (x * y)\ng : ι → M\ns : Finset ι\nhs_nonempty : Finset.Nonempty s\nhs : ∀ (i : ι), i ∈ s → p (g i)\n⊢ Multiset.prod (Multiset.map (f ∘ fun i => g i) s.val) ≤ ∏ i in s, f (g i)",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nι : Type u_3\nα : Type ?u.59\nβ : Type ?u.62\nM : Type u_2\nN : Type u_1\nG : Type ?u.71\nk : Type ?u.74\nR : Type ?u.77\ninst✝¹ : CommMonoid M\ninst✝ : OrderedCommMonoid N\nf : M → N\np : M → Prop\nh_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y\nhp_mul : ∀ (x y : M), p x → p y → p (x * y)\ng : ι → M\ns : Finset ι\nhs_nonempty : Finset.Nonempty s\nhs : ∀ (i : ι), i ∈ s → p (g i)\n⊢ Multiset.map (fun i => g i) s.val ≠ ∅",
"tactic": "simp [hs_nonempty.ne_empty]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nι : Type u_3\nα : Type ?u.59\nβ : Type ?u.62\nM : Type u_2\nN : Type u_1\nG : Type ?u.71\nk : Type ?u.74\nR : Type ?u.77\ninst✝¹ : CommMonoid M\ninst✝ : OrderedCommMonoid N\nf : M → N\np : M → Prop\nh_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y\nhp_mul : ∀ (x y : M), p x → p y → p (x * y)\ng : ι → M\ns : Finset ι\nhs_nonempty : Finset.Nonempty s\nhs : ∀ (i : ι), i ∈ s → p (g i)\n⊢ ∀ (a : M), a ∈ Multiset.map (fun i => g i) s.val → p a",
"tactic": "exact Multiset.forall_mem_map_iff.mpr hs"
}
] |
[
49,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
41,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Constructions.lean
|
Pmf.mem_support_normalize_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.141337\nγ : Type ?u.141340\nf : α → ℝ≥0∞\nhf0 : tsum f ≠ 0\nhf : tsum f ≠ ⊤\na : α\n⊢ a ∈ support (normalize f hf0 hf) ↔ f a ≠ 0",
"tactic": "simp"
}
] |
[
259,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
259,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
Submodule.span_singleton_eq_range
|
[] |
[
432,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
431,
1
] |
Mathlib/Data/Int/Interval.lean
|
Int.Ioc_eq_finset_map
|
[] |
[
97,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Data/Vector/Mem.lean
|
Vector.mem_iff_get
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.160\nn : ℕ\na a' : α\nv : Vector α n\n⊢ (∃ i h, List.get (toList v) { val := i, isLt := h } = a) ↔\n ∃ i h, List.get (toList v) (↑(Fin.cast (_ : n = List.length (toList v))) { val := i, isLt := h }) = a",
"state_before": "α : Type u_1\nβ : Type ?u.160\nn : ℕ\na a' : α\nv : Vector α n\n⊢ a ∈ toList v ↔ ∃ i, get v i = a",
"tactic": "simp only [List.mem_iff_get, Fin.exists_iff, Vector.get_eq_get]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.160\nn : ℕ\na a' : α\nv : Vector α n\n⊢ (∃ i h, List.get (toList v) { val := i, isLt := h } = a) ↔\n ∃ i h, List.get (toList v) (↑(Fin.cast (_ : n = List.length (toList v))) { val := i, isLt := h }) = a",
"tactic": "exact\n ⟨fun ⟨i, hi, h⟩ => ⟨i, by rwa [toList_length] at hi, h⟩, fun ⟨i, hi, h⟩ =>\n ⟨i, by rwa [toList_length], h⟩⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.160\nn : ℕ\na a' : α\nv : Vector α n\nx✝ : ∃ i h, List.get (toList v) { val := i, isLt := h } = a\ni : ℕ\nhi : i < List.length (toList v)\nh : List.get (toList v) { val := i, isLt := hi } = a\n⊢ i < n",
