file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/Finset/Lattice.lean
|
List.foldr_inf_eq_inf_toFinset
|
[
{
"state_after": "F : Type ?u.129296\nα : Type u_1\nβ : Type ?u.129302\nγ : Type ?u.129305\nι : Type ?u.129308\nκ : Type ?u.129311\ninst✝² : SemilatticeInf α\ninst✝¹ : OrderTop α\ns s₁ s₂ : Finset β\nf g : β → α\na : α\ninst✝ : DecidableEq α\nl : List α\n⊢ Multiset.fold (fun x x_1 => x ⊓ x_1) ⊤ (dedup ↑l) = Multiset.inf (dedup ↑l)",
"state_before": "F : Type ?u.129296\nα : Type u_1\nβ : Type ?u.129302\nγ : Type ?u.129305\nι : Type ?u.129308\nκ : Type ?u.129311\ninst✝² : SemilatticeInf α\ninst✝¹ : OrderTop α\ns s₁ s₂ : Finset β\nf g : β → α\na : α\ninst✝ : DecidableEq α\nl : List α\n⊢ List.foldr (fun x x_1 => x ⊓ x_1) ⊤ l = inf (List.toFinset l) id",
"tactic": "rw [← coe_fold_r, ← Multiset.fold_dedup_idem, inf_def, ← List.toFinset_coe, toFinset_val,\n Multiset.map_id]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.129296\nα : Type u_1\nβ : Type ?u.129302\nγ : Type ?u.129305\nι : Type ?u.129308\nκ : Type ?u.129311\ninst✝² : SemilatticeInf α\ninst✝¹ : OrderTop α\ns s₁ s₂ : Finset β\nf g : β → α\na : α\ninst✝ : DecidableEq α\nl : List α\n⊢ Multiset.fold (fun x x_1 => x ⊓ x_1) ⊤ (dedup ↑l) = Multiset.inf (dedup ↑l)",
"tactic": "rfl"
}
] |
[
462,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
458,
1
] |
Mathlib/Analysis/Calculus/MeanValue.lean
|
mul_sub_le_image_sub_of_le_deriv
|
[] |
[
840,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
837,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
Filter.EventuallyEq.fderivWithin
|
[] |
[
957,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
955,
11
] |
Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean
|
Complex.one_cpow
|
[
{
"state_after": "x : ℂ\n⊢ (if 1 = 0 then if x = 0 then 1 else 0 else exp (log 1 * x)) = 1",
"state_before": "x : ℂ\n⊢ 1 ^ x = 1",
"tactic": "rw [cpow_def]"
},
{
"state_after": "no goals",
"state_before": "x : ℂ\n⊢ (if 1 = 0 then if x = 0 then 1 else 0 else exp (log 1 * x)) = 1",
"tactic": "split_ifs <;> simp_all [one_ne_zero]"
}
] |
[
91,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/Geometry/Manifold/ChartedSpace.lean
|
OpenEmbedding.singleton_hasGroupoid
|
[] |
[
1042,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1040,
1
] |
Mathlib/Algebra/Star/Unitary.lean
|
unitary.coe_star_mul_self
|
[] |
[
87,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/Combinatorics/SimpleGraph/Coloring.lean
|
SimpleGraph.Coloring.valid
|
[] |
[
75,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
|
NonUnitalSubsemiring.center_le_centralizer
|
[] |
[
536,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
535,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
TensorProduct.lid_tmul
|
[] |
[
633,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
632,
1
] |
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieModule.lowerCentralSeries_succ
|
[] |
[
93,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
AffineSubspace.map_iSup
|
[] |
[
1671,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1669,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.single_smul
|
[
{
"state_after": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : Monoid γ\ninst✝¹ : (i : ι) → AddMonoid (β i)\ninst✝ : (i : ι) → DistribMulAction γ (β i)\ni✝ : ι\nc : γ\nx : β i✝\ni : ι\n⊢ (if h : i✝ = i then (_ : i✝ = i) ▸ (c • x) else 0) = c • if h : i✝ = i then (_ : i✝ = i) ▸ x else 0",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : Monoid γ\ninst✝¹ : (i : ι) → AddMonoid (β i)\ninst✝ : (i : ι) → DistribMulAction γ (β i)\ni✝ : ι\nc : γ\nx : β i✝\ni : ι\n⊢ ↑(single i✝ (c • x)) i = ↑(c • single i✝ x) i",
"tactic": "simp only [smul_apply, single_apply]"
},
{
"state_after": "case inl\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : Monoid γ\ninst✝¹ : (i : ι) → AddMonoid (β i)\ninst✝ : (i : ι) → DistribMulAction γ (β i)\ni✝ : ι\nc : γ\nx : β i✝\ni : ι\nh : i✝ = i\n⊢ (_ : i✝ = i) ▸ (c • x) = c • (_ : i✝ = i) ▸ x\n\ncase inr\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : Monoid γ\ninst✝¹ : (i : ι) → AddMonoid (β i)\ninst✝ : (i : ι) → DistribMulAction γ (β i)\ni✝ : ι\nc : γ\nx : β i✝\ni : ι\nh : ¬i✝ = i\n⊢ 0 = c • 0",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : Monoid γ\ninst✝¹ : (i : ι) → AddMonoid (β i)\ninst✝ : (i : ι) → DistribMulAction γ (β i)\ni✝ : ι\nc : γ\nx : β i✝\ni : ι\n⊢ (if h : i✝ = i then (_ : i✝ = i) ▸ (c • x) else 0) = c • if h : i✝ = i then (_ : i✝ = i) ▸ x else 0",
"tactic": "split_ifs with h"
},
{
"state_after": "case inl.refl\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : Monoid γ\ninst✝¹ : (i : ι) → AddMonoid (β i)\ninst✝ : (i : ι) → DistribMulAction γ (β i)\ni : ι\nc : γ\nx : β i\n⊢ (_ : i = i) ▸ (c • x) = c • (_ : i = i) ▸ x",
"state_before": "case inl\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : Monoid γ\ninst✝¹ : (i : ι) → AddMonoid (β i)\ninst✝ : (i : ι) → DistribMulAction γ (β i)\ni✝ : ι\nc : γ\nx : β i✝\ni : ι\nh : i✝ = i\n⊢ (_ : i✝ = i) ▸ (c • x) = c • (_ : i✝ = i) ▸ x",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case inl.refl\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : Monoid γ\ninst✝¹ : (i : ι) → AddMonoid (β i)\ninst✝ : (i : ι) → DistribMulAction γ (β i)\ni : ι\nc : γ\nx : β i\n⊢ (_ : i = i) ▸ (c • x) = c • (_ : i = i) ▸ x",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case inr\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : Monoid γ\ninst✝¹ : (i : ι) → AddMonoid (β i)\ninst✝ : (i : ι) → DistribMulAction γ (β i)\ni✝ : ι\nc : γ\nx : β i✝\ni : ι\nh : ¬i✝ = i\n⊢ 0 = c • 0",
"tactic": "rw [smul_zero]"
}
] |
[
1083,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1078,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean
|
linearIndependent_sUnion_of_directed
|
[
{
"state_after": "ι : Type u'\nι' : Type ?u.216784\nR : Type u_2\nK : Type ?u.216790\nM : Type u_1\nM' : Type ?u.216796\nM'' : Type ?u.216799\nV : Type u\nV' : Type ?u.216804\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\ns : Set (Set M)\nhs : DirectedOn (fun x x_1 => x ⊆ x_1) s\nh : ∀ (a : Set M), a ∈ s → LinearIndependent R Subtype.val\n⊢ LinearIndependent R fun x => ↑x",
"state_before": "ι : Type u'\nι' : Type ?u.216784\nR : Type u_2\nK : Type ?u.216790\nM : Type u_1\nM' : Type ?u.216796\nM'' : Type ?u.216799\nV : Type u\nV' : Type ?u.216804\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\ns : Set (Set M)\nhs : DirectedOn (fun x x_1 => x ⊆ x_1) s\nh : ∀ (a : Set M), a ∈ s → LinearIndependent R Subtype.val\n⊢ LinearIndependent R fun x => ↑x",
"tactic": "rw [sUnion_eq_iUnion]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.216784\nR : Type u_2\nK : Type ?u.216790\nM : Type u_1\nM' : Type ?u.216796\nM'' : Type ?u.216799\nV : Type u\nV' : Type ?u.216804\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\ns : Set (Set M)\nhs : DirectedOn (fun x x_1 => x ⊆ x_1) s\nh : ∀ (a : Set M), a ∈ s → LinearIndependent R Subtype.val\n⊢ LinearIndependent R fun x => ↑x",
"tactic": "exact linearIndependent_iUnion_of_directed hs.directed_val (by simpa using h)"
},
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.216784\nR : Type u_2\nK : Type ?u.216790\nM : Type u_1\nM' : Type ?u.216796\nM'' : Type ?u.216799\nV : Type u\nV' : Type ?u.216804\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\ns : Set (Set M)\nhs : DirectedOn (fun x x_1 => x ⊆ x_1) s\nh : ∀ (a : Set M), a ∈ s → LinearIndependent R Subtype.val\n⊢ ∀ (i : ↑s), LinearIndependent R fun x => ↑x",
"tactic": "simpa using h"
}
] |
[
451,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
447,
1
] |
Mathlib/MeasureTheory/Integral/MeanInequalities.lean
|
ENNReal.funMulInvSnorm_rpow
|
[
{
"state_after": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nhp0 : 0 < p\nf : α → ℝ≥0∞\na : α\n⊢ f a ^ p * ((∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p))⁻¹ ^ p = f a ^ p * (∫⁻ (c : α), f c ^ p ∂μ)⁻¹",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nhp0 : 0 < p\nf : α → ℝ≥0∞\na : α\n⊢ funMulInvSnorm f p μ a ^ p = f a ^ p * (∫⁻ (c : α), f c ^ p ∂μ)⁻¹",
"tactic": "rw [funMulInvSnorm, mul_rpow_of_nonneg _ _ (le_of_lt hp0)]"
},
{
"state_after": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nhp0 : 0 < p\nf : α → ℝ≥0∞\na : α\nh_inv_rpow : ((∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p))⁻¹ ^ p = (∫⁻ (c : α), f c ^ p ∂μ)⁻¹\n⊢ f a ^ p * ((∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p))⁻¹ ^ p = f a ^ p * (∫⁻ (c : α), f c ^ p ∂μ)⁻¹\n\ncase h_inv_rpow\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nhp0 : 0 < p\nf : α → ℝ≥0∞\na : α\n⊢ ((∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p))⁻¹ ^ p = (∫⁻ (c : α), f c ^ p ∂μ)⁻¹",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nhp0 : 0 < p\nf : α → ℝ≥0∞\na : α\n⊢ f a ^ p * ((∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p))⁻¹ ^ p = f a ^ p * (∫⁻ (c : α), f c ^ p ∂μ)⁻¹",
"tactic": "suffices h_inv_rpow : ((∫⁻ c : α, f c ^ p ∂μ) ^ (1 / p))⁻¹ ^ p = (∫⁻ c : α, f c ^ p ∂μ)⁻¹"
},
{
"state_after": "no goals",
"state_before": "case h_inv_rpow\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nhp0 : 0 < p\nf : α → ℝ≥0∞\na : α\n⊢ ((∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p))⁻¹ ^ p = (∫⁻ (c : α), f c ^ p ∂μ)⁻¹",
"tactic": "rw [inv_rpow, ← rpow_mul, one_div_mul_cancel hp0.ne', rpow_one]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nhp0 : 0 < p\nf : α → ℝ≥0∞\na : α\nh_inv_rpow : ((∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p))⁻¹ ^ p = (∫⁻ (c : α), f c ^ p ∂μ)⁻¹\n⊢ f a ^ p * ((∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p))⁻¹ ^ p = f a ^ p * (∫⁻ (c : α), f c ^ p ∂μ)⁻¹",
"tactic": "rw [h_inv_rpow]"
}
] |
[
96,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Algebra/Lie/CartanSubalgebra.lean
|
LieSubalgebra.ucs_eq_self_of_isCartanSubalgebra
|
[
{
"state_after": "case zero\nR : Type u\nL : Type v\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nH✝ H : LieSubalgebra R L\ninst✝ : IsCartanSubalgebra H\n⊢ LieSubmodule.ucs Nat.zero (toLieSubmodule H) = toLieSubmodule H\n\ncase succ\nR : Type u\nL : Type v\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nH✝ H : LieSubalgebra R L\ninst✝ : IsCartanSubalgebra H\nk : ℕ\nih : LieSubmodule.ucs k (toLieSubmodule H) = toLieSubmodule H\n⊢ LieSubmodule.ucs (Nat.succ k) (toLieSubmodule H) = toLieSubmodule H",
"state_before": "R : Type u\nL : Type v\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nH✝ H : LieSubalgebra R L\ninst✝ : IsCartanSubalgebra H\nk : ℕ\n⊢ LieSubmodule.ucs k (toLieSubmodule H) = toLieSubmodule H",
"tactic": "induction' k with k ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u\nL : Type v\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nH✝ H : LieSubalgebra R L\ninst✝ : IsCartanSubalgebra H\n⊢ LieSubmodule.ucs Nat.zero (toLieSubmodule H) = toLieSubmodule H",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : Type u\nL : Type v\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nH✝ H : LieSubalgebra R L\ninst✝ : IsCartanSubalgebra H\nk : ℕ\nih : LieSubmodule.ucs k (toLieSubmodule H) = toLieSubmodule H\n⊢ LieSubmodule.ucs (Nat.succ k) (toLieSubmodule H) = toLieSubmodule H",
"tactic": "simp [ih]"
}
] |
[
71,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
hasFTaylorSeriesUpTo_top_iff
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\n⊢ HasFTaylorSeriesUpTo ⊤ f p ↔ ∀ (n : ℕ), HasFTaylorSeriesUpTo (↑n) f p",
"tactic": "simp only [← hasFTaylorSeriesUpToOn_univ_iff, hasFTaylorSeriesUpToOn_top_iff]"
}
] |
[
1272,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1270,
1
] |
Mathlib/Data/List/Sigma.lean
|
List.sizeOf_dedupKeys
|
[
{
"state_after": "α✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\nxs : List (Sigma β)\n⊢ rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys xs) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs",
"state_before": "α✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\nxs : List (Sigma β)\n⊢ sizeOf (dedupKeys xs) ≤ sizeOf xs",
"tactic": "simp only [SizeOf.sizeOf, _sizeOf_1]"
},
{
"state_after": "case nil\nα✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\n⊢ rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys []) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) []\n\ncase cons\nα✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\nx : Sigma β\nxs : List (Sigma β)\ntail_ih✝ :\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys xs) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n⊢ rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys (x :: xs)) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (x :: xs)",
"state_before": "α✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\nxs : List (Sigma β)\n⊢ rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys xs) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs",
"tactic": "induction' xs with x xs"
},
{
"state_after": "no goals",
"state_before": "case nil\nα✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\n⊢ rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys []) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) []",
"tactic": "simp [dedupKeys]"
},
{
"state_after": "case cons\nα✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\nx : Sigma β\nxs : List (Sigma β)\ntail_ih✝ :\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys xs) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n⊢ rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x.fst (dedupKeys xs)) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs",
"state_before": "case cons\nα✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\nx : Sigma β\nxs : List (Sigma β)\ntail_ih✝ :\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys xs) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n⊢ rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys (x :: xs)) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (x :: xs)",
"tactic": "simp only [dedupKeys_cons, kinsert_def, add_le_add_iff_left, Sigma.eta]"
},
{
"state_after": "α✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\nx : Sigma β\nxs : List (Sigma β)\ntail_ih✝ :\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys xs) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n⊢ rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x.fst (dedupKeys xs)) ≤ ?m.358104\n\nα✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\nx : Sigma β\nxs : List (Sigma β)\ntail_ih✝ :\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys xs) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n⊢ ?m.358104 ≤ rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n\nα✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\nx : Sigma β\nxs : List (Sigma β)\ntail_ih✝ :\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys xs) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n⊢ ℕ",
"state_before": "case cons\nα✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\nx : Sigma β\nxs : List (Sigma β)\ntail_ih✝ :\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys xs) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n⊢ rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x.fst (dedupKeys xs)) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs",
"tactic": "trans"
},
{
"state_after": "α✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\nx : Sigma β\nxs : List (Sigma β)\ntail_ih✝ :\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys xs) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n⊢ sizeOf (dedupKeys xs) ≤ rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs",
"state_before": "α✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\nx : Sigma β\nxs : List (Sigma β)\ntail_ih✝ :\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys xs) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n⊢ rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x.fst (dedupKeys xs)) ≤ ?m.358104\n\nα✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\nx : Sigma β\nxs : List (Sigma β)\ntail_ih✝ :\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys xs) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n⊢ ?m.358104 ≤ rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n\nα✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\nx : Sigma β\nxs : List (Sigma β)\ntail_ih✝ :\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys xs) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n⊢ ℕ",
"tactic": "apply sizeOf_kerase"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\nx : Sigma β\nxs : List (Sigma β)\ntail_ih✝ :\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys xs) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n⊢ sizeOf (dedupKeys xs) ≤ rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs",
"tactic": "assumption"
}
] |
[
671,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
663,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
HasSum.prod_mk
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type ?u.796690\ninst✝³ : AddCommMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : AddCommMonoid γ\ninst✝ : TopologicalSpace γ\nf : β → α\ng : β → γ\na : α\nb : γ\nhf : HasSum f a\nhg : HasSum g b\n⊢ HasSum (fun x => (f x, g x)) (a, b)",
"tactic": "simp [HasSum, ← prod_mk_sum, Filter.Tendsto.prod_mk_nhds hf hg]"
}
] |
[
1303,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1301,
1
] |
Std/Data/String/Lemmas.lean
|
Substring.ValidFor.stopPos
|
[] |
[
810,
17
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
809,
1
] |
Mathlib/CategoryTheory/Sites/Grothendieck.lean
|
CategoryTheory.GrothendieckTopology.Cover.Relation.map_snd
|
[] |
[
556,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
554,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.Cofork.app_one_eq_π
|
[] |
[
336,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
335,
1
] |
Mathlib/Data/List/Perm.lean
|
List.perm_cons
|
[] |
[
650,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
