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/Analysis/SpecialFunctions/Stirling.lean
|
Stirling.stirlingSeq'_antitone
|
[] |
[
201,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/MeasureTheory/PiSystem.lean
|
IsPiSystem.insert_univ
|
[
{
"state_after": "α : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns : Set α\nhs : s ∈ insert univ S\nt : Set α\nht : t ∈ insert univ S\nhst : Set.Nonempty (s ∩ t)\n⊢ s ∩ t ∈ insert univ S",
"state_before": "α : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\n⊢ IsPiSystem (insert univ S)",
"tactic": "intro s hs t ht hst"
},
{
"state_after": "case inl\nα : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns t : Set α\nht : t ∈ insert univ S\nhst : Set.Nonempty (s ∩ t)\nhs : s = univ\n⊢ s ∩ t ∈ insert univ S\n\ncase inr\nα : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns t : Set α\nht : t ∈ insert univ S\nhst : Set.Nonempty (s ∩ t)\nhs : s ∈ S\n⊢ s ∩ t ∈ insert univ S",
"state_before": "α : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns : Set α\nhs : s ∈ insert univ S\nt : Set α\nht : t ∈ insert univ S\nhst : Set.Nonempty (s ∩ t)\n⊢ s ∩ t ∈ insert univ S",
"tactic": "cases' hs with hs hs"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns t : Set α\nht : t ∈ insert univ S\nhst : Set.Nonempty (s ∩ t)\nhs : s = univ\n⊢ s ∩ t ∈ insert univ S",
"tactic": "cases' ht with ht ht <;> simp [hs, ht]"
},
{
"state_after": "case inr.inl\nα : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns t : Set α\nhst : Set.Nonempty (s ∩ t)\nhs : s ∈ S\nht : t = univ\n⊢ s ∩ t ∈ insert univ S\n\ncase inr.inr\nα : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns t : Set α\nhst : Set.Nonempty (s ∩ t)\nhs : s ∈ S\nht : t ∈ S\n⊢ s ∩ t ∈ insert univ S",
"state_before": "case inr\nα : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns t : Set α\nht : t ∈ insert univ S\nhst : Set.Nonempty (s ∩ t)\nhs : s ∈ S\n⊢ s ∩ t ∈ insert univ S",
"tactic": "cases' ht with ht ht"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nα : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns t : Set α\nhst : Set.Nonempty (s ∩ t)\nhs : s ∈ S\nht : t = univ\n⊢ s ∩ t ∈ insert univ S",
"tactic": "simp [hs, ht]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nα : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns t : Set α\nhst : Set.Nonempty (s ∩ t)\nhs : s ∈ S\nht : t ∈ S\n⊢ s ∩ t ∈ insert univ S",
"tactic": "exact Set.mem_insert_of_mem _ (h_pi s hs t ht hst)"
}
] |
[
104,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/LinearAlgebra/BilinearMap.lean
|
LinearMap.lcomp_apply
|
[] |
[
282,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
282,
1
] |
Mathlib/NumberTheory/LucasLehmer.lean
|
LucasLehmer.X.closed_form
|
[
{
"state_after": "case zero\nq : ℕ+\n⊢ ↑(s zero) = ω ^ 2 ^ zero + ωb ^ 2 ^ zero\n\ncase succ\nq : ℕ+\ni : ℕ\nih : ↑(s i) = ω ^ 2 ^ i + ωb ^ 2 ^ i\n⊢ ↑(s (succ i)) = ω ^ 2 ^ succ i + ωb ^ 2 ^ succ i",
"state_before": "q : ℕ+\ni : ℕ\n⊢ ↑(s i) = ω ^ 2 ^ i + ωb ^ 2 ^ i",
"tactic": "induction' i with i ih"
},
{
"state_after": "case zero\nq : ℕ+\n⊢ ↑4 = (2, 1) ^ 1 + (2, -1) ^ 1",
"state_before": "case zero\nq : ℕ+\n⊢ ↑(s zero) = ω ^ 2 ^ zero + ωb ^ 2 ^ zero",
"tactic": "dsimp [s, ω, ωb]"
},
{
"state_after": "no goals",
"state_before": "case zero\nq : ℕ+\n⊢ ↑4 = (2, 1) ^ 1 + (2, -1) ^ 1",
"tactic": "ext <;> norm_num"
},
{
"state_after": "no goals",
"state_before": "case succ\nq : ℕ+\ni : ℕ\nih : ↑(s i) = ω ^ 2 ^ i + ωb ^ 2 ^ i\n⊢ ↑(s (succ i)) = ω ^ 2 ^ succ i + ωb ^ 2 ^ succ i",
"tactic": "calc\n (s (i + 1) : X q) = (s i ^ 2 - 2 : ℤ) := rfl\n _ = (s i : X q) ^ 2 - 2 := by push_cast; rfl\n _ = (ω ^ 2 ^ i + ωb ^ 2 ^ i) ^ 2 - 2 := by rw [ih]\n _ = (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2 + 2 * (ωb ^ 2 ^ i * ω ^ 2 ^ i) - 2 := by ring\n _ = (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2 := by\n rw [← mul_pow ωb ω, ωb_mul_ω, one_pow, mul_one, add_sub_cancel]\n _ = ω ^ 2 ^ (i + 1) + ωb ^ 2 ^ (i + 1) := by rw [← pow_mul, ← pow_mul, _root_.pow_succ']"
},
{
"state_after": "q : ℕ+\ni : ℕ\nih : ↑(s i) = ω ^ 2 ^ i + ωb ^ 2 ^ i\n⊢ ↑(s i) ^ 2 - 2 = ↑(s i) ^ 2 - 2",
"state_before": "q : ℕ+\ni : ℕ\nih : ↑(s i) = ω ^ 2 ^ i + ωb ^ 2 ^ i\n⊢ ↑(s i ^ 2 - 2) = ↑(s i) ^ 2 - 2",
"tactic": "push_cast"
},
{
"state_after": "no goals",
"state_before": "q : ℕ+\ni : ℕ\nih : ↑(s i) = ω ^ 2 ^ i + ωb ^ 2 ^ i\n⊢ ↑(s i) ^ 2 - 2 = ↑(s i) ^ 2 - 2",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "q : ℕ+\ni : ℕ\nih : ↑(s i) = ω ^ 2 ^ i + ωb ^ 2 ^ i\n⊢ ↑(s i) ^ 2 - 2 = (ω ^ 2 ^ i + ωb ^ 2 ^ i) ^ 2 - 2",
"tactic": "rw [ih]"
},
{
"state_after": "no goals",
"state_before": "q : ℕ+\ni : ℕ\nih : ↑(s i) = ω ^ 2 ^ i + ωb ^ 2 ^ i\n⊢ (ω ^ 2 ^ i + ωb ^ 2 ^ i) ^ 2 - 2 = (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2 + 2 * (ωb ^ 2 ^ i * ω ^ 2 ^ i) - 2",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "q : ℕ+\ni : ℕ\nih : ↑(s i) = ω ^ 2 ^ i + ωb ^ 2 ^ i\n⊢ (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2 + 2 * (ωb ^ 2 ^ i * ω ^ 2 ^ i) - 2 = (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2",
"tactic": "rw [← mul_pow ωb ω, ωb_mul_ω, one_pow, mul_one, add_sub_cancel]"
},
{
"state_after": "no goals",
"state_before": "q : ℕ+\ni : ℕ\nih : ↑(s i) = ω ^ 2 ^ i + ωb ^ 2 ^ i\n⊢ (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2 = ω ^ 2 ^ (i + 1) + ωb ^ 2 ^ (i + 1)",
"tactic": "rw [← pow_mul, ← pow_mul, _root_.pow_succ']"
}
] |
[
391,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
380,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.add_div
|
[] |
[
1687,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1686,
11
] |
Std/Data/Int/DivMod.lean
|
Int.emod_eq_emod_iff_emod_sub_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "m n k : Int\n⊢ (m - k) % n = (k - k) % n ↔ (m - k) % n = 0",
"tactic": "simp [Int.sub_self]"
}
] |
[
451,
65
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
450,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.factorization_one_right
|
[] |
[
183,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
UniformContinuous₂.bicompl
|
[] |
[
1729,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1726,
1
] |
Mathlib/Data/Polynomial/HasseDeriv.lean
|
Polynomial.hasseDeriv_coeff
|
[
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn : ℕ\n⊢ (if n + k - k = n then ↑(choose (n + k) k) * coeff f (n + k) else 0) = ↑(choose (n + k) k) * coeff f (n + k)\n\ncase h₀\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn : ℕ\n⊢ ∀ (b : ℕ), b ∈ support f → b ≠ n + k → coeff (↑(monomial (b - k)) (↑(choose b k) * coeff f b)) n = 0\n\ncase h₁\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn : ℕ\n⊢ ¬n + k ∈ support f → coeff (↑(monomial (n + k - k)) (↑(choose (n + k) k) * coeff f (n + k))) n = 0",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn : ℕ\n⊢ coeff (↑(hasseDeriv k) f) n = ↑(choose (n + k) k) * coeff f (n + k)",
"tactic": "rw [hasseDeriv_apply, coeff_sum, sum_def, Finset.sum_eq_single (n + k), coeff_monomial]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn : ℕ\n⊢ (if n + k - k = n then ↑(choose (n + k) k) * coeff f (n + k) else 0) = ↑(choose (n + k) k) * coeff f (n + k)",
"tactic": "simp only [if_true, add_tsub_cancel_right, eq_self_iff_true]"
},
{
"state_after": "case h₀\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn i : ℕ\n_hi : i ∈ support f\nhink : i ≠ n + k\n⊢ coeff (↑(monomial (i - k)) (↑(choose i k) * coeff f i)) n = 0",
"state_before": "case h₀\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn : ℕ\n⊢ ∀ (b : ℕ), b ∈ support f → b ≠ n + k → coeff (↑(monomial (b - k)) (↑(choose b k) * coeff f b)) n = 0",
"tactic": "intro i _hi hink"
},
{
"state_after": "case h₀\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn i : ℕ\n_hi : i ∈ support f\nhink : i ≠ n + k\n⊢ (if i - k = n then ↑(choose i k) * coeff f i else 0) = 0",
"state_before": "case h₀\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn i : ℕ\n_hi : i ∈ support f\nhink : i ≠ n + k\n⊢ coeff (↑(monomial (i - k)) (↑(choose i k) * coeff f i)) n = 0",
"tactic": "rw [coeff_monomial]"
},
{
"state_after": "case pos\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn i : ℕ\n_hi : i ∈ support f\nhink : i ≠ n + k\nhik : i < k\n⊢ (if i - k = n then ↑(choose i k) * coeff f i else 0) = 0\n\ncase neg\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn i : ℕ\n_hi : i ∈ support f\nhink : i ≠ n + k\nhik : ¬i < k\n⊢ (if i - k = n then ↑(choose i k) * coeff f i else 0) = 0",
"state_before": "case h₀\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn i : ℕ\n_hi : i ∈ support f\nhink : i ≠ n + k\n⊢ (if i - k = n then ↑(choose i k) * coeff f i else 0) = 0",
"tactic": "by_cases hik : i < k"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn i : ℕ\n_hi : i ∈ support f\nhink : i ≠ n + k\nhik : i < k\n⊢ (if i - k = n then ↑(choose i k) * coeff f i else 0) = 0",
"tactic": "simp only [Nat.choose_eq_zero_of_lt hik, ite_self, Nat.cast_zero, MulZeroClass.zero_mul]"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn i : ℕ\n_hi : i ∈ support f\nhink : i ≠ n + k\nhik : k ≤ i\n⊢ (if i - k = n then ↑(choose i k) * coeff f i else 0) = 0",
"state_before": "case neg\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn i : ℕ\n_hi : i ∈ support f\nhink : i ≠ n + k\nhik : ¬i < k\n⊢ (if i - k = n then ↑(choose i k) * coeff f i else 0) = 0",
"tactic": "push_neg at hik"
},
{
"state_after": "case neg.hnc\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn i : ℕ\n_hi : i ∈ support f\nhink : i ≠ n + k\nhik : k ≤ i\n⊢ ¬i - k = n",
"state_before": "case neg\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn i : ℕ\n_hi : i ∈ support f\nhink : i ≠ n + k\nhik : k ≤ i\n⊢ (if i - k = n then ↑(choose i k) * coeff f i else 0) = 0",
"tactic": "rw [if_neg]"
},
{
"state_after": "case neg.hnc\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn i : ℕ\n_hi : i ∈ support f\nhik : k ≤ i\nhink : i - k = n\n⊢ i = n + k",
"state_before": "case neg.hnc\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn i : ℕ\n_hi : i ∈ support f\nhink : i ≠ n + k\nhik : k ≤ i\n⊢ ¬i - k = n",
"tactic": "contrapose! hink"
},
{
"state_after": "no goals",
"state_before": "case neg.hnc\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn i : ℕ\n_hi : i ∈ support f\nhik : k ≤ i\nhink : i - k = n\n⊢ i = n + k",
"tactic": "exact (tsub_eq_iff_eq_add_of_le hik).mp hink"
},
{
"state_after": "case h₁\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn : ℕ\nh : ¬n + k ∈ support f\n⊢ coeff (↑(monomial (n + k - k)) (↑(choose (n + k) k) * coeff f (n + k))) n = 0",
"state_before": "case h₁\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn : ℕ\n⊢ ¬n + k ∈ support f → coeff (↑(monomial (n + k - k)) (↑(choose (n + k) k) * coeff f (n + k))) n = 0",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case h₁\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn : ℕ\nh : ¬n + k ∈ support f\n⊢ coeff (↑(monomial (n + k - k)) (↑(choose (n + k) k) * coeff f (n + k))) n = 0",
"tactic": "simp only [not_mem_support_iff.mp h, monomial_zero_right, MulZeroClass.mul_zero, coeff_zero]"
}
] |
[
85,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Algebra/Lie/Subalgebra.lean
|
LieSubalgebra.homOfLe_injective
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\nL₂ : Type w\ninst✝¹ : LieRing L₂\ninst✝ : LieAlgebra R L₂\nf : L →ₗ⁅R⁆ L₂\nK K' : LieSubalgebra R L\nK₂ : LieSubalgebra R L₂\nh : K ≤ K'\nx y : { x // x ∈ K }\n⊢ ↑(homOfLe h) x = ↑(homOfLe h) y → x = y",
"tactic": "simp only [homOfLe_apply, imp_self, Subtype.mk_eq_mk, SetLike.coe_eq_coe]"
}
] |
[
614,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
613,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
not_bddBelow_iff
|
[] |
[
143,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.sin_neg_of_neg_of_neg_pi_lt
|
[] |
[
429,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
428,
1
] |
Mathlib/Algebra/CharZero/Lemmas.lean
|
RingHom.charZero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type u_2\ninst✝¹ : NonAssocSemiring R\ninst✝ : NonAssocSemiring S\nϕ : R →+* S\nhS : CharZero S\na b : ℕ\nh : ↑a = ↑b\n⊢ ↑a = ↑b",
"tactic": "rw [← map_natCast ϕ, ← map_natCast ϕ, h]"
}
] |
[
192,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/GroupTheory/Submonoid/Pointwise.lean
|
Submonoid.pointwise_smul_le_pointwise_smul_iff₀
|
[] |
[
330,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Std/Data/List/Lemmas.lean
|
List.append_eq_nil
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\np q : List α✝\n⊢ p ++ q = [] ↔ p = [] ∧ q = []",
