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/Geometry/Manifold/ChartedSpace.lean
|
LocalHomeomorph.singletonChartedSpace_chartAt_source
|
[] |
[
1000,
4
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
998,
1
] |
Mathlib/Data/Finset/Card.lean
|
Finset.card_eq_one
|
[
{
"state_after": "case mk\nα : Type u_1\nβ : Type ?u.46876\nt : Finset α\nf : α → β\nn : ℕ\nval✝ : Multiset α\nnodup✝ : Nodup val✝\n⊢ card { val := val✝, nodup := nodup✝ } = 1 ↔ ∃ a, { val := val✝, nodup := nodup✝ } = {a}",
"state_before": "α : Type u_1\nβ : Type ?u.46876\ns t : Finset α\nf : α → β\nn : ℕ\n⊢ card s = 1 ↔ ∃ a, s = {a}",
"tactic": "cases s"
},
{
"state_after": "no goals",
"state_before": "case mk\nα : Type u_1\nβ : Type ?u.46876\nt : Finset α\nf : α → β\nn : ℕ\nval✝ : Multiset α\nnodup✝ : Nodup val✝\n⊢ card { val := val✝, nodup := nodup✝ } = 1 ↔ ∃ a, { val := val✝, nodup := nodup✝ } = {a}",
"tactic": "simp only [Multiset.card_eq_one, Finset.card, ← val_inj, singleton_val]"
}
] |
[
512,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
510,
1
] |
Mathlib/Analysis/Calculus/TangentCone.lean
|
UniqueDiffWithinAt.pi
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.151877\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type ?u.151972\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.152062\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace ℝ G\nx✝ y : E✝\ns✝ t : Set E✝\nι : Type u_1\ninst✝² : Finite ι\nE : ι → Type u_2\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\ns : (i : ι) → Set (E i)\nx : (i : ι) → E i\nI : Set ι\nh : ∀ (i : ι), i ∈ I → UniqueDiffWithinAt 𝕜 (s i) (x i)\n⊢ UniqueDiffWithinAt 𝕜 (Set.pi I s) x",
"tactic": "classical\nrw [← Set.univ_pi_piecewise_univ]\nrefine' UniqueDiffWithinAt.univ_pi ι E _ _ fun i => _\nby_cases hi : i ∈ I <;> simp [*, uniqueDiffWithinAt_univ]"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.151877\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type ?u.151972\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.152062\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace ℝ G\nx✝ y : E✝\ns✝ t : Set E✝\nι : Type u_1\ninst✝² : Finite ι\nE : ι → Type u_2\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\ns : (i : ι) → Set (E i)\nx : (i : ι) → E i\nI : Set ι\nh : ∀ (i : ι), i ∈ I → UniqueDiffWithinAt 𝕜 (s i) (x i)\n⊢ UniqueDiffWithinAt 𝕜 (Set.pi univ (piecewise I s fun x => univ)) x",
"state_before": "𝕜 : Type u_3\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.151877\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type ?u.151972\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.152062\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace ℝ G\nx✝ y : E✝\ns✝ t : Set E✝\nι : Type u_1\ninst✝² : Finite ι\nE : ι → Type u_2\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\ns : (i : ι) → Set (E i)\nx : (i : ι) → E i\nI : Set ι\nh : ∀ (i : ι), i ∈ I → UniqueDiffWithinAt 𝕜 (s i) (x i)\n⊢ UniqueDiffWithinAt 𝕜 (Set.pi I s) x",
"tactic": "rw [← Set.univ_pi_piecewise_univ]"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.151877\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type ?u.151972\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.152062\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace ℝ G\nx✝ y : E✝\ns✝ t : Set E✝\nι : Type u_1\ninst✝² : Finite ι\nE : ι → Type u_2\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\ns : (i : ι) → Set (E i)\nx : (i : ι) → E i\nI : Set ι\nh : ∀ (i : ι), i ∈ I → UniqueDiffWithinAt 𝕜 (s i) (x i)\ni : ι\n⊢ UniqueDiffWithinAt 𝕜 (piecewise I s (fun x => univ) i) (x i)",
"state_before": "𝕜 : Type u_3\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.151877\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type ?u.151972\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.152062\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace ℝ G\nx✝ y : E✝\ns✝ t : Set E✝\nι : Type u_1\ninst✝² : Finite ι\nE : ι → Type u_2\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\ns : (i : ι) → Set (E i)\nx : (i : ι) → E i\nI : Set ι\nh : ∀ (i : ι), i ∈ I → UniqueDiffWithinAt 𝕜 (s i) (x i)\n⊢ UniqueDiffWithinAt 𝕜 (Set.pi univ (piecewise I s fun x => univ)) x",
"tactic": "refine' UniqueDiffWithinAt.univ_pi ι E _ _ fun i => _"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁹ : NontriviallyNormedField 𝕜\nE✝ : Type ?u.151877\ninst✝⁸ : NormedAddCommGroup E✝\ninst✝⁷ : NormedSpace 𝕜 E✝\nF : Type ?u.151972\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.152062\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace ℝ G\nx✝ y : E✝\ns✝ t : Set E✝\nι : Type u_1\ninst✝² : Finite ι\nE : ι → Type u_2\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\ns : (i : ι) → Set (E i)\nx : (i : ι) → E i\nI : Set ι\nh : ∀ (i : ι), i ∈ I → UniqueDiffWithinAt 𝕜 (s i) (x i)\ni : ι\n⊢ UniqueDiffWithinAt 𝕜 (piecewise I s (fun x => univ) i) (x i)",
"tactic": "by_cases hi : i ∈ I <;> simp [*, uniqueDiffWithinAt_univ]"
}
] |
[
366,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
359,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.sum_add_index
|
[
{
"state_after": "case ofFinsupp\nR : Type u\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q✝ : R[X]\nS : Type u_1\ninst✝ : AddCommMonoid S\nq : R[X]\nf : ℕ → R → S\nhf : ∀ (i : ℕ), f i 0 = 0\nh_add : ∀ (a : ℕ) (b₁ b₂ : R), f a (b₁ + b₂) = f a b₁ + f a b₂\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ sum ({ toFinsupp := toFinsupp✝ } + q) f = sum { toFinsupp := toFinsupp✝ } f + sum q f",
"state_before": "R : Type u\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q✝ : R[X]\nS : Type u_1\ninst✝ : AddCommMonoid S\np q : R[X]\nf : ℕ → R → S\nhf : ∀ (i : ℕ), f i 0 = 0\nh_add : ∀ (a : ℕ) (b₁ b₂ : R), f a (b₁ + b₂) = f a b₁ + f a b₂\n⊢ sum (p + q) f = sum p f + sum q f",
"tactic": "rcases p with ⟨⟩"
},
{
"state_after": "case ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q : R[X]\nS : Type u_1\ninst✝ : AddCommMonoid S\nf : ℕ → R → S\nhf : ∀ (i : ℕ), f i 0 = 0\nh_add : ∀ (a : ℕ) (b₁ b₂ : R), f a (b₁ + b₂) = f a b₁ + f a b₂\ntoFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ sum ({ toFinsupp := toFinsupp✝¹ } + { toFinsupp := toFinsupp✝ }) f =\n sum { toFinsupp := toFinsupp✝¹ } f + sum { toFinsupp := toFinsupp✝ } f",
"state_before": "case ofFinsupp\nR : Type u\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q✝ : R[X]\nS : Type u_1\ninst✝ : AddCommMonoid S\nq : R[X]\nf : ℕ → R → S\nhf : ∀ (i : ℕ), f i 0 = 0\nh_add : ∀ (a : ℕ) (b₁ b₂ : R), f a (b₁ + b₂) = f a b₁ + f a b₂\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ sum ({ toFinsupp := toFinsupp✝ } + q) f = sum { toFinsupp := toFinsupp✝ } f + sum q f",
"tactic": "rcases q with ⟨⟩"
},
{
"state_after": "case ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q : R[X]\nS : Type u_1\ninst✝ : AddCommMonoid S\nf : ℕ → R → S\nhf : ∀ (i : ℕ), f i 0 = 0\nh_add : ∀ (a : ℕ) (b₁ b₂ : R), f a (b₁ + b₂) = f a b₁ + f a b₂\ntoFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ sum { toFinsupp := toFinsupp✝¹ + toFinsupp✝ } f =\n sum { toFinsupp := toFinsupp✝¹ } f + sum { toFinsupp := toFinsupp✝ } f",
"state_before": "case ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q : R[X]\nS : Type u_1\ninst✝ : AddCommMonoid S\nf : ℕ → R → S\nhf : ∀ (i : ℕ), f i 0 = 0\nh_add : ∀ (a : ℕ) (b₁ b₂ : R), f a (b₁ + b₂) = f a b₁ + f a b₂\ntoFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ sum ({ toFinsupp := toFinsupp✝¹ } + { toFinsupp := toFinsupp✝ }) f =\n sum { toFinsupp := toFinsupp✝¹ } f + sum { toFinsupp := toFinsupp✝ } f",
"tactic": "simp only [← ofFinsupp_add, Sum, support, coeff, Pi.add_apply, coe_add]"
},
{
"state_after": "no goals",
"state_before": "case ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q : R[X]\nS : Type u_1\ninst✝ : AddCommMonoid S\nf : ℕ → R → S\nhf : ∀ (i : ℕ), f i 0 = 0\nh_add : ∀ (a : ℕ) (b₁ b₂ : R), f a (b₁ + b₂) = f a b₁ + f a b₂\ntoFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ sum { toFinsupp := toFinsupp✝¹ + toFinsupp✝ } f =\n sum { toFinsupp := toFinsupp✝¹ } f + sum { toFinsupp := toFinsupp✝ } f",
"tactic": "exact Finsupp.sum_add_index' hf h_add"
}
] |
[
990,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
985,
1
] |
Mathlib/Algebra/Group/TypeTags.lean
|
ofMul_inv
|
[] |
[
301,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
300,
1
] |
Mathlib/CategoryTheory/Whiskering.lean
|
CategoryTheory.whiskerRight_comp
|
[] |
[
151,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Real.lean
|
dist_le_tsum_of_dist_le_of_tendsto₀
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f (Nat.succ n)) ≤ d n\nhd : Summable d\nha : Tendsto f atTop (𝓝 a)\n⊢ dist (f 0) a ≤ tsum d",
"tactic": "simpa only [zero_add] using dist_le_tsum_of_dist_le_of_tendsto d hf hd ha 0"
}
] |
[
87,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.eventually_of_mem
|
[] |
[
1090,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1088,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.min_eq_left
|
[
{
"state_after": "no goals",
"state_before": "a b : Nat\nh : a ≤ b\n⊢ min a b = a",
"tactic": "simp [Nat.min_def, h]"
}
] |
[
187,
96
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
187,
11
] |
Mathlib/Algebra/Associated.lean
|
Prime.not_dvd_one
|
[] |
[
44,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.toFinsupp_eq_one
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\na✝ b : R\nm n : ℕ\ninst✝ : Semiring R\np q a : R[X]\n⊢ a.toFinsupp = 1 ↔ a = 1",
"tactic": "rw [← toFinsupp_one, toFinsupp_inj]"
}
] |
[
271,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
270,
1
] |
Mathlib/Data/Multiset/Powerset.lean
|
Multiset.revzip_powersetAux
|
[
{
"state_after": "α : Type u_1\nl : List α\nx : Multiset α × Multiset α\nh : ∃ a, a ∈ revzip (sublists l) ∧ Prod.map ofList ofList a = x\n⊢ x.fst + x.snd = ↑l",
"state_before": "α : Type u_1\nl : List α\nx : Multiset α × Multiset α\nh : x ∈ revzip (powersetAux l)\n⊢ x.fst + x.snd = ↑l",
"tactic": "rw [revzip, powersetAux_eq_map_coe, ← map_reverse, zip_map, ← revzip, List.mem_map] at h"
},
{
"state_after": "α : Type u_1\nl : List α\nx : Multiset α × Multiset α\nh : ∃ a b, (a, b) ∈ revzip (sublists l) ∧ (↑a, ↑b) = x\n⊢ x.fst + x.snd = ↑l",
"state_before": "α : Type u_1\nl : List α\nx : Multiset α × Multiset α\nh : ∃ a, a ∈ revzip (sublists l) ∧ Prod.map ofList ofList a = x\n⊢ x.fst + x.snd = ↑l",
"tactic": "simp only [Prod_map, Prod.exists] at h"
},
{
"state_after": "case intro.intro.intro.refl\nα : Type u_1\nl l₁ l₂ : List α\nh : (l₁, l₂) ∈ revzip (sublists l)\n⊢ (↑l₁, ↑l₂).fst + (↑l₁, ↑l₂).snd = ↑l",
"state_before": "α : Type u_1\nl : List α\nx : Multiset α × Multiset α\nh : ∃ a b, (a, b) ∈ revzip (sublists l) ∧ (↑a, ↑b) = x\n⊢ x.fst + x.snd = ↑l",
"tactic": "rcases h with ⟨l₁, l₂, h, rfl, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.refl\nα : Type u_1\nl l₁ l₂ : List α\nh : (l₁, l₂) ∈ revzip (sublists l)\n⊢ (↑l₁, ↑l₂).fst + (↑l₁, ↑l₂).snd = ↑l",
"tactic": "exact Quot.sound (revzip_sublists _ _ _ h)"
}
] |
[
131,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Dynamics/OmegaLimit.lean
|
mem_omegaLimit_iff_frequently
|
[
{
"state_after": "τ : Type u_2\nα : Type u_3\nβ : Type u_1\nι : Type ?u.28145\ninst✝ : TopologicalSpace β\nf : Filter τ\nϕ : τ → α → β\ns s₁ s₂ : Set α\ny : β\n⊢ (∀ (i : Set τ), i ∈ f → ∀ (t : Set β), t ∈ 𝓝 y → Set.Nonempty (t ∩ image2 ϕ i s)) ↔\n ∀ (n : Set β), n ∈ 𝓝 y → ∀ {U : Set τ}, U ∈ f → ∃ x, x ∈ U ∧ Set.Nonempty (s ∩ ϕ x ⁻¹' n)",
"state_before": "τ : Type u_2\nα : Type u_3\nβ : Type u_1\nι : Type ?u.28145\ninst✝ : TopologicalSpace β\nf : Filter τ\nϕ : τ → α → β\ns s₁ s₂ : Set α\ny : β\n⊢ y ∈ ω f ϕ s ↔ ∀ (n : Set β), n ∈ 𝓝 y → ∃ᶠ (t : τ) in f, Set.Nonempty (s ∩ ϕ t ⁻¹' n)",
"tactic": "simp_rw [frequently_iff, omegaLimit_def, mem_iInter, mem_closure_iff_nhds]"
},
{
"state_after": "case mp\nτ : Type u_2\nα : Type u_3\nβ : Type u_1\nι : Type ?u.28145\ninst✝ : TopologicalSpace β\nf : Filter τ\nϕ : τ → α → β\ns s₁ s₂ : Set α\ny : β\n⊢ (∀ (i : Set τ), i ∈ f → ∀ (t : Set β), t ∈ 𝓝 y → Set.Nonempty (t ∩ image2 ϕ i s)) →\n ∀ (n : Set β), n ∈ 𝓝 y → ∀ {U : Set τ}, U ∈ f → ∃ x, x ∈ U ∧ Set.Nonempty (s ∩ ϕ x ⁻¹' n)\n\ncase mpr\nτ : Type u_2\nα : Type u_3\nβ : Type u_1\nι : Type ?u.28145\ninst✝ : TopologicalSpace β\nf : Filter τ\nϕ : τ → α → β\ns s₁ s₂ : Set α\ny : β\n⊢ (∀ (n : Set β), n ∈ 𝓝 y → ∀ {U : Set τ}, U ∈ f → ∃ x, x ∈ U ∧ Set.Nonempty (s ∩ ϕ x ⁻¹' n)) →\n ∀ (i : Set τ), i ∈ f → ∀ (t : Set β), t ∈ 𝓝 y → Set.Nonempty (t ∩ image2 ϕ i s)",