649,
1
] |
Mathlib/Logic/IsEmpty.lean
|
Function.isEmpty
|
[] |
[
44,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
11
] |
Mathlib/Topology/UniformSpace/Separation.lean
|
separationRel_iff_inseparable
|
[] |
[
122,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/CategoryTheory/Monoidal/End.lean
|
CategoryTheory.obj_μ_zero_app
|
[
{
"state_after": "C : Type u\ninst✝² : Category C\nM : Type u_2\ninst✝¹ : Category M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nm₁ m₂ : M\nX : C\n⊢ (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor (𝟙_ M) m₂).app ((F.obj m₁).obj X) ≫\n (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor m₁ (𝟙_ M ⊗ m₂)).app X ≫\n (F.map (α_ m₁ (𝟙_ M) m₂).inv).app X ≫ (MonoidalFunctor.μIso F (m₁ ⊗ 𝟙_ M) m₂).inv.app X =\n (F.obj m₂).map ((MonoidalFunctor.εIso F).inv.app ((F.obj m₁).obj X) ≫ (F.map (ρ_ m₁).inv).app X)",
"state_before": "C : Type u\ninst✝² : Category C\nM : Type u_2\ninst✝¹ : Category M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nm₁ m₂ : M\nX : C\n⊢ (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor (𝟙_ M) m₂).app ((F.obj m₁).obj X) ≫\n (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor m₁ (𝟙_ M ⊗ m₂)).app X ≫\n (F.map (α_ m₁ (𝟙_ M) m₂).inv).app X ≫ (MonoidalFunctor.μIso F (m₁ ⊗ 𝟙_ M) m₂).inv.app X =\n (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor (𝟙_ M) m₂).app ((F.obj m₁).obj X) ≫\n (F.map (λ_ m₂).hom).app ((F.obj m₁).obj X) ≫ (F.obj m₂).map ((F.map (ρ_ m₁).inv).app X)",
"tactic": "rw [← obj_ε_inv_app_assoc, ← Functor.map_comp]"
},
{
"state_after": "C : Type u\ninst✝² : Category C\nM : Type u_2\ninst✝¹ : Category M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nm₁ m₂ : M\nX : C\n⊢ (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor (𝟙_ M) m₂).app ((F.obj m₁).obj X) ≫\n (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor m₁ (𝟙_ M ⊗ m₂)).app X ≫\n (F.map (α_ m₁ (𝟙_ M) m₂).inv).app X ≫ (MonoidalFunctor.μIso F (m₁ ⊗ 𝟙_ M) m₂).inv.app X =\n (F.obj m₂).map ((MonoidalFunctor.εIso F).inv.app ((F.obj m₁).obj X) ≫ (F.map (ρ_ m₁).inv).app X)",
"state_before": "C : Type u\ninst✝² : Category C\nM : Type u_2\ninst✝¹ : Category M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nm₁ m₂ : M\nX : C\n⊢ (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor (𝟙_ M) m₂).app ((F.obj m₁).obj X) ≫\n (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor m₁ (𝟙_ M ⊗ m₂)).app X ≫\n (F.map (α_ m₁ (𝟙_ M) m₂).inv).app X ≫ (MonoidalFunctor.μIso F (m₁ ⊗ 𝟙_ M) m₂).inv.app X =\n (F.obj m₂).map ((MonoidalFunctor.εIso F).inv.app ((F.obj m₁).obj X) ≫ (F.map (ρ_ m₁).inv).app X)",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nM : Type u_2\ninst✝¹ : Category M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nm₁ m₂ : M\nX : C\n⊢ (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor (𝟙_ M) m₂).app ((F.obj m₁).obj X) ≫\n (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor m₁ (𝟙_ M ⊗ m₂)).app X ≫\n (F.map (α_ m₁ (𝟙_ M) m₂).inv).app X ≫ (MonoidalFunctor.μIso F (m₁ ⊗ 𝟙_ M) m₂).inv.app X =\n (F.obj m₂).map ((MonoidalFunctor.εIso F).inv.app ((F.obj m₁).obj X) ≫ (F.map (ρ_ m₁).inv).app X)",
"tactic": "simp"
}
] |
[
298,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
292,
1
] |
Mathlib/RingTheory/Polynomial/Basic.lean
|
MvPolynomial.mem_map_C_iff
|
[
{
"state_after": "case mp\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\n⊢ f ∈ Ideal.map C I → ∀ (m : σ →₀ ℕ), coeff m f ∈ I\n\ncase mpr\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\n⊢ (∀ (m : σ →₀ ℕ), coeff m f ∈ I) → f ∈ Ideal.map C I",
"state_before": "R : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\n⊢ f ∈ Ideal.map C I ↔ ∀ (m : σ →₀ ℕ), coeff m f ∈ I",
"tactic": "constructor"
},
{
"state_after": "case mp\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : f ∈ Ideal.map C I\n⊢ ∀ (m : σ →₀ ℕ), coeff m f ∈ I",
"state_before": "case mp\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\n⊢ f ∈ Ideal.map C I → ∀ (m : σ →₀ ℕ), coeff m f ∈ I",
"tactic": "intro hf"
},
{
"state_after": "case mp.Hs\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : f ∈ Ideal.map C I\n⊢ ∀ (x : MvPolynomial σ R), x ∈ ↑C '' ↑I → ∀ (m : σ →₀ ℕ), coeff m x ∈ I\n\ncase mp.H0\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : f ∈ Ideal.map C I\n⊢ ∀ (m : σ →₀ ℕ), coeff m 0 ∈ I\n\ncase mp.H1\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : f ∈ Ideal.map C I\n⊢ ∀ (x y : MvPolynomial σ R),\n (∀ (m : σ →₀ ℕ), coeff m x ∈ I) → (∀ (m : σ →₀ ℕ), coeff m y ∈ I) → ∀ (m : σ →₀ ℕ), coeff m (x + y) ∈ I\n\ncase mp.H2\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : f ∈ Ideal.map C I\n⊢ ∀ (a x : MvPolynomial σ R), (∀ (m : σ →₀ ℕ), coeff m x ∈ I) → ∀ (m : σ →₀ ℕ), coeff m (a • x) ∈ I",
"state_before": "case mp\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : f ∈ Ideal.map C I\n⊢ ∀ (m : σ →₀ ℕ), coeff m f ∈ I",
"tactic": "apply @Submodule.span_induction _ _ _ _ Semiring.toModule f _ _ hf"
},
{
"state_after": "case mp.Hs\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf✝ : MvPolynomial σ R\nhf✝ : f✝ ∈ Ideal.map C I\nf : MvPolynomial σ R\nhf : f ∈ ↑C '' ↑I\nn : σ →₀ ℕ\n⊢ coeff n f ∈ I",
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"tactic": "intro f hf n"
},
{
"state_after": "case mp.Hs.intro\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf✝ : MvPolynomial σ R\nhf✝ : f✝ ∈ Ideal.map C I\nf : MvPolynomial σ R\nhf : f ∈ ↑C '' ↑I\nn : σ →₀ ℕ\nx : R\nhx : x ∈ ↑I ∧ ↑C x = f\n⊢ coeff n f ∈ I",
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"tactic": "cases' (Set.mem_image _ _ _).mp hf with x hx"
},
{
"state_after": "case mp.Hs.intro\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf✝ : MvPolynomial σ R\nhf✝ : f✝ ∈ Ideal.map C I\nf : MvPolynomial σ R\nhf : f ∈ ↑C '' ↑I\nn : σ →₀ ℕ\nx : R\nhx : x ∈ ↑I ∧ ↑C x = f\n⊢ (if 0 = n then x else 0) ∈ I",
"state_before": "case mp.Hs.intro\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf✝ : MvPolynomial σ R\nhf✝ : f✝ ∈ Ideal.map C I\nf : MvPolynomial σ R\nhf : f ∈ ↑C '' ↑I\nn : σ →₀ ℕ\nx : R\nhx : x ∈ ↑I ∧ ↑C x = f\n⊢ coeff n f ∈ I",
"tactic": "rw [← hx.right, coeff_C]"
},
{
"state_after": "case pos\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf✝ : MvPolynomial σ R\nhf✝ : f✝ ∈ Ideal.map C I\nf : MvPolynomial σ R\nhf : f ∈ ↑C '' ↑I\nn : σ →₀ ℕ\nx : R\nhx : x ∈ ↑I ∧ ↑C x = f\nh : n = 0\n⊢ (if 0 = n then x else 0) ∈ I\n\ncase neg\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf✝ : MvPolynomial σ R\nhf✝ : f✝ ∈ Ideal.map C I\nf : MvPolynomial σ R\nhf : f ∈ ↑C '' ↑I\nn : σ →₀ ℕ\nx : R\nhx : x ∈ ↑I ∧ ↑C x = f\nh : ¬n = 0\n⊢ (if 0 = n then x else 0) ∈ I",
"state_before": "case mp.Hs.intro\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf✝ : MvPolynomial σ R\nhf✝ : f✝ ∈ Ideal.map C I\nf : MvPolynomial σ R\nhf : f ∈ ↑C '' ↑I\nn : σ →₀ ℕ\nx : R\nhx : x ∈ ↑I ∧ ↑C x = f\n⊢ (if 0 = n then x else 0) ∈ I",
"tactic": "by_cases h : n = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf✝ : MvPolynomial σ R\nhf✝ : f✝ ∈ Ideal.map C I\nf : MvPolynomial σ R\nhf : f ∈ ↑C '' ↑I\nn : σ →₀ ℕ\nx : R\nhx : x ∈ ↑I ∧ ↑C x = f\nh : n = 0\n⊢ (if 0 = n then x else 0) ∈ I",
"tactic": "simpa [h] using hx.left"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf✝ : MvPolynomial σ R\nhf✝ : f✝ ∈ Ideal.map C I\nf : MvPolynomial σ R\nhf : f ∈ ↑C '' ↑I\nn : σ →₀ ℕ\nx : R\nhx : x ∈ ↑I ∧ ↑C x = f\nh : ¬n = 0\n⊢ (if 0 = n then x else 0) ∈ I",
"tactic": "simp [Ne.symm h]"
},
{
"state_after": "no goals",
"state_before": "case mp.H0\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : f ∈ Ideal.map C I\n⊢ ∀ (m : σ →₀ ℕ), coeff m 0 ∈ I",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case mp.H1\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : f ∈ Ideal.map C I\n⊢ ∀ (x y : MvPolynomial σ R),\n (∀ (m : σ →₀ ℕ), coeff m x ∈ I) → (∀ (m : σ →₀ ℕ), coeff m y ∈ I) → ∀ (m : σ →₀ ℕ), coeff m (x + y) ∈ I",
"tactic": "exact fun f g hf hg n => by simp [I.add_mem (hf n) (hg n)]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf✝ : MvPolynomial σ R\nhf✝ : f✝ ∈ Ideal.map C I\nf g : MvPolynomial σ R\nhf : ∀ (m : σ →₀ ℕ), coeff m f ∈ I\nhg : ∀ (m : σ →₀ ℕ), coeff m g ∈ I\nn : σ →₀ ℕ\n⊢ coeff n (f + g) ∈ I",
"tactic": "simp [I.add_mem (hf n) (hg n)]"
},
{
"state_after": "case mp.H2\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf✝ : MvPolynomial σ R\nhf : f✝ ∈ Ideal.map C I\nf g : MvPolynomial σ R\nhg : ∀ (m : σ →₀ ℕ), coeff m g ∈ I\nn : σ →₀ ℕ\n⊢ coeff n (f • g) ∈ I",
"state_before": "case mp.H2\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : f ∈ Ideal.map C I\n⊢ ∀ (a x : MvPolynomial σ R), (∀ (m : σ →₀ ℕ), coeff m x ∈ I) → ∀ (m : σ →₀ ℕ), coeff m (a • x) ∈ I",
"tactic": "refine' fun f g hg n => _"
},
{
"state_after": "case mp.H2\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf✝ : MvPolynomial σ R\nhf : f✝ ∈ Ideal.map C I\nf g : MvPolynomial σ R\nhg : ∀ (m : σ →₀ ℕ), coeff m g ∈ I\nn : σ →₀ ℕ\n⊢ ∑ x in Finsupp.antidiagonal n, coeff x.fst f * coeff x.snd g ∈ I",
"state_before": "case mp.H2\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf✝ : MvPolynomial σ R\nhf : f✝ ∈ Ideal.map C I\nf g : MvPolynomial σ R\nhg : ∀ (m : σ →₀ ℕ), coeff m g ∈ I\nn : σ →₀ ℕ\n⊢ coeff n (f • g) ∈ I",
"tactic": "rw [smul_eq_mul, coeff_mul]"
},
{
"state_after": "no goals",
"state_before": "case mp.H2\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf✝ : MvPolynomial σ R\nhf : f✝ ∈ Ideal.map C I\nf g : MvPolynomial σ R\nhg : ∀ (m : σ →₀ ℕ), coeff m g ∈ I\nn : σ →₀ ℕ\n⊢ ∑ x in Finsupp.antidiagonal n, coeff x.fst f * coeff x.snd g ∈ I",
"tactic": "exact I.sum_mem fun c _ => I.mul_mem_left (f.coeff c.fst) (hg c.snd)"
},
{
"state_after": "case mpr\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : ∀ (m : σ →₀ ℕ), coeff m f ∈ I\n⊢ f ∈ Ideal.map C I",
"state_before": "case mpr\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\n⊢ (∀ (m : σ →₀ ℕ), coeff m f ∈ I) → f ∈ Ideal.map C I",
"tactic": "intro hf"
},
{
"state_after": "case mpr\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : ∀ (m : σ →₀ ℕ), coeff m f ∈ I\n⊢ ∑ v in support f, ↑(monomial v) (coeff v f) ∈ Ideal.map C I",
"state_before": "case mpr\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : ∀ (m : σ →₀ ℕ), coeff m f ∈ I\n⊢ f ∈ Ideal.map C I",
"tactic": "rw [as_sum f]"
},
{
"state_after": "case mpr\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : ∀ (m : σ →₀ ℕ), coeff m f ∈ I\n⊢ ∀ (m : σ →₀ ℕ), m ∈ support f → ↑(monomial m) (coeff m f) ∈ Ideal.map C I",
"state_before": "case mpr\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : ∀ (m : σ →₀ ℕ), coeff m f ∈ I\n⊢ ∑ v in support f, ↑(monomial v) (coeff v f) ∈ Ideal.map C I",
"tactic": "suffices ∀ m ∈ f.support, monomial m (coeff m f) ∈ (Ideal.map C I : Ideal (MvPolynomial σ R)) by\n exact Submodule.sum_mem _ this"
},
{
"state_after": "case mpr\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : ∀ (m : σ →₀ ℕ), coeff m f ∈ I\nm : σ →₀ ℕ\na✝ : m ∈ support f\n⊢ ↑(monomial m) (coeff m f) ∈ Ideal.map C I",
"state_before": "case mpr\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : ∀ (m : σ →₀ ℕ), coeff m f ∈ I\n⊢ ∀ (m : σ →₀ ℕ), m ∈ support f → ↑(monomial m) (coeff m f) ∈ Ideal.map C I",
"tactic": "intro m _"
},
{
"state_after": "case mpr\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : ∀ (m : σ →₀ ℕ), coeff m f ∈ I\nm : σ →₀ ℕ\na✝ : m ∈ support f\n⊢ ↑C (coeff m f) * ↑(monomial m) 1 ∈ Ideal.map C I",
"state_before": "case mpr\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : ∀ (m : σ →₀ ℕ), coeff m f ∈ I\nm : σ →₀ ℕ\na✝ : m ∈ support f\n⊢ ↑(monomial m) (coeff m f) ∈ Ideal.map C I",
"tactic": "rw [← mul_one (coeff m f), ← C_mul_monomial]"
},
{
"state_after": "case mpr\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : ∀ (m : σ →₀ ℕ), coeff m f ∈ I\nm : σ →₀ ℕ\na✝ : m ∈ support f\n⊢ ↑C (coeff m f) ∈ Ideal.map C I",
"state_before": "case mpr\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : ∀ (m : σ →₀ ℕ), coeff m f ∈ I\nm : σ →₀ ℕ\na✝ : m ∈ support f\n⊢ ↑C (coeff m f) * ↑(monomial m) 1 ∈ Ideal.map C I",
"tactic": "suffices C (coeff m f) ∈ (Ideal.map C I : Ideal (MvPolynomial σ R)) by\n exact Ideal.mul_mem_right _ _ this"
},
{
"state_after": "case mpr\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : ∀ (m : σ →₀ ℕ), coeff m f ∈ I\nm : σ →₀ ℕ\na✝ : m ∈ support f\n⊢ coeff m f ∈ I",
"state_before": "case mpr\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : ∀ (m : σ →₀ ℕ), coeff m f ∈ I\nm : σ →₀ ℕ\na✝ : m ∈ support f\n⊢ ↑C (coeff m f) ∈ Ideal.map C I",
"tactic": "apply Ideal.mem_map_of_mem _"
},
{
"state_after": "no goals",
"state_before": "case mpr\nR : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : ∀ (m : σ →₀ ℕ), coeff m f ∈ I\nm : σ →₀ ℕ\na✝ : m ∈ support f\n⊢ coeff m f ∈ I",
"tactic": "exact hf m"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : ∀ (m : σ →₀ ℕ), coeff m f ∈ I\nthis : ∀ (m : σ →₀ ℕ), m ∈ support f → ↑(monomial m) (coeff m f) ∈ Ideal.map C I\n⊢ ∑ v in support f, ↑(monomial v) (coeff v f) ∈ Ideal.map C I",
"tactic": "exact Submodule.sum_mem _ this"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type ?u.979377\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nf : MvPolynomial σ R\nhf : ∀ (m : σ →₀ ℕ), coeff m f ∈ I\nm : σ →₀ ℕ\na✝ : m ∈ support f\nthis : ↑C (coeff m f) ∈ Ideal.map C I\n⊢ ↑C (coeff m f) * ↑(monomial m) 1 ∈ Ideal.map C I",
"tactic": "exact Ideal.mul_mem_right _ _ this"
}
] |
[
1190,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1164,
1
] |
Mathlib/CategoryTheory/Adjunction/Basic.lean
|
CategoryTheory.Adjunction.homEquiv_naturality_left
|
[
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nf : X' ⟶ X\ng : F.obj X ⟶ Y\n⊢ F.map f ≫ g = ↑(homEquiv adj X' Y).symm (f ≫ ↑(homEquiv adj X Y) g)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nf : X' ⟶ X\ng : F.obj X ⟶ Y\n⊢ ↑(homEquiv adj X' Y) (F.map f ≫ g) = f ≫ ↑(homEquiv adj X Y) g",
"tactic": "rw [← Equiv.eq_symm_apply]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nf : X' ⟶ X\ng : F.obj X ⟶ Y\n⊢ F.map f ≫ g = ↑(homEquiv adj X' Y).symm (f ≫ ↑(homEquiv adj X Y) g)",
"tactic": "simp only [Equiv.symm_apply_apply,eq_self_iff_true,homEquiv_naturality_left_symm]"
}
] |
[
161,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Analysis/Calculus/Deriv/Add.lean
|
HasDerivAt.add_const
|
[] |
[
103,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
8
] |
Mathlib/Analysis/Convex/Slope.lean
|
ConvexOn.secant_mono_aux1
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\n⊢ (z - x) * f y ≤ (z - y) * f x + (y - x) * f z",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\n⊢ (z - x) * f y ≤ (z - y) * f x + (y - x) * f z",
"tactic": "have hxy' : 0 < y - x := by linarith"
},
{
"state_after": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\n⊢ (z - x) * f y ≤ (z - y) * f x + (y - x) * f z",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\n⊢ (z - x) * f y ≤ (z - y) * f x + (y - x) * f z",
"tactic": "have hyz' : 0 < z - y := by linarith"
},
{
"state_after": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\n⊢ (z - x) * f y ≤ (z - y) * f x + (y - x) * f z",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\n⊢ (z - x) * f y ≤ (z - y) * f x + (y - x) * f z",
"tactic": "have hxz' : 0 < z - x := by linarith"
},
{
"state_after": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\n⊢ f y ≤ ((z - y) * f x + (y - x) * f z) / (z - x)",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\n⊢ (z - x) * f y ≤ (z - y) * f x + (y - x) * f z",
"tactic": "rw [← le_div_iff' hxz']"
},
{
"state_after": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 ≤ (z - y) / (z - x)\n⊢ f y ≤ ((z - y) * f x + (y - x) * f z) / (z - x)",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\n⊢ f y ≤ ((z - y) * f x + (y - x) * f z) / (z - x)",
"tactic": "have ha : 0 ≤ (z - y) / (z - x) := by positivity"
},
{
"state_after": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 ≤ (z - y) / (z - x)\nhb : 0 ≤ (y - x) / (z - x)\n⊢ f y ≤ ((z - y) * f x + (y - x) * f z) / (z - x)",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 ≤ (z - y) / (z - x)\n⊢ f y ≤ ((z - y) * f x + (y - x) * f z) / (z - x)",
"tactic": "have hb : 0 ≤ (y - x) / (z - x) := by positivity"
},
{
"state_after": "case calc_1\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 ≤ (z - y) / (z - x)\nhb : 0 ≤ (y - x) / (z - x)\n⊢ f y = f ((z - y) / (z - x) * x + (y - x) / (z - x) * z)\n\ncase calc_2\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 ≤ (z - y) / (z - x)\nhb : 0 ≤ (y - x) / (z - x)\n⊢ Div.div (z - y) (z - x) + Div.div (y - x) (z - x) = 1\n\ncase calc_3\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 ≤ (z - y) / (z - x)\nhb : 0 ≤ (y - x) / (z - x)\n⊢ (z - y) / (z - x) * f x + (y - x) / (z - x) * f z = ((z - y) * f x + (y - x) * f z) / (z - x)",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 ≤ (z - y) / (z - x)\nhb : 0 ≤ (y - x) / (z - x)\n⊢ f y ≤ ((z - y) * f x + (y - x) * f z) / (z - x)",
"tactic": "calc\n f y = f ((z - y) / (z - x) * x + (y - x) / (z - x) * z) := ?_\n _ ≤ (z - y) / (z - x) * f x + (y - x) / (z - x) * f z := hf.2 hx hz ha hb ?_\n _ = ((z - y) * f x + (y - x) * f z) / (z - x) := ?_"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\n⊢ 0 < y - x",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\n⊢ 0 < z - y",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\n⊢ 0 < z - x",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\n⊢ 0 ≤ (z - y) / (z - x)",
"tactic": "positivity"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 ≤ (z - y) / (z - x)\n⊢ 0 ≤ (y - x) / (z - x)",
"tactic": "positivity"
},
{
"state_after": "case calc_1.e_a\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 ≤ (z - y) / (z - x)\nhb : 0 ≤ (y - x) / (z - x)\n⊢ y = (z - y) / (z - x) * x + (y - x) / (z - x) * z",
"state_before": "case calc_1\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 ≤ (z - y) / (z - x)\nhb : 0 ≤ (y - x) / (z - x)\n⊢ f y = f ((z - y) / (z - x) * x + (y - x) / (z - x) * z)",