"tactic": "cases p <;> simp"
}
] |
[
111,
19
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
110,
9
] |
Mathlib/Data/List/Chain.lean
|
List.Chain'.right_of_append
|
[] |
[
306,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
305,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.map_def
|
[] |
[
2040,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2039,
1
] |
Mathlib/Algebra/GroupPower/Order.lean
|
one_lt_sq_iff
|
[] |
[
597,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
596,
1
] |
Std/Logic.lean
|
ne_of_mem_of_not_mem
|
[] |
[
677,
78
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
677,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.SimpleFunc.norm_setToSimpleFunc_le_sum_mul_norm
|
[
{
"state_after": "case h.h\nα : Type u_1\nE : Type ?u.398702\nF : Type u_2\nF' : Type u_3\nG : Type ?u.398711\n𝕜 : Type ?u.398714\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nC : ℝ\nhT_norm : ∀ (s : Set α), MeasurableSet s → ‖T s‖ ≤ C * ENNReal.toReal (↑↑μ s)\nf : α →ₛ F\ni✝ : F\na✝ : i✝ ∈ SimpleFunc.range f\n⊢ ‖T (↑f ⁻¹' {i✝})‖ ≤ C * ENNReal.toReal (↑↑μ (↑f ⁻¹' {i✝}))",
"state_before": "α : Type u_1\nE : Type ?u.398702\nF : Type u_2\nF' : Type u_3\nG : Type ?u.398711\n𝕜 : Type ?u.398714\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nC : ℝ\nhT_norm : ∀ (s : Set α), MeasurableSet s → ‖T s‖ ≤ C * ENNReal.toReal (↑↑μ s)\nf : α →ₛ F\n⊢ ∑ x in SimpleFunc.range f, ‖T (↑f ⁻¹' {x})‖ * ‖x‖ ≤\n ∑ x in SimpleFunc.range f, C * ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) * ‖x‖",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "case h.h\nα : Type u_1\nE : Type ?u.398702\nF : Type u_2\nF' : Type u_3\nG : Type ?u.398711\n𝕜 : Type ?u.398714\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nC : ℝ\nhT_norm : ∀ (s : Set α), MeasurableSet s → ‖T s‖ ≤ C * ENNReal.toReal (↑↑μ s)\nf : α →ₛ F\ni✝ : F\na✝ : i✝ ∈ SimpleFunc.range f\n⊢ ‖T (↑f ⁻¹' {i✝})‖ ≤ C * ENNReal.toReal (↑↑μ (↑f ⁻¹' {i✝}))",
"tactic": "exact hT_norm _ <| SimpleFunc.measurableSet_fiber _ _"
},
{
"state_after": "α : Type u_1\nE : Type ?u.398702\nF : Type u_2\nF' : Type u_3\nG : Type ?u.398711\n𝕜 : Type ?u.398714\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nC : ℝ\nhT_norm : ∀ (s : Set α), MeasurableSet s → ‖T s‖ ≤ C * ENNReal.toReal (↑↑μ s)\nf : α →ₛ F\n⊢ ∑ x in SimpleFunc.range f, C * ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) * ‖x‖ ≤\n ∑ x in SimpleFunc.range f, C * ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) * ‖x‖",
"state_before": "α : Type u_1\nE : Type ?u.398702\nF : Type u_2\nF' : Type u_3\nG : Type ?u.398711\n𝕜 : Type ?u.398714\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nC : ℝ\nhT_norm : ∀ (s : Set α), MeasurableSet s → ‖T s‖ ≤ C * ENNReal.toReal (↑↑μ s)\nf : α →ₛ F\n⊢ ∑ x in SimpleFunc.range f, C * ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) * ‖x‖ ≤\n C * ∑ x in SimpleFunc.range f, ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) * ‖x‖",
"tactic": "simp_rw [mul_sum, ← mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.398702\nF : Type u_2\nF' : Type u_3\nG : Type ?u.398711\n𝕜 : Type ?u.398714\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nC : ℝ\nhT_norm : ∀ (s : Set α), MeasurableSet s → ‖T s‖ ≤ C * ENNReal.toReal (↑↑μ s)\nf : α →ₛ F\n⊢ ∑ x in SimpleFunc.range f, C * ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) * ‖x‖ ≤\n ∑ x in SimpleFunc.range f, C * ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) * ‖x‖",
"tactic": "rfl"
}
] |
[
587,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
578,
1
] |
Mathlib/Combinatorics/SetFamily/LYM.lean
|
Finset.slice_subset_falling
|
[] |
[
145,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/LinearAlgebra/AdicCompletion.lean
|
adicCompletion.coe_eval
|
[] |
[
235,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.adjoin.range_algebraMap_subset
|
[
{
"state_after": "F : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nx : E\nhx : x ∈ Set.range ↑(algebraMap F E)\n⊢ x ∈ ↑(adjoin F S)",
"state_before": "F : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\n⊢ Set.range ↑(algebraMap F E) ⊆ ↑(adjoin F S)",
"tactic": "intro x hx"
},
{
"state_after": "case intro\nF : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nx : E\nf : F\nhf : ↑(algebraMap F E) f = x\n⊢ x ∈ ↑(adjoin F S)",
"state_before": "F : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nx : E\nhx : x ∈ Set.range ↑(algebraMap F E)\n⊢ x ∈ ↑(adjoin F S)",
"tactic": "cases' hx with f hf"
},
{
"state_after": "case intro\nF : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nx : E\nf : F\nhf : ↑(algebraMap F E) f = x\n⊢ ↑(algebraMap F E) f ∈ ↑(adjoin F S)",
"state_before": "case intro\nF : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nx : E\nf : F\nhf : ↑(algebraMap F E) f = x\n⊢ x ∈ ↑(adjoin F S)",
"tactic": "rw [← hf]"
},
{
"state_after": "no goals",
"state_before": "case intro\nF : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nx : E\nf : F\nhf : ↑(algebraMap F E) f = x\n⊢ ↑(algebraMap F E) f ∈ ↑(adjoin F S)",
"tactic": "exact adjoin.algebraMap_mem F S f"
}
] |
[
314,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
310,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean
|
LinearIndependent.not_mem_span_image
|
[
{
"state_after": "ι : Type u'\nι' : Type ?u.344052\nR : Type u_1\nK : Type ?u.344058\nM : Type u_2\nM' : Type ?u.344064\nM'' : Type ?u.344067\nV : Type u\nV' : Type ?u.344072\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx✝ y : M\ninst✝ : Nontrivial R\nhv : LinearIndependent R v\ns : Set ι\nx : ι\nh : ¬x ∈ s\nh' : v x ∈ span R (v '' {x})\n⊢ ¬v x ∈ span R (v '' s)",
"state_before": "ι : Type u'\nι' : Type ?u.344052\nR : Type u_1\nK : Type ?u.344058\nM : Type u_2\nM' : Type ?u.344064\nM'' : Type ?u.344067\nV : Type u\nV' : Type ?u.344072\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx✝ y : M\ninst✝ : Nontrivial R\nhv : LinearIndependent R v\ns : Set ι\nx : ι\nh : ¬x ∈ s\n⊢ ¬v x ∈ span R (v '' s)",
"tactic": "have h' : v x ∈ Submodule.span R (v '' {x}) := by\n rw [Set.image_singleton]\n exact mem_span_singleton_self (v x)"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.344052\nR : Type u_1\nK : Type ?u.344058\nM : Type u_2\nM' : Type ?u.344064\nM'' : Type ?u.344067\nV : Type u\nV' : Type ?u.344072\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx✝ y : M\ninst✝ : Nontrivial R\nhv : LinearIndependent R v\ns : Set ι\nx : ι\nh : ¬x ∈ s\nh' : v x ∈ span R (v '' {x})\nw : v x ∈ span R (v '' s)\n⊢ False",
"state_before": "ι : Type u'\nι' : Type ?u.344052\nR : Type u_1\nK : Type ?u.344058\nM : Type u_2\nM' : Type ?u.344064\nM'' : Type ?u.344067\nV : Type u\nV' : Type ?u.344072\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx✝ y : M\ninst✝ : Nontrivial R\nhv : LinearIndependent R v\ns : Set ι\nx : ι\nh : ¬x ∈ s\nh' : v x ∈ span R (v '' {x})\n⊢ ¬v x ∈ span R (v '' s)",
"tactic": "intro w"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.344052\nR : Type u_1\nK : Type ?u.344058\nM : Type u_2\nM' : Type ?u.344064\nM'' : Type ?u.344067\nV : Type u\nV' : Type ?u.344072\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx✝ y : M\ninst✝ : Nontrivial R\nhv : LinearIndependent R v\ns : Set ι\nx : ι\nh : ¬x ∈ s\nh' : v x ∈ span R (v '' {x})\nw : v x ∈ span R (v '' s)\n⊢ v x = 0",
"state_before": "ι : Type u'\nι' : Type ?u.344052\nR : Type u_1\nK : Type ?u.344058\nM : Type u_2\nM' : Type ?u.344064\nM'' : Type ?u.344067\nV : Type u\nV' : Type ?u.344072\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx✝ y : M\ninst✝ : Nontrivial R\nhv : LinearIndependent R v\ns : Set ι\nx : ι\nh : ¬x ∈ s\nh' : v x ∈ span R (v '' {x})\nw : v x ∈ span R (v '' s)\n⊢ False",
"tactic": "apply LinearIndependent.ne_zero x hv"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.344052\nR : Type u_1\nK : Type ?u.344058\nM : Type u_2\nM' : Type ?u.344064\nM'' : Type ?u.344067\nV : Type u\nV' : Type ?u.344072\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx✝ y : M\ninst✝ : Nontrivial R\nhv : LinearIndependent R v\ns : Set ι\nx : ι\nh : ¬x ∈ s\nh' : v x ∈ span R (v '' {x})\nw : v x ∈ span R (v '' s)\n⊢ Disjoint {x} s",
"state_before": "ι : Type u'\nι' : Type ?u.344052\nR : Type u_1\nK : Type ?u.344058\nM : Type u_2\nM' : Type ?u.344064\nM'' : Type ?u.344067\nV : Type u\nV' : Type ?u.344072\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx✝ y : M\ninst✝ : Nontrivial R\nhv : LinearIndependent R v\ns : Set ι\nx : ι\nh : ¬x ∈ s\nh' : v x ∈ span R (v '' {x})\nw : v x ∈ span R (v '' s)\n⊢ v x = 0",
"tactic": "refine' disjoint_def.1 (hv.disjoint_span_image _) (v x) h' w"
},
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.344052\nR : Type u_1\nK : Type ?u.344058\nM : Type u_2\nM' : Type ?u.344064\nM'' : Type ?u.344067\nV : Type u\nV' : Type ?u.344072\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx✝ y : M\ninst✝ : Nontrivial R\nhv : LinearIndependent R v\ns : Set ι\nx : ι\nh : ¬x ∈ s\nh' : v x ∈ span R (v '' {x})\nw : v x ∈ span R (v '' s)\n⊢ Disjoint {x} s",
"tactic": "simpa using h"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.344052\nR : Type u_1\nK : Type ?u.344058\nM : Type u_2\nM' : Type ?u.344064\nM'' : Type ?u.344067\nV : Type u\nV' : Type ?u.344072\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx✝ y : M\ninst✝ : Nontrivial R\nhv : LinearIndependent R v\ns : Set ι\nx : ι\nh : ¬x ∈ s\n⊢ v x ∈ span R {v x}",
"state_before": "ι : Type u'\nι' : Type ?u.344052\nR : Type u_1\nK : Type ?u.344058\nM : Type u_2\nM' : Type ?u.344064\nM'' : Type ?u.344067\nV : Type u\nV' : Type ?u.344072\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx✝ y : M\ninst✝ : Nontrivial R\nhv : LinearIndependent R v\ns : Set ι\nx : ι\nh : ¬x ∈ s\n⊢ v x ∈ span R (v '' {x})",
"tactic": "rw [Set.image_singleton]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.344052\nR : Type u_1\nK : Type ?u.344058\nM : Type u_2\nM' : Type ?u.344064\nM'' : Type ?u.344067\nV : Type u\nV' : Type ?u.344072\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx✝ y : M\ninst✝ : Nontrivial R\nhv : LinearIndependent R v\ns : Set ι\nx : ι\nh : ¬x ∈ s\n⊢ v x ∈ span R {v x}",
"tactic": "exact mem_span_singleton_self (v x)"
}
] |
[
628,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
620,
1
] |
Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
|
finite_set_of_fin_dim_affineIndependent
|
[] |
[
96,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.neg_pi_div_two_ne_zero
|
[
{
"state_after": "⊢ -π / 2 ≠ 0",
"state_before": "⊢ ↑(-π / 2) ≠ 0",
"tactic": "rw [← toReal_injective.ne_iff, toReal_neg_pi_div_two, toReal_zero]"
},
{
"state_after": "no goals",
"state_before": "⊢ -π / 2 ≠ 0",
"tactic": "exact div_ne_zero (neg_ne_zero.2 Real.pi_ne_zero) two_ne_zero"
}
] |
[
639,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
637,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.range_coe_ennreal
|
[] |
[
547,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
543,
9
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.isCycle_copy
|
[
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu' : V\np : Walk G u' u'\n⊢ IsCycle (Walk.copy p (_ : u' = u') (_ : u' = u')) ↔ IsCycle p",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu u' : V\np : Walk G u u\nhu : u = u'\n⊢ IsCycle (Walk.copy p hu hu) ↔ IsCycle p",
"tactic": "subst_vars"
},
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu' : V\np : Walk G u' u'\n⊢ IsCycle (Walk.copy p (_ : u' = u') (_ : u' = u')) ↔ IsCycle p",
"tactic": "rfl"
}
] |
[
914,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
911,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.sup_mem
|
[] |
[
274,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
272,
1
] |
Mathlib/Order/BoundedOrder.lean
|
exists_ge_and_iff_exists
|
[] |
[
603,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
601,
1
] |
Mathlib/Algebra/Algebra/Unitization.lean
|
Unitization.inl_one
|
[] |
[
386,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
385,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.type_fintype
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.240276\nγ : Type ?u.240279\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : Fintype α\n⊢ type r = ↑(Fintype.card α)",
"tactic": "rw [← card_eq_nat, card_type, mk_fintype]"
}
] |
[
1587,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1586,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Fin.univ_succ