"state_before": "τ : Type u_2\nα : Type u_3\nβ : Type u_1\nι : Type ?u.28145\ninst✝ : TopologicalSpace β\nf : Filter τ\nϕ : τ → α → β\ns s₁ s₂ : Set α\ny : β\n⊢ (∀ (i : Set τ), i ∈ f → ∀ (t : Set β), t ∈ 𝓝 y → Set.Nonempty (t ∩ image2 ϕ i s)) ↔\n ∀ (n : Set β), n ∈ 𝓝 y → ∀ {U : Set τ}, U ∈ f → ∃ x, x ∈ U ∧ Set.Nonempty (s ∩ ϕ x ⁻¹' n)",
"tactic": "constructor"
},
{
"state_after": "case mp\nτ : Type u_2\nα : Type u_3\nβ : Type u_1\nι : Type ?u.28145\ninst✝ : TopologicalSpace β\nf : Filter τ\nϕ : τ → α → β\ns s₁ s₂ : Set α\ny : β\nh : ∀ (i : Set τ), i ∈ f → ∀ (t : Set β), t ∈ 𝓝 y → Set.Nonempty (t ∩ image2 ϕ i s)\nn✝ : Set β\nhn : n✝ ∈ 𝓝 y\nU✝ : Set τ\nhu : U✝ ∈ f\n⊢ ∃ x, x ∈ U✝ ∧ Set.Nonempty (s ∩ ϕ x ⁻¹' n✝)",
"state_before": "case mp\nτ : Type u_2\nα : Type u_3\nβ : Type u_1\nι : Type ?u.28145\ninst✝ : TopologicalSpace β\nf : Filter τ\nϕ : τ → α → β\ns s₁ s₂ : Set α\ny : β\n⊢ (∀ (i : Set τ), i ∈ f → ∀ (t : Set β), t ∈ 𝓝 y → Set.Nonempty (t ∩ image2 ϕ i s)) →\n ∀ (n : Set β), n ∈ 𝓝 y → ∀ {U : Set τ}, U ∈ f → ∃ x, x ∈ U ∧ Set.Nonempty (s ∩ ϕ x ⁻¹' n)",
"tactic": "intro h _ hn _ hu"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.intro\nτ : Type u_2\nα : Type u_3\nβ : Type u_1\nι : Type ?u.28145\ninst✝ : TopologicalSpace β\nf : Filter τ\nϕ : τ → α → β\ns s₁ s₂ : Set α\ny : β\nh : ∀ (i : Set τ), i ∈ f → ∀ (t : Set β), t ∈ 𝓝 y → Set.Nonempty (t ∩ image2 ϕ i s)\nn✝ : Set β\nhn : n✝ ∈ 𝓝 y\nU✝ : Set τ\nhu : U✝ ∈ f\nw✝² : β\nleft✝ : w✝² ∈ n✝\nw✝¹ : τ\nw✝ : α\nht : w✝¹ ∈ U✝\nhx : w✝ ∈ s\nhϕtx : ϕ w✝¹ w✝ = w✝²\n⊢ ∃ x, x ∈ U✝ ∧ Set.Nonempty (s ∩ ϕ x ⁻¹' n✝)",
"state_before": "case mp\nτ : Type u_2\nα : Type u_3\nβ : Type u_1\nι : Type ?u.28145\ninst✝ : TopologicalSpace β\nf : Filter τ\nϕ : τ → α → β\ns s₁ s₂ : Set α\ny : β\nh : ∀ (i : Set τ), i ∈ f → ∀ (t : Set β), t ∈ 𝓝 y → Set.Nonempty (t ∩ image2 ϕ i s)\nn✝ : Set β\nhn : n✝ ∈ 𝓝 y\nU✝ : Set τ\nhu : U✝ ∈ f\n⊢ ∃ x, x ∈ U✝ ∧ Set.Nonempty (s ∩ ϕ x ⁻¹' n✝)",
"tactic": "rcases h _ hu _ hn with ⟨_, _, _, _, ht, hx, hϕtx⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro.intro.intro\nτ : Type u_2\nα : Type u_3\nβ : Type u_1\nι : Type ?u.28145\ninst✝ : TopologicalSpace β\nf : Filter τ\nϕ : τ → α → β\ns s₁ s₂ : Set α\ny : β\nh : ∀ (i : Set τ), i ∈ f → ∀ (t : Set β), t ∈ 𝓝 y → Set.Nonempty (t ∩ image2 ϕ i s)\nn✝ : Set β\nhn : n✝ ∈ 𝓝 y\nU✝ : Set τ\nhu : U✝ ∈ f\nw✝² : β\nleft✝ : w✝² ∈ n✝\nw✝¹ : τ\nw✝ : α\nht : w✝¹ ∈ U✝\nhx : w✝ ∈ s\nhϕtx : ϕ w✝¹ w✝ = w✝²\n⊢ ∃ x, x ∈ U✝ ∧ Set.Nonempty (s ∩ ϕ x ⁻¹' n✝)",
"tactic": "exact ⟨_, ht, _, hx, by rwa [mem_preimage, hϕtx]⟩"
},
{
"state_after": "no goals",
"state_before": "τ : Type u_2\nα : Type u_3\nβ : Type u_1\nι : Type ?u.28145\ninst✝ : TopologicalSpace β\nf : Filter τ\nϕ : τ → α → β\ns s₁ s₂ : Set α\ny : β\nh : ∀ (i : Set τ), i ∈ f → ∀ (t : Set β), t ∈ 𝓝 y → Set.Nonempty (t ∩ image2 ϕ i s)\nn✝ : Set β\nhn : n✝ ∈ 𝓝 y\nU✝ : Set τ\nhu : U✝ ∈ f\nw✝² : β\nleft✝ : w✝² ∈ n✝\nw✝¹ : τ\nw✝ : α\nht : w✝¹ ∈ U✝\nhx : w✝ ∈ s\nhϕtx : ϕ w✝¹ w✝ = w✝²\n⊢ w✝ ∈ ϕ w✝¹ ⁻¹' n✝",
"tactic": "rwa [mem_preimage, hϕtx]"
},
{
"state_after": "case mpr\nτ : Type u_2\nα : Type u_3\nβ : Type u_1\nι : Type ?u.28145\ninst✝ : TopologicalSpace β\nf : Filter τ\nϕ : τ → α → β\ns s₁ s₂ : Set α\ny : β\nh : ∀ (n : Set β), n ∈ 𝓝 y → ∀ {U : Set τ}, U ∈ f → ∃ x, x ∈ U ∧ Set.Nonempty (s ∩ ϕ x ⁻¹' n)\ni✝ : Set τ\nhu : i✝ ∈ f\nt✝ : Set β\nhn : t✝ ∈ 𝓝 y\n⊢ Set.Nonempty (t✝ ∩ image2 ϕ i✝ s)",
"state_before": "case mpr\nτ : Type u_2\nα : Type u_3\nβ : Type u_1\nι : Type ?u.28145\ninst✝ : TopologicalSpace β\nf : Filter τ\nϕ : τ → α → β\ns s₁ s₂ : Set α\ny : β\n⊢ (∀ (n : Set β), n ∈ 𝓝 y → ∀ {U : Set τ}, U ∈ f → ∃ x, x ∈ U ∧ Set.Nonempty (s ∩ ϕ x ⁻¹' n)) →\n ∀ (i : Set τ), i ∈ f → ∀ (t : Set β), t ∈ 𝓝 y → Set.Nonempty (t ∩ image2 ϕ i s)",
"tactic": "intro h _ hu _ hn"
},
{
"state_after": "case mpr.intro.intro.intro.intro\nτ : Type u_2\nα : Type u_3\nβ : Type u_1\nι : Type ?u.28145\ninst✝ : TopologicalSpace β\nf : Filter τ\nϕ : τ → α → β\ns s₁ s₂ : Set α\ny : β\nh : ∀ (n : Set β), n ∈ 𝓝 y → ∀ {U : Set τ}, U ∈ f → ∃ x, x ∈ U ∧ Set.Nonempty (s ∩ ϕ x ⁻¹' n)\ni✝ : Set τ\nhu : i✝ ∈ f\nt✝ : Set β\nhn : t✝ ∈ 𝓝 y\nw✝¹ : τ\nht : w✝¹ ∈ i✝\nw✝ : α\nhx : w✝ ∈ s\nhϕtx : w✝ ∈ ϕ w✝¹ ⁻¹' t✝\n⊢ Set.Nonempty (t✝ ∩ image2 ϕ i✝ s)",
"state_before": "case mpr\nτ : Type u_2\nα : Type u_3\nβ : Type u_1\nι : Type ?u.28145\ninst✝ : TopologicalSpace β\nf : Filter τ\nϕ : τ → α → β\ns s₁ s₂ : Set α\ny : β\nh : ∀ (n : Set β), n ∈ 𝓝 y → ∀ {U : Set τ}, U ∈ f → ∃ x, x ∈ U ∧ Set.Nonempty (s ∩ ϕ x ⁻¹' n)\ni✝ : Set τ\nhu : i✝ ∈ f\nt✝ : Set β\nhn : t✝ ∈ 𝓝 y\n⊢ Set.Nonempty (t✝ ∩ image2 ϕ i✝ s)",
"tactic": "rcases h _ hn hu with ⟨_, ht, _, hx, hϕtx⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro.intro\nτ : Type u_2\nα : Type u_3\nβ : Type u_1\nι : Type ?u.28145\ninst✝ : TopologicalSpace β\nf : Filter τ\nϕ : τ → α → β\ns s₁ s₂ : Set α\ny : β\nh : ∀ (n : Set β), n ∈ 𝓝 y → ∀ {U : Set τ}, U ∈ f → ∃ x, x ∈ U ∧ Set.Nonempty (s ∩ ϕ x ⁻¹' n)\ni✝ : Set τ\nhu : i✝ ∈ f\nt✝ : Set β\nhn : t✝ ∈ 𝓝 y\nw✝¹ : τ\nht : w✝¹ ∈ i✝\nw✝ : α\nhx : w✝ ∈ s\nhϕtx : w✝ ∈ ϕ w✝¹ ⁻¹' t✝\n⊢ Set.Nonempty (t✝ ∩ image2 ϕ i✝ s)",
"tactic": "exact ⟨_, hϕtx, _, _, ht, hx, rfl⟩"
}
] |
[
144,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.range_castLE
|
[] |
[
1033,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1032,
1
] |
Mathlib/Algebra/Order/Monoid/WithTop.lean
|
WithBot.add_le_add_iff_left
|
[] |
[
696,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
694,
11
] |
Mathlib/RingTheory/ClassGroup.lean
|
coe_toPrincipalIdeal
|
[
{
"state_after": "R : Type u_2\nK : Type u_1\nL : Type ?u.37953\ninst✝⁹ : CommRing R\ninst✝⁸ : Field K\ninst✝⁷ : Field L\ninst✝⁶ : DecidableEq L\ninst✝⁵ : Algebra R K\ninst✝⁴ : IsFractionRing R K\ninst✝³ : Algebra K L\ninst✝² : FiniteDimensional K L\ninst✝¹ : Algebra R L\ninst✝ : IsScalarTower R K L\nx : Kˣ\n⊢ ↑(↑{\n toOneHom :=\n {\n toFun := fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) },\n map_one' :=\n (_ :\n (fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : Kˣ),\n OneHom.toFun\n {\n toFun := fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) },\n map_one' :=\n (_ :\n (fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) },\n map_one' :=\n (_ :\n (fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) },\n map_one' :=\n (_ :\n (fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) })\n 1 =\n 1) }\n y) }\n x) =\n spanSingleton R⁰ ↑x",
"state_before": "R : Type u_2\nK : Type u_1\nL : Type ?u.37953\ninst✝⁹ : CommRing R\ninst✝⁸ : Field K\ninst✝⁷ : Field L\ninst✝⁶ : DecidableEq L\ninst✝⁵ : Algebra R K\ninst✝⁴ : IsFractionRing R K\ninst✝³ : Algebra K L\ninst✝² : FiniteDimensional K L\ninst✝¹ : Algebra R L\ninst✝ : IsScalarTower R K L\nx : Kˣ\n⊢ ↑(↑(toPrincipalIdeal R K) x) = spanSingleton R⁰ ↑x",
"tactic": "simp only [toPrincipalIdeal]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nK : Type u_1\nL : Type ?u.37953\ninst✝⁹ : CommRing R\ninst✝⁸ : Field K\ninst✝⁷ : Field L\ninst✝⁶ : DecidableEq L\ninst✝⁵ : Algebra R K\ninst✝⁴ : IsFractionRing R K\ninst✝³ : Algebra K L\ninst✝² : FiniteDimensional K L\ninst✝¹ : Algebra R L\ninst✝ : IsScalarTower R K L\nx : Kˣ\n⊢ ↑(↑{\n toOneHom :=\n {\n toFun := fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) },\n map_one' :=\n (_ :\n (fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : Kˣ),\n OneHom.toFun\n {\n toFun := fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) },\n map_one' :=\n (_ :\n (fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) },\n map_one' :=\n (_ :\n (fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) },\n map_one' :=\n (_ :\n (fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) })\n 1 =\n 1) }\n y) }\n x) =\n spanSingleton R⁰ ↑x",
"tactic": "rfl"
}
] |
[
70,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/Topology/Algebra/Ring/Basic.lean
|
Subsemiring.le_topologicalClosure
|
[] |
[
118,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/Topology/LocalExtr.lean
|
IsLocalMax.bicomp_mono
|
[] |
[
285,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
282,
8
] |
Mathlib/Topology/LocalHomeomorph.lean
|
TopologicalSpace.Opens.localHomeomorphSubtypeCoe_target
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.112631\nγ : Type ?u.112634\nδ : Type ?u.112637\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\n⊢ {x | x ∈ s.carrier} = ↑s",
"state_before": "α : Type u_1\nβ : Type ?u.112631\nγ : Type ?u.112634\nδ : Type ?u.112637\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\n⊢ (localHomeomorphSubtypeCoe s).toLocalEquiv.target = ↑s",
"tactic": "simp only [localHomeomorphSubtypeCoe, Subtype.range_coe_subtype, mfld_simps]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.112631\nγ : Type ?u.112634\nδ : Type ?u.112637\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\n⊢ {x | x ∈ s.carrier} = ↑s",
"tactic": "rfl"
}
] |
[
1348,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1346,
1
] |
Mathlib/Order/Hom/Basic.lean
|
OrderHom.comp_mono
|
[] |
[
362,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/Algebra/Ring/Divisibility.lean
|
dvd_add_right
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.9814\ninst✝ : NonUnitalRing α\na b c : α\nh : a ∣ b\n⊢ a ∣ c + b ↔ a ∣ c",
"state_before": "α : Type u_1\nβ : Type ?u.9814\ninst✝ : NonUnitalRing α\na b c : α\nh : a ∣ b\n⊢ a ∣ b + c ↔ a ∣ c",
"tactic": "rw [add_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.9814\ninst✝ : NonUnitalRing α\na b c : α\nh : a ∣ b\n⊢ a ∣ c + b ↔ a ∣ c",
"tactic": "exact dvd_add_left h"
}
] |
[
101,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean
|
InnerProductGeometry.norm_sub_eq_add_norm_iff_angle_eq_pi
|
[
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\nhx : x ≠ 0\nhy : y ≠ 0\nh : ‖x - y‖ = ‖x‖ + ‖y‖\n⊢ angle x y = π",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ ‖x - y‖ = ‖x‖ + ‖y‖ ↔ angle x y = π",
"tactic": "refine' ⟨fun h => _, norm_sub_eq_add_norm_of_angle_eq_pi⟩"
},
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\nhx : x ≠ 0\nhy : y ≠ 0\nh : ‖x - y‖ = ‖x‖ + ‖y‖\n⊢ inner x y = -(‖x‖ * ‖y‖)",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\nhx : x ≠ 0\nhy : y ≠ 0\nh : ‖x - y‖ = ‖x‖ + ‖y‖\n⊢ angle x y = π",
"tactic": "rw [← inner_eq_neg_mul_norm_iff_angle_eq_pi hx hy]"
},
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\nhx : x ≠ 0\nhy : y ≠ 0\nh : ‖x - y‖ = ‖x‖ + ‖y‖\nhxy₁ : 0 ≤ ‖x - y‖\nhxy₂ : 0 ≤ ‖x‖ + ‖y‖\n⊢ inner x y = -(‖x‖ * ‖y‖)",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\nhx : x ≠ 0\nhy : y ≠ 0\nh : ‖x - y‖ = ‖x‖ + ‖y‖\n⊢ inner x y = -(‖x‖ * ‖y‖)",
"tactic": "obtain ⟨hxy₁, hxy₂⟩ := norm_nonneg (x - y), add_nonneg (norm_nonneg x) (norm_nonneg y)"
},
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\nhx : x ≠ 0\nhy : y ≠ 0\nh✝ : ‖x - y‖ ^ 2 = (‖x‖ + ‖y‖) ^ 2\nh : ‖x‖ ^ 2 - 2 * inner x y + ‖y‖ ^ 2 = (‖x‖ + ‖y‖) ^ 2\nhxy₁ : 0 ≤ ‖x - y‖\nhxy₂ : 0 ≤ ‖x‖ + ‖y‖\n⊢ inner x y = -(‖x‖ * ‖y‖)",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\nhx : x ≠ 0\nhy : y ≠ 0\nh : ‖x - y‖ = ‖x‖ + ‖y‖\nhxy₁ : 0 ≤ ‖x - y‖\nhxy₂ : 0 ≤ ‖x‖ + ‖y‖\n⊢ inner x y = -(‖x‖ * ‖y‖)",
"tactic": "rw [← sq_eq_sq hxy₁ hxy₂, norm_sub_pow_two_real] at h"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\nhx : x ≠ 0\nhy : y ≠ 0\nh✝ : ‖x - y‖ ^ 2 = (‖x‖ + ‖y‖) ^ 2\nh : ‖x‖ ^ 2 - 2 * inner x y + ‖y‖ ^ 2 = (‖x‖ + ‖y‖) ^ 2\nhxy₁ : 0 ≤ ‖x - y‖\nhxy₂ : 0 ≤ ‖x‖ + ‖y‖\n⊢ inner x y = -(‖x‖ * ‖y‖)",
"tactic": "calc\n ⟪x, y⟫ = (‖x‖ ^ 2 + ‖y‖ ^ 2 - (‖x‖ + ‖y‖) ^ 2) / 2 := by linarith\n _ = -(‖x‖ * ‖y‖) := by ring"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\nhx : x ≠ 0\nhy : y ≠ 0\nh✝ : ‖x - y‖ ^ 2 = (‖x‖ + ‖y‖) ^ 2\nh : ‖x‖ ^ 2 - 2 * inner x y + ‖y‖ ^ 2 = (‖x‖ + ‖y‖) ^ 2\nhxy₁ : 0 ≤ ‖x - y‖\nhxy₂ : 0 ≤ ‖x‖ + ‖y‖\n⊢ inner x y = (‖x‖ ^ 2 + ‖y‖ ^ 2 - (‖x‖ + ‖y‖) ^ 2) / 2",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\nhx : x ≠ 0\nhy : y ≠ 0\nh✝ : ‖x - y‖ ^ 2 = (‖x‖ + ‖y‖) ^ 2\nh : ‖x‖ ^ 2 - 2 * inner x y + ‖y‖ ^ 2 = (‖x‖ + ‖y‖) ^ 2\nhxy₁ : 0 ≤ ‖x - y‖\nhxy₂ : 0 ≤ ‖x‖ + ‖y‖\n⊢ (‖x‖ ^ 2 + ‖y‖ ^ 2 - (‖x‖ + ‖y‖) ^ 2) / 2 = -(‖x‖ * ‖y‖)",