"tactic": "congr 1"
},
{
"state_after": "case calc_1.e_a\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 ≤ (z - y) / (z - x)\nhb : 0 ≤ (y - x) / (z - x)\n⊢ y * (z - x) = (z - y) * x + (y - x) * z",
"state_before": "case calc_1.e_a\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 ≤ (z - y) / (z - x)\nhb : 0 ≤ (y - x) / (z - x)\n⊢ y = (z - y) / (z - x) * x + (y - x) / (z - x) * z",
"tactic": "field_simp [hxy'.ne', hyz'.ne', hxz'.ne']"
},
{
"state_after": "no goals",
"state_before": "case calc_1.e_a\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 ≤ (z - y) / (z - x)\nhb : 0 ≤ (y - x) / (z - x)\n⊢ y * (z - x) = (z - y) * x + (y - x) * z",
"tactic": "ring"
},
{
"state_after": "case calc_2\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 ≤ (z - y) / (z - x)\nhb : 0 ≤ (y - x) / (z - x)\n⊢ (z - y) / (z - x) + (y - x) / (z - x) = 1",
"state_before": "case calc_2\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 ≤ (z - y) / (z - x)\nhb : 0 ≤ (y - x) / (z - x)\n⊢ Div.div (z - y) (z - x) + Div.div (y - x) (z - x) = 1",
"tactic": "show (z - y) / (z - x) + (y - x) / (z - x) = 1"
},
{
"state_after": "no goals",
"state_before": "case calc_2\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 ≤ (z - y) / (z - x)\nhb : 0 ≤ (y - x) / (z - x)\n⊢ (z - y) / (z - x) + (y - x) / (z - x) = 1",
"tactic": "field_simp [hxy'.ne', hyz'.ne', hxz'.ne']"
},
{
"state_after": "no goals",
"state_before": "case calc_3\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 ≤ (z - y) / (z - x)\nhb : 0 ≤ (y - x) / (z - x)\n⊢ (z - y) / (z - x) * f x + (y - x) / (z - x) * f z = ((z - y) * f x + (y - x) * f z) / (z - x)",
"tactic": "field_simp [hxy'.ne', hyz'.ne', hxz'.ne']"
}
] |
[
258,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/LinearAlgebra/Matrix/Determinant.lean
|
Matrix.det_eq_one_of_card_eq_zero
|
[] |
[
114,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/GroupTheory/Submonoid/Pointwise.lean
|
AddSubmonoid.le_pointwise_smul_iff
|
[] |
[
438,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
437,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Pairwise.exists_mem_filter_basis_of_disjoint
|
[
{
"state_after": "case intro.intro\nα : Type u_2\nβ : Type ?u.53092\nγ : Type ?u.53095\nι✝ : Sort ?u.53098\nι' : Sort ?u.53101\nl✝ l' : Filter α\np✝ : ι✝ → Prop\ns✝ : ι✝ → Set α\nt✝ : Set α\ni : ι✝\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nI : Type u_1\ninst✝ : Finite I\nl : I → Filter α\nι : I → Sort u_3\np : (i : I) → ι i → Prop\ns : (i : I) → ι i → Set α\nhd✝ : Pairwise (Disjoint on l)\nh : ∀ (i : I), HasBasis (l i) (p i) (s i)\nt : I → Set α\nhtl : ∀ (i : I), t i ∈ l i\nhd : Pairwise (Disjoint on t)\n⊢ ∃ ind, (∀ (i : I), p i (ind i)) ∧ Pairwise (Disjoint on fun i => s i (ind i))",
"state_before": "α : Type u_2\nβ : Type ?u.53092\nγ : Type ?u.53095\nι✝ : Sort ?u.53098\nι' : Sort ?u.53101\nl✝ l' : Filter α\np✝ : ι✝ → Prop\ns✝ : ι✝ → Set α\nt : Set α\ni : ι✝\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nI : Type u_1\ninst✝ : Finite I\nl : I → Filter α\nι : I → Sort u_3\np : (i : I) → ι i → Prop\ns : (i : I) → ι i → Set α\nhd : Pairwise (Disjoint on l)\nh : ∀ (i : I), HasBasis (l i) (p i) (s i)\n⊢ ∃ ind, (∀ (i : I), p i (ind i)) ∧ Pairwise (Disjoint on fun i => s i (ind i))",
"tactic": "rcases hd.exists_mem_filter_of_disjoint with ⟨t, htl, hd⟩"
},
{
"state_after": "case intro.intro\nα : Type u_2\nβ : Type ?u.53092\nγ : Type ?u.53095\nι✝ : Sort ?u.53098\nι' : Sort ?u.53101\nl✝ l' : Filter α\np✝ : ι✝ → Prop\ns✝ : ι✝ → Set α\nt✝ : Set α\ni : ι✝\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nI : Type u_1\ninst✝ : Finite I\nl : I → Filter α\nι : I → Sort u_3\np : (i : I) → ι i → Prop\ns : (i : I) → ι i → Set α\nhd✝ : Pairwise (Disjoint on l)\nh : ∀ (i : I), HasBasis (l i) (p i) (s i)\nt : I → Set α\nhtl : ∀ (i : I), t i ∈ l i\nhd : Pairwise (Disjoint on t)\nind : (i : I) → ι i\nhp : ∀ (i : I), p i (ind i)\nht : ∀ (i : I), s i (ind i) ⊆ t i\n⊢ ∃ ind, (∀ (i : I), p i (ind i)) ∧ Pairwise (Disjoint on fun i => s i (ind i))",
"state_before": "case intro.intro\nα : Type u_2\nβ : Type ?u.53092\nγ : Type ?u.53095\nι✝ : Sort ?u.53098\nι' : Sort ?u.53101\nl✝ l' : Filter α\np✝ : ι✝ → Prop\ns✝ : ι✝ → Set α\nt✝ : Set α\ni : ι✝\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nI : Type u_1\ninst✝ : Finite I\nl : I → Filter α\nι : I → Sort u_3\np : (i : I) → ι i → Prop\ns : (i : I) → ι i → Set α\nhd✝ : Pairwise (Disjoint on l)\nh : ∀ (i : I), HasBasis (l i) (p i) (s i)\nt : I → Set α\nhtl : ∀ (i : I), t i ∈ l i\nhd : Pairwise (Disjoint on t)\n⊢ ∃ ind, (∀ (i : I), p i (ind i)) ∧ Pairwise (Disjoint on fun i => s i (ind i))",
"tactic": "choose ind hp ht using fun i => (h i).mem_iff.1 (htl i)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_2\nβ : Type ?u.53092\nγ : Type ?u.53095\nι✝ : Sort ?u.53098\nι' : Sort ?u.53101\nl✝ l' : Filter α\np✝ : ι✝ → Prop\ns✝ : ι✝ → Set α\nt✝ : Set α\ni : ι✝\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nI : Type u_1\ninst✝ : Finite I\nl : I → Filter α\nι : I → Sort u_3\np : (i : I) → ι i → Prop\ns : (i : I) → ι i → Set α\nhd✝ : Pairwise (Disjoint on l)\nh : ∀ (i : I), HasBasis (l i) (p i) (s i)\nt : I → Set α\nhtl : ∀ (i : I), t i ∈ l i\nhd : Pairwise (Disjoint on t)\nind : (i : I) → ι i\nhp : ∀ (i : I), p i (ind i)\nht : ∀ (i : I), s i (ind i) ⊆ t i\n⊢ ∃ ind, (∀ (i : I), p i (ind i)) ∧ Pairwise (Disjoint on fun i => s i (ind i))",
"tactic": "exact ⟨ind, hp, hd.mono fun i j hij => hij.mono (ht _) (ht _)⟩"
}
] |
[
665,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
659,
1
] |
Mathlib/GroupTheory/Coset.lean
|
QuotientGroup.leftRel_eq
|
[
{
"state_after": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\n⊢ ∀ (a b : α), Setoid.r a b ↔ a⁻¹ * b ∈ s",
"state_before": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\n⊢ ∀ (a b : α), Setoid.r a b = (a⁻¹ * b ∈ s)",
"tactic": "simp only [eq_iff_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\n⊢ ∀ (a b : α), Setoid.r a b ↔ a⁻¹ * b ∈ s",
"tactic": "apply leftRel_apply"
}
] |
[
340,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
337,
1
] |
Mathlib/Data/List/Sort.lean
|
List.Sorted.rel_get_of_le
|
[
{
"state_after": "case inl\nα : Type uu\nr : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : IsRefl α r\nl : List α\nh : Sorted r l\na : Fin (length l)\nhab : a ≤ a\n⊢ r (get l a) (get l a)\n\ncase inr\nα : Type uu\nr : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : IsRefl α r\nl : List α\nh : Sorted r l\na b : Fin (length l)\nhab : a ≤ b\nhlt : a < b\n⊢ r (get l a) (get l b)",
"state_before": "α : Type uu\nr : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : IsRefl α r\nl : List α\nh : Sorted r l\na b : Fin (length l)\nhab : a ≤ b\n⊢ r (get l a) (get l b)",
"tactic": "rcases hab.eq_or_lt with (rfl | hlt)"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type uu\nr : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : IsRefl α r\nl : List α\nh : Sorted r l\na : Fin (length l)\nhab : a ≤ a\n⊢ r (get l a) (get l a)\n\ncase inr\nα : Type uu\nr : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : IsRefl α r\nl : List α\nh : Sorted r l\na b : Fin (length l)\nhab : a ≤ b\nhlt : a < b\n⊢ r (get l a) (get l b)",
"tactic": "exacts [refl _, h.rel_get_of_lt hlt]"
}
] |
[
125,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
Associates.eq_pow_count_factors_of_dvd_pow
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh : a ∣ p ^ n\n✝ : Nontrivial α\n⊢ a = p ^ count p (factors a)",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh : a ∣ p ^ n\n⊢ a = p ^ count p (factors a)",
"tactic": "nontriviality α"
},
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh : a ∣ p ^ n\n✝ : Nontrivial α\nhph : p ^ n ≠ 0\n⊢ a = p ^ count p (factors a)",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh : a ∣ p ^ n\n✝ : Nontrivial α\n⊢ a = p ^ count p (factors a)",
"tactic": "have hph := pow_ne_zero n hp.ne_zero"
},
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh : a ∣ p ^ n\n✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\n⊢ a = p ^ count p (factors a)",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh : a ∣ p ^ n\n✝ : Nontrivial α\nhph : p ^ n ≠ 0\n⊢ a = p ^ count p (factors a)",
"tactic": "have ha := ne_zero_of_dvd_ne_zero hph h"
},
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh : a ∣ p ^ n\n✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\n⊢ ∀ (p_1 : Associates α), Irreducible p_1 → count p_1 (factors a) = count p_1 (factors (p ^ count p (factors a)))",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh : a ∣ p ^ n\n✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\n⊢ a = p ^ count p (factors a)",
"tactic": "apply eq_of_eq_counts ha (pow_ne_zero _ hp.ne_zero)"
},
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh : a ∣ p ^ n\n✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\neq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → count q (factors a) = 0\n⊢ ∀ (p_1 : Associates α), Irreducible p_1 → count p_1 (factors a) = count p_1 (factors (p ^ count p (factors a)))",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh : a ∣ p ^ n\n✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\n⊢ ∀ (p_1 : Associates α), Irreducible p_1 → count p_1 (factors a) = count p_1 (factors (p ^ count p (factors a)))",
"tactic": "have eq_zero_of_ne : ∀ q : Associates α, Irreducible q → q ≠ p → _ = 0 := fun q hq h' =>\n Nat.eq_zero_of_le_zero <| by\n convert count_le_count_of_le hph hq h\n symm\n rw [count_pow hp.ne_zero hq, count_eq_zero_of_ne hq hp h', MulZeroClass.mul_zero]"
},
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh : a ∣ p ^ n\n✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\neq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → count q (factors a) = 0\nq : Associates α\nhq : Irreducible q\n⊢ count q (factors a) = count q (factors (p ^ count p (factors a)))",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh : a ∣ p ^ n\n✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\neq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → count q (factors a) = 0\n⊢ ∀ (p_1 : Associates α), Irreducible p_1 → count p_1 (factors a) = count p_1 (factors (p ^ count p (factors a)))",
"tactic": "intro q hq"
},
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh : a ∣ p ^ n\n✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\neq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → count q (factors a) = 0\nq : Associates α\nhq : Irreducible q\n⊢ count q (factors a) = count p (factors a) * count q (factors p)",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh : a ∣ p ^ n\n✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\neq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → count q (factors a) = 0\nq : Associates α\nhq : Irreducible q\n⊢ count q (factors a) = count q (factors (p ^ count p (factors a)))",
"tactic": "rw [count_pow hp.ne_zero hq]"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh✝ : a ∣ p ^ n\n✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\neq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → count q (factors a) = 0\nq : Associates α\nhq : Irreducible q\nh : q = p\n⊢ count q (factors a) = count p (factors a) * count q (factors p)\n\ncase neg\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh✝ : a ∣ p ^ n\n✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\neq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → count q (factors a) = 0\nq : Associates α\nhq : Irreducible q\nh : ¬q = p\n⊢ count q (factors a) = count p (factors a) * count q (factors p)",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh : a ∣ p ^ n\n✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\neq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → count q (factors a) = 0\nq : Associates α\nhq : Irreducible q\n⊢ count q (factors a) = count p (factors a) * count q (factors p)",
"tactic": "by_cases h : q = p"
},
{
"state_after": "case h.e'_4\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh : a ∣ p ^ n\n✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\nq : Associates α\nhq : Irreducible q\nh' : q ≠ p\n⊢ 0 = count q (factors (p ^ n))",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh : a ∣ p ^ n\n✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\nq : Associates α\nhq : Irreducible q\nh' : q ≠ p\n⊢ ?m.2897509 q ≤ 0",
"tactic": "convert count_le_count_of_le hph hq h"
},
{
"state_after": "case h.e'_4\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh : a ∣ p ^ n\n✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\nq : Associates α\nhq : Irreducible q\nh' : q ≠ p\n⊢ count q (factors (p ^ n)) = 0",
"state_before": "case h.e'_4\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh : a ∣ p ^ n\n✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\nq : Associates α\nhq : Irreducible q\nh' : q ≠ p\n⊢ 0 = count q (factors (p ^ n))",
"tactic": "symm"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh : a ∣ p ^ n\n✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\nq : Associates α\nhq : Irreducible q\nh' : q ≠ p\n⊢ count q (factors (p ^ n)) = 0",
"tactic": "rw [count_pow hp.ne_zero hq, count_eq_zero_of_ne hq hp h', MulZeroClass.mul_zero]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh✝ : a ∣ p ^ n\n✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\neq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → count q (factors a) = 0\nq : Associates α\nhq : Irreducible q\nh : q = p\n⊢ count q (factors a) = count p (factors a) * count q (factors p)",
"tactic": "rw [h, count_self hp, mul_one]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh✝ : a ∣ p ^ n\n✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\neq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → count q (factors a) = 0\nq : Associates α\nhq : Irreducible q\nh : ¬q = p\n⊢ count q (factors a) = count p (factors a) * count q (factors p)",
"tactic": "rw [count_eq_zero_of_ne hq hp h, MulZeroClass.mul_zero, eq_zero_of_ne q hq h]"
}
] |
[
1852,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1837,
1
] |
Mathlib/Data/Nat/Prime.lean
|
Nat.Prime.ne_zero
|
[] |
[
66,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/Order/Interval.lean
|
Interval.bot_ne_pure
|
[] |
[
410,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
409,
1
] |
Std/Data/HashMap/WF.lean
|
Std.HashMap.Imp.insert_WF
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\n⊢ Bucket.WF\n (match\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] with\n | true =>\n { size := m.size,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.replace k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }\n | false =>\n if numBucketsForCapacity (m.size + 1) ≤ Array.size m.buckets.val then\n { size := m.size + 1,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.cons k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }\n else\n expand (m.size + 1)\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.cons k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val))).buckets",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\n⊢ Bucket.WF (insert m k v).buckets",
"tactic": "dsimp [insert, cond]"
},
{
"state_after": "case h_1\nα : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\nc✝ : Bool\nheq✝ :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n true\n⊢ Bucket.WF\n { size := m.size,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.replace k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }.buckets\n\ncase h_2\nα : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\nc✝ : Bool\nheq✝ :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n false\n⊢ Bucket.WF\n (if numBucketsForCapacity (m.size + 1) ≤ Array.size m.buckets.val then\n { size := m.size + 1,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.cons k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }\n else\n expand (m.size + 1)\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.cons k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val))).buckets",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\n⊢ Bucket.WF\n (match\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] with\n | true =>\n { size := m.size,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.replace k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }\n | false =>\n if numBucketsForCapacity (m.size + 1) ≤ Array.size m.buckets.val then\n { size := m.size + 1,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.cons k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }\n else\n expand (m.size + 1)\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.cons k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val))).buckets",
"tactic": "split"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∃ x,\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ∧\n (x.fst == k) = true\n⊢ Bucket.WF\n { size := m.size,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.replace k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }.buckets",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n true\n⊢ Bucket.WF\n { size := m.size,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.replace k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }.buckets",
"tactic": "simp at h₁"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∃ x,\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ∧\n (x.fst == k) = true\nx : α × β\nhx₁ :\n x ∈\n AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nhx₂ : (x.fst == k) = true\n⊢ Bucket.WF\n { size := m.size,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.replace k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }.buckets",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∃ x,\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ∧\n (x.fst == k) = true\n⊢ Bucket.WF\n { size := m.size,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.replace k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }.buckets",