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.103890\nβ : Type ?u.103893\nγ : Type ?u.103896\nn : ℕ\n⊢ ¬0 ∈ map { toFun := succ, inj' := (_ : Injective succ) } univ",
"tactic": "simp [map_eq_image]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.103890\nβ : Type ?u.103893\nγ : Type ?u.103896\nn : ℕ\n⊢ univ =\n cons 0 (map { toFun := succ, inj' := (_ : Injective succ) } univ)\n (_ : ¬0 ∈ map { toFun := succ, inj' := (_ : Injective succ) } univ)",
"tactic": "simp [map_eq_image]"
}
] |
[
836,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
833,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
AddSubmonoid.mem_iSup_iff_exists_dfinsupp
|
[] |
[
1981,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1979,
1
] |
Std/Data/Int/DivMod.lean
|
Int.zero_mod
|
[
{
"state_after": "no goals",
"state_before": "b : Int\n⊢ mod 0 b = 0",
"tactic": "cases b <;> simp [mod]"
}
] |
[
248,
78
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
248,
9
] |
Mathlib/Order/Cover.lean
|
Covby.ne'
|
[] |
[
281,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
1
] |
Mathlib/Order/Monotone/Union.lean
|
StrictAntiOn.Iic_union_Ici
|
[] |
[
76,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
11
] |
Mathlib/LinearAlgebra/Basis/Bilinear.lean
|
LinearMap.ext_basis
|
[] |
[
48,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
46,
1
] |
Mathlib/Data/Polynomial/Monic.lean
|
Polynomial.monic_of_isUnit_leadingCoeff_inv_smul
|
[
{
"state_after": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : IsUnit (leadingCoeff p)\n⊢ ↑(IsUnit.unit h)⁻¹ • leadingCoeff p = 1",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : IsUnit (leadingCoeff p)\n⊢ Monic ((IsUnit.unit h)⁻¹ • p)",
"tactic": "rw [Monic.def, leadingCoeff_smul_of_smul_regular _ (isSMulRegular_of_group _), Units.smul_def]"
},
{
"state_after": "case intro\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nk : Rˣ\nhk : ↑k = leadingCoeff p\n⊢ ↑(IsUnit.unit (_ : ∃ u, ↑u = leadingCoeff p))⁻¹ • leadingCoeff p = 1",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : IsUnit (leadingCoeff p)\n⊢ ↑(IsUnit.unit h)⁻¹ • leadingCoeff p = 1",
"tactic": "obtain ⟨k, hk⟩ := h"
},
{
"state_after": "case intro\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nk : Rˣ\nhk : ↑k = leadingCoeff p\n⊢ k = IsUnit.unit (_ : IsUnit ↑k) * 1",
"state_before": "case intro\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nk : Rˣ\nhk : ↑k = leadingCoeff p\n⊢ ↑(IsUnit.unit (_ : ∃ u, ↑u = leadingCoeff p))⁻¹ • leadingCoeff p = 1",
"tactic": "simp only [← hk, smul_eq_mul, ← Units.val_mul, Units.val_eq_one, inv_mul_eq_iff_eq_mul]"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nk : Rˣ\nhk : ↑k = leadingCoeff p\n⊢ k = IsUnit.unit (_ : IsUnit ↑k) * 1",
"tactic": "simp [Units.ext_iff, IsUnit.unit_spec]"
}
] |
[
521,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
516,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
span_eq_top_of_linearIndependent_of_card_eq_finrank
|
[
{
"state_after": "case pos\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\nhι : Nonempty ι\ninst✝ : Fintype ι\nb : ι → V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card ι = finrank K V\nfin : FiniteDimensional K V\n⊢ span K (Set.range b) = ⊤\n\ncase neg\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\nhι : Nonempty ι\ninst✝ : Fintype ι\nb : ι → V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card ι = finrank K V\nfin : ¬FiniteDimensional K V\n⊢ span K (Set.range b) = ⊤",
"state_before": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\nhι : Nonempty ι\ninst✝ : Fintype ι\nb : ι → V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card ι = finrank K V\n⊢ span K (Set.range b) = ⊤",
"tactic": "by_cases fin : FiniteDimensional K V"
},
{
"state_after": "case pos\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\nhι : Nonempty ι\ninst✝ : Fintype ι\nb : ι → V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card ι = finrank K V\nfin : FiniteDimensional K V\n⊢ span K (Set.range b) = ⊤",
"state_before": "case pos\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\nhι : Nonempty ι\ninst✝ : Fintype ι\nb : ι → V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card ι = finrank K V\nfin : FiniteDimensional K V\n⊢ span K (Set.range b) = ⊤",
"tactic": "replace fin : FiniteDimensional _ _ := fin"
},
{
"state_after": "case pos\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\nhι : Nonempty ι\ninst✝ : Fintype ι\nb : ι → V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card ι = finrank K V\nfin : FiniteDimensional K V\nne_top : ¬span K (Set.range b) = ⊤\n⊢ False",
"state_before": "case pos\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\nhι : Nonempty ι\ninst✝ : Fintype ι\nb : ι → V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card ι = finrank K V\nfin : FiniteDimensional K V\n⊢ span K (Set.range b) = ⊤",
"tactic": "by_contra ne_top"
},
{
"state_after": "case pos\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\nhι : Nonempty ι\ninst✝ : Fintype ι\nb : ι → V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card ι = finrank K V\nfin : FiniteDimensional K V\nne_top : ¬span K (Set.range b) = ⊤\nlt_top : span K (Set.range b) < ⊤\n⊢ False",
"state_before": "case pos\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\nhι : Nonempty ι\ninst✝ : Fintype ι\nb : ι → V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card ι = finrank K V\nfin : FiniteDimensional K V\nne_top : ¬span K (Set.range b) = ⊤\n⊢ False",
"tactic": "have lt_top : span K (Set.range b) < ⊤ := lt_of_le_of_ne le_top ne_top"
},
{
"state_after": "no goals",
"state_before": "case pos\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\nhι : Nonempty ι\ninst✝ : Fintype ι\nb : ι → V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card ι = finrank K V\nfin : FiniteDimensional K V\nne_top : ¬span K (Set.range b) = ⊤\nlt_top : span K (Set.range b) < ⊤\n⊢ False",
"tactic": "exact ne_of_lt (Submodule.finrank_lt lt_top)\n (_root_.trans (finrank_span_eq_card lin_ind) card_eq)"
},
{
"state_after": "case neg.h\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\nhι : Nonempty ι\ninst✝ : Fintype ι\nb : ι → V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card ι = finrank K V\nfin : ¬FiniteDimensional K V\n⊢ False",
"state_before": "case neg\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\nhι : Nonempty ι\ninst✝ : Fintype ι\nb : ι → V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card ι = finrank K V\nfin : ¬FiniteDimensional K V\n⊢ span K (Set.range b) = ⊤",
"tactic": "exfalso"
},
{
"state_after": "case neg.h\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\nhι : Nonempty ι\ninst✝ : Fintype ι\nb : ι → V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card ι = finrank K V\nfin : ¬FiniteDimensional K V\n⊢ 0 = Fintype.card ι",
"state_before": "case neg.h\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\nhι : Nonempty ι\ninst✝ : Fintype ι\nb : ι → V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card ι = finrank K V\nfin : ¬FiniteDimensional K V\n⊢ False",
"tactic": "apply ne_of_lt (Fintype.card_pos_iff.mpr hι)"
},
{
"state_after": "case neg.h\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\nhι : Nonempty ι\ninst✝ : Fintype ι\nb : ι → V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card ι = finrank K V\nfin : ¬FiniteDimensional K V\n⊢ Fintype.card ι = 0",
"state_before": "case neg.h\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\nhι : Nonempty ι\ninst✝ : Fintype ι\nb : ι → V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card ι = finrank K V\nfin : ¬FiniteDimensional K V\n⊢ 0 = Fintype.card ι",
"tactic": "symm"
},
{
"state_after": "case neg.h\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\nhι : Nonempty ι\ninst✝ : Fintype ι\nb : ι → V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card ι = finrank K V\nfin : ¬IsNoetherian K V\n⊢ Fintype.card ι = 0",
"state_before": "case neg.h\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\nhι : Nonempty ι\ninst✝ : Fintype ι\nb : ι → V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card ι = finrank K V\nfin : ¬FiniteDimensional K V\n⊢ Fintype.card ι = 0",
"tactic": "replace fin := (not_iff_not.2 IsNoetherian.iff_fg).2 fin"
},
{
"state_after": "no goals",
"state_before": "case neg.h\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\nhι : Nonempty ι\ninst✝ : Fintype ι\nb : ι → V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card ι = finrank K V\nfin : ¬IsNoetherian K V\n⊢ Fintype.card ι = 0",
"tactic": "calc\n Fintype.card ι = finrank K V := card_eq\n _ = 0 := dif_neg (mt IsNoetherian.iff_rank_lt_aleph0.mpr fin)"
}
] |
[
1185,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1167,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
MulEquiv.map_prod
|
[] |
[
227,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
11
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.trans_source'
|
[] |
[
826,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
825,
1
] |
Mathlib/Tactic/LinearCombination.lean
|
Mathlib.Tactic.LinearCombination.c_div_pf
|
[] |
[
49,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/Data/Nat/Cast/Basic.lean
|
Nat.cast_commute
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.6256\ninst✝ : NonAssocSemiring α\nn : ℕ\nx : α\n⊢ Commute (↑n) x",
"tactic": "induction n with\n| zero => rw [Nat.cast_zero]; exact Commute.zero_left x\n| succ n ihn => rw [Nat.cast_succ]; exact ihn.add_left (Commute.one_left x)"
},
{
"state_after": "case zero\nα : Type u_1\nβ : Type ?u.6256\ninst✝ : NonAssocSemiring α\nx : α\n⊢ Commute 0 x",
"state_before": "case zero\nα : Type u_1\nβ : Type ?u.6256\ninst✝ : NonAssocSemiring α\nx : α\n⊢ Commute (↑zero) x",
"tactic": "rw [Nat.cast_zero]"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u_1\nβ : Type ?u.6256\ninst✝ : NonAssocSemiring α\nx : α\n⊢ Commute 0 x",
"tactic": "exact Commute.zero_left x"
},
{
"state_after": "case succ\nα : Type u_1\nβ : Type ?u.6256\ninst✝ : NonAssocSemiring α\nx : α\nn : ℕ\nihn : Commute (↑n) x\n⊢ Commute (↑n + 1) x",
"state_before": "case succ\nα : Type u_1\nβ : Type ?u.6256\ninst✝ : NonAssocSemiring α\nx : α\nn : ℕ\nihn : Commute (↑n) x\n⊢ Commute (↑(succ n)) x",
"tactic": "rw [Nat.cast_succ]"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type u_1\nβ : Type ?u.6256\ninst✝ : NonAssocSemiring α\nx : α\nn : ℕ\nihn : Commute (↑n) x\n⊢ Commute (↑n + 1) x",
"tactic": "exact ihn.add_left (Commute.one_left x)"
}
] |
[
70,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.cos_add_sin_mul_I_pow
|
[
{
"state_after": "x y : ℂ\nn : ℕ\nz : ℂ\n⊢ exp (z * I) ^ n = exp (↑n * z * I)",
"state_before": "x y : ℂ\nn : ℕ\nz : ℂ\n⊢ (cos z + sin z * I) ^ n = cos (↑n * z) + sin (↑n * z) * I",
"tactic": "rw [← exp_mul_I, ← exp_mul_I]"
},
{
"state_after": "case zero\nx y z : ℂ\n⊢ exp (z * I) ^ Nat.zero = exp (↑Nat.zero * z * I)\n\ncase succ\nx y z : ℂ\nn : ℕ\nih : exp (z * I) ^ n = exp (↑n * z * I)\n⊢ exp (z * I) ^ Nat.succ n = exp (↑(Nat.succ n) * z * I)",
"state_before": "x y : ℂ\nn : ℕ\nz : ℂ\n⊢ exp (z * I) ^ n = exp (↑n * z * I)",
"tactic": "induction' n with n ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nx y z : ℂ\n⊢ exp (z * I) ^ Nat.zero = exp (↑Nat.zero * z * I)",
"tactic": "rw [pow_zero, Nat.cast_zero, zero_mul, zero_mul, exp_zero]"
},
{
"state_after": "no goals",
"state_before": "case succ\nx y z : ℂ\nn : ℕ\nih : exp (z * I) ^ n = exp (↑n * z * I)\n⊢ exp (z * I) ^ Nat.succ n = exp (↑(Nat.succ n) * z * I)",
"tactic": "rw [pow_succ', ih, Nat.cast_succ, add_mul, add_mul, one_mul, exp_add]"
}
] |
[
1121,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1116,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.closedBall_zero'
|
[] |
[
692,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
691,
1
] |
Mathlib/Order/Filter/ENNReal.lean
|
ENNReal.limsup_liminf_le_liminf_limsup
|
[
{
"state_after": "α : Type u_2\nf✝ : Filter α\nβ : Type u_1\ninst✝¹ : Countable β\nf : Filter α\ninst✝ : CountableInterFilter f\ng : Filter β\nu : α → β → ℝ≥0∞\n⊢ ∀ (i : β), ∀ᶠ (x : α) in f, u x i ≤ limsup (fun a' => u a' i) f",
"state_before": "α : Type u_2\nf✝ : Filter α\nβ : Type u_1\ninst✝¹ : Countable β\nf : Filter α\ninst✝ : CountableInterFilter f\ng : Filter β\nu : α → β → ℝ≥0∞\n⊢ ∀ᶠ (a : α) in f, ∀ (b : β), u a b ≤ limsup (fun a' => u a' b) f",
"tactic": "rw [eventually_countable_forall]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nf✝ : Filter α\nβ : Type u_1\ninst✝¹ : Countable β\nf : Filter α\ninst✝ : CountableInterFilter f\ng : Filter β\nu : α → β → ℝ≥0∞\n⊢ ∀ (i : β), ∀ᶠ (x : α) in f, u x i ≤ limsup (fun a' => u a' i) f",
