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/LinearAlgebra/PiTensorProduct.lean
|
PiTensorProduct.lift_reindex
|
[] |
[
498,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
496,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.symm_mapsTo
|
[] |
[
175,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
174,
11
] |
Mathlib/Analysis/NormedSpace/Units.lean
|
NormedRing.inverse_add_norm_diff_second_order
|
[
{
"state_after": "case h.e'_7.h\nR : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : Rˣ\nx✝ : R\n⊢ inverse (↑x + x✝) - ↑x⁻¹ + ↑x⁻¹ * x✝ * ↑x⁻¹ = inverse (↑x + x✝) - (∑ i in range 2, (-↑x⁻¹ * x✝) ^ i) * ↑x⁻¹",
"state_before": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : Rˣ\n⊢ (fun t => inverse (↑x + t) - ↑x⁻¹ + ↑x⁻¹ * t * ↑x⁻¹) =O[𝓝 0] fun t => ‖t‖ ^ 2",
"tactic": "convert inverse_add_norm_diff_nth_order x 2 using 2"
},
{
"state_after": "no goals",
"state_before": "case h.e'_7.h\nR : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : Rˣ\nx✝ : R\n⊢ inverse (↑x + x✝) - ↑x⁻¹ + ↑x⁻¹ * x✝ * ↑x⁻¹ = inverse (↑x + x✝) - (∑ i in range 2, (-↑x⁻¹ * x✝) ^ i) * ↑x⁻¹",
"tactic": "simp only [sum_range_succ, sum_range_zero, zero_add, pow_zero, pow_one, add_mul, one_mul,\n ← sub_sub, neg_mul, sub_neg_eq_add]"
}
] |
[
213,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
209,
1
] |
Mathlib/Data/Polynomial/Expand.lean
|
Polynomial.map_expand_pow_char
|
[
{
"state_after": "case zero\nR : Type u\ninst✝² : CommSemiring R\nS : Type v\ninst✝¹ : CommSemiring S\np q : ℕ\ninst✝ : CharP R p\nhp : Fact (Nat.Prime p)\nf : R[X]\n⊢ map (frobenius R p ^ Nat.zero) (↑(expand R (p ^ Nat.zero)) f) = f ^ p ^ Nat.zero\n\ncase succ\nR : Type u\ninst✝² : CommSemiring R\nS : Type v\ninst✝¹ : CommSemiring S\np q : ℕ\ninst✝ : CharP R p\nhp : Fact (Nat.Prime p)\nf : R[X]\nn✝ : ℕ\nn_ih : map (frobenius R p ^ n✝) (↑(expand R (p ^ n✝)) f) = f ^ p ^ n✝\n⊢ map (frobenius R p ^ Nat.succ n✝) (↑(expand R (p ^ Nat.succ n✝)) f) = f ^ p ^ Nat.succ n✝",
"state_before": "R : Type u\ninst✝² : CommSemiring R\nS : Type v\ninst✝¹ : CommSemiring S\np q : ℕ\ninst✝ : CharP R p\nhp : Fact (Nat.Prime p)\nf : R[X]\nn : ℕ\n⊢ map (frobenius R p ^ n) (↑(expand R (p ^ n)) f) = f ^ p ^ n",
"tactic": "induction' n with _ n_ih"
},
{
"state_after": "case succ\nR : Type u\ninst✝² : CommSemiring R\nS : Type v\ninst✝¹ : CommSemiring S\np q : ℕ\ninst✝ : CharP R p\nhp : Fact (Nat.Prime p)\nf : R[X]\nn✝ : ℕ\nn_ih : map (frobenius R p ^ n✝) (↑(expand R (p ^ n✝)) f) = f ^ p ^ n✝\n⊢ f ^ p ^ Nat.succ n✝ = map (frobenius R p ^ Nat.succ n✝) (↑(expand R (p ^ Nat.succ n✝)) f)",
"state_before": "case succ\nR : Type u\ninst✝² : CommSemiring R\nS : Type v\ninst✝¹ : CommSemiring S\np q : ℕ\ninst✝ : CharP R p\nhp : Fact (Nat.Prime p)\nf : R[X]\nn✝ : ℕ\nn_ih : map (frobenius R p ^ n✝) (↑(expand R (p ^ n✝)) f) = f ^ p ^ n✝\n⊢ map (frobenius R p ^ Nat.succ n✝) (↑(expand R (p ^ Nat.succ n✝)) f) = f ^ p ^ Nat.succ n✝",
"tactic": "symm"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : Type u\ninst✝² : CommSemiring R\nS : Type v\ninst✝¹ : CommSemiring S\np q : ℕ\ninst✝ : CharP R p\nhp : Fact (Nat.Prime p)\nf : R[X]\nn✝ : ℕ\nn_ih : map (frobenius R p ^ n✝) (↑(expand R (p ^ n✝)) f) = f ^ p ^ n✝\n⊢ f ^ p ^ Nat.succ n✝ = map (frobenius R p ^ Nat.succ n✝) (↑(expand R (p ^ Nat.succ n✝)) f)",
"tactic": "rw [pow_succ', pow_mul, ← n_ih, ← expand_char, pow_succ, RingHom.mul_def, ← map_map, mul_comm,\n expand_mul, ← map_expand]"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u\ninst✝² : CommSemiring R\nS : Type v\ninst✝¹ : CommSemiring S\np q : ℕ\ninst✝ : CharP R p\nhp : Fact (Nat.Prime p)\nf : R[X]\n⊢ map (frobenius R p ^ Nat.zero) (↑(expand R (p ^ Nat.zero)) f) = f ^ p ^ Nat.zero",
"tactic": "simp [RingHom.one_def]"
}
] |
[
262,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
1
] |
Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
|
affineIndependent_iff_not_collinear_set
|
[
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type ?u.277216\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np₁ p₂ p₃ : P\n⊢ ¬Collinear k (Set.range ![p₁, p₂, p₃]) ↔ ¬Collinear k {p₁, p₂, p₃}",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type ?u.277216\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np₁ p₂ p₃ : P\n⊢ AffineIndependent k ![p₁, p₂, p₃] ↔ ¬Collinear k {p₁, p₂, p₃}",
"tactic": "rw [affineIndependent_iff_not_collinear]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type ?u.277216\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np₁ p₂ p₃ : P\n⊢ ¬Collinear k (Set.range ![p₁, p₂, p₃]) ↔ ¬Collinear k {p₁, p₂, p₃}",
"tactic": "simp_rw [Matrix.range_cons, Matrix.range_empty, Set.singleton_union, insert_emptyc_eq]"
}
] |
[
464,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
461,
1
] |
Mathlib/Data/Nat/ModEq.lean
|
Nat.ModEq.mul_right'
|
[
{
"state_after": "m n a b c✝ d c : ℕ\nh : a ≡ b [MOD n]\n⊢ c * a ≡ c * b [MOD c * n]",
"state_before": "m n a b c✝ d c : ℕ\nh : a ≡ b [MOD n]\n⊢ a * c ≡ b * c [MOD n * c]",
"tactic": "rw [mul_comm a, mul_comm b, mul_comm n]"
},
{
"state_after": "no goals",
"state_before": "m n a b c✝ d c : ℕ\nh : a ≡ b [MOD n]\n⊢ c * a ≡ c * b [MOD c * n]",
"tactic": "exact h.mul_left' c"
}
] |
[
122,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
11
] |
Mathlib/Data/List/Infix.lean
|
List.mem_of_mem_suffix
|
[] |
[
510,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
509,
1
] |
Mathlib/CategoryTheory/Idempotents/Karoubi.lean
|
CategoryTheory.Idempotents.Karoubi.comp_f
|
[
{
"state_after": "no goals",
"state_before": "C : Type u_1\ninst✝ : Category C\nP Q R : Karoubi C\nf : P ⟶ Q\ng : Q ⟶ R\n⊢ (f ≫ g).f = f.f ≫ g.f",
"tactic": "rfl"
}
] |
[
124,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Topology/Category/Compactum.lean
|
Compactum.basic_inter
|
[
{
"state_after": "case h\nX : Compactum\nA B : Set X.A\nG : Ultrafilter X.A\n⊢ G ∈ Compactum.basic (A ∩ B) ↔ G ∈ Compactum.basic A ∩ Compactum.basic B",
"state_before": "X : Compactum\nA B : Set X.A\n⊢ Compactum.basic (A ∩ B) = Compactum.basic A ∩ Compactum.basic B",
"tactic": "ext G"
},
{
"state_after": "case h.mp\nX : Compactum\nA B : Set X.A\nG : Ultrafilter X.A\n⊢ G ∈ Compactum.basic (A ∩ B) → G ∈ Compactum.basic A ∩ Compactum.basic B\n\ncase h.mpr\nX : Compactum\nA B : Set X.A\nG : Ultrafilter X.A\n⊢ G ∈ Compactum.basic A ∩ Compactum.basic B → G ∈ Compactum.basic (A ∩ B)",
"state_before": "case h\nX : Compactum\nA B : Set X.A\nG : Ultrafilter X.A\n⊢ G ∈ Compactum.basic (A ∩ B) ↔ G ∈ Compactum.basic A ∩ Compactum.basic B",
"tactic": "constructor"
},
{
"state_after": "case h.mp\nX : Compactum\nA B : Set X.A\nG : Ultrafilter X.A\nhG : G ∈ Compactum.basic (A ∩ B)\n⊢ G ∈ Compactum.basic A ∩ Compactum.basic B",
"state_before": "case h.mp\nX : Compactum\nA B : Set X.A\nG : Ultrafilter X.A\n⊢ G ∈ Compactum.basic (A ∩ B) → G ∈ Compactum.basic A ∩ Compactum.basic B",
"tactic": "intro hG"
},
{
"state_after": "case h\nX : Compactum\nA B : Set X.A\nG : Ultrafilter X.A\nhG : G ∈ Compactum.basic (A ∩ B)\na✝ : X.A\n⊢ a✝ ∈ A ∩ B → a✝ ∈ A\n\ncase h\nX : Compactum\nA B : Set X.A\nG : Ultrafilter X.A\nhG : G ∈ Compactum.basic (A ∩ B)\na✝ : X.A\n⊢ a✝ ∈ A ∩ B → a✝ ∈ B",
"state_before": "case h.mp\nX : Compactum\nA B : Set X.A\nG : Ultrafilter X.A\nhG : G ∈ Compactum.basic (A ∩ B)\n⊢ G ∈ Compactum.basic A ∩ Compactum.basic B",
"tactic": "constructor <;> filter_upwards [hG]with _"
},
{
"state_after": "no goals",
"state_before": "case h\nX : Compactum\nA B : Set X.A\nG : Ultrafilter X.A\nhG : G ∈ Compactum.basic (A ∩ B)\na✝ : X.A\n⊢ a✝ ∈ A ∩ B → a✝ ∈ A\n\ncase h\nX : Compactum\nA B : Set X.A\nG : Ultrafilter X.A\nhG : G ∈ Compactum.basic (A ∩ B)\na✝ : X.A\n⊢ a✝ ∈ A ∩ B → a✝ ∈ B",
"tactic": "exacts [And.left, And.right]"
},
{
"state_after": "case h.mpr.intro\nX : Compactum\nA B : Set X.A\nG : Ultrafilter X.A\nh1 : G ∈ Compactum.basic A\nh2 : G ∈ Compactum.basic B\n⊢ G ∈ Compactum.basic (A ∩ B)",
"state_before": "case h.mpr\nX : Compactum\nA B : Set X.A\nG : Ultrafilter X.A\n⊢ G ∈ Compactum.basic A ∩ Compactum.basic B → G ∈ Compactum.basic (A ∩ B)",
"tactic": "rintro ⟨h1, h2⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mpr.intro\nX : Compactum\nA B : Set X.A\nG : Ultrafilter X.A\nh1 : G ∈ Compactum.basic A\nh2 : G ∈ Compactum.basic B\n⊢ G ∈ Compactum.basic (A ∩ B)",