"tactic": "rwa [toList_length] at hi"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.160\nn : ℕ\na a' : α\nv : Vector α n\nx✝ : ∃ i h, List.get (toList v) (↑(Fin.cast (_ : n = List.length (toList v))) { val := i, isLt := h }) = a\ni : ℕ\nhi : i < n\nh : List.get (toList v) (↑(Fin.cast (_ : n = List.length (toList v))) { val := i, isLt := hi }) = a\n⊢ i < List.length (toList v)",
"tactic": "rwa [toList_length]"
}
] |
[
38,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
34,
1
] |
Std/Data/List/Lemmas.lean
|
List.get_mem
|
[] |
[
527,
45
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
525,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.pushoutCoconeOfRightIso_ι_app_none
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nW X Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\ninst✝ : IsIso g\n⊢ (pushoutCoconeOfRightIso f g).ι.app none = f",
"tactic": "simp"
}
] |
[
1871,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1870,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
intervalIntegrable_iff_integrable_Ioc_of_le
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.462744\n𝕜 : Type ?u.462747\nE : Type u_1\nF : Type ?u.462753\nA : Type ?u.462756\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhab : a ≤ b\n⊢ IntervalIntegrable f μ a b ↔ IntegrableOn f (Ioc a b)",
"tactic": "rw [intervalIntegrable_iff, uIoc_of_le hab]"
}
] |
[
96,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Order/Height.lean
|
Set.nil_mem_subchain
|
[] |
[
68,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/Analysis/Convex/Segment.lean
|
mem_segment_iff_sameRay
|
[
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.214396\nG : Type ?u.214399\nι : Type ?u.214402\nπ : ι → Type ?u.214407\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx y z : E\nh : SameRay 𝕜 (x - y) (z - x)\n⊢ x ∈ [y-[𝕜]z]",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.214396\nG : Type ?u.214399\nι : Type ?u.214402\nπ : ι → Type ?u.214407\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx y z : E\n⊢ x ∈ [y-[𝕜]z] ↔ SameRay 𝕜 (x - y) (z - x)",
"tactic": "refine' ⟨sameRay_of_mem_segment, fun h => _⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.214396\nG : Type ?u.214399\nι : Type ?u.214402\nπ : ι → Type ?u.214407\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx y z : E\nh : SameRay 𝕜 (x - y) (z - x)\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhxy : x - y = a • (x - y + (z - x))\nhzx : z - x = b • (x - y + (z - x))\n⊢ x ∈ [y-[𝕜]z]",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.214396\nG : Type ?u.214399\nι : Type ?u.214402\nπ : ι → Type ?u.214407\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx y z : E\nh : SameRay 𝕜 (x - y) (z - x)\n⊢ x ∈ [y-[𝕜]z]",
