file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/Prod/Basic.lean
|
Prod.exists
|
[] |
[
39,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
38,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.lf_of_lt
|
[] |
[
544,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
543,
1
] |
Mathlib/Analysis/InnerProductSpace/Calculus.lean
|
HasStrictFDerivAt.inner
|
[] |
[
107,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/RingTheory/Polynomial/Bernstein.lean
|
bernsteinPolynomial.linearIndependent_aux
|
[
{
"state_after": "case zero\nR : Type ?u.195275\ninst✝ : CommRing R\nn k : ℕ\nh✝ : k ≤ n + 1\nh : Nat.zero ≤ n + 1\n⊢ LinearIndependent ℚ fun ν => bernsteinPolynomial ℚ n ↑ν\n\ncase succ\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nih : k ≤ n + 1 → LinearIndependent ℚ fun ν => bernsteinPolynomial ℚ n ↑ν\nh : Nat.succ k ≤ n + 1\n⊢ LinearIndependent ℚ fun ν => bernsteinPolynomial ℚ n ↑ν",
"state_before": "R : Type ?u.195275\ninst✝ : CommRing R\nn k : ℕ\nh : k ≤ n + 1\n⊢ LinearIndependent ℚ fun ν => bernsteinPolynomial ℚ n ↑ν",
"tactic": "induction' k with k ih"
},
{
"state_after": "case zero\nR : Type ?u.195275\ninst✝ : CommRing R\nn k : ℕ\nh✝ : k ≤ n + 1\nh : Nat.zero ≤ n + 1\n⊢ LinearIndependent ℚ fun ν => bernsteinPolynomial ℚ n ↑ν",
"state_before": "case zero\nR : Type ?u.195275\ninst✝ : CommRing R\nn k : ℕ\nh✝ : k ≤ n + 1\nh : Nat.zero ≤ n + 1\n⊢ LinearIndependent ℚ fun ν => bernsteinPolynomial ℚ n ↑ν",
"tactic": "simp [Nat.zero_eq]"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type ?u.195275\ninst✝ : CommRing R\nn k : ℕ\nh✝ : k ≤ n + 1\nh : Nat.zero ≤ n + 1\n⊢ LinearIndependent ℚ fun ν => bernsteinPolynomial ℚ n ↑ν",
"tactic": "apply linearIndependent_empty_type"
},
{
"state_after": "case succ\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nih : k ≤ n + 1 → LinearIndependent ℚ fun ν => bernsteinPolynomial ℚ n ↑ν\nh : Nat.succ k ≤ n + 1\n⊢ LinearIndependent ℚ (Fin.init fun ν => bernsteinPolynomial ℚ n ↑ν) ∧\n ¬bernsteinPolynomial ℚ n ↑(Fin.last k) ∈ span ℚ (Set.range (Fin.init fun ν => bernsteinPolynomial ℚ n ↑ν))",
"state_before": "case succ\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nih : k ≤ n + 1 → LinearIndependent ℚ fun ν => bernsteinPolynomial ℚ n ↑ν\nh : Nat.succ k ≤ n + 1\n⊢ LinearIndependent ℚ fun ν => bernsteinPolynomial ℚ n ↑ν",
"tactic": "apply linearIndependent_fin_succ'.mpr"
},
{
"state_after": "case succ.left\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nih : k ≤ n + 1 → LinearIndependent ℚ fun ν => bernsteinPolynomial ℚ n ↑ν\nh : Nat.succ k ≤ n + 1\n⊢ LinearIndependent ℚ (Fin.init fun ν => bernsteinPolynomial ℚ n ↑ν)\n\ncase succ.right\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nih : k ≤ n + 1 → LinearIndependent ℚ fun ν => bernsteinPolynomial ℚ n ↑ν\nh : Nat.succ k ≤ n + 1\n⊢ ¬bernsteinPolynomial ℚ n ↑(Fin.last k) ∈ span ℚ (Set.range (Fin.init fun ν => bernsteinPolynomial ℚ n ↑ν))",
"state_before": "case succ\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nih : k ≤ n + 1 → LinearIndependent ℚ fun ν => bernsteinPolynomial ℚ n ↑ν\nh : Nat.succ k ≤ n + 1\n⊢ LinearIndependent ℚ (Fin.init fun ν => bernsteinPolynomial ℚ n ↑ν) ∧\n ¬bernsteinPolynomial ℚ n ↑(Fin.last k) ∈ span ℚ (Set.range (Fin.init fun ν => bernsteinPolynomial ℚ n ↑ν))",
"tactic": "fconstructor"
},
{
"state_after": "no goals",
"state_before": "case succ.left\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nih : k ≤ n + 1 → LinearIndependent ℚ fun ν => bernsteinPolynomial ℚ n ↑ν\nh : Nat.succ k ≤ n + 1\n⊢ LinearIndependent ℚ (Fin.init fun ν => bernsteinPolynomial ℚ n ↑ν)",
"tactic": "exact ih (le_of_lt h)"
},
{
"state_after": "case succ.right\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : Nat.succ k ≤ n + 1\n⊢ ¬bernsteinPolynomial ℚ n ↑(Fin.last k) ∈ span ℚ (Set.range (Fin.init fun ν => bernsteinPolynomial ℚ n ↑ν))",
"state_before": "case succ.right\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nih : k ≤ n + 1 → LinearIndependent ℚ fun ν => bernsteinPolynomial ℚ n ↑ν\nh : Nat.succ k ≤ n + 1\n⊢ ¬bernsteinPolynomial ℚ n ↑(Fin.last k) ∈ span ℚ (Set.range (Fin.init fun ν => bernsteinPolynomial ℚ n ↑ν))",
"tactic": "clear ih"
},
{
"state_after": "case succ.right\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\n⊢ ¬bernsteinPolynomial ℚ n ↑(Fin.last k) ∈ span ℚ (Set.range (Fin.init fun ν => bernsteinPolynomial ℚ n ↑ν))",
"state_before": "case succ.right\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : Nat.succ k ≤ n + 1\n⊢ ¬bernsteinPolynomial ℚ n ↑(Fin.last k) ∈ span ℚ (Set.range (Fin.init fun ν => bernsteinPolynomial ℚ n ↑ν))",
"tactic": "simp only [Nat.succ_eq_add_one, add_le_add_iff_right] at h"
},
{
"state_after": "case succ.right\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\n⊢ ¬bernsteinPolynomial ℚ n k ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑(↑Fin.castSucc k_1))",
"state_before": "case succ.right\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\n⊢ ¬bernsteinPolynomial ℚ n ↑(Fin.last k) ∈ span ℚ (Set.range (Fin.init fun ν => bernsteinPolynomial ℚ n ↑ν))",
"tactic": "simp only [Fin.val_last, Fin.init_def]"
},
{
"state_after": "case succ.right\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\n⊢ ¬bernsteinPolynomial ℚ n k ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)",
"state_before": "case succ.right\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\n⊢ ¬bernsteinPolynomial ℚ n k ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑(↑Fin.castSucc k_1))",
"tactic": "dsimp"
},
{
"state_after": "case succ.right\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\n⊢ ¬↑(derivative ^ (n - k)) (bernsteinPolynomial ℚ n k) ∈\n span ℚ (↑(derivative ^ (n - k)) '' Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)",
"state_before": "case succ.right\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\n⊢ ¬bernsteinPolynomial ℚ n k ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)",
"tactic": "apply not_mem_span_of_apply_not_mem_span_image (@Polynomial.derivative ℚ _ ^ (n - k))"
},
{
"state_after": "case succ.right\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\n⊢ ∀ (x : ℚ[X]),\n x ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1) →\n ¬↑(derivative ^ (n - k)) x = ↑(derivative ^ (n - k)) (bernsteinPolynomial ℚ n k)",
"state_before": "case succ.right\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\n⊢ ¬↑(derivative ^ (n - k)) (bernsteinPolynomial ℚ n k) ∈\n span ℚ (↑(derivative ^ (n - k)) '' Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)",
"tactic": "simp only [not_exists, not_and, Submodule.mem_map, Submodule.span_image]"
},
{
"state_after": "case succ.right\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\n⊢ ¬↑(derivative ^ (n - k)) p = ↑(derivative ^ (n - k)) (bernsteinPolynomial ℚ n k)",
"state_before": "case succ.right\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\n⊢ ∀ (x : ℚ[X]),\n x ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1) →\n ¬↑(derivative ^ (n - k)) x = ↑(derivative ^ (n - k)) (bernsteinPolynomial ℚ n k)",
"tactic": "intro p m"
},
{
"state_after": "R : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\n⊢ eval 1 (↑(derivative ^ (n - k)) p) ≠ eval 1 (↑(derivative ^ (n - k)) (bernsteinPolynomial ℚ n k))",
"state_before": "case succ.right\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\n⊢ ¬↑(derivative ^ (n - k)) p = ↑(derivative ^ (n - k)) (bernsteinPolynomial ℚ n k)",
"tactic": "apply_fun Polynomial.eval (1 : ℚ)"
},
{
"state_after": "R : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\n⊢ eval 1 ((↑derivative^[n - k]) p) ≠ eval 1 ((↑derivative^[n - k]) (bernsteinPolynomial ℚ n k))",
"state_before": "R : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\n⊢ eval 1 (↑(derivative ^ (n - k)) p) ≠ eval 1 (↑(derivative ^ (n - k)) (bernsteinPolynomial ℚ n k))",
"tactic": "simp only [LinearMap.pow_apply]"
},
{
"state_after": "R : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\n⊢ eval 1 ((↑derivative^[n - k]) p) = 0",
"state_before": "R : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\n⊢ eval 1 ((↑derivative^[n - k]) p) ≠ eval 1 ((↑derivative^[n - k]) (bernsteinPolynomial ℚ n k))",
"tactic": "suffices ((Polynomial.derivative^[n - k]) p).eval 1 = 0 by\n rw [this]\n exact (iterate_derivative_at_1_ne_zero ℚ n k h).symm"
},
{
"state_after": "case refine_1\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\n⊢ ∀ (x : ℚ[X]), (x ∈ Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1) → eval 1 ((↑derivative^[n - k]) x) = 0\n\ncase refine_2\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\n⊢ eval 1 ((↑derivative^[n - k]) 0) = 0\n\ncase refine_3\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\n⊢ ∀ (x y : ℚ[X]),\n eval 1 ((↑derivative^[n - k]) x) = 0 →\n eval 1 ((↑derivative^[n - k]) y) = 0 → eval 1 ((↑derivative^[n - k]) (x + y)) = 0\n\ncase refine_4\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\n⊢ ∀ (a : ℚ) (x : ℚ[X]), eval 1 ((↑derivative^[n - k]) x) = 0 → eval 1 ((↑derivative^[n - k]) (a • x)) = 0",
"state_before": "R : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\n⊢ eval 1 ((↑derivative^[n - k]) p) = 0",
"tactic": "refine span_induction m ?_ ?_ ?_ ?_"
},
{
"state_after": "R : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\nthis : eval 1 ((↑derivative^[n - k]) p) = 0\n⊢ 0 ≠ eval 1 ((↑derivative^[n - k]) (bernsteinPolynomial ℚ n k))",
"state_before": "R : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\nthis : eval 1 ((↑derivative^[n - k]) p) = 0\n⊢ eval 1 ((↑derivative^[n - k]) p) ≠ eval 1 ((↑derivative^[n - k]) (bernsteinPolynomial ℚ n k))",
"tactic": "rw [this]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\nthis : eval 1 ((↑derivative^[n - k]) p) = 0\n⊢ 0 ≠ eval 1 ((↑derivative^[n - k]) (bernsteinPolynomial ℚ n k))",
"tactic": "exact (iterate_derivative_at_1_ne_zero ℚ n k h).symm"
},
{
"state_after": "case refine_1\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\n⊢ ∀ (a : Fin k), eval 1 ((↑derivative^[n - k]) (bernsteinPolynomial ℚ n ↑a)) = 0",
"state_before": "case refine_1\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\n⊢ ∀ (x : ℚ[X]), (x ∈ Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1) → eval 1 ((↑derivative^[n - k]) x) = 0",
"tactic": "simp"
},
{
"state_after": "case refine_1.mk\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\na : ℕ\nw : a < k\n⊢ eval 1 ((↑derivative^[n - k]) (bernsteinPolynomial ℚ n ↑{ val := a, isLt := w })) = 0",
"state_before": "case refine_1\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\n⊢ ∀ (a : Fin k), eval 1 ((↑derivative^[n - k]) (bernsteinPolynomial ℚ n ↑a)) = 0",
"tactic": "rintro ⟨a, w⟩"
},
{
"state_after": "case refine_1.mk\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\na : ℕ\nw : a < k\n⊢ eval 1 ((↑derivative^[n - k]) (bernsteinPolynomial ℚ n a)) = 0",
"state_before": "case refine_1.mk\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\na : ℕ\nw : a < k\n⊢ eval 1 ((↑derivative^[n - k]) (bernsteinPolynomial ℚ n ↑{ val := a, isLt := w })) = 0",
"tactic": "simp only [Fin.val_mk]"
},
{
"state_after": "no goals",
"state_before": "case refine_1.mk\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\na : ℕ\nw : a < k\n⊢ eval 1 ((↑derivative^[n - k]) (bernsteinPolynomial ℚ n a)) = 0",
"tactic": "rw [iterate_derivative_at_1_eq_zero_of_lt ℚ n ((tsub_lt_tsub_iff_left_of_le h).mpr w)]"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\n⊢ eval 1 ((↑derivative^[n - k]) 0) = 0",
"tactic": "simp"
},
{
"state_after": "case refine_3\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\nx y : ℚ[X]\nhx : eval 1 ((↑derivative^[n - k]) x) = 0\nhy : eval 1 ((↑derivative^[n - k]) y) = 0\n⊢ eval 1 ((↑derivative^[n - k]) (x + y)) = 0",
"state_before": "case refine_3\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\n⊢ ∀ (x y : ℚ[X]),\n eval 1 ((↑derivative^[n - k]) x) = 0 →\n eval 1 ((↑derivative^[n - k]) y) = 0 → eval 1 ((↑derivative^[n - k]) (x + y)) = 0",
"tactic": "intro x y hx hy"
},
{
"state_after": "no goals",