"tactic": "exact inter_mem h1 h2"
}
] |
[
217,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
9
] |
Mathlib/Topology/UniformSpace/Completion.lean
|
UniformSpace.Completion.coe_injective
|
[] |
[
459,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
458,
1
] |
Mathlib/GroupTheory/Abelianization.lean
|
abelianizationCongr_of
|
[] |
[
227,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Data/Set/Image.lean
|
Subtype.range_val
|
[] |
[
1400,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1399,
1
] |
Mathlib/Algebra/GroupWithZero/Units/Basic.lean
|
Ring.inverse_zero
|
[
{
"state_after": "α : Type ?u.9084\nM₀ : Type u_1\nG₀ : Type ?u.9090\nM₀' : Type ?u.9093\nG₀' : Type ?u.9096\nF : Type ?u.9099\nF' : Type ?u.9102\ninst✝ : MonoidWithZero M₀\n✝ : Nontrivial M₀\n⊢ inverse 0 = 0",
"state_before": "α : Type ?u.9084\nM₀ : Type u_1\nG₀ : Type ?u.9090\nM₀' : Type ?u.9093\nG₀' : Type ?u.9096\nF : Type ?u.9099\nF' : Type ?u.9102\ninst✝ : MonoidWithZero M₀\n⊢ inverse 0 = 0",
"tactic": "nontriviality"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.9084\nM₀ : Type u_1\nG₀ : Type ?u.9090\nM₀' : Type ?u.9093\nG₀' : Type ?u.9096\nF : Type ?u.9099\nF' : Type ?u.9102\ninst✝ : MonoidWithZero M₀\n✝ : Nontrivial M₀\n⊢ inverse 0 = 0",
"tactic": "exact inverse_non_unit _ not_isUnit_zero"
}
] |
[
152,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Topology/Bases.lean
|
TopologicalSpace.isOpen_sUnion_countable
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nt : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nS : Set (Set α)\nH : ∀ (s : Set α), s ∈ S → IsOpen s\nT : Set ↑S\ncT : Set.Countable T\nhT : (⋃ (i : ↑S) (_ : i ∈ T), ↑i) = ⋃ (i : ↑S), ↑i\n⊢ ⋃₀ (Subtype.val '' T) = ⋃₀ S",
"tactic": "rwa [sUnion_image, sUnion_eq_iUnion]"
}
] |
[
738,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
734,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.coe_spanSingleton
|
[
{
"state_after": "R : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1296088\ninst✝³ : CommRing R₁\nK : Type ?u.1296094\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx : P\n⊢ ↑{ val := span R {x}, property := (_ : IsFractional S (span R {x})) } = span R {x}",
"state_before": "R : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1296088\ninst✝³ : CommRing R₁\nK : Type ?u.1296094\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx : P\n⊢ ↑(spanSingleton S x) = span R {x}",
"tactic": "rw [spanSingleton]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1296088\ninst✝³ : CommRing R₁\nK : Type ?u.1296094\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx : P\n⊢ ↑{ val := span R {x}, property := (_ : IsFractional S (span R {x})) } = span R {x}",
"tactic": "rfl"
}
] |
[
1302,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1300,
1
] |
Mathlib/Topology/MetricSpace/Antilipschitz.lean
|
AntilipschitzWith.of_subsingleton
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.14379\ninst✝³ : PseudoEMetricSpace α\ninst✝² : PseudoEMetricSpace β\ninst✝¹ : PseudoEMetricSpace γ\nK✝ : ℝ≥0\nf : α → β\ninst✝ : Subsingleton α\nK : ℝ≥0\nx y : α\n⊢ edist x y ≤ ↑K * edist (f x) (f y)",
"tactic": "simp only [Subsingleton.elim x y, edist_self, zero_le]"
}
] |
[
209,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.sin_zero
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\n⊢ sin 0 = 0",
"tactic": "simp [sin]"
}
] |
[
1177,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1177,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIocDiv_neg
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIocDiv hp a (-b) = -(toIcoDiv hp (-a) b + 1)",
"tactic": "rw [← neg_neg b, toIcoDiv_neg, neg_neg, neg_neg, neg_add', neg_neg, add_sub_cancel]"
}
] |
[
397,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
396,
1
] |
Mathlib/Algebra/Star/Subalgebra.lean
|
StarSubalgebra.adjoin_induction'
|
[
{
"state_after": "F : Type ?u.594069\nR : Type u_2\nA : Type u_1\nB : Type ?u.594078\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\ns : Set A\np : { x // x ∈ adjoin R s } → Prop\na : { x // x ∈ adjoin R s }\nHs : ∀ (x : A) (h : x ∈ s), p { val := x, property := (_ : x ∈ ↑(adjoin R s)) }\nHalg : ∀ (r : R), p (↑(algebraMap R { x // x ∈ adjoin R s }) r)\nHadd : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x + y)\nHmul : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x * y)\nHstar : ∀ (x : { x // x ∈ adjoin R s }), p x → p (star x)\nb : A\nhb : b ∈ adjoin R s\n⊢ ∃ x, p { val := b, property := x }",
"state_before": "F : Type ?u.594069\nR : Type u_2\nA : Type u_1\nB : Type ?u.594078\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\ns : Set A\np : { x // x ∈ adjoin R s } → Prop\na : { x // x ∈ adjoin R s }\nHs : ∀ (x : A) (h : x ∈ s), p { val := x, property := (_ : x ∈ ↑(adjoin R s)) }\nHalg : ∀ (r : R), p (↑(algebraMap R { x // x ∈ adjoin R s }) r)\nHadd : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x + y)\nHmul : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x * y)\nHstar : ∀ (x : { x // x ∈ adjoin R s }), p x → p (star x)\nb : A\nhb : b ∈ adjoin R s\n⊢ p { val := b, property := hb }",
"tactic": "refine' Exists.elim _ fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc"
},
{
"state_after": "case Hs\nF : Type ?u.594069\nR : Type u_2\nA : Type u_1\nB : Type ?u.594078\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\ns : Set A\np : { x // x ∈ adjoin R s } → Prop\na : { x // x ∈ adjoin R s }\nHs : ∀ (x : A) (h : x ∈ s), p { val := x, property := (_ : x ∈ ↑(adjoin R s)) }\nHalg : ∀ (r : R), p (↑(algebraMap R { x // x ∈ adjoin R s }) r)\nHadd : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x + y)\nHmul : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x * y)\nHstar : ∀ (x : { x // x ∈ adjoin R s }), p x → p (star x)\nb : A\nhb : b ∈ adjoin R s\n⊢ ∀ (x : A), x ∈ s → ∃ x_1, p { val := x, property := x_1 }\n\ncase Halg\nF : Type ?u.594069\nR : Type u_2\nA : Type u_1\nB : Type ?u.594078\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\ns : Set A\np : { x // x ∈ adjoin R s } → Prop\na : { x // x ∈ adjoin R s }\nHs : ∀ (x : A) (h : x ∈ s), p { val := x, property := (_ : x ∈ ↑(adjoin R s)) }\nHalg : ∀ (r : R), p (↑(algebraMap R { x // x ∈ adjoin R s }) r)\nHadd : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x + y)\nHmul : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x * y)\nHstar : ∀ (x : { x // x ∈ adjoin R s }), p x → p (star x)\nb : A\nhb : b ∈ adjoin R s\n⊢ ∀ (r : R), ∃ x, p { val := ↑(algebraMap R A) r, property := x }\n\ncase Hadd\nF : Type ?u.594069\nR : Type u_2\nA : Type u_1\nB : Type ?u.594078\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\ns : Set A\np : { x // x ∈ adjoin R s } → Prop\na : { x // x ∈ adjoin R s }\nHs : ∀ (x : A) (h : x ∈ s), p { val := x, property := (_ : x ∈ ↑(adjoin R s)) }\nHalg : ∀ (r : R), p (↑(algebraMap R { x // x ∈ adjoin R s }) r)\nHadd : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x + y)\nHmul : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x * y)\nHstar : ∀ (x : { x // x ∈ adjoin R s }), p x → p (star x)\nb : A\nhb : b ∈ adjoin R s\n⊢ ∀ (x y : A),\n (∃ x_1, p { val := x, property := x_1 }) →\n (∃ x, p { val := y, property := x }) → ∃ x_1, p { val := x + y, property := x_1 }\n\ncase Hmul\nF : Type ?u.594069\nR : Type u_2\nA : Type u_1\nB : Type ?u.594078\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\ns : Set A\np : { x // x ∈ adjoin R s } → Prop\na : { x // x ∈ adjoin R s }\nHs : ∀ (x : A) (h : x ∈ s), p { val := x, property := (_ : x ∈ ↑(adjoin R s)) }\nHalg : ∀ (r : R), p (↑(algebraMap R { x // x ∈ adjoin R s }) r)\nHadd : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x + y)\nHmul : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x * y)\nHstar : ∀ (x : { x // x ∈ adjoin R s }), p x → p (star x)\nb : A\nhb : b ∈ adjoin R s\n⊢ ∀ (x y : A),\n (∃ x_1, p { val := x, property := x_1 }) →\n (∃ x, p { val := y, property := x }) → ∃ x_1, p { val := x * y, property := x_1 }\n\ncase Hstar\nF : Type ?u.594069\nR : Type u_2\nA : Type u_1\nB : Type ?u.594078\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\ns : Set A\np : { x // x ∈ adjoin R s } → Prop\na : { x // x ∈ adjoin R s }\nHs : ∀ (x : A) (h : x ∈ s), p { val := x, property := (_ : x ∈ ↑(adjoin R s)) }\nHalg : ∀ (r : R), p (↑(algebraMap R { x // x ∈ adjoin R s }) r)\nHadd : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x + y)\nHmul : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x * y)\nHstar : ∀ (x : { x // x ∈ adjoin R s }), p x → p (star x)\nb : A\nhb : b ∈ adjoin R s\n⊢ ∀ (x : A), (∃ x_1, p { val := x, property := x_1 }) → ∃ x_1, p { val := star x, property := x_1 }",