"tactic": "rcases h.exists_eq_smul_add with ⟨a, b, ha, hb, hab, hxy, hzx⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.214396\nG : Type ?u.214399\nι : Type ?u.214402\nπ : ι → Type ?u.214407\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx y z : E\nh : SameRay 𝕜 (x - y) (z - x)\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhxy : x - y = a • (z - y)\nhzx : z - x = b • (z - y)\n⊢ x ∈ [y-[𝕜]z]",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.214396\nG : Type ?u.214399\nι : Type ?u.214402\nπ : ι → Type ?u.214407\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx y z : E\nh : SameRay 𝕜 (x - y) (z - x)\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhxy : x - y = a • (x - y + (z - x))\nhzx : z - x = b • (x - y + (z - x))\n⊢ x ∈ [y-[𝕜]z]",
"tactic": "rw [add_comm, sub_add_sub_cancel] at hxy hzx"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.214396\nG : Type ?u.214399\nι : Type ?u.214402\nπ : ι → Type ?u.214407\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx y z : E\nh : SameRay 𝕜 (x - y) (z - x)\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhxy : x - y = a • (z - y)\nhzx : z - x = b • (z - y)\n⊢ 0 ∈ [-x + y-[𝕜]-x + z]",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.214396\nG : Type ?u.214399\nι : Type ?u.214402\nπ : ι → Type ?u.214407\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx y z : E\nh : SameRay 𝕜 (x - y) (z - x)\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhxy : x - y = a • (z - y)\nhzx : z - x = b • (z - y)\n⊢ x ∈ [y-[𝕜]z]",
"tactic": "rw [← mem_segment_translate _ (-x), neg_add_self]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.214396\nG : Type ?u.214399\nι : Type ?u.214402\nπ : ι → Type ?u.214407\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx y z : E\nh : SameRay 𝕜 (x - y) (z - x)\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhxy : x - y = a • (z - y)\nhzx : z - x = b • (z - y)\n⊢ b • (-x + y) + a • (-x + z) = 0",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.214396\nG : Type ?u.214399\nι : Type ?u.214402\nπ : ι → Type ?u.214407\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx y z : E\nh : SameRay 𝕜 (x - y) (z - x)\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhxy : x - y = a • (z - y)\nhzx : z - x = b • (z - y)\n⊢ 0 ∈ [-x + y-[𝕜]-x + z]",
"tactic": "refine' ⟨b, a, hb, ha, add_comm a b ▸ hab, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.214396\nG : Type ?u.214399\nι : Type ?u.214402\nπ : ι → Type ?u.214407\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx y z : E\nh : SameRay 𝕜 (x - y) (z - x)\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhxy : x - y = a • (z - y)\nhzx : z - x = b • (z - y)\n⊢ b • (-x + y) + a • (-x + z) = 0",
"tactic": "rw [← sub_eq_neg_add, ← neg_sub, hxy, ← sub_eq_neg_add, hzx, smul_neg, smul_comm, neg_add_self]"
}
] |
[
380,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
374,
1
] |
Mathlib/Control/Fold.lean
|
Traversable.foldrm_toList
|
[
{
"state_after": "α β γ : Type u\nt : Type u → Type u\ninst✝³ : Traversable t\ninst✝² : IsLawfulTraversable t\nm : Type u → Type u\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nf : α → β → m β\nx : β\nxs : t α\n⊢ foldrm f x xs = ↑(foldrM.ofFreeMonoid f) (↑FreeMonoid.ofList (toList xs)) x",
"state_before": "α β γ : Type u\nt : Type u → Type u\ninst✝³ : Traversable t\ninst✝² : IsLawfulTraversable t\nm : Type u → Type u\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nf : α → β → m β\nx : β\nxs : t α\n⊢ foldrm f x xs = List.foldrM f x (toList xs)",