"state_before": "case refine_3\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\nx y : ℚ[X]\nhx : eval 1 ((↑derivative^[n - k]) x) = 0\nhy : eval 1 ((↑derivative^[n - k]) y) = 0\n⊢ eval 1 ((↑derivative^[n - k]) (x + y)) = 0",
"tactic": "simp [hx, hy]"
},
{
"state_after": "case refine_4\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝¹ : k✝ ≤ n + 1\nk : ℕ\nh✝ : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\na : ℚ\nx : ℚ[X]\nh : eval 1 ((↑derivative^[n - k]) x) = 0\n⊢ eval 1 ((↑derivative^[n - k]) (a • x)) = 0",
"state_before": "case refine_4\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝ : k✝ ≤ n + 1\nk : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\n⊢ ∀ (a : ℚ) (x : ℚ[X]), eval 1 ((↑derivative^[n - k]) x) = 0 → eval 1 ((↑derivative^[n - k]) (a • x)) = 0",
"tactic": "intro a x h"
},
{
"state_after": "no goals",
"state_before": "case refine_4\nR : Type ?u.195275\ninst✝ : CommRing R\nn k✝ : ℕ\nh✝¹ : k✝ ≤ n + 1\nk : ℕ\nh✝ : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)\na : ℚ\nx : ℚ[X]\nh : eval 1 ((↑derivative^[n - k]) x) = 0\n⊢ eval 1 ((↑derivative^[n - k]) (a • x)) = 0",
"tactic": "simp [h]"
}
] |
[
281,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
250,
1
] |
Mathlib/Topology/Sets/Closeds.lean
|
TopologicalSpace.Closeds.compl_bijective
|
[] |
[
218,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Data/Subtype.lean
|
Subtype.trans
|
[] |
[
247,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
11
] |
Mathlib/Algebra/DualQuaternion.lean
|
Quaternion.fst_imI_dualNumberEquiv_symm
|
[] |
[
121,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/RingTheory/Valuation/Basic.lean
|
AddValuation.top_iff
|
[] |
[
738,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
737,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.inv_strictAnti
|
[
{
"state_after": "α : Type ?u.265925\nβ : Type ?u.265928\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na b : ℝ≥0∞\nh : a < b\n⊢ b⁻¹ < a⁻¹",
"state_before": "α : Type ?u.265925\nβ : Type ?u.265928\na b c d : ℝ≥0∞\nr p q : ℝ≥0\n⊢ StrictAnti Inv.inv",
"tactic": "intro a b h"
},
{
"state_after": "case intro\nα : Type ?u.265925\nβ : Type ?u.265928\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\nb : ℝ≥0∞\na : ℝ≥0\nh : ↑a < b\n⊢ b⁻¹ < (↑a)⁻¹",
"state_before": "α : Type ?u.265925\nβ : Type ?u.265928\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na b : ℝ≥0∞\nh : a < b\n⊢ b⁻¹ < a⁻¹",
"tactic": "lift a to ℝ≥0 using h.ne_top"
},
{
"state_after": "case intro.top\nα : Type ?u.265925\nβ : Type ?u.265928\na✝ b c d : ℝ≥0∞\nr p q a : ℝ≥0\nh : ↑a < ⊤\n⊢ ⊤⁻¹ < (↑a)⁻¹\n\ncase intro.coe\nα : Type ?u.265925\nβ : Type ?u.265928\na✝ b c d : ℝ≥0∞\nr p q a x✝ : ℝ≥0\nh : ↑a < ↑x✝\n⊢ (↑x✝)⁻¹ < (↑a)⁻¹",
"state_before": "case intro\nα : Type ?u.265925\nβ : Type ?u.265928\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\nb : ℝ≥0∞\na : ℝ≥0\nh : ↑a < b\n⊢ b⁻¹ < (↑a)⁻¹",
"tactic": "induction b using recTopCoe"
},
{
"state_after": "case intro.coe\nα : Type ?u.265925\nβ : Type ?u.265928\na✝ b c d : ℝ≥0∞\nr p q a x✝ : ℝ≥0\nh : a < x✝\n⊢ (↑x✝)⁻¹ < (↑a)⁻¹",
"state_before": "case intro.coe\nα : Type ?u.265925\nβ : Type ?u.265928\na✝ b c d : ℝ≥0∞\nr p q a x✝ : ℝ≥0\nh : ↑a < ↑x✝\n⊢ (↑x✝)⁻¹ < (↑a)⁻¹",
"tactic": "rw [coe_lt_coe] at h"
},
{
"state_after": "case intro.coe.inl\nα : Type ?u.265925\nβ : Type ?u.265928\na b c d : ℝ≥0∞\nr p q x✝ : ℝ≥0\nh : 0 < x✝\n⊢ (↑x✝)⁻¹ < (↑0)⁻¹\n\ncase intro.coe.inr\nα : Type ?u.265925\nβ : Type ?u.265928\na✝ b c d : ℝ≥0∞\nr p q a x✝ : ℝ≥0\nh : a < x✝\nha : a ≠ 0\n⊢ (↑x✝)⁻¹ < (↑a)⁻¹",
"state_before": "case intro.coe\nα : Type ?u.265925\nβ : Type ?u.265928\na✝ b c d : ℝ≥0∞\nr p q a x✝ : ℝ≥0\nh : a < x✝\n⊢ (↑x✝)⁻¹ < (↑a)⁻¹",
"tactic": "rcases eq_or_ne a 0 with (rfl | ha)"
},
{
"state_after": "case intro.coe.inr\nα : Type ?u.265925\nβ : Type ?u.265928\na✝ b c d : ℝ≥0∞\nr p q a x✝ : ℝ≥0\nh : a < x✝\nha : a ≠ 0\n⊢ x✝⁻¹ < a⁻¹",
"state_before": "case intro.coe.inr\nα : Type ?u.265925\nβ : Type ?u.265928\na✝ b c d : ℝ≥0∞\nr p q a x✝ : ℝ≥0\nh : a < x✝\nha : a ≠ 0\n⊢ (↑x✝)⁻¹ < (↑a)⁻¹",
"tactic": "rw [← coe_inv h.ne_bot, ← coe_inv ha, coe_lt_coe]"
},
{
"state_after": "no goals",
"state_before": "case intro.coe.inr\nα : Type ?u.265925\nβ : Type ?u.265928\na✝ b c d : ℝ≥0∞\nr p q a x✝ : ℝ≥0\nh : a < x✝\nha : a ≠ 0\n⊢ x✝⁻¹ < a⁻¹",
"tactic": "exact NNReal.inv_lt_inv ha h"
},
{
"state_after": "no goals",
"state_before": "case intro.top\nα : Type ?u.265925\nβ : Type ?u.265928\na✝ b c d : ℝ≥0∞\nr p q a : ℝ≥0\nh : ↑a < ⊤\n⊢ ⊤⁻¹ < (↑a)⁻¹",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case intro.coe.inl\nα : Type ?u.265925\nβ : Type ?u.265928\na b c d : ℝ≥0∞\nr p q x✝ : ℝ≥0\nh : 0 < x✝\n⊢ (↑x✝)⁻¹ < (↑0)⁻¹",
"tactic": "simp [h]"
}
] |
[
1474,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1467,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isBigO_neg_right
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.131381\nE : Type u_2\nF : Type ?u.131387\nG : Type ?u.131390\nE' : Type ?u.131393\nF' : Type u_3\nG' : Type ?u.131399\nE'' : Type ?u.131402\nF'' : Type ?u.131405\nG'' : Type ?u.131408\nR : Type ?u.131411\nR' : Type ?u.131414\n𝕜 : Type ?u.131417\n𝕜' : Type ?u.131420\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\n⊢ (∃ c, IsBigOWith c l f fun x => -g' x) ↔ ∃ c, IsBigOWith c l f g'",
"state_before": "α : Type u_1\nβ : Type ?u.131381\nE : Type u_2\nF : Type ?u.131387\nG : Type ?u.131390\nE' : Type ?u.131393\nF' : Type u_3\nG' : Type ?u.131399\nE'' : Type ?u.131402\nF'' : Type ?u.131405\nG'' : Type ?u.131408\nR : Type ?u.131411\nR' : Type ?u.131414\n𝕜 : Type ?u.131417\n𝕜' : Type ?u.131420\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\n⊢ (f =O[l] fun x => -g' x) ↔ f =O[l] g'",
"tactic": "simp only [IsBigO_def]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.131381\nE : Type u_2\nF : Type ?u.131387\nG : Type ?u.131390\nE' : Type ?u.131393\nF' : Type u_3\nG' : Type ?u.131399\nE'' : Type ?u.131402\nF'' : Type ?u.131405\nG'' : Type ?u.131408\nR : Type ?u.131411\nR' : Type ?u.131414\n𝕜 : Type ?u.131417\n𝕜' : Type ?u.131420\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\n⊢ (∃ c, IsBigOWith c l f fun x => -g' x) ↔ ∃ c, IsBigOWith c l f g'",
"tactic": "exact exists_congr fun _ => isBigOWith_neg_right"
}
] |
[
874,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
872,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.filter_erase
|
[
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.390807\nγ : Type ?u.390810\np q : α → Prop\ninst✝² : DecidablePred p\ninst✝¹ : DecidablePred q\ninst✝ : DecidableEq α\na : α\ns : Finset α\nx : α\n⊢ x ∈ filter p (erase s a) ↔ x ∈ erase (filter p s) a",
"state_before": "α : Type u_1\nβ : Type ?u.390807\nγ : Type ?u.390810\np q : α → Prop\ninst✝² : DecidablePred p\ninst✝¹ : DecidablePred q\ninst✝ : DecidableEq α\na : α\ns : Finset α\n⊢ filter p (erase s a) = erase (filter p s) a",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.390807\nγ : Type ?u.390810\np q : α → Prop\ninst✝² : DecidablePred p\ninst✝¹ : DecidablePred q\ninst✝ : DecidableEq α\na : α\ns : Finset α\nx : α\n⊢ x ∈ filter p (erase s a) ↔ x ∈ erase (filter p s) a",
"tactic": "simp only [and_assoc, mem_filter, iff_self_iff, mem_erase]"
}
] |
[
2847,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2845,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.frequently_atBot'
|
[] |
[
340,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.sin_zero
|
[
{
"state_after": "no goals",
"state_before": "⊢ sin 0 = 0",
"tactic": "rw [← coe_zero, sin_coe, Real.sin_zero]"
}
] |
[
358,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
358,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Mul.lean
|
HasFDerivWithinAt.mul
|
[
{
"state_after": "case h.e'_10.h.e'_6\n𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type ?u.715008\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type ?u.715103\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nG' : Type ?u.715198\ninst✝⁵ : NormedAddCommGroup G'\ninst✝⁴ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\n𝔸 : Type ?u.717333\n𝔸' : Type u_3\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedCommRing 𝔸'\ninst✝¹ : NormedAlgebra 𝕜 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸'\na b : E → 𝔸\na' b' : E →L[𝕜] 𝔸\nc d : E → 𝔸'\nc' d' : E →L[𝕜] 𝔸'\nhc : HasFDerivWithinAt c c' s x\nhd : HasFDerivWithinAt d d' s x\n⊢ d x • c' = smulRight c' (d x)",
"state_before": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type ?u.715008\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type ?u.715103\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nG' : Type ?u.715198\ninst✝⁵ : NormedAddCommGroup G'\ninst✝⁴ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\n𝔸 : Type ?u.717333\n𝔸' : Type u_3\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedCommRing 𝔸'\ninst✝¹ : NormedAlgebra 𝕜 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸'\na b : E → 𝔸\na' b' : E →L[𝕜] 𝔸\nc d : E → 𝔸'\nc' d' : E →L[𝕜] 𝔸'\nhc : HasFDerivWithinAt c c' s x\nhd : HasFDerivWithinAt d d' s x\n⊢ HasFDerivWithinAt (fun y => c y * d y) (c x • d' + d x • c') s x",
"tactic": "convert hc.mul' hd"
},
{
"state_after": "case h.e'_10.h.e'_6.h\n𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type ?u.715008\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type ?u.715103\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nG' : Type ?u.715198\ninst✝⁵ : NormedAddCommGroup G'\ninst✝⁴ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\n𝔸 : Type ?u.717333\n𝔸' : Type u_3\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedCommRing 𝔸'\ninst✝¹ : NormedAlgebra 𝕜 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸'\na b : E → 𝔸\na' b' : E →L[𝕜] 𝔸\nc d : E → 𝔸'\nc' d' : E →L[𝕜] 𝔸'\nhc : HasFDerivWithinAt c c' s x\nhd : HasFDerivWithinAt d d' s x\nz : E\n⊢ ↑(d x • c') z = ↑(smulRight c' (d x)) z",
"state_before": "case h.e'_10.h.e'_6\n𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type ?u.715008\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type ?u.715103\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nG' : Type ?u.715198\ninst✝⁵ : NormedAddCommGroup G'\ninst✝⁴ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\n𝔸 : Type ?u.717333\n𝔸' : Type u_3\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedCommRing 𝔸'\ninst✝¹ : NormedAlgebra 𝕜 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸'\na b : E → 𝔸\na' b' : E →L[𝕜] 𝔸\nc d : E → 𝔸'\nc' d' : E →L[𝕜] 𝔸'\nhc : HasFDerivWithinAt c c' s x\nhd : HasFDerivWithinAt d d' s x\n⊢ d x • c' = smulRight c' (d x)",
"tactic": "ext z"
},
{
"state_after": "no goals",
"state_before": "case h.e'_10.h.e'_6.h\n𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type ?u.715008\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type ?u.715103\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nG' : Type ?u.715198\ninst✝⁵ : NormedAddCommGroup G'\ninst✝⁴ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\n𝔸 : Type ?u.717333\n𝔸' : Type u_3\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedCommRing 𝔸'\ninst✝¹ : NormedAlgebra 𝕜 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸'\na b : E → 𝔸\na' b' : E →L[𝕜] 𝔸\nc d : E → 𝔸'\nc' d' : E →L[𝕜] 𝔸'\nhc : HasFDerivWithinAt c c' s x\nhd : HasFDerivWithinAt d d' s x\nz : E\n⊢ ↑(d x • c') z = ↑(smulRight c' (d x)) z",
"tactic": "apply mul_comm"
}
] |
[
307,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
303,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean
|
AffineIsometry.map_ne
|
[] |
[
173,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/Topology/Algebra/Order/Group.lean
|
continuous_abs
|
[] |
[
63,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
Wbtw.left_ne_right_of_ne_right
|
[
{
"state_after": "R : Type u_1\nV : Type u_2\nV' : Type ?u.162563\nP : Type u_3\nP' : Type ?u.162569\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y : P\nh : Wbtw R x y x\nhne : y ≠ x\n⊢ False",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.162563\nP : Type u_3\nP' : Type ?u.162569\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y z : P\nh : Wbtw R x y z\nhne : y ≠ z\n⊢ x ≠ z",
"tactic": "rintro rfl"
},
{