"tactic": "have ⟨x, hx₁, hx₂⟩ := h₁"
},
{
"state_after": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝³ : BEq α\ninst✝² : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∃ x,\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ∧\n (x.fst == k) = true\nx : α × β\nhx₁ :\n x ∈\n AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nhx₂ : (x.fst == k) = true\ninst✝¹ : PartialEquivBEq α\ninst✝ : LawfulHashable α\nH :\n List.Pairwise (fun a b => ¬(a.fst == b.fst) = true)\n (AssocList.toList m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n⊢ List.Pairwise (fun a b => ¬(a.fst == b.fst) = true)\n (AssocList.toList\n (AssocList.replace k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]))\n\ncase refine_2\nα : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh✝ : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∃ x,\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ∧\n (x.fst == k) = true\nx : α × β\nhx₁ :\n x ∈\n AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nhx₂ : (x.fst == k) = true\nH :\n AssocList.All\n (fun k_1 x =>\n USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)\n m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\na : α × β\nh :\n a ∈\n AssocList.toList\n (AssocList.replace k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n⊢ (fun k_1 x =>\n USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)\n a.fst a.snd",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∃ x,\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ∧\n (x.fst == k) = true\nx : α × β\nhx₁ :\n x ∈\n AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nhx₂ : (x.fst == k) = true\n⊢ Bucket.WF\n { size := m.size,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.replace k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }.buckets",
"tactic": "refine h.update (fun H => ?_) (fun H a h => ?_)"
},
{
"state_after": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝³ : BEq α\ninst✝² : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∃ x,\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ∧\n (x.fst == k) = true\nx : α × β\nhx₁ :\n x ∈\n AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nhx₂ : (x.fst == k) = true\ninst✝¹ : PartialEquivBEq α\ninst✝ : LawfulHashable α\nH :\n List.Pairwise (fun a b => ¬(a.fst == b.fst) = true)\n (AssocList.toList m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n⊢ List.Pairwise (fun a b => ¬(a.fst == b.fst) = true)\n (List.replaceF (fun x => bif x.fst == k then some (k, v) else none)\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]))",
"state_before": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝³ : BEq α\ninst✝² : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∃ x,\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ∧\n (x.fst == k) = true\nx : α × β\nhx₁ :\n x ∈\n AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nhx₂ : (x.fst == k) = true\ninst✝¹ : PartialEquivBEq α\ninst✝ : LawfulHashable α\nH :\n List.Pairwise (fun a b => ¬(a.fst == b.fst) = true)\n (AssocList.toList m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n⊢ List.Pairwise (fun a b => ¬(a.fst == b.fst) = true)\n (AssocList.toList\n (AssocList.replace k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]))",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝³ : BEq α\ninst✝² : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∃ x,\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ∧\n (x.fst == k) = true\nx : α × β\nhx₁ :\n x ∈\n AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nhx₂ : (x.fst == k) = true\ninst✝¹ : PartialEquivBEq α\ninst✝ : LawfulHashable α\nH :\n List.Pairwise (fun a b => ¬(a.fst == b.fst) = true)\n (AssocList.toList m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n⊢ List.Pairwise (fun a b => ¬(a.fst == b.fst) = true)\n (List.replaceF (fun x => bif x.fst == k then some (k, v) else none)\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]))",
"tactic": "exact pairwise_replaceF H"
},
{
"state_after": "case refine_2\nα : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh✝ : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∃ x,\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ∧\n (x.fst == k) = true\nx : α × β\nhx₁ :\n x ∈\n AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nhx₂ : (x.fst == k) = true\nH :\n ∀ (a : α × β),\n a ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] →\n USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) =\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\na : α × β\nh :\n a ∈\n List.replaceF (fun x => bif x.fst == k then some (k, v) else none)\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n⊢ USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) =\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val",
"state_before": "case refine_2\nα : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh✝ : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∃ x,\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ∧\n (x.fst == k) = true\nx : α × β\nhx₁ :\n x ∈\n AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nhx₂ : (x.fst == k) = true\nH :\n AssocList.All\n (fun k_1 x =>\n USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)\n m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\na : α × β\nh :\n a ∈\n AssocList.toList\n (AssocList.replace k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n⊢ (fun k_1 x =>\n USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)\n a.fst a.snd",
"tactic": "simp [AssocList.All] at H h ⊢"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nα : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh✝ : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∃ x,\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ∧\n (x.fst == k) = true\nx : α × β\nhx₁ :\n x ∈\n AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nhx₂ : (x.fst == k) = true\nH :\n ∀ (a : α × β),\n a ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] →\n USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) =\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\na : α × β\nh :\n a ∈\n List.replaceF (fun x => bif x.fst == k then some (k, v) else none)\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n⊢ USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) =\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val",
"tactic": "match mem_replaceF h with\n| .inl rfl => rfl\n| .inr h => exact H _ h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nv : β\nh✝ : Bucket.WF m.buckets\nc✝ : Bool\nx a : α × β\nh₁ :\n ∃ x,\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash a.fst))).val] ∧\n (x.fst == a.fst) = true\nhx₁ :\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash a.fst))).val]\nhx₂ : (x.fst == a.fst) = true\nH :\n ∀ (a_1 : α × β),\n a_1 ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash a.fst))).val] →\n USize.toNat (UInt64.toUSize (hash a_1.fst) % Array.size m.buckets.val) =\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash a.fst))).val\nh :\n a ∈\n List.replaceF (fun x => bif x.fst == a.fst then some (a.fst, v) else none)\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash a.fst))).val])\n⊢ USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) =\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash a.fst))).val",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nv : β\nh✝¹ : Bucket.WF m.buckets\nc✝ : Bool\nx a : α × β\nk : α\nh :\n a ∈\n AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nh₁ :\n ∃ x,\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ∧\n (x.fst == k) = true\nhx₁ :\n x ∈\n AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nhx₂ : (x.fst == k) = true\nH :\n ∀ (a : α × β),\n a ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] →\n USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) =\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\nh✝ :\n a ∈\n List.replaceF (fun x => bif x.fst == k then some (k, v) else none)\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n⊢ USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) =\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val",
"tactic": "exact H _ h"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ≠\n true\n⊢ Bucket.WF\n (if numBucketsForCapacity (m.size + 1) ≤ Array.size m.buckets.val then\n { size := m.size + 1,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.cons k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }\n else\n expand (m.size + 1)\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.cons k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val))).buckets",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n false\n⊢ Bucket.WF\n (if numBucketsForCapacity (m.size + 1) ≤ Array.size m.buckets.val then\n { size := m.size + 1,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.cons k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }\n else\n expand (m.size + 1)\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.cons k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val))).buckets",
"tactic": "rw [Bool.eq_false_iff] at h₁"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∀ (x : α × β),\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] →\n ¬(x.fst == k) = true\n⊢ Bucket.WF\n (if numBucketsForCapacity (m.size + 1) ≤ Array.size m.buckets.val then\n { size := m.size + 1,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.cons k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }\n else\n expand (m.size + 1)\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.cons k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val))).buckets",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ≠\n true\n⊢ Bucket.WF\n (if numBucketsForCapacity (m.size + 1) ≤ Array.size m.buckets.val then\n { size := m.size + 1,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.cons k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }\n else\n expand (m.size + 1)\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.cons k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val))).buckets",
"tactic": "simp at h₁"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∀ (x : α × β),\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] →\n ¬(x.fst == k) = true\n⊢ Bucket.WF\n { size := m.size + 1,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.cons k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }.buckets",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∀ (x : α × β),\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] →\n ¬(x.fst == k) = true\n⊢ Bucket.WF\n (if numBucketsForCapacity (m.size + 1) ≤ Array.size m.buckets.val then\n { size := m.size + 1,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.cons k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }\n else\n expand (m.size + 1)\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.cons k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val))).buckets",
"tactic": "suffices _ by split <;> [exact this; refine expand_WF this]"
},
{
"state_after": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝³ : BEq α\ninst✝² : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∀ (x : α × β),\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] →\n ¬(x.fst == k) = true\ninst✝¹ : PartialEquivBEq α\ninst✝ : LawfulHashable α\n⊢ ∀ (a' : α × β),\n a' ∈ AssocList.toList m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] →\n ¬((k, v).fst == a'.fst) = true\n\ncase refine_2\nα : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh✝ : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∀ (x : α × β),\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] →\n ¬(x.fst == k) = true\nH :\n AssocList.All\n (fun k_1 x =>\n USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)\n m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\na : α × β\nh :\n a ∈\n AssocList.toList\n (AssocList.cons k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n⊢ (fun k_1 x =>\n USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)\n a.fst a.snd",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∀ (x : α × β),\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] →\n ¬(x.fst == k) = true\n⊢ Bucket.WF\n { size := m.size + 1,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.cons k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }.buckets",
"tactic": "refine h.update (.cons ?_) (fun H a h => ?_)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∀ (x : α × β),\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] →\n ¬(x.fst == k) = true\nthis : ?m.87772\n⊢ Bucket.WF\n (if numBucketsForCapacity (m.size + 1) ≤ Array.size m.buckets.val then\n { size := m.size + 1,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.cons k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }\n else\n expand (m.size + 1)\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.cons k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val))).buckets",
"tactic": "split <;> [exact this; refine expand_WF this]"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝³ : BEq α\ninst✝² : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∀ (x : α × β),\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] →\n ¬(x.fst == k) = true\ninst✝¹ : PartialEquivBEq α\ninst✝ : LawfulHashable α\n⊢ ∀ (a' : α × β),\n a' ∈ AssocList.toList m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] →\n ¬((k, v).fst == a'.fst) = true",
"tactic": "exact fun a h h' => h₁ a h (PartialEquivBEq.symm h')"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nα : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh✝ : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∀ (x : α × β),\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] →\n ¬(x.fst == k) = true\nH :\n AssocList.All\n (fun k_1 x =>\n USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)\n m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\na : α × β\nh :\n a ∈\n AssocList.toList\n (AssocList.cons k v\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n⊢ (fun k_1 x =>\n USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)\n a.fst a.snd",
"tactic": "cases h with\n| head => rfl\n| tail _ h => exact H _ h"
},
{
"state_after": "no goals",
"state_before": "case refine_2.head\nα : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∀ (x : α × β),\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] →\n ¬(x.fst == k) = true\nH :\n AssocList.All\n (fun k_1 x =>\n USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)\n m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\n⊢ USize.toNat (UInt64.toUSize (hash (k, v).fst) % Array.size m.buckets.val) =\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case refine_2.tail\nα : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nv : β\nh✝ : Bucket.WF m.buckets\nc✝ : Bool\nh₁ :\n ∀ (x : α × β),\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] →\n ¬(x.fst == k) = true\nH :\n AssocList.All\n (fun k_1 x =>\n USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)\n m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\na : α × β\nh :\n List.Mem a\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n⊢ USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) =\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val",
"tactic": "exact H _ h"
}
] |
[
239,
32
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
221,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
Finset.isCompact_biUnion
|
[] |
[
417,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
415,
1
] |
Mathlib/Topology/MetricSpace/Lipschitz.lean
|
LipschitzWith.min_const
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : PseudoEMetricSpace α\nf g : α → ℝ\nKf Kg : ℝ≥0\nhf : LipschitzWith Kf f\na : ℝ\n⊢ LipschitzWith Kf fun x => min (f x) a",
"tactic": "simpa only [max_eq_left (zero_le Kf)] using hf.min (LipschitzWith.const a)"
}
] |
[
460,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
459,
1
] |
Mathlib/Order/Monotone/Basic.lean
|
Monotone.strictMono_iff_injective
|
[] |
[
880,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
879,
1
] |
Mathlib/Data/Seq/Seq.lean
|
Stream'.Seq.destruct_eq_nil
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns : Seq α\n⊢ Option.map (fun a' => (a', tail s)) (get? s 0) = none → s = nil",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Seq α\n⊢ destruct s = none → s = nil",
"tactic": "dsimp [destruct]"
},
{
"state_after": "case none\nα : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nx✝ : Option α\nf0✝ : get? s 0 = x✝\nf0 : get? s 0 = none\nh : Option.map (fun a' => (a', tail s)) none = none\n⊢ s = nil\n\ncase some\nα : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nx✝ : Option α\nf0✝ : get? s 0 = x✝\nval✝ : α\nf0 : get? s 0 = some val✝\nh : Option.map (fun a' => (a', tail s)) (some val✝) = none\n⊢ s = nil",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Seq α\n⊢ Option.map (fun a' => (a', tail s)) (get? s 0) = none → s = nil",
"tactic": "induction' f0 : get? s 0 <;> intro h"
},
{
"state_after": "case none.a\nα : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nx✝ : Option α\nf0✝ : get? s 0 = x✝\nf0 : get? s 0 = none\nh : Option.map (fun a' => (a', tail s)) none = none\n⊢ ↑s = ↑nil",
"state_before": "case none\nα : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nx✝ : Option α\nf0✝ : get? s 0 = x✝\nf0 : get? s 0 = none\nh : Option.map (fun a' => (a', tail s)) none = none\n⊢ s = nil",
"tactic": "apply Subtype.eq"
},
{
"state_after": "case none.a.h\nα : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nx✝ : Option α\nf0✝ : get? s 0 = x✝\nf0 : get? s 0 = none\nh : Option.map (fun a' => (a', tail s)) none = none\nn : ℕ\n⊢ ↑s n = ↑nil n",
"state_before": "case none.a\nα : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nx✝ : Option α\nf0✝ : get? s 0 = x✝\nf0 : get? s 0 = none\nh : Option.map (fun a' => (a', tail s)) none = none\n⊢ ↑s = ↑nil",