"state_before": "F : Type ?u.594069\nR : Type u_2\nA : Type u_1\nB : Type ?u.594078\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\ns : Set A\np : { x // x ∈ adjoin R s } → Prop\na : { x // x ∈ adjoin R s }\nHs : ∀ (x : A) (h : x ∈ s), p { val := x, property := (_ : x ∈ ↑(adjoin R s)) }\nHalg : ∀ (r : R), p (↑(algebraMap R { x // x ∈ adjoin R s }) r)\nHadd : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x + y)\nHmul : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x * y)\nHstar : ∀ (x : { x // x ∈ adjoin R s }), p x → p (star x)\nb : A\nhb : b ∈ adjoin R s\n⊢ ∃ x, p { val := b, property := x }",
"tactic": "apply adjoin_induction hb"
},
{
"state_after": "no goals",
"state_before": "case Hs\nF : Type ?u.594069\nR : Type u_2\nA : Type u_1\nB : Type ?u.594078\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\ns : Set A\np : { x // x ∈ adjoin R s } → Prop\na : { x // x ∈ adjoin R s }\nHs : ∀ (x : A) (h : x ∈ s), p { val := x, property := (_ : x ∈ ↑(adjoin R s)) }\nHalg : ∀ (r : R), p (↑(algebraMap R { x // x ∈ adjoin R s }) r)\nHadd : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x + y)\nHmul : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x * y)\nHstar : ∀ (x : { x // x ∈ adjoin R s }), p x → p (star x)\nb : A\nhb : b ∈ adjoin R s\n⊢ ∀ (x : A), x ∈ s → ∃ x_1, p { val := x, property := x_1 }\n\ncase Halg\nF : Type ?u.594069\nR : Type u_2\nA : Type u_1\nB : Type ?u.594078\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\ns : Set A\np : { x // x ∈ adjoin R s } → Prop\na : { x // x ∈ adjoin R s }\nHs : ∀ (x : A) (h : x ∈ s), p { val := x, property := (_ : x ∈ ↑(adjoin R s)) }\nHalg : ∀ (r : R), p (↑(algebraMap R { x // x ∈ adjoin R s }) r)\nHadd : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x + y)\nHmul : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x * y)\nHstar : ∀ (x : { x // x ∈ adjoin R s }), p x → p (star x)\nb : A\nhb : b ∈ adjoin R s\n⊢ ∀ (r : R), ∃ x, p { val := ↑(algebraMap R A) r, property := x }\n\ncase Hadd\nF : Type ?u.594069\nR : Type u_2\nA : Type u_1\nB : Type ?u.594078\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\ns : Set A\np : { x // x ∈ adjoin R s } → Prop\na : { x // x ∈ adjoin R s }\nHs : ∀ (x : A) (h : x ∈ s), p { val := x, property := (_ : x ∈ ↑(adjoin R s)) }\nHalg : ∀ (r : R), p (↑(algebraMap R { x // x ∈ adjoin R s }) r)\nHadd : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x + y)\nHmul : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x * y)\nHstar : ∀ (x : { x // x ∈ adjoin R s }), p x → p (star x)\nb : A\nhb : b ∈ adjoin R s\n⊢ ∀ (x y : A),\n (∃ x_1, p { val := x, property := x_1 }) →\n (∃ x, p { val := y, property := x }) → ∃ x_1, p { val := x + y, property := x_1 }\n\ncase Hmul\nF : Type ?u.594069\nR : Type u_2\nA : Type u_1\nB : Type ?u.594078\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\ns : Set A\np : { x // x ∈ adjoin R s } → Prop\na : { x // x ∈ adjoin R s }\nHs : ∀ (x : A) (h : x ∈ s), p { val := x, property := (_ : x ∈ ↑(adjoin R s)) }\nHalg : ∀ (r : R), p (↑(algebraMap R { x // x ∈ adjoin R s }) r)\nHadd : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x + y)\nHmul : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x * y)\nHstar : ∀ (x : { x // x ∈ adjoin R s }), p x → p (star x)\nb : A\nhb : b ∈ adjoin R s\n⊢ ∀ (x y : A),\n (∃ x_1, p { val := x, property := x_1 }) →\n (∃ x, p { val := y, property := x }) → ∃ x_1, p { val := x * y, property := x_1 }\n\ncase Hstar\nF : Type ?u.594069\nR : Type u_2\nA : Type u_1\nB : Type ?u.594078\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\ns : Set A\np : { x // x ∈ adjoin R s } → Prop\na : { x // x ∈ adjoin R s }\nHs : ∀ (x : A) (h : x ∈ s), p { val := x, property := (_ : x ∈ ↑(adjoin R s)) }\nHalg : ∀ (r : R), p (↑(algebraMap R { x // x ∈ adjoin R s }) r)\nHadd : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x + y)\nHmul : ∀ (x y : { x // x ∈ adjoin R s }), p x → p y → p (x * y)\nHstar : ∀ (x : { x // x ∈ adjoin R s }), p x → p (star x)\nb : A\nhb : b ∈ adjoin R s\n⊢ ∀ (x : A), (∃ x_1, p { val := x, property := x_1 }) → ∃ x_1, p { val := star x, property := x_1 }",
"tactic": "exacts [fun x hx => ⟨subset_adjoin R s hx, Hs x hx⟩, fun r =>\n ⟨StarSubalgebra.algebraMap_mem _ r, Halg r⟩, fun x y hx hy =>\n Exists.elim hx fun hx' hx => Exists.elim hy fun hy' hy => ⟨add_mem hx' hy', Hadd _ _ hx hy⟩,\n fun x y hx hy =>\n Exists.elim hx fun hx' hx => Exists.elim hy fun hy' hy => ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩,\n fun x hx => Exists.elim hx fun hx' hx => ⟨star_mem hx', Hstar _ hx⟩]"
}
] |
[
542,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
530,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.exp_le_exp
|
[] |
[
1525,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1524,
1
] |
Mathlib/Combinatorics/SimpleGraph/Coloring.lean
|
SimpleGraph.Colorable.mono
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\nG : SimpleGraph V\nα : Type v\nC : Coloring G α\nn m : ℕ\nh : n ≤ m\nhc : Colorable G n\n⊢ Fintype.card (Fin n) ≤ Fintype.card (Fin m)",
"tactic": "simp [h]"
}
] |
[
203,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean
|
MvPolynomial.coeff_weightedHomogeneousComponent
|
[] |
[
341,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
1
] |
Mathlib/Data/Polynomial/Monic.lean
|
Polynomial.Monic.natDegree_mul_comm
|
[
{
"state_after": "case pos\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q✝ r : R[X]\nhp : Monic p\nq : R[X]\nh : q = 0\n⊢ natDegree (p * q) = natDegree (q * p)\n\ncase neg\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q✝ r : R[X]\nhp : Monic p\nq : R[X]\nh : ¬q = 0\n⊢ natDegree (p * q) = natDegree (q * p)",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q✝ r : R[X]\nhp : Monic p\nq : R[X]\n⊢ natDegree (p * q) = natDegree (q * p)",
"tactic": "by_cases h : q = 0"
},
{
"state_after": "case neg\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q✝ r : R[X]\nhp : Monic p\nq : R[X]\nh : ¬q = 0\n⊢ Polynomial.leadingCoeff q * Polynomial.leadingCoeff p ≠ 0",
"state_before": "case neg\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q✝ r : R[X]\nhp : Monic p\nq : R[X]\nh : ¬q = 0\n⊢ natDegree (p * q) = natDegree (q * p)",
"tactic": "rw [hp.natDegree_mul' h, Polynomial.natDegree_mul', add_comm]"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q✝ r : R[X]\nhp : Monic p\nq : R[X]\nh : ¬q = 0\n⊢ Polynomial.leadingCoeff q * Polynomial.leadingCoeff p ≠ 0",
"tactic": "simpa [hp.leadingCoeff, leadingCoeff_ne_zero]"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q✝ r : R[X]\nhp : Monic p\nq : R[X]\nh : q = 0\n⊢ natDegree (p * q) = natDegree (q * p)",
"tactic": "simp [h]"
}
] |
[
197,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Algebra/Module/Basic.lean
|
Int.smul_one_eq_coe
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.325888\nR✝ : Type ?u.325891\nk : Type ?u.325894\nS : Type ?u.325897\nM : Type ?u.325900\nM₂ : Type ?u.325903\nM₃ : Type ?u.325906\nι : Type ?u.325909\nR : Type u_1\ninst✝ : Ring R\nm : ℤ\n⊢ m • 1 = ↑m",
"tactic": "rw [zsmul_eq_mul, mul_one]"
}
] |
[
767,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
766,
1
] |
Mathlib/Data/Finset/Image.lean
|
Finset.forall_mem_map
|
[
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.6251\nf✝ : α ↪ β\ns✝ : Finset α\nf : α ↪ β\ns : Finset α\np : (a : β) → a ∈ map f s → Prop\nh : ∀ (x : α) (H : x ∈ s), p (↑f x) (_ : ↑f x ∈ map f s)\ny : α\nhy : y ∈ s\nhx : ↑f y ∈ map f s\n⊢ p (↑f y) hx",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.6251\nf✝ : α ↪ β\ns✝ : Finset α\nf : α ↪ β\ns : Finset α\np : (a : β) → a ∈ map f s → Prop\nh : ∀ (x : α) (H : x ∈ s), p (↑f x) (_ : ↑f x ∈ map f s)\nx : β\nhx : x ∈ map f s\n⊢ p x hx",
"tactic": "obtain ⟨y, hy, rfl⟩ := mem_map.1 hx"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.6251\nf✝ : α ↪ β\ns✝ : Finset α\nf : α ↪ β\ns : Finset α\np : (a : β) → a ∈ map f s → Prop\nh : ∀ (x : α) (H : x ∈ s), p (↑f x) (_ : ↑f x ∈ map f s)\ny : α\nhy : y ∈ s\nhx : ↑f y ∈ map f s\n⊢ p (↑f y) hx",