"state_after": "R : Type u_1\nV : Type u_2\nV' : Type ?u.162563\nP : Type u_3\nP' : Type ?u.162569\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y : P\nh : y = x\nhne : y ≠ x\n⊢ False",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.162563\nP : Type u_3\nP' : Type ?u.162569\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y : P\nh : Wbtw R x y x\nhne : y ≠ x\n⊢ False",
"tactic": "rw [wbtw_self_iff] at h"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.162563\nP : Type u_3\nP' : Type ?u.162569\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y : P\nh : y = x\nhne : y ≠ x\n⊢ False",
"tactic": "exact hne h"
}
] |
[
331,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Mathlib/RingTheory/Localization/LocalizationLocalization.lean
|
IsLocalization.isLocalization_of_is_exists_mul_mem
|
[
{
"state_after": "case intro\nR : Type u_1\ninst✝⁶ : CommRing R\nM✝ : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nP : Type ?u.656657\ninst✝³ : CommRing P\nN✝ : Submonoid S\nT : Type ?u.656821\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh : M ≤ N\nh' : ∀ (x : { x // x ∈ N }), ∃ m, m * ↑x ∈ M\ny : { x // x ∈ N }\nm : R\nhm : m * ↑y ∈ M\n⊢ IsUnit (↑(algebraMap R S) ↑y)",
"state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nM✝ : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nP : Type ?u.656657\ninst✝³ : CommRing P\nN✝ : Submonoid S\nT : Type ?u.656821\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh : M ≤ N\nh' : ∀ (x : { x // x ∈ N }), ∃ m, m * ↑x ∈ M\ny : { x // x ∈ N }\n⊢ IsUnit (↑(algebraMap R S) ↑y)",
"tactic": "obtain ⟨m, hm⟩ := h' y"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝⁶ : CommRing R\nM✝ : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nP : Type ?u.656657\ninst✝³ : CommRing P\nN✝ : Submonoid S\nT : Type ?u.656821\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh : M ≤ N\nh' : ∀ (x : { x // x ∈ N }), ∃ m, m * ↑x ∈ M\ny : { x // x ∈ N }\nm : R\nhm : m * ↑y ∈ M\nthis : IsUnit (↑(algebraMap R S) ↑{ val := m * ↑y, property := hm })\n⊢ IsUnit (↑(algebraMap R S) ↑y)",
"state_before": "case intro\nR : Type u_1\ninst✝⁶ : CommRing R\nM✝ : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nP : Type ?u.656657\ninst✝³ : CommRing P\nN✝ : Submonoid S\nT : Type ?u.656821\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh : M ≤ N\nh' : ∀ (x : { x // x ∈ N }), ∃ m, m * ↑x ∈ M\ny : { x // x ∈ N }\nm : R\nhm : m * ↑y ∈ M\n⊢ IsUnit (↑(algebraMap R S) ↑y)",
"tactic": "have := IsLocalization.map_units S ⟨_, hm⟩"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝⁶ : CommRing R\nM✝ : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nP : Type ?u.656657\ninst✝³ : CommRing P\nN✝ : Submonoid S\nT : Type ?u.656821\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh : M ≤ N\nh' : ∀ (x : { x // x ∈ N }), ∃ m, m * ↑x ∈ M\ny : { x // x ∈ N }\nm : R\nhm : m * ↑y ∈ M\nthis : IsUnit (↑(algebraMap R S) m * ↑(algebraMap R S) ↑y)\n⊢ IsUnit (↑(algebraMap R S) ↑y)",
"state_before": "case intro\nR : Type u_1\ninst✝⁶ : CommRing R\nM✝ : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nP : Type ?u.656657\ninst✝³ : CommRing P\nN✝ : Submonoid S\nT : Type ?u.656821\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh : M ≤ N\nh' : ∀ (x : { x // x ∈ N }), ∃ m, m * ↑x ∈ M\ny : { x // x ∈ N }\nm : R\nhm : m * ↑y ∈ M\nthis : IsUnit (↑(algebraMap R S) ↑{ val := m * ↑y, property := hm })\n⊢ IsUnit (↑(algebraMap R S) ↑y)",
"tactic": "erw [map_mul] at this"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u_1\ninst✝⁶ : CommRing R\nM✝ : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nP : Type ?u.656657\ninst✝³ : CommRing P\nN✝ : Submonoid S\nT : Type ?u.656821\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh : M ≤ N\nh' : ∀ (x : { x // x ∈ N }), ∃ m, m * ↑x ∈ M\ny : { x // x ∈ N }\nm : R\nhm : m * ↑y ∈ M\nthis : IsUnit (↑(algebraMap R S) m * ↑(algebraMap R S) ↑y)\n⊢ IsUnit (↑(algebraMap R S) ↑y)",
"tactic": "exact (IsUnit.mul_iff.mp this).2"
},
{
"state_after": "case intro.mk\nR : Type u_1\ninst✝⁶ : CommRing R\nM✝ : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nP : Type ?u.656657\ninst✝³ : CommRing P\nN✝ : Submonoid S\nT : Type ?u.656821\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh : M ≤ N\nh' : ∀ (x : { x // x ∈ N }), ∃ m, m * ↑x ∈ M\nz : S\ny : R\ns : { x // x ∈ M }\ne : z * ↑(algebraMap R S) ↑(y, s).snd = ↑(algebraMap R S) (y, s).fst\n⊢ ∃ x, z * ↑(algebraMap R S) ↑x.snd = ↑(algebraMap R S) x.fst",
"state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nM✝ : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nP : Type ?u.656657\ninst✝³ : CommRing P\nN✝ : Submonoid S\nT : Type ?u.656821\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh : M ≤ N\nh' : ∀ (x : { x // x ∈ N }), ∃ m, m * ↑x ∈ M\nz : S\n⊢ ∃ x, z * ↑(algebraMap R S) ↑x.snd = ↑(algebraMap R S) x.fst",
"tactic": "obtain ⟨⟨y, s⟩, e⟩ := IsLocalization.surj M z"
},
{
"state_after": "no goals",
"state_before": "case intro.mk\nR : Type u_1\ninst✝⁶ : CommRing R\nM✝ : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nP : Type ?u.656657\ninst✝³ : CommRing P\nN✝ : Submonoid S\nT : Type ?u.656821\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh : M ≤ N\nh' : ∀ (x : { x // x ∈ N }), ∃ m, m * ↑x ∈ M\nz : S\ny : R\ns : { x // x ∈ M }\ne : z * ↑(algebraMap R S) ↑(y, s).snd = ↑(algebraMap R S) (y, s).fst\n⊢ ∃ x, z * ↑(algebraMap R S) ↑x.snd = ↑(algebraMap R S) x.fst",
"tactic": "exact ⟨⟨y, _, h s.prop⟩, e⟩"
},
{
"state_after": "R : Type u_1\ninst✝⁶ : CommRing R\nM✝ : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nP : Type ?u.656657\ninst✝³ : CommRing P\nN✝ : Submonoid S\nT : Type ?u.656821\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh : M ≤ N\nh' : ∀ (x : { x // x ∈ N }), ∃ m, m * ↑x ∈ M\nx✝¹ x✝ : R\n⊢ (∃ c, ↑c * x✝¹ = ↑c * x✝) ↔ ∃ c, ↑c * x✝¹ = ↑c * x✝",
"state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nM✝ : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nP : Type ?u.656657\ninst✝³ : CommRing P\nN✝ : Submonoid S\nT : Type ?u.656821\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh : M ≤ N\nh' : ∀ (x : { x // x ∈ N }), ∃ m, m * ↑x ∈ M\nx✝¹ x✝ : R\n⊢ ↑(algebraMap R S) x✝¹ = ↑(algebraMap R S) x✝ ↔ ∃ c, ↑c * x✝¹ = ↑c * x✝",
"tactic": "rw [IsLocalization.eq_iff_exists M]"
},
{
"state_after": "R : Type u_1\ninst✝⁶ : CommRing R\nM✝ : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nP : Type ?u.656657\ninst✝³ : CommRing P\nN✝ : Submonoid S\nT : Type ?u.656821\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh : M ≤ N\nh' : ∀ (x : { x // x ∈ N }), ∃ m, m * ↑x ∈ M\nx✝¹ x✝ : R\n⊢ (∃ c, ↑c * x✝¹ = ↑c * x✝) → ∃ c, ↑c * x✝¹ = ↑c * x✝",
"state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nM✝ : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nP : Type ?u.656657\ninst✝³ : CommRing P\nN✝ : Submonoid S\nT : Type ?u.656821\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh : M ≤ N\nh' : ∀ (x : { x // x ∈ N }), ∃ m, m * ↑x ∈ M\nx✝¹ x✝ : R\n⊢ (∃ c, ↑c * x✝¹ = ↑c * x✝) ↔ ∃ c, ↑c * x✝¹ = ↑c * x✝",
"tactic": "refine ⟨fun ⟨x, hx⟩ => ⟨⟨_, h x.prop⟩, hx⟩, ?_⟩"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝⁶ : CommRing R\nM✝ : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nP : Type ?u.656657\ninst✝³ : CommRing P\nN✝ : Submonoid S\nT : Type ?u.656821\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh✝ : M ≤ N\nh' : ∀ (x : { x // x ∈ N }), ∃ m, m * ↑x ∈ M\nx✝¹ x✝ : R\nx : { x // x ∈ N }\nh : ↑x * x✝¹ = ↑x * x✝\n⊢ ∃ c, ↑c * x✝¹ = ↑c * x✝",
"state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nM✝ : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nP : Type ?u.656657\ninst✝³ : CommRing P\nN✝ : Submonoid S\nT : Type ?u.656821\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh : M ≤ N\nh' : ∀ (x : { x // x ∈ N }), ∃ m, m * ↑x ∈ M\nx✝¹ x✝ : R\n⊢ (∃ c, ↑c * x✝¹ = ↑c * x✝) → ∃ c, ↑c * x✝¹ = ↑c * x✝",
"tactic": "rintro ⟨x, h⟩"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nM✝ : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nP : Type ?u.656657\ninst✝³ : CommRing P\nN✝ : Submonoid S\nT : Type ?u.656821\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh✝ : M ≤ N\nh' : ∀ (x : { x // x ∈ N }), ∃ m, m * ↑x ∈ M\nx✝¹ x✝ : R\nx : { x // x ∈ N }\nh : ↑x * x✝¹ = ↑x * x✝\nm : R\nhm : m * ↑x ∈ M\n⊢ ∃ c, ↑c * x✝¹ = ↑c * x✝",
"state_before": "case intro\nR : Type u_1\ninst✝⁶ : CommRing R\nM✝ : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nP : Type ?u.656657\ninst✝³ : CommRing P\nN✝ : Submonoid S\nT : Type ?u.656821\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh✝ : M ≤ N\nh' : ∀ (x : { x // x ∈ N }), ∃ m, m * ↑x ∈ M\nx✝¹ x✝ : R\nx : { x // x ∈ N }\nh : ↑x * x✝¹ = ↑x * x✝\n⊢ ∃ c, ↑c * x✝¹ = ↑c * x✝",
"tactic": "obtain ⟨m, hm⟩ := h' x"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nM✝ : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nP : Type ?u.656657\ninst✝³ : CommRing P\nN✝ : Submonoid S\nT : Type ?u.656821\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh✝ : M ≤ N\nh' : ∀ (x : { x // x ∈ N }), ∃ m, m * ↑x ∈ M\nx✝¹ x✝ : R\nx : { x // x ∈ N }\nh : ↑x * x✝¹ = ↑x * x✝\nm : R\nhm : m * ↑x ∈ M\n⊢ ↑{ val := m * ↑x, property := hm } * x✝¹ = ↑{ val := m * ↑x, property := hm } * x✝",
"state_before": "case intro.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nM✝ : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nP : Type ?u.656657\ninst✝³ : CommRing P\nN✝ : Submonoid S\nT : Type ?u.656821\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh✝ : M ≤ N\nh' : ∀ (x : { x // x ∈ N }), ∃ m, m * ↑x ∈ M\nx✝¹ x✝ : R\nx : { x // x ∈ N }\nh : ↑x * x✝¹ = ↑x * x✝\nm : R\nhm : m * ↑x ∈ M\n⊢ ∃ c, ↑c * x✝¹ = ↑c * x✝",
"tactic": "refine ⟨⟨_, hm⟩, ?_⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nM✝ : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nP : Type ?u.656657\ninst✝³ : CommRing P\nN✝ : Submonoid S\nT : Type ?u.656821\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh✝ : M ≤ N\nh' : ∀ (x : { x // x ∈ N }), ∃ m, m * ↑x ∈ M\nx✝¹ x✝ : R\nx : { x // x ∈ N }\nh : ↑x * x✝¹ = ↑x * x✝\nm : R\nhm : m * ↑x ∈ M\n⊢ ↑{ val := m * ↑x, property := hm } * x✝¹ = ↑{ val := m * ↑x, property := hm } * x✝",
"tactic": "simp [h, mul_assoc]"
}
] |
[
266,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
250,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.Hom.map_adj
|
[] |
[
1686,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1685,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Split.lean
|
BoxIntegral.Box.disjoint_splitLower_splitUpper
|
[
{
"state_after": "ι : Type u_1\nM : Type ?u.20595\nn : ℕ\nI✝ : Box ι\ni✝ : ι\nx✝ : ℝ\ny : ι → ℝ\nI : Box ι\ni : ι\nx : ℝ\n⊢ Disjoint (↑I ∩ {y | y i ≤ x}) (↑I ∩ {y | x < y i})",
"state_before": "ι : Type u_1\nM : Type ?u.20595\nn : ℕ\nI✝ : Box ι\ni✝ : ι\nx✝ : ℝ\ny : ι → ℝ\nI : Box ι\ni : ι\nx : ℝ\n⊢ Disjoint (splitLower I i x) (splitUpper I i x)",
"tactic": "rw [← disjoint_withBotCoe, coe_splitLower, coe_splitUpper]"
},
{
"state_after": "ι : Type u_1\nM : Type ?u.20595\nn : ℕ\nI✝ : Box ι\ni✝ : ι\nx✝ : ℝ\ny : ι → ℝ\nI : Box ι\ni : ι\nx : ℝ\n⊢ Disjoint {y | y i ≤ x} {y | x < y i}",
"state_before": "ι : Type u_1\nM : Type ?u.20595\nn : ℕ\nI✝ : Box ι\ni✝ : ι\nx✝ : ℝ\ny : ι → ℝ\nI : Box ι\ni : ι\nx : ℝ\n⊢ Disjoint (↑I ∩ {y | y i ≤ x}) (↑I ∩ {y | x < y i})",
"tactic": "refine' (Disjoint.inf_left' _ _).inf_right' _"
},
{
"state_after": "ι : Type u_1\nM : Type ?u.20595\nn : ℕ\nI✝ : Box ι\ni✝ : ι\nx✝ : ℝ\ny : ι → ℝ\nI : Box ι\ni : ι\nx : ℝ\n⊢ ∀ ⦃a : ι → ℝ⦄, a ∈ {y | y i ≤ x} → ¬a ∈ {y | x < y i}",
"state_before": "ι : Type u_1\nM : Type ?u.20595\nn : ℕ\nI✝ : Box ι\ni✝ : ι\nx✝ : ℝ\ny : ι → ℝ\nI : Box ι\ni : ι\nx : ℝ\n⊢ Disjoint {y | y i ≤ x} {y | x < y i}",
"tactic": "rw [Set.disjoint_left]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nM : Type ?u.20595\nn : ℕ\nI✝ : Box ι\ni✝ : ι\nx✝ : ℝ\ny : ι → ℝ\nI : Box ι\ni : ι\nx : ℝ\n⊢ ∀ ⦃a : ι → ℝ⦄, a ∈ {y | y i ≤ x} → ¬a ∈ {y | x < y i}",
"tactic": "exact fun y (hle : y i ≤ x) hlt => not_lt_of_le hle hlt"
}
] |
[
144,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/GroupTheory/FreeProduct.lean
|
FreeProduct.of_injective