"tactic": "funext n"
},
{
"state_after": "case none.a.h.zero\nα : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nx✝ : Option α\nf0✝ : get? s 0 = x✝\nf0 : get? s 0 = none\nh : Option.map (fun a' => (a', tail s)) none = none\n⊢ ↑s Nat.zero = ↑nil Nat.zero\n\ncase none.a.h.succ\nα : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nx✝ : Option α\nf0✝ : get? s 0 = x✝\nf0 : get? s 0 = none\nh : Option.map (fun a' => (a', tail s)) none = none\nn : ℕ\nIH : ↑s n = ↑nil n\n⊢ ↑s (Nat.succ n) = ↑nil (Nat.succ n)",
"state_before": "case none.a.h\nα : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nx✝ : Option α\nf0✝ : get? s 0 = x✝\nf0 : get? s 0 = none\nh : Option.map (fun a' => (a', tail s)) none = none\nn : ℕ\n⊢ ↑s n = ↑nil n",
"tactic": "induction' n with n IH"
},
{
"state_after": "no goals",
"state_before": "case none.a.h.zero\nα : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nx✝ : Option α\nf0✝ : get? s 0 = x✝\nf0 : get? s 0 = none\nh : Option.map (fun a' => (a', tail s)) none = none\n⊢ ↑s Nat.zero = ↑nil Nat.zero\n\ncase none.a.h.succ\nα : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nx✝ : Option α\nf0✝ : get? s 0 = x✝\nf0 : get? s 0 = none\nh : Option.map (fun a' => (a', tail s)) none = none\nn : ℕ\nIH : ↑s n = ↑nil n\n⊢ ↑s (Nat.succ n) = ↑nil (Nat.succ n)",
"tactic": "exacts [f0, s.2 IH]"
},
{
"state_after": "no goals",
"state_before": "case some\nα : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nx✝ : Option α\nf0✝ : get? s 0 = x✝\nval✝ : α\nf0 : get? s 0 = some val✝\nh : Option.map (fun a' => (a', tail s)) (some val✝) = none\n⊢ s = nil",
"tactic": "contradiction"
}
] |
[
220,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.maps_univ_to
|
[] |
[
528,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
527,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Finset.univ_nonempty_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1860\nγ : Type ?u.1863\ninst✝ : Fintype α\ns t : Finset α\n⊢ Finset.Nonempty univ ↔ Nonempty α",
"tactic": "rw [← coe_nonempty, coe_univ, Set.nonempty_iff_univ_nonempty]"
}
] |
[
105,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/Data/Set/Function.lean
|
Function.Surjective.surjOn
|
[] |
[
1600,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1599,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.snormEssSup_measure_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.951145\nF : Type u_2\nG : Type ?u.951151\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : MeasurableSpace α\nf : α → F\n⊢ snormEssSup f 0 = 0",
"tactic": "simp [snormEssSup]"
}
] |
[
242,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
241,
1
] |
Mathlib/Computability/Reduce.lean
|
ManyOneEquiv.symm
|
[] |
[
170,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/LinearAlgebra/Pi.lean
|
LinearEquiv.sumArrowLequivProdArrow_symm_apply_inl
|
[] |
[
464,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
462,
1
] |
Mathlib/Analysis/Complex/Basic.lean
|
Complex.dist_conj_conj
|
[] |
[
331,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
330,
1
] |
Mathlib/Order/LocallyFinite.lean
|
Multiset.mem_Ioi
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.33793\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrderTop α\na x : α\n⊢ x ∈ Ioi a ↔ a < x",
"tactic": "rw [Ioi, ← Finset.mem_def, Finset.mem_Ioi]"
}
] |
[
592,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
592,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.blockDiagonal'_transpose
|
[
{
"state_after": "case a.mk.h.mk\nl : Type ?u.198919\nm : Type ?u.198922\nn : Type ?u.198925\no : Type u_4\np : Type ?u.198931\nq : Type ?u.198934\nm' : o → Type u_1\nn' : o → Type u_2\np' : o → Type ?u.198949\nR : Type ?u.198952\nS : Type ?u.198955\nα : Type u_3\nβ : Type ?u.198961\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : (i : o) → Matrix (m' i) (n' i) α\nii : o\nix : n' ii\nji : o\njx : m' ji\n⊢ (blockDiagonal' M)ᵀ { fst := ii, snd := ix } { fst := ji, snd := jx } =\n blockDiagonal' (fun k => (M k)ᵀ) { fst := ii, snd := ix } { fst := ji, snd := jx }",
"state_before": "l : Type ?u.198919\nm : Type ?u.198922\nn : Type ?u.198925\no : Type u_4\np : Type ?u.198931\nq : Type ?u.198934\nm' : o → Type u_1\nn' : o → Type u_2\np' : o → Type ?u.198949\nR : Type ?u.198952\nS : Type ?u.198955\nα : Type u_3\nβ : Type ?u.198961\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : (i : o) → Matrix (m' i) (n' i) α\n⊢ (blockDiagonal' M)ᵀ = blockDiagonal' fun k => (M k)ᵀ",
"tactic": "ext (⟨ii, ix⟩⟨ji, jx⟩)"
},
{
"state_after": "case a.mk.h.mk\nl : Type ?u.198919\nm : Type ?u.198922\nn : Type ?u.198925\no : Type u_4\np : Type ?u.198931\nq : Type ?u.198934\nm' : o → Type u_1\nn' : o → Type u_2\np' : o → Type ?u.198949\nR : Type ?u.198952\nS : Type ?u.198955\nα : Type u_3\nβ : Type ?u.198961\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : (i : o) → Matrix (m' i) (n' i) α\nii : o\nix : n' ii\nji : o\njx : m' ji\n⊢ (if h : ji = ii then M ji jx (cast (_ : n' { fst := ii, snd := ix }.fst = n' { fst := ji, snd := jx }.fst) ix)\n else 0) =\n if h : ii = ji then M ii (cast (_ : m' { fst := ji, snd := jx }.fst = m' { fst := ii, snd := ix }.fst) jx) ix else 0",
"state_before": "case a.mk.h.mk\nl : Type ?u.198919\nm : Type ?u.198922\nn : Type ?u.198925\no : Type u_4\np : Type ?u.198931\nq : Type ?u.198934\nm' : o → Type u_1\nn' : o → Type u_2\np' : o → Type ?u.198949\nR : Type ?u.198952\nS : Type ?u.198955\nα : Type u_3\nβ : Type ?u.198961\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : (i : o) → Matrix (m' i) (n' i) α\nii : o\nix : n' ii\nji : o\njx : m' ji\n⊢ (blockDiagonal' M)ᵀ { fst := ii, snd := ix } { fst := ji, snd := jx } =\n blockDiagonal' (fun k => (M k)ᵀ) { fst := ii, snd := ix } { fst := ji, snd := jx }",
"tactic": "simp only [transpose_apply, blockDiagonal'_apply]"
},
{
"state_after": "case a.mk.h.mk.inl.inl\nl : Type ?u.198919\nm : Type ?u.198922\nn : Type ?u.198925\no : Type u_4\np : Type ?u.198931\nq : Type ?u.198934\nm' : o → Type u_1\nn' : o → Type u_2\np' : o → Type ?u.198949\nR : Type ?u.198952\nS : Type ?u.198955\nα : Type u_3\nβ : Type ?u.198961\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : (i : o) → Matrix (m' i) (n' i) α\nii : o\nix : n' ii\nji : o\njx : m' ji\nh : ji = ii\nh✝ : ii = ji\n⊢ M ji jx (cast (_ : n' { fst := ii, snd := ix }.fst = n' { fst := ji, snd := jx }.fst) ix) =\n M ii (cast (_ : m' { fst := ji, snd := jx }.fst = m' { fst := ii, snd := ix }.fst) jx) ix\n\ncase a.mk.h.mk.inl.inr\nl : Type ?u.198919\nm : Type ?u.198922\nn : Type ?u.198925\no : Type u_4\np : Type ?u.198931\nq : Type ?u.198934\nm' : o → Type u_1\nn' : o → Type u_2\np' : o → Type ?u.198949\nR : Type ?u.198952\nS : Type ?u.198955\nα : Type u_3\nβ : Type ?u.198961\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : (i : o) → Matrix (m' i) (n' i) α\nii : o\nix : n' ii\nji : o\njx : m' ji\nh : ji = ii\nh✝ : ¬ii = ji\n⊢ M ji jx (cast (_ : n' { fst := ii, snd := ix }.fst = n' { fst := ji, snd := jx }.fst) ix) = 0\n\ncase a.mk.h.mk.inr.inl\nl : Type ?u.198919\nm : Type ?u.198922\nn : Type ?u.198925\no : Type u_4\np : Type ?u.198931\nq : Type ?u.198934\nm' : o → Type u_1\nn' : o → Type u_2\np' : o → Type ?u.198949\nR : Type ?u.198952\nS : Type ?u.198955\nα : Type u_3\nβ : Type ?u.198961\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : (i : o) → Matrix (m' i) (n' i) α\nii : o\nix : n' ii\nji : o\njx : m' ji\nh : ¬ji = ii\nh✝ : ii = ji\n⊢ 0 = M ii (cast (_ : m' { fst := ji, snd := jx }.fst = m' { fst := ii, snd := ix }.fst) jx) ix\n\ncase a.mk.h.mk.inr.inr\nl : Type ?u.198919\nm : Type ?u.198922\nn : Type ?u.198925\no : Type u_4\np : Type ?u.198931\nq : Type ?u.198934\nm' : o → Type u_1\nn' : o → Type u_2\np' : o → Type ?u.198949\nR : Type ?u.198952\nS : Type ?u.198955\nα : Type u_3\nβ : Type ?u.198961\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : (i : o) → Matrix (m' i) (n' i) α\nii : o\nix : n' ii\nji : o\njx : m' ji\nh : ¬ji = ii\nh✝ : ¬ii = ji\n⊢ 0 = 0",
"state_before": "case a.mk.h.mk\nl : Type ?u.198919\nm : Type ?u.198922\nn : Type ?u.198925\no : Type u_4\np : Type ?u.198931\nq : Type ?u.198934\nm' : o → Type u_1\nn' : o → Type u_2\np' : o → Type ?u.198949\nR : Type ?u.198952\nS : Type ?u.198955\nα : Type u_3\nβ : Type ?u.198961\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : (i : o) → Matrix (m' i) (n' i) α\nii : o\nix : n' ii\nji : o\njx : m' ji\n⊢ (if h : ji = ii then M ji jx (cast (_ : n' { fst := ii, snd := ix }.fst = n' { fst := ji, snd := jx }.fst) ix)\n else 0) =\n if h : ii = ji then M ii (cast (_ : m' { fst := ji, snd := jx }.fst = m' { fst := ii, snd := ix }.fst) jx) ix else 0",
"tactic": "split_ifs with h"
},
{
"state_after": "case a.mk.h.mk.inl.inl\nl : Type ?u.198919\nm : Type ?u.198922\nn : Type ?u.198925\no : Type u_4\np : Type ?u.198931\nq : Type ?u.198934\nm' : o → Type u_1\nn' : o → Type u_2\np' : o → Type ?u.198949\nR : Type ?u.198952\nS : Type ?u.198955\nα : Type u_3\nβ : Type ?u.198961\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : (i : o) → Matrix (m' i) (n' i) α\nji : o\njx : m' ji\nix : n' ji\nh✝ : ji = ji\n⊢ M ji jx (cast (_ : n' { fst := ji, snd := ix }.fst = n' { fst := ji, snd := jx }.fst) ix) =\n M ji (cast (_ : m' { fst := ji, snd := jx }.fst = m' { fst := ji, snd := ix }.fst) jx) ix",
"state_before": "case a.mk.h.mk.inl.inl\nl : Type ?u.198919\nm : Type ?u.198922\nn : Type ?u.198925\no : Type u_4\np : Type ?u.198931\nq : Type ?u.198934\nm' : o → Type u_1\nn' : o → Type u_2\np' : o → Type ?u.198949\nR : Type ?u.198952\nS : Type ?u.198955\nα : Type u_3\nβ : Type ?u.198961\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : (i : o) → Matrix (m' i) (n' i) α\nii : o\nix : n' ii\nji : o\njx : m' ji\nh : ji = ii\nh✝ : ii = ji\n⊢ M ji jx (cast (_ : n' { fst := ii, snd := ix }.fst = n' { fst := ji, snd := jx }.fst) ix) =\n M ii (cast (_ : m' { fst := ji, snd := jx }.fst = m' { fst := ii, snd := ix }.fst) jx) ix",
"tactic": "subst h"
},
{
"state_after": "no goals",
"state_before": "case a.mk.h.mk.inl.inl\nl : Type ?u.198919\nm : Type ?u.198922\nn : Type ?u.198925\no : Type u_4\np : Type ?u.198931\nq : Type ?u.198934\nm' : o → Type u_1\nn' : o → Type u_2\np' : o → Type ?u.198949\nR : Type ?u.198952\nS : Type ?u.198955\nα : Type u_3\nβ : Type ?u.198961\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : (i : o) → Matrix (m' i) (n' i) α\nji : o\njx : m' ji\nix : n' ji\nh✝ : ji = ji\n⊢ M ji jx (cast (_ : n' { fst := ji, snd := ix }.fst = n' { fst := ji, snd := jx }.fst) ix) =\n M ji (cast (_ : m' { fst := ji, snd := jx }.fst = m' { fst := ji, snd := ix }.fst) jx) ix",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case a.mk.h.mk.inl.inr\nl : Type ?u.198919\nm : Type ?u.198922\nn : Type ?u.198925\no : Type u_4\np : Type ?u.198931\nq : Type ?u.198934\nm' : o → Type u_1\nn' : o → Type u_2\np' : o → Type ?u.198949\nR : Type ?u.198952\nS : Type ?u.198955\nα : Type u_3\nβ : Type ?u.198961\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : (i : o) → Matrix (m' i) (n' i) α\nii : o\nix : n' ii\nji : o\njx : m' ji\nh : ji = ii\nh✝ : ¬ii = ji\n⊢ M ji jx (cast (_ : n' { fst := ii, snd := ix }.fst = n' { fst := ji, snd := jx }.fst) ix) = 0",
"tactic": "simp_all only [not_true]"
},
{
"state_after": "no goals",
"state_before": "case a.mk.h.mk.inr.inl\nl : Type ?u.198919\nm : Type ?u.198922\nn : Type ?u.198925\no : Type u_4\np : Type ?u.198931\nq : Type ?u.198934\nm' : o → Type u_1\nn' : o → Type u_2\np' : o → Type ?u.198949\nR : Type ?u.198952\nS : Type ?u.198955\nα : Type u_3\nβ : Type ?u.198961\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : (i : o) → Matrix (m' i) (n' i) α\nii : o\nix : n' ii\nji : o\njx : m' ji\nh : ¬ji = ii\nh✝ : ii = ji\n⊢ 0 = M ii (cast (_ : m' { fst := ji, snd := jx }.fst = m' { fst := ii, snd := ix }.fst) jx) ix",
"tactic": "simp_all only [not_true]"
},
{
"state_after": "no goals",
"state_before": "case a.mk.h.mk.inr.inr\nl : Type ?u.198919\nm : Type ?u.198922\nn : Type ?u.198925\no : Type u_4\np : Type ?u.198931\nq : Type ?u.198934\nm' : o → Type u_1\nn' : o → Type u_2\np' : o → Type ?u.198949\nR : Type ?u.198952\nS : Type ?u.198955\nα : Type u_3\nβ : Type ?u.198961\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : (i : o) → Matrix (m' i) (n' i) α\nii : o\nix : n' ii\nji : o\njx : m' ji\nh : ¬ji = ii\nh✝ : ¬ii = ji\n⊢ 0 = 0",
"tactic": "rfl"
}
] |
[
684,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
676,
1
] |
Mathlib/Data/Set/Pairwise/Basic.lean
|
Set.PairwiseDisjoint.subset
|
[] |
[
250,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
249,
1
] |
Mathlib/LinearAlgebra/Dfinsupp.lean
|
Dfinsupp.coprodMap_apply
|
[] |
[
277,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
273,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Connected.set_univ_walk_nonempty
|
[] |
[
2209,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2207,
1
] |
Mathlib/Order/Disjointed.lean
|
disjointed_le_id
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.1032\ninst✝ : GeneralizedBooleanAlgebra α\nf : ℕ → α\nn : ℕ\n⊢ disjointed f n ≤ id f n",
"state_before": "α : Type u_1\nβ : Type ?u.1032\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ disjointed ≤ id",
"tactic": "rintro f n"
},
{
"state_after": "case zero\nα : Type u_1\nβ : Type ?u.1032\ninst✝ : GeneralizedBooleanAlgebra α\nf : ℕ → α\n⊢ disjointed f Nat.zero ≤ id f Nat.zero\n\ncase succ\nα : Type u_1\nβ : Type ?u.1032\ninst✝ : GeneralizedBooleanAlgebra α\nf : ℕ → α\nn✝ : ℕ\n⊢ disjointed f (Nat.succ n✝) ≤ id f (Nat.succ n✝)",
"state_before": "α : Type u_1\nβ : Type ?u.1032\ninst✝ : GeneralizedBooleanAlgebra α\nf : ℕ → α\nn : ℕ\n⊢ disjointed f n ≤ id f n",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u_1\nβ : Type ?u.1032\ninst✝ : GeneralizedBooleanAlgebra α\nf : ℕ → α\n⊢ disjointed f Nat.zero ≤ id f Nat.zero",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type u_1\nβ : Type ?u.1032\ninst✝ : GeneralizedBooleanAlgebra α\nf : ℕ → α\nn✝ : ℕ\n⊢ disjointed f (Nat.succ n✝) ≤ id f (Nat.succ n✝)",
"tactic": "exact sdiff_le"
}
] |
[
70,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ioo_subset_Ioi_self
|
[] |
[
533,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
533,
1
] |
Mathlib/Topology/Order.lean
|
continuous_inf_dom_left
|
[] |
[
779,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
777,
1
] |
Mathlib/GroupTheory/Subgroup/Pointwise.lean
|
Subgroup.pointwise_smul_subset_iff
|
[] |
[
358,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
357,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.count_apply_eq_top
|
[
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.378850\nγ : Type ?u.378853\nδ : Type ?u.378856\nι : Type ?u.378859\nR : Type ?u.378862\nR' : Type ?u.378865\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nhs : Set.Finite s\n⊢ ↑↑count s = ⊤ ↔ Set.Infinite s\n\ncase neg\nα : Type u_1\nβ : Type ?u.378850\nγ : Type ?u.378853\nδ : Type ?u.378856\nι : Type ?u.378859\nR : Type ?u.378862\nR' : Type ?u.378865\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nhs : ¬Set.Finite s\n⊢ ↑↑count s = ⊤ ↔ Set.Infinite s",
"state_before": "α : Type u_1\nβ : Type ?u.378850\nγ : Type ?u.378853\nδ : Type ?u.378856\nι : Type ?u.378859\nR : Type ?u.378862\nR' : Type ?u.378865\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\n⊢ ↑↑count s = ⊤ ↔ Set.Infinite s",
"tactic": "by_cases hs : s.Finite"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.378850\nγ : Type ?u.378853\nδ : Type ?u.378856\nι : Type ?u.378859\nR : Type ?u.378862\nR' : Type ?u.378865\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nhs : Set.Finite s\n⊢ ↑↑count s = ⊤ ↔ Set.Infinite s",
"tactic": "exact count_apply_eq_top' hs.measurableSet"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.378850\nγ : Type ?u.378853\nδ : Type ?u.378856\nι : Type ?u.378859\nR : Type ?u.378862\nR' : Type ?u.378865\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nhs : Set.Infinite s\n⊢ ↑↑count s = ⊤ ↔ Set.Infinite s",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.378850\nγ : Type ?u.378853\nδ : Type ?u.378856\nι : Type ?u.378859\nR : Type ?u.378862\nR' : Type ?u.378865\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nhs : ¬Set.Finite s\n⊢ ↑↑count s = ⊤ ↔ Set.Infinite s",
"tactic": "change s.Infinite at hs"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.378850\nγ : Type ?u.378853\nδ : Type ?u.378856\nι : Type ?u.378859\nR : Type ?u.378862\nR' : Type ?u.378865\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nhs : Set.Infinite s\n⊢ ↑↑count s = ⊤ ↔ Set.Infinite s",
"tactic": "simp [hs, count_apply_infinite]"
}
] |
[
2254,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2250,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.symm_piecewise
|
[] |
[
1009,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1006,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
upperClosure_anti
|
[] |
[
1383,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1382,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
|
Real.sinh_log
|
[
{
"state_after": "no goals",
"state_before": "x✝ y x : ℝ\nhx : 0 < x\n⊢ sinh (log x) = (x - x⁻¹) / 2",
"tactic": "rw [sinh_eq, exp_neg, exp_log hx]"
}
] |
[
118,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/LinearAlgebra/Dimension.lean
|
rank_range_add_rank_ker
|
[
{
"state_after": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.580352\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V'\ninst✝² : Module K V'\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K V₁\nf : V →ₗ[K] V₁\nthis : (p : Submodule K V) → DecidableEq (V ⧸ p)\n⊢ Module.rank K { x // x ∈ LinearMap.range f } + Module.rank K { x // x ∈ LinearMap.ker f } = Module.rank K V",