"tactic": "exact fun b => ENNReal.eventually_le_limsup fun a => u a b"
}
] |
[
97,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
ENNReal.rpow_eq_pow
|
[] |
[
300,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
299,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
|
AffineMap.decomp
|
[
{
"state_after": "case h\nk : Type u_1\nV1 : Type u_2\nP1 : Type ?u.585564\nV2 : Type u_3\nP2 : Type ?u.585570\nV3 : Type ?u.585573\nP3 : Type ?u.585576\nV4 : Type ?u.585579\nP4 : Type ?u.585582\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\nf : V1 →ᵃ[k] V2\nx : V1\n⊢ ↑f x = (↑f.linear + fun x => ↑f 0) x",
"state_before": "k : Type u_1\nV1 : Type u_2\nP1 : Type ?u.585564\nV2 : Type u_3\nP2 : Type ?u.585570\nV3 : Type ?u.585573\nP3 : Type ?u.585576\nV4 : Type ?u.585579\nP4 : Type ?u.585582\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\nf : V1 →ᵃ[k] V2\n⊢ ↑f = ↑f.linear + fun x => ↑f 0",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nk : Type u_1\nV1 : Type u_2\nP1 : Type ?u.585564\nV2 : Type u_3\nP2 : Type ?u.585570\nV3 : Type ?u.585573\nP3 : Type ?u.585576\nV4 : Type ?u.585579\nP4 : Type ?u.585582\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\nf : V1 →ᵃ[k] V2\nx : V1\n⊢ ↑f x = (↑f.linear + fun x => ↑f 0) x",
"tactic": "calc\n f x = f.linear x +ᵥ f 0 := by rw [← f.map_vadd, vadd_eq_add, add_zero]\n _ = (f.linear + fun _ : V1 => f 0) x := rfl"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV1 : Type u_2\nP1 : Type ?u.585564\nV2 : Type u_3\nP2 : Type ?u.585570\nV3 : Type ?u.585573\nP3 : Type ?u.585576\nV4 : Type ?u.585579\nP4 : Type ?u.585582\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\nf : V1 →ᵃ[k] V2\nx : V1\n⊢ ↑f x = ↑f.linear x +ᵥ ↑f 0",
"tactic": "rw [← f.map_vadd, vadd_eq_add, add_zero]"
}
] |
[
670,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
666,
1
] |
Mathlib/Topology/Homotopy/HomotopyGroup.lean
|
genLoopOneEquivPathSelf_homotopic_iff
|
[
{
"state_after": "no goals",
"state_before": "X : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx : X\nf g : GenLoop 1 x\n⊢ Path.Homotopic (↑genLoopOneEquivPathSelf f) (↑genLoopOneEquivPathSelf g) ↔ GenLoop.Homotopic f g",
"tactic": "rw [← genLoopOneEquivPathSelf_symm_homotopic_iff, Equiv.symm_apply_apply, Equiv.symm_apply_apply]"
}
] |
[
278,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
276,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean
|
EulerSine.integral_cos_mul_cos_pow_even
|
[
{
"state_after": "case h.e'_2.h.e'_5.h.e'_6\nz : ℂ\nn✝ n : ℕ\nhz : z ≠ 0\n⊢ z ^ 2 / (↑n + 1) ^ 2 = 4 * z ^ 2 / ↑(2 * n + 2) ^ 2\n\ncase h.e'_3.h.e'_5.h.e'_5\nz : ℂ\nn✝ n : ℕ\nhz : z ≠ 0\n⊢ 2 * ↑n + 1 = ↑(2 * n + 2) - 1\n\ncase h.e'_3.h.e'_5.h.e'_6\nz : ℂ\nn✝ n : ℕ\nhz : z ≠ 0\n⊢ 2 * ↑n + 2 = ↑(2 * n + 2)",
"state_before": "z : ℂ\nn✝ n : ℕ\nhz : z ≠ 0\n⊢ ((1 - z ^ 2 / (↑n + 1) ^ 2) * ∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n + 2)) =\n (2 * ↑n + 1) / (2 * ↑n + 2) * ∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)",
"tactic": "convert integral_cos_mul_cos_pow (by linarith : 2 ≤ 2 * n + 2) hz using 3"
},
{
"state_after": "no goals",
"state_before": "z : ℂ\nn✝ n : ℕ\nhz : z ≠ 0\n⊢ 2 ≤ 2 * n + 2",
"tactic": "linarith"
},
{
"state_after": "case h.e'_2.h.e'_5.h.e'_6\nz : ℂ\nn✝ n : ℕ\nhz : z ≠ 0\n⊢ z ^ 2 / (↑n + 1) ^ 2 = 4 * z ^ 2 / (2 * ↑n + 2) ^ 2",
"state_before": "case h.e'_2.h.e'_5.h.e'_6\nz : ℂ\nn✝ n : ℕ\nhz : z ≠ 0\n⊢ z ^ 2 / (↑n + 1) ^ 2 = 4 * z ^ 2 / ↑(2 * n + 2) ^ 2",
"tactic": "simp only [Nat.cast_add, Nat.cast_mul, Nat.cast_two]"
},
{
"state_after": "case h.e'_2.h.e'_5.h.e'_6\nz : ℂ\nn✝ n : ℕ\nhz : z ≠ 0\n⊢ z ^ 2 / (↑n + 1) ^ 2 = 4 * z ^ 2 / (2 * ↑n + 2 * 1) ^ 2",
"state_before": "case h.e'_2.h.e'_5.h.e'_6\nz : ℂ\nn✝ n : ℕ\nhz : z ≠ 0\n⊢ z ^ 2 / (↑n + 1) ^ 2 = 4 * z ^ 2 / (2 * ↑n + 2) ^ 2",
"tactic": "nth_rw 2 [← mul_one (2 : ℂ)]"
},
{
"state_after": "case h.e'_2.h.e'_5.h.e'_6\nz : ℂ\nn✝ n : ℕ\nhz : z ≠ 0\n⊢ z ^ 2 / (↑n + 1) ^ 2 = 4 * z ^ 2 / 2 ^ 2 / (↑n + 1) ^ 2",
"state_before": "case h.e'_2.h.e'_5.h.e'_6\nz : ℂ\nn✝ n : ℕ\nhz : z ≠ 0\n⊢ z ^ 2 / (↑n + 1) ^ 2 = 4 * z ^ 2 / (2 * ↑n + 2 * 1) ^ 2",
"tactic": "rw [← mul_add, mul_pow, ← div_div]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_5.h.e'_6\nz : ℂ\nn✝ n : ℕ\nhz : z ≠ 0\n⊢ z ^ 2 / (↑n + 1) ^ 2 = 4 * z ^ 2 / 2 ^ 2 / (↑n + 1) ^ 2",
"tactic": "ring"
},
{
"state_after": "case h.e'_3.h.e'_5.h.e'_5\nz : ℂ\nn✝ n : ℕ\nhz : z ≠ 0\n⊢ 2 * ↑n + 1 = 2 * ↑n + 2 - 1",
"state_before": "case h.e'_3.h.e'_5.h.e'_5\nz : ℂ\nn✝ n : ℕ\nhz : z ≠ 0\n⊢ 2 * ↑n + 1 = ↑(2 * n + 2) - 1",
"tactic": "push_cast"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_5.h.e'_5\nz : ℂ\nn✝ n : ℕ\nhz : z ≠ 0\n⊢ 2 * ↑n + 1 = 2 * ↑n + 2 - 1",
"tactic": "ring"
},
{
"state_after": "case h.e'_3.h.e'_5.h.e'_6\nz : ℂ\nn✝ n : ℕ\nhz : z ≠ 0\n⊢ 2 * ↑n + 2 = 2 * ↑n + 2",
"state_before": "case h.e'_3.h.e'_5.h.e'_6\nz : ℂ\nn✝ n : ℕ\nhz : z ≠ 0\n⊢ 2 * ↑n + 2 = ↑(2 * n + 2)",
"tactic": "push_cast"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_5.h.e'_6\nz : ℂ\nn✝ n : ℕ\nhz : z ≠ 0\n⊢ 2 * ↑n + 2 = 2 * ↑n + 2",
"tactic": "ring"
}
] |
[
183,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
hasSum_extend_zero
|
[
{
"state_after": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.24186\ninst✝¹ : AddCommMonoid α\ninst✝ : TopologicalSpace α\nf g✝ : β → α\na b : α\ns : Finset β\ng : β → γ\nhg : Injective g\n⊢ ∀ (x : γ), ¬x ∈ Set.range g → extend g f 0 x = 0",
"state_before": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.24186\ninst✝¹ : AddCommMonoid α\ninst✝ : TopologicalSpace α\nf g✝ : β → α\na b : α\ns : Finset β\ng : β → γ\nhg : Injective g\n⊢ HasSum (extend g f 0) a ↔ HasSum f a",
"tactic": "rw [← hg.hasSum_iff, extend_comp hg]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.24186\ninst✝¹ : AddCommMonoid α\ninst✝ : TopologicalSpace α\nf g✝ : β → α\na b : α\ns : Finset β\ng : β → γ\nhg : Injective g\n⊢ ∀ (x : γ), ¬x ∈ Set.range g → extend g f 0 x = 0",
"tactic": "exact extend_apply' _ _"
}
] |
[
146,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
9
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.adjoin_finite_isCompactElement
|
[] |
[
626,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
625,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.diff_subset_comm
|
[] |
[
1951,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1950,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
div_nonpos_iff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.139274\nα : Type u_1\nβ : Type ?u.139280\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\n⊢ a / b ≤ 0 ↔ 0 ≤ a ∧ b ≤ 0 ∨ a ≤ 0 ∧ 0 ≤ b",
"tactic": "simp [division_def, mul_nonpos_iff]"
}
] |
[
693,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
692,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Function.piCongrLeft'_symm_update
|
[
{
"state_after": "no goals",
"state_before": "α : Sort u_1\nβ : Sort u_2\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nP : α → Sort u_3\ne : α ≃ β\nf : (b : β) → P (↑e.symm b)\nb : β\nx : P (↑e.symm b)\n⊢ ↑(Equiv.piCongrLeft' P e).symm (update f b x) = update (↑(Equiv.piCongrLeft' P e).symm f) (↑e.symm b) x",
"tactic": "simp [(e.piCongrLeft' P).symm_apply_eq, piCongrLeft'_update]"
}
] |
[
1949,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1946,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.iUnion_pi_of_monotone
|
[
{
"state_after": "α✝ : Type u\nβ : Type v\nι✝ : Sort w\nγ : Type x\nι : Type u_1\nι' : Type u_2\ninst✝¹ : LinearOrder ι'\ninst✝ : Nonempty ι'\nα : ι → Type u_3\nI : Set ι\ns : (i : ι) → ι' → Set (α i)\nhI : Set.Finite I\nhs : ∀ (i : ι), i ∈ I → Monotone (s i)\n⊢ (⋃ (j : ι'), ⋂ (x : ↑I), eval ↑x ⁻¹' s (↑x) j) = ⋂ (x : ↑I), ⋃ (i : ι'), eval ↑x ⁻¹' s (↑x) i",
"state_before": "α✝ : Type u\nβ : Type v\nι✝ : Sort w\nγ : Type x\nι : Type u_1\nι' : Type u_2\ninst✝¹ : LinearOrder ι'\ninst✝ : Nonempty ι'\nα : ι → Type u_3\nI : Set ι\ns : (i : ι) → ι' → Set (α i)\nhI : Set.Finite I\nhs : ∀ (i : ι), i ∈ I → Monotone (s i)\n⊢ (⋃ (j : ι'), pi I fun i => s i j) = pi I fun i => ⋃ (j : ι'), s i j",
"tactic": "simp only [pi_def, biInter_eq_iInter, preimage_iUnion]"
},
{
"state_after": "α✝ : Type u\nβ : Type v\nι✝ : Sort w\nγ : Type x\nι : Type u_1\nι' : Type u_2\ninst✝¹ : LinearOrder ι'\ninst✝ : Nonempty ι'\nα : ι → Type u_3\nI : Set ι\ns : (i : ι) → ι' → Set (α i)\nhI : Set.Finite I\nhs : ∀ (i : ι), i ∈ I → Monotone (s i)\nthis : Finite ↑I\n⊢ (⋃ (j : ι'), ⋂ (x : ↑I), eval ↑x ⁻¹' s (↑x) j) = ⋂ (x : ↑I), ⋃ (i : ι'), eval ↑x ⁻¹' s (↑x) i",
"state_before": "α✝ : Type u\nβ : Type v\nι✝ : Sort w\nγ : Type x\nι : Type u_1\nι' : Type u_2\ninst✝¹ : LinearOrder ι'\ninst✝ : Nonempty ι'\nα : ι → Type u_3\nI : Set ι\ns : (i : ι) → ι' → Set (α i)\nhI : Set.Finite I\nhs : ∀ (i : ι), i ∈ I → Monotone (s i)\n⊢ (⋃ (j : ι'), ⋂ (x : ↑I), eval ↑x ⁻¹' s (↑x) j) = ⋂ (x : ↑I), ⋃ (i : ι'), eval ↑x ⁻¹' s (↑x) i",
"tactic": "haveI := hI.fintype.finite"
},
{
"state_after": "α✝ : Type u\nβ : Type v\nι✝ : Sort w\nγ : Type x\nι : Type u_1\nι' : Type u_2\ninst✝¹ : LinearOrder ι'\ninst✝ : Nonempty ι'\nα : ι → Type u_3\nI : Set ι\ns : (i : ι) → ι' → Set (α i)\nhI : Set.Finite I\nhs : ∀ (i : ι), i ∈ I → Monotone (s i)\nthis : Finite ↑I\ni : ↑I\nj₁ j₂ : ι'\nh : j₁ ≤ j₂\n⊢ eval ↑i ⁻¹' s (↑i) j₁ ≤ eval ↑i ⁻¹' s (↑i) j₂",
"state_before": "α✝ : Type u\nβ : Type v\nι✝ : Sort w\nγ : Type x\nι : Type u_1\nι' : Type u_2\ninst✝¹ : LinearOrder ι'\ninst✝ : Nonempty ι'\nα : ι → Type u_3\nI : Set ι\ns : (i : ι) → ι' → Set (α i)\nhI : Set.Finite I\nhs : ∀ (i : ι), i ∈ I → Monotone (s i)\nthis : Finite ↑I\n⊢ (⋃ (j : ι'), ⋂ (x : ↑I), eval ↑x ⁻¹' s (↑x) j) = ⋂ (x : ↑I), ⋃ (i : ι'), eval ↑x ⁻¹' s (↑x) i",
"tactic": "refine' iUnion_iInter_of_monotone (ι' := ι') (fun (i : I) j₁ j₂ h => _)"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nβ : Type v\nι✝ : Sort w\nγ : Type x\nι : Type u_1\nι' : Type u_2\ninst✝¹ : LinearOrder ι'\ninst✝ : Nonempty ι'\nα : ι → Type u_3\nI : Set ι\ns : (i : ι) → ι' → Set (α i)\nhI : Set.Finite I\nhs : ∀ (i : ι), i ∈ I → Monotone (s i)\nthis : Finite ↑I\ni : ↑I\nj₁ j₂ : ι'\nh : j₁ ≤ j₂\n⊢ eval ↑i ⁻¹' s (↑i) j₁ ≤ eval ↑i ⁻¹' s (↑i) j₂",
"tactic": "exact preimage_mono <| hs i i.2 h"
}
] |
[
1556,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1550,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.biUnion_null_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nR : Type ?u.7093\nR' : Type ?u.7096\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : Set β\nhs : Set.Countable s\nt : β → Set α\nthis : Encodable ↑s\n⊢ ↑m (⋃ (i : β) (_ : i ∈ s), t i) = 0 ↔ ∀ (i : β), i ∈ s → ↑m (t i) = 0",
"state_before": "α : Type u_1\nβ : Type u_2\nR : Type ?u.7093\nR' : Type ?u.7096\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : Set β\nhs : Set.Countable s\nt : β → Set α\n⊢ ↑m (⋃ (i : β) (_ : i ∈ s), t i) = 0 ↔ ∀ (i : β), i ∈ s → ↑m (t i) = 0",
"tactic": "haveI := hs.toEncodable"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nR : Type ?u.7093\nR' : Type ?u.7096\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : Set β\nhs : Set.Countable s\nt : β → Set α\nthis : Encodable ↑s\n⊢ ↑m (⋃ (i : β) (_ : i ∈ s), t i) = 0 ↔ ∀ (i : β), i ∈ s → ↑m (t i) = 0",
"tactic": "rw [biUnion_eq_iUnion, iUnion_null_iff, SetCoe.forall']"
}
] |
[
141,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
Complex.hasDerivAt_sinh
|
[] |
[
116,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Order/Closure.lean
|
LowerAdjoint.idempotent
|
[] |
[
371,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
370,
1
] |
Mathlib/Topology/Algebra/Field.lean
|
Filter.tendsto_cocompact_mul_left₀
|
[] |
[
30,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
28,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.quot_lift_mk
|
[] |
[
505,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
503,
1
] |
Mathlib/Topology/DiscreteQuotient.lean
|
DiscreteQuotient.ofLE_proj
|
[] |
[
242,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
241,
1
] |
Mathlib/Topology/PartitionOfUnity.lean
|
BumpCovering.locallyFinite_tsupport
|
[] |
[
240,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
239,
1
] |
Mathlib/LinearAlgebra/BilinearForm.lean
|
BilinForm.IsAlt.isRefl
|
[
{