"tactic": "ring"
}
] |
[
305,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
297,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.pushoutCoconeOfLeftIso_inr
|
[] |
[
1790,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1790,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.mem_cons_of_mem
|
[] |
[
964,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
963,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.monomial_single_add
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\n⊢ ↑(monomial (Finsupp.single n e + s)) a = X n ^ e * ↑(monomial s) a",
"tactic": "rw [X_pow_eq_monomial, monomial_mul, one_mul]"
}
] |
[
331,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
330,
1
] |
Mathlib/GroupTheory/Submonoid/Operations.lean
|
Submonoid.map_surjective_of_surjective
|
[] |
[
459,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
458,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.union_le
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.168188\nγ : Type ?u.168191\ninst✝ : DecidableEq α\ns t u : Multiset α\na b : α\nh₁ : s ≤ u\nh₂ : t ≤ u\n⊢ s ∪ t ≤ u ∪ t",
"state_before": "α : Type u_1\nβ : Type ?u.168188\nγ : Type ?u.168191\ninst✝ : DecidableEq α\ns t u : Multiset α\na b : α\nh₁ : s ≤ u\nh₂ : t ≤ u\n⊢ s ∪ t ≤ u",
"tactic": "rw [← eq_union_left h₂]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.168188\nγ : Type ?u.168191\ninst✝ : DecidableEq α\ns t u : Multiset α\na b : α\nh₁ : s ≤ u\nh₂ : t ≤ u\n⊢ s ∪ t ≤ u ∪ t",
"tactic": "exact union_le_union_right h₁ t"
}
] |
[
1715,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1714,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
le_mul_iff_one_le_left'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.18226\ninst✝³ : MulOneClass α\ninst✝² : LE α\ninst✝¹ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : ContravariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b : α\n⊢ a ≤ b * a ↔ 1 * a ≤ b * a",
"tactic": "rw [one_mul]"
}
] |
[
410,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
407,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.nat_cast_inj
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ\n⊢ ↑m = ↑n ↔ m = n",
"tactic": "simp only [le_antisymm_iff, nat_cast_le]"
}
] |
[
2316,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2315,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
|
MeasureTheory.Measure.prod_add
|
[
{
"state_after": "α : Type u_2\nα' : Type ?u.4495945\nβ : Type u_1\nβ' : Type ?u.4495951\nγ : Type ?u.4495954\nE : Type ?u.4495957\ninst✝⁸ : MeasurableSpace α\ninst✝⁷ : MeasurableSpace α'\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : MeasurableSpace β'\ninst✝⁴ : MeasurableSpace γ\nμ μ' : Measure α\nν ν'✝ : Measure β\nτ : Measure γ\ninst✝³ : NormedAddCommGroup E\ninst✝² : SigmaFinite ν\ninst✝¹ : SigmaFinite μ\nν' : Measure β\ninst✝ : SigmaFinite ν'\ns : Set α\nt : Set β\nx✝¹ : MeasurableSet s\nx✝ : MeasurableSet t\n⊢ ↑↑(Measure.prod μ ν + Measure.prod μ ν') (s ×ˢ t) = ↑↑μ s * ↑↑(ν + ν') t",
"state_before": "α : Type u_2\nα' : Type ?u.4495945\nβ : Type u_1\nβ' : Type ?u.4495951\nγ : Type ?u.4495954\nE : Type ?u.4495957\ninst✝⁸ : MeasurableSpace α\ninst✝⁷ : MeasurableSpace α'\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : MeasurableSpace β'\ninst✝⁴ : MeasurableSpace γ\nμ μ' : Measure α\nν ν'✝ : Measure β\nτ : Measure γ\ninst✝³ : NormedAddCommGroup E\ninst✝² : SigmaFinite ν\ninst✝¹ : SigmaFinite μ\nν' : Measure β\ninst✝ : SigmaFinite ν'\n⊢ Measure.prod μ (ν + ν') = Measure.prod μ ν + Measure.prod μ ν'",
"tactic": "refine' prod_eq fun s t _ _ => _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nα' : Type ?u.4495945\nβ : Type u_1\nβ' : Type ?u.4495951\nγ : Type ?u.4495954\nE : Type ?u.4495957\ninst✝⁸ : MeasurableSpace α\ninst✝⁷ : MeasurableSpace α'\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : MeasurableSpace β'\ninst✝⁴ : MeasurableSpace γ\nμ μ' : Measure α\nν ν'✝ : Measure β\nτ : Measure γ\ninst✝³ : NormedAddCommGroup E\ninst✝² : SigmaFinite ν\ninst✝¹ : SigmaFinite μ\nν' : Measure β\ninst✝ : SigmaFinite ν'\ns : Set α\nt : Set β\nx✝¹ : MeasurableSet s\nx✝ : MeasurableSet t\n⊢ ↑↑(Measure.prod μ ν + Measure.prod μ ν') (s ×ˢ t) = ↑↑μ s * ↑↑(ν + ν') t",
"tactic": "simp_rw [add_apply, prod_prod, left_distrib]"
}
] |
[
585,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
583,
1
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
EsakiaHom.cancel_left
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.96586\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.96598\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : Preorder α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : Preorder β\ninst✝³ : TopologicalSpace γ\ninst✝² : Preorder γ\ninst✝¹ : TopologicalSpace δ\ninst✝ : Preorder δ\ng : EsakiaHom β γ\nf₁ f₂ : EsakiaHom α β\nhg : Injective ↑g\nh : comp g f₁ = comp g f₂\na : α\n⊢ ↑g (↑f₁ a) = ↑g (↑f₂ a)",
"tactic": "rw [← comp_apply, h, comp_apply]"
}
] |
[
357,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
355,
1
] |
Mathlib/Topology/FiberBundle/Basic.lean
|
FiberBundleCore.localTrivAsLocalEquiv_source
|
[] |
[
664,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
662,
1
] |
Mathlib/RingTheory/Subring/Pointwise.lean
|
Subring.mem_inv_pointwise_smul_iff₀
|
[] |
[
163,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/CategoryTheory/Sites/DenseSubsite.lean
|
CategoryTheory.CoverDense.iso_of_restrict_iso
|
[
{
"state_after": "case h.e'_5\nC : Type u_5\ninst✝⁴ : Category C\nD : Type u_3\ninst✝³ : Category D\nE : Type ?u.139299\ninst✝² : Category E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nA : Type u_4\ninst✝¹ : Category A\nG : C ⥤ D\nH : CoverDense K G\ninst✝ : Full G\nℱ✝ : Dᵒᵖ ⥤ A\nℱ'✝ ℱ ℱ' : Sheaf K A\nα : ℱ ⟶ ℱ'\ni : IsIso (whiskerLeft (Functor.op G) α.val)\n⊢ α = (sheafIso H (asIso (whiskerLeft (Functor.op G) α.val))).hom",
"state_before": "C : Type u_5\ninst✝⁴ : Category C\nD : Type u_3\ninst✝³ : Category D\nE : Type ?u.139299\ninst✝² : Category E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nA : Type u_4\ninst✝¹ : Category A\nG : C ⥤ D\nH : CoverDense K G\ninst✝ : Full G\nℱ✝ : Dᵒᵖ ⥤ A\nℱ'✝ ℱ ℱ' : Sheaf K A\nα : ℱ ⟶ ℱ'\ni : IsIso (whiskerLeft (Functor.op G) α.val)\n⊢ IsIso α",
"tactic": "convert IsIso.of_iso (sheafIso H (asIso (whiskerLeft G.op α.val))) using 1"
},
{
"state_after": "case h.e'_5.h\nC : Type u_5\ninst✝⁴ : Category C\nD : Type u_3\ninst✝³ : Category D\nE : Type ?u.139299\ninst✝² : Category E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nA : Type u_4\ninst✝¹ : Category A\nG : C ⥤ D\nH : CoverDense K G\ninst✝ : Full G\nℱ✝ : Dᵒᵖ ⥤ A\nℱ'✝ ℱ ℱ' : Sheaf K A\nα : ℱ ⟶ ℱ'\ni : IsIso (whiskerLeft (Functor.op G) α.val)\n⊢ α.val = (sheafIso H (asIso (whiskerLeft (Functor.op G) α.val))).hom.val",
"state_before": "case h.e'_5\nC : Type u_5\ninst✝⁴ : Category C\nD : Type u_3\ninst✝³ : Category D\nE : Type ?u.139299\ninst✝² : Category E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nA : Type u_4\ninst✝¹ : Category A\nG : C ⥤ D\nH : CoverDense K G\ninst✝ : Full G\nℱ✝ : Dᵒᵖ ⥤ A\nℱ'✝ ℱ ℱ' : Sheaf K A\nα : ℱ ⟶ ℱ'\ni : IsIso (whiskerLeft (Functor.op G) α.val)\n⊢ α = (sheafIso H (asIso (whiskerLeft (Functor.op G) α.val))).hom",
"tactic": "ext1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h\nC : Type u_5\ninst✝⁴ : Category C\nD : Type u_3\ninst✝³ : Category D\nE : Type ?u.139299\ninst✝² : Category E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nA : Type u_4\ninst✝¹ : Category A\nG : C ⥤ D\nH : CoverDense K G\ninst✝ : Full G\nℱ✝ : Dᵒᵖ ⥤ A\nℱ'✝ ℱ ℱ' : Sheaf K A\nα : ℱ ⟶ ℱ'\ni : IsIso (whiskerLeft (Functor.op G) α.val)\n⊢ α.val = (sheafIso H (asIso (whiskerLeft (Functor.op G) α.val))).hom.val",
"tactic": "apply (sheafHom_eq H _).symm"
}
] |
[
468,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
464,
1
] |
Mathlib/Algebra/Order/LatticeGroup.lean
|
LatticeOrderedCommGroup.pos_eq_neg_inv
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝¹ : Lattice α\ninst✝ : CommGroup α\na : α\n⊢ a⁺ = a⁻¹⁻",
"tactic": "rw [neg_eq_pos_inv, inv_inv]"
}
] |
[
266,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.LiftRel.imp
|
[] |
[
1108,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1100,
1
] |
Mathlib/LinearAlgebra/PiTensorProduct.lean
|
PiTensorProduct.liftAux_tprodCoeff
|
[] |
[
380,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
379,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.isCircuit_def
|
[] |
[
895,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
894,
1
] |
Mathlib/AlgebraicTopology/DoldKan/Projections.lean
|
AlgebraicTopology.DoldKan.Q_succ
|
[
{
"state_after": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nq : ℕ\n⊢ 𝟙 K[X] - (P q + P q ≫ Hσ q) = 𝟙 K[X] - P q - P q ≫ Hσ q",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nq : ℕ\n⊢ Q (q + 1) = Q q - P q ≫ Hσ q",
"tactic": "simp only [Q, P_succ, comp_add, comp_id]"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nq : ℕ\n⊢ 𝟙 K[X] - (P q + P q ≫ Hσ q) = 𝟙 K[X] - P q - P q ≫ Hσ q",
"tactic": "abel"
}
] |
[
96,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Order/Iterate.lean
|
StrictMono.strictAnti_iterate_of_map_lt
|
[] |
[
266,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
264,
1
] |
Mathlib/Algebra/Hom/Centroid.lean
|
CentroidHom.comp_mul_comm
|
[
{
"state_after": "F : Type ?u.56322\nα : Type u_1\ninst✝ : NonUnitalNonAssocSemiring α\nT S : CentroidHom α\na b : α\n⊢ ↑T (↑S (a * b)) = ↑S (↑T (a * b))",
"state_before": "F : Type ?u.56322\nα : Type u_1\ninst✝ : NonUnitalNonAssocSemiring α\nT S : CentroidHom α\na b : α\n⊢ (↑T ∘ ↑S) (a * b) = (↑S ∘ ↑T) (a * b)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.56322\nα : Type u_1\ninst✝ : NonUnitalNonAssocSemiring α\nT S : CentroidHom α\na b : α\n⊢ ↑T (↑S (a * b)) = ↑S (↑T (a * b))",
"tactic": "rw [map_mul_right, map_mul_left, ← map_mul_right, ← map_mul_left]"
}
] |
[
402,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
400,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
ConvexOn.sup
|
[
{
"state_after": "case refine'_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.374344\nα : Type ?u.374347\nβ : Type u_3\nι : Type ?u.374353\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nhg : ConvexOn 𝕜 s g\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ f (a • x + b • y) ≤ a • (f ⊔ g) x + b • (f ⊔ g) y\n\ncase refine'_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.374344\nα : Type ?u.374347\nβ : Type u_3\nι : Type ?u.374353\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nhg : ConvexOn 𝕜 s g\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ g (a • x + b • y) ≤ a • (f ⊔ g) x + b • (f ⊔ g) y",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.374344\nα : Type ?u.374347\nβ : Type u_3\nι : Type ?u.374353\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nhg : ConvexOn 𝕜 s g\n⊢ ConvexOn 𝕜 s (f ⊔ g)",
"tactic": "refine' ⟨hf.left, fun x hx y hy a b ha hb hab => sup_le _ _⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.374344\nα : Type ?u.374347\nβ : Type u_3\nι : Type ?u.374353\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nhg : ConvexOn 𝕜 s g\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ f (a • x + b • y) ≤ a • (f ⊔ g) x + b • (f ⊔ g) y",
"tactic": "calc\n f (a • x + b • y) ≤ a • f x + b • f y := hf.right hx hy ha hb hab\n _ ≤ a • (f x ⊔ g x) + b • (f y ⊔ g y) := by gcongr <;> apply le_sup_left"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.374344\nα : Type ?u.374347\nβ : Type u_3\nι : Type ?u.374353\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nhg : ConvexOn 𝕜 s g\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a • f x + b • f y ≤ a • (f x ⊔ g x) + b • (f y ⊔ g y)",
"tactic": "gcongr <;> apply le_sup_left"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.374344\nα : Type ?u.374347\nβ : Type u_3\nι : Type ?u.374353\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nhg : ConvexOn 𝕜 s g\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ g (a • x + b • y) ≤ a • (f ⊔ g) x + b • (f ⊔ g) y",
"tactic": "calc\n g (a • x + b • y) ≤ a • g x + b • g y := hg.right hx hy ha hb hab\n _ ≤ a • (f x ⊔ g x) + b • (f y ⊔ g y) := by gcongr <;> apply le_sup_right"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.374344\nα : Type ?u.374347\nβ : Type u_3\nι : Type ?u.374353\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nhg : ConvexOn 𝕜 s g\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a • g x + b • g y ≤ a • (f x ⊔ g x) + b • (f y ⊔ g y)",