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nM : ι → Type u_1\ninst✝¹ : (i : ι) → Monoid (M i)\nN : Type ?u.240831\ninst✝ : Monoid N\ni : ι\n⊢ Function.Injective ↑of",
"tactic": "classical exact (of_leftInverse i).injective"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nM : ι → Type u_1\ninst✝¹ : (i : ι) → Monoid (M i)\nN : Type ?u.240831\ninst✝ : Monoid N\ni : ι\n⊢ Function.Injective ↑of",
"tactic": "exact (of_leftInverse i).injective"
}
] |
[
192,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/Init/CcLemmas.lean
|
imp_eq_of_eq_true_left
|
[] |
[
52,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.span_map_fst
|
[] |
[
237,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.count_singleton'
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.385073\nγ : Type ?u.385076\nδ : Type ?u.385079\nι : Type ?u.385082\nR : Type ?u.385085\nR' : Type ?u.385088\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝ : MeasurableSpace α\na : α\nha : MeasurableSet {a}\n⊢ ↑(Finset.card (toFinset {a})) = 1",
"state_before": "α : Type u_1\nβ : Type ?u.385073\nγ : Type ?u.385076\nδ : Type ?u.385079\nι : Type ?u.385082\nR : Type ?u.385085\nR' : Type ?u.385088\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝ : MeasurableSpace α\na : α\nha : MeasurableSet {a}\n⊢ ↑↑count {a} = 1",
"tactic": "rw [count_apply_finite' (Set.finite_singleton a) ha, Set.Finite.toFinset]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.385073\nγ : Type ?u.385076\nδ : Type ?u.385079\nι : Type ?u.385082\nR : Type ?u.385085\nR' : Type ?u.385088\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝ : MeasurableSpace α\na : α\nha : MeasurableSet {a}\n⊢ ↑(Finset.card (toFinset {a})) = 1",
"tactic": "simp [@toFinset_card _ _ (Set.finite_singleton a).fintype,\n @Fintype.card_unique _ _ (Set.finite_singleton a).fintype]"
}
] |
[
2313,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2310,
1
] |
Mathlib/Data/Nat/Choose/Basic.lean
|
Nat.choose_le_choose
|
[] |
[
327,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
326,
1
] |
Mathlib/Algebra/Homology/HomotopyCategory.lean
|
HomotopyCategory.eq_of_homotopy
|
[] |
[
93,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
HasFDerivAtFilter.sub_const
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.483317\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.483412\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nhf : HasFDerivAtFilter f f' x L\nc : F\n⊢ HasFDerivAtFilter (fun x => f x - c) f' x L",
"tactic": "simpa only [sub_eq_add_neg] using hf.add_const (-c)"
}
] |
[
534,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
532,
1
] |
Mathlib/Topology/Instances/AddCircle.lean
|
continuous_left_toIocMod
|
[
{
"state_after": "𝕜 : Type u_1\nB : Type ?u.10078\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\n⊢ ContinuousWithinAt ((fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg) (Iic x) x",
"state_before": "𝕜 : Type u_1\nB : Type ?u.10078\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\n⊢ ContinuousWithinAt (toIocMod hp a) (Iic x) x",
"tactic": "rw [(funext fun y => Eq.trans (by rw [neg_neg]) <| toIocMod_neg _ _ _ :\n toIocMod hp a = (fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg)]"
},
{
"state_after": "𝕜 : Type u_1\nB : Type ?u.10078\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\nthis : ContinuousNeg 𝕜\n⊢ ContinuousWithinAt ((fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg) (Iic x) x",
"state_before": "𝕜 : Type u_1\nB : Type ?u.10078\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\n⊢ ContinuousWithinAt ((fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg) (Iic x) x",
"tactic": "have : ContinuousNeg 𝕜 := TopologicalAddGroup.toContinuousNeg"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nB : Type ?u.10078\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\nthis : ContinuousNeg 𝕜\n⊢ ContinuousWithinAt ((fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg) (Iic x) x",
"tactic": "exact\n (continuous_sub_left _).continuousAt.comp_continuousWithinAt <|\n (continuous_right_toIcoMod _ _ _).comp continuous_neg.continuousWithinAt fun y => neg_le_neg"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nB : Type ?u.10078\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x y : 𝕜\n⊢ toIocMod hp a y = toIocMod hp a (- -y)",
"tactic": "rw [neg_neg]"
}
] |
[
93,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
|
DifferentiableWithinAt.rpow_const
|
[] |
[
469,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
467,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
differentiableWithinAt_Ioi_iff_Ici
|
[] |
[
545,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
542,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.cos_toReal
|
[
{
"state_after": "no goals",
"state_before": "θ : Angle\n⊢ Real.cos (toReal θ) = cos θ",
"tactic": "conv_rhs => rw [← coe_toReal θ, cos_coe]"
}
] |
[
733,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
732,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Complex.cos_periodic
|
[] |
[
1204,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1203,
1
] |
Mathlib/Algebra/Algebra/Equiv.lean
|
AlgEquiv.coe_ringHom_commutes
|
[] |
[
270,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
269,
1
] |
Std/Logic.lean
|
Decidable.iff_not_comm
|
[
{
"state_after": "a b : Prop\ninst✝¹ : Decidable a\ninst✝ : Decidable b\n⊢ (a → ¬b) ∧ (¬b → a) ↔ (b → ¬a) ∧ (¬a → b)",
"state_before": "a b : Prop\ninst✝¹ : Decidable a\ninst✝ : Decidable b\n⊢ (a ↔ ¬b) ↔ (b ↔ ¬a)",
"tactic": "rw [@iff_def a, @iff_def b]"
},
{
"state_after": "no goals",
"state_before": "a b : Prop\ninst✝¹ : Decidable a\ninst✝ : Decidable b\n⊢ (a → ¬b) ∧ (¬b → a) ↔ (b → ¬a) ∧ (¬a → b)",
"tactic": "exact and_congr imp_not_comm not_imp_comm"
}
] |
[
586,
73
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
585,
1
] |
Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean
|
div_eq_iff
|
[] |
[
91,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Topology/Homeomorph.lean
|
Homeomorph.isClosed_image
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.64614\nδ : Type ?u.64617\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nh : α ≃ₜ β\ns : Set α\n⊢ IsClosed (↑h '' s) ↔ IsClosed s",
"tactic": "rw [← preimage_symm, isClosed_preimage]"
}
] |
[
325,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
324,
1
] |
Mathlib/Analysis/InnerProductSpace/Orthogonal.lean
|
Submodule.le_orthogonal_orthogonal
|
[] |
[
170,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.IsCycle.pow_eq_one_iff
|
[
{
"state_after": "case intro\nι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\n⊢ f ^ n = 1 ↔ ∃ x, ↑f x ≠ x ∧ ↑(f ^ n) x = x",
"state_before": "ι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\n⊢ f ^ n = 1 ↔ ∃ x, ↑f x ≠ x ∧ ↑(f ^ n) x = x",
"tactic": "cases nonempty_fintype β"
},
{
"state_after": "case intro.mp\nι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\n⊢ f ^ n = 1 → ∃ x, ↑f x ≠ x ∧ ↑(f ^ n) x = x\n\ncase intro.mpr\nι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\n⊢ (∃ x, ↑f x ≠ x ∧ ↑(f ^ n) x = x) → f ^ n = 1",
"state_before": "case intro\nι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\n⊢ f ^ n = 1 ↔ ∃ x, ↑f x ≠ x ∧ ↑(f ^ n) x = x",
"tactic": "constructor"
},
{
"state_after": "case intro.mp\nι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : f ^ n = 1\n⊢ ∃ x, ↑f x ≠ x ∧ ↑(f ^ n) x = x",
"state_before": "case intro.mp\nι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\n⊢ f ^ n = 1 → ∃ x, ↑f x ≠ x ∧ ↑(f ^ n) x = x",
"tactic": "intro h"
},
{
"state_after": "case intro.mp.intro.intro\nι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx✝ y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : f ^ n = 1\nx : β\nhx : ↑f x ≠ x\n⊢ ∃ x, ↑f x ≠ x ∧ ↑(f ^ n) x = x",
"state_before": "case intro.mp\nι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : f ^ n = 1\n⊢ ∃ x, ↑f x ≠ x ∧ ↑(f ^ n) x = x",
"tactic": "obtain ⟨x, hx, -⟩ := id hf"
},
{
"state_after": "no goals",
"state_before": "case intro.mp.intro.intro\nι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx✝ y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : f ^ n = 1\nx : β\nhx : ↑f x ≠ x\n⊢ ∃ x, ↑f x ≠ x ∧ ↑(f ^ n) x = x",
"tactic": "exact ⟨x, hx, by simp [h]⟩"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx✝ y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : f ^ n = 1\nx : β\nhx : ↑f x ≠ x\n⊢ ↑(f ^ n) x = x",
"tactic": "simp [h]"
},
{
"state_after": "case intro.mpr.intro.intro\nι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx✝ y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nx : β\nhx : ↑f x ≠ x\nhx' : ↑(f ^ n) x = x\n⊢ f ^ n = 1",
"state_before": "case intro.mpr\nι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\n⊢ (∃ x, ↑f x ≠ x ∧ ↑(f ^ n) x = x) → f ^ n = 1",
"tactic": "rintro ⟨x, hx, hx'⟩"
},
{
"state_after": "case pos\nι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx✝ y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nx : β\nhx : ↑f x ≠ x\nhx' : ↑(f ^ n) x = x\nh : support (f ^ n) = support f\n⊢ f ^ n = 1\n\ncase neg\nι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx✝ y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nx : β\nhx : ↑f x ≠ x\nhx' : ↑(f ^ n) x = x\nh : ¬support (f ^ n) = support f\n⊢ f ^ n = 1",
"state_before": "case intro.mpr.intro.intro\nι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx✝ y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nx : β\nhx : ↑f x ≠ x\nhx' : ↑(f ^ n) x = x\n⊢ f ^ n = 1",
"tactic": "by_cases h : support (f ^ n) = support f"
},
{
"state_after": "case pos\nι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx✝ y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nx : β\nhx✝ : ↑f x ≠ x\nhx : ↑(f ^ n) x ≠ x\nhx' : ↑(f ^ n) x = x\nh : support (f ^ n) = support f\n⊢ f ^ n = 1",
"state_before": "case pos\nι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx✝ y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nx : β\nhx : ↑f x ≠ x\nhx' : ↑(f ^ n) x = x\nh : support (f ^ n) = support f\n⊢ f ^ n = 1",
"tactic": "rw [← mem_support, ← h, mem_support] at hx"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx✝ y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nx : β\nhx✝ : ↑f x ≠ x\nhx : ↑(f ^ n) x ≠ x\nhx' : ↑(f ^ n) x = x\nh : support (f ^ n) = support f\n⊢ f ^ n = 1",
"tactic": "contradiction"
},
{
"state_after": "case neg\nι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx✝ y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nx : β\nhx : ↑f x ≠ x\nhx' : ↑(f ^ n) x = x\nh : orderOf f ∣ n\n⊢ f ^ n = 1",
"state_before": "case neg\nι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx✝ y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nx : β\nhx : ↑f x ≠ x\nhx' : ↑(f ^ n) x = x\nh : ¬support (f ^ n) = support f\n⊢ f ^ n = 1",
"tactic": "rw [hf.support_pow_eq_iff, Classical.not_not] at h"
},
{
"state_after": "case neg.intro\nι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx✝ y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nval✝ : Fintype β\nx : β\nhx : ↑f x ≠ x\nk : ℕ\nhx' : ↑(f ^ (orderOf f * k)) x = x\n⊢ f ^ (orderOf f * k) = 1",
"state_before": "case neg\nι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx✝ y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nx : β\nhx : ↑f x ≠ x\nhx' : ↑(f ^ n) x = x\nh : orderOf f ∣ n\n⊢ f ^ n = 1",
"tactic": "obtain ⟨k, rfl⟩ := h"
},
{
"state_after": "no goals",
"state_before": "case neg.intro\nι : Type ?u.1228455\nα : Type ?u.1228458\nβ : Type u_1\nf✝ g : Perm α\nx✝ y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nval✝ : Fintype β\nx : β\nhx : ↑f x ≠ x\nk : ℕ\nhx' : ↑(f ^ (orderOf f * k)) x = x\n⊢ f ^ (orderOf f * k) = 1",
"tactic": "rw [pow_mul, pow_orderOf_eq_one, one_pow]"
}
] |
[
685,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
671,
1
] |
Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
|
ContinuousMap.exists_mem_subalgebra_near_continuous_of_separatesPoints
|
[
{
"state_after": "case intro\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nw : Subalgebra.SeparatesPoints A\nf : X → ℝ\nc : Continuous f\nε : ℝ\npos : 0 < ε\ng : { x // x ∈ A }\nb : ‖↑g - mk f‖ < ε\n⊢ ∃ g, ∀ (x : X), ‖↑↑g x - f x‖ < ε",