"state_after": "R : Type ?u.965534\nM : Type ?u.965537\ninst✝¹⁸ : Semiring R\ninst✝¹⁷ : AddCommMonoid M\ninst✝¹⁶ : Module R M\nR₁ : Type u_1\nM₁ : Type u_2\ninst✝¹⁵ : Ring R₁\ninst✝¹⁴ : AddCommGroup M₁\ninst✝¹³ : Module R₁ M₁\nR₂ : Type ?u.966185\nM₂ : Type ?u.966188\ninst✝¹² : CommSemiring R₂\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : Module R₂ M₂\nR₃ : Type ?u.966375\nM₃ : Type ?u.966378\ninst✝⁹ : CommRing R₃\ninst✝⁸ : AddCommGroup M₃\ninst✝⁷ : Module R₃ M₃\nV : Type ?u.966966\nK : Type ?u.966969\ninst✝⁶ : Field K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.968183\nM₂'' : Type ?u.968186\ninst✝³ : AddCommMonoid M₂'\ninst✝² : AddCommMonoid M₂''\ninst✝¹ : Module R₂ M₂'\ninst✝ : Module R₂ M₂''\nH : IsAlt B₁\nx y : M₁\nh : bilin B₁ x y = 0\n⊢ bilin B₁ y x = 0",
"state_before": "R : Type ?u.965534\nM : Type ?u.965537\ninst✝¹⁸ : Semiring R\ninst✝¹⁷ : AddCommMonoid M\ninst✝¹⁶ : Module R M\nR₁ : Type u_1\nM₁ : Type u_2\ninst✝¹⁵ : Ring R₁\ninst✝¹⁴ : AddCommGroup M₁\ninst✝¹³ : Module R₁ M₁\nR₂ : Type ?u.966185\nM₂ : Type ?u.966188\ninst✝¹² : CommSemiring R₂\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : Module R₂ M₂\nR₃ : Type ?u.966375\nM₃ : Type ?u.966378\ninst✝⁹ : CommRing R₃\ninst✝⁸ : AddCommGroup M₃\ninst✝⁷ : Module R₃ M₃\nV : Type ?u.966966\nK : Type ?u.966969\ninst✝⁶ : Field K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.968183\nM₂'' : Type ?u.968186\ninst✝³ : AddCommMonoid M₂'\ninst✝² : AddCommMonoid M₂''\ninst✝¹ : Module R₂ M₂'\ninst✝ : Module R₂ M₂''\nH : IsAlt B₁\n⊢ IsRefl B₁",
"tactic": "intro x y h"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.965534\nM : Type ?u.965537\ninst✝¹⁸ : Semiring R\ninst✝¹⁷ : AddCommMonoid M\ninst✝¹⁶ : Module R M\nR₁ : Type u_1\nM₁ : Type u_2\ninst✝¹⁵ : Ring R₁\ninst✝¹⁴ : AddCommGroup M₁\ninst✝¹³ : Module R₁ M₁\nR₂ : Type ?u.966185\nM₂ : Type ?u.966188\ninst✝¹² : CommSemiring R₂\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : Module R₂ M₂\nR₃ : Type ?u.966375\nM₃ : Type ?u.966378\ninst✝⁹ : CommRing R₃\ninst✝⁸ : AddCommGroup M₃\ninst✝⁷ : Module R₃ M₃\nV : Type ?u.966966\nK : Type ?u.966969\ninst✝⁶ : Field K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.968183\nM₂'' : Type ?u.968186\ninst✝³ : AddCommMonoid M₂'\ninst✝² : AddCommMonoid M₂''\ninst✝¹ : Module R₂ M₂'\ninst✝ : Module R₂ M₂''\nH : IsAlt B₁\nx y : M₁\nh : bilin B₁ x y = 0\n⊢ bilin B₁ y x = 0",
"tactic": "rw [← neg_eq H, h, neg_zero]"
}
] |
[
973,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
971,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.map_isPrime_of_surjective
|
[
{
"state_after": "case refine'_1\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝¹ : Ring R\ninst✝ : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f ≤ I\nh : map f I = ⊤\n⊢ ⊤ ≤ I\n\ncase refine'_2\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝¹ : Ring R\ninst✝ : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f ≤ I\nx y : S\n⊢ x * y ∈ map f I → x ∈ map f I ∨ y ∈ map f I",
"state_before": "R : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝¹ : Ring R\ninst✝ : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f ≤ I\n⊢ IsPrime (map f I)",
"tactic": "refine' ⟨fun h => H.ne_top (eq_top_iff.2 _), fun {x y} => _⟩"
},
{
"state_after": "case refine'_1\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝¹ : Ring R\ninst✝ : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f ≤ I\nh : comap f (map f I) = comap f ⊤\n⊢ ⊤ ≤ I",
"state_before": "case refine'_1\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝¹ : Ring R\ninst✝ : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f ≤ I\nh : map f I = ⊤\n⊢ ⊤ ≤ I",
"tactic": "replace h := congr_arg (comap f) h"
},
{
"state_after": "case refine'_1\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝¹ : Ring R\ninst✝ : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f ≤ I\nh : I ⊔ comap f ⊥ = ⊤\n⊢ ⊤ ≤ I",
"state_before": "case refine'_1\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝¹ : Ring R\ninst✝ : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f ≤ I\nh : comap f (map f I) = comap f ⊤\n⊢ ⊤ ≤ I",
"tactic": "rw [comap_map_of_surjective _ hf, comap_top] at h"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝¹ : Ring R\ninst✝ : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f ≤ I\nh : I ⊔ comap f ⊥ = ⊤\n⊢ ⊤ ≤ I",
"tactic": "exact h ▸ sup_le (le_of_eq rfl) hk"
},
{
"state_after": "case refine'_2\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝¹ : Ring R\ninst✝ : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f ≤ I\nx y : S\nhxy : x * y ∈ map f I\na : R\nha : ↑f a = x\nb : R\nhb : ↑f b = y\n⊢ x ∈ map f I ∨ y ∈ map f I",
"state_before": "case refine'_2\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝¹ : Ring R\ninst✝ : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f ≤ I\nx y : S\n⊢ x * y ∈ map f I → x ∈ map f I ∨ y ∈ map f I",
"tactic": "refine' fun hxy => (hf x).recOn fun a ha => (hf y).recOn fun b hb => _"
},
{
"state_after": "case refine'_2\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝¹ : Ring R\ninst✝ : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f ≤ I\nx y : S\na : R\nha : ↑f a = x\nb : R\nhxy : ∃ x, x ∈ I ∧ ↑f x = ↑f (a * b)\nhb : ↑f b = y\n⊢ x ∈ map f I ∨ y ∈ map f I",
"state_before": "case refine'_2\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝¹ : Ring R\ninst✝ : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f ≤ I\nx y : S\nhxy : x * y ∈ map f I\na : R\nha : ↑f a = x\nb : R\nhb : ↑f b = y\n⊢ x ∈ map f I ∨ y ∈ map f I",
"tactic": "rw [← ha, ← hb, ← _root_.map_mul f, mem_map_iff_of_surjective _ hf] at hxy"
},
{
"state_after": "case refine'_2.intro.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝¹ : Ring R\ninst✝ : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f ≤ I\nx y : S\na : R\nha : ↑f a = x\nb : R\nhb : ↑f b = y\nc : R\nhc : c ∈ I\nhc' : ↑f c = ↑f (a * b)\n⊢ x ∈ map f I ∨ y ∈ map f I",
"state_before": "case refine'_2\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝¹ : Ring R\ninst✝ : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f ≤ I\nx y : S\na : R\nha : ↑f a = x\nb : R\nhxy : ∃ x, x ∈ I ∧ ↑f x = ↑f (a * b)\nhb : ↑f b = y\n⊢ x ∈ map f I ∨ y ∈ map f I",
"tactic": "rcases hxy with ⟨c, hc, hc'⟩"
},
{
"state_after": "case refine'_2.intro.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝¹ : Ring R\ninst✝ : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f ≤ I\nx y : S\na : R\nha : ↑f a = x\nb : R\nhb : ↑f b = y\nc : R\nhc : c ∈ I\nhc'✝ : ↑f c = ↑f (a * b)\nhc' : ↑f (c - a * b) = 0\n⊢ x ∈ map f I ∨ y ∈ map f I",
"state_before": "case refine'_2.intro.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝¹ : Ring R\ninst✝ : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f ≤ I\nx y : S\na : R\nha : ↑f a = x\nb : R\nhb : ↑f b = y\nc : R\nhc : c ∈ I\nhc' : ↑f c = ↑f (a * b)\n⊢ x ∈ map f I ∨ y ∈ map f I",
"tactic": "rw [← sub_eq_zero, ← map_sub] at hc'"
},
{
"state_after": "case refine'_2.intro.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝¹ : Ring R\ninst✝ : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f ≤ I\nx y : S\na : R\nha : ↑f a = x\nb : R\nhb : ↑f b = y\nc : R\nhc : c ∈ I\nhc'✝ : ↑f c = ↑f (a * b)\nhc' : ↑f (c - a * b) = 0\nthis : a * b ∈ I\n⊢ x ∈ map f I ∨ y ∈ map f I",
"state_before": "case refine'_2.intro.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝¹ : Ring R\ninst✝ : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f ≤ I\nx y : S\na : R\nha : ↑f a = x\nb : R\nhb : ↑f b = y\nc : R\nhc : c ∈ I\nhc'✝ : ↑f c = ↑f (a * b)\nhc' : ↑f (c - a * b) = 0\n⊢ x ∈ map f I ∨ y ∈ map f I",
"tactic": "have : a * b ∈ I := by\n convert I.sub_mem hc (hk (hc' : c - a * b ∈ RingHom.ker f)) using 1\n abel"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝¹ : Ring R\ninst✝ : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f ≤ I\nx y : S\na : R\nha : ↑f a = x\nb : R\nhb : ↑f b = y\nc : R\nhc : c ∈ I\nhc'✝ : ↑f c = ↑f (a * b)\nhc' : ↑f (c - a * b) = 0\nthis : a * b ∈ I\n⊢ x ∈ map f I ∨ y ∈ map f I",
"tactic": "exact\n (H.mem_or_mem this).imp (fun h => ha ▸ mem_map_of_mem f h) fun h => hb ▸ mem_map_of_mem f h"
},
{
"state_after": "case h.e'_4\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝¹ : Ring R\ninst✝ : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f ≤ I\nx y : S\na : R\nha : ↑f a = x\nb : R\nhb : ↑f b = y\nc : R\nhc : c ∈ I\nhc'✝ : ↑f c = ↑f (a * b)\nhc' : ↑f (c - a * b) = 0\n⊢ a * b = c - (c - a * b)",
"state_before": "R : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝¹ : Ring R\ninst✝ : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f ≤ I\nx y : S\na : R\nha : ↑f a = x\nb : R\nhb : ↑f b = y\nc : R\nhc : c ∈ I\nhc'✝ : ↑f c = ↑f (a * b)\nhc' : ↑f (c - a * b) = 0\n⊢ a * b ∈ I",
"tactic": "convert I.sub_mem hc (hk (hc' : c - a * b ∈ RingHom.ker f)) using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝¹ : Ring R\ninst✝ : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f ≤ I\nx y : S\na : R\nha : ↑f a = x\nb : R\nhb : ↑f b = y\nc : R\nhc : c ∈ I\nhc'✝ : ↑f c = ↑f (a * b)\nhc' : ↑f (c - a * b) = 0\n⊢ a * b = c - (c - a * b)",
"tactic": "abel"
}
] |
[
2145,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2131,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.pi_ne_zero
|
[
{
"state_after": "⊢ π ≠ 0",
"state_before": "⊢ ↑π ≠ 0",
"tactic": "rw [← toReal_injective.ne_iff, toReal_pi, toReal_zero]"
},
{
"state_after": "no goals",
"state_before": "⊢ π ≠ 0",
"tactic": "exact Real.pi_ne_zero"
}
] |
[
609,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
607,
1
] |
Mathlib/Algebra/Hom/Ring.lean
|
NonUnitalRingHom.coe_copy
|
[] |
[
165,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiffAt.mul
|
[] |
[
1393,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1391,
8
] |
Mathlib/CategoryTheory/Monoidal/Free/Basic.lean
|
CategoryTheory.FreeMonoidalCategory.mk_α_hom
|
[] |
[
206,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean
|
IsCoercive.ker_eq_bot
|
[
{
"state_after": "V : Type u\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : CompleteSpace V\nB : V →L[ℝ] V →L[ℝ] ℝ\ncoercive : IsCoercive B\n⊢ Function.Injective ↑(continuousLinearMapOfBilin B)",
"state_before": "V : Type u\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : CompleteSpace V\nB : V →L[ℝ] V →L[ℝ] ℝ\ncoercive : IsCoercive B\n⊢ ker (continuousLinearMapOfBilin B) = ⊥",
"tactic": "rw [LinearMapClass.ker_eq_bot]"
},
{
"state_after": "case intro.intro\nV : Type u\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : CompleteSpace V\nB : V →L[ℝ] V →L[ℝ] ℝ\ncoercive : IsCoercive B\nw✝ : ℝ≥0\nleft✝ : 0 < w✝\nantilipschitz : AntilipschitzWith w✝ ↑(continuousLinearMapOfBilin B)\n⊢ Function.Injective ↑(continuousLinearMapOfBilin B)",
"state_before": "V : Type u\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : CompleteSpace V\nB : V →L[ℝ] V →L[ℝ] ℝ\ncoercive : IsCoercive B\n⊢ Function.Injective ↑(continuousLinearMapOfBilin B)",
"tactic": "rcases coercive.antilipschitz with ⟨_, _, antilipschitz⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nV : Type u\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : CompleteSpace V\nB : V →L[ℝ] V →L[ℝ] ℝ\ncoercive : IsCoercive B\nw✝ : ℝ≥0\nleft✝ : 0 < w✝\nantilipschitz : AntilipschitzWith w✝ ↑(continuousLinearMapOfBilin B)\n⊢ Function.Injective ↑(continuousLinearMapOfBilin B)",
"tactic": "exact antilipschitz.injective"
}
] |
[
85,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/Topology/Compactification/OnePoint.lean
|
OnePoint.tendsto_nhds_infty'
|
[
{
"state_after": "no goals",
"state_before": "X : Type u_2\ninst✝ : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nα : Type u_1\nf : OnePoint X → α\nl : Filter α\n⊢ Tendsto f (𝓝 ∞) l ↔ Tendsto f (pure ∞) l ∧ Tendsto (f ∘ some) (coclosedCompact X) l",
"tactic": "simp [nhds_infty_eq, and_comm]"
}
] |
[
362,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
360,
1
] |
Mathlib/Algebra/Support.lean
|
Function.nmem_mulSupport
|
[] |
[
58,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/Data/Fintype/Card.lean
|
Fintype.card_quotient_lt
|
[] |
[
882,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
879,
1
] |
Mathlib/Init/Algebra/Order.lean
|
le_or_gt
|
[] |
[
353,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
352,
1
] |
Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean
|
AlgebraicTopology.DoldKan.Γ₀.Obj.map_on_summand'
|
[
{
"state_after": "case fac\nC : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝ Δ'✝ Δ'' : SimplexCategory\ninst✝ : HasFiniteCoproducts C\nΔ Δ' : SimplexCategoryᵒᵖ\nA : Splitting.IndexSet Δ\nθ : Δ ⟶ Δ'\n⊢ factorThruImage (θ.unop ≫ Splitting.IndexSet.e A) ≫ image.ι (θ.unop ≫ Splitting.IndexSet.e A) =\n θ.unop ≫ Splitting.IndexSet.e A",