"tactic": "exact h _ hy"
}
] |
[
96,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.comap_inf
|
[] |
[
1471,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1470,
1
] |
Mathlib/Topology/Algebra/Module/CharacterSpace.lean
|
WeakDual.CharacterSpace.coe_toNonUnitalAlgHom
|
[] |
[
108,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Std/Data/RBMap/Alter.lean
|
Std.RBNode.Path.zoom_zoomed₂
|
[
{
"state_after": "α✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\nhp : Zoomed cut path\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\n⊢ zoom cut (node c✝ l✝ v✝ r✝) path = (t', path') → Zoomed cut path'",
"state_before": "α✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne✝ : zoom cut t path = (t'✝, path'✝)\nhp : Zoomed cut path\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\ne : zoom cut (node c✝ l✝ v✝ r✝) path = (t', path')\n⊢ Zoomed cut path'",
"tactic": "revert e"
},
{
"state_after": "α✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\nhp : Zoomed cut path\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\n⊢ (match cut v✝ with\n | Ordering.lt => zoom cut l✝ (left c✝ path v✝ r✝)\n | Ordering.gt => zoom cut r✝ (right c✝ l✝ v✝ path)\n | Ordering.eq => (node c✝ l✝ v✝ r✝, path)) =\n (t', path') →\n Zoomed cut path'",
"state_before": "α✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\nhp : Zoomed cut path\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\n⊢ zoom cut (node c✝ l✝ v✝ r✝) path = (t', path') → Zoomed cut path'",
"tactic": "unfold zoom"
},
{
"state_after": "case h_1\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\nhp : Zoomed cut path\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.lt\n⊢ zoom cut l✝ (left c✝ path v✝ r✝) = (t', path') → Zoomed cut path'\n\ncase h_2\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\nhp : Zoomed cut path\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.gt\n⊢ zoom cut r✝ (right c✝ l✝ v✝ path) = (t', path') → Zoomed cut path'\n\ncase h_3\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\nhp : Zoomed cut path\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\n⊢ (node c✝ l✝ v✝ r✝, path) = (t', path') → Zoomed cut path'",
"state_before": "α✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\nhp : Zoomed cut path\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\n⊢ (match cut v✝ with\n | Ordering.lt => zoom cut l✝ (left c✝ path v✝ r✝)\n | Ordering.gt => zoom cut r✝ (right c✝ l✝ v✝ path)\n | Ordering.eq => (node c✝ l✝ v✝ r✝, path)) =\n (t', path') →\n Zoomed cut path'",
"tactic": "split"
},
{
"state_after": "no goals",
"state_before": "case h_1\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\nhp : Zoomed cut path\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.lt\n⊢ zoom cut l✝ (left c✝ path v✝ r✝) = (t', path') → Zoomed cut path'",
"tactic": "next h => exact fun e => zoom_zoomed₂ e ⟨h, hp⟩"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\nhp : Zoomed cut path\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nh : cut v✝ = Ordering.lt\n⊢ zoom cut l✝ (left c✝ path v✝ r✝) = (t', path') → Zoomed cut path'",
"tactic": "exact fun e => zoom_zoomed₂ e ⟨h, hp⟩"
},
{
"state_after": "no goals",
"state_before": "case h_2\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\nhp : Zoomed cut path\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.gt\n⊢ zoom cut r✝ (right c✝ l✝ v✝ path) = (t', path') → Zoomed cut path'",
"tactic": "next h => exact fun e => zoom_zoomed₂ e ⟨h, hp⟩"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\nhp : Zoomed cut path\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nh : cut v✝ = Ordering.gt\n⊢ zoom cut r✝ (right c✝ l✝ v✝ path) = (t', path') → Zoomed cut path'",
"tactic": "exact fun e => zoom_zoomed₂ e ⟨h, hp⟩"
},
{
"state_after": "case h_3\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne✝ : zoom cut t path = (t'✝, path'✝)\nhp : Zoomed cut path\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\ne : (node c✝ l✝ v✝ r✝, path) = (t', path')\n⊢ Zoomed cut path'",
"state_before": "case h_3\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\nhp : Zoomed cut path\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\n⊢ (node c✝ l✝ v✝ r✝, path) = (t', path') → Zoomed cut path'",
"tactic": "intro e"
},
{
"state_after": "case h_3.refl\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\ne : zoom cut t path = (t', path')\nhp : Zoomed cut path\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\n⊢ Zoomed cut path",
"state_before": "case h_3\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne✝ : zoom cut t path = (t'✝, path'✝)\nhp : Zoomed cut path\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\ne : (node c✝ l✝ v✝ r✝, path) = (t', path')\n⊢ Zoomed cut path'",
"tactic": "cases e"
},
{
"state_after": "no goals",
"state_before": "case h_3.refl\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\ne : zoom cut t path = (t', path')\nhp : Zoomed cut path\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\n⊢ Zoomed cut path",
"tactic": "exact hp"
}
] |
[
235,
33
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
227,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIcoMod_sub_self
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIcoMod hp a b - b = -toIcoDiv hp a b • p",
"tactic": "rw [toIcoMod, sub_sub_cancel_left, neg_smul]"
}
] |
[
136,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
ClosedEmbedding.compactSpace
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.96598\nπ : ι → Type ?u.96603\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nh : CompactSpace β\nf : α → β\nhf : ClosedEmbedding f\n⊢ IsCompact (range f)",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.96598\nπ : ι → Type ?u.96603\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nh : CompactSpace β\nf : α → β\nhf : ClosedEmbedding f\n⊢ IsCompact univ",
"tactic": "rw [← hf.toInducing.isCompact_iff, image_univ]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.96598\nπ : ι → Type ?u.96603\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nh : CompactSpace β\nf : α → β\nhf : ClosedEmbedding f\n⊢ IsCompact (range f)",
"tactic": "exact hf.closed_range.isCompact"
}
] |
[
940,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
938,
11
] |
Mathlib/Data/Set/Intervals/Infinite.lean
|
Set.Iio_infinite
|
[] |
[
81,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/Analysis/Convex/Quasiconvex.lean
|
Monotone.quasiconcaveOn
|
[] |
[
213,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Algebra/Order/Monoid/WithTop.lean
|
WithBot.coe_eq_one
|
[] |
[
491,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
490,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
exists_measurable_piecewise
|
[
{
"state_after": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\n⊢ ∃ f, Measurable f ∧ ∀ (n : ι), EqOn f (g n) (t n)",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\n⊢ ∃ f, Measurable f ∧ ∀ (n : ι), EqOn f (g n) (t n)",
"tactic": "inhabit ι"
},
{
"state_after": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\ng' : (i : ι) → ↑(t i) → β := fun i => g i ∘ Subtype.val\n⊢ ∃ f, Measurable f ∧ ∀ (n : ι), EqOn f (g n) (t n)",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\n⊢ ∃ f, Measurable f ∧ ∀ (n : ι), EqOn f (g n) (t n)",
"tactic": "set g' : (i : ι) → t i → β := fun i => g i ∘ (↑)"
},
{
"state_after": "case ht'\nα : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\ng' : (i : ι) → ↑(t i) → β := fun i => g i ∘ Subtype.val\n⊢ ∀ (i j : ι) (x : α) (hxi : x ∈ t i) (hxj : x ∈ t j),\n g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\n\nα : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\ng' : (i : ι) → ↑(t i) → β := fun i => g i ∘ Subtype.val\nht' :\n ∀ (i j : ι) (x : α) (hxi : x ∈ t i) (hxj : x ∈ t j),\n g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\n⊢ ∃ f, Measurable f ∧ ∀ (n : ι), EqOn f (g n) (t n)",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\ng' : (i : ι) → ↑(t i) → β := fun i => g i ∘ Subtype.val\n⊢ ∃ f, Measurable f ∧ ∀ (n : ι), EqOn f (g n) (t n)",
"tactic": "have ht' : ∀ (i j) (x : α) (hxi : x ∈ t i) (hxj : x ∈ t j), g' i ⟨x, hxi⟩ = g' j ⟨x, hxj⟩"
},
{