"tactic": "change _ = foldrM.ofFreeMonoid f (FreeMonoid.ofList <| toList xs) x"
},
{
"state_after": "no goals",
"state_before": "α β γ : Type u\nt : Type u → Type u\ninst✝³ : Traversable t\ninst✝² : IsLawfulTraversable t\nm : Type u → Type u\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nf : α → β → m β\nx : β\nxs : t α\n⊢ foldrm f x xs = ↑(foldrM.ofFreeMonoid f) (↑FreeMonoid.ofList (toList xs)) x",
"tactic": "simp only [foldrm, toList_spec, foldMap_hom_free (foldrM.ofFreeMonoid f),\n foldrm.ofFreeMonoid_comp_of, foldrM.get, FreeMonoid.ofList_toList]"
}
] |
[
420,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
416,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.exists_succAbove_eq_iff
|
[
{
"state_after": "n m : ℕ\nx y : Fin (n + 1)\n⊢ (∃ z, ↑(succAbove x) z = y) → y ≠ x",
"state_before": "n m : ℕ\nx y : Fin (n + 1)\n⊢ (∃ z, ↑(succAbove x) z = y) ↔ y ≠ x",
"tactic": "refine' ⟨_, exists_succAbove_eq⟩"
},
{
"state_after": "case intro\nn m : ℕ\nx : Fin (n + 1)\ny : Fin n\n⊢ ↑(succAbove x) y ≠ x",
"state_before": "n m : ℕ\nx y : Fin (n + 1)\n⊢ (∃ z, ↑(succAbove x) z = y) → y ≠ x",
"tactic": "rintro ⟨y, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nn m : ℕ\nx : Fin (n + 1)\ny : Fin n\n⊢ ↑(succAbove x) y ≠ x",
"tactic": "exact succAbove_ne _ _"
}
] |
[
2164,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2161,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean
|
BoxIntegral.TaggedPrepartition.mem_single
|
[] |
[
302,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Integral.lean
|
MeasureTheory.StronglyMeasurable.integral_prod_right'
|
[
{
"state_after": "α : Type u_2\nα' : Type ?u.2378632\nβ : Type u_1\nβ' : Type ?u.2378638\nγ : Type ?u.2378641\nE : Type u_3\ninst✝⁸ : MeasurableSpace α\ninst✝⁷ : MeasurableSpace α'\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : MeasurableSpace β'\ninst✝⁴ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\ninst✝ : SigmaFinite ν\nf : α × β → E\nhf : StronglyMeasurable (uncurry (curry f))\n⊢ StronglyMeasurable fun x => ∫ (y : β), f (x, y) ∂ν",
"state_before": "α : Type u_2\nα' : Type ?u.2378632\nβ : Type u_1\nβ' : Type ?u.2378638\nγ : Type ?u.2378641\nE : Type u_3\ninst✝⁸ : MeasurableSpace α\ninst✝⁷ : MeasurableSpace α'\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : MeasurableSpace β'\ninst✝⁴ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\ninst✝ : SigmaFinite ν\nf : α × β → E\nhf : StronglyMeasurable f\n⊢ StronglyMeasurable fun x => ∫ (y : β), f (x, y) ∂ν",
"tactic": "rw [← uncurry_curry f] at hf"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nα' : Type ?u.2378632\nβ : Type u_1\nβ' : Type ?u.2378638\nγ : Type ?u.2378641\nE : Type u_3\ninst✝⁸ : MeasurableSpace α\ninst✝⁷ : MeasurableSpace α'\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : MeasurableSpace β'\ninst✝⁴ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\ninst✝ : SigmaFinite ν\nf : α × β → E\nhf : StronglyMeasurable (uncurry (curry f))\n⊢ StronglyMeasurable fun x => ∫ (y : β), f (x, y) ∂ν",
"tactic": "exact hf.integral_prod_right"
}
] |
[
130,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.prod_eq_prod_fintype