"tactic": "gcongr <;> apply le_sup_right"
}
] |
[
606,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
599,
1
] |
Mathlib/Data/Polynomial/Taylor.lean
|
Polynomial.taylor_taylor
|
[
{
"state_after": "no goals",
"state_before": "R✝ : Type ?u.213757\ninst✝¹ : Semiring R✝\nr✝ : R✝\nf✝ : R✝[X]\nR : Type u_1\ninst✝ : CommSemiring R\nf : R[X]\nr s : R\n⊢ ↑(taylor r) (↑(taylor s) f) = ↑(taylor (r + s)) f",
"tactic": "simp only [taylor_apply, comp_assoc, map_add, add_comp, X_comp, C_comp, C_add, add_assoc]"
}
] |
[
121,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
EMetric.closedBall_prod_same
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.220280\ninst✝¹ : PseudoEMetricSpace α\nx✝ y✝ z✝ : α\nε ε₁ ε₂ : ℝ≥0∞\ns t : Set α\ninst✝ : PseudoEMetricSpace β\nx : α\ny : β\nr : ℝ≥0∞\nz : α × β\n⊢ z ∈ closedBall x r ×ˢ closedBall y r ↔ z ∈ closedBall (x, y) r",
"tactic": "simp [Prod.edist_eq]"
}
] |
[
715,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
713,
1
] |
Mathlib/Analysis/Convex/Segment.lean
|
mem_segment_sub_add
|
[
{
"state_after": "case h.e'_4\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.162050\nG : Type ?u.162053\nι : Type ?u.162056\nπ : ι → Type ?u.162061\ninst✝³ : LinearOrderedRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\nx✝ y✝ : E\ninst✝ : Invertible 2\nx y : E\n⊢ x = midpoint 𝕜 (x - y) (x + y)",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.162050\nG : Type ?u.162053\nι : Type ?u.162056\nπ : ι → Type ?u.162061\ninst✝³ : LinearOrderedRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\nx✝ y✝ : E\ninst✝ : Invertible 2\nx y : E\n⊢ x ∈ [x - y-[𝕜]x + y]",
"tactic": "convert @midpoint_mem_segment 𝕜 _ _ _ _ _ _ _"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.162050\nG : Type ?u.162053\nι : Type ?u.162056\nπ : ι → Type ?u.162061\ninst✝³ : LinearOrderedRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\nx✝ y✝ : E\ninst✝ : Invertible 2\nx y : E\n⊢ x = midpoint 𝕜 (x - y) (x + y)",
"tactic": "rw [midpoint_sub_add]"
}
] |
[
313,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
311,
1
] |
Mathlib/Control/Traversable/Instances.lean
|
Option.traverse_eq_map_id
|
[
{
"state_after": "no goals",
"state_before": "F G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : Applicative G\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nα β : Type u_1\nf : α → β\nx : Option α\n⊢ Option.traverse (pure ∘ f) x = pure (f <$> x)",
"tactic": "cases x <;> rfl"
}
] |
[
47,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
46,
1
] |
Mathlib/Data/List/Range.lean
|
List.range'_one
|
[] |
[
34,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
34,
9
] |
Mathlib/LinearAlgebra/SymplecticGroup.lean
|
SymplecticGroup.transpose_mem
|
[
{
"state_after": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ⬝ J l R ⬝ Aᵀ = J l R\n⊢ Aᵀ ⬝ J l R ⬝ Aᵀᵀ = J l R",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ∈ symplecticGroup l R\n⊢ Aᵀ ∈ symplecticGroup l R",
"tactic": "rw [mem_iff] at hA⊢"
},
{
"state_after": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ⬝ J l R ⬝ Aᵀ = J l R\n⊢ Aᵀ ⬝ J l R ⬝ A = J l R",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ⬝ J l R ⬝ Aᵀ = J l R\n⊢ Aᵀ ⬝ J l R ⬝ Aᵀᵀ = J l R",
"tactic": "rw [transpose_transpose]"
},
{
"state_after": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ⬝ J l R ⬝ Aᵀ = J l R\nhuA : IsUnit (det A)\n⊢ Aᵀ ⬝ J l R ⬝ A = J l R",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ⬝ J l R ⬝ Aᵀ = J l R\n⊢ Aᵀ ⬝ J l R ⬝ A = J l R",
"tactic": "have huA := symplectic_det hA"
},
{
"state_after": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ⬝ J l R ⬝ Aᵀ = J l R\nhuA : IsUnit (det A)\nhuAT : IsUnit (det Aᵀ)\n⊢ Aᵀ ⬝ J l R ⬝ A = J l R",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ⬝ J l R ⬝ Aᵀ = J l R\nhuA : IsUnit (det A)\n⊢ Aᵀ ⬝ J l R ⬝ A = J l R",
"tactic": "have huAT : IsUnit Aᵀ.det := by\n rw [Matrix.det_transpose]\n exact huA"
},
{
"state_after": "no goals",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ⬝ J l R ⬝ Aᵀ = J l R\nhuA : IsUnit (det A)\nhuAT : IsUnit (det Aᵀ)\n⊢ Aᵀ ⬝ J l R ⬝ A = J l R",
"tactic": "calc\n Aᵀ ⬝ J l R ⬝ A = (-Aᵀ) ⬝ (J l R)⁻¹ ⬝ A := by\n rw [J_inv]\n simp\n _ = (-Aᵀ) ⬝ (A ⬝ J l R ⬝ Aᵀ)⁻¹ ⬝ A := by rw [hA]\n _ = (-Aᵀ ⬝ (Aᵀ⁻¹ ⬝ (J l R)⁻¹)) ⬝ A⁻¹ ⬝ A := by\n simp only [Matrix.mul_inv_rev, Matrix.mul_assoc, Matrix.neg_mul]\n _ = -(J l R)⁻¹ := by\n rw [mul_nonsing_inv_cancel_left _ _ huAT, nonsing_inv_mul_cancel_right _ _ huA]\n _ = J l R := by simp [J_inv]"
},
{
"state_after": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ⬝ J l R ⬝ Aᵀ = J l R\nhuA : IsUnit (det A)\n⊢ IsUnit (det A)",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ⬝ J l R ⬝ Aᵀ = J l R\nhuA : IsUnit (det A)\n⊢ IsUnit (det Aᵀ)",
"tactic": "rw [Matrix.det_transpose]"
},
{
"state_after": "no goals",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ⬝ J l R ⬝ Aᵀ = J l R\nhuA : IsUnit (det A)\n⊢ IsUnit (det A)",
"tactic": "exact huA"
},
{
"state_after": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ⬝ J l R ⬝ Aᵀ = J l R\nhuA : IsUnit (det A)\nhuAT : IsUnit (det Aᵀ)\n⊢ Aᵀ ⬝ J l R ⬝ A = (-Aᵀ) ⬝ (-J l R) ⬝ A",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ⬝ J l R ⬝ Aᵀ = J l R\nhuA : IsUnit (det A)\nhuAT : IsUnit (det Aᵀ)\n⊢ Aᵀ ⬝ J l R ⬝ A = (-Aᵀ) ⬝ (J l R)⁻¹ ⬝ A",
"tactic": "rw [J_inv]"
},
{
"state_after": "no goals",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ⬝ J l R ⬝ Aᵀ = J l R\nhuA : IsUnit (det A)\nhuAT : IsUnit (det Aᵀ)\n⊢ Aᵀ ⬝ J l R ⬝ A = (-Aᵀ) ⬝ (-J l R) ⬝ A",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ⬝ J l R ⬝ Aᵀ = J l R\nhuA : IsUnit (det A)\nhuAT : IsUnit (det Aᵀ)\n⊢ (-Aᵀ) ⬝ (J l R)⁻¹ ⬝ A = (-Aᵀ) ⬝ (A ⬝ J l R ⬝ Aᵀ)⁻¹ ⬝ A",
"tactic": "rw [hA]"
},
{
"state_after": "no goals",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ⬝ J l R ⬝ Aᵀ = J l R\nhuA : IsUnit (det A)\nhuAT : IsUnit (det Aᵀ)\n⊢ (-Aᵀ) ⬝ (A ⬝ J l R ⬝ Aᵀ)⁻¹ ⬝ A = (-Aᵀ ⬝ (Aᵀ⁻¹ ⬝ (J l R)⁻¹)) ⬝ A⁻¹ ⬝ A",
"tactic": "simp only [Matrix.mul_inv_rev, Matrix.mul_assoc, Matrix.neg_mul]"
},
{
"state_after": "no goals",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ⬝ J l R ⬝ Aᵀ = J l R\nhuA : IsUnit (det A)\nhuAT : IsUnit (det Aᵀ)\n⊢ (-Aᵀ ⬝ (Aᵀ⁻¹ ⬝ (J l R)⁻¹)) ⬝ A⁻¹ ⬝ A = -(J l R)⁻¹",
"tactic": "rw [mul_nonsing_inv_cancel_left _ _ huAT, nonsing_inv_mul_cancel_right _ _ huA]"
},
{
"state_after": "no goals",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ⬝ J l R ⬝ Aᵀ = J l R\nhuA : IsUnit (det A)\nhuAT : IsUnit (det Aᵀ)\n⊢ -(J l R)⁻¹ = J l R",
"tactic": "simp [J_inv]"
}
] |
[
173,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.bind_pure'
|
[
{
"state_after": "case bisim\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\n⊢ IsBisimulation fun c₁ c₂ => c₁ = c₂ ∨ ∃ s, c₁ = bind s pure ∧ c₂ = s\n\ncase r\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\n⊢ bind s pure = s ∨ ∃ s_1, bind s pure = bind s_1 pure ∧ s = s_1",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\n⊢ bind s pure = s",
"tactic": "apply eq_of_bisim fun c₁ c₂ => c₁ = c₂ ∨ ∃ s, c₁ = bind s (pure) ∧ c₂ = s"
},
{
"state_after": "case r\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\n⊢ bind s pure = s ∨ ∃ s_1, bind s pure = bind s_1 pure ∧ s = s_1",
"state_before": "case bisim\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\n⊢ IsBisimulation fun c₁ c₂ => c₁ = c₂ ∨ ∃ s, c₁ = bind s pure ∧ c₂ = s\n\ncase r\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\n⊢ bind s pure = s ∨ ∃ s_1, bind s pure = bind s_1 pure ∧ s = s_1",
"tactic": ". intro c₁ c₂ h\n exact\n match c₁, c₂, h with\n | _, c₂, Or.inl (Eq.refl _) => by cases' destruct c₂ with b cb <;> simp\n | _, _, Or.inr ⟨s, rfl, rfl⟩ => by\n apply recOn s <;> intro s <;> simp"
},
{
"state_after": "no goals",
"state_before": "case r\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\n⊢ bind s pure = s ∨ ∃ s_1, bind s pure = bind s_1 pure ∧ s = s_1",
"tactic": ". exact Or.inr ⟨s, rfl, rfl⟩"
},
{
"state_after": "case bisim\nα : Type u\nβ : Type v\nγ : Type w\ns c₁ c₂ : Computation α\nh : c₁ = c₂ ∨ ∃ s, c₁ = bind s pure ∧ c₂ = s\n⊢ BisimO (fun c₁ c₂ => c₁ = c₂ ∨ ∃ s, c₁ = bind s pure ∧ c₂ = s) (destruct c₁) (destruct c₂)",
"state_before": "case bisim\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\n⊢ IsBisimulation fun c₁ c₂ => c₁ = c₂ ∨ ∃ s, c₁ = bind s pure ∧ c₂ = s",
"tactic": "intro c₁ c₂ h"
},
{
"state_after": "no goals",
"state_before": "case bisim\nα : Type u\nβ : Type v\nγ : Type w\ns c₁ c₂ : Computation α\nh : c₁ = c₂ ∨ ∃ s, c₁ = bind s pure ∧ c₂ = s\n⊢ BisimO (fun c₁ c₂ => c₁ = c₂ ∨ ∃ s, c₁ = bind s pure ∧ c₂ = s) (destruct c₁) (destruct c₂)",
"tactic": "exact\n match c₁, c₂, h with\n | _, c₂, Or.inl (Eq.refl _) => by cases' destruct c₂ with b cb <;> simp\n | _, _, Or.inr ⟨s, rfl, rfl⟩ => by\n apply recOn s <;> intro s <;> simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns c₁ c₂✝ : Computation α\nh : c₁ = c₂✝ ∨ ∃ s, c₁ = bind s pure ∧ c₂✝ = s\nc₂ : Computation α\n⊢ BisimO (fun c₁ c₂ => c₁ = c₂ ∨ ∃ s, c₁ = bind s pure ∧ c₂ = s) (destruct c₂) (destruct c₂)",
"tactic": "cases' destruct c₂ with b cb <;> simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns✝ c₁ c₂ : Computation α\nh : c₁ = c₂ ∨ ∃ s, c₁ = bind s pure ∧ c₂ = s\ns : Computation α\n⊢ BisimO (fun c₁ c₂ => c₁ = c₂ ∨ ∃ s, c₁ = bind s pure ∧ c₂ = s) (destruct (bind s pure)) (destruct s)",
"tactic": "apply recOn s <;> intro s <;> simp"
},
{
"state_after": "no goals",
"state_before": "case r\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\n⊢ bind s pure = s ∨ ∃ s_1, bind s pure = bind s_1 pure ∧ s = s_1",
"tactic": "exact Or.inr ⟨s, rfl, rfl⟩"
}
] |
[
770,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
762,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
PSet.Resp.eval_val
|
[] |
[
619,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
618,
1
] |
Mathlib/Topology/Algebra/Order/ProjIcc.lean
|
continuous_projIcc
|
[] |
[
43,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
42,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.subset_union_elim
|
[
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.400981\nγ : Type ?u.400984\np q : α → Prop\ninst✝² : DecidablePred p\ninst✝¹ : DecidablePred q\ninst✝ : DecidableEq α\ns : Finset α\nt₁ t₂ : Set α\nh : ↑s ⊆ t₁ ∪ t₂\n⊢ filter (fun x => x ∈ t₁) s ∪ filter (fun x => ¬x ∈ t₁) s = s",
"tactic": "simp [filter_union_right, em]"
},
{
"state_after": "case refine'_2\nα : Type u_1\nβ : Type ?u.400981\nγ : Type ?u.400984\np q : α → Prop\ninst✝² : DecidablePred p\ninst✝¹ : DecidablePred q\ninst✝ : DecidableEq α\ns : Finset α\nt₁ t₂ : Set α\nh : ↑s ⊆ t₁ ∪ t₂\nx : α\n⊢ x ∈ ↑(filter (fun x => x ∈ t₁) s) → x ∈ t₁",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.400981\nγ : Type ?u.400984\np q : α → Prop\ninst✝² : DecidablePred p\ninst✝¹ : DecidablePred q\ninst✝ : DecidableEq α\ns : Finset α\nt₁ t₂ : Set α\nh : ↑s ⊆ t₁ ∪ t₂\n⊢ ↑(filter (fun x => x ∈ t₁) s) ⊆ t₁",
"tactic": "intro x"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.400981\nγ : Type ?u.400984\np q : α → Prop\ninst✝² : DecidablePred p\ninst✝¹ : DecidablePred q\ninst✝ : DecidableEq α\ns : Finset α\nt₁ t₂ : Set α\nh : ↑s ⊆ t₁ ∪ t₂\nx : α\n⊢ x ∈ ↑(filter (fun x => x ∈ t₁) s) → x ∈ t₁",
"tactic": "simp"
},
{
"state_after": "case refine'_3\nα : Type u_1\nβ : Type ?u.400981\nγ : Type ?u.400984\np q : α → Prop\ninst✝² : DecidablePred p\ninst✝¹ : DecidablePred q\ninst✝ : DecidableEq α\ns : Finset α\nt₁ t₂ : Set α\nh : ↑s ⊆ t₁ ∪ t₂\nx : α\n⊢ x ∈ ↑(filter (fun x => ¬x ∈ t₁) s) → x ∈ t₂ \\ t₁",
"state_before": "case refine'_3\nα : Type u_1\nβ : Type ?u.400981\nγ : Type ?u.400984\np q : α → Prop\ninst✝² : DecidablePred p\ninst✝¹ : DecidablePred q\ninst✝ : DecidableEq α\ns : Finset α\nt₁ t₂ : Set α\nh : ↑s ⊆ t₁ ∪ t₂\n⊢ ↑(filter (fun x => ¬x ∈ t₁) s) ⊆ t₂ \\ t₁",
"tactic": "intro x"
},
{
"state_after": "case refine'_3\nα : Type u_1\nβ : Type ?u.400981\nγ : Type ?u.400984\np q : α → Prop\ninst✝² : DecidablePred p\ninst✝¹ : DecidablePred q\ninst✝ : DecidableEq α\ns : Finset α\nt₁ t₂ : Set α\nh : ↑s ⊆ t₁ ∪ t₂\nx : α\n⊢ x ∈ s → ¬x ∈ t₁ → x ∈ t₂ ∧ ¬x ∈ t₁",