"state_before": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝ Δ'✝ Δ'' : SimplexCategory\ninst✝ : HasFiniteCoproducts C\nΔ Δ' : SimplexCategoryᵒᵖ\nA : Splitting.IndexSet Δ\nθ : Δ ⟶ Δ'\n⊢ Splitting.ιSummand (splitting K) A ≫ (obj K).map θ =\n Termwise.mapMono K (image.ι (θ.unop ≫ Splitting.IndexSet.e A)) ≫\n Splitting.ιSummand (splitting K) (Splitting.IndexSet.pull A θ)",
"tactic": "apply Obj.map_on_summand"
},
{
"state_after": "no goals",
"state_before": "case fac\nC : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝ Δ'✝ Δ'' : SimplexCategory\ninst✝ : HasFiniteCoproducts C\nΔ Δ' : SimplexCategoryᵒᵖ\nA : Splitting.IndexSet Δ\nθ : Δ ⟶ Δ'\n⊢ factorThruImage (θ.unop ≫ Splitting.IndexSet.e A) ≫ image.ι (θ.unop ≫ Splitting.IndexSet.e A) =\n θ.unop ≫ Splitting.IndexSet.e A",
"tactic": "apply image.fac"
}
] |
[
280,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
276,
1
] |
Std/Logic.lean
|
Decidable.imp_or
|
[
{
"state_after": "no goals",
"state_before": "a b c : Prop\ninst✝ : Decidable a\n⊢ a → b ∨ c ↔ (a → b) ∨ (a → c)",
"tactic": "by_cases a <;> simp_all"
}
] |
[
562,
26
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
561,
1
] |
Mathlib/RingTheory/Polynomial/Basic.lean
|
Ideal.polynomial_not_isField
|
[
{
"state_after": "R : Type u\nS : Type ?u.190529\ninst✝ : Ring R\n✝ : Nontrivial R\n⊢ ¬IsField R[X]",
"state_before": "R : Type u\nS : Type ?u.190529\ninst✝ : Ring R\n⊢ ¬IsField R[X]",
"tactic": "nontriviality R"
},
{
"state_after": "R : Type u\nS : Type ?u.190529\ninst✝ : Ring R\n✝ : Nontrivial R\nhR : IsField R[X]\n⊢ False",
"state_before": "R : Type u\nS : Type ?u.190529\ninst✝ : Ring R\n✝ : Nontrivial R\n⊢ ¬IsField R[X]",
"tactic": "intro hR"
},
{
"state_after": "case intro\nR : Type u\nS : Type ?u.190529\ninst✝ : Ring R\n✝ : Nontrivial R\nhR : IsField R[X]\np : R[X]\nhp : X * p = 1\n⊢ False",
"state_before": "R : Type u\nS : Type ?u.190529\ninst✝ : Ring R\n✝ : Nontrivial R\nhR : IsField R[X]\n⊢ False",
"tactic": "obtain ⟨p, hp⟩ := hR.mul_inv_cancel X_ne_zero"
},
{
"state_after": "case intro\nR : Type u\nS : Type ?u.190529\ninst✝ : Ring R\n✝ : Nontrivial R\nhR : IsField R[X]\np : R[X]\nhp : X * p = 1\nhp0 : p ≠ 0\n⊢ False",
"state_before": "case intro\nR : Type u\nS : Type ?u.190529\ninst✝ : Ring R\n✝ : Nontrivial R\nhR : IsField R[X]\np : R[X]\nhp : X * p = 1\n⊢ False",
"tactic": "have hp0 : p ≠ 0 := right_ne_zero_of_mul_eq_one hp"
},
{
"state_after": "case intro\nR : Type u\nS : Type ?u.190529\ninst✝ : Ring R\n✝ : Nontrivial R\nhR : IsField R[X]\np : R[X]\nhp : X * p = 1\nhp0 : p ≠ 0\nthis : degree p < degree (p * X)\n⊢ False",
"state_before": "case intro\nR : Type u\nS : Type ?u.190529\ninst✝ : Ring R\n✝ : Nontrivial R\nhR : IsField R[X]\np : R[X]\nhp : X * p = 1\nhp0 : p ≠ 0\n⊢ False",
"tactic": "have := degree_lt_degree_mul_X hp0"
},
{
"state_after": "case intro\nR : Type u\nS : Type ?u.190529\ninst✝ : Ring R\n✝ : Nontrivial R\nhR : IsField R[X]\np : R[X]\nhp : X * p = 1\nhp0 : p ≠ 0\nthis : p = 0\n⊢ False",
"state_before": "case intro\nR : Type u\nS : Type ?u.190529\ninst✝ : Ring R\n✝ : Nontrivial R\nhR : IsField R[X]\np : R[X]\nhp : X * p = 1\nhp0 : p ≠ 0\nthis : degree p < degree (p * X)\n⊢ False",
"tactic": "rw [← X_mul, congr_arg degree hp, degree_one, Nat.WithBot.lt_zero_iff, degree_eq_bot] at this"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u\nS : Type ?u.190529\ninst✝ : Ring R\n✝ : Nontrivial R\nhR : IsField R[X]\np : R[X]\nhp : X * p = 1\nhp0 : p ≠ 0\nthis : p = 0\n⊢ False",
"tactic": "exact hp0 this"
}
] |
[
650,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
643,
1
] |
Mathlib/Data/Real/GoldenRatio.lean
|
fib_isSol_fibRec
|
[
{
"state_after": "α : Type u_1\ninst✝ : CommSemiring α\n⊢ LinearRecurrence.IsSolution { order := 2, coeffs := ![1, 1] } fun x => ↑(Nat.fib x)",
"state_before": "α : Type u_1\ninst✝ : CommSemiring α\n⊢ LinearRecurrence.IsSolution fibRec fun x => ↑(Nat.fib x)",
"tactic": "rw [fibRec]"
},
{
"state_after": "α : Type u_1\ninst✝ : CommSemiring α\nn : ℕ\n⊢ (fun x => ↑(Nat.fib x)) (n + { order := 2, coeffs := ![1, 1] }.order) =\n Finset.sum Finset.univ fun i =>\n LinearRecurrence.coeffs { order := 2, coeffs := ![1, 1] } i * (fun x => ↑(Nat.fib x)) (n + ↑i)",
"state_before": "α : Type u_1\ninst✝ : CommSemiring α\n⊢ LinearRecurrence.IsSolution { order := 2, coeffs := ![1, 1] } fun x => ↑(Nat.fib x)",
"tactic": "intro n"
},
{
"state_after": "α : Type u_1\ninst✝ : CommSemiring α\nn : ℕ\n⊢ ↑(Nat.fib (n + 2)) = Finset.sum Finset.univ fun x => Matrix.vecCons 1 ![1] x * ↑(Nat.fib (n + ↑x))",
"state_before": "α : Type u_1\ninst✝ : CommSemiring α\nn : ℕ\n⊢ (fun x => ↑(Nat.fib x)) (n + { order := 2, coeffs := ![1, 1] }.order) =\n Finset.sum Finset.univ fun i =>\n LinearRecurrence.coeffs { order := 2, coeffs := ![1, 1] } i * (fun x => ↑(Nat.fib x)) (n + ↑i)",
"tactic": "simp only"
},
{
"state_after": "α : Type u_1\ninst✝ : CommSemiring α\nn : ℕ\n⊢ ↑(Nat.fib (n + 1) + Nat.fib n) = Finset.sum Finset.univ fun x => Matrix.vecCons 1 ![1] x * ↑(Nat.fib (n + ↑x))",
"state_before": "α : Type u_1\ninst✝ : CommSemiring α\nn : ℕ\n⊢ ↑(Nat.fib (n + 2)) = Finset.sum Finset.univ fun x => Matrix.vecCons 1 ![1] x * ↑(Nat.fib (n + ↑x))",
"tactic": "rw [Nat.fib_add_two, add_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : CommSemiring α\nn : ℕ\n⊢ ↑(Nat.fib (n + 1) + Nat.fib n) = Finset.sum Finset.univ fun x => Matrix.vecCons 1 ![1] x * ↑(Nat.fib (n + ↑x))",
"tactic": "simp [Finset.sum_fin_eq_sum_range, Finset.sum_range_succ']"
}
] |
[
194,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/RingTheory/MvPolynomial/Homogeneous.lean
|
MvPolynomial.homogeneousComponent_zero
|
[
{
"state_after": "case a\nσ : Type u_1\nτ : Type ?u.182889\nR : Type u_2\nS : Type ?u.182895\ninst✝ : CommSemiring R\nn : ℕ\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\n⊢ coeff d (↑(homogeneousComponent 0) φ) = coeff d (↑C (coeff 0 φ))",
"state_before": "σ : Type u_1\nτ : Type ?u.182889\nR : Type u_2\nS : Type ?u.182895\ninst✝ : CommSemiring R\nn : ℕ\nφ ψ : MvPolynomial σ R\n⊢ ↑(homogeneousComponent 0) φ = ↑C (coeff 0 φ)",
"tactic": "ext1 d"
},
{
"state_after": "case a.inl\nσ : Type u_1\nτ : Type ?u.182889\nR : Type u_2\nS : Type ?u.182895\ninst✝ : CommSemiring R\nn : ℕ\nφ ψ : MvPolynomial σ R\n⊢ coeff 0 (↑(homogeneousComponent 0) φ) = coeff 0 (↑C (coeff 0 φ))\n\ncase a.inr\nσ : Type u_1\nτ : Type ?u.182889\nR : Type u_2\nS : Type ?u.182895\ninst✝ : CommSemiring R\nn : ℕ\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\nhd : ¬d = 0\n⊢ coeff d (↑(homogeneousComponent 0) φ) = coeff d (↑C (coeff 0 φ))",
"state_before": "case a\nσ : Type u_1\nτ : Type ?u.182889\nR : Type u_2\nS : Type ?u.182895\ninst✝ : CommSemiring R\nn : ℕ\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\n⊢ coeff d (↑(homogeneousComponent 0) φ) = coeff d (↑C (coeff 0 φ))",
"tactic": "rcases em (d = 0) with (rfl | hd)"
},
{
"state_after": "no goals",
"state_before": "case a.inl\nσ : Type u_1\nτ : Type ?u.182889\nR : Type u_2\nS : Type ?u.182895\ninst✝ : CommSemiring R\nn : ℕ\nφ ψ : MvPolynomial σ R\n⊢ coeff 0 (↑(homogeneousComponent 0) φ) = coeff 0 (↑C (coeff 0 φ))",
"tactic": "simp only [coeff_homogeneousComponent, sum_eq_zero_iff, Finsupp.zero_apply, if_true, coeff_C,\n eq_self_iff_true, forall_true_iff]"
},
{
"state_after": "case a.inr.hnc\nσ : Type u_1\nτ : Type ?u.182889\nR : Type u_2\nS : Type ?u.182895\ninst✝ : CommSemiring R\nn : ℕ\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\nhd : ¬d = 0\n⊢ ¬∑ i in d.support, ↑d i = 0",
"state_before": "case a.inr\nσ : Type u_1\nτ : Type ?u.182889\nR : Type u_2\nS : Type ?u.182895\ninst✝ : CommSemiring R\nn : ℕ\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\nhd : ¬d = 0\n⊢ coeff d (↑(homogeneousComponent 0) φ) = coeff d (↑C (coeff 0 φ))",
"tactic": "rw [coeff_homogeneousComponent, if_neg, coeff_C, if_neg (Ne.symm hd)]"
},
{
"state_after": "case a.inr.hnc\nσ : Type u_1\nτ : Type ?u.182889\nR : Type u_2\nS : Type ?u.182895\ninst✝ : CommSemiring R\nn : ℕ\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\nhd : ¬∀ (x : σ), ↑d x = 0\n⊢ ¬∑ i in d.support, ↑d i = 0",
"state_before": "case a.inr.hnc\nσ : Type u_1\nτ : Type ?u.182889\nR : Type u_2\nS : Type ?u.182895\ninst✝ : CommSemiring R\nn : ℕ\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\nhd : ¬d = 0\n⊢ ¬∑ i in d.support, ↑d i = 0",
"tactic": "simp only [FunLike.ext_iff, Finsupp.zero_apply] at hd"
},
{
"state_after": "no goals",
"state_before": "case a.inr.hnc\nσ : Type u_1\nτ : Type ?u.182889\nR : Type u_2\nS : Type ?u.182895\ninst✝ : CommSemiring R\nn : ℕ\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\nhd : ¬∀ (x : σ), ↑d x = 0\n⊢ ¬∑ i in d.support, ↑d i = 0",
"tactic": "simp [hd]"
}
] |
[
282,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
274,
1
] |
Mathlib/Control/ULift.lean
|
PLift.bind_up
|
[] |
[
59,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/Data/Nat/Order/Basic.lean
|
Nat.add_mod_eq_ite
|
[
{
"state_after": "case zero\nm n l : ℕ\n⊢ (m + n) % zero = if zero ≤ m % zero + n % zero then m % zero + n % zero - zero else m % zero + n % zero\n\ncase succ\nm n l n✝ : ℕ\n⊢ (m + n) % succ n✝ =\n if succ n✝ ≤ m % succ n✝ + n % succ n✝ then m % succ n✝ + n % succ n✝ - succ n✝ else m % succ n✝ + n % succ n✝",
"state_before": "m n k l : ℕ\n⊢ (m + n) % k = if k ≤ m % k + n % k then m % k + n % k - k else m % k + n % k",
"tactic": "cases k"
},
{
"state_after": "case succ\nm n l n✝ : ℕ\n⊢ (m + n) % succ n✝ =\n if succ n✝ ≤ m % succ n✝ + n % succ n✝ then m % succ n✝ + n % succ n✝ - succ n✝ else m % succ n✝ + n % succ n✝",
"state_before": "case zero\nm n l : ℕ\n⊢ (m + n) % zero = if zero ≤ m % zero + n % zero then m % zero + n % zero - zero else m % zero + n % zero\n\ncase succ\nm n l n✝ : ℕ\n⊢ (m + n) % succ n✝ =\n if succ n✝ ≤ m % succ n✝ + n % succ n✝ then m % succ n✝ + n % succ n✝ - succ n✝ else m % succ n✝ + n % succ n✝",
"tactic": "simp [mod_zero]"
},
{
"state_after": "case succ\nm n l n✝ : ℕ\n⊢ (m % succ n✝ + n % succ n✝) % succ n✝ =\n if succ n✝ ≤ m % succ n✝ + n % succ n✝ then m % succ n✝ + n % succ n✝ - succ n✝ else m % succ n✝ + n % succ n✝",
"state_before": "case succ\nm n l n✝ : ℕ\n⊢ (m + n) % succ n✝ =\n if succ n✝ ≤ m % succ n✝ + n % succ n✝ then m % succ n✝ + n % succ n✝ - succ n✝ else m % succ n✝ + n % succ n✝",
"tactic": "rw [Nat.add_mod]"
},
{
"state_after": "case succ.inl\nm n l n✝ : ℕ\nh : succ n✝ ≤ m % succ n✝ + n % succ n✝\n⊢ (m % succ n✝ + n % succ n✝) % succ n✝ = m % succ n✝ + n % succ n✝ - succ n✝\n\ncase succ.inr\nm n l n✝ : ℕ\nh : ¬succ n✝ ≤ m % succ n✝ + n % succ n✝\n⊢ (m % succ n✝ + n % succ n✝) % succ n✝ = m % succ n✝ + n % succ n✝",
"state_before": "case succ\nm n l n✝ : ℕ\n⊢ (m % succ n✝ + n % succ n✝) % succ n✝ =\n if succ n✝ ≤ m % succ n✝ + n % succ n✝ then m % succ n✝ + n % succ n✝ - succ n✝ else m % succ n✝ + n % succ n✝",
"tactic": "split_ifs with h"
},
{
"state_after": "case succ.inl\nm n l n✝ : ℕ\nh : succ n✝ ≤ m % succ n✝ + n % succ n✝\n⊢ m % succ n✝ + n % succ n✝ - succ n✝ < succ n✝",
"state_before": "case succ.inl\nm n l n✝ : ℕ\nh : succ n✝ ≤ m % succ n✝ + n % succ n✝\n⊢ (m % succ n✝ + n % succ n✝) % succ n✝ = m % succ n✝ + n % succ n✝ - succ n✝",
"tactic": "rw [Nat.mod_eq_sub_mod h, Nat.mod_eq_of_lt]"
},
{
"state_after": "no goals",
"state_before": "case succ.inl\nm n l n✝ : ℕ\nh : succ n✝ ≤ m % succ n✝ + n % succ n✝\n⊢ m % succ n✝ + n % succ n✝ - succ n✝ < succ n✝",
"tactic": "exact\n (tsub_lt_iff_right h).mpr (Nat.add_lt_add (m.mod_lt (zero_lt_succ _))\n (n.mod_lt (zero_lt_succ _)))"
},
{
"state_after": "no goals",
"state_before": "case succ.inr\nm n l n✝ : ℕ\nh : ¬succ n✝ ≤ m % succ n✝ + n % succ n✝\n⊢ (m % succ n✝ + n % succ n✝) % succ n✝ = m % succ n✝ + n % succ n✝",
"tactic": "exact Nat.mod_eq_of_lt (lt_of_not_ge h)"
}
] |
[
514,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
505,
1
] |
Mathlib/Topology/Constructions.lean
|
DenseRange.quotient
|
[] |
[
200,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/Order/SymmDiff.lean
|
ofDual_bihimp
|
[] |
[
114,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
|
UniformFun.postcomp_uniformContinuous
|
[
{
"state_after": "α : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.39939\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : γ → β\nhf : UniformContinuous f\n⊢ uniformSpace α γ ≤ UniformSpace.comap (↑ofFun ∘ (fun x => f ∘ x) ∘ ↑toFun) (uniformSpace α β)",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.39939\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : γ → β\nhf : UniformContinuous f\n⊢ UniformContinuous (↑ofFun ∘ (fun x => f ∘ x) ∘ ↑toFun)",