|
[
{
"state_after": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝³ : Fintype ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nv : Π₀ (i : ι), β i\nf : (i : ι) → β i → γ\nhf : ∀ (i : ι), f i 0 = 1\n⊢ ∏ i in support v, f i (↑v i) = ∏ i : ι, f i (↑v i)",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝³ : Fintype ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nv : Π₀ (i : ι), β i\nf : (i : ι) → β i → γ\nhf : ∀ (i : ι), f i 0 = 1\n⊢ prod v f = ∏ i : ι, f i (↑equivFunOnFintype v i)",
"tactic": "suffices (∏ i in v.support, f i (v i)) = ∏ i, f i (v i) by simp [Dfinsupp.prod, this]"
},
{
"state_after": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝³ : Fintype ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nv : Π₀ (i : ι), β i\nf : (i : ι) → β i → γ\nhf : ∀ (i : ι), f i 0 = 1\n⊢ ∀ (x : ι), x ∈ Finset.univ → ¬x ∈ support v → f x (↑v x) = 1",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝³ : Fintype ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nv : Π₀ (i : ι), β i\nf : (i : ι) → β i → γ\nhf : ∀ (i : ι), f i 0 = 1\n⊢ ∏ i in support v, f i (↑v i) = ∏ i : ι, f i (↑v i)",
"tactic": "apply Finset.prod_subset v.support.subset_univ"
},
{
"state_after": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝³ : Fintype ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nv : Π₀ (i : ι), β i\nf : (i : ι) → β i → γ\nhf : ∀ (i : ι), f i 0 = 1\ni : ι\na✝ : i ∈ Finset.univ\nhi : ¬i ∈ support v\n⊢ f i (↑v i) = 1",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝³ : Fintype ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nv : Π₀ (i : ι), β i\nf : (i : ι) → β i → γ\nhf : ∀ (i : ι), f i 0 = 1\n⊢ ∀ (x : ι), x ∈ Finset.univ → ¬x ∈ support v → f x (↑v x) = 1",
"tactic": "intro i _ hi"
},
{
"state_after": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝³ : Fintype ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nv : Π₀ (i : ι), β i\nf : (i : ι) → β i → γ\nhf : ∀ (i : ι), f i 0 = 1\ni : ι\na✝ : i ∈ Finset.univ\nhi : ↑v i = 0\n⊢ f i (↑v i) = 1",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝³ : Fintype ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nv : Π₀ (i : ι), β i\nf : (i : ι) → β i → γ\nhf : ∀ (i : ι), f i 0 = 1\ni : ι\na✝ : i ∈ Finset.univ\nhi : ¬i ∈ support v\n⊢ f i (↑v i) = 1",
"tactic": "rw [mem_support_iff, not_not] at hi"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝³ : Fintype ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nv : Π₀ (i : ι), β i\nf : (i : ι) → β i → γ\nhf : ∀ (i : ι), f i 0 = 1\ni : ι\na✝ : i ∈ Finset.univ\nhi : ↑v i = 0\n⊢ f i (↑v i) = 1",
"tactic": "rw [hi, hf]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝³ : Fintype ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nv : Π₀ (i : ι), β i\nf : (i : ι) → β i → γ\nhf : ∀ (i : ι), f i 0 = 1\nthis : ∏ i in support v, f i (↑v i) = ∏ i : ι, f i (↑v i)\n⊢ prod v f = ∏ i : ι, f i (↑equivFunOnFintype v i)",
"tactic": "simp [Dfinsupp.prod, this]"
}
] |
[
1857,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1849,
1
] |
Mathlib/MeasureTheory/Measure/VectorMeasure.lean
|