"state_before": "case refine'_3\nα : Type u_1\nβ : Type ?u.400981\nγ : Type ?u.400984\np q : α → Prop\ninst✝² : DecidablePred p\ninst✝¹ : DecidablePred q\ninst✝ : DecidableEq α\ns : Finset α\nt₁ t₂ : Set α\nh : ↑s ⊆ t₁ ∪ t₂\nx : α\n⊢ x ∈ ↑(filter (fun x => ¬x ∈ t₁) s) → x ∈ t₂ \\ t₁",
"tactic": "simp"
},
{
"state_after": "case refine'_3\nα : Type u_1\nβ : Type ?u.400981\nγ : Type ?u.400984\np q : α → Prop\ninst✝² : DecidablePred p\ninst✝¹ : DecidablePred q\ninst✝ : DecidableEq α\ns : Finset α\nt₁ t₂ : Set α\nh : ↑s ⊆ t₁ ∪ t₂\nx : α\nhx : x ∈ s\nhx₂ : ¬x ∈ t₁\n⊢ x ∈ t₂ ∧ ¬x ∈ t₁",
"state_before": "case refine'_3\nα : Type u_1\nβ : Type ?u.400981\nγ : Type ?u.400984\np q : α → Prop\ninst✝² : DecidablePred p\ninst✝¹ : DecidablePred q\ninst✝ : DecidableEq α\ns : Finset α\nt₁ t₂ : Set α\nh : ↑s ⊆ t₁ ∪ t₂\nx : α\n⊢ x ∈ s → ¬x ∈ t₁ → x ∈ t₂ ∧ ¬x ∈ t₁",
"tactic": "intro hx hx₂"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nα : Type u_1\nβ : Type ?u.400981\nγ : Type ?u.400984\np q : α → Prop\ninst✝² : DecidablePred p\ninst✝¹ : DecidablePred q\ninst✝ : DecidableEq α\ns : Finset α\nt₁ t₂ : Set α\nh : ↑s ⊆ t₁ ∪ t₂\nx : α\nhx : x ∈ s\nhx₂ : ¬x ∈ t₁\n⊢ x ∈ t₂ ∧ ¬x ∈ t₁",
"tactic": "refine' ⟨Or.resolve_left (h hx) hx₂, hx₂⟩"
}
] |
[
2882,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2872,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
upperClosure_eq_top_iff
|
[] |
[
1424,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1423,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.unitsSMul_apply
|
[] |
[
1249,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1247,
1
] |
Mathlib/Algebra/Algebra/Unitization.lean
|
Unitization.inl_neg
|
[] |
[
269,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
268,
1
] |
Mathlib/MeasureTheory/Covering/VitaliFamily.lean
|
VitaliFamily.FineSubfamilyOn.measure_diff_biUnion
|
[] |
[
155,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Topology/LocalExtr.lean
|
IsLocalMinOn.min
|
[] |
[
516,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
514,
8
] |
Std/Data/Int/Lemmas.lean
|
Int.toNat_lt
|
[
{
"state_after": "no goals",
"state_before": "n : Nat\nz : Int\nh : 0 ≤ z\n⊢ toNat z < n ↔ z < ↑n",
"tactic": "rw [← Int.not_le, ← Nat.not_le, Int.le_toNat h]"
}
] |
[
1384,
50
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1383,
9
] |
Mathlib/Algebra/Algebra/Subalgebra/Tower.lean
|
Subalgebra.restrictScalars_toSubmodule
|
[] |
[
112,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/RingTheory/MvPolynomial/Symmetric.lean
|
MvPolynomial.IsSymmetric.add
|
[] |
[
124,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/LinearAlgebra/Matrix/Determinant.lean
|
Matrix.det_reindex_self
|
[] |
[
268,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
Mathlib/RingTheory/RootsOfUnity/Basic.lean
|
IsPrimitiveRoot.zmodEquivZpowers_symm_apply_pow
|
[
{
"state_after": "no goals",
"state_before": "M : Type ?u.3141531\nN : Type ?u.3141534\nG : Type ?u.3141537\nR : Type u_1\nS : Type ?u.3141543\nF : Type ?u.3141546\ninst✝³ : CommMonoid M\ninst✝² : CommMonoid N\ninst✝¹ : DivisionCommMonoid G\nk l : ℕ\ninst✝ : CommRing R\nζ : Rˣ\nh : IsPrimitiveRoot ζ k\ni : ℕ\n⊢ ↑(AddEquiv.symm (zmodEquivZpowers h))\n (↑Additive.ofMul { val := ζ ^ i, property := (_ : ∃ y, (fun x x_1 => x ^ x_1) ζ y = ζ ^ i) }) =\n ↑i",
"tactic": "rw [← h.zmodEquivZpowers.symm_apply_apply i, zmodEquivZpowers_apply_coe_nat]"
}
] |
[
732,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
730,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
innerSL_apply_norm
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3288929\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx : E\n⊢ ‖x‖ ≤ ‖↑(innerSL 𝕜) x‖",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3288929\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx : E\n⊢ ‖↑(innerSL 𝕜) x‖ = ‖x‖",
"tactic": "refine'\n le_antisymm ((innerSL 𝕜 x).op_norm_le_bound (norm_nonneg _) fun y => norm_inner_le_norm _ _) _"
},
{
"state_after": "case inl\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3288929\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\n⊢ ‖0‖ ≤ ‖↑(innerSL 𝕜) 0‖\n\ncase inr\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3288929\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx : E\nh : x ≠ 0\n⊢ ‖x‖ ≤ ‖↑(innerSL 𝕜) x‖",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3288929\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx : E\n⊢ ‖x‖ ≤ ‖↑(innerSL 𝕜) x‖",
"tactic": "rcases eq_or_ne x 0 with (rfl | h)"
},
{
"state_after": "no goals",
"state_before": "case inl\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3288929\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\n⊢ ‖0‖ ≤ ‖↑(innerSL 𝕜) 0‖",
"tactic": "simp"
},
{
"state_after": "case inr\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3288929\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx : E\nh : x ≠ 0\n⊢ ‖x‖ * ‖x‖ ≤ ‖↑(innerSL 𝕜) x‖ * ‖x‖",
"state_before": "case inr\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3288929\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx : E\nh : x ≠ 0\n⊢ ‖x‖ ≤ ‖↑(innerSL 𝕜) x‖",
"tactic": "refine' (mul_le_mul_right (norm_pos_iff.2 h)).mp _"
},
{
"state_after": "no goals",
"state_before": "case inr\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3288929\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx : E\nh : x ≠ 0\n⊢ ‖x‖ * ‖x‖ ≤ ‖↑(innerSL 𝕜) x‖ * ‖x‖",
"tactic": "calc\n ‖x‖ * ‖x‖ = ‖(⟪x, x⟫ : 𝕜)‖ := by\n rw [← sq, inner_self_eq_norm_sq_to_K, norm_pow, norm_ofReal, abs_norm]\n _ ≤ ‖innerSL 𝕜 x‖ * ‖x‖ := (innerSL 𝕜 x).le_op_norm _"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3288929\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx : E\nh : x ≠ 0\n⊢ ‖x‖ * ‖x‖ = ‖inner x x‖",
"tactic": "rw [← sq, inner_self_eq_norm_sq_to_K, norm_pow, norm_ofReal, abs_norm]"
}
] |
[
1798,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1789,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.zero_mem_smul_iff
|
[
{
"state_after": "case mp\nF : Type ?u.93331\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93340\ninst✝³ : Zero α\ninst✝² : Zero β\ninst✝¹ : SMulWithZero α β\ns : Set α\nt : Set β\ninst✝ : NoZeroSMulDivisors α β\na : α\n⊢ 0 ∈ s • t → 0 ∈ s ∧ Set.Nonempty t ∨ 0 ∈ t ∧ Set.Nonempty s\n\ncase mpr\nF : Type ?u.93331\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93340\ninst✝³ : Zero α\ninst✝² : Zero β\ninst✝¹ : SMulWithZero α β\ns : Set α\nt : Set β\ninst✝ : NoZeroSMulDivisors α β\na : α\n⊢ 0 ∈ s ∧ Set.Nonempty t ∨ 0 ∈ t ∧ Set.Nonempty s → 0 ∈ s • t",
"state_before": "F : Type ?u.93331\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93340\ninst✝³ : Zero α\ninst✝² : Zero β\ninst✝¹ : SMulWithZero α β\ns : Set α\nt : Set β\ninst✝ : NoZeroSMulDivisors α β\na : α\n⊢ 0 ∈ s • t ↔ 0 ∈ s ∧ Set.Nonempty t ∨ 0 ∈ t ∧ Set.Nonempty s",
"tactic": "constructor"
},
{
"state_after": "case mp.intro.intro.intro.intro\nF : Type ?u.93331\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93340\ninst✝³ : Zero α\ninst✝² : Zero β\ninst✝¹ : SMulWithZero α β\ns : Set α\nt : Set β\ninst✝ : NoZeroSMulDivisors α β\na✝ a : α\nb : β\nha : a ∈ s\nhb : b ∈ t\nh : (fun x x_1 => x • x_1) a b = 0\n⊢ 0 ∈ s ∧ Set.Nonempty t ∨ 0 ∈ t ∧ Set.Nonempty s",
"state_before": "case mp\nF : Type ?u.93331\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93340\ninst✝³ : Zero α\ninst✝² : Zero β\ninst✝¹ : SMulWithZero α β\ns : Set α\nt : Set β\ninst✝ : NoZeroSMulDivisors α β\na : α\n⊢ 0 ∈ s • t → 0 ∈ s ∧ Set.Nonempty t ∨ 0 ∈ t ∧ Set.Nonempty s",
"tactic": "rintro ⟨a, b, ha, hb, h⟩"
},
{
"state_after": "case mp.intro.intro.intro.intro.inl\nF : Type ?u.93331\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93340\ninst✝³ : Zero α\ninst✝² : Zero β\ninst✝¹ : SMulWithZero α β\ns : Set α\nt : Set β\ninst✝ : NoZeroSMulDivisors α β\na : α\nb : β\nhb : b ∈ t\nha : 0 ∈ s\nh : (fun x x_1 => x • x_1) 0 b = 0\n⊢ 0 ∈ s ∧ Set.Nonempty t ∨ 0 ∈ t ∧ Set.Nonempty s\n\ncase mp.intro.intro.intro.intro.inr\nF : Type ?u.93331\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93340\ninst✝³ : Zero α\ninst✝² : Zero β\ninst✝¹ : SMulWithZero α β\ns : Set α\nt : Set β\ninst✝ : NoZeroSMulDivisors α β\na✝ a : α\nha : a ∈ s\nhb : 0 ∈ t\nh : (fun x x_1 => x • x_1) a 0 = 0\n⊢ 0 ∈ s ∧ Set.Nonempty t ∨ 0 ∈ t ∧ Set.Nonempty s",
"state_before": "case mp.intro.intro.intro.intro\nF : Type ?u.93331\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93340\ninst✝³ : Zero α\ninst✝² : Zero β\ninst✝¹ : SMulWithZero α β\ns : Set α\nt : Set β\ninst✝ : NoZeroSMulDivisors α β\na✝ a : α\nb : β\nha : a ∈ s\nhb : b ∈ t\nh : (fun x x_1 => x • x_1) a b = 0\n⊢ 0 ∈ s ∧ Set.Nonempty t ∨ 0 ∈ t ∧ Set.Nonempty s",
"tactic": "obtain rfl | rfl := eq_zero_or_eq_zero_of_smul_eq_zero h"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro.inl\nF : Type ?u.93331\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93340\ninst✝³ : Zero α\ninst✝² : Zero β\ninst✝¹ : SMulWithZero α β\ns : Set α\nt : Set β\ninst✝ : NoZeroSMulDivisors α β\na : α\nb : β\nhb : b ∈ t\nha : 0 ∈ s\nh : (fun x x_1 => x • x_1) 0 b = 0\n⊢ 0 ∈ s ∧ Set.Nonempty t ∨ 0 ∈ t ∧ Set.Nonempty s",
"tactic": "exact Or.inl ⟨ha, b, hb⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro.inr\nF : Type ?u.93331\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93340\ninst✝³ : Zero α\ninst✝² : Zero β\ninst✝¹ : SMulWithZero α β\ns : Set α\nt : Set β\ninst✝ : NoZeroSMulDivisors α β\na✝ a : α\nha : a ∈ s\nhb : 0 ∈ t\nh : (fun x x_1 => x • x_1) a 0 = 0\n⊢ 0 ∈ s ∧ Set.Nonempty t ∨ 0 ∈ t ∧ Set.Nonempty s",
"tactic": "exact Or.inr ⟨hb, a, ha⟩"
},
{
"state_after": "case mpr.inl.intro.intro\nF : Type ?u.93331\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93340\ninst✝³ : Zero α\ninst✝² : Zero β\ninst✝¹ : SMulWithZero α β\ns : Set α\nt : Set β\ninst✝ : NoZeroSMulDivisors α β\na : α\nhs : 0 ∈ s\nb : β\nhb : b ∈ t\n⊢ 0 ∈ s • t\n\ncase mpr.inr.intro.intro\nF : Type ?u.93331\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93340\ninst✝³ : Zero α\ninst✝² : Zero β\ninst✝¹ : SMulWithZero α β\ns : Set α\nt : Set β\ninst✝ : NoZeroSMulDivisors α β\na✝ : α\nht : 0 ∈ t\na : α\nha : a ∈ s\n⊢ 0 ∈ s • t",
"state_before": "case mpr\nF : Type ?u.93331\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93340\ninst✝³ : Zero α\ninst✝² : Zero β\ninst✝¹ : SMulWithZero α β\ns : Set α\nt : Set β\ninst✝ : NoZeroSMulDivisors α β\na : α\n⊢ 0 ∈ s ∧ Set.Nonempty t ∨ 0 ∈ t ∧ Set.Nonempty s → 0 ∈ s • t",
"tactic": "rintro (⟨hs, b, hb⟩ | ⟨ht, a, ha⟩)"
},
{
"state_after": "no goals",
"state_before": "case mpr.inl.intro.intro\nF : Type ?u.93331\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93340\ninst✝³ : Zero α\ninst✝² : Zero β\ninst✝¹ : SMulWithZero α β\ns : Set α\nt : Set β\ninst✝ : NoZeroSMulDivisors α β\na : α\nhs : 0 ∈ s\nb : β\nhb : b ∈ t\n⊢ 0 ∈ s • t",
"tactic": "exact ⟨0, b, hs, hb, zero_smul _ _⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.inr.intro.intro\nF : Type ?u.93331\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93340\ninst✝³ : Zero α\ninst✝² : Zero β\ninst✝¹ : SMulWithZero α β\ns : Set α\nt : Set β\ninst✝ : NoZeroSMulDivisors α β\na✝ : α\nht : 0 ∈ t\na : α\nha : a ∈ s\n⊢ 0 ∈ s • t",
"tactic": "exact ⟨a, 0, ha, ht, smul_zero _⟩"
}
] |
[
837,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
828,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.absorbent_ball_zero
|
[] |
[
986,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
983,
11
] |
Std/Logic.lean
|
imp_true_iff
|
[] |
[
116,
89
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
116,
1
] |
Mathlib/Control/Basic.lean
|
map_seq
|
[
{
"state_after": "α β γ : Type u\nF : Type u → Type v\ninst✝¹ : Applicative F\ninst✝ : LawfulApplicative F\nf : β → γ\nx : F (α → β)\ny : F α\n⊢ (Seq.seq (pure f) fun x_1 => Seq.seq x fun x => y) = Seq.seq (Seq.seq (pure fun x => f ∘ x) fun x_1 => x) fun x => y",
"state_before": "α β γ : Type u\nF : Type u → Type v\ninst✝¹ : Applicative F\ninst✝ : LawfulApplicative F\nf : β → γ\nx : F (α → β)\ny : F α\n⊢ (f <$> Seq.seq x fun x => y) = Seq.seq ((fun x => f ∘ x) <$> x) fun x => y",
"tactic": "simp only [← pure_seq]"
},
{
"state_after": "no goals",