"state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.580352\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V'\ninst✝² : Module K V'\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K V₁\nf : V →ₗ[K] V₁\n⊢ Module.rank K { x // x ∈ LinearMap.range f } + Module.rank K { x // x ∈ LinearMap.ker f } = Module.rank K V",
"tactic": "haveI := fun p : Submodule K V => Classical.decEq (V ⧸ p)"
},
{
"state_after": "no goals",
"state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.580352\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V'\ninst✝² : Module K V'\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K V₁\nf : V →ₗ[K] V₁\nthis : (p : Submodule K V) → DecidableEq (V ⧸ p)\n⊢ Module.rank K { x // x ∈ LinearMap.range f } + Module.rank K { x // x ∈ LinearMap.ker f } = Module.rank K V",
"tactic": "rw [← f.quotKerEquivRange.rank_eq, rank_quotient_add_rank]"
}
] |
[
1093,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1090,
1
] |
Mathlib/Topology/Algebra/ContinuousAffineMap.lean
|
ContinuousAffineMap.to_continuousMap_injective
|
[
{
"state_after": "case h\nR : Type u_1\nV : Type u_2\nW : Type u_3\nP : Type u_4\nQ : Type u_5\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : TopologicalSpace P\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup W\ninst✝² : Module R W\ninst✝¹ : TopologicalSpace Q\ninst✝ : AddTorsor W Q\nf g : P →A[R] Q\nh : ContinuousMap.mk ↑f = ContinuousMap.mk ↑g\na : P\n⊢ ↑f a = ↑g a",
"state_before": "R : Type u_1\nV : Type u_2\nW : Type u_3\nP : Type u_4\nQ : Type u_5\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : TopologicalSpace P\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup W\ninst✝² : Module R W\ninst✝¹ : TopologicalSpace Q\ninst✝ : AddTorsor W Q\nf g : P →A[R] Q\nh : ContinuousMap.mk ↑f = ContinuousMap.mk ↑g\n⊢ f = g",
"tactic": "ext a"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\nV : Type u_2\nW : Type u_3\nP : Type u_4\nQ : Type u_5\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : TopologicalSpace P\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup W\ninst✝² : Module R W\ninst✝¹ : TopologicalSpace Q\ninst✝ : AddTorsor W Q\nf g : P →A[R] Q\nh : ContinuousMap.mk ↑f = ContinuousMap.mk ↑g\na : P\n⊢ ↑f a = ↑g a",
"tactic": "exact ContinuousMap.congr_fun h a"
}
] |
[
119,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Data/Finset/NAry.lean
|
Finset.image₂_curry
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nα' : Type ?u.65130\nβ : Type u_2\nβ' : Type ?u.65136\nγ : Type u_3\nγ' : Type ?u.65142\nδ : Type ?u.65145\nδ' : Type ?u.65148\nε : Type ?u.65151\nε' : Type ?u.65154\nζ : Type ?u.65157\nζ' : Type ?u.65160\nν : Type ?u.65163\ninst✝⁷ : DecidableEq α'\ninst✝⁶ : DecidableEq β'\ninst✝⁵ : DecidableEq γ\ninst✝⁴ : DecidableEq γ'\ninst✝³ : DecidableEq δ\ninst✝² : DecidableEq δ'\ninst✝¹ : DecidableEq ε\ninst✝ : DecidableEq ε'\nf✝ f' : α → β → γ\ng g' : α → β → γ → δ\ns✝ s' : Finset α\nt✝ t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\nf : α × β → γ\ns : Finset α\nt : Finset β\n⊢ image₂ (curry f) s t = image f (s ×ˢ t)",
"tactic": "classical rw [← image₂_mk_eq_product, image_image₂]; dsimp [curry]"
},
{
"state_after": "α : Type u_1\nα' : Type ?u.65130\nβ : Type u_2\nβ' : Type ?u.65136\nγ : Type u_3\nγ' : Type ?u.65142\nδ : Type ?u.65145\nδ' : Type ?u.65148\nε : Type ?u.65151\nε' : Type ?u.65154\nζ : Type ?u.65157\nζ' : Type ?u.65160\nν : Type ?u.65163\ninst✝⁷ : DecidableEq α'\ninst✝⁶ : DecidableEq β'\ninst✝⁵ : DecidableEq γ\ninst✝⁴ : DecidableEq γ'\ninst✝³ : DecidableEq δ\ninst✝² : DecidableEq δ'\ninst✝¹ : DecidableEq ε\ninst✝ : DecidableEq ε'\nf✝ f' : α → β → γ\ng g' : α → β → γ → δ\ns✝ s' : Finset α\nt✝ t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\nf : α × β → γ\ns : Finset α\nt : Finset β\n⊢ image₂ (curry f) s t = image₂ (fun a b => f (a, b)) s t",
"state_before": "α : Type u_1\nα' : Type ?u.65130\nβ : Type u_2\nβ' : Type ?u.65136\nγ : Type u_3\nγ' : Type ?u.65142\nδ : Type ?u.65145\nδ' : Type ?u.65148\nε : Type ?u.65151\nε' : Type ?u.65154\nζ : Type ?u.65157\nζ' : Type ?u.65160\nν : Type ?u.65163\ninst✝⁷ : DecidableEq α'\ninst✝⁶ : DecidableEq β'\ninst✝⁵ : DecidableEq γ\ninst✝⁴ : DecidableEq γ'\ninst✝³ : DecidableEq δ\ninst✝² : DecidableEq δ'\ninst✝¹ : DecidableEq ε\ninst✝ : DecidableEq ε'\nf✝ f' : α → β → γ\ng g' : α → β → γ → δ\ns✝ s' : Finset α\nt✝ t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\nf : α × β → γ\ns : Finset α\nt : Finset β\n⊢ image₂ (curry f) s t = image f (s ×ˢ t)",
"tactic": "rw [← image₂_mk_eq_product, image_image₂]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nα' : Type ?u.65130\nβ : Type u_2\nβ' : Type ?u.65136\nγ : Type u_3\nγ' : Type ?u.65142\nδ : Type ?u.65145\nδ' : Type ?u.65148\nε : Type ?u.65151\nε' : Type ?u.65154\nζ : Type ?u.65157\nζ' : Type ?u.65160\nν : Type ?u.65163\ninst✝⁷ : DecidableEq α'\ninst✝⁶ : DecidableEq β'\ninst✝⁵ : DecidableEq γ\ninst✝⁴ : DecidableEq γ'\ninst✝³ : DecidableEq δ\ninst✝² : DecidableEq δ'\ninst✝¹ : DecidableEq ε\ninst✝ : DecidableEq ε'\nf✝ f' : α → β → γ\ng g' : α → β → γ → δ\ns✝ s' : Finset α\nt✝ t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\nf : α × β → γ\ns : Finset α\nt : Finset β\n⊢ image₂ (curry f) s t = image₂ (fun a b => f (a, b)) s t",
"tactic": "dsimp [curry]"
}
] |
[
342,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
340,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.subset_singleton_iff'
|
[] |
[
793,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
792,
1
] |
Mathlib/Data/Polynomial/Expand.lean
|
Polynomial.expand_eval
|
[
{
"state_after": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q p : ℕ\nP : R[X]\nr : R\nf g : R[X]\nhf : eval r (↑(expand R p) f) = eval (r ^ p) f\nhg : eval r (↑(expand R p) g) = eval (r ^ p) g\n⊢ eval r (↑(expand R p) (f + g)) = eval (r ^ p) (f + g)",
"state_before": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q p : ℕ\nP : R[X]\nr : R\n⊢ eval r (↑(expand R p) P) = eval (r ^ p) P",
"tactic": "refine' Polynomial.induction_on P (fun a => by simp) (fun f g hf hg => _) fun n a _ => by simp"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q p : ℕ\nP : R[X]\nr : R\nf g : R[X]\nhf : eval r (↑(expand R p) f) = eval (r ^ p) f\nhg : eval r (↑(expand R p) g) = eval (r ^ p) g\n⊢ eval r (↑(expand R p) (f + g)) = eval (r ^ p) (f + g)",
"tactic": "rw [AlgHom.map_add, eval_add, eval_add, hf, hg]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q p : ℕ\nP : R[X]\nr a : R\n⊢ eval r (↑(expand R p) (↑C a)) = eval (r ^ p) (↑C a)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q p : ℕ\nP : R[X]\nr : R\nn : ℕ\na : R\nx✝ : eval r (↑(expand R p) (↑C a * X ^ n)) = eval (r ^ p) (↑C a * X ^ n)\n⊢ eval r (↑(expand R p) (↑C a * X ^ (n + 1))) = eval (r ^ p) (↑C a * X ^ (n + 1))",
"tactic": "simp"
}
] |
[
193,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/Topology/Separation.lean
|
Set.EqOn.closure
|
[] |
[
1191,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1189,
11
] |
Mathlib/Topology/Sober.lean
|
IsGenericPoint.specializes
|
[] |
[
67,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
11
] |
Mathlib/MeasureTheory/Integral/CircleIntegral.lean
|
lipschitzWith_circleMap
|
[
{
"state_after": "no goals",
"state_before": "E : Type ?u.27048\ninst✝ : NormedAddCommGroup E\nc : ℂ\nR θ : ℝ\n⊢ ↑‖deriv (circleMap c R) θ‖₊ ≤ ↑(↑Real.nnabs R)",
"tactic": "simp"
}
] |
[
213,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
1
] |
Mathlib/LinearAlgebra/Pi.lean
|
LinearMap.apply_single
|
[] |
[
124,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.Dart.symm_mk
|
[] |
[
747,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
746,
1
] |
Std/Logic.lean
|
and_iff_right_iff_imp
|
[] |
[
212,
49
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
211,
9
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.bilinearComp_apply
|
[] |
[
1364,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1362,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Type.lean
|
Equiv.Perm.isCycle_of_prime_order''
|
[
{
"state_after": "α : Type u_1\ninst✝ : Fintype α\nσ : Perm α\nh1 : Nat.Prime (Fintype.card α)\nh2 : orderOf σ = Fintype.card α\n⊢ 1 < 2",
"state_before": "α : Type u_1\ninst✝ : Fintype α\nσ : Perm α\nh1 : Nat.Prime (Fintype.card α)\nh2 : orderOf σ = Fintype.card α\n⊢ Fintype.card α < 2 * orderOf σ",
"tactic": "rw [← one_mul (Fintype.card α), ← h2, mul_lt_mul_right (orderOf_pos σ)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Fintype α\nσ : Perm α\nh1 : Nat.Prime (Fintype.card α)\nh2 : orderOf σ = Fintype.card α\n⊢ 1 < 2",
"tactic": "exact one_lt_two"
}
] |
[
365,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean
|
AffineIsometryEquiv.ediam_image
|
[] |
[
669,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
668,
1
] |
Std/Data/Fin/Lemmas.lean
|
USize.le_def
|
[] |
[
26,
73
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
26,
9
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.adjoin_int
|
[
{
"state_after": "no goals",
"state_before": "F : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nS : Set E\nn : ℤ\n⊢ F⟮↑n⟯ = ⊥",
"tactic": "refine' adjoin_simple_eq_bot_iff.mpr (coe_int_mem ⊥ n)"
}
] |
[
697,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
696,
1
] |
src/lean/Init/Data/Nat/Linear.lean
|
Nat.Linear.Poly.of_denote_eq_cancel
|
[
{
"state_after": "ctx : Context\nm₁ m₂ : Poly\nh : denote_eq ctx (cancel m₁ m₂)\n⊢ denote_eq ctx (m₁, m₂)",
"state_before": "ctx : Context\nm₁ m₂ : Poly\nh : denote_eq ctx (cancel m₁ m₂)\n⊢ denote_eq ctx (m₁, m₂)",
"tactic": "simp at h"
},
{
"state_after": "ctx : Context\nm₁ m₂ : Poly\nh : denote_eq ctx (cancel m₁ m₂)\nthis : denote_eq ctx (List.reverse [] ++ m₁, List.reverse [] ++ m₂)\n⊢ denote_eq ctx (m₁, m₂)",
"state_before": "ctx : Context\nm₁ m₂ : Poly\nh : denote_eq ctx (cancel m₁ m₂)\n⊢ denote_eq ctx (m₁, m₂)",
"tactic": "have := Poly.of_denote_eq_cancelAux (h := h)"
},
{
"state_after": "ctx : Context\nm₁ m₂ : Poly\nh : denote_eq ctx (cancel m₁ m₂)\nthis : denote_eq ctx (m₁, m₂)\n⊢ denote_eq ctx (m₁, m₂)",
"state_before": "ctx : Context\nm₁ m₂ : Poly\nh : denote_eq ctx (cancel m₁ m₂)\nthis : denote_eq ctx (List.reverse [] ++ m₁, List.reverse [] ++ m₂)\n⊢ denote_eq ctx (m₁, m₂)",
"tactic": "simp at this"
},
{
"state_after": "no goals",
"state_before": "ctx : Context\nm₁ m₂ : Poly\nh : denote_eq ctx (cancel m₁ m₂)\nthis : denote_eq ctx (m₁, m₂)\n⊢ denote_eq ctx (m₁, m₂)",
"tactic": "assumption"
}
] |
[
422,
13
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
418,
1
] |
Mathlib/MeasureTheory/Measure/Lebesgue/Complex.lean
|
Complex.volume_preserving_equiv_pi
|
[] |
[
48,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
1
] |
Mathlib/LinearAlgebra/BilinearMap.lean
|
LinearMap.compl₁₂_inj
|
[
{
"state_after": "case mp\nR : Type u_1\ninst✝²³ : CommSemiring R\nR₂ : Type ?u.489611\ninst✝²² : CommSemiring R₂\nR₃ : Type ?u.489617\ninst✝²¹ : CommSemiring R₃\nR₄ : Type ?u.489623\ninst✝²⁰ : CommSemiring R₄\nM : Type ?u.489629\nN : Type ?u.489632\nP : Type ?u.489635\nQ : Type ?u.489638\nMₗ : Type u_2\nNₗ : Type u_4\nPₗ : Type u_3\nQₗ : Type u_5\nQₗ' : Type u_6\ninst✝¹⁹ : AddCommMonoid M\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : AddCommMonoid P\ninst✝¹⁶ : AddCommMonoid Q\ninst✝¹⁵ : AddCommMonoid Mₗ\ninst✝¹⁴ : AddCommMonoid Nₗ\ninst✝¹³ : AddCommMonoid Pₗ\ninst✝¹² : AddCommMonoid Qₗ\ninst✝¹¹ : AddCommMonoid Qₗ'\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R₂ N\ninst✝⁸ : Module R₃ P\ninst✝⁷ : Module R₄ Q\ninst✝⁶ : Module R Mₗ\ninst✝⁵ : Module R Nₗ\ninst✝⁴ : Module R Pₗ\ninst✝³ : Module R Qₗ\ninst✝² : Module R Qₗ'\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nσ₄₂ : R₄ →+* R₂\nσ₄₃ : R₄ →+* R₃\ninst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝ : RingHomCompTriple σ₄₂ σ₂₃ σ₄₃\nf : M →ₛₗ[σ₁₃] N →ₛₗ[σ₂₃] P\nf₁ f₂ : Mₗ →ₗ[R] Nₗ →ₗ[R] Pₗ\ng : Qₗ →ₗ[R] Mₗ\ng' : Qₗ' →ₗ[R] Nₗ\nhₗ : Function.Surjective ↑g\nhᵣ : Function.Surjective ↑g'\nh : compl₁₂ f₁ g g' = compl₁₂ f₂ g g'\n⊢ f₁ = f₂\n\ncase mpr\nR : Type u_1\ninst✝²³ : CommSemiring R\nR₂ : Type ?u.489611\ninst✝²² : CommSemiring R₂\nR₃ : Type ?u.489617\ninst✝²¹ : CommSemiring R₃\nR₄ : Type ?u.489623\ninst✝²⁰ : CommSemiring R₄\nM : Type ?u.489629\nN : Type ?u.489632\nP : Type ?u.489635\nQ : Type ?u.489638\nMₗ : Type u_2\nNₗ : Type u_4\nPₗ : Type u_3\nQₗ : Type u_5\nQₗ' : Type u_6\ninst✝¹⁹ : AddCommMonoid M\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : AddCommMonoid P\ninst✝¹⁶ : AddCommMonoid Q\ninst✝¹⁵ : AddCommMonoid Mₗ\ninst✝¹⁴ : AddCommMonoid Nₗ\ninst✝¹³ : AddCommMonoid Pₗ\ninst✝¹² : AddCommMonoid Qₗ\ninst✝¹¹ : AddCommMonoid Qₗ'\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R₂ N\ninst✝⁸ : Module R₃ P\ninst✝⁷ : Module R₄ Q\ninst✝⁶ : Module R Mₗ\ninst✝⁵ : Module R Nₗ\ninst✝⁴ : Module R Pₗ\ninst✝³ : Module R Qₗ\ninst✝² : Module R Qₗ'\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nσ₄₂ : R₄ →+* R₂\nσ₄₃ : R₄ →+* R₃\ninst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝ : RingHomCompTriple σ₄₂ σ₂₃ σ₄₃\nf : M →ₛₗ[σ₁₃] N →ₛₗ[σ₂₃] P\nf₁ f₂ : Mₗ →ₗ[R] Nₗ →ₗ[R] Pₗ\ng : Qₗ →ₗ[R] Mₗ\ng' : Qₗ' →ₗ[R] Nₗ\nhₗ : Function.Surjective ↑g\nhᵣ : Function.Surjective ↑g'\nh : f₁ = f₂\n⊢ compl₁₂ f₁ g g' = compl₁₂ f₂ g g'",
"state_before": "R : Type u_1\ninst✝²³ : CommSemiring R\nR₂ : Type ?u.489611\ninst✝²² : CommSemiring R₂\nR₃ : Type ?u.489617\ninst✝²¹ : CommSemiring R₃\nR₄ : Type ?u.489623\ninst✝²⁰ : CommSemiring R₄\nM : Type ?u.489629\nN : Type ?u.489632\nP : Type ?u.489635\nQ : Type ?u.489638\nMₗ : Type u_2\nNₗ : Type u_4\nPₗ : Type u_3\nQₗ : Type u_5\nQₗ' : Type u_6\ninst✝¹⁹ : AddCommMonoid M\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : AddCommMonoid P\ninst✝¹⁶ : AddCommMonoid Q\ninst✝¹⁵ : AddCommMonoid Mₗ\ninst✝¹⁴ : AddCommMonoid Nₗ\ninst✝¹³ : AddCommMonoid Pₗ\ninst✝¹² : AddCommMonoid Qₗ\ninst✝¹¹ : AddCommMonoid Qₗ'\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R₂ N\ninst✝⁸ : Module R₃ P\ninst✝⁷ : Module R₄ Q\ninst✝⁶ : Module R Mₗ\ninst✝⁵ : Module R Nₗ\ninst✝⁴ : Module R Pₗ\ninst✝³ : Module R Qₗ\ninst✝² : Module R Qₗ'\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nσ₄₂ : R₄ →+* R₂\nσ₄₃ : R₄ →+* R₃\ninst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝ : RingHomCompTriple σ₄₂ σ₂₃ σ₄₃\nf : M →ₛₗ[σ₁₃] N →ₛₗ[σ₂₃] P\nf₁ f₂ : Mₗ →ₗ[R] Nₗ →ₗ[R] Pₗ\ng : Qₗ →ₗ[R] Mₗ\ng' : Qₗ' →ₗ[R] Nₗ\nhₗ : Function.Surjective ↑g\nhᵣ : Function.Surjective ↑g'\n⊢ compl₁₂ f₁ g g' = compl₁₂ f₂ g g' ↔ f₁ = f₂",
"tactic": "constructor <;> intro h"
},
{
"state_after": "case mp.h.h\nR : Type u_1\ninst✝²³ : CommSemiring R\nR₂ : Type ?u.489611\ninst✝²² : CommSemiring R₂\nR₃ : Type ?u.489617\ninst✝²¹ : CommSemiring R₃\nR₄ : Type ?u.489623\ninst✝²⁰ : CommSemiring R₄\nM : Type ?u.489629\nN : Type ?u.489632\nP : Type ?u.489635\nQ : Type ?u.489638\nMₗ : Type u_2\nNₗ : Type u_4\nPₗ : Type u_3\nQₗ : Type u_5\nQₗ' : Type u_6\ninst✝¹⁹ : AddCommMonoid M\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : AddCommMonoid P\ninst✝¹⁶ : AddCommMonoid Q\ninst✝¹⁵ : AddCommMonoid Mₗ\ninst✝¹⁴ : AddCommMonoid Nₗ\ninst✝¹³ : AddCommMonoid Pₗ\ninst✝¹² : AddCommMonoid Qₗ\ninst✝¹¹ : AddCommMonoid Qₗ'\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R₂ N\ninst✝⁸ : Module R₃ P\ninst✝⁷ : Module R₄ Q\ninst✝⁶ : Module R Mₗ\ninst✝⁵ : Module R Nₗ\ninst✝⁴ : Module R Pₗ\ninst✝³ : Module R Qₗ\ninst✝² : Module R Qₗ'\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nσ₄₂ : R₄ →+* R₂\nσ₄₃ : R₄ →+* R₃\ninst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝ : RingHomCompTriple σ₄₂ σ₂₃ σ₄₃\nf : M →ₛₗ[σ₁₃] N →ₛₗ[σ₂₃] P\nf₁ f₂ : Mₗ →ₗ[R] Nₗ →ₗ[R] Pₗ\ng : Qₗ →ₗ[R] Mₗ\ng' : Qₗ' →ₗ[R] Nₗ\nhₗ : Function.Surjective ↑g\nhᵣ : Function.Surjective ↑g'\nh : compl₁₂ f₁ g g' = compl₁₂ f₂ g g'\nx : Mₗ\ny : Nₗ\n⊢ ↑(↑f₁ x) y = ↑(↑f₂ x) y",
"state_before": "case mp\nR : Type u_1\ninst✝²³ : CommSemiring R\nR₂ : Type ?u.489611\ninst✝²² : CommSemiring R₂\nR₃ : Type ?u.489617\ninst✝²¹ : CommSemiring R₃\nR₄ : Type ?u.489623\ninst✝²⁰ : CommSemiring R₄\nM : Type ?u.489629\nN : Type ?u.489632\nP : Type ?u.489635\nQ : Type ?u.489638\nMₗ : Type u_2\nNₗ : Type u_4\nPₗ : Type u_3\nQₗ : Type u_5\nQₗ' : Type u_6\ninst✝¹⁹ : AddCommMonoid M\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : AddCommMonoid P\ninst✝¹⁶ : AddCommMonoid Q\ninst✝¹⁵ : AddCommMonoid Mₗ\ninst✝¹⁴ : AddCommMonoid Nₗ\ninst✝¹³ : AddCommMonoid Pₗ\ninst✝¹² : AddCommMonoid Qₗ\ninst✝¹¹ : AddCommMonoid Qₗ'\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R₂ N\ninst✝⁸ : Module R₃ P\ninst✝⁷ : Module R₄ Q\ninst✝⁶ : Module R Mₗ\ninst✝⁵ : Module R Nₗ\ninst✝⁴ : Module R Pₗ\ninst✝³ : Module R Qₗ\ninst✝² : Module R Qₗ'\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nσ₄₂ : R₄ →+* R₂\nσ₄₃ : R₄ →+* R₃\ninst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝ : RingHomCompTriple σ₄₂ σ₂₃ σ₄₃\nf : M →ₛₗ[σ₁₃] N →ₛₗ[σ₂₃] P\nf₁ f₂ : Mₗ →ₗ[R] Nₗ →ₗ[R] Pₗ\ng : Qₗ →ₗ[R] Mₗ\ng' : Qₗ' →ₗ[R] Nₗ\nhₗ : Function.Surjective ↑g\nhᵣ : Function.Surjective ↑g'\nh : compl₁₂ f₁ g g' = compl₁₂ f₂ g g'\n⊢ f₁ = f₂",
"tactic": "ext (x y)"
},
{
"state_after": "case mp.h.h.intro\nR : Type u_1\ninst✝²³ : CommSemiring R\nR₂ : Type ?u.489611\ninst✝²² : CommSemiring R₂\nR₃ : Type ?u.489617\ninst✝²¹ : CommSemiring R₃\nR₄ : Type ?u.489623\ninst✝²⁰ : CommSemiring R₄\nM : Type ?u.489629\nN : Type ?u.489632\nP : Type ?u.489635\nQ : Type ?u.489638\nMₗ : Type u_2\nNₗ : Type u_4\nPₗ : Type u_3\nQₗ : Type u_5\nQₗ' : Type u_6\ninst✝¹⁹ : AddCommMonoid M\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : AddCommMonoid P\ninst✝¹⁶ : AddCommMonoid Q\ninst✝¹⁵ : AddCommMonoid Mₗ\ninst✝¹⁴ : AddCommMonoid Nₗ\ninst✝¹³ : AddCommMonoid Pₗ\ninst✝¹² : AddCommMonoid Qₗ\ninst✝¹¹ : AddCommMonoid Qₗ'\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R₂ N\ninst✝⁸ : Module R₃ P\ninst✝⁷ : Module R₄ Q\ninst✝⁶ : Module R Mₗ\ninst✝⁵ : Module R Nₗ\ninst✝⁴ : Module R Pₗ\ninst✝³ : Module R Qₗ\ninst✝² : Module R Qₗ'\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nσ₄₂ : R₄ →+* R₂\nσ₄₃ : R₄ →+* R₃\ninst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝ : RingHomCompTriple σ₄₂ σ₂₃ σ₄₃\nf : M →ₛₗ[σ₁₃] N →ₛₗ[σ₂₃] P\nf₁ f₂ : Mₗ →ₗ[R] Nₗ →ₗ[R] Pₗ\ng : Qₗ →ₗ[R] Mₗ\ng' : Qₗ' →ₗ[R] Nₗ\nhₗ : Function.Surjective ↑g\nhᵣ : Function.Surjective ↑g'\nh : compl₁₂ f₁ g g' = compl₁₂ f₂ g g'\nx : Mₗ\ny : Nₗ\nx' : Qₗ\nhx : ↑g x' = x\n⊢ ↑(↑f₁ x) y = ↑(↑f₂ x) y",