MeasureTheory.VectorMeasure.AbsolutelyContinuous.neg_right
|
[
{
"state_after": "α : Type u_2\nβ : Type ?u.628486\nm : MeasurableSpace α\nL : Type ?u.628492\nM : Type u_3\nN✝ : Type ?u.628498\ninst✝⁸ : AddCommMonoid L\ninst✝⁷ : TopologicalSpace L\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommMonoid N✝\ninst✝³ : TopologicalSpace N✝\nv✝ : VectorMeasure α M\nw✝ : VectorMeasure α N✝\nN : Type u_1\ninst✝² : AddCommGroup N\ninst✝¹ : TopologicalSpace N\ninst✝ : TopologicalAddGroup N\nv : VectorMeasure α M\nw : VectorMeasure α N\nh : v ≪ᵥ w\ns : Set α\nhs : ↑(-w) s = 0\n⊢ ↑v s = 0",
"state_before": "α : Type u_2\nβ : Type ?u.628486\nm : MeasurableSpace α\nL : Type ?u.628492\nM : Type u_3\nN✝ : Type ?u.628498\ninst✝⁸ : AddCommMonoid L\ninst✝⁷ : TopologicalSpace L\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommMonoid N✝\ninst✝³ : TopologicalSpace N✝\nv✝ : VectorMeasure α M\nw✝ : VectorMeasure α N✝\nN : Type u_1\ninst✝² : AddCommGroup N\ninst✝¹ : TopologicalSpace N\ninst✝ : TopologicalAddGroup N\nv : VectorMeasure α M\nw : VectorMeasure α N\nh : v ≪ᵥ w\n⊢ v ≪ᵥ -w",
"tactic": "intro s hs"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.628486\nm : MeasurableSpace α\nL : Type ?u.628492\nM : Type u_3\nN✝ : Type ?u.628498\ninst✝⁸ : AddCommMonoid L\ninst✝⁷ : TopologicalSpace L\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommMonoid N✝\ninst✝³ : TopologicalSpace N✝\nv✝ : VectorMeasure α M\nw✝ : VectorMeasure α N✝\nN : Type u_1\ninst✝² : AddCommGroup N\ninst✝¹ : TopologicalSpace N\ninst✝ : TopologicalAddGroup N\nv : VectorMeasure α M\nw : VectorMeasure α N\nh : v ≪ᵥ w\ns : Set α\nhs : ↑w s = 0\n⊢ ↑v s = 0",
"state_before": "α : Type u_2\nβ : Type ?u.628486\nm : MeasurableSpace α\nL : Type ?u.628492\nM : Type u_3\nN✝ : Type ?u.628498\ninst✝⁸ : AddCommMonoid L\ninst✝⁷ : TopologicalSpace L\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommMonoid N✝\ninst✝³ : TopologicalSpace N✝\nv✝ : VectorMeasure α M\nw✝ : VectorMeasure α N✝\nN : Type u_1\ninst✝² : AddCommGroup N\ninst✝¹ : TopologicalSpace N\ninst✝ : TopologicalAddGroup N\nv : VectorMeasure α M\nw : VectorMeasure α N\nh : v ≪ᵥ w\ns : Set α\nhs : ↑(-w) s = 0\n⊢ ↑v s = 0",
"tactic": "rw [neg_apply, neg_eq_zero] at hs"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.628486\nm : MeasurableSpace α\nL : Type ?u.628492\nM : Type u_3\nN✝ : Type ?u.628498\ninst✝⁸ : AddCommMonoid L\ninst✝⁷ : TopologicalSpace L\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommMonoid N✝\ninst✝³ : TopologicalSpace N✝\nv✝ : VectorMeasure α M\nw✝ : VectorMeasure α N✝\nN : Type u_1\ninst✝² : AddCommGroup N\ninst✝¹ : TopologicalSpace N\ninst✝ : TopologicalAddGroup N\nv : VectorMeasure α M\nw : VectorMeasure α N\nh : v ≪ᵥ w\ns : Set α\nhs : ↑w s = 0\n⊢ ↑v s = 0",
"tactic": "exact h hs"
}
] |
[
1124,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1120,
1
] |
Mathlib/Data/PNat/Prime.lean
|
PNat.Coprime.gcd_mul_left_cancel
|
[
{
"state_after": "m n k : ℕ+\nh : Coprime k n\n⊢ gcd (k * m) n = gcd m n",
"state_before": "m n k : ℕ+\n⊢ Coprime k n → gcd (k * m) n = gcd m n",
"tactic": "intro h"
},
{