"state_before": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nw : Subalgebra.SeparatesPoints A\nf : X → ℝ\nc : Continuous f\nε : ℝ\npos : 0 < ε\n⊢ ∃ g, ∀ (x : X), ‖↑↑g x - f x‖ < ε",
"tactic": "obtain ⟨g, b⟩ := exists_mem_subalgebra_near_continuousMap_of_separatesPoints A w ⟨f, c⟩ ε pos"
},
{
"state_after": "case intro\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nw : Subalgebra.SeparatesPoints A\nf : X → ℝ\nc : Continuous f\nε : ℝ\npos : 0 < ε\ng : { x // x ∈ A }\nb : ‖↑g - mk f‖ < ε\n⊢ ∀ (x : X), ‖↑↑g x - f x‖ < ε",
"state_before": "case intro\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nw : Subalgebra.SeparatesPoints A\nf : X → ℝ\nc : Continuous f\nε : ℝ\npos : 0 < ε\ng : { x // x ∈ A }\nb : ‖↑g - mk f‖ < ε\n⊢ ∃ g, ∀ (x : X), ‖↑↑g x - f x‖ < ε",
"tactic": "use g"
},
{
"state_after": "no goals",
"state_before": "case intro\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nw : Subalgebra.SeparatesPoints A\nf : X → ℝ\nc : Continuous f\nε : ℝ\npos : 0 < ε\ng : { x // x ∈ A }\nb : ‖↑g - mk f‖ < ε\n⊢ ∀ (x : X), ‖↑↑g x - f x‖ < ε",
"tactic": "rwa [norm_lt_iff _ pos] at b"
}
] |
[
329,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
324,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.exists_mem_insert
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.438273\nγ : Type ?u.438276\ninst✝ : DecidableEq α\na : α\ns : Finset α\np : α → Prop\n⊢ (∃ x, x ∈ insert a s ∧ p x) ↔ p a ∨ ∃ x, x ∈ s ∧ p x",
"tactic": "simp only [mem_insert, or_and_right, exists_or, exists_eq_left]"
}
] |
[
3073,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3071,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.inter_congr_left
|
[] |
[
985,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
984,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Finset.top_eq_univ
|
[] |
[
138,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Topology/PathConnected.lean
|
IsPathConnected.union
|
[
{
"state_after": "case intro.intro\nX : Type u_1\nY : Type ?u.646605\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type ?u.646620\nF U V : Set X\nhU : IsPathConnected U\nhV : IsPathConnected V\nx : X\nxU : x ∈ U\nxV : x ∈ V\n⊢ IsPathConnected (U ∪ V)",
"state_before": "X : Type u_1\nY : Type ?u.646605\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.646620\nF U V : Set X\nhU : IsPathConnected U\nhV : IsPathConnected V\nhUV : Set.Nonempty (U ∩ V)\n⊢ IsPathConnected (U ∪ V)",
"tactic": "rcases hUV with ⟨x, xU, xV⟩"
},
{
"state_after": "case intro.intro\nX : Type u_1\nY : Type ?u.646605\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type ?u.646620\nF U V : Set X\nhU : IsPathConnected U\nhV : IsPathConnected V\nx : X\nxU : x ∈ U\nxV : x ∈ V\n⊢ ∀ {y : X}, y ∈ U ∪ V → JoinedIn (U ∪ V) x y",
"state_before": "case intro.intro\nX : Type u_1\nY : Type ?u.646605\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type ?u.646620\nF U V : Set X\nhU : IsPathConnected U\nhV : IsPathConnected V\nx : X\nxU : x ∈ U\nxV : x ∈ V\n⊢ IsPathConnected (U ∪ V)",
"tactic": "use x, Or.inl xU"
},
{
"state_after": "case intro.intro.inl\nX : Type u_1\nY : Type ?u.646605\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y✝ z : X\nι : Type ?u.646620\nF U V : Set X\nhU : IsPathConnected U\nhV : IsPathConnected V\nx : X\nxU : x ∈ U\nxV : x ∈ V\ny : X\nyU : y ∈ U\n⊢ JoinedIn (U ∪ V) x y\n\ncase intro.intro.inr\nX : Type u_1\nY : Type ?u.646605\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y✝ z : X\nι : Type ?u.646620\nF U V : Set X\nhU : IsPathConnected U\nhV : IsPathConnected V\nx : X\nxU : x ∈ U\nxV : x ∈ V\ny : X\nyV : y ∈ V\n⊢ JoinedIn (U ∪ V) x y",
"state_before": "case intro.intro\nX : Type u_1\nY : Type ?u.646605\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type ?u.646620\nF U V : Set X\nhU : IsPathConnected U\nhV : IsPathConnected V\nx : X\nxU : x ∈ U\nxV : x ∈ V\n⊢ ∀ {y : X}, y ∈ U ∪ V → JoinedIn (U ∪ V) x y",
"tactic": "rintro y (yU | yV)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.inl\nX : Type u_1\nY : Type ?u.646605\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y✝ z : X\nι : Type ?u.646620\nF U V : Set X\nhU : IsPathConnected U\nhV : IsPathConnected V\nx : X\nxU : x ∈ U\nxV : x ∈ V\ny : X\nyU : y ∈ U\n⊢ JoinedIn (U ∪ V) x y",
"tactic": "exact (hU.joinedIn x xU y yU).mono (subset_union_left U V)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.inr\nX : Type u_1\nY : Type ?u.646605\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y✝ z : X\nι : Type ?u.646620\nF U V : Set X\nhU : IsPathConnected U\nhV : IsPathConnected V\nx : X\nxU : x ∈ U\nxV : x ∈ V\ny : X\nyV : y ∈ V\n⊢ JoinedIn (U ∪ V) x y",
"tactic": "exact (hV.joinedIn x xV y yV).mono (subset_union_right U V)"
}
] |
[
1003,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
997,
1
] |
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
reflection_trans_reflection
|
[] |
[
684,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
682,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
inv_le_of_neg
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.150435\nα : Type u_1\nβ : Type ?u.150441\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\nha : a < 0\nhb : b < 0\n⊢ a⁻¹ ≤ b ↔ b⁻¹ ≤ a",
"tactic": "rw [← inv_le_inv_of_neg hb (inv_lt_zero.2 ha), inv_inv]"
}
] |
[
764,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
763,
1
] |
Mathlib/Topology/Basic.lean
|
Continuous.comp'
|
[] |
[
1644,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1643,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.cos_zero
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\n⊢ cos 0 = 1",
"tactic": "simp [cos]"
}
] |
[
1189,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1189,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.Hom.mapDart_apply
|
[] |
[
1717,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1716,
1
] |
Mathlib/CategoryTheory/Abelian/Transfer.lean
|
CategoryTheory.AbelianOfAdjunction.coimageIsoImage_hom
|
[
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\nD : Type u₂\ninst✝⁶ : Category D\ninst✝⁵ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝⁴ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\ninst✝³ : HasCokernels C\ninst✝² : HasKernels C\ninst✝¹ : PreservesFiniteLimits G\ninst✝ : Functor.PreservesZeroMorphisms F\nX Y : C\nf : X ⟶ Y\nthis : ∀ (X' Y' : C) (f' : X' ⟶ Y'), HasCokernel f'\n⊢ (coimageIsoImage F G i adj f).hom = Abelian.coimageImageComparison f",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\nD : Type u₂\ninst✝⁶ : Category D\ninst✝⁵ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝⁴ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\ninst✝³ : HasCokernels C\ninst✝² : HasKernels C\ninst✝¹ : PreservesFiniteLimits G\ninst✝ : Functor.PreservesZeroMorphisms F\nX Y : C\nf : X ⟶ Y\n⊢ (coimageIsoImage F G i adj f).hom = Abelian.coimageImageComparison f",
"tactic": "have : ∀ (X' Y' : C) (f' : X' ⟶ Y'), HasCokernel f' := inferInstance"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\nD : Type u₂\ninst✝⁶ : Category D\ninst✝⁵ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝⁴ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\ninst✝³ : HasCokernels C\ninst✝² : HasKernels C\ninst✝¹ : PreservesFiniteLimits G\ninst✝ : Functor.PreservesZeroMorphisms F\nX Y : C\nf : X ⟶ Y\nthis✝ : ∀ (X' Y' : C) (f' : X' ⟶ Y'), HasCokernel f'\nthis : ∀ (X' Y' : C) (f' : X' ⟶ Y'), HasKernel f'\n⊢ (coimageIsoImage F G i adj f).hom = Abelian.coimageImageComparison f",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\nD : Type u₂\ninst✝⁶ : Category D\ninst✝⁵ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝⁴ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\ninst✝³ : HasCokernels C\ninst✝² : HasKernels C\ninst✝¹ : PreservesFiniteLimits G\ninst✝ : Functor.PreservesZeroMorphisms F\nX Y : C\nf : X ⟶ Y\nthis : ∀ (X' Y' : C) (f' : X' ⟶ Y'), HasCokernel f'\n⊢ (coimageIsoImage F G i adj f).hom = Abelian.coimageImageComparison f",
"tactic": "have : ∀ (X' Y' : C) (f' : X' ⟶ Y'), HasKernel f' := inferInstance"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\nD : Type u₂\ninst✝⁶ : Category D\ninst✝⁵ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝⁴ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\ninst✝³ : HasCokernels C\ninst✝² : HasKernels C\ninst✝¹ : PreservesFiniteLimits G\ninst✝ : Functor.PreservesZeroMorphisms F\nX Y : C\nf : X ⟶ Y\nthis✝¹ : ∀ (X' Y' : C) (f' : X' ⟶ Y'), HasCokernel f'\nthis✝ : ∀ (X' Y' : C) (f' : X' ⟶ Y'), HasKernel f'\nthis : ∀ (X' Y' : D) (f' : X' ⟶ Y'), HasCokernel f'\n⊢ (coimageIsoImage F G i adj f).hom = Abelian.coimageImageComparison f",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\nD : Type u₂\ninst✝⁶ : Category D\ninst✝⁵ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝⁴ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\ninst✝³ : HasCokernels C\ninst✝² : HasKernels C\ninst✝¹ : PreservesFiniteLimits G\ninst✝ : Functor.PreservesZeroMorphisms F\nX Y : C\nf : X ⟶ Y\nthis✝ : ∀ (X' Y' : C) (f' : X' ⟶ Y'), HasCokernel f'\nthis : ∀ (X' Y' : C) (f' : X' ⟶ Y'), HasKernel f'\n⊢ (coimageIsoImage F G i adj f).hom = Abelian.coimageImageComparison f",
"tactic": "have : ∀ (X' Y' : D) (f' : X' ⟶ Y'), HasCokernel f' := inferInstance"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\nD : Type u₂\ninst✝⁶ : Category D\ninst✝⁵ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝⁴ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\ninst✝³ : HasCokernels C\ninst✝² : HasKernels C\ninst✝¹ : PreservesFiniteLimits G\ninst✝ : Functor.PreservesZeroMorphisms F\nX Y : C\nf : X ⟶ Y\nthis✝² : ∀ (X' Y' : C) (f' : X' ⟶ Y'), HasCokernel f'\nthis✝¹ : ∀ (X' Y' : C) (f' : X' ⟶ Y'), HasKernel f'\nthis✝ : ∀ (X' Y' : D) (f' : X' ⟶ Y'), HasCokernel f'\nthis : ∀ (X' Y' : D) (f' : X' ⟶ Y'), HasKernel f'\n⊢ (coimageIsoImage F G i adj f).hom = Abelian.coimageImageComparison f",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\nD : Type u₂\ninst✝⁶ : Category D\ninst✝⁵ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝⁴ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\ninst✝³ : HasCokernels C\ninst✝² : HasKernels C\ninst✝¹ : PreservesFiniteLimits G\ninst✝ : Functor.PreservesZeroMorphisms F\nX Y : C\nf : X ⟶ Y\nthis✝¹ : ∀ (X' Y' : C) (f' : X' ⟶ Y'), HasCokernel f'\nthis✝ : ∀ (X' Y' : C) (f' : X' ⟶ Y'), HasKernel f'\nthis : ∀ (X' Y' : D) (f' : X' ⟶ Y'), HasCokernel f'\n⊢ (coimageIsoImage F G i adj f).hom = Abelian.coimageImageComparison f",