"state_after": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\ng' : (i : ι) → ↑(t i) → β := fun i => g i ∘ Subtype.val\nht' :\n ∀ (i j : ι) (x : α) (hxi : x ∈ t i) (hxj : x ∈ t j),\n g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\nf : ↑(⋃ (i : ι), t i) → β := iUnionLift t g' ht' (⋃ (i : ι), t i) (_ : (⋃ (i : ι), t i) ⊆ ⋃ (i : ι), t i)\n⊢ ∃ f, Measurable f ∧ ∀ (n : ι), EqOn f (g n) (t n)",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\ng' : (i : ι) → ↑(t i) → β := fun i => g i ∘ Subtype.val\nht' :\n ∀ (i j : ι) (x : α) (hxi : x ∈ t i) (hxj : x ∈ t j),\n g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\n⊢ ∃ f, Measurable f ∧ ∀ (n : ι), EqOn f (g n) (t n)",
"tactic": "set f : (⋃ i, t i) → β := iUnionLift t g' ht' _ Subset.rfl"
},
{
"state_after": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\ng' : (i : ι) → ↑(t i) → β := fun i => g i ∘ Subtype.val\nht' :\n ∀ (i j : ι) (x : α) (hxi : x ∈ t i) (hxj : x ∈ t j),\n g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\nf : ↑(⋃ (i : ι), t i) → β := iUnionLift t g' ht' (⋃ (i : ι), t i) (_ : (⋃ (i : ι), t i) ⊆ ⋃ (i : ι), t i)\nhfm : Measurable f\n⊢ ∃ f, Measurable f ∧ ∀ (n : ι), EqOn f (g n) (t n)",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\ng' : (i : ι) → ↑(t i) → β := fun i => g i ∘ Subtype.val\nht' :\n ∀ (i j : ι) (x : α) (hxi : x ∈ t i) (hxj : x ∈ t j),\n g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\nf : ↑(⋃ (i : ι), t i) → β := iUnionLift t g' ht' (⋃ (i : ι), t i) (_ : (⋃ (i : ι), t i) ⊆ ⋃ (i : ι), t i)\n⊢ ∃ f, Measurable f ∧ ∀ (n : ι), EqOn f (g n) (t n)",
"tactic": "have hfm : Measurable f := measurable_iUnionLift _ _ t_meas\n (fun i => (hg i).comp measurable_subtype_coe)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\ng' : (i : ι) → ↑(t i) → β := fun i => g i ∘ Subtype.val\nht' :\n ∀ (i j : ι) (x : α) (hxi : x ∈ t i) (hxj : x ∈ t j),\n g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\nf : ↑(⋃ (i : ι), t i) → β := iUnionLift t g' ht' (⋃ (i : ι), t i) (_ : (⋃ (i : ι), t i) ⊆ ⋃ (i : ι), t i)\nhfm : Measurable f\n⊢ ∃ f, Measurable f ∧ ∀ (n : ι), EqOn f (g n) (t n)",
"tactic": "classical\n refine ⟨fun x => if hx : x ∈ ⋃ i, t i then f ⟨x, hx⟩ else g default x,\n hfm.dite ((hg default).comp measurable_subtype_coe) (.iUnion t_meas), fun i x hx => ?_⟩\n simp only [dif_pos (mem_iUnion.2 ⟨i, hx⟩)]\n exact iUnionLift_of_mem ⟨x, mem_iUnion.2 ⟨i, hx⟩⟩ hx"
},
{
"state_after": "case ht'\nα : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\ng' : (i : ι) → ↑(t i) → β := fun i => g i ∘ Subtype.val\ni j : ι\nx : α\nhxi : x ∈ t i\nhxj : x ∈ t j\n⊢ g' i { val := x, property := hxi } = g' j { val := x, property := hxj }",
"state_before": "case ht'\nα : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\ng' : (i : ι) → ↑(t i) → β := fun i => g i ∘ Subtype.val\n⊢ ∀ (i j : ι) (x : α) (hxi : x ∈ t i) (hxj : x ∈ t j),\n g' i { val := x, property := hxi } = g' j { val := x, property := hxj }",
"tactic": "intro i j x hxi hxj"
},
{
"state_after": "case ht'.inl\nα : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\ng' : (i : ι) → ↑(t i) → β := fun i => g i ∘ Subtype.val\ni : ι\nx : α\nhxi hxj : x ∈ t i\n⊢ g' i { val := x, property := hxi } = g' i { val := x, property := hxj }\n\ncase ht'.inr\nα : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\ng' : (i : ι) → ↑(t i) → β := fun i => g i ∘ Subtype.val\ni j : ι\nx : α\nhxi : x ∈ t i\nhxj : x ∈ t j\nhij : i ≠ j\n⊢ g' i { val := x, property := hxi } = g' j { val := x, property := hxj }",
"state_before": "case ht'\nα : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\ng' : (i : ι) → ↑(t i) → β := fun i => g i ∘ Subtype.val\ni j : ι\nx : α\nhxi : x ∈ t i\nhxj : x ∈ t j\n⊢ g' i { val := x, property := hxi } = g' j { val := x, property := hxj }",
"tactic": "rcases eq_or_ne i j with rfl | hij"
},
{
"state_after": "no goals",
"state_before": "case ht'.inl\nα : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\ng' : (i : ι) → ↑(t i) → β := fun i => g i ∘ Subtype.val\ni : ι\nx : α\nhxi hxj : x ∈ t i\n⊢ g' i { val := x, property := hxi } = g' i { val := x, property := hxj }",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case ht'.inr\nα : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\ng' : (i : ι) → ↑(t i) → β := fun i => g i ∘ Subtype.val\ni j : ι\nx : α\nhxi : x ∈ t i\nhxj : x ∈ t j\nhij : i ≠ j\n⊢ g' i { val := x, property := hxi } = g' j { val := x, property := hxj }",
"tactic": "exact ht hij ⟨hxi, hxj⟩"
},
{
"state_after": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\ng' : (i : ι) → ↑(t i) → β := fun i => g i ∘ Subtype.val\nht' :\n ∀ (i j : ι) (x : α) (hxi : x ∈ t i) (hxj : x ∈ t j),\n g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\nf : ↑(⋃ (i : ι), t i) → β := iUnionLift t g' ht' (⋃ (i : ι), t i) (_ : (⋃ (i : ι), t i) ⊆ ⋃ (i : ι), t i)\nhfm : Measurable f\ni : ι\nx : α\nhx : x ∈ t i\n⊢ (fun x => if hx : x ∈ ⋃ (i : ι), t i then f { val := x, property := hx } else g default x) x = g i x",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\ng' : (i : ι) → ↑(t i) → β := fun i => g i ∘ Subtype.val\nht' :\n ∀ (i j : ι) (x : α) (hxi : x ∈ t i) (hxj : x ∈ t j),\n g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\nf : ↑(⋃ (i : ι), t i) → β := iUnionLift t g' ht' (⋃ (i : ι), t i) (_ : (⋃ (i : ι), t i) ⊆ ⋃ (i : ι), t i)\nhfm : Measurable f\n⊢ ∃ f, Measurable f ∧ ∀ (n : ι), EqOn f (g n) (t n)",
"tactic": "refine ⟨fun x => if hx : x ∈ ⋃ i, t i then f ⟨x, hx⟩ else g default x,\n hfm.dite ((hg default).comp measurable_subtype_coe) (.iUnion t_meas), fun i x hx => ?_⟩"
},
{
"state_after": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\ng' : (i : ι) → ↑(t i) → β := fun i => g i ∘ Subtype.val\nht' :\n ∀ (i j : ι) (x : α) (hxi : x ∈ t i) (hxj : x ∈ t j),\n g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\nf : ↑(⋃ (i : ι), t i) → β := iUnionLift t g' ht' (⋃ (i : ι), t i) (_ : (⋃ (i : ι), t i) ⊆ ⋃ (i : ι), t i)\nhfm : Measurable f\ni : ι\nx : α\nhx : x ∈ t i\n⊢ iUnionLift t (fun i => g i ∘ Subtype.val) ht' (⋃ (i : ι), t i) (_ : (⋃ (i : ι), t i) ⊆ ⋃ (i : ι), t i)\n { val := x, property := (_ : x ∈ ⋃ (i : ι), t i) } =\n g i x",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\ng' : (i : ι) → ↑(t i) → β := fun i => g i ∘ Subtype.val\nht' :\n ∀ (i j : ι) (x : α) (hxi : x ∈ t i) (hxj : x ∈ t j),\n g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\nf : ↑(⋃ (i : ι), t i) → β := iUnionLift t g' ht' (⋃ (i : ι), t i) (_ : (⋃ (i : ι), t i) ⊆ ⋃ (i : ι), t i)\nhfm : Measurable f\ni : ι\nx : α\nhx : x ∈ t i\n⊢ (fun x => if hx : x ∈ ⋃ (i : ι), t i then f { val := x, property := hx } else g default x) x = g i x",
"tactic": "simp only [dif_pos (mem_iUnion.2 ⟨i, hx⟩)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.93731\nδ : Type ?u.93734\nδ' : Type ?u.93737\nι✝ : Sort uι\ns t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_1\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι\ng' : (i : ι) → ↑(t i) → β := fun i => g i ∘ Subtype.val\nht' :\n ∀ (i j : ι) (x : α) (hxi : x ∈ t i) (hxj : x ∈ t j),\n g' i { val := x, property := hxi } = g' j { val := x, property := hxj }\nf : ↑(⋃ (i : ι), t i) → β := iUnionLift t g' ht' (⋃ (i : ι), t i) (_ : (⋃ (i : ι), t i) ⊆ ⋃ (i : ι), t i)\nhfm : Measurable f\ni : ι\nx : α\nhx : x ∈ t i\n⊢ iUnionLift t (fun i => g i ∘ Subtype.val) ht' (⋃ (i : ι), t i) (_ : (⋃ (i : ι), t i) ⊆ ⋃ (i : ι), t i)\n { val := x, property := (_ : x ∈ ⋃ (i : ι), t i) } =\n g i x",
"tactic": "exact iUnionLift_of_mem ⟨x, mem_iUnion.2 ⟨i, hx⟩⟩ hx"
}
] |
[
816,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
797,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.mem_singleton_self
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.25331\nγ : Type ?u.25334\na : α\n⊢ a ∈ a ::ₘ 0",
"state_before": "α : Type u_1\nβ : Type ?u.25331\nγ : Type ?u.25334\na : α\n⊢ a ∈ {a}",
"tactic": "rw [← cons_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.25331\nγ : Type ?u.25334\na : α\n⊢ a ∈ a ::ₘ 0",
"tactic": "exact mem_cons_self _ _"
}
] |
[
344,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
342,
1
] |
Mathlib/LinearAlgebra/Projection.lean
|
Submodule.coe_prodEquivOfIsCompl'
|
[] |
[
118,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.mapRange.addMonoidHom_comp