"state_before": "α β γ : Type u\nF : Type u → Type v\ninst✝¹ : Applicative F\ninst✝ : LawfulApplicative F\nf : β → γ\nx : F (α → β)\ny : F α\n⊢ (Seq.seq (pure f) fun x_1 => Seq.seq x fun x => y) = Seq.seq (Seq.seq (pure fun x => f ∘ x) fun x_1 => x) fun x => y",
"tactic": "simp [seq_assoc]"
}
] |
[
77,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Algebra/Module/Submodule/Basic.lean
|
Submodule.not_mem_of_ortho
|
[
{
"state_after": "G : Type u''\nS : Type u'\nR : Type u\nM : Type v\nι : Type w\ninst✝³ : Ring R\ninst✝² : IsDomain R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nb : ι → M\nx : M\nN : Submodule R M\northo : ∀ (c : R) (y : M), y ∈ N → c • x + y = 0 → c = 0\nhx : x ∈ N\n⊢ False",
"state_before": "G : Type u''\nS : Type u'\nR : Type u\nM : Type v\nι : Type w\ninst✝³ : Ring R\ninst✝² : IsDomain R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nb : ι → M\nx : M\nN : Submodule R M\northo : ∀ (c : R) (y : M), y ∈ N → c • x + y = 0 → c = 0\n⊢ ¬x ∈ N",
"tactic": "intro hx"
},
{
"state_after": "no goals",
"state_before": "G : Type u''\nS : Type u'\nR : Type u\nM : Type v\nι : Type w\ninst✝³ : Ring R\ninst✝² : IsDomain R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nb : ι → M\nx : M\nN : Submodule R M\northo : ∀ (c : R) (y : M), y ∈ N → c • x + y = 0 → c = 0\nhx : x ∈ N\n⊢ False",
"tactic": "simpa using ortho (-1) x hx"
}
] |
[
608,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
605,
1
] |
Mathlib/Order/Filter/NAry.lean
|
Filter.map_map₂_antidistrib_left
|
[] |
[
388,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieIdeal.ker_incl
|
[
{
"state_after": "case h\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nm✝ : { x // x ∈ ↑I }\n⊢ m✝ ∈ LieHom.ker (incl I) ↔ m✝ ∈ ⊥",
"state_before": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\n⊢ LieHom.ker (incl I) = ⊥",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nm✝ : { x // x ∈ ↑I }\n⊢ m✝ ∈ LieHom.ker (incl I) ↔ m✝ ∈ ⊥",
"tactic": "simp"
}
] |
[
1132,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1132,
1
] |
Mathlib/LinearAlgebra/Matrix/Block.lean
|
Matrix.twoBlockTriangular_det'
|
[
{
"state_after": "α : Type ?u.96201\nβ : Type ?u.96204\nm : Type u_1\nn : Type ?u.96210\no : Type ?u.96213\nm' : α → Type ?u.96218\nn' : α → Type ?u.96223\nR : Type v\ninst✝⁵ : CommRing R\nM✝ N : Matrix m m R\nb : m → α\ninst✝⁴ : DecidableEq m\ninst✝³ : Fintype m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nM : Matrix m m R\np : m → Prop\ninst✝ : DecidablePred p\nh : ∀ (i : m), p i → ∀ (j : m), ¬p j → M i j = 0\n⊢ det (toSquareBlockProp M fun i => ¬¬p i) * det (toSquareBlockProp M fun i => ¬p i) =\n det (toSquareBlockProp M p) * det (toSquareBlockProp M fun i => ¬p i)\n\nα : Type ?u.96201\nβ : Type ?u.96204\nm : Type u_1\nn : Type ?u.96210\no : Type ?u.96213\nm' : α → Type ?u.96218\nn' : α → Type ?u.96223\nR : Type v\ninst✝⁵ : CommRing R\nM✝ N : Matrix m m R\nb : m → α\ninst✝⁴ : DecidableEq m\ninst✝³ : Fintype m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nM : Matrix m m R\np : m → Prop\ninst✝ : DecidablePred p\nh : ∀ (i : m), p i → ∀ (j : m), ¬p j → M i j = 0\n⊢ ∀ (i : m), ¬¬p i → ∀ (j : m), ¬p j → M i j = 0",
"state_before": "α : Type ?u.96201\nβ : Type ?u.96204\nm : Type u_1\nn : Type ?u.96210\no : Type ?u.96213\nm' : α → Type ?u.96218\nn' : α → Type ?u.96223\nR : Type v\ninst✝⁵ : CommRing R\nM✝ N : Matrix m m R\nb : m → α\ninst✝⁴ : DecidableEq m\ninst✝³ : Fintype m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nM : Matrix m m R\np : m → Prop\ninst✝ : DecidablePred p\nh : ∀ (i : m), p i → ∀ (j : m), ¬p j → M i j = 0\n⊢ det M = det (toSquareBlockProp M p) * det (toSquareBlockProp M fun i => ¬p i)",
"tactic": "rw [M.twoBlockTriangular_det fun i => ¬p i, mul_comm]"
},
{
"state_after": "case e_a\nα : Type ?u.96201\nβ : Type ?u.96204\nm : Type u_1\nn : Type ?u.96210\no : Type ?u.96213\nm' : α → Type ?u.96218\nn' : α → Type ?u.96223\nR : Type v\ninst✝⁵ : CommRing R\nM✝ N : Matrix m m R\nb : m → α\ninst✝⁴ : DecidableEq m\ninst✝³ : Fintype m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nM : Matrix m m R\np : m → Prop\ninst✝ : DecidablePred p\nh : ∀ (i : m), p i → ∀ (j : m), ¬p j → M i j = 0\n⊢ det (toSquareBlockProp M fun i => ¬¬p i) = det (toSquareBlockProp M p)\n\nα : Type ?u.96201\nβ : Type ?u.96204\nm : Type u_1\nn : Type ?u.96210\no : Type ?u.96213\nm' : α → Type ?u.96218\nn' : α → Type ?u.96223\nR : Type v\ninst✝⁵ : CommRing R\nM✝ N : Matrix m m R\nb : m → α\ninst✝⁴ : DecidableEq m\ninst✝³ : Fintype m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nM : Matrix m m R\np : m → Prop\ninst✝ : DecidablePred p\nh : ∀ (i : m), p i → ∀ (j : m), ¬p j → M i j = 0\n⊢ ∀ (i : m), ¬¬p i → ∀ (j : m), ¬p j → M i j = 0",
"state_before": "α : Type ?u.96201\nβ : Type ?u.96204\nm : Type u_1\nn : Type ?u.96210\no : Type ?u.96213\nm' : α → Type ?u.96218\nn' : α → Type ?u.96223\nR : Type v\ninst✝⁵ : CommRing R\nM✝ N : Matrix m m R\nb : m → α\ninst✝⁴ : DecidableEq m\ninst✝³ : Fintype m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nM : Matrix m m R\np : m → Prop\ninst✝ : DecidablePred p\nh : ∀ (i : m), p i → ∀ (j : m), ¬p j → M i j = 0\n⊢ det (toSquareBlockProp M fun i => ¬¬p i) * det (toSquareBlockProp M fun i => ¬p i) =\n det (toSquareBlockProp M p) * det (toSquareBlockProp M fun i => ¬p i)\n\nα : Type ?u.96201\nβ : Type ?u.96204\nm : Type u_1\nn : Type ?u.96210\no : Type ?u.96213\nm' : α → Type ?u.96218\nn' : α → Type ?u.96223\nR : Type v\ninst✝⁵ : CommRing R\nM✝ N : Matrix m m R\nb : m → α\ninst✝⁴ : DecidableEq m\ninst✝³ : Fintype m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nM : Matrix m m R\np : m → Prop\ninst✝ : DecidablePred p\nh : ∀ (i : m), p i → ∀ (j : m), ¬p j → M i j = 0\n⊢ ∀ (i : m), ¬¬p i → ∀ (j : m), ¬p j → M i j = 0",
"tactic": "congr 1"
},
{
"state_after": "α : Type ?u.96201\nβ : Type ?u.96204\nm : Type u_1\nn : Type ?u.96210\no : Type ?u.96213\nm' : α → Type ?u.96218\nn' : α → Type ?u.96223\nR : Type v\ninst✝⁵ : CommRing R\nM✝ N : Matrix m m R\nb : m → α\ninst✝⁴ : DecidableEq m\ninst✝³ : Fintype m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nM : Matrix m m R\np : m → Prop\ninst✝ : DecidablePred p\nh : ∀ (i : m), p i → ∀ (j : m), ¬p j → M i j = 0\n⊢ ∀ (i : m), ¬¬p i → ∀ (j : m), ¬p j → M i j = 0",
"state_before": "case e_a\nα : Type ?u.96201\nβ : Type ?u.96204\nm : Type u_1\nn : Type ?u.96210\no : Type ?u.96213\nm' : α → Type ?u.96218\nn' : α → Type ?u.96223\nR : Type v\ninst✝⁵ : CommRing R\nM✝ N : Matrix m m R\nb : m → α\ninst✝⁴ : DecidableEq m\ninst✝³ : Fintype m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nM : Matrix m m R\np : m → Prop\ninst✝ : DecidablePred p\nh : ∀ (i : m), p i → ∀ (j : m), ¬p j → M i j = 0\n⊢ det (toSquareBlockProp M fun i => ¬¬p i) = det (toSquareBlockProp M p)\n\nα : Type ?u.96201\nβ : Type ?u.96204\nm : Type u_1\nn : Type ?u.96210\no : Type ?u.96213\nm' : α → Type ?u.96218\nn' : α → Type ?u.96223\nR : Type v\ninst✝⁵ : CommRing R\nM✝ N : Matrix m m R\nb : m → α\ninst✝⁴ : DecidableEq m\ninst✝³ : Fintype m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nM : Matrix m m R\np : m → Prop\ninst✝ : DecidablePred p\nh : ∀ (i : m), p i → ∀ (j : m), ¬p j → M i j = 0\n⊢ ∀ (i : m), ¬¬p i → ∀ (j : m), ¬p j → M i j = 0",
"tactic": "exact equiv_block_det _ fun _ => not_not.symm"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.96201\nβ : Type ?u.96204\nm : Type u_1\nn : Type ?u.96210\no : Type ?u.96213\nm' : α → Type ?u.96218\nn' : α → Type ?u.96223\nR : Type v\ninst✝⁵ : CommRing R\nM✝ N : Matrix m m R\nb : m → α\ninst✝⁴ : DecidableEq m\ninst✝³ : Fintype m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nM : Matrix m m R\np : m → Prop\ninst✝ : DecidablePred p\nh : ∀ (i : m), p i → ∀ (j : m), ¬p j → M i j = 0\n⊢ ∀ (i : m), ¬¬p i → ∀ (j : m), ¬p j → M i j = 0",
"tactic": "simpa only [Classical.not_not] using h"
}
] |
[
196,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
190,
1
] |
Mathlib/NumberTheory/Fermat4.lean
|
Fermat42.mul
|
[
{
"state_after": "a b c k : ℤ\nhk0 : k ≠ 0\n⊢ a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2 ↔ k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2",
"state_before": "a b c k : ℤ\nhk0 : k ≠ 0\n⊢ Fermat42 a b c ↔ Fermat42 (k * a) (k * b) (k ^ 2 * c)",
"tactic": "delta Fermat42"
},
{
"state_after": "case mp\na b c k : ℤ\nhk0 : k ≠ 0\n⊢ a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2 → k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2\n\ncase mpr\na b c k : ℤ\nhk0 : k ≠ 0\n⊢ k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2 → a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2",
"state_before": "a b c k : ℤ\nhk0 : k ≠ 0\n⊢ a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2 ↔ k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2",
"tactic": "constructor"
},
{
"state_after": "case mp\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2\n⊢ k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2",
"state_before": "case mp\na b c k : ℤ\nhk0 : k ≠ 0\n⊢ a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2 → k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2",
"tactic": "intro f42"
},
{
"state_after": "case mp.left\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2\n⊢ k * a ≠ 0\n\ncase mp.right\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2\n⊢ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2",
"state_before": "case mp\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2\n⊢ k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2",
"tactic": "constructor"
},
{
"state_after": "case mp.right.left\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2\n⊢ k * b ≠ 0\n\ncase mp.right.right\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2\n⊢ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2",
"state_before": "case mp.right\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2\n⊢ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case mp.left\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2\n⊢ k * a ≠ 0",
"tactic": "exact mul_ne_zero hk0 f42.1"
},
{
"state_after": "no goals",
"state_before": "case mp.right.left\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2\n⊢ k * b ≠ 0",
"tactic": "exact mul_ne_zero hk0 f42.2.1"
},
{
"state_after": "case mp.right.right\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2\nH : a ^ 4 + b ^ 4 = c ^ 2\n⊢ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2",
"state_before": "case mp.right.right\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2\n⊢ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2",
"tactic": "have H : a ^ 4 + b ^ 4 = c ^ 2 := f42.2.2"
},
{
"state_after": "no goals",
"state_before": "case mp.right.right\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2\nH : a ^ 4 + b ^ 4 = c ^ 2\n⊢ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2",
"tactic": "linear_combination k ^ 4 * H"
},
{
"state_after": "case mpr\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2\n⊢ a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2",
"state_before": "case mpr\na b c k : ℤ\nhk0 : k ≠ 0\n⊢ k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2 → a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2",
"tactic": "intro f42"
},
{
"state_after": "case mpr.left\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2\n⊢ a ≠ 0\n\ncase mpr.right\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2\n⊢ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2",
"state_before": "case mpr\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2\n⊢ a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2",
"tactic": "constructor"
},
{
"state_after": "case mpr.right.left\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2\n⊢ b ≠ 0\n\ncase mpr.right.right\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2\n⊢ a ^ 4 + b ^ 4 = c ^ 2",
"state_before": "case mpr.right\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2\n⊢ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2",
"tactic": "constructor"
},
{
"state_after": "case mpr.right.right\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2\n⊢ k ^ 4 * (a ^ 4 + b ^ 4) = k ^ 4 * c ^ 2",
"state_before": "case mpr.right.right\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2\n⊢ a ^ 4 + b ^ 4 = c ^ 2",
"tactic": "apply (mul_right_inj' (pow_ne_zero 4 hk0)).mp"
},
{
"state_after": "no goals",