"tactic": "have : ∀ (X' Y' : D) (f' : X' ⟶ Y'), HasKernel f' := inferInstance"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\nD : Type u₂\ninst✝⁶ : Category D\ninst✝⁵ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝⁴ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\ninst✝³ : HasCokernels C\ninst✝² : HasKernels C\ninst✝¹ : PreservesFiniteLimits G\ninst✝ : Functor.PreservesZeroMorphisms F\nX Y : C\nf : X ⟶ Y\nthis✝² : ∀ (X' Y' : C) (f' : X' ⟶ Y'), HasCokernel f'\nthis✝¹ : ∀ (X' Y' : C) (f' : X' ⟶ Y'), HasKernel f'\nthis✝ : ∀ (X' Y' : D) (f' : X' ⟶ Y'), HasCokernel f'\nthis : ∀ (X' Y' : D) (f' : X' ⟶ Y'), HasKernel f'\n⊢ ((((((((𝟙 (Abelian.coimage f) ≫\n cokernel.desc (kernel.ι f) (i.inv.app X ≫ cokernel.π (kernel.ι f ≫ i.inv.app X))\n (_ : kernel.ι f ≫ i.inv.app X ≫ cokernel.π (kernel.ι f ≫ i.inv.app X) = 0) ≫\n cokernel.desc (kernel.ι f ≫ i.inv.app X)\n (cokernel.π (i.hom.app (kernel f) ≫ kernel.ι f ≫ i.inv.app X))\n (_ :\n (kernel.ι f ≫ i.inv.app X) ≫\n cokernel.π (i.hom.app (kernel f) ≫ kernel.ι f ≫ i.inv.app X) =\n 0) ≫\n (cokernelIsoOfEq\n (_ :\n (F ⋙ G).map (kernel.ι f) =\n i.hom.app (kernel f) ≫ (𝟭 C).map (kernel.ι f) ≫ i.inv.app X)).inv ≫\n cokernelComparison (F.map (kernel.ι f)) G) ≫\n G.map (cokernelIsoOfEq (_ : F.map (kernel.ι f) = kernelComparison f F ≫ kernel.ι (F.map f))).hom) ≫\n G.map\n (cokernel.desc (kernelComparison f F ≫ kernel.ι (F.map f)) (cokernel.π (kernel.ι (F.map f)))\n (_ : (kernelComparison f F ≫ kernel.ι (F.map f)) ≫ cokernel.π (kernel.ι (F.map f)) = 0))) ≫\n 𝟙 (G.obj (cokernel (kernel.ι (F.map f))))) ≫\n G.map (Abelian.coimageIsoImage (F.map f)).hom) ≫\n 𝟙 (G.obj (Abelian.image (F.map f)))) ≫\n (PreservesKernel.iso G (cokernel.π (F.map f))).hom) ≫\n ((((((((kernelIsoOfEq\n (_ :\n G.map (cokernel.π (F.map f)) =\n cokernel.π (G.map (F.map f)) ≫ cokernelComparison (F.map f) G)).hom ≫\n kernel.lift (cokernel.π (G.map (F.map f)))\n (kernel.ι (cokernel.π (G.map (F.map f)) ≫ cokernelComparison (F.map f) G))\n (_ :\n kernel.ι (cokernel.π (G.map (F.map f)) ≫ cokernelComparison (F.map f) G) ≫\n cokernel.π (G.map (F.map f)) =\n 0)) ≫\n (kernelIsoOfEq\n (_ :\n cokernel.π ((F ⋙ G).map f) =\n cokernel.π (i.hom.app X ≫ (𝟭 C).map f ≫ i.inv.app Y) ≫\n (cokernelIsoOfEq\n (_ : i.hom.app X ≫ (𝟭 C).map f ≫ i.inv.app Y = (F ⋙ G).map f)).hom)).hom) ≫\n kernel.lift (cokernel.π (i.hom.app X ≫ f ≫ i.inv.app Y))\n (kernel.ι\n (cokernel.π (i.hom.app X ≫ f ≫ i.inv.app Y) ≫\n (cokernelIsoOfEq (_ : i.hom.app X ≫ (𝟭 C).map f ≫ i.inv.app Y = (F ⋙ G).map f)).hom))\n (_ :\n kernel.ι\n (cokernel.π (i.hom.app X ≫ f ≫ i.inv.app Y) ≫\n (cokernelIsoOfEq (_ : i.hom.app X ≫ (𝟭 C).map f ≫ i.inv.app Y = (F ⋙ G).map f)).hom) ≫\n cokernel.π (i.hom.app X ≫ f ≫ i.inv.app Y) =\n 0)) ≫\n (kernelIsoOfEq\n (_ :\n cokernel.π (i.hom.app X ≫ f ≫ i.inv.app Y) =\n cokernel.π (f ≫ i.inv.app Y) ≫\n cokernel.desc (f ≫ i.inv.app Y) (cokernel.π (i.hom.app X ≫ f ≫ i.inv.app Y))\n (_ : (f ≫ i.inv.app Y) ≫ cokernel.π (i.hom.app X ≫ f ≫ i.inv.app Y) = 0))).hom) ≫\n kernel.lift (cokernel.π (f ≫ i.inv.app Y))\n (kernel.ι\n (cokernel.π (f ≫ i.inv.app Y) ≫\n cokernel.desc (f ≫ i.inv.app Y) (cokernel.π (i.hom.app X ≫ f ≫ i.inv.app Y))\n (_ : (f ≫ i.inv.app Y) ≫ cokernel.π (i.hom.app X ≫ f ≫ i.inv.app Y) = 0)))\n (_ :\n kernel.ι\n (cokernel.π (f ≫ i.inv.app Y) ≫\n cokernel.desc (f ≫ i.inv.app Y) (cokernel.π (i.hom.app X ≫ f ≫ i.inv.app Y))\n (_ : (f ≫ i.inv.app Y) ≫ cokernel.π (i.hom.app X ≫ f ≫ i.inv.app Y) = 0)) ≫\n cokernel.π (f ≫ i.inv.app Y) =\n 0)) ≫\n (kernelIsoOfEq\n (_ :\n cokernel.π (f ≫ i.inv.app Y) =\n inv (i.inv.app Y) ≫\n cokernel.π f ≫\n cokernel.desc f (i.inv.app Y ≫ cokernel.π (f ≫ i.inv.app Y))\n (_ : f ≫ i.inv.app Y ≫ cokernel.π (f ≫ i.inv.app Y) = 0))).hom) ≫\n (kernelIsIsoComp (inv (i.inv.app Y))\n (cokernel.π f ≫\n cokernel.desc f (i.inv.app Y ≫ cokernel.π (f ≫ i.inv.app Y))\n (_ : f ≫ i.inv.app Y ≫ cokernel.π (f ≫ i.inv.app Y) = 0))).hom) ≫\n kernel.lift (cokernel.π f)\n (kernel.ι\n (cokernel.π f ≫\n cokernel.desc f (i.inv.app Y ≫ cokernel.π (f ≫ i.inv.app Y))\n (_ : f ≫ i.inv.app Y ≫ cokernel.π (f ≫ i.inv.app Y) = 0)))\n (_ :\n kernel.ι\n (cokernel.π f ≫\n cokernel.desc f (i.inv.app Y ≫ cokernel.π (f ≫ i.inv.app Y))\n (_ : f ≫ i.inv.app Y ≫ cokernel.π (f ≫ i.inv.app Y) = 0)) ≫\n cokernel.π f =\n 0)) ≫\n 𝟙 (kernel (cokernel.π f)) =\n Abelian.coimageImageComparison f",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\nD : Type u₂\ninst✝⁶ : Category D\ninst✝⁵ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝⁴ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\ninst✝³ : HasCokernels C\ninst✝² : HasKernels C\ninst✝¹ : PreservesFiniteLimits G\ninst✝ : Functor.PreservesZeroMorphisms F\nX Y : C\nf : X ⟶ Y\nthis✝² : ∀ (X' Y' : C) (f' : X' ⟶ Y'), HasCokernel f'\nthis✝¹ : ∀ (X' Y' : C) (f' : X' ⟶ Y'), HasKernel f'\nthis✝ : ∀ (X' Y' : D) (f' : X' ⟶ Y'), HasCokernel f'\nthis : ∀ (X' Y' : D) (f' : X' ⟶ Y'), HasKernel f'\n⊢ (coimageIsoImage F G i adj f).hom = Abelian.coimageImageComparison f",
"tactic": "dsimp only [coimageIsoImage, Iso.instTransIso_trans, Iso.refl, Iso.trans, Iso.symm,\n Functor.mapIso, cokernelEpiComp, cokernelIso, cokernelCompIsIso_inv,\n asIso, coimageIsoImageAux, kernelCompMono]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\nD : Type u₂\ninst✝⁶ : Category D\ninst✝⁵ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝⁴ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\ninst✝³ : HasCokernels C\ninst✝² : HasKernels C\ninst✝¹ : PreservesFiniteLimits G\ninst✝ : Functor.PreservesZeroMorphisms F\nX Y : C\nf : X ⟶ Y\nthis✝² : ∀ (X' Y' : C) (f' : X' ⟶ Y'), HasCokernel f'\nthis✝¹ : ∀ (X' Y' : C) (f' : X' ⟶ Y'), HasKernel f'\nthis✝ : ∀ (X' Y' : D) (f' : X' ⟶ Y'), HasCokernel f'\nthis : ∀ (X' Y' : D) (f' : X' ⟶ Y'), HasKernel f'\n⊢ ((((((((𝟙 (Abelian.coimage f) ≫\n cokernel.desc (kernel.ι f) (i.inv.app X ≫ cokernel.π (kernel.ι f ≫ i.inv.app X))\n (_ : kernel.ι f ≫ i.inv.app X ≫ cokernel.π (kernel.ι f ≫ i.inv.app X) = 0) ≫\n cokernel.desc (kernel.ι f ≫ i.inv.app X)\n (cokernel.π (i.hom.app (kernel f) ≫ kernel.ι f ≫ i.inv.app X))\n (_ :\n (kernel.ι f ≫ i.inv.app X) ≫\n cokernel.π (i.hom.app (kernel f) ≫ kernel.ι f ≫ i.inv.app X) =\n 0) ≫\n (cokernelIsoOfEq\n (_ :\n (F ⋙ G).map (kernel.ι f) =\n i.hom.app (kernel f) ≫ (𝟭 C).map (kernel.ι f) ≫ i.inv.app X)).inv ≫\n cokernelComparison (F.map (kernel.ι f)) G) ≫\n G.map (cokernelIsoOfEq (_ : F.map (kernel.ι f) = kernelComparison f F ≫ kernel.ι (F.map f))).hom) ≫\n G.map\n (cokernel.desc (kernelComparison f F ≫ kernel.ι (F.map f)) (cokernel.π (kernel.ι (F.map f)))\n (_ : (kernelComparison f F ≫ kernel.ι (F.map f)) ≫ cokernel.π (kernel.ι (F.map f)) = 0))) ≫\n 𝟙 (G.obj (cokernel (kernel.ι (F.map f))))) ≫\n G.map (Abelian.coimageIsoImage (F.map f)).hom) ≫\n 𝟙 (G.obj (Abelian.image (F.map f)))) ≫\n (PreservesKernel.iso G (cokernel.π (F.map f))).hom) ≫\n ((((((((kernelIsoOfEq\n (_ :\n G.map (cokernel.π (F.map f)) =\n cokernel.π (G.map (F.map f)) ≫ cokernelComparison (F.map f) G)).hom ≫\n kernel.lift (cokernel.π (G.map (F.map f)))\n (kernel.ι (cokernel.π (G.map (F.map f)) ≫ cokernelComparison (F.map f) G))\n (_ :\n kernel.ι (cokernel.π (G.map (F.map f)) ≫ cokernelComparison (F.map f) G) ≫\n cokernel.π (G.map (F.map f)) =\n 0)) ≫\n (kernelIsoOfEq\n (_ :\n cokernel.π ((F ⋙ G).map f) =\n cokernel.π (i.hom.app X ≫ (𝟭 C).map f ≫ i.inv.app Y) ≫\n (cokernelIsoOfEq\n (_ : i.hom.app X ≫ (𝟭 C).map f ≫ i.inv.app Y = (F ⋙ G).map f)).hom)).hom) ≫\n kernel.lift (cokernel.π (i.hom.app X ≫ f ≫ i.inv.app Y))\n (kernel.ι\n (cokernel.π (i.hom.app X ≫ f ≫ i.inv.app Y) ≫\n (cokernelIsoOfEq (_ : i.hom.app X ≫ (𝟭 C).map f ≫ i.inv.app Y = (F ⋙ G).map f)).hom))\n (_ :\n kernel.ι\n (cokernel.π (i.hom.app X ≫ f ≫ i.inv.app Y) ≫\n (cokernelIsoOfEq (_ : i.hom.app X ≫ (𝟭 C).map f ≫ i.inv.app Y = (F ⋙ G).map f)).hom) ≫\n cokernel.π (i.hom.app X ≫ f ≫ i.inv.app Y) =\n 0)) ≫\n (kernelIsoOfEq\n (_ :\n cokernel.π (i.hom.app X ≫ f ≫ i.inv.app Y) =\n cokernel.π (f ≫ i.inv.app Y) ≫\n cokernel.desc (f ≫ i.inv.app Y) (cokernel.π (i.hom.app X ≫ f ≫ i.inv.app Y))\n (_ : (f ≫ i.inv.app Y) ≫ cokernel.π (i.hom.app X ≫ f ≫ i.inv.app Y) = 0))).hom) ≫\n kernel.lift (cokernel.π (f ≫ i.inv.app Y))\n (kernel.ι\n (cokernel.π (f ≫ i.inv.app Y) ≫\n cokernel.desc (f ≫ i.inv.app Y) (cokernel.π (i.hom.app X ≫ f ≫ i.inv.app Y))\n (_ : (f ≫ i.inv.app Y) ≫ cokernel.π (i.hom.app X ≫ f ≫ i.inv.app Y) = 0)))\n (_ :\n kernel.ι\n (cokernel.π (f ≫ i.inv.app Y) ≫\n cokernel.desc (f ≫ i.inv.app Y) (cokernel.π (i.hom.app X ≫ f ≫ i.inv.app Y))\n (_ : (f ≫ i.inv.app Y) ≫ cokernel.π (i.hom.app X ≫ f ≫ i.inv.app Y) = 0)) ≫\n cokernel.π (f ≫ i.inv.app Y) =\n 0)) ≫\n (kernelIsoOfEq\n (_ :\n cokernel.π (f ≫ i.inv.app Y) =\n inv (i.inv.app Y) ≫\n cokernel.π f ≫\n cokernel.desc f (i.inv.app Y ≫ cokernel.π (f ≫ i.inv.app Y))\n (_ : f ≫ i.inv.app Y ≫ cokernel.π (f ≫ i.inv.app Y) = 0))).hom) ≫\n (kernelIsIsoComp (inv (i.inv.app Y))\n (cokernel.π f ≫\n cokernel.desc f (i.inv.app Y ≫ cokernel.π (f ≫ i.inv.app Y))\n (_ : f ≫ i.inv.app Y ≫ cokernel.π (f ≫ i.inv.app Y) = 0))).hom) ≫\n kernel.lift (cokernel.π f)\n (kernel.ι\n (cokernel.π f ≫\n cokernel.desc f (i.inv.app Y ≫ cokernel.π (f ≫ i.inv.app Y))\n (_ : f ≫ i.inv.app Y ≫ cokernel.π (f ≫ i.inv.app Y) = 0)))\n (_ :\n kernel.ι\n (cokernel.π f ≫\n cokernel.desc f (i.inv.app Y ≫ cokernel.π (f ≫ i.inv.app Y))\n (_ : f ≫ i.inv.app Y ≫ cokernel.π (f ≫ i.inv.app Y) = 0)) ≫\n cokernel.π f =\n 0)) ≫\n 𝟙 (kernel (cokernel.π f)) =\n Abelian.coimageImageComparison f",
"tactic": "simpa only [← cancel_mono (Abelian.image.ι f), ← cancel_epi (Abelian.coimage.π f),\n Category.assoc, Category.id_comp, cokernel.π_desc_assoc,\n π_comp_cokernelIsoOfEq_inv_assoc, PreservesKernel.iso_hom,\n π_comp_cokernelComparison_assoc, ← G.map_comp_assoc, kernel.lift_ι,\n Abelian.coimage_image_factorisation, lift_comp_kernelIsoOfEq_hom_assoc,\n kernelIsIsoComp_hom, kernel.lift_ι_assoc, kernelIsoOfEq_hom_comp_ι_assoc,\n kernelComparison_comp_ι_assoc, π_comp_cokernelIsoOfEq_hom_assoc,\n asIso_hom, NatIso.inv_inv_app] using NatIso.naturality_1 i f"
}
] |
[
166,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Algebra/Hom/Freiman.lean
|
FreimanHom.mk_coe
|
[] |
[
211,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.zero_le
|
[] |
[
387,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
11
] |
Mathlib/Data/List/Sigma.lean
|
List.NodupKeys.sublist
|
[] |
[
137,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/CategoryTheory/Monoidal/Free/Coherence.lean
|
CategoryTheory.FreeMonoidalCategory.tensorFunc_map_app
|
[] |
[
173,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/Data/Set/Finite.lean
|
Multiset.finite_toSet
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns : Multiset α\n⊢ Set.Finite {x | x ∈ s}",
"tactic": "classical simpa only [← Multiset.mem_toFinset] using s.toFinset.finite_toSet"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns : Multiset α\n⊢ Set.Finite {x | x ∈ s}",
"tactic": "simpa only [← Multiset.mem_toFinset] using s.toFinset.finite_toSet"
}
] |
[
549,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
548,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.mapsTo
|
[] |
[
234,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
11
] |
Mathlib/Data/List/Chain.lean
|
List.chain'_reverse
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nR r : α → α → Prop\nl l₁ l₂ : List α\na✝ b a : α\n⊢ Chain' R (reverse [a]) ↔ Chain' (flip R) [a]",
"tactic": "simp only [chain'_singleton, reverse_singleton]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nR r : α → α → Prop\nl✝ l₁ l₂ : List α\na✝ b✝ a b : α\nl : List α\n⊢ Chain' R (reverse (a :: b :: l)) ↔ Chain' (flip R) (a :: b :: l)",
"tactic": "rw [chain'_cons, reverse_cons, reverse_cons, append_assoc, cons_append, nil_append,\n chain'_split, ← reverse_cons, @chain'_reverse (b :: l), and_comm, chain'_pair, flip]"
}
] |
[
348,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
343,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
Finsupp.lhom_ext'
|
[] |
[
80,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Computability/Reduce.lean
|
manyOneEquiv_refl
|
[] |
[
164,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/Data/Polynomial/Module.lean
|
PolynomialModule.comp_single
|
[
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.629765\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type ?u.632417\nM' : Type ?u.632420\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\np : R[X]\ni : ℕ\nm : M\n⊢ ↑(eval p) (↑(map R[X] (lsingle R 0)) (↑(single R i) m)) = p ^ i • ↑(single R 0) m",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.629765\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type ?u.632417\nM' : Type ?u.632420\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\np : R[X]\ni : ℕ\nm : M\n⊢ ↑(comp p) (↑(single R i) m) = p ^ i • ↑(single R 0) m",
"tactic": "rw [comp_apply]"
},
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.629765\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type ?u.632417\nM' : Type ?u.632420\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\np : R[X]\ni : ℕ\nm : M\n⊢ p ^ i • ↑(lsingle R 0) m = p ^ i • ↑(single R 0) m",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.629765\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type ?u.632417\nM' : Type ?u.632420\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\np : R[X]\ni : ℕ\nm : M\n⊢ ↑(eval p) (↑(map R[X] (lsingle R 0)) (↑(single R i) m)) = p ^ i • ↑(single R 0) m",
"tactic": "erw [map_single, eval_single]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.629765\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type ?u.632417\nM' : Type ?u.632420\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\np : R[X]\ni : ℕ\nm : M\n⊢ p ^ i • ↑(lsingle R 0) m = p ^ i • ↑(single R 0) m",