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_4\nβ : Type ?u.49269\nγ : Type ?u.49272\nι : Type ?u.49275\nM : Type u_3\nM' : Type ?u.49281\nN : Type u_1\nP : Type u_2\nG : Type ?u.49290\nH : Type ?u.49293\nR : Type ?u.49296\nS : Type ?u.49299\ninst✝² : AddCommMonoid M\ninst✝¹ : AddCommMonoid N\ninst✝ : AddCommMonoid P\nf : N →+ P\nf₂ : M →+ N\n⊢ (↑f ∘ ↑f₂) 0 = 0",
"tactic": "simp only [comp_apply, map_zero]"
}
] |
[
228,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
224,
1
] |
Mathlib/Topology/Connected.lean
|
Inducing.isPreconnected_image
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.24377\nπ : ι → Type ?u.24382\ninst✝¹ : TopologicalSpace α\ns✝ t u v : Set α\ninst✝ : TopologicalSpace β\ns : Set α\nf : α → β\nhf : Inducing f\nh : IsPreconnected (f '' s)\n⊢ IsPreconnected s",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.24377\nπ : ι → Type ?u.24382\ninst✝¹ : TopologicalSpace α\ns✝ t u v : Set α\ninst✝ : TopologicalSpace β\ns : Set α\nf : α → β\nhf : Inducing f\n⊢ IsPreconnected (f '' s) ↔ IsPreconnected s",
"tactic": "refine' ⟨fun h => _, fun h => h.image _ hf.continuous.continuousOn⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.24377\nπ : ι → Type ?u.24382\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set α\nf : α → β\nhf : Inducing f\nh : IsPreconnected (f '' s)\nu v : Set α\nhu' : IsOpen u\nhv' : IsOpen v\nhuv : s ⊆ u ∪ v\nx : α\nhxs : x ∈ s\nhxu : x ∈ u\ny : α\nhys : y ∈ s\nhyv : y ∈ v\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.24377\nπ : ι → Type ?u.24382\ninst✝¹ : TopologicalSpace α\ns✝ t u v : Set α\ninst✝ : TopologicalSpace β\ns : Set α\nf : α → β\nhf : Inducing f\nh : IsPreconnected (f '' s)\n⊢ IsPreconnected s",
"tactic": "rintro u v hu' hv' huv ⟨x, hxs, hxu⟩ ⟨y, hys, hyv⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.24377\nπ : ι → Type ?u.24382\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set α\nf : α → β\nhf : Inducing f\nh : IsPreconnected (f '' s)\nv : Set α\nhv' : IsOpen v\nx : α\nhxs : x ∈ s\ny : α\nhys : y ∈ s\nhyv : y ∈ v\nu : Set β\nhu : IsOpen u\nhu' : IsOpen (f ⁻¹' u)\nhuv : s ⊆ f ⁻¹' u ∪ v\nhxu : x ∈ f ⁻¹' u\n⊢ Set.Nonempty (s ∩ (f ⁻¹' u ∩ v))",
"state_before": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.24377\nπ : ι → Type ?u.24382\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set α\nf : α → β\nhf : Inducing f\nh : IsPreconnected (f '' s)\nu v : Set α\nhu' : IsOpen u\nhv' : IsOpen v\nhuv : s ⊆ u ∪ v\nx : α\nhxs : x ∈ s\nhxu : x ∈ u\ny : α\nhys : y ∈ s\nhyv : y ∈ v\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"tactic": "rcases hf.isOpen_iff.1 hu' with ⟨u, hu, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.24377\nπ : ι → Type ?u.24382\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set α\nf : α → β\nhf : Inducing f\nh : IsPreconnected (f '' s)\nx : α\nhxs : x ∈ s\ny : α\nhys : y ∈ s\nu : Set β\nhu : IsOpen u\nhu' : IsOpen (f ⁻¹' u)\nhxu : x ∈ f ⁻¹' u\nv : Set β\nhv : IsOpen v\nhv' : IsOpen (f ⁻¹' v)\nhyv : y ∈ f ⁻¹' v\nhuv : s ⊆ f ⁻¹' u ∪ f ⁻¹' v\n⊢ Set.Nonempty (s ∩ (f ⁻¹' u ∩ f ⁻¹' v))",
"state_before": "case intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.24377\nπ : ι → Type ?u.24382\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set α\nf : α → β\nhf : Inducing f\nh : IsPreconnected (f '' s)\nv : Set α\nhv' : IsOpen v\nx : α\nhxs : x ∈ s\ny : α\nhys : y ∈ s\nhyv : y ∈ v\nu : Set β\nhu : IsOpen u\nhu' : IsOpen (f ⁻¹' u)\nhuv : s ⊆ f ⁻¹' u ∪ v\nhxu : x ∈ f ⁻¹' u\n⊢ Set.Nonempty (s ∩ (f ⁻¹' u ∩ v))",
"tactic": "rcases hf.isOpen_iff.1 hv' with ⟨v, hv, rfl⟩"
},
{
"state_after": "case huv\nα : Type u\nβ : Type v\nι : Type ?u.24377\nπ : ι → Type ?u.24382\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set α\nf : α → β\nhf : Inducing f\nh : IsPreconnected (f '' s)\nx : α\nhxs : x ∈ s\ny : α\nhys : y ∈ s\nu : Set β\nhu : IsOpen u\nhu' : IsOpen (f ⁻¹' u)\nhxu : x ∈ f ⁻¹' u\nv : Set β\nhv : IsOpen v\nhv' : IsOpen (f ⁻¹' v)\nhyv : y ∈ f ⁻¹' v\nhuv : s ⊆ f ⁻¹' u ∪ f ⁻¹' v\n⊢ f '' s ⊆ u ∪ v\n\ncase intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.24377\nπ : ι → Type ?u.24382\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set α\nf : α → β\nhf : Inducing f\nh : IsPreconnected (f '' s)\nx : α\nhxs : x ∈ s\ny : α\nhys : y ∈ s\nu : Set β\nhu : IsOpen u\nhu' : IsOpen (f ⁻¹' u)\nhxu : x ∈ f ⁻¹' u\nv : Set β\nhv : IsOpen v\nhv' : IsOpen (f ⁻¹' v)\nhyv : y ∈ f ⁻¹' v\nhuv : f '' s ⊆ u ∪ v\n⊢ Set.Nonempty (s ∩ (f ⁻¹' u ∩ f ⁻¹' v))",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.24377\nπ : ι → Type ?u.24382\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set α\nf : α → β\nhf : Inducing f\nh : IsPreconnected (f '' s)\nx : α\nhxs : x ∈ s\ny : α\nhys : y ∈ s\nu : Set β\nhu : IsOpen u\nhu' : IsOpen (f ⁻¹' u)\nhxu : x ∈ f ⁻¹' u\nv : Set β\nhv : IsOpen v\nhv' : IsOpen (f ⁻¹' v)\nhyv : y ∈ f ⁻¹' v\nhuv : s ⊆ f ⁻¹' u ∪ f ⁻¹' v\n⊢ Set.Nonempty (s ∩ (f ⁻¹' u ∩ f ⁻¹' v))",
"tactic": "replace huv : f '' s ⊆ u ∪ v"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.24377\nπ : ι → Type ?u.24382\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set α\nf : α → β\nhf : Inducing f\nh : IsPreconnected (f '' s)\nx : α\nhxs : x ∈ s\ny : α\nhys : y ∈ s\nu : Set β\nhu : IsOpen u\nhu' : IsOpen (f ⁻¹' u)\nhxu : x ∈ f ⁻¹' u\nv : Set β\nhv : IsOpen v\nhv' : IsOpen (f ⁻¹' v)\nhyv : y ∈ f ⁻¹' v\nhuv : f '' s ⊆ u ∪ v\nz : α\nhzs : z ∈ s\nhzuv : f z ∈ u ∩ v\n⊢ Set.Nonempty (s ∩ (f ⁻¹' u ∩ f ⁻¹' v))",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.24377\nπ : ι → Type ?u.24382\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set α\nf : α → β\nhf : Inducing f\nh : IsPreconnected (f '' s)\nx : α\nhxs : x ∈ s\ny : α\nhys : y ∈ s\nu : Set β\nhu : IsOpen u\nhu' : IsOpen (f ⁻¹' u)\nhxu : x ∈ f ⁻¹' u\nv : Set β\nhv : IsOpen v\nhv' : IsOpen (f ⁻¹' v)\nhyv : y ∈ f ⁻¹' v\nhuv : f '' s ⊆ u ∪ v\n⊢ Set.Nonempty (s ∩ (f ⁻¹' u ∩ f ⁻¹' v))",
"tactic": "rcases h u v hu hv huv ⟨f x, mem_image_of_mem _ hxs, hxu⟩ ⟨f y, mem_image_of_mem _ hys, hyv⟩ with\n ⟨_, ⟨z, hzs, rfl⟩, hzuv⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.24377\nπ : ι → Type ?u.24382\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set α\nf : α → β\nhf : Inducing f\nh : IsPreconnected (f '' s)\nx : α\nhxs : x ∈ s\ny : α\nhys : y ∈ s\nu : Set β\nhu : IsOpen u\nhu' : IsOpen (f ⁻¹' u)\nhxu : x ∈ f ⁻¹' u\nv : Set β\nhv : IsOpen v\nhv' : IsOpen (f ⁻¹' v)\nhyv : y ∈ f ⁻¹' v\nhuv : f '' s ⊆ u ∪ v\nz : α\nhzs : z ∈ s\nhzuv : f z ∈ u ∩ v\n⊢ Set.Nonempty (s ∩ (f ⁻¹' u ∩ f ⁻¹' v))",
"tactic": "exact ⟨z, hzs, hzuv⟩"
},
{
"state_after": "no goals",
"state_before": "case huv\nα : Type u\nβ : Type v\nι : Type ?u.24377\nπ : ι → Type ?u.24382\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set α\nf : α → β\nhf : Inducing f\nh : IsPreconnected (f '' s)\nx : α\nhxs : x ∈ s\ny : α\nhys : y ∈ s\nu : Set β\nhu : IsOpen u\nhu' : IsOpen (f ⁻¹' u)\nhxu : x ∈ f ⁻¹' u\nv : Set β\nhv : IsOpen v\nhv' : IsOpen (f ⁻¹' v)\nhyv : y ∈ f ⁻¹' v\nhuv : s ⊆ f ⁻¹' u ∪ f ⁻¹' v\n⊢ f '' s ⊆ u ∪ v",
"tactic": "rwa [image_subset_iff]"
}
] |
[
375,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
366,
1
] |
Mathlib/Topology/Constructions.lean
|
nhds_toAdd
|
[] |
[
139,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/Analysis/SpecificLimits/Normed.lean
|
tsum_geometric_of_abs_lt_1
|
[] |
[
314,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
313,
1
] |
Mathlib/Topology/Instances/RealVectorSpace.lean
|
map_real_smul
|
[] |
[
30,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
26,
1
] |
Mathlib/Order/Minimal.lean
|
maximals_singleton
|
[
{
"state_after": "α : Type u_1\nr r₁ r₂ : α → α → Prop\ns t : Set α\nb✝ b : α\n⊢ r b b → r b b",
"state_before": "α : Type u_1\nr r₁ r₂ : α → α → Prop\ns t : Set α\na b : α\n⊢ ∀ ⦃b : α⦄, b ∈ {a} → r a b → r b a",
"tactic": "rintro b (rfl : b = a)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nr r₁ r₂ : α → α → Prop\ns t : Set α\nb✝ b : α\n⊢ r b b → r b b",
"tactic": "exact id"
}
] |
[
68,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/Analysis/Convex/Segment.lean
|
segment_eq_uIcc
|
[] |
[
530,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
529,
1
] |
Mathlib/Data/PNat/Xgcd.lean
|
PNat.XgcdType.finish_isSpecial
|
[
{
"state_after": "u : XgcdType\nhs : u.wp + u.zp + u.wp * u.zp = u.x * u.y\n⊢ u.wp + (u.y * qp u + u.zp) + u.wp * (u.y * qp u + u.zp) = ((u.wp + 1) * qp u + u.x) * u.y",