"state_before": "case mp.h.h\nR : Type u_1\ninst✝²³ : CommSemiring R\nR₂ : Type ?u.489611\ninst✝²² : CommSemiring R₂\nR₃ : Type ?u.489617\ninst✝²¹ : CommSemiring R₃\nR₄ : Type ?u.489623\ninst✝²⁰ : CommSemiring R₄\nM : Type ?u.489629\nN : Type ?u.489632\nP : Type ?u.489635\nQ : Type ?u.489638\nMₗ : Type u_2\nNₗ : Type u_4\nPₗ : Type u_3\nQₗ : Type u_5\nQₗ' : Type u_6\ninst✝¹⁹ : AddCommMonoid M\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : AddCommMonoid P\ninst✝¹⁶ : AddCommMonoid Q\ninst✝¹⁵ : AddCommMonoid Mₗ\ninst✝¹⁴ : AddCommMonoid Nₗ\ninst✝¹³ : AddCommMonoid Pₗ\ninst✝¹² : AddCommMonoid Qₗ\ninst✝¹¹ : AddCommMonoid Qₗ'\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R₂ N\ninst✝⁸ : Module R₃ P\ninst✝⁷ : Module R₄ Q\ninst✝⁶ : Module R Mₗ\ninst✝⁵ : Module R Nₗ\ninst✝⁴ : Module R Pₗ\ninst✝³ : Module R Qₗ\ninst✝² : Module R Qₗ'\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nσ₄₂ : R₄ →+* R₂\nσ₄₃ : R₄ →+* R₃\ninst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝ : RingHomCompTriple σ₄₂ σ₂₃ σ₄₃\nf : M →ₛₗ[σ₁₃] N →ₛₗ[σ₂₃] P\nf₁ f₂ : Mₗ →ₗ[R] Nₗ →ₗ[R] Pₗ\ng : Qₗ →ₗ[R] Mₗ\ng' : Qₗ' →ₗ[R] Nₗ\nhₗ : Function.Surjective ↑g\nhᵣ : Function.Surjective ↑g'\nh : compl₁₂ f₁ g g' = compl₁₂ f₂ g g'\nx : Mₗ\ny : Nₗ\n⊢ ↑(↑f₁ x) y = ↑(↑f₂ x) y",
"tactic": "cases' hₗ x with x' hx"
},
{
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361,
1
] |
Mathlib/Computability/Reduce.lean
|
ManyOneDegree.add_le
|
[
{
"state_after": "case h\nα : Type u\ninst✝⁵ : Primcodable α\ninst✝⁴ : Inhabited α\nβ : Type v\ninst✝³ : Primcodable β\ninst✝² : Inhabited β\nγ : Type w\ninst✝¹ : Primcodable γ\ninst✝ : Inhabited γ\nd₂ d₃ : ManyOneDegree\np✝ : Set ℕ\n⊢ of p✝ + d₂ ≤ d₃ ↔ of p✝ ≤ d₃ ∧ d₂ ≤ d₃",
"state_before": "α : Type u\ninst✝⁵ : Primcodable α\ninst✝⁴ : Inhabited α\nβ : Type v\ninst✝³ : Primcodable β\ninst✝² : Inhabited β\nγ : Type w\ninst✝¹ : Primcodable γ\ninst✝ : Inhabited γ\nd₁ d₂ d₃ : ManyOneDegree\n⊢ d₁ + d₂ ≤ d₃ ↔ d₁ ≤ d₃ ∧ d₂ ≤ d₃",
"tactic": "induction d₁ using ManyOneDegree.ind_on"
},
{
"state_after": "case h.h\nα : Type u\ninst✝⁵ : Primcodable α\ninst✝⁴ : Inhabited α\nβ : Type v\ninst✝³ : Primcodable β\ninst✝² : Inhabited β\nγ : Type w\ninst✝¹ : Primcodable γ\ninst✝ : Inhabited γ\nd₃ : ManyOneDegree\np✝¹ p✝ : Set ℕ\n⊢ of p✝¹ + of p✝ ≤ d₃ ↔ of p✝¹ ≤ d₃ ∧ of p✝ ≤ d₃",
"state_before": "case h\nα : Type u\ninst✝⁵ : Primcodable α\ninst✝⁴ : Inhabited α\nβ : Type v\ninst✝³ : Primcodable β\ninst✝² : Inhabited β\nγ : Type w\ninst✝¹ : Primcodable γ\ninst✝ : Inhabited γ\nd₂ d₃ : ManyOneDegree\np✝ : Set ℕ\n⊢ of p✝ + d₂ ≤ d₃ ↔ of p✝ ≤ d₃ ∧ d₂ ≤ d₃",
"tactic": "induction d₂ using ManyOneDegree.ind_on"
},
{
"state_after": "case h.h.h\nα : Type u\ninst✝⁵ : Primcodable α\ninst✝⁴ : Inhabited α\nβ : Type v\ninst✝³ : Primcodable β\ninst✝² : Inhabited β\nγ : Type w\ninst✝¹ : Primcodable γ\ninst✝ : Inhabited γ\np✝² p✝¹ p✝ : Set ℕ\n⊢ of p✝² + of p✝¹ ≤ of p✝ ↔ of p✝² ≤ of p✝ ∧ of p✝¹ ≤ of p✝",
"state_before": "case h.h\nα : Type u\ninst✝⁵ : Primcodable α\ninst✝⁴ : Inhabited α\nβ : Type v\ninst✝³ : Primcodable β\ninst✝² : Inhabited β\nγ : Type w\ninst✝¹ : Primcodable γ\ninst✝ : Inhabited γ\nd₃ : ManyOneDegree\np✝¹ p✝ : Set ℕ\n⊢ of p✝¹ + of p✝ ≤ d₃ ↔ of p✝¹ ≤ d₃ ∧ of p✝ ≤ d₃",
"tactic": "induction d₃ using ManyOneDegree.ind_on"
},
{
"state_after": "no goals",
"state_before": "case h.h.h\nα : Type u\ninst✝⁵ : Primcodable α\ninst✝⁴ : Inhabited α\nβ : Type v\ninst✝³ : Primcodable β\ninst✝² : Inhabited β\nγ : Type w\ninst✝¹ : Primcodable γ\ninst✝ : Inhabited γ\np✝² p✝¹ p✝ : Set ℕ\n⊢ of p✝² + of p✝¹ ≤ of p✝ ↔ of p✝² ≤ of p✝ ∧ of p✝¹ ≤ of p✝",
"tactic": "simpa only [← add_of, of_le_of] using disjoin_le"
}
] |
[
495,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
491,
11
] |
Mathlib/Analysis/Calculus/FDeriv/Mul.lean
|
HasFDerivAt.mul_const
|
[
{
"state_after": "case h.e'_10\n𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type ?u.934607\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type ?u.934702\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nG' : Type ?u.934797\ninst✝⁵ : NormedAddCommGroup G'\ninst✝⁴ : 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\n𝔸 : Type ?u.936932\n𝔸' : Type u_3\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedCommRing 𝔸'\ninst✝¹ : NormedAlgebra 𝕜 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸'\na b : E → 𝔸\na' b' : E →L[𝕜] 𝔸\nc d✝ : E → 𝔸'\nc' d' : E →L[𝕜] 𝔸'\nhc : HasFDerivAt c c' x\nd : 𝔸'\n⊢ d • c' = smulRight c' d",
"state_before": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type ?u.934607\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type ?u.934702\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nG' : Type ?u.934797\ninst✝⁵ : NormedAddCommGroup G'\ninst✝⁴ : 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\n𝔸 : Type ?u.936932\n𝔸' : Type u_3\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedCommRing 𝔸'\ninst✝¹ : NormedAlgebra 𝕜 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸'\na b : E → 𝔸\na' b' : E →L[𝕜] 𝔸\nc d✝ : E → 𝔸'\nc' d' : E →L[𝕜] 𝔸'\nhc : HasFDerivAt c c' x\nd : 𝔸'\n⊢ HasFDerivAt (fun y => c y * d) (d • c') x",
"tactic": "convert hc.mul_const' d"
},
{
"state_after": "case h.e'_10.h\n𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type ?u.934607\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type ?u.934702\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nG' : Type ?u.934797\ninst✝⁵ : NormedAddCommGroup G'\ninst✝⁴ : 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\n𝔸 : Type ?u.936932\n𝔸' : Type u_3\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedCommRing 𝔸'\ninst✝¹ : NormedAlgebra 𝕜 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸'\na b : E → 𝔸\na' b' : E →L[𝕜] 𝔸\nc d✝ : E → 𝔸'\nc' d' : E →L[𝕜] 𝔸'\nhc : HasFDerivAt c c' x\nd : 𝔸'\nz : E\n⊢ ↑(d • c') z = ↑(smulRight c' d) z",
"state_before": "case h.e'_10\n𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type ?u.934607\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type ?u.934702\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nG' : Type ?u.934797\ninst✝⁵ : NormedAddCommGroup G'\ninst✝⁴ : 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\n𝔸 : Type ?u.936932\n𝔸' : Type u_3\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedCommRing 𝔸'\ninst✝¹ : NormedAlgebra 𝕜 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸'\na b : E → 𝔸\na' b' : E →L[𝕜] 𝔸\nc d✝ : E → 𝔸'\nc' d' : E →L[𝕜] 𝔸'\nhc : HasFDerivAt c c' x\nd : 𝔸'\n⊢ d • c' = smulRight c' d",
"tactic": "ext z"
},
{
"state_after": "no goals",
"state_before": "case h.e'_10.h\n𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type ?u.934607\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type ?u.934702\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nG' : Type ?u.934797\ninst✝⁵ : NormedAddCommGroup G'\ninst✝⁴ : 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\n𝔸 : Type ?u.936932\n𝔸' : Type u_3\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedCommRing 𝔸'\ninst✝¹ : NormedAlgebra 𝕜 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸'\na b : E → 𝔸\na' b' : E →L[𝕜] 𝔸\nc d✝ : E → 𝔸'\nc' d' : E →L[𝕜] 𝔸'\nhc : HasFDerivAt c c' x\nd : 𝔸'\nz : E\n⊢ ↑(d • c') z = ↑(smulRight c' d) z",
"tactic": "apply mul_comm"
}
] |
[
420,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
416,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.pow_antitone_exp
|
[] |
[
726,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
725,
1
] |
Mathlib/Data/Holor.lean
|
Holor.mul_zero
|
[] |
[
226,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
8
] |
Std/Data/RBMap/Lemmas.lean
|
Std.RBNode.Ordered.lowerBound?_least_lb
|
[
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nlb : Option α\nh : Ordered cmp nil\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) nil\n⊢ x ∈ lowerBound? cut nil lb → y ∈ nil → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt",
"tactic": "intro."
},
{
"state_after": "case node\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\n⊢ (match cut v✝ with\n | Ordering.lt => lowerBound? cut l✝ lb\n | Ordering.gt => lowerBound? cut r✝ (some v✝)\n | Ordering.eq => some v✝) =\n some x →\n y = v✝ ∨ y ∈ l✝ ∨ y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt",
"state_before": "case node\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\n⊢ x ∈ lowerBound? cut (node c✝ l✝ v✝ r✝) lb →\n y ∈ node c✝ l✝ v✝ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt",
"tactic": "simp [lowerBound?]"
},
{
"state_after": "case node.h_1.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ r✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nx✝ : Ordering\nh₁ : lowerBound? cut l✝ lb = some x\nh : Ordered cmp (node c✝ l✝ y r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ y r✝)\nhv : cut y = Ordering.lt\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ cut y = Ordering.lt\n\ncase node.h_1.inr.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\nx✝ : Ordering\nhv : cut v✝ = Ordering.lt\nh₁ : lowerBound? cut l✝ lb = some x\nhy' : y ∈ l✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ cut y = Ordering.lt\n\ncase node.h_1.inr.inr\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\nx✝ : Ordering\nhv : cut v✝ = Ordering.lt\nh₁ : lowerBound? cut l✝ lb = some x\nhy' : y ∈ r✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ cut y = Ordering.lt\n\ncase node.h_2.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ r✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nx✝ : Ordering\nh : Ordered cmp (node c✝ l✝ y r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ y r✝)\nhv : cut y = Ordering.gt\nh₁ : lowerBound? cut r✝ (some y) = some x\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ cut y = Ordering.lt\n\ncase node.h_2.inr.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\nx✝ : Ordering\nhv : cut v✝ = Ordering.gt\nh₁ : lowerBound? cut r✝ (some v✝) = some x\nhy' : y ∈ l✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ cut y = Ordering.lt\n\ncase node.h_2.inr.inr\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\nx✝ : Ordering\nhv : cut v✝ = Ordering.gt\nh₁ : lowerBound? cut r✝ (some v✝) = some x\nhy' : y ∈ r✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ cut y = Ordering.lt\n\ncase node.h_3.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ r✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nx✝ : Ordering\nh : Ordered cmp (node c✝ l✝ y r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ y r✝)\nhv : cut y = Ordering.eq\nh₁ : some y = some x\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ cut y = Ordering.lt\n\ncase node.h_3.inr.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\nx✝ : Ordering\nhv : cut v✝ = Ordering.eq\nh₁ : some v✝ = some x\nhy' : y ∈ l✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ cut y = Ordering.lt\n\ncase node.h_3.inr.inr\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\nx✝ : Ordering\nhv : cut v✝ = Ordering.eq\nh₁ : some v✝ = some x\nhy' : y ∈ r✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ cut y = Ordering.lt",
"state_before": "case node\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\n⊢ (match cut v✝ with\n | Ordering.lt => lowerBound? cut l✝ lb\n | Ordering.gt => lowerBound? cut r✝ (some v✝)\n | Ordering.eq => some v✝) =\n some x →\n y = v✝ ∨ y ∈ l✝ ∨ y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt",
"tactic": "split <;> rename_i hv <;> rintro h₁ (rfl | hy' | hy') hx h₂"
},
{
"state_after": "no goals",
"state_before": "case node.h_1.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ r✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nx✝ : Ordering\nh₁ : lowerBound? cut l✝ lb = some x\nh : Ordered cmp (node c✝ l✝ y r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ y r✝)\nhv : cut y = Ordering.lt\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ cut y = Ordering.lt",
"tactic": "exact hv"
},
{
"state_after": "no goals",
"state_before": "case node.h_1.inr.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\nx✝ : Ordering\nhv : cut v✝ = Ordering.lt\nh₁ : lowerBound? cut l✝ lb = some x\nhy' : y ∈ l✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ cut y = Ordering.lt",
"tactic": "exact ihl h.2.2.1 (fun h => (hlb h).2.1) h₁ hy' hx h₂"
},
{
"state_after": "no goals",
"state_before": "case node.h_1.inr.inr\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\nx✝ : Ordering\nhv : cut v✝ = Ordering.lt\nh₁ : lowerBound? cut l✝ lb = some x\nhy' : y ∈ r✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ cut y = Ordering.lt",
"tactic": "exact IsCut.lt_trans (cut := cut) (cmp := cmp) (All_def.1 h.2.1 _ hy').1 hv"
},
{
"state_after": "case node.h_2.inl.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ r✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nx✝ : Ordering\nh : Ordered cmp (node c✝ l✝ y r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ y r✝)\nhv : cut y = Ordering.gt\nh₁✝ : lowerBound? cut r✝ (some y) = some x\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\nh₁ : x ∈ r✝\n⊢ cut y = Ordering.lt\n\ncase node.h_2.inl.inr.refl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ r✝ : RBNode α\nlb : Option α\nx✝ : Ordering\nhx : cut x = Ordering.gt\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → x ∈ l✝ → cut x = Ordering.gt → cmp x x = Ordering.lt → cut x = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → x ∈ r✝ → cut x = Ordering.gt → cmp x x = Ordering.lt → cut x = Ordering.lt\nh : Ordered cmp (node c✝ l✝ x r✝)\nhlb : ∀ {x_1 : α}, x_1 ∈ lb → All (fun x => cmpLT cmp x_1 x) (node c✝ l✝ x r✝)\nhv : cut x = Ordering.gt\nh₁ : lowerBound? cut r✝ (some x) = some x\nh₂ : cmp x x = Ordering.lt\n⊢ cut x = Ordering.lt",
"state_before": "case node.h_2.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ r✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nx✝ : Ordering\nh : Ordered cmp (node c✝ l✝ y r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ y r✝)\nhv : cut y = Ordering.gt\nh₁ : lowerBound? cut r✝ (some y) = some x\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ cut y = Ordering.lt",
"tactic": "rcases lowerBound?_mem_lb h₁ with h₁ | ⟨⟨⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case node.h_2.inl.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ r✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nx✝ : Ordering\nh : Ordered cmp (node c✝ l✝ y r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ y r✝)\nhv : cut y = Ordering.gt\nh₁✝ : lowerBound? cut r✝ (some y) = some x\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\nh₁ : x ∈ r✝\n⊢ cut y = Ordering.lt",
"tactic": "cases TransCmp.lt_asymm h₂ (All_def.1 h.2.1 _ h₁).1"
},
{
"state_after": "no goals",
"state_before": "case node.h_2.inl.inr.refl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ r✝ : RBNode α\nlb : Option α\nx✝ : Ordering\nhx : cut x = Ordering.gt\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → x ∈ l✝ → cut x = Ordering.gt → cmp x x = Ordering.lt → cut x = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → x ∈ r✝ → cut x = Ordering.gt → cmp x x = Ordering.lt → cut x = Ordering.lt\nh : Ordered cmp (node c✝ l✝ x r✝)\nhlb : ∀ {x_1 : α}, x_1 ∈ lb → All (fun x => cmpLT cmp x_1 x) (node c✝ l✝ x r✝)\nhv : cut x = Ordering.gt\nh₁ : lowerBound? cut r✝ (some x) = some x\nh₂ : cmp x x = Ordering.lt\n⊢ cut x = Ordering.lt",
"tactic": "cases TransCmp.lt_asymm h₂ h₂"
},
{
"state_after": "case node.h_2.inr.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\nx✝ : Ordering\nhv : cut v✝ = Ordering.gt\nh₁ : lowerBound? cut r✝ (some v✝) = some x\nhy' : y ∈ l✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ cmp y x = Ordering.lt",
"state_before": "case node.h_2.inr.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\nx✝ : Ordering\nhv : cut v✝ = Ordering.gt\nh₁ : lowerBound? cut r✝ (some v✝) = some x\nhy' : y ∈ l✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ cut y = Ordering.lt",
"tactic": "refine (TransCmp.lt_asymm h₂ ?_).elim"
},
{
"state_after": "case node.h_2.inr.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\nx✝ : Ordering\nhv : cut v✝ = Ordering.gt\nh₁ : lowerBound? cut r✝ (some v✝) = some x\nhy' : y ∈ l✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\nthis : cmp y v✝ = Ordering.lt\n⊢ cmp y x = Ordering.lt",
"state_before": "case node.h_2.inr.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\nx✝ : Ordering\nhv : cut v✝ = Ordering.gt\nh₁ : lowerBound? cut r✝ (some v✝) = some x\nhy' : y ∈ l✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ cmp y x = Ordering.lt",
"tactic": "have := (All_def.1 h.1 _ hy').1"
},
{
"state_after": "case node.h_2.inr.inl.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\nx✝ : Ordering\nhv : cut v✝ = Ordering.gt\nh₁✝ : lowerBound? cut r✝ (some v✝) = some x\nhy' : y ∈ l✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\nthis : cmp y v✝ = Ordering.lt\nh₁ : x ∈ r✝\n⊢ cmp y x = Ordering.lt\n\ncase node.h_2.inr.inl.inr.refl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ r✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nx✝ : Ordering\nhy' : y ∈ l✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\nh : Ordered cmp (node c✝ l✝ x r✝)\nhlb : ∀ {x_1 : α}, x_1 ∈ lb → All (fun x => cmpLT cmp x_1 x) (node c✝ l✝ x r✝)\nhv : cut x = Ordering.gt\nh₁ : lowerBound? cut r✝ (some x) = some x\nthis : cmp y x = Ordering.lt\n⊢ cmp y x = Ordering.lt",
"state_before": "case node.h_2.inr.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\nx✝ : Ordering\nhv : cut v✝ = Ordering.gt\nh₁ : lowerBound? cut r✝ (some v✝) = some x\nhy' : y ∈ l✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\nthis : cmp y v✝ = Ordering.lt\n⊢ cmp y x = Ordering.lt",