"tactic": "rfl"
}
] |
[
334,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
331,
1
] |
Mathlib/Analysis/SpecificLimits/Normed.lean
|
Monotone.cauchySeq_alternating_series_of_tendsto_zero
|
[
{
"state_after": "α : Type ?u.1705167\nβ : Type ?u.1705170\nι : Type ?u.1705173\nE : Type ?u.1705176\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nb : ℝ\nf : ℕ → ℝ\nz : ℕ → E\nhfa : Monotone f\nhf0 : Tendsto f atTop (𝓝 0)\n⊢ CauchySeq fun n => ∑ x in Finset.range (n + 1), f x * (-1) ^ x",
"state_before": "α : Type ?u.1705167\nβ : Type ?u.1705170\nι : Type ?u.1705173\nE : Type ?u.1705176\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nb : ℝ\nf : ℕ → ℝ\nz : ℕ → E\nhfa : Monotone f\nhf0 : Tendsto f atTop (𝓝 0)\n⊢ CauchySeq fun n => ∑ i in Finset.range (n + 1), (-1) ^ i * f i",
"tactic": "simp_rw [mul_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.1705167\nβ : Type ?u.1705170\nι : Type ?u.1705173\nE : Type ?u.1705176\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nb : ℝ\nf : ℕ → ℝ\nz : ℕ → E\nhfa : Monotone f\nhf0 : Tendsto f atTop (𝓝 0)\n⊢ CauchySeq fun n => ∑ x in Finset.range (n + 1), f x * (-1) ^ x",
"tactic": "exact hfa.cauchySeq_series_mul_of_tendsto_zero_of_bounded hf0 norm_sum_neg_one_pow_le"
}
] |
[
623,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
620,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.BalancedSz.symm
|
[
{
"state_after": "α : Type ?u.7857\nl r : ℕ\n⊢ r + l ≤ 1 → r + l ≤ 1",
"state_before": "α : Type ?u.7857\nl r : ℕ\n⊢ l + r ≤ 1 → r + l ≤ 1",
"tactic": "rw [add_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.7857\nl r : ℕ\n⊢ r + l ≤ 1 → r + l ≤ 1",
"tactic": "exact id"
}
] |
[
194,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
Finset.weightedVSubOfPoint_congr
|
[
{
"state_after": "k : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.22097\ns₂ : Finset ι₂\nw₁ w₂ : ι → k\nhw : ∀ (i : ι), i ∈ s → w₁ i = w₂ i\np₁ p₂ : ι → P\nhp : ∀ (i : ι), i ∈ s → p₁ i = p₂ i\nb : P\n⊢ ∑ i in s, w₁ i • (p₁ i -ᵥ b) = ∑ i in s, w₂ i • (p₂ i -ᵥ b)",
"state_before": "k : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.22097\ns₂ : Finset ι₂\nw₁ w₂ : ι → k\nhw : ∀ (i : ι), i ∈ s → w₁ i = w₂ i\np₁ p₂ : ι → P\nhp : ∀ (i : ι), i ∈ s → p₁ i = p₂ i\nb : P\n⊢ ↑(weightedVSubOfPoint s p₁ b) w₁ = ↑(weightedVSubOfPoint s p₂ b) w₂",
"tactic": "simp_rw [weightedVSubOfPoint_apply]"
},
{
"state_after": "k : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.22097\ns₂ : Finset ι₂\nw₁ w₂ : ι → k\nhw : ∀ (i : ι), i ∈ s → w₁ i = w₂ i\np₁ p₂ : ι → P\nhp : ∀ (i : ι), i ∈ s → p₁ i = p₂ i\nb : P\ni : ι\nhi : i ∈ s\n⊢ w₁ i • (p₁ i -ᵥ b) = w₂ i • (p₂ i -ᵥ b)",
"state_before": "k : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.22097\ns₂ : Finset ι₂\nw₁ w₂ : ι → k\nhw : ∀ (i : ι), i ∈ s → w₁ i = w₂ i\np₁ p₂ : ι → P\nhp : ∀ (i : ι), i ∈ s → p₁ i = p₂ i\nb : P\n⊢ ∑ i in s, w₁ i • (p₁ i -ᵥ b) = ∑ i in s, w₂ i • (p₂ i -ᵥ b)",
"tactic": "refine sum_congr rfl fun i hi => ?_"
},
{
"state_after": "no goals",
"state_before": "k : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.22097\ns₂ : Finset ι₂\nw₁ w₂ : ι → k\nhw : ∀ (i : ι), i ∈ s → w₁ i = w₂ i\np₁ p₂ : ι → P\nhp : ∀ (i : ι), i ∈ s → p₁ i = p₂ i\nb : P\ni : ι\nhi : i ∈ s\n⊢ w₁ i • (p₁ i -ᵥ b) = w₂ i • (p₂ i -ᵥ b)",
"tactic": "rw [hw i hi, hp i hi]"
}
] |
[
100,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean
|
MvPolynomial.le_weightedTotalDegree
|
[] |
[
126,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Data/Sum/Basic.lean
|
Sum.update_inl_comp_inl
|
[] |
[
291,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.aleph0_le_add_iff
|
[
{
"state_after": "no goals",
"state_before": "α β : Type u\na b : Cardinal\n⊢ ℵ₀ ≤ a + b ↔ ℵ₀ ≤ a ∨ ℵ₀ ≤ b",
"tactic": "simp only [← not_lt, add_lt_aleph0_iff, not_and_or]"
}
] |
[
1535,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1534,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.smul_mem_smul_finset_iff
|
[] |
[
1952,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1951,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.cast_int_cast'
|
[] |
[
404,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
403,
1
] |
Mathlib/Computability/Ackermann.lean
|
ack_inj_right
|
[] |
[
170,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Std/Data/List/Lemmas.lean
|
List.filterMap_nil
|
[] |
[
1148,
78
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1148,
9
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.lsub_sum
|
[] |
[
1705,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1702,
1
] |
Std/Data/List/Basic.lean
|
List.pairwise_cons
|
[] |
[
1081,
62
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1080,
9
] |
Mathlib/Data/Nat/PartENat.lean
|
PartENat.get_eq_iff_eq_coe
|
[
{
"state_after": "a : PartENat\nha : a.Dom\nb : ℕ\n⊢ a = ↑b ↔ a = ↑b",
"state_before": "a : PartENat\nha : a.Dom\nb : ℕ\n⊢ Part.get a ha = b ↔ a = ↑b",
"tactic": "rw [get_eq_iff_eq_some]"
},
{
"state_after": "no goals",
"state_before": "a : PartENat\nha : a.Dom\nb : ℕ\n⊢ a = ↑b ↔ a = ↑b",
"tactic": "rfl"
}
] |
[
210,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Analysis/Complex/LocallyUniformLimit.lean
|
TendstoLocallyUniformlyOn.deriv
|
[
{
"state_after": "E : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\n⊢ ∀ (K : Set ℂ), K ⊆ U → IsCompact K → TendstoUniformlyOn (_root_.deriv ∘ F) (_root_.deriv f) φ K",
"state_before": "E : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\n⊢ TendstoLocallyUniformlyOn (_root_.deriv ∘ F) (_root_.deriv f) φ U",
"tactic": "rw [tendstoLocallyUniformlyOn_iff_forall_isCompact hU]"
},
{
"state_after": "case pos\nE : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nh : φ = ⊥\n⊢ ∀ (K : Set ℂ), K ⊆ U → IsCompact K → TendstoUniformlyOn (_root_.deriv ∘ F) (_root_.deriv f) φ K\n\ncase neg\nE : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nh : ¬φ = ⊥\n⊢ ∀ (K : Set ℂ), K ⊆ U → IsCompact K → TendstoUniformlyOn (_root_.deriv ∘ F) (_root_.deriv f) φ K",
"state_before": "E : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\n⊢ ∀ (K : Set ℂ), K ⊆ U → IsCompact K → TendstoUniformlyOn (_root_.deriv ∘ F) (_root_.deriv f) φ K",
"tactic": "by_cases φ = ⊥"
},
{
"state_after": "case neg\nE : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nh : ¬φ = ⊥\nthis : NeBot φ\n⊢ ∀ (K : Set ℂ), K ⊆ U → IsCompact K → TendstoUniformlyOn (_root_.deriv ∘ F) (_root_.deriv f) φ K",
"state_before": "case neg\nE : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nh : ¬φ = ⊥\n⊢ ∀ (K : Set ℂ), K ⊆ U → IsCompact K → TendstoUniformlyOn (_root_.deriv ∘ F) (_root_.deriv f) φ K",
"tactic": "haveI : φ.NeBot := neBot_iff.2 h"
},
{
"state_after": "case neg\nE : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nh : ¬φ = ⊥\nthis : NeBot φ\nK : Set ℂ\nhKU : K ⊆ U\nhK : IsCompact K\n⊢ TendstoUniformlyOn (_root_.deriv ∘ F) (_root_.deriv f) φ K",
"state_before": "case neg\nE : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nh : ¬φ = ⊥\nthis : NeBot φ\n⊢ ∀ (K : Set ℂ), K ⊆ U → IsCompact K → TendstoUniformlyOn (_root_.deriv ∘ F) (_root_.deriv f) φ K",
"tactic": "rintro K hKU hK"
},
{
"state_after": "case neg.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nh✝ : ¬φ = ⊥\nthis : NeBot φ\nK : Set ℂ\nhKU : K ⊆ U\nhK : IsCompact K\nδ : ℝ\nhδ : δ > 0\nhK4 : cthickening δ K ⊆ U\nh : TendstoUniformlyOn (_root_.deriv ∘ F) (cderiv δ f) φ K\n⊢ TendstoUniformlyOn (_root_.deriv ∘ F) (_root_.deriv f) φ K",
"state_before": "case neg\nE : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nh : ¬φ = ⊥\nthis : NeBot φ\nK : Set ℂ\nhKU : K ⊆ U\nhK : IsCompact K\n⊢ TendstoUniformlyOn (_root_.deriv ∘ F) (_root_.deriv f) φ K",
"tactic": "obtain ⟨δ, hδ, hK4, h⟩ := exists_cthickening_tendstoUniformlyOn hf hF hK hU hKU"
},
{
"state_after": "case neg.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K✝ : Set ℂ\nz✝ : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nh✝ : ¬φ = ⊥\nthis : NeBot φ\nK : Set ℂ\nhKU : K ⊆ U\nhK : IsCompact K\nδ : ℝ\nhδ : δ > 0\nhK4 : cthickening δ K ⊆ U\nh : TendstoUniformlyOn (_root_.deriv ∘ F) (cderiv δ f) φ K\nz : ℂ\nhz : z ∈ K\n⊢ closedBall z δ ⊆ U",
"state_before": "case neg.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nh✝ : ¬φ = ⊥\nthis : NeBot φ\nK : Set ℂ\nhKU : K ⊆ U\nhK : IsCompact K\nδ : ℝ\nhδ : δ > 0\nhK4 : cthickening δ K ⊆ U\nh : TendstoUniformlyOn (_root_.deriv ∘ F) (cderiv δ f) φ K\n⊢ TendstoUniformlyOn (_root_.deriv ∘ F) (_root_.deriv f) φ K",
"tactic": "refine' h.congr_right fun z hz => cderiv_eq_deriv hU (hf.differentiableOn hF hU) hδ _"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K✝ : Set ℂ\nz✝ : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nh✝ : ¬φ = ⊥\nthis : NeBot φ\nK : Set ℂ\nhKU : K ⊆ U\nhK : IsCompact K\nδ : ℝ\nhδ : δ > 0\nhK4 : cthickening δ K ⊆ U\nh : TendstoUniformlyOn (_root_.deriv ∘ F) (cderiv δ f) φ K\nz : ℂ\nhz : z ∈ K\n⊢ closedBall z δ ⊆ U",
"tactic": "exact (closedBall_subset_cthickening hz δ).trans hK4"
},
{
"state_after": "no goals",
"state_before": "case pos\nE : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nh : φ = ⊥\n⊢ ∀ (K : Set ℂ), K ⊆ U → IsCompact K → TendstoUniformlyOn (_root_.deriv ∘ F) (_root_.deriv f) φ K",
"tactic": "simp only [h, TendstoUniformlyOn, eventually_bot, imp_true_iff]"
}
] |
[
176,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/Topology/Basic.lean
|
exists_open_set_nhds
|
[] |
[
944,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
942,
1
] |
Mathlib/Topology/UniformSpace/AbstractCompletion.lean
|
AbstractCompletion.inverse_compare
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : UniformSpace α\npkg : AbstractCompletion α\nβ : Type ?u.20377\ninst✝ : UniformSpace β\npkg' : AbstractCompletion α\nuc : UniformContinuous (compare pkg pkg')\n⊢ compare pkg pkg' ∘ compare pkg' pkg = id",
"state_before": "α : Type u_1\ninst✝¹ : UniformSpace α\npkg : AbstractCompletion α\nβ : Type ?u.20377\ninst✝ : UniformSpace β\npkg' : AbstractCompletion α\n⊢ compare pkg pkg' ∘ compare pkg' pkg = id",
"tactic": "have uc := pkg.uniformContinuous_compare pkg'"
},
{
"state_after": "α : Type u_1\ninst✝¹ : UniformSpace α\npkg : AbstractCompletion α\nβ : Type ?u.20377\ninst✝ : UniformSpace β\npkg' : AbstractCompletion α\nuc : UniformContinuous (compare pkg pkg')\nuc' : UniformContinuous (compare pkg' pkg)\n⊢ compare pkg pkg' ∘ compare pkg' pkg = id",
"state_before": "α : Type u_1\ninst✝¹ : UniformSpace α\npkg : AbstractCompletion α\nβ : Type ?u.20377\ninst✝ : UniformSpace β\npkg' : AbstractCompletion α\nuc : UniformContinuous (compare pkg pkg')\n⊢ compare pkg pkg' ∘ compare pkg' pkg = id",
"tactic": "have uc' := pkg'.uniformContinuous_compare pkg"
},
{
"state_after": "α : Type u_1\ninst✝¹ : UniformSpace α\npkg : AbstractCompletion α\nβ : Type ?u.20377\ninst✝ : UniformSpace β\npkg' : AbstractCompletion α\nuc : UniformContinuous (compare pkg pkg')\nuc' : UniformContinuous (compare pkg' pkg)\n⊢ ∀ (a : α), (compare pkg pkg' ∘ compare pkg' pkg) (coe pkg' a) = id (coe pkg' a)",
"state_before": "α : Type u_1\ninst✝¹ : UniformSpace α\npkg : AbstractCompletion α\nβ : Type ?u.20377\ninst✝ : UniformSpace β\npkg' : AbstractCompletion α\nuc : UniformContinuous (compare pkg pkg')\nuc' : UniformContinuous (compare pkg' pkg)\n⊢ compare pkg pkg' ∘ compare pkg' pkg = id",
"tactic": "apply pkg'.funext (uc.comp uc').continuous continuous_id"
},
{
"state_after": "α : Type u_1\ninst✝¹ : UniformSpace α\npkg : AbstractCompletion α\nβ : Type ?u.20377\ninst✝ : UniformSpace β\npkg' : AbstractCompletion α\nuc : UniformContinuous (compare pkg pkg')\nuc' : UniformContinuous (compare pkg' pkg)\na : α\n⊢ (compare pkg pkg' ∘ compare pkg' pkg) (coe pkg' a) = id (coe pkg' a)",
"state_before": "α : Type u_1\ninst✝¹ : UniformSpace α\npkg : AbstractCompletion α\nβ : Type ?u.20377\ninst✝ : UniformSpace β\npkg' : AbstractCompletion α\nuc : UniformContinuous (compare pkg pkg')\nuc' : UniformContinuous (compare pkg' pkg)\n⊢ ∀ (a : α), (compare pkg pkg' ∘ compare pkg' pkg) (coe pkg' a) = id (coe pkg' a)",
"tactic": "intro a"
},
{