"state_before": "case mpr.right.right\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2\n⊢ k ^ 4 * (a ^ 4 + b ^ 4) = k ^ 4 * c ^ 2",
"tactic": "linear_combination f42.2.2"
},
{
"state_after": "no goals",
"state_before": "case mpr.left\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2\n⊢ a ≠ 0",
"tactic": "exact right_ne_zero_of_mul f42.1"
},
{
"state_after": "no goals",
"state_before": "case mpr.right.left\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2\n⊢ b ≠ 0",
"tactic": "exact right_ne_zero_of_mul f42.2.1"
}
] |
[
57,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Mathlib/Analysis/InnerProductSpace/PiL2.lean
|
OrthonormalBasis.coe_reindex
|
[] |
[
608,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
606,
11
] |
Mathlib/Order/Closure.lean
|
ClosureOperator.mem_closed_iff
|
[] |
[
174,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.val_add_eq_ite
|
[
{
"state_after": "no goals",
"state_before": "n✝ m n : ℕ\na b : Fin n\n⊢ ↑(a + b) = if n ≤ ↑a + ↑b then ↑a + ↑b - n else ↑a + ↑b",
"tactic": "rw [Fin.val_add, Nat.add_mod_eq_ite, Nat.mod_eq_of_lt (show ↑a < n from a.2),\n Nat.mod_eq_of_lt (show ↑b < n from b.2)]"
}
] |
[
712,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
709,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
lub_Iio_le
|
[] |
[
551,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
550,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.factorization_disjoint_of_coprime
|
[
{
"state_after": "no goals",
"state_before": "a b : ℕ\nhab : coprime a b\n⊢ _root_.Disjoint (factorization a).support (factorization b).support",
"tactic": "simpa only [support_factorization] using\n disjoint_toFinset_iff_disjoint.mpr (coprime_factors_disjoint hab)"
}
] |
[
784,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
781,
1
] |
Mathlib/Algebra/Star/StarAlgHom.lean
|
StarAlgHom.comp_assoc
|
[] |
[
460,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
458,
1
] |
Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean
|
GeneralizedContinuedFraction.succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squashSeq
|
[
{
"state_after": "case none\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = none\n⊢ convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1)\n\ncase some\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\n⊢ convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1)",
"state_before": "K : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\n⊢ convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1)",
"tactic": "cases' s_succ_nth_eq : s.get? <| n + 1 with gp_succ_n"
},
{
"state_after": "case some\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\n⊢ convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1)",
"state_before": "case none\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = none\n⊢ convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1)\n\ncase some\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\n⊢ convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1)",
"tactic": "case none =>\n rw [squashSeq_eq_self_of_terminated s_succ_nth_eq,\n convergents'Aux_stable_step_of_terminated s_succ_nth_eq]"
},
{
"state_after": "no goals",
"state_before": "case some\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\n⊢ convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1)",
"tactic": "case some =>\n induction' n with m IH generalizing s gp_succ_n\n case zero =>\n obtain ⟨gp_head, s_head_eq⟩ : ∃ gp_head, s.head = some gp_head\n exact s.ge_stable zero_le_one s_succ_nth_eq\n have : (squashSeq s 0).head = some ⟨gp_head.a, gp_head.b + gp_succ_n.a / gp_succ_n.b⟩ :=\n squashSeq_nth_of_not_terminated s_head_eq s_succ_nth_eq\n simp_all [convergents'Aux, Stream'.Seq.head, Stream'.Seq.get?_tail]\n case succ =>\n obtain ⟨gp_head, s_head_eq⟩ : ∃ gp_head, s.head = some gp_head\n exact s.ge_stable (m + 2).zero_le s_succ_nth_eq\n suffices\n gp_head.a / (gp_head.b + convergents'Aux s.tail (m + 2)) =\n convergents'Aux (squashSeq s (m + 1)) (m + 2)\n by simpa only [convergents'Aux, s_head_eq]\n have : convergents'Aux s.tail (m + 2) = convergents'Aux (squashSeq s.tail m) (m + 1) := by\n refine' IH gp_succ_n _\n simpa [Stream'.Seq.get?_tail] using s_succ_nth_eq\n have : (squashSeq s (m + 1)).head = some gp_head :=\n (squashSeq_nth_of_lt m.succ_pos).trans s_head_eq\n simp_all [convergents'Aux, squashSeq_succ_n_tail_eq_squashSeq_tail_n]"
},
{
"state_after": "no goals",
"state_before": "K : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = none\n⊢ convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1)",
"tactic": "rw [squashSeq_eq_self_of_terminated s_succ_nth_eq,\n convergents'Aux_stable_step_of_terminated s_succ_nth_eq]"
},
{
"state_after": "case zero\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.zero + 1) = some gp_succ_n\n⊢ convergents'Aux s (Nat.zero + 2) = convergents'Aux (squashSeq s Nat.zero) (Nat.zero + 1)\n\ncase succ\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n\n⊢ convergents'Aux s (Nat.succ m + 2) = convergents'Aux (squashSeq s (Nat.succ m)) (Nat.succ m + 1)",
"state_before": "K : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\n⊢ convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1)",
"tactic": "induction' n with m IH generalizing s gp_succ_n"
},
{
"state_after": "case succ\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n\n⊢ convergents'Aux s (Nat.succ m + 2) = convergents'Aux (squashSeq s (Nat.succ m)) (Nat.succ m + 1)",
"state_before": "case zero\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.zero + 1) = some gp_succ_n\n⊢ convergents'Aux s (Nat.zero + 2) = convergents'Aux (squashSeq s Nat.zero) (Nat.zero + 1)\n\ncase succ\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n\n⊢ convergents'Aux s (Nat.succ m + 2) = convergents'Aux (squashSeq s (Nat.succ m)) (Nat.succ m + 1)",
"tactic": "case zero =>\n obtain ⟨gp_head, s_head_eq⟩ : ∃ gp_head, s.head = some gp_head\n exact s.ge_stable zero_le_one s_succ_nth_eq\n have : (squashSeq s 0).head = some ⟨gp_head.a, gp_head.b + gp_succ_n.a / gp_succ_n.b⟩ :=\n squashSeq_nth_of_not_terminated s_head_eq s_succ_nth_eq\n simp_all [convergents'Aux, Stream'.Seq.head, Stream'.Seq.get?_tail]"
},
{
"state_after": "no goals",
"state_before": "case succ\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n\n⊢ convergents'Aux s (Nat.succ m + 2) = convergents'Aux (squashSeq s (Nat.succ m)) (Nat.succ m + 1)",
"tactic": "case succ =>\n obtain ⟨gp_head, s_head_eq⟩ : ∃ gp_head, s.head = some gp_head\n exact s.ge_stable (m + 2).zero_le s_succ_nth_eq\n suffices\n gp_head.a / (gp_head.b + convergents'Aux s.tail (m + 2)) =\n convergents'Aux (squashSeq s (m + 1)) (m + 2)\n by simpa only [convergents'Aux, s_head_eq]\n have : convergents'Aux s.tail (m + 2) = convergents'Aux (squashSeq s.tail m) (m + 1) := by\n refine' IH gp_succ_n _\n simpa [Stream'.Seq.get?_tail] using s_succ_nth_eq\n have : (squashSeq s (m + 1)).head = some gp_head :=\n (squashSeq_nth_of_lt m.succ_pos).trans s_head_eq\n simp_all [convergents'Aux, squashSeq_succ_n_tail_eq_squashSeq_tail_n]"
},
{
"state_after": "K : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.zero + 1) = some gp_succ_n\n⊢ ∃ gp_head, Stream'.Seq.head s = some gp_head\n\ncase intro\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.zero + 1) = some gp_succ_n\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n⊢ convergents'Aux s (Nat.zero + 2) = convergents'Aux (squashSeq s Nat.zero) (Nat.zero + 1)",
"state_before": "K : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.zero + 1) = some gp_succ_n\n⊢ convergents'Aux s (Nat.zero + 2) = convergents'Aux (squashSeq s Nat.zero) (Nat.zero + 1)",
"tactic": "obtain ⟨gp_head, s_head_eq⟩ : ∃ gp_head, s.head = some gp_head"
},
{
"state_after": "case intro\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.zero + 1) = some gp_succ_n\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n⊢ convergents'Aux s (Nat.zero + 2) = convergents'Aux (squashSeq s Nat.zero) (Nat.zero + 1)",
"state_before": "K : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.zero + 1) = some gp_succ_n\n⊢ ∃ gp_head, Stream'.Seq.head s = some gp_head\n\ncase intro\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.zero + 1) = some gp_succ_n\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n⊢ convergents'Aux s (Nat.zero + 2) = convergents'Aux (squashSeq s Nat.zero) (Nat.zero + 1)",
"tactic": "exact s.ge_stable zero_le_one s_succ_nth_eq"
},
{
"state_after": "case intro\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.zero + 1) = some gp_succ_n\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\nthis : Stream'.Seq.head (squashSeq s 0) = some { a := gp_head.a, b := gp_head.b + gp_succ_n.a / gp_succ_n.b }\n⊢ convergents'Aux s (Nat.zero + 2) = convergents'Aux (squashSeq s Nat.zero) (Nat.zero + 1)",
"state_before": "case intro\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.zero + 1) = some gp_succ_n\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n⊢ convergents'Aux s (Nat.zero + 2) = convergents'Aux (squashSeq s Nat.zero) (Nat.zero + 1)",
"tactic": "have : (squashSeq s 0).head = some ⟨gp_head.a, gp_head.b + gp_succ_n.a / gp_succ_n.b⟩ :=\n squashSeq_nth_of_not_terminated s_head_eq s_succ_nth_eq"
},
{
"state_after": "no goals",
"state_before": "case intro\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.zero + 1) = some gp_succ_n\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\nthis : Stream'.Seq.head (squashSeq s 0) = some { a := gp_head.a, b := gp_head.b + gp_succ_n.a / gp_succ_n.b }\n⊢ convergents'Aux s (Nat.zero + 2) = convergents'Aux (squashSeq s Nat.zero) (Nat.zero + 1)",
"tactic": "simp_all [convergents'Aux, Stream'.Seq.head, Stream'.Seq.get?_tail]"
},
{
"state_after": "K : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n\n⊢ ∃ gp_head, Stream'.Seq.head s = some gp_head\n\ncase intro\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n⊢ convergents'Aux s (Nat.succ m + 2) = convergents'Aux (squashSeq s (Nat.succ m)) (Nat.succ m + 1)",
"state_before": "K : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n\n⊢ convergents'Aux s (Nat.succ m + 2) = convergents'Aux (squashSeq s (Nat.succ m)) (Nat.succ m + 1)",
"tactic": "obtain ⟨gp_head, s_head_eq⟩ : ∃ gp_head, s.head = some gp_head"
},
{
"state_after": "case intro\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n⊢ convergents'Aux s (Nat.succ m + 2) = convergents'Aux (squashSeq s (Nat.succ m)) (Nat.succ m + 1)",
"state_before": "K : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n\n⊢ ∃ gp_head, Stream'.Seq.head s = some gp_head\n\ncase intro\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n⊢ convergents'Aux s (Nat.succ m + 2) = convergents'Aux (squashSeq s (Nat.succ m)) (Nat.succ m + 1)",
"tactic": "exact s.ge_stable (m + 2).zero_le s_succ_nth_eq"
},
{
"state_after": "case intro\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n⊢ gp_head.a / (gp_head.b + convergents'Aux (Stream'.Seq.tail s) (m + 2)) = convergents'Aux (squashSeq s (m + 1)) (m + 2)",
"state_before": "case intro\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n⊢ convergents'Aux s (Nat.succ m + 2) = convergents'Aux (squashSeq s (Nat.succ m)) (Nat.succ m + 1)",
"tactic": "suffices\n gp_head.a / (gp_head.b + convergents'Aux s.tail (m + 2)) =\n convergents'Aux (squashSeq s (m + 1)) (m + 2)\n by simpa only [convergents'Aux, s_head_eq]"
},
{
"state_after": "case intro\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\nthis : convergents'Aux (Stream'.Seq.tail s) (m + 2) = convergents'Aux (squashSeq (Stream'.Seq.tail s) m) (m + 1)\n⊢ gp_head.a / (gp_head.b + convergents'Aux (Stream'.Seq.tail s) (m + 2)) = convergents'Aux (squashSeq s (m + 1)) (m + 2)",
"state_before": "case intro\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n⊢ gp_head.a / (gp_head.b + convergents'Aux (Stream'.Seq.tail s) (m + 2)) = convergents'Aux (squashSeq s (m + 1)) (m + 2)",
"tactic": "have : convergents'Aux s.tail (m + 2) = convergents'Aux (squashSeq s.tail m) (m + 1) := by\n refine' IH gp_succ_n _\n simpa [Stream'.Seq.get?_tail] using s_succ_nth_eq"
},
{