"state_before": "u : XgcdType\nhs : IsSpecial u\n⊢ IsSpecial (finish u)",
"tactic": "dsimp [IsSpecial, finish] at hs⊢"
},
{
"state_after": "u : XgcdType\nhs : u.wp + u.zp + u.wp * u.zp = u.x * u.y\n⊢ u.wp + (u.y * qp u + u.zp) + u.wp * (u.y * qp u + u.zp) = u.wp + u.zp + u.wp * u.zp + (u.wp + 1) * qp u * u.y",
"state_before": "u : XgcdType\nhs : u.wp + u.zp + u.wp * u.zp = u.x * u.y\n⊢ u.wp + (u.y * qp u + u.zp) + u.wp * (u.y * qp u + u.zp) = ((u.wp + 1) * qp u + u.x) * u.y",
"tactic": "rw [add_mul _ _ u.y, add_comm _ (u.x * u.y), ← hs]"
},
{
"state_after": "no goals",
"state_before": "u : XgcdType\nhs : u.wp + u.zp + u.wp * u.zp = u.x * u.y\n⊢ u.wp + (u.y * qp u + u.zp) + u.wp * (u.y * qp u + u.zp) = u.wp + u.zp + u.wp * u.zp + (u.wp + 1) * qp u * u.y",
"tactic": "ring"
}
] |
[
286,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/Data/Set/Countable.lean
|
Set.Countable.biUnion_iff
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ns : Set α\nt : (a : α) → a ∈ s → Set β\nhs : Set.Countable s\nthis : Countable ↑s\n⊢ Set.Countable (⋃ (a : α) (h : a ∈ s), t a h) ↔ ∀ (a : α) (ha : a ∈ s), Set.Countable (t a ha)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ns : Set α\nt : (a : α) → a ∈ s → Set β\nhs : Set.Countable s\n⊢ Set.Countable (⋃ (a : α) (h : a ∈ s), t a h) ↔ ∀ (a : α) (ha : a ∈ s), Set.Countable (t a ha)",
"tactic": "haveI := hs.to_subtype"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ns : Set α\nt : (a : α) → a ∈ s → Set β\nhs : Set.Countable s\nthis : Countable ↑s\n⊢ Set.Countable (⋃ (a : α) (h : a ∈ s), t a h) ↔ ∀ (a : α) (ha : a ∈ s), Set.Countable (t a ha)",
"tactic": "rw [biUnion_eq_iUnion, countable_iUnion_iff, SetCoe.forall']"
}
] |
[
201,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/Data/Finset/Card.lean
|
Finset.card_le_one_iff_subset_singleton
|
[
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type ?u.54709\ns t : Finset α\nf : α → β\nn : ℕ\ninst✝ : Nonempty α\nH : card s ≤ 1\n⊢ ∃ x, s ⊆ {x}\n\ncase refine'_2\nα : Type u_1\nβ : Type ?u.54709\ns t : Finset α\nf : α → β\nn : ℕ\ninst✝ : Nonempty α\n⊢ (∃ x, s ⊆ {x}) → card s ≤ 1",
"state_before": "α : Type u_1\nβ : Type ?u.54709\ns t : Finset α\nf : α → β\nn : ℕ\ninst✝ : Nonempty α\n⊢ card s ≤ 1 ↔ ∃ x, s ⊆ {x}",
"tactic": "refine' ⟨fun H => _, _⟩"
},
{
"state_after": "case refine'_1.inl\nα : Type u_1\nβ : Type ?u.54709\nt : Finset α\nf : α → β\nn : ℕ\ninst✝ : Nonempty α\nH : card ∅ ≤ 1\n⊢ ∃ x, ∅ ⊆ {x}\n\ncase refine'_1.inr.intro\nα : Type u_1\nβ : Type ?u.54709\ns t : Finset α\nf : α → β\nn : ℕ\ninst✝ : Nonempty α\nH : card s ≤ 1\nx : α\nhx : x ∈ s\n⊢ ∃ x, s ⊆ {x}",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.54709\ns t : Finset α\nf : α → β\nn : ℕ\ninst✝ : Nonempty α\nH : card s ≤ 1\n⊢ ∃ x, s ⊆ {x}",
"tactic": "obtain rfl | ⟨x, hx⟩ := s.eq_empty_or_nonempty"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.inl\nα : Type u_1\nβ : Type ?u.54709\nt : Finset α\nf : α → β\nn : ℕ\ninst✝ : Nonempty α\nH : card ∅ ≤ 1\n⊢ ∃ x, ∅ ⊆ {x}",
"tactic": "exact ⟨Classical.arbitrary α, empty_subset _⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.inr.intro\nα : Type u_1\nβ : Type ?u.54709\ns t : Finset α\nf : α → β\nn : ℕ\ninst✝ : Nonempty α\nH : card s ≤ 1\nx : α\nhx : x ∈ s\n⊢ ∃ x, s ⊆ {x}",
"tactic": "exact ⟨x, fun y hy => by rw [card_le_one.1 H y hy x hx, mem_singleton]⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.54709\ns t : Finset α\nf : α → β\nn : ℕ\ninst✝ : Nonempty α\nH : card s ≤ 1\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\n⊢ y ∈ {x}",
"tactic": "rw [card_le_one.1 H y hy x hx, mem_singleton]"
},
{
"state_after": "case refine'_2.intro\nα : Type u_1\nβ : Type ?u.54709\ns t : Finset α\nf : α → β\nn : ℕ\ninst✝ : Nonempty α\nx : α\nhx : s ⊆ {x}\n⊢ card s ≤ 1",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.54709\ns t : Finset α\nf : α → β\nn : ℕ\ninst✝ : Nonempty α\n⊢ (∃ x, s ⊆ {x}) → card s ≤ 1",
"tactic": "rintro ⟨x, hx⟩"
},
{
"state_after": "case refine'_2.intro\nα : Type u_1\nβ : Type ?u.54709\ns t : Finset α\nf : α → β\nn : ℕ\ninst✝ : Nonempty α\nx : α\nhx : s ⊆ {x}\n⊢ card s ≤ card {x}",
"state_before": "case refine'_2.intro\nα : Type u_1\nβ : Type ?u.54709\ns t : Finset α\nf : α → β\nn : ℕ\ninst✝ : Nonempty α\nx : α\nhx : s ⊆ {x}\n⊢ card s ≤ 1",
"tactic": "rw [← card_singleton x]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro\nα : Type u_1\nβ : Type ?u.54709\ns t : Finset α\nf : α → β\nn : ℕ\ninst✝ : Nonempty α\nx : α\nhx : s ⊆ {x}\n⊢ card s ≤ card {x}",
"tactic": "exact card_le_of_subset hx"
}
] |
[
551,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
544,
1
] |
Mathlib/Geometry/Manifold/ChartedSpace.lean
|
OpenEmbedding.singletonChartedSpace_chartAt_eq
|
[] |
[
1037,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1035,
1
] |
Mathlib/Analysis/ODE/PicardLindelof.lean
|
PicardLindelof.continuous_proj
|
[] |
[
157,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Data/Nat/PartENat.lean
|
PartENat.add_lt_add_right
|
[
{
"state_after": "case intro\ny z : PartENat\nhz : z ≠ ⊤\nm : ℕ\nh : ↑m < y\n⊢ ↑m + z < y + z",
"state_before": "x y z : PartENat\nh : x < y\nhz : z ≠ ⊤\n⊢ x + z < y + z",
"tactic": "rcases ne_top_iff.mp (ne_top_of_lt h) with ⟨m, rfl⟩"
},
{
"state_after": "case intro.intro\ny : PartENat\nm : ℕ\nh : ↑m < y\nk : ℕ\nhz : ↑k ≠ ⊤\n⊢ ↑m + ↑k < y + ↑k",
"state_before": "case intro\ny z : PartENat\nhz : z ≠ ⊤\nm : ℕ\nh : ↑m < y\n⊢ ↑m + z < y + z",
"tactic": "rcases ne_top_iff.mp hz with ⟨k, rfl⟩"
},
{
"state_after": "case intro.intro.a\ny : PartENat\nm : ℕ\nh✝ : ↑m < y\nk : ℕ\nhz : ↑k ≠ ⊤\nh : ↑m < ⊤\n⊢ ↑m + ↑k < ⊤ + ↑k\n\ncase intro.intro.a\ny : PartENat\nm : ℕ\nh✝ : ↑m < y\nk : ℕ\nhz : ↑k ≠ ⊤\nn : ℕ\nh : ↑m < ↑n\n⊢ ↑m + ↑k < ↑n + ↑k",
"state_before": "case intro.intro\ny : PartENat\nm : ℕ\nh : ↑m < y\nk : ℕ\nhz : ↑k ≠ ⊤\n⊢ ↑m + ↑k < y + ↑k",
"tactic": "induction' y using PartENat.casesOn with n"
},
{
"state_after": "case intro.intro.a\ny : PartENat\nm : ℕ\nh✝ : ↑m < y\nk : ℕ\nhz : ↑k ≠ ⊤\nn : ℕ\nh : m < n\n⊢ ↑m + ↑k < ↑n + ↑k",
"state_before": "case intro.intro.a\ny : PartENat\nm : ℕ\nh✝ : ↑m < y\nk : ℕ\nhz : ↑k ≠ ⊤\nn : ℕ\nh : ↑m < ↑n\n⊢ ↑m + ↑k < ↑n + ↑k",
"tactic": "norm_cast at h"
},
{
"state_after": "case intro.intro.a\ny : PartENat\nm : ℕ\nh✝ : ↑m < y\nk : ℕ\nhz : ↑k ≠ ⊤\nn : ℕ\nh : m < n\n⊢ m + k < n + k",
"state_before": "case intro.intro.a\ny : PartENat\nm : ℕ\nh✝ : ↑m < y\nk : ℕ\nhz : ↑k ≠ ⊤\nn : ℕ\nh : m < n\n⊢ ↑m + ↑k < ↑n + ↑k",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.a\ny : PartENat\nm : ℕ\nh✝ : ↑m < y\nk : ℕ\nhz : ↑k ≠ ⊤\nn : ℕ\nh : m < n\n⊢ m + k < n + k",
"tactic": "apply add_lt_add_right h"
},
{
"state_after": "case intro.intro.a\ny : PartENat\nm : ℕ\nh✝ : ↑m < y\nk : ℕ\nhz : ↑k ≠ ⊤\nh : ↑m < ⊤\n⊢ ↑m + ↑k < ⊤",
"state_before": "case intro.intro.a\ny : PartENat\nm : ℕ\nh✝ : ↑m < y\nk : ℕ\nhz : ↑k ≠ ⊤\nh : ↑m < ⊤\n⊢ ↑m + ↑k < ⊤ + ↑k",
"tactic": "rw [top_add]"
},
{
"state_after": "case intro.intro.a\ny : PartENat\nm : ℕ\nh✝ : ↑m < y\nk : ℕ\nhz : ↑k ≠ ⊤\nh : ↑m < ⊤\n⊢ ↑(m + k) < ⊤",
"state_before": "case intro.intro.a\ny : PartENat\nm : ℕ\nh✝ : ↑m < y\nk : ℕ\nhz : ↑k ≠ ⊤\nh : ↑m < ⊤\n⊢ ↑m + ↑k < ⊤",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.a\ny : PartENat\nm : ℕ\nh✝ : ↑m < y\nk : ℕ\nhz : ↑k ≠ ⊤\nh : ↑m < ⊤\n⊢ ↑(m + k) < ⊤",
"tactic": "apply natCast_lt_top"
}
] |
[
469,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
460,
11
] |
Mathlib/Combinatorics/SimpleGraph/Hasse.lean
|
SimpleGraph.hasse_preconnected_of_succ
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.5179\ninst✝² : LinearOrder α\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\na b : α\n⊢ ReflTransGen (hasse α).Adj a b",
"state_before": "α : Type u_1\nβ : Type ?u.5179\ninst✝² : LinearOrder α\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\na b : α\n⊢ Reachable (hasse α) a b",
"tactic": "rw [reachable_iff_reflTransGen]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.5179\ninst✝² : LinearOrder α\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\na b : α\n⊢ ReflTransGen (hasse α).Adj a b",
"tactic": "exact\n reflTransGen_of_succ _ (fun c hc => Or.inl <| covby_succ_of_not_isMax hc.2.not_isMax)\n fun c hc => Or.inr <| covby_succ_of_not_isMax hc.2.not_isMax"
}
] |
[