"tactic": "refine uniformContinuous_iff.mpr ?_"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.39939\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : γ → β\nhf : UniformContinuous f\n⊢ uniformSpace α γ ≤ UniformSpace.comap (↑ofFun ∘ (fun x => f ∘ x) ∘ ↑toFun) (uniformSpace α β)",
"tactic": "exact (UniformFun.mono (uniformContinuous_iff.mp hf)).trans_eq UniformFun.comap_eq"
}
] |
[
418,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
413,
11
] |
Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean
|
Polynomial.IsWeaklyEisensteinAt.map
|
[
{
"state_after": "R : Type u\ninst✝¹ : CommSemiring R\n𝓟 : Ideal R\nf : R[X]\nhf : IsWeaklyEisensteinAt f 𝓟\nA : Type v\ninst✝ : CommRing A\nφ : R →+* A\nn✝ : ℕ\nhn : n✝ < natDegree (Polynomial.map φ f)\n⊢ coeff (Polynomial.map φ f) n✝ ∈ Ideal.map φ 𝓟",
"state_before": "R : Type u\ninst✝¹ : CommSemiring R\n𝓟 : Ideal R\nf : R[X]\nhf : IsWeaklyEisensteinAt f 𝓟\nA : Type v\ninst✝ : CommRing A\nφ : R →+* A\n⊢ IsWeaklyEisensteinAt (Polynomial.map φ f) (Ideal.map φ 𝓟)",
"tactic": "refine' (IsWeaklyEisensteinAt_iff _ _).2 fun hn => _"
},
{
"state_after": "R : Type u\ninst✝¹ : CommSemiring R\n𝓟 : Ideal R\nf : R[X]\nhf : IsWeaklyEisensteinAt f 𝓟\nA : Type v\ninst✝ : CommRing A\nφ : R →+* A\nn✝ : ℕ\nhn : n✝ < natDegree (Polynomial.map φ f)\n⊢ ↑φ (coeff f n✝) ∈ Ideal.map φ 𝓟",
"state_before": "R : Type u\ninst✝¹ : CommSemiring R\n𝓟 : Ideal R\nf : R[X]\nhf : IsWeaklyEisensteinAt f 𝓟\nA : Type v\ninst✝ : CommRing A\nφ : R →+* A\nn✝ : ℕ\nhn : n✝ < natDegree (Polynomial.map φ f)\n⊢ coeff (Polynomial.map φ f) n✝ ∈ Ideal.map φ 𝓟",
"tactic": "rw [coeff_map]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : CommSemiring R\n𝓟 : Ideal R\nf : R[X]\nhf : IsWeaklyEisensteinAt f 𝓟\nA : Type v\ninst✝ : CommRing A\nφ : R →+* A\nn✝ : ℕ\nhn : n✝ < natDegree (Polynomial.map φ f)\n⊢ ↑φ (coeff f n✝) ∈ Ideal.map φ 𝓟",
"tactic": "exact mem_map_of_mem _ (hf.mem (lt_of_lt_of_le hn (natDegree_map_le _ _)))"
}
] |
[
72,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/CategoryTheory/Arrow.lean
|
CategoryTheory.Arrow.square_to_iso_invert
|
[
{
"state_after": "no goals",
"state_before": "T : Type u\ninst✝ : Category T\ni : Arrow T\nX Y : T\np : X ≅ Y\nsq : i ⟶ mk p.hom\n⊢ i.hom ≫ sq.right ≫ p.inv = sq.left",
"tactic": "simpa only [Category.assoc] using (Iso.comp_inv_eq p).mpr (Arrow.w_mk_right sq).symm"
}
] |
[
261,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
259,
1
] |
Mathlib/Topology/FiberBundle/Trivialization.lean
|
Pretrivialization.symm_trans_target_eq
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.12276\nB : Type u_1\nF : Type u_2\nE : B → Type ?u.12287\nZ : Type u_3\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nproj : Z → B\ne✝ : Pretrivialization F proj\nx : Z\ne e' : Pretrivialization F proj\n⊢ (LocalEquiv.trans (LocalEquiv.symm e.toLocalEquiv) e'.toLocalEquiv).target = (e.baseSet ∩ e'.baseSet) ×ˢ univ",
"tactic": "rw [← LocalEquiv.symm_source, symm_trans_symm, symm_trans_source_eq, inter_comm]"
}
] |
[
224,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
222,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Constructions.lean
|
Pmf.mem_support_bernoulli_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.170166\nβ : Type ?u.170169\nγ : Type ?u.170172\np : ℝ≥0∞\nh : p ≤ 1\nb : Bool\n⊢ b ∈ support (bernoulli p h) ↔ bif b then p ≠ 0 else p ≠ 1",
"tactic": "simp"
}
] |
[
324,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
324,
1
] |
Mathlib/Data/Nat/Basic.lean
|
Nat.strongRecOn'_beta
|
[
{
"state_after": "m n✝ k : ℕ\nP : ℕ → Sort u_1\nh : (n : ℕ) → ((m : ℕ) → m < n → P m) → P n\nn : ℕ\n⊢ Nat.strongRec' h n = h n fun m x => Nat.strongRec' h m",
"state_before": "m n✝ k : ℕ\nP : ℕ → Sort u_1\nh : (n : ℕ) → ((m : ℕ) → m < n → P m) → P n\nn : ℕ\n⊢ strongRecOn' n h = h n fun m x => strongRecOn' m h",
"tactic": "simp only [strongRecOn']"
},
{
"state_after": "no goals",
"state_before": "m n✝ k : ℕ\nP : ℕ → Sort u_1\nh : (n : ℕ) → ((m : ℕ) → m < n → P m) → P n\nn : ℕ\n⊢ Nat.strongRec' h n = h n fun m x => Nat.strongRec' h m",
"tactic": "rw [Nat.strongRec']"
}
] |
[
475,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
472,
1
] |
Mathlib/Combinatorics/Quiver/SingleObj.lean
|
Quiver.SingleObj.pathToList_listToPath
|
[
{
"state_after": "case nil\nα : Type u_1\nβ : Type ?u.5108\nγ : Type ?u.5111\n⊢ pathToList (listToPath []) = []\n\ncase cons\nα : Type u_1\nβ : Type ?u.5108\nγ : Type ?u.5111\na : α\nl : List α\nih : pathToList (listToPath l) = l\n⊢ pathToList (listToPath (a :: l)) = a :: l",
"state_before": "α : Type u_1\nβ : Type ?u.5108\nγ : Type ?u.5111\nl : List α\n⊢ pathToList (listToPath l) = l",
"tactic": "induction' l with a l ih"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\nβ : Type ?u.5108\nγ : Type ?u.5111\n⊢ pathToList (listToPath []) = []",
"tactic": "rfl"
},
{
"state_after": "case cons\nα : Type u_1\nβ : Type ?u.5108\nγ : Type ?u.5111\na : α\nl : List α\nih : pathToList (listToPath l) = l\n⊢ a :: pathToList (listToPath l) = a :: l",
"state_before": "case cons\nα : Type u_1\nβ : Type ?u.5108\nγ : Type ?u.5111\na : α\nl : List α\nih : pathToList (listToPath l) = l\n⊢ pathToList (listToPath (a :: l)) = a :: l",
"tactic": "change a :: pathToList (listToPath l) = a :: l"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_1\nβ : Type ?u.5108\nγ : Type ?u.5111\na : α\nl : List α\nih : pathToList (listToPath l) = l\n⊢ a :: pathToList (listToPath l) = a :: l",
"tactic": "rw [ih]"
}
] |
[
148,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Nat.ceil_add_le
|
[
{
"state_after": "F : Type ?u.92323\nα : Type u_1\nβ : Type ?u.92329\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na✝ : α\nn : ℕ\na b : α\n⊢ a + b ≤ ↑⌈a⌉₊ + ↑⌈b⌉₊",
"state_before": "F : Type ?u.92323\nα : Type u_1\nβ : Type ?u.92329\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na✝ : α\nn : ℕ\na b : α\n⊢ ⌈a + b⌉₊ ≤ ⌈a⌉₊ + ⌈b⌉₊",
"tactic": "rw [ceil_le, Nat.cast_add]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.92323\nα : Type u_1\nβ : Type ?u.92329\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na✝ : α\nn : ℕ\na b : α\n⊢ a + b ≤ ↑⌈a⌉₊ + ↑⌈b⌉₊",
"tactic": "exact _root_.add_le_add (le_ceil _) (le_ceil _)"
}
] |
[
488,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
486,
1
] |
Mathlib/Order/Ideal.lean
|
Order.mem_idealOfCofinals
|
[] |
[
596,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
595,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
|
CategoryTheory.Limits.coprod.map_inl_inr_codiag
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nX✝ Y✝ X Y : C\ninst✝¹ : HasBinaryCoproduct X Y\ninst✝ : HasBinaryCoproduct (X ⨿ Y) (X ⨿ Y)\n⊢ map inl inr ≫ codiag (X ⨿ Y) = 𝟙 (X ⨿ Y)",
"tactic": "simp"
}
] |
[
943,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
941,
1
] |
Mathlib/CategoryTheory/Monad/Algebra.lean
|
CategoryTheory.Comonad.Coalgebra.id_eq_id
|
[] |
[
406,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
405,
1
] |
Mathlib/Analysis/Calculus/IteratedDeriv.lean
|
iteratedDeriv_eq_iteratedFDeriv
|
[] |
[
221,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/Order/SymmDiff.lean
|
bihimp_eq
|
[] |
[
717,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
717,
1
] |
Mathlib/Data/PFun.lean
|
PFun.core_mono
|
[] |
[
480,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
479,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
measurable_liminf
|
[] |
[
1333,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1331,
1
] |
Std/Data/HashMap/WF.lean
|
Std.HashMap.Imp.erase_size
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\n⊢ (match\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] with\n | true =>\n { size := m.size - 1,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }\n | false => m).size =\n Bucket.size\n (match\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] with\n | true =>\n { size := m.size - 1,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }\n | false => m).buckets",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\n⊢ (erase m k).size = Bucket.size (erase m k).buckets",
"tactic": "dsimp [erase, cond]"
},
{
"state_after": "case h_1\nα : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nheq✝ :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n true\n⊢ { size := m.size - 1,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }.size =\n Bucket.size\n { size := m.size - 1,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }.buckets\n\ncase h_2\nα : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nheq✝ :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n false\n⊢ m.size = Bucket.size m.buckets",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\n⊢ (match\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] with\n | true =>\n { size := m.size - 1,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }\n | false => m).size =\n Bucket.size\n (match\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] with\n | true =>\n { size := m.size - 1,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }\n | false => m).buckets",
"tactic": "split"
},
{
"state_after": "no goals",
"state_before": "case h_1\nα : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nheq✝ :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n true\n⊢ { size := m.size - 1,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }.size =\n Bucket.size\n { size := m.size - 1,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }.buckets",
"tactic": "next H =>\nsimp [h, Bucket.size]\nrefine have ⟨_, _, h₁, _, eq⟩ := Bucket.exists_of_update ..; eq ▸ ?_\nsimp [h, h₁, Bucket.size_eq]\nrw [(_ : List.length _ = _ + 1), Nat.add_right_comm]; {rfl}\nclear h₁ eq\nsimp [AssocList.contains_eq] at H\nhave ⟨a, h₁, h₂⟩ := H\nrefine have ⟨_, _, _, _, _, h, eq⟩ := List.exists_of_eraseP h₁ h₂; eq ▸ ?_\nsimp [h]; rfl"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nH :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n true\n⊢ Nat.sum (List.map (fun x => List.length (AssocList.toList x)) m.buckets.val.data) - 1 =\n Nat.sum\n (List.map (fun x => List.length (AssocList.toList x))\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val.data)",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nH :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n true\n⊢ { size := m.size - 1,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }.size =\n Bucket.size\n { size := m.size - 1,\n buckets :=\n Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val) }.buckets",
"tactic": "simp [h, Bucket.size]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nH :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n true\nw✝¹ w✝ : List (AssocList α β)\nh₁ :\n m.buckets.val.data =\n w✝¹ ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w✝\nleft✝ : List.length w✝¹ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val.data =\n w✝¹ ++\n AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n w✝\n⊢ Nat.sum (List.map (fun x => List.length (AssocList.toList x)) m.buckets.val.data) - 1 =\n Nat.sum\n (List.map (fun x => List.length (AssocList.toList x))\n (w✝¹ ++\n AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n w✝))",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nH :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n true\n⊢ Nat.sum (List.map (fun x => List.length (AssocList.toList x)) m.buckets.val.data) - 1 =\n Nat.sum\n (List.map (fun x => List.length (AssocList.toList x))\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val.data)",
"tactic": "refine have ⟨_, _, h₁, _, eq⟩ := Bucket.exists_of_update ..; eq ▸ ?_"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nH :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n true\nw✝¹ w✝ : List (AssocList α β)\nh₁ :\n m.buckets.val.data =\n w✝¹ ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w✝\nleft✝ : List.length w✝¹ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val.data =\n w✝¹ ++\n AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n w✝\n⊢ Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w✝¹) +\n (List.length\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) +\n Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w✝)) -\n 1 =\n Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w✝¹) +\n (List.length\n (List.eraseP (fun x => x.fst == k)\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w✝))",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nH :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n true\nw✝¹ w✝ : List (AssocList α β)\nh₁ :\n m.buckets.val.data =\n w✝¹ ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w✝\nleft✝ : List.length w✝¹ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val.data =\n w✝¹ ++\n AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n w✝\n⊢ Nat.sum (List.map (fun x => List.length (AssocList.toList x)) m.buckets.val.data) - 1 =\n Nat.sum\n (List.map (fun x => List.length (AssocList.toList x))\n (w✝¹ ++\n AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n w✝))",