"state_after": "α : Type u_1\ninst✝¹ : UniformSpace α\npkg : AbstractCompletion α\nβ : Type ?u.20377\ninst✝ : UniformSpace β\npkg' : AbstractCompletion α\nuc : UniformContinuous (compare pkg pkg')\nuc' : UniformContinuous (compare pkg' pkg)\na : α\n⊢ coe pkg' a = id (coe pkg' a)",
"state_before": "α : Type u_1\ninst✝¹ : UniformSpace α\npkg : AbstractCompletion α\nβ : Type ?u.20377\ninst✝ : UniformSpace β\npkg' : AbstractCompletion α\nuc : UniformContinuous (compare pkg pkg')\nuc' : UniformContinuous (compare pkg' pkg)\na : α\n⊢ (compare pkg pkg' ∘ compare pkg' pkg) (coe pkg' a) = id (coe pkg' a)",
"tactic": "rw [comp_apply, pkg'.compare_coe pkg, pkg.compare_coe pkg']"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : UniformSpace α\npkg : AbstractCompletion α\nβ : Type ?u.20377\ninst✝ : UniformSpace β\npkg' : AbstractCompletion α\nuc : UniformContinuous (compare pkg pkg')\nuc' : UniformContinuous (compare pkg' pkg)\na : α\n⊢ coe pkg' a = id (coe pkg' a)",
"tactic": "rfl"
}
] |
[
267,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/Algebra/Support.lean
|
Function.support_prod_subset
|
[] |
[
404,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
402,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Int.floor_toNat
|
[] |
[
1583,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1582,
1
] |
Mathlib/Tactic/NormNum/Basic.lean
|
Mathlib.Meta.NormNum.isRat_inv_neg_one
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DivisionRing α\n⊢ (↑(Int.negOfNat 1))⁻¹ = ↑(Int.negOfNat 1)",
"tactic": "simp [inv_neg_one]"
}
] |
[
501,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
499,
1
] |
Mathlib/Computability/EpsilonNFA.lean
|
εNFA.εClosure_univ
|
[] |
[
69,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/Data/Finsupp/ToDfinsupp.lean
|
finsuppLequivDfinsupp_apply_apply
|
[
{
"state_after": "ι : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝⁴ : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : (m : M) → Decidable (m ≠ 0)\ninst✝ : Module R M\n⊢ ↑(finsuppLequivDfinsupp R) = Finsupp.toDfinsupp",
"state_before": "ι : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝⁴ : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : (m : M) → Decidable (m ≠ 0)\ninst✝ : Module R M\n⊢ ↑(finsuppLequivDfinsupp R) = Finsupp.toDfinsupp",
"tactic": "simp only [@LinearEquiv.coe_coe]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝⁴ : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : (m : M) → Decidable (m ≠ 0)\ninst✝ : Module R M\n⊢ ↑(finsuppLequivDfinsupp R) = Finsupp.toDfinsupp",
"tactic": "rfl"
}
] |
[
266,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.blockDiagonal_apply'
|
[] |
[
353,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
351,
1
] |
Mathlib/FieldTheory/Separable.lean
|
IsSeparable.separable
|
[] |
[
510,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
509,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
Class.coe_sep
|
[] |
[
1639,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1637,
1
] |
Mathlib/Algebra/Ring/Defs.lean
|
neg_mul_eq_neg_mul
|
[] |
[
305,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
304,
1
] |
Mathlib/Algebra/Ring/CompTypeclasses.lean
|
RingHomInvPair.of_ringEquiv
|
[] |
[
126,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.closure_induction₂
|
[] |
[
938,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
928,
1
] |
Mathlib/CategoryTheory/Adjunction/Basic.lean
|
CategoryTheory.Adjunction.CoreHomEquiv.homEquiv_naturality_right_symm
|
[
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : CoreHomEquiv F G\nX' X : C\nY Y' : D\nf : X ⟶ G.obj Y\ng : Y ⟶ Y'\n⊢ f ≫ G.map g = ↑(homEquiv adj X Y') (↑(homEquiv adj X Y).symm f ≫ g)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : CoreHomEquiv F G\nX' X : C\nY Y' : D\nf : X ⟶ G.obj Y\ng : Y ⟶ Y'\n⊢ ↑(homEquiv adj X Y').symm (f ≫ G.map g) = ↑(homEquiv adj X Y).symm f ≫ g",
"tactic": "rw [Equiv.symm_apply_eq]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : CoreHomEquiv F G\nX' X : C\nY Y' : D\nf : X ⟶ G.obj Y\ng : Y ⟶ Y'\n⊢ f ≫ G.map g = ↑(homEquiv adj X Y') (↑(homEquiv adj X Y).symm f ≫ g)",
"tactic": "simp"
}
] |
[
298,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
296,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
measurable_ciInf
|
[] |
[
1376,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1374,
1
] |
Std/Data/List/Init/Lemmas.lean
|
List.append_inj_left
|
[] |
[
66,
25
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
65,
1
] |
Mathlib/Data/Nat/Lattice.lean
|
Nat.iSup_lt_succ
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : CompleteLattice α\nu : ℕ → α\nn : ℕ\n⊢ (⨆ (k : ℕ) (_ : k < n + 1), u k) = (⨆ (k : ℕ) (_ : k < n), u k) ⊔ u n",
"tactic": "simp [Nat.lt_succ_iff_lt_or_eq, iSup_or, iSup_sup_eq]"
}
] |
[
181,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.diagonal_mul_diagonal
|
[
{
"state_after": "case a.h\nl : Type ?u.274619\nm : Type ?u.274622\nn : Type u_1\no : Type ?u.274628\nm' : o → Type ?u.274633\nn' : o → Type ?u.274638\nR : Type ?u.274641\nS : Type ?u.274644\nα : Type v\nβ : Type w\nγ : Type ?u.274651\ninst✝² : NonUnitalNonAssocSemiring α\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nd₁ d₂ : n → α\ni j : n\n⊢ (diagonal d₁ ⬝ diagonal d₂) i j = diagonal (fun i => d₁ i * d₂ i) i j",
"state_before": "l : Type ?u.274619\nm : Type ?u.274622\nn : Type u_1\no : Type ?u.274628\nm' : o → Type ?u.274633\nn' : o → Type ?u.274638\nR : Type ?u.274641\nS : Type ?u.274644\nα : Type v\nβ : Type w\nγ : Type ?u.274651\ninst✝² : NonUnitalNonAssocSemiring α\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nd₁ d₂ : n → α\n⊢ diagonal d₁ ⬝ diagonal d₂ = diagonal fun i => d₁ i * d₂ i",
"tactic": "ext i j"
},
{
"state_after": "no goals",
"state_before": "case a.h\nl : Type ?u.274619\nm : Type ?u.274622\nn : Type u_1\no : Type ?u.274628\nm' : o → Type ?u.274633\nn' : o → Type ?u.274638\nR : Type ?u.274641\nS : Type ?u.274644\nα : Type v\nβ : Type w\nγ : Type ?u.274651\ninst✝² : NonUnitalNonAssocSemiring α\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nd₁ d₂ : n → α\ni j : n\n⊢ (diagonal d₁ ⬝ diagonal d₂) i j = diagonal (fun i => d₁ i * d₂ i) i j",
"tactic": "by_cases i = j <;>\nsimp [h]"
}
] |
[
1040,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1036,
1
] |
Mathlib/Logic/Nontrivial.lean
|
nontrivial_of_ne
|
[] |
[
62,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
|
Real.isLittleO_log_id_atTop
|
[] |
[
370,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
369,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.coeff_index_single_self_X
|
[] |
[
413,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
412,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.coe_le_coe_iff
|
[] |
[
103,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
11
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.tmpUniformSpace_eq
|
[
{
"state_after": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.578081\nE : Type u_1\nEₗ : Type ?u.578087\nF : Type u_2\nFₗ : Type ?u.578093\nG : Type ?u.578096\nGₗ : Type ?u.578099\n𝓕 : Type ?u.578102\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx : E\n⊢ TopologicalAddGroup.toUniformSpace (E →SL[σ₁₂] F) = TopologicalAddGroup.toUniformSpace (E →SL[σ₁₂] F)",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.578081\nE : Type u_1\nEₗ : Type ?u.578087\nF : Type u_2\nFₗ : Type ?u.578093\nG : Type ?u.578096\nGₗ : Type ?u.578099\n𝓕 : Type ?u.578102\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx : E\n⊢ ContinuousLinearMap.tmpUniformSpace = uniformSpace",
"tactic": "rw [← @UniformAddGroup.toUniformSpace_eq _ ContinuousLinearMap.tmpUniformSpace, ←\n @UniformAddGroup.toUniformSpace_eq _ ContinuousLinearMap.uniformSpace]"
},
{
"state_after": "case h.e'_3\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.578081\nE : Type u_1\nEₗ : Type ?u.578087\nF : Type u_2\nFₗ : Type ?u.578093\nG : Type ?u.578096\nGₗ : Type ?u.578099\n𝓕 : Type ?u.578102\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx : E\n⊢ UniformSpace.toTopologicalSpace = UniformSpace.toTopologicalSpace",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.578081\nE : Type u_1\nEₗ : Type ?u.578087\nF : Type u_2\nFₗ : Type ?u.578093\nG : Type ?u.578096\nGₗ : Type ?u.578099\n𝓕 : Type ?u.578102\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx : E\n⊢ TopologicalAddGroup.toUniformSpace (E →SL[σ₁₂] F) = TopologicalAddGroup.toUniformSpace (E →SL[σ₁₂] F)",
"tactic": "congr! 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.578081\nE : Type u_1\nEₗ : Type ?u.578087\nF : Type u_2\nFₗ : Type ?u.578093\nG : Type ?u.578096\nGₗ : Type ?u.578099\n𝓕 : Type ?u.578102\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx : E\n⊢ UniformSpace.toTopologicalSpace = UniformSpace.toTopologicalSpace",
"tactic": "exact ContinuousLinearMap.tmp_topology_eq"
}
] |
[
394,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
388,
11
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.card_algHom_adjoin_integral
|
[
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\nα : E\nK : Type u_3\ninst✝¹ : Field K\ninst✝ : Algebra F K\nh : IsIntegral F α\nh_sep : Separable (minpoly F α)\nh_splits : Splits (algebraMap F K) (minpoly F α)\n⊢ Fintype.card ({ x // x ∈ F⟮α⟯ } →ₐ[F] K) = natDegree (minpoly F α)",
"tactic": "rw [AlgHom.card_of_powerBasis] <;>\n simp only [adjoin.powerBasis_dim, adjoin.powerBasis_gen, minpoly_gen h, h_sep, h_splits]"
}
] |
[
908,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
904,
1
] |
Mathlib/Data/Multiset/Bind.lean
|
Multiset.join_add
|
[] |
[
61,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
LinearMap.rTensor_comp
|
[
{
"state_after": "case H.h.h\nR : Type u_4\ninst✝¹⁴ : CommSemiring R\nR' : Type ?u.1245030\ninst✝¹³ : Monoid R'\nR'' : Type ?u.1245036\ninst✝¹² : Semiring R''\nM : Type u_1\nN : Type u_2\nP : Type u_5\nQ : Type u_3\nS : Type ?u.1245054\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid N\ninst✝⁹ : AddCommMonoid P\ninst✝⁸ : AddCommMonoid Q\ninst✝⁷ : AddCommMonoid S\ninst✝⁶ : Module R M\ninst✝⁵ : Module R N\ninst✝⁴ : Module R P\ninst✝³ : Module R Q\ninst✝² : Module R S\ninst✝¹ : DistribMulAction R' M\ninst✝ : Module R'' M\ng : P →ₗ[R] Q\nf : N →ₗ[R] P\nm : N\nn : M\n⊢ ↑(↑(compr₂ (TensorProduct.mk R N M) (rTensor M (comp g f))) m) n =\n ↑(↑(compr₂ (TensorProduct.mk R N M) (comp (rTensor M g) (rTensor M f))) m) n",
"state_before": "R : Type u_4\ninst✝¹⁴ : CommSemiring R\nR' : Type ?u.1245030\ninst✝¹³ : Monoid R'\nR'' : Type ?u.1245036\ninst✝¹² : Semiring R''\nM : Type u_1\nN : Type u_2\nP : Type u_5\nQ : Type u_3\nS : Type ?u.1245054\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid N\ninst✝⁹ : AddCommMonoid P\ninst✝⁸ : AddCommMonoid Q\ninst✝⁷ : AddCommMonoid S\ninst✝⁶ : Module R M\ninst✝⁵ : Module R N\ninst✝⁴ : Module R P\ninst✝³ : Module R Q\ninst✝² : Module R S\ninst✝¹ : DistribMulAction R' M\ninst✝ : Module R'' M\ng : P →ₗ[R] Q\nf : N →ₗ[R] P\n⊢ rTensor M (comp g f) = comp (rTensor M g) (rTensor M f)",
"tactic": "ext (m n)"
},
{
"state_after": "no goals",
"state_before": "case H.h.h\nR : Type u_4\ninst✝¹⁴ : CommSemiring R\nR' : Type ?u.1245030\ninst✝¹³ : Monoid R'\nR'' : Type ?u.1245036\ninst✝¹² : Semiring R''\nM : Type u_1\nN : Type u_2\nP : Type u_5\nQ : Type u_3\nS : Type ?u.1245054\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid N\ninst✝⁹ : AddCommMonoid P\ninst✝⁸ : AddCommMonoid Q\ninst✝⁷ : AddCommMonoid S\ninst✝⁶ : Module R M\ninst✝⁵ : Module R N\ninst✝⁴ : Module R P\ninst✝³ : Module R Q\ninst✝² : Module R S\ninst✝¹ : DistribMulAction R' M\ninst✝ : Module R'' M\ng : P →ₗ[R] Q\nf : N →ₗ[R] P\nm : N\nn : M\n⊢ ↑(↑(compr₂ (TensorProduct.mk R N M) (rTensor M (comp g f))) m) n =\n ↑(↑(compr₂ (TensorProduct.mk R N M) (comp (rTensor M g) (rTensor M f))) m) n",
"tactic": "simp only [compr₂_apply, mk_apply, comp_apply, rTensor_tmul]"
}
] |
[
1077,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1075,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.coe_mul
|
[] |
[
432,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
431,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.