"state_after": "case a\nm n k : ℕ+\nh : Coprime k n\n⊢ ↑(gcd (k * m) n) = ↑(gcd m n)",
"state_before": "m n k : ℕ+\nh : Coprime k n\n⊢ gcd (k * m) n = gcd m n",
"tactic": "apply eq"
},
{
"state_after": "case a\nm n k : ℕ+\nh : Coprime k n\n⊢ Nat.gcd (↑k * ↑m) ↑n = Nat.gcd ↑m ↑n",
"state_before": "case a\nm n k : ℕ+\nh : Coprime k n\n⊢ ↑(gcd (k * m) n) = ↑(gcd m n)",
"tactic": "simp only [gcd_coe, mul_coe]"
},
{
"state_after": "case a.H\nm n k : ℕ+\nh : Coprime k n\n⊢ coprime ↑k ↑n",
"state_before": "case a\nm n k : ℕ+\nh : Coprime k n\n⊢ Nat.gcd (↑k * ↑m) ↑n = Nat.gcd ↑m ↑n",
"tactic": "apply Nat.coprime.gcd_mul_left_cancel"
},
{
"state_after": "no goals",
"state_before": "case a.H\nm n k : ℕ+\nh : Coprime k n\n⊢ coprime ↑k ↑n",
"tactic": "simpa"
}
] |
[
210,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
|
coplanar_of_fact_finrank_eq_two
|
[] |
[
744,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
743,
1
] |
src/lean/Init/SimpLemmas.lean
|
beq_self_eq_true
|
[] |
[
148,
98
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
148,
9
] |
Mathlib/Algebra/Hom/Commute.lean
|
SemiconjBy.map
|
[
{
"state_after": "no goals",
"state_before": "F : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝² : Mul M\ninst✝¹ : Mul N\na x y : M\ninst✝ : MulHomClass F M N\nh : SemiconjBy a x y\nf : F\n⊢ SemiconjBy (↑f a) (↑f x) (↑f y)",
"tactic": "simpa only [SemiconjBy, map_mul] using congr_arg f h"
}
] |
[
27,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
26,
11
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
span_singleton_dvd_span_singleton_iff_dvd
|
[] |
[
1414,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1411,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.embDomain_mapRange
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.236087\nι : Type ?u.236090\nM : Type u_3\nM' : Type ?u.236096\nN : Type u_4\nP : Type ?u.236102\nG : Type ?u.236105\nH : Type ?u.236108\nR : Type ?u.236111\nS : Type ?u.236114\ninst✝¹ : Zero M\ninst✝ : Zero N\nf : α ↪ β\ng : M → N\np : α →₀ M\nhg : g 0 = 0\na : β\n⊢ ↑(embDomain f (mapRange g hg p)) a = ↑(mapRange g hg (embDomain f p)) a",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.236087\nι : Type ?u.236090\nM : Type u_3\nM' : Type ?u.236096\nN : Type u_4\nP : Type ?u.236102\nG : Type ?u.236105\nH : Type ?u.236108\nR : Type ?u.236111\nS : Type ?u.236114\ninst✝¹ : Zero M\ninst✝ : Zero N\nf : α ↪ β\ng : M → N\np : α →₀ M\nhg : g 0 = 0\n⊢ embDomain f (mapRange g hg p) = mapRange g hg (embDomain f p)",
"tactic": "ext a"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.236087\nι : Type ?u.236090\nM : Type u_3\nM' : Type ?u.236096\nN : Type u_4\nP : Type ?u.236102\nG : Type ?u.236105\nH : Type ?u.236108\nR : Type ?u.236111\nS : Type ?u.236114\ninst✝¹ : Zero M\ninst✝ : Zero N\nf : α ↪ β\ng : M → N\np : α →₀ M\nhg : g 0 = 0\na : β\nh : a ∈ Set.range ↑f\n⊢ ↑(embDomain f (mapRange g hg p)) a = ↑(mapRange g hg (embDomain f p)) a\n\ncase neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.236087\nι : Type ?u.236090\nM : Type u_3\nM' : Type ?u.236096\nN : Type u_4\nP : Type ?u.236102\nG : Type ?u.236105\nH : Type ?u.236108\nR : Type ?u.236111\nS : Type ?u.236114\ninst✝¹ : Zero M\ninst✝ : Zero N\nf : α ↪ β\ng : M → N\np : α →₀ M\nhg : g 0 = 0\na : β\nh : ¬a ∈ Set.range ↑f\n⊢ ↑(embDomain f (mapRange g hg p)) a = ↑(mapRange g hg (embDomain f p)) a",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.236087\nι : Type ?u.236090\nM : Type u_3\nM' : Type ?u.236096\nN : Type u_4\nP : Type ?u.236102\nG : Type ?u.236105\nH : Type ?u.236108\nR : Type ?u.236111\nS : Type ?u.236114\ninst✝¹ : Zero M\ninst✝ : Zero N\nf : α ↪ β\ng : M → N\np : α →₀ M\nhg : g 0 = 0\na : β\n⊢ ↑(embDomain f (mapRange g hg p)) a = ↑(mapRange g hg (embDomain f p)) a",
"tactic": "by_cases h : a ∈ Set.range f"
},
{
"state_after": "case pos.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.236087\nι : Type ?u.236090\nM : Type u_3\nM' : Type ?u.236096\nN : Type u_4\nP : Type ?u.236102\nG : Type ?u.236105\nH : Type ?u.236108\nR : Type ?u.236111\nS : Type ?u.236114\ninst✝¹ : Zero M\ninst✝ : Zero N\nf : α ↪ β\ng : M → N\np : α →₀ M\nhg : g 0 = 0\na' : α\n⊢ ↑(embDomain f (mapRange g hg p)) (↑f a') = ↑(mapRange g hg (embDomain f p)) (↑f a')",
"state_before": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.236087\nι : Type ?u.236090\nM : Type u_3\nM' : Type ?u.236096\nN : Type u_4\nP : Type ?u.236102\nG : Type ?u.236105\nH : Type ?u.236108\nR : Type ?u.236111\nS : Type ?u.236114\ninst✝¹ : Zero M\ninst✝ : Zero N\nf : α ↪ β\ng : M → N\np : α →₀ M\nhg : g 0 = 0\na : β\nh : a ∈ Set.range ↑f\n⊢ ↑(embDomain f (mapRange g hg p)) a = ↑(mapRange g hg (embDomain f p)) a",
"tactic": "rcases h with ⟨a', rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case pos.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.236087\nι : Type ?u.236090\nM : Type u_3\nM' : Type ?u.236096\nN : Type u_4\nP : Type ?u.236102\nG : Type ?u.236105\nH : Type ?u.236108\nR : Type ?u.236111\nS : Type ?u.236114\ninst✝¹ : Zero M\ninst✝ : Zero N\nf : α ↪ β\ng : M → N\np : α →₀ M\nhg : g 0 = 0\na' : α\n⊢ ↑(embDomain f (mapRange g hg p)) (↑f a') = ↑(mapRange g hg (embDomain f p)) (↑f a')",
"tactic": "rw [mapRange_apply, embDomain_apply, embDomain_apply, mapRange_apply]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.236087\nι : Type ?u.236090\nM : Type u_3\nM' : Type ?u.236096\nN : Type u_4\nP : Type ?u.236102\nG : Type ?u.236105\nH : Type ?u.236108\nR : Type ?u.236111\nS : Type ?u.236114\ninst✝¹ : Zero M\ninst✝ : Zero N\nf : α ↪ β\ng : M → N\np : α →₀ M\nhg : g 0 = 0\na : β\nh : ¬a ∈ Set.range ↑f\n⊢ ↑(embDomain f (mapRange g hg p)) a = ↑(mapRange g hg (embDomain f p)) a",
"tactic": "rw [mapRange_apply, embDomain_notin_range, embDomain_notin_range, ← hg] <;> assumption"
}
] |
[
891,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
885,
1
] |
Mathlib/Data/Rat/Cast.lean
|
eq_ratCast
|
[
{
"state_after": "no goals",
"state_before": "F : Type u_2\nι : Type ?u.81845\nα : Type ?u.81848\nβ : Type ?u.81851\nk : Type u_1\ninst✝¹ : DivisionRing k\ninst✝ : RingHomClass F ℚ k\nf : F\nr : ℚ\n⊢ ↑f r = ↑r",
"tactic": "rw [← map_ratCast f, Rat.cast_id]"
}
] |
[
444,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
443,
1
] |
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