"tactic": "simp [h, h₁, Bucket.size_eq]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nH :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n true\nw✝¹ w✝ : List (AssocList α β)\nh₁ :\n m.buckets.val.data =\n w✝¹ ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w✝\nleft✝ : List.length w✝¹ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val.data =\n w✝¹ ++\n AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n w✝\n⊢ Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w✝¹) +\n (?n + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w✝) + 1) -\n 1 =\n Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w✝¹) +\n (List.length\n (List.eraseP (fun x => x.fst == k)\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w✝))\n\ncase n\nα : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nH :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n true\nw✝¹ w✝ : List (AssocList α β)\nh₁ :\n m.buckets.val.data =\n w✝¹ ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w✝\nleft✝ : List.length w✝¹ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val.data =\n w✝¹ ++\n AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n w✝\n⊢ Nat\n\nα : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nH :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n true\nw✝¹ w✝ : List (AssocList α β)\nh₁ :\n m.buckets.val.data =\n w✝¹ ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w✝\nleft✝ : List.length w✝¹ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val.data =\n w✝¹ ++\n AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n w✝\n⊢ List.length\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n ?n + 1",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nH :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n true\nw✝¹ w✝ : List (AssocList α β)\nh₁ :\n m.buckets.val.data =\n w✝¹ ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w✝\nleft✝ : List.length w✝¹ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val.data =\n w✝¹ ++\n AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n w✝\n⊢ Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w✝¹) +\n (List.length\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) +\n Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w✝)) -\n 1 =\n Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w✝¹) +\n (List.length\n (List.eraseP (fun x => x.fst == k)\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w✝))",
"tactic": "rw [(_ : List.length _ = _ + 1), Nat.add_right_comm]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nH :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n true\nw✝¹ w✝ : List (AssocList α β)\nh₁ :\n m.buckets.val.data =\n w✝¹ ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w✝\nleft✝ : List.length w✝¹ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val.data =\n w✝¹ ++\n AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n w✝\n⊢ List.length\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n List.length\n (List.eraseP (fun x => x.fst == k)\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n 1",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nH :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n true\nw✝¹ w✝ : List (AssocList α β)\nh₁ :\n m.buckets.val.data =\n w✝¹ ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w✝\nleft✝ : List.length w✝¹ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val.data =\n w✝¹ ++\n AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n w✝\n⊢ Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w✝¹) +\n (?n + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w✝) + 1) -\n 1 =\n Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w✝¹) +\n (List.length\n (List.eraseP (fun x => x.fst == k)\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w✝))\n\ncase n\nα : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nH :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n true\nw✝¹ w✝ : List (AssocList α β)\nh₁ :\n m.buckets.val.data =\n w✝¹ ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w✝\nleft✝ : List.length w✝¹ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val.data =\n w✝¹ ++\n AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n w✝\n⊢ Nat\n\nα : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nH :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n true\nw✝¹ w✝ : List (AssocList α β)\nh₁ :\n m.buckets.val.data =\n w✝¹ ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w✝\nleft✝ : List.length w✝¹ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val.data =\n w✝¹ ++\n AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n w✝\n⊢ List.length\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n ?n + 1",
"tactic": "{rfl}"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nH :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n true\nw✝¹ w✝ : List (AssocList α β)\nleft✝ : List.length w✝¹ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n⊢ List.length\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n List.length\n (List.eraseP (fun x => x.fst == k)\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n 1",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nH :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n true\nw✝¹ w✝ : List (AssocList α β)\nh₁ :\n m.buckets.val.data =\n w✝¹ ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w✝\nleft✝ : List.length w✝¹ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val.data =\n w✝¹ ++\n AssocList.erase k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n w✝\n⊢ List.length\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n List.length\n (List.eraseP (fun x => x.fst == k)\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n 1",
"tactic": "clear h₁ eq"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nw✝¹ w✝ : List (AssocList α β)\nleft✝ : List.length w✝¹ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\nH :\n ∃ x,\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ∧\n (x.fst == k) = true\n⊢ List.length\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n List.length\n (List.eraseP (fun x => x.fst == k)\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n 1",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nH :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n true\nw✝¹ w✝ : List (AssocList α β)\nleft✝ : List.length w✝¹ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n⊢ List.length\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n List.length\n (List.eraseP (fun x => x.fst == k)\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n 1",
"tactic": "simp [AssocList.contains_eq] at H"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nw✝¹ w✝ : List (AssocList α β)\nleft✝ : List.length w✝¹ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\nH :\n ∃ x,\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ∧\n (x.fst == k) = true\na : α × β\nh₁ :\n a ∈\n AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nh₂ : (a.fst == k) = true\n⊢ List.length\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n List.length\n (List.eraseP (fun x => x.fst == k)\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n 1",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nw✝¹ w✝ : List (AssocList α β)\nleft✝ : List.length w✝¹ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\nH :\n ∃ x,\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ∧\n (x.fst == k) = true\n⊢ List.length\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n List.length\n (List.eraseP (fun x => x.fst == k)\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n 1",
"tactic": "have ⟨a, h₁, h₂⟩ := H"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh✝ : m.size = Bucket.size m.buckets\nc✝ : Bool\nw✝⁴ w✝³ : List (AssocList α β)\nleft✝² : List.length w✝⁴ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\nH :\n ∃ x,\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ∧\n (x.fst == k) = true\na : α × β\nh₁ :\n a ∈\n AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nh₂ : (a.fst == k) = true\nw✝² : α × β\nw✝¹ w✝ : List (α × β)\nleft✝¹ : ∀ (b : α × β), b ∈ w✝¹ → ¬(b.fst == k) = true\nleft✝ : (w✝².fst == k) = true\nh :\n AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n w✝¹ ++ w✝² :: w✝\neq :\n List.eraseP (fun x => x.fst == k)\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n w✝¹ ++ w✝\n⊢ List.length\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n List.length (w✝¹ ++ w✝) + 1",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nw✝¹ w✝ : List (AssocList α β)\nleft✝ : List.length w✝¹ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\nH :\n ∃ x,\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ∧\n (x.fst == k) = true\na : α × β\nh₁ :\n a ∈\n AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nh₂ : (a.fst == k) = true\n⊢ List.length\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n List.length\n (List.eraseP (fun x => x.fst == k)\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +\n 1",
"tactic": "refine have ⟨_, _, _, _, _, h, eq⟩ := List.exists_of_eraseP h₁ h₂; eq ▸ ?_"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh✝ : m.size = Bucket.size m.buckets\nc✝ : Bool\nw✝⁴ w✝³ : List (AssocList α β)\nleft✝² : List.length w✝⁴ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\nH :\n ∃ x,\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ∧\n (x.fst == k) = true\na : α × β\nh₁ :\n a ∈\n AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nh₂ : (a.fst == k) = true\nw✝² : α × β\nw✝¹ w✝ : List (α × β)\nleft✝¹ : ∀ (b : α × β), b ∈ w✝¹ → ¬(b.fst == k) = true\nleft✝ : (w✝².fst == k) = true\nh :\n AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n w✝¹ ++ w✝² :: w✝\neq :\n List.eraseP (fun x => x.fst == k)\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n w✝¹ ++ w✝\n⊢ List.length w✝¹ + Nat.succ (List.length w✝) = List.length w✝¹ + List.length w✝ + 1",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh✝ : m.size = Bucket.size m.buckets\nc✝ : Bool\nw✝⁴ w✝³ : List (AssocList α β)\nleft✝² : List.length w✝⁴ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\nH :\n ∃ x,\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ∧\n (x.fst == k) = true\na : α × β\nh₁ :\n a ∈\n AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nh₂ : (a.fst == k) = true\nw✝² : α × β\nw✝¹ w✝ : List (α × β)\nleft✝¹ : ∀ (b : α × β), b ∈ w✝¹ → ¬(b.fst == k) = true\nleft✝ : (w✝².fst == k) = true\nh :\n AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n w✝¹ ++ w✝² :: w✝\neq :\n List.eraseP (fun x => x.fst == k)\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n w✝¹ ++ w✝\n⊢ List.length\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n List.length (w✝¹ ++ w✝) + 1",
"tactic": "simp [h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh✝ : m.size = Bucket.size m.buckets\nc✝ : Bool\nw✝⁴ w✝³ : List (AssocList α β)\nleft✝² : List.length w✝⁴ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\nH :\n ∃ x,\n x ∈\n AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ∧\n (x.fst == k) = true\na : α × β\nh₁ :\n a ∈\n AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]\nh₂ : (a.fst == k) = true\nw✝² : α × β\nw✝¹ w✝ : List (α × β)\nleft✝¹ : ∀ (b : α × β), b ∈ w✝¹ → ¬(b.fst == k) = true\nleft✝ : (w✝².fst == k) = true\nh :\n AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n w✝¹ ++ w✝² :: w✝\neq :\n List.eraseP (fun x => x.fst == k)\n (AssocList.toList\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =\n w✝¹ ++ w✝\n⊢ List.length w✝¹ + Nat.succ (List.length w✝) = List.length w✝¹ + List.length w✝ + 1",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case h_2\nα : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nc✝ : Bool\nheq✝ :\n AssocList.contains k\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =\n false\n⊢ m.size = Bucket.size m.buckets",
"tactic": "exact h"
}
] |
[
255,
12
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
241,
1
] |
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