"state_after": "case intro\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\nthis✝ : convergents'Aux (Stream'.Seq.tail s) (m + 2) = convergents'Aux (squashSeq (Stream'.Seq.tail s) m) (m + 1)\nthis : Stream'.Seq.head (squashSeq s (m + 1)) = some gp_head\n⊢ gp_head.a / (gp_head.b + convergents'Aux (Stream'.Seq.tail s) (m + 2)) = convergents'Aux (squashSeq s (m + 1)) (m + 2)",
"state_before": "case intro\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\nthis : convergents'Aux (Stream'.Seq.tail s) (m + 2) = convergents'Aux (squashSeq (Stream'.Seq.tail s) m) (m + 1)\n⊢ gp_head.a / (gp_head.b + convergents'Aux (Stream'.Seq.tail s) (m + 2)) = convergents'Aux (squashSeq s (m + 1)) (m + 2)",
"tactic": "have : (squashSeq s (m + 1)).head = some gp_head :=\n (squashSeq_nth_of_lt m.succ_pos).trans s_head_eq"
},
{
"state_after": "no goals",
"state_before": "case intro\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\nthis✝ : convergents'Aux (Stream'.Seq.tail s) (m + 2) = convergents'Aux (squashSeq (Stream'.Seq.tail s) m) (m + 1)\nthis : Stream'.Seq.head (squashSeq s (m + 1)) = some gp_head\n⊢ gp_head.a / (gp_head.b + convergents'Aux (Stream'.Seq.tail s) (m + 2)) = convergents'Aux (squashSeq s (m + 1)) (m + 2)",
"tactic": "simp_all [convergents'Aux, squashSeq_succ_n_tail_eq_squashSeq_tail_n]"
},
{
"state_after": "no goals",
"state_before": "K : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\nthis :\n gp_head.a / (gp_head.b + convergents'Aux (Stream'.Seq.tail s) (m + 2)) = convergents'Aux (squashSeq s (m + 1)) (m + 2)\n⊢ convergents'Aux s (Nat.succ m + 2) = convergents'Aux (squashSeq s (Nat.succ m)) (Nat.succ m + 1)",
"tactic": "simpa only [convergents'Aux, s_head_eq]"
},
{
"state_after": "K : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n⊢ Stream'.Seq.get? (Stream'.Seq.tail s) (m + 1) = some gp_succ_n",
"state_before": "K : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n⊢ convergents'Aux (Stream'.Seq.tail s) (m + 2) = convergents'Aux (squashSeq (Stream'.Seq.tail s) m) (m + 1)",
"tactic": "refine' IH gp_succ_n _"
},
{
"state_after": "no goals",
"state_before": "K : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns✝ : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_n✝ : Pair K\ns_succ_nth_eq✝ : Stream'.Seq.get? s✝ (n + 1) = some gp_succ_n✝\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n Stream'.Seq.get? s (m + 1) = some gp_succ_n → convergents'Aux s (m + 2) = convergents'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (Nat.succ m + 1) = some gp_succ_n\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n⊢ Stream'.Seq.get? (Stream'.Seq.tail s) (m + 1) = some gp_succ_n",
"tactic": "simpa [Stream'.Seq.get?_tail] using s_succ_nth_eq"
}
] |
[
186,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/Algebra/Ring/Equiv.lean
|
RingEquiv.surjective
|
[] |
[
335,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
11
] |
Mathlib/Order/Hom/Basic.lean
|
OrderIso.map_sup
|
[] |
[
1205,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1203,
1
] |
Mathlib/Data/List/Lex.lean
|
List.Lex.imp
|
[] |
[
148,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Algebra/Algebra/Spectrum.lean
|
spectrum.neg_eq
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nA : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nx : R\n⊢ x ∈ -σ a ↔ x ∈ σ (-a)",
"tactic": "simp only [mem_neg, mem_iff, map_neg, ← neg_add', IsUnit.neg_iff, sub_neg_eq_add]"
}
] |
[
311,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
309,
1
] |
Mathlib/Data/PNat/Factors.lean
|
PrimeMultiset.prod_ofNatMultiset
|
[
{
"state_after": "no goals",
"state_before": "v : Multiset ℕ\nh : ∀ (p : ℕ), p ∈ v → Nat.Prime p\n⊢ ↑(prod (ofNatMultiset v h)) = Multiset.prod v",
"tactic": "rw [coe_prod, to_ofNatMultiset]"
}
] |
[
170,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Algebra/Order/Interval.lean
|
Interval.length_zero
|
[] |
[
701,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
700,
1
] |
Mathlib/Algebra/Quaternion.lean
|
Quaternion.coe_normSq_add
|
[
{
"state_after": "no goals",
"state_before": "S : Type ?u.795476\nT : Type ?u.795479\nR : Type u_1\ninst✝ : CommRing R\nr x y z : R\na b c : ℍ[R]\n⊢ ↑(↑normSq (a + b)) = ↑(↑normSq a) + a * star b + b * star a + ↑(↑normSq b)",
"tactic": "simp only [star_add, ← self_mul_star, mul_add, add_mul, add_assoc, add_left_comm]"
}
] |
[
1239,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1238,
1
] |
Mathlib/Data/Nat/Bitwise.lean
|
Nat.lxor'_cancel_left
|
[
{
"state_after": "no goals",
"state_before": "n m : ℕ\n⊢ lxor' n (lxor' n m) = m",
"tactic": "rw [← lxor'_assoc, lxor'_self, zero_lxor']"
}
] |
[
248,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Ordinal.le_cof_type
|
[
{
"state_after": "case intro.intro\nα : Type u_1\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nc : Cardinal\nH : ∀ (S : Set α), Unbounded r S → c ≤ (#↑S)\nS : Set α\nh : Unbounded r S\n⊢ c ≤ (#↑S)",
"state_before": "α : Type u_1\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nc : Cardinal\n⊢ (∀ (S : Set α), Unbounded r S → c ≤ (#↑S)) → ∀ (b : Cardinal), b ∈ {c | ∃ S, Unbounded r S ∧ (#↑S) = c} → c ≤ b",
"tactic": "rintro H d ⟨S, h, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nc : Cardinal\nH : ∀ (S : Set α), Unbounded r S → c ≤ (#↑S)\nS : Set α\nh : Unbounded r S\n⊢ c ≤ (#↑S)",
"tactic": "exact H _ h"
}
] |
[
167,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
not_sbtw_self
|
[] |
[
353,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
352,
1
] |
Mathlib/Data/Stream/Init.lean
|
Stream'.nth_odd
|
[
{
"state_after": "α : Type u\nβ : Type v\nδ : Type w\nn : ℕ\ns : Stream' α\n⊢ nth (tail s) (2 * n) = nth s (2 * n + 1)",
"state_before": "α : Type u\nβ : Type v\nδ : Type w\nn : ℕ\ns : Stream' α\n⊢ nth (odd s) n = nth s (2 * n + 1)",
"tactic": "rw [odd_eq, nth_even]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nδ : Type w\nn : ℕ\ns : Stream' α\n⊢ nth (tail s) (2 * n) = nth s (2 * n + 1)",
"tactic": "rfl"
}
] |
[
511,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
510,
1
] |
Mathlib/Algebra/Quaternion.lean
|
QuaternionAlgebra.coe_imI
|
[] |
[
148,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
RingHom.range_top_of_surjective
|
[] |
[
1220,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1218,
1
] |
Mathlib/GroupTheory/Submonoid/Operations.lean
|
MonoidHom.mrange_eq_map
|
[] |
[
1058,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1057,
1
] |
Mathlib/RingTheory/Finiteness.lean
|
Module.Finite.iff_addGroup_fg
|
[] |
[
551,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
549,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.tsum_lt_tsum
|
[] |
[
1281,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1279,
1
] |
Mathlib/LinearAlgebra/Lagrange.lean
|
Lagrange.interpolate_singleton
|
[
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\n⊢ ↑(interpolate {i} v) r = ↑C (r i)",
"tactic": "rw [interpolate_apply, sum_singleton, basis_singleton, mul_one]"
}
] |
[
315,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
314,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
|
Real.log_zpow
|
[
{
"state_after": "case ofNat\nx✝ y x : ℝ\na✝ : ℕ\n⊢ log (x ^ Int.ofNat a✝) = ↑(Int.ofNat a✝) * log x\n\ncase negSucc\nx✝ y x : ℝ\na✝ : ℕ\n⊢ log (x ^ Int.negSucc a✝) = ↑(Int.negSucc a✝) * log x",
"state_before": "x✝ y x : ℝ\nn : ℤ\n⊢ log (x ^ n) = ↑n * log x",
"tactic": "induction n"
},
{
"state_after": "no goals",
"state_before": "case negSucc\nx✝ y x : ℝ\na✝ : ℕ\n⊢ log (x ^ Int.negSucc a✝) = ↑(Int.negSucc a✝) * log x",
"tactic": "rw [zpow_negSucc, log_inv, log_pow, Int.cast_negSucc, Nat.cast_add_one, neg_mul_eq_neg_mul]"
},
{
"state_after": "no goals",
"state_before": "case ofNat\nx✝ y x : ℝ\na✝ : ℕ\n⊢ log (x ^ Int.ofNat a✝) = ↑(Int.ofNat a✝) * log x",
"tactic": "rw [Int.ofNat_eq_coe, zpow_ofNat, log_pow, Int.cast_ofNat]"
}
] |
[
263,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.comp_neg
|
[
{
"state_after": "case h\nR : Type u_1\ninst✝¹⁶ : Ring R\nR₂ : Type u_2\ninst✝¹⁵ : Ring R₂\nR₃ : Type u_3\ninst✝¹⁴ : Ring R₃\nM : Type u_6\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : AddCommGroup M\nM₂ : Type u_4\ninst✝¹¹ : TopologicalSpace M₂\ninst✝¹⁰ : AddCommGroup M₂\nM₃ : Type u_5\ninst✝⁹ : TopologicalSpace M₃\ninst✝⁸ : AddCommGroup M₃\nM₄ : Type ?u.836875\ninst✝⁷ : TopologicalSpace M₄\ninst✝⁶ : AddCommGroup M₄\ninst✝⁵ : Module R M\ninst✝⁴ : Module R₂ M₂\ninst✝³ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : TopologicalAddGroup M₂\ninst✝ : TopologicalAddGroup M₃\ng : M₂ →SL[σ₂₃] M₃\nf : M →SL[σ₁₂] M₂\nx : M\n⊢ ↑(comp g (-f)) x = ↑(-comp g f) x",
"state_before": "R : Type u_1\ninst✝¹⁶ : Ring R\nR₂ : Type u_2\ninst✝¹⁵ : Ring R₂\nR₃ : Type u_3\ninst✝¹⁴ : Ring R₃\nM : Type u_6\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : AddCommGroup M\nM₂ : Type u_4\ninst✝¹¹ : TopologicalSpace M₂\ninst✝¹⁰ : AddCommGroup M₂\nM₃ : Type u_5\ninst✝⁹ : TopologicalSpace M₃\ninst✝⁸ : AddCommGroup M₃\nM₄ : Type ?u.836875\ninst✝⁷ : TopologicalSpace M₄\ninst✝⁶ : AddCommGroup M₄\ninst✝⁵ : Module R M\ninst✝⁴ : Module R₂ M₂\ninst✝³ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : TopologicalAddGroup M₂\ninst✝ : TopologicalAddGroup M₃\ng : M₂ →SL[σ₂₃] M₃\nf : M →SL[σ₁₂] M₂\n⊢ comp g (-f) = -comp g f",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\ninst✝¹⁶ : Ring R\nR₂ : Type u_2\ninst✝¹⁵ : Ring R₂\nR₃ : Type u_3\ninst✝¹⁴ : Ring R₃\nM : Type u_6\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : AddCommGroup M\nM₂ : Type u_4\ninst✝¹¹ : TopologicalSpace M₂\ninst✝¹⁰ : AddCommGroup M₂\nM₃ : Type u_5\ninst✝⁹ : TopologicalSpace M₃\ninst✝⁸ : AddCommGroup M₃\nM₄ : Type ?u.836875\ninst✝⁷ : TopologicalSpace M₄\ninst✝⁶ : AddCommGroup M₄\ninst✝⁵ : Module R M\ninst✝⁴ : Module R₂ M₂\ninst✝³ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : TopologicalAddGroup M₂\ninst✝ : TopologicalAddGroup M₃\ng : M₂ →SL[σ₂₃] M₃\nf : M →SL[σ₁₂] M₂\nx : M\n⊢ ↑(comp g (-f)) x = ↑(-comp g f) x",
"tactic": "simp"
}
] |
[
1401,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1398,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
Matrix.toLin'_apply
|
[] |
[
353,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
352,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
ContinuousMultilinearMap.norm_compContinuous_linearIsometry_le
|
[
{
"state_after": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\ng : ContinuousMultilinearMap 𝕜 E₁ G\nf : (i : ι) → E i →ₗᵢ[𝕜] E₁ i\nm : (i : ι) → E i\n⊢ ‖↑(compContinuousLinearMap g fun i => LinearIsometry.toContinuousLinearMap (f i)) m‖ ≤ ‖g‖ * ∏ i : ι, ‖m i‖",
"state_before": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\ng : ContinuousMultilinearMap 𝕜 E₁ G\nf : (i : ι) → E i →ₗᵢ[𝕜] E₁ i\n⊢ ‖compContinuousLinearMap g fun i => LinearIsometry.toContinuousLinearMap (f i)‖ ≤ ‖g‖",
"tactic": "refine op_norm_le_bound _ (norm_nonneg _) fun m => ?_"
},
{
"state_after": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\ng : ContinuousMultilinearMap 𝕜 E₁ G\nf : (i : ι) → E i →ₗᵢ[𝕜] E₁ i\nm : (i : ι) → E i\n⊢ ‖g‖ * ∏ i : ι, ‖↑((fun i => ↑((fun i => LinearIsometry.toContinuousLinearMap (f i)) i)) i) (m i)‖ ≤\n ‖g‖ * ∏ i : ι, ‖m i‖",
"state_before": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\ng : ContinuousMultilinearMap 𝕜 E₁ G\nf : (i : ι) → E i →ₗᵢ[𝕜] E₁ i\nm : (i : ι) → E i\n⊢ ‖↑(compContinuousLinearMap g fun i => LinearIsometry.toContinuousLinearMap (f i)) m‖ ≤ ‖g‖ * ∏ i : ι, ‖m i‖",
"tactic": "apply (g.le_op_norm _).trans _"
},
{
"state_after": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\ng : ContinuousMultilinearMap 𝕜 E₁ G\nf : (i : ι) → E i →ₗᵢ[𝕜] E₁ i\nm : (i : ι) → E i\n⊢ ‖g‖ * ∏ x : ι, ‖m x‖ ≤ ‖g‖ * ∏ i : ι, ‖m i‖",
"state_before": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\ng : ContinuousMultilinearMap 𝕜 E₁ G\nf : (i : ι) → E i →ₗᵢ[𝕜] E₁ i\nm : (i : ι) → E i\n⊢ ‖g‖ * ∏ i : ι, ‖↑((fun i => ↑((fun i => LinearIsometry.toContinuousLinearMap (f i)) i)) i) (m i)‖ ≤\n ‖g‖ * ∏ i : ι, ‖m i‖",
"tactic": "simp only [ContinuousLinearMap.coe_coe, LinearIsometry.coe_toContinuousLinearMap,\n LinearIsometry.norm_map]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\ng : ContinuousMultilinearMap 𝕜 E₁ G\nf : (i : ι) → E i →ₗᵢ[𝕜] E₁ i\nm : (i : ι) → E i\n⊢ ‖g‖ * ∏ x : ι, ‖m x‖ ≤ ‖g‖ * ∏ i : ι, ‖m i‖",
"tactic": "rfl"
}
] |
[
1160,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1154,
1
] |
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