"tactic": "rcases lowerBound?_mem_lb h₁ with h₁ | ⟨⟨⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case node.h_2.inr.inl.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\nx✝ : Ordering\nhv : cut v✝ = Ordering.gt\nh₁✝ : lowerBound? cut r✝ (some v✝) = some x\nhy' : y ∈ l✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\nthis : cmp y v✝ = Ordering.lt\nh₁ : x ∈ r✝\n⊢ cmp y x = Ordering.lt",
"tactic": "exact TransCmp.lt_trans this (All_def.1 h.2.1 _ h₁).1"
},
{
"state_after": "no goals",
"state_before": "case node.h_2.inr.inl.inr.refl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ r✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nx✝ : Ordering\nhy' : y ∈ l✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\nh : Ordered cmp (node c✝ l✝ x r✝)\nhlb : ∀ {x_1 : α}, x_1 ∈ lb → All (fun x => cmpLT cmp x_1 x) (node c✝ l✝ x r✝)\nhv : cut x = Ordering.gt\nh₁ : lowerBound? cut r✝ (some x) = some x\nthis : cmp y x = Ordering.lt\n⊢ cmp y x = Ordering.lt",
"tactic": "exact this"
},
{
"state_after": "no goals",
"state_before": "case node.h_2.inr.inr\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\nx✝ : Ordering\nhv : cut v✝ = Ordering.gt\nh₁ : lowerBound? cut r✝ (some v✝) = some x\nhy' : y ∈ r✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ cut y = Ordering.lt",
"tactic": "exact ihr h.2.2.2 (by rintro _ ⟨⟨⟩⟩; exact h.2.1) h₁ hy' hx h₂"
},
{
"state_after": "case refl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\nx✝ : Ordering\nhv : cut v✝ = Ordering.gt\nh₁ : lowerBound? cut r✝ (some v✝) = some x\nhy' : y ∈ r✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ All (fun x => cmpLT cmp v✝ x) r✝",
"state_before": "α : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\nx✝ : Ordering\nhv : cut v✝ = Ordering.gt\nh₁ : lowerBound? cut r✝ (some v✝) = some x\nhy' : y ∈ r✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ ∀ {x : α}, x ∈ some v✝ → All (fun x_1 => cmpLT cmp x x_1) r✝",
"tactic": "rintro _ ⟨⟨⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case refl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\nx✝ : Ordering\nhv : cut v✝ = Ordering.gt\nh₁ : lowerBound? cut r✝ (some v✝) = some x\nhy' : y ∈ r✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ All (fun x => cmpLT cmp v✝ x) r✝",
"tactic": "exact h.2.1"
},
{
"state_after": "case node.h_3.inl.refl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ r✝ : RBNode α\nlb : Option α\nx✝ : Ordering\nhx : cut x = Ordering.gt\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → x ∈ l✝ → cut x = Ordering.gt → cmp x x = Ordering.lt → cut x = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → x ∈ r✝ → cut x = Ordering.gt → cmp x x = Ordering.lt → cut x = Ordering.lt\nh : Ordered cmp (node c✝ l✝ x r✝)\nhlb : ∀ {x_1 : α}, x_1 ∈ lb → All (fun x => cmpLT cmp x_1 x) (node c✝ l✝ x r✝)\nhv : cut x = Ordering.eq\nh₂ : cmp x x = Ordering.lt\n⊢ cut x = Ordering.lt",
"state_before": "case node.h_3.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ r✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nx✝ : Ordering\nh : Ordered cmp (node c✝ l✝ y r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ y r✝)\nhv : cut y = Ordering.eq\nh₁ : some y = some x\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ cut y = Ordering.lt",
"tactic": "cases h₁"
},
{
"state_after": "no goals",
"state_before": "case node.h_3.inl.refl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ r✝ : RBNode α\nlb : Option α\nx✝ : Ordering\nhx : cut x = Ordering.gt\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → x ∈ l✝ → cut x = Ordering.gt → cmp x x = Ordering.lt → cut x = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → x ∈ r✝ → cut x = Ordering.gt → cmp x x = Ordering.lt → cut x = Ordering.lt\nh : Ordered cmp (node c✝ l✝ x r✝)\nhlb : ∀ {x_1 : α}, x_1 ∈ lb → All (fun x => cmpLT cmp x_1 x) (node c✝ l✝ x r✝)\nhv : cut x = Ordering.eq\nh₂ : cmp x x = Ordering.lt\n⊢ cut x = Ordering.lt",
"tactic": "cases TransCmp.lt_asymm h₂ h₂"
},
{
"state_after": "case node.h_3.inr.inl.refl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ r✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nx✝ : Ordering\nhy' : y ∈ l✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\nh : Ordered cmp (node c✝ l✝ x r✝)\nhlb : ∀ {x_1 : α}, x_1 ∈ lb → All (fun x => cmpLT cmp x_1 x) (node c✝ l✝ x r✝)\nhv : cut x = Ordering.eq\n⊢ cut y = Ordering.lt",
"state_before": "case node.h_3.inr.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\nx✝ : Ordering\nhv : cut v✝ = Ordering.eq\nh₁ : some v✝ = some x\nhy' : y ∈ l✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ cut y = Ordering.lt",
"tactic": "cases h₁"
},
{
"state_after": "no goals",
"state_before": "case node.h_3.inr.inl.refl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ r✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nx✝ : Ordering\nhy' : y ∈ l✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\nh : Ordered cmp (node c✝ l✝ x r✝)\nhlb : ∀ {x_1 : α}, x_1 ∈ lb → All (fun x => cmpLT cmp x_1 x) (node c✝ l✝ x r✝)\nhv : cut x = Ordering.eq\n⊢ cut y = Ordering.lt",
"tactic": "cases hx.symm.trans hv"
},
{
"state_after": "case node.h_3.inr.inr.refl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ r✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nx✝ : Ordering\nhy' : y ∈ r✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\nh : Ordered cmp (node c✝ l✝ x r✝)\nhlb : ∀ {x_1 : α}, x_1 ∈ lb → All (fun x => cmpLT cmp x_1 x) (node c✝ l✝ x r✝)\nhv : cut x = Ordering.eq\n⊢ cut y = Ordering.lt",
"state_before": "case node.h_3.inr.inr\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nh : Ordered cmp (node c✝ l✝ v✝ r✝)\nhlb : ∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) (node c✝ l✝ v✝ r✝)\nx✝ : Ordering\nhv : cut v✝ = Ordering.eq\nh₁ : some v✝ = some x\nhy' : y ∈ r✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\n⊢ cut y = Ordering.lt",
"tactic": "cases h₁"
},
{
"state_after": "no goals",
"state_before": "case node.h_3.inr.inr.refl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nx y : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl✝ r✝ : RBNode α\nihl :\n ∀ {lb : Option α},\n Ordered cmp l✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) l✝) →\n x ∈ lowerBound? cut l✝ lb → y ∈ l✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nihr :\n ∀ {lb : Option α},\n Ordered cmp r✝ →\n (∀ {x : α}, x ∈ lb → All (fun x_1 => cmpLT cmp x x_1) r✝) →\n x ∈ lowerBound? cut r✝ lb → y ∈ r✝ → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt\nlb : Option α\nx✝ : Ordering\nhy' : y ∈ r✝\nhx : cut x = Ordering.gt\nh₂ : cmp x y = Ordering.lt\nh : Ordered cmp (node c✝ l✝ x r✝)\nhlb : ∀ {x_1 : α}, x_1 ∈ lb → All (fun x => cmpLT cmp x_1 x) (node c✝ l✝ x r✝)\nhv : cut x = Ordering.eq\n⊢ cut y = Ordering.lt",
"tactic": "cases hx.symm.trans hv"
}
] |
[
291,
39
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
271,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.preimage_empty
|
[] |
[
63,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Algebra/Hom/Units.lean
|
IsUnit.liftRight_inv_mul
|
[
{
"state_after": "F : Type ?u.44210\nG : Type ?u.44213\nα : Type ?u.44216\nM : Type u_1\nN : Type u_2\ninst✝¹ : Monoid M\ninst✝ : Monoid N\nf : M →* N\nh : ∀ (x : M), IsUnit (↑f x)\nx x✝ : M\n⊢ ↑(IsUnit.unit (_ : IsUnit (↑f x✝))) = ↑f x✝",
"state_before": "F : Type ?u.44210\nG : Type ?u.44213\nα : Type ?u.44216\nM : Type u_1\nN : Type u_2\ninst✝¹ : Monoid M\ninst✝ : Monoid N\nf : M →* N\nh : ∀ (x : M), IsUnit (↑f x)\nx : M\n⊢ ∀ (x : M), ↑(IsUnit.unit (_ : IsUnit (↑f x))) = ↑f x",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.44210\nG : Type ?u.44213\nα : Type ?u.44216\nM : Type u_1\nN : Type u_2\ninst✝¹ : Monoid M\ninst✝ : Monoid N\nf : M →* N\nh : ∀ (x : M), IsUnit (↑f x)\nx x✝ : M\n⊢ ↑(IsUnit.unit (_ : IsUnit (↑f x✝))) = ↑f x✝",
"tactic": "rfl"
}
] |
[
266,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.nonempty_coe_sort
|
[] |
[
497,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
496,
1
] |
Mathlib/Order/Closure.lean
|
ClosureOperator.le_closure_iff
|
[] |
[
166,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.IsNormal.strictMono
|
[] |
[
417,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
413,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.tendsto_mul_const_atBot_of_pos
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.227494\nι' : Type ?u.227497\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.227506\ninst✝ : LinearOrderedField α\nl : Filter β\nf : β → α\nr : α\nhr : 0 < r\n⊢ Tendsto (fun x => f x * r) l atBot ↔ Tendsto f l atBot",
"tactic": "simpa only [mul_comm] using tendsto_const_mul_atBot_of_pos hr"
}
] |
[
1086,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1084,
1
] |
Mathlib/Data/Fin/Tuple/Basic.lean
|
Fin.snoc_comp_castSucc
|
[
{
"state_after": "no goals",
"state_before": "m n✝ : ℕ\nα✝ : Fin (n✝ + 1) → Type u\nx : α✝ (last n✝)\nq : (i : Fin (n✝ + 1)) → α✝ i\np : (i : Fin n✝) → α✝ (↑castSucc i)\ni✝ : Fin n✝\ny : α✝ (↑castSucc i✝)\nz : α✝ (last n✝)\nn : ℕ\nα : Type u_1\na : α\nf : Fin n → α\ni : Fin n\n⊢ (snoc f a ∘ ↑castSucc) i = f i",
"tactic": "rw [Function.comp_apply, snoc_castSucc]"
}
] |
[
467,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
465,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.cos_add
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\n⊢ ↑(cos (x + y)) = ↑(cos x * cos y - sin x * sin y)",
"tactic": "simp [cos_add]"
}
] |
[
1202,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1201,
8
] |
Mathlib/Data/Set/Accumulate.lean
|
Set.subset_accumulate
|
[] |
[
39,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
1
] |
Mathlib/RingTheory/Coprime/Basic.lean
|
IsCoprime.of_add_mul_right_right
|
[
{
"state_after": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime x (y + x * z)\n⊢ IsCoprime x y",
"state_before": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime x (y + z * x)\n⊢ IsCoprime x y",
"tactic": "rw [mul_comm] at h"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime x (y + x * z)\n⊢ IsCoprime x y",
"tactic": "exact h.of_add_mul_left_right"
}
] |
[
201,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
199,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Equiv.lean
|
ContinuousLinearEquiv.comp_hasFDerivWithinAt_iff
|
[
{
"state_after": "𝕜 : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_1\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.81289\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\niso : E ≃L[𝕜] F\nf : G → E\ns : Set G\nx : G\nf' : G →L[𝕜] E\nH : HasFDerivWithinAt (↑iso ∘ f) (comp (↑iso) f') s x\n⊢ HasFDerivWithinAt f f' s x",
"state_before": "𝕜 : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_1\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.81289\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\niso : E ≃L[𝕜] F\nf : G → E\ns : Set G\nx : G\nf' : G →L[𝕜] E\n⊢ HasFDerivWithinAt (↑iso ∘ f) (comp (↑iso) f') s x ↔ HasFDerivWithinAt f f' s x",
"tactic": "refine' ⟨fun H => _, fun H => iso.hasFDerivAt.comp_hasFDerivWithinAt x H⟩"
},
{
"state_after": "𝕜 : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_1\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.81289\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\niso : E ≃L[𝕜] F\nf : G → E\ns : Set G\nx : G\nf' : G →L[𝕜] E\nH : HasFDerivWithinAt (↑iso ∘ f) (comp (↑iso) f') s x\nA : f = ↑(ContinuousLinearEquiv.symm iso) ∘ ↑iso ∘ f\n⊢ HasFDerivWithinAt f f' s x",
"state_before": "𝕜 : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_1\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.81289\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\niso : E ≃L[𝕜] F\nf : G → E\ns : Set G\nx : G\nf' : G →L[𝕜] E\nH : HasFDerivWithinAt (↑iso ∘ f) (comp (↑iso) f') s x\n⊢ HasFDerivWithinAt f f' s x",
"tactic": "have A : f = iso.symm ∘ iso ∘ f := by\n rw [← Function.comp.assoc, iso.symm_comp_self]\n rfl"
},
{
"state_after": "𝕜 : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_1\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.81289\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\niso : E ≃L[𝕜] F\nf : G → E\ns : Set G\nx : G\nf' : G →L[𝕜] E\nH : HasFDerivWithinAt (↑iso ∘ f) (comp (↑iso) f') s x\nA : f = ↑(ContinuousLinearEquiv.symm iso) ∘ ↑iso ∘ f\nB : f' = comp (↑(ContinuousLinearEquiv.symm iso)) (comp (↑iso) f')\n⊢ HasFDerivWithinAt f f' s x",
"state_before": "𝕜 : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_1\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.81289\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\niso : E ≃L[𝕜] F\nf : G → E\ns : Set G\nx : G\nf' : G →L[𝕜] E\nH : HasFDerivWithinAt (↑iso ∘ f) (comp (↑iso) f') s x\nA : f = ↑(ContinuousLinearEquiv.symm iso) ∘ ↑iso ∘ f\n⊢ HasFDerivWithinAt f f' s x",
"tactic": "have B : f' = (iso.symm : F →L[𝕜] E).comp ((iso : E →L[𝕜] F).comp f') := by\n rw [← ContinuousLinearMap.comp_assoc, iso.coe_symm_comp_coe, ContinuousLinearMap.id_comp]"
},
{
"state_after": "𝕜 : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_1\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.81289\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\niso : E ≃L[𝕜] F\nf : G → E\ns : Set G\nx : G\nf' : G →L[𝕜] E\nH : HasFDerivWithinAt (↑iso ∘ f) (comp (↑iso) f') s x\nA : f = ↑(ContinuousLinearEquiv.symm iso) ∘ ↑iso ∘ f\nB : f' = comp (↑(ContinuousLinearEquiv.symm iso)) (comp (↑iso) f')\n⊢ HasFDerivWithinAt (↑(ContinuousLinearEquiv.symm iso) ∘ ↑iso ∘ f)\n (comp (↑(ContinuousLinearEquiv.symm iso)) (comp (↑iso) f')) s x",
"state_before": "𝕜 : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_1\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.81289\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\niso : E ≃L[𝕜] F\nf : G → E\ns : Set G\nx : G\nf' : G →L[𝕜] E\nH : HasFDerivWithinAt (↑iso ∘ f) (comp (↑iso) f') s x\nA : f = ↑(ContinuousLinearEquiv.symm iso) ∘ ↑iso ∘ f\nB : f' = comp (↑(ContinuousLinearEquiv.symm iso)) (comp (↑iso) f')\n⊢ HasFDerivWithinAt f f' s x",
"tactic": "rw [A, B]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_1\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.81289\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\niso : E ≃L[𝕜] F\nf : G → E\ns : Set G\nx : G\nf' : G →L[𝕜] E\nH : HasFDerivWithinAt (↑iso ∘ f) (comp (↑iso) f') s x\nA : f = ↑(ContinuousLinearEquiv.symm iso) ∘ ↑iso ∘ f\nB : f' = comp (↑(ContinuousLinearEquiv.symm iso)) (comp (↑iso) f')\n⊢ HasFDerivWithinAt (↑(ContinuousLinearEquiv.symm iso) ∘ ↑iso ∘ f)\n (comp (↑(ContinuousLinearEquiv.symm iso)) (comp (↑iso) f')) s x",
"tactic": "exact iso.symm.hasFDerivAt.comp_hasFDerivWithinAt x H"
},
{
"state_after": "𝕜 : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_1\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.81289\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\niso : E ≃L[𝕜] F\nf : G → E\ns : Set G\nx : G\nf' : G →L[𝕜] E\nH : HasFDerivWithinAt (↑iso ∘ f) (comp (↑iso) f') s x\n⊢ f = _root_.id ∘ f",
"state_before": "𝕜 : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_1\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.81289\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\niso : E ≃L[𝕜] F\nf : G → E\ns : Set G\nx : G\nf' : G →L[𝕜] E\nH : HasFDerivWithinAt (↑iso ∘ f) (comp (↑iso) f') s x\n⊢ f = ↑(ContinuousLinearEquiv.symm iso) ∘ ↑iso ∘ f",
"tactic": "rw [← Function.comp.assoc, iso.symm_comp_self]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_1\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.81289\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\niso : E ≃L[𝕜] F\nf : G → E\ns : Set G\nx : G\nf' : G →L[𝕜] E\nH : HasFDerivWithinAt (↑iso ∘ f) (comp (↑iso) f') s x\n⊢ f = _root_.id ∘ f",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_1\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.81289\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\niso : E ≃L[𝕜] F\nf : G → E\ns : Set G\nx : G\nf' : G →L[𝕜] E\nH : HasFDerivWithinAt (↑iso ∘ f) (comp (↑iso) f') s x\nA : f = ↑(ContinuousLinearEquiv.symm iso) ∘ ↑iso ∘ f\n⊢ f' = comp (↑(ContinuousLinearEquiv.symm iso)) (comp (↑iso) f')",
"tactic": "rw [← ContinuousLinearMap.comp_assoc, iso.coe_symm_comp_coe, ContinuousLinearMap.id_comp]"
}
] |
[
132,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.TM1to0.tr_eval
|
[
{
"state_after": "Γ : Type u_1\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_3\ninst✝ : Inhabited σ\nM : Λ → Stmt₁\nl : List Γ\n⊢ Part.map ((fun c => Tape.right₀ c.Tape) ∘ fun a => trCfg M a)\n (eval (TM1.step M) { l := some default, var := default, Tape := Tape.mk₁ l }) =\n Part.map (fun c => Tape.right₀ c.Tape) (eval (TM1.step M) (TM1.init l))",
"state_before": "Γ : Type u_1\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_3\ninst✝ : Inhabited σ\nM : Λ → Stmt₁\nl : List Γ\n⊢ Part.map (fun c => Tape.right₀ c.Tape)\n ((fun a => trCfg M a) <$> eval (TM1.step M) { l := some default, var := default, Tape := Tape.mk₁ l }) =\n TM1.eval M l",
"tactic": "rw [Part.map_eq_map, Part.map_map, TM1.eval]"
},
{
"state_after": "no goals",
"state_before": "Γ : Type u_1\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_3\ninst✝ : Inhabited σ\nM : Λ → Stmt₁\nl : List Γ\n⊢ Part.map ((fun c => Tape.right₀ c.Tape) ∘ fun a => trCfg M a)\n (eval (TM1.step M) { l := some default, var := default, Tape := Tape.mk₁ l }) =\n Part.map (fun c => Tape.right₀ c.Tape) (eval (TM1.step M) (TM1.init l))",
"tactic": "congr with ⟨⟩"
}
] |
[
1523,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1519,
1
] |
Mathlib/Topology/Instances/Matrix.lean
|
continuousAt_matrix_inv
|
[] |
[
237,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
235,
1
] |
Mathlib/Analysis/Asymptotics/Theta.lean
|
Asymptotics.IsBigO.antisymm
|
[] |
[
58,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
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