91,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Std/Data/List/Lemmas.lean
|
List.pairwise_append
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nR : α → α → Prop\nl₁ l₂ : List α\n⊢ Pairwise R (l₁ ++ l₂) ↔ Pairwise R l₁ ∧ Pairwise R l₂ ∧ ∀ (a : α), a ∈ l₁ → ∀ (b : α), b ∈ l₂ → R a b",
"tactic": "induction l₁ <;> simp [*, or_imp, forall_and, and_assoc, and_left_comm]"
}
] |
[
1289,
74
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1287,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
contDiffWithinAt_iff_forall_nat_le
|
[] |
[
436,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
434,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
LinearEquiv.fst_comp_prodComm
|
[
{
"state_after": "no goals",
"state_before": "N : Type u_1\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.401591\nM₆ : Type ?u.401594\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R M\ninst✝ : Module R N\n⊢ LinearMap.comp (LinearMap.fst R N M) ↑(prodComm R M N) = LinearMap.snd R M N",
"tactic": "ext <;> simp"
}
] |
[
755,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
753,
1
] |
Mathlib/Algebra/Homology/Augment.lean
|
CochainComplex.augment_d_succ_succ
|
[] |
[
283,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
281,
1
] |
Mathlib/Computability/TMToPartrec.lean
|
Turing.PartrecToTM2.tr_ret_cons₂
|
[] |
[
1140,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1139,
1
] |
Mathlib/Algebra/Quaternion.lean
|
QuaternionAlgebra.neg_imK
|
[] |
[
252,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
9
] |
Mathlib/Algebra/Homology/HomologicalComplex.lean
|
CochainComplex.mk_d_1_0
|
[
{
"state_after": "ι : Type ?u.323151\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nX₀ X₁ X₂ : V\nd₀ : X₀ ⟶ X₁\nd₁ : X₁ ⟶ X₂\ns : d₀ ≫ d₁ = 0\nsucc :\n (t : (X₀ : V) ×' (X₁ : V) ×' (X₂ : V) ×' (d₀ : X₀ ⟶ X₁) ×' (d₁ : X₁ ⟶ X₂) ×' d₀ ≫ d₁ = 0) →\n (X₃ : V) ×' (d₂ : t.snd.snd.fst ⟶ X₃) ×' t.snd.snd.snd.snd.fst ≫ d₂ = 0\n⊢ (if 1 = 0 + 1 then d₀ ≫ 𝟙 X₁ else 0) = d₀",
"state_before": "ι : Type ?u.323151\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nX₀ X₁ X₂ : V\nd₀ : X₀ ⟶ X₁\nd₁ : X₁ ⟶ X₂\ns : d₀ ≫ d₁ = 0\nsucc :\n (t : (X₀ : V) ×' (X₁ : V) ×' (X₂ : V) ×' (d₀ : X₀ ⟶ X₁) ×' (d₁ : X₁ ⟶ X₂) ×' d₀ ≫ d₁ = 0) →\n (X₃ : V) ×' (d₂ : t.snd.snd.fst ⟶ X₃) ×' t.snd.snd.snd.snd.fst ≫ d₂ = 0\n⊢ HomologicalComplex.d (mk X₀ X₁ X₂ d₀ d₁ s succ) 0 1 = d₀",
"tactic": "change ite (1 = 0 + 1) (d₀ ≫ 𝟙 X₁) 0 = d₀"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.323151\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nX₀ X₁ X₂ : V\nd₀ : X₀ ⟶ X₁\nd₁ : X₁ ⟶ X₂\ns : d₀ ≫ d₁ = 0\nsucc :\n (t : (X₀ : V) ×' (X₁ : V) ×' (X₂ : V) ×' (d₀ : X₀ ⟶ X₁) ×' (d₁ : X₁ ⟶ X₂) ×' d₀ ≫ d₁ = 0) →\n (X₃ : V) ×' (d₂ : t.snd.snd.fst ⟶ X₃) ×' t.snd.snd.snd.snd.fst ≫ d₂ = 0\n⊢ (if 1 = 0 + 1 then d₀ ≫ 𝟙 X₁ else 0) = d₀",
"tactic": "rw [if_pos rfl, Category.comp_id]"
}
] |
[
1010,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1008,
1
] |
Mathlib/Algebra/MonoidAlgebra/Grading.lean
|
AddMonoidAlgebra.mem_grade_iff'
|
[
{
"state_after": "M : Type u_2\nι : Type ?u.7453\nR : Type u_1\ninst✝¹ : DecidableEq M\ninst✝ : CommSemiring R\nm : M\na : AddMonoidAlgebra R M\n⊢ (∃ b, a = Finsupp.single m b) ↔ a ∈ LinearMap.range (Finsupp.lsingle m)",
"state_before": "M : Type u_2\nι : Type ?u.7453\nR : Type u_1\ninst✝¹ : DecidableEq M\ninst✝ : CommSemiring R\nm : M\na : AddMonoidAlgebra R M\n⊢ a ∈ grade R m ↔ a ∈ LinearMap.range (Finsupp.lsingle m)",
"tactic": "rw [mem_grade_iff, Finsupp.support_subset_singleton']"
},
{
"state_after": "case h\nM : Type u_2\nι : Type ?u.7453\nR : Type u_1\ninst✝¹ : DecidableEq M\ninst✝ : CommSemiring R\nm : M\na : AddMonoidAlgebra R M\n⊢ ∀ (a_1 : R), a = Finsupp.single m a_1 ↔ ↑(Finsupp.lsingle m) a_1 = a",
"state_before": "M : Type u_2\nι : Type ?u.7453\nR : Type u_1\ninst✝¹ : DecidableEq M\ninst✝ : CommSemiring R\nm : M\na : AddMonoidAlgebra R M\n⊢ (∃ b, a = Finsupp.single m b) ↔ a ∈ LinearMap.range (Finsupp.lsingle m)",
"tactic": "apply exists_congr"
},
{
"state_after": "case h\nM : Type u_2\nι : Type ?u.7453\nR : Type u_1\ninst✝¹ : DecidableEq M\ninst✝ : CommSemiring R\nm : M\na : AddMonoidAlgebra R M\nr : R\n⊢ a = Finsupp.single m r ↔ ↑(Finsupp.lsingle m) r = a",
"state_before": "case h\nM : Type u_2\nι : Type ?u.7453\nR : Type u_1\ninst✝¹ : DecidableEq M\ninst✝ : CommSemiring R\nm : M\na : AddMonoidAlgebra R M\n⊢ ∀ (a_1 : R), a = Finsupp.single m a_1 ↔ ↑(Finsupp.lsingle m) a_1 = a",
"tactic": "intro r"
},
{
"state_after": "no goals",
"state_before": "case h\nM : Type u_2\nι : Type ?u.7453\nR : Type u_1\ninst✝¹ : DecidableEq M\ninst✝ : CommSemiring R\nm : M\na : AddMonoidAlgebra R M\nr : R\n⊢ a = Finsupp.single m r ↔ ↑(Finsupp.lsingle m) r = a",
"tactic": "constructor <;> exact Eq.symm"
}
] |
[
80,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Data/List/BigOperators/Basic.lean
|
List.prod_take_mul_prod_drop
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.51930\nα : Type ?u.51933\nM : Type u_1\nN : Type ?u.51939\nP : Type ?u.51942\nM₀ : Type ?u.51945\nG : Type ?u.51948\nR : Type ?u.51951\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl l₁ l₂ : List M\na : M\ni : ℕ\n⊢ prod (take i []) * prod (drop i []) = prod []",
"tactic": "simp [Nat.zero_le]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.51930\nα : Type ?u.51933\nM : Type u_1\nN : Type ?u.51939\nP : Type ?u.51942\nM₀ : Type ?u.51945\nG : Type ?u.51948\nR : Type ?u.51951\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl l₁ l₂ : List M\na : M\nL : List M\n⊢ prod (take 0 L) * prod (drop 0 L) = prod L",
"tactic": "simp"
},
{
"state_after": "ι : Type ?u.51930\nα : Type ?u.51933\nM : Type u_1\nN : Type ?u.51939\nP : Type ?u.51942\nM₀ : Type ?u.51945\nG : Type ?u.51948\nR : Type ?u.51951\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl l₁ l₂ : List M\na h : M\nt : List M\nn : ℕ\n⊢ prod (h :: take n t) * prod (drop n t) = prod (h :: t)",
"state_before": "ι : Type ?u.51930\nα : Type ?u.51933\nM : Type u_1\nN : Type ?u.51939\nP : Type ?u.51942\nM₀ : Type ?u.51945\nG : Type ?u.51948\nR : Type ?u.51951\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl l₁ l₂ : List M\na h : M\nt : List M\nn : ℕ\n⊢ prod (take (n + 1) (h :: t)) * prod (drop (n + 1) (h :: t)) = prod (h :: t)",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.51930\nα : Type ?u.51933\nM : Type u_1\nN : Type ?u.51939\nP : Type ?u.51942\nM₀ : Type ?u.51945\nG : Type ?u.51948\nR : Type ?u.51951\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl l₁ l₂ : List M\na h : M\nt : List M\nn : ℕ\n⊢ prod (h :: take n t) * prod (drop n t) = prod (h :: t)",
"tactic": "rw [prod_cons, prod_cons, mul_assoc, prod_take_mul_prod_drop t]"
}
] |
[
170,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.Memℒp.of_le
|
[] |
[
537,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
535,
1
] |
Mathlib/SetTheory/Cardinal/Divisibility.lean
|
Cardinal.dvd_of_le_of_aleph0_le
|
[] |
[
75,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.of_graph
|
[
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.161236\nγ : Type ?u.161239\nδ : Type ?u.161242\nσ : Type ?u.161245\ninst✝⁴ : Primcodable α\ninst✝³ : Primcodable β\ninst✝² : Primcodable γ\ninst✝¹ : Primcodable δ\ninst✝ : Primcodable σ\nf : α → ℕ\nh₂ : PrimrecRel fun a b => f a = b\ng : α → ℕ\npg : Primrec g\nhg : ∀ (x : α), f x ≤ g x\n⊢ Primrec f",
"state_before": "α : Type u_1\nβ : Type ?u.161236\nγ : Type ?u.161239\nδ : Type ?u.161242\nσ : Type ?u.161245\ninst✝⁴ : Primcodable α\ninst✝³ : Primcodable β\ninst✝² : Primcodable γ\ninst✝¹ : Primcodable δ\ninst✝ : Primcodable σ\nf : α → ℕ\nh₁ : PrimrecBounded f\nh₂ : PrimrecRel fun a b => f a = b\n⊢ Primrec f",
"tactic": "rcases h₁ with ⟨g, pg, hg : ∀ x, f x ≤ g x⟩"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.161236\nγ : Type ?u.161239\nδ : Type ?u.161242\nσ : Type ?u.161245\ninst✝⁴ : Primcodable α\ninst✝³ : Primcodable β\ninst✝² : Primcodable γ\ninst✝¹ : Primcodable δ\ninst✝ : Primcodable σ\nf : α → ℕ\nh₂ : PrimrecRel fun a b => f a = b\ng : α → ℕ\npg : Primrec g\nhg : ∀ (x : α), f x ≤ g x\nn : α\n⊢ Nat.findGreatest (fun b => f n = b) (g n) = f n",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.161236\nγ : Type ?u.161239\nδ : Type ?u.161242\nσ : Type ?u.161245\ninst✝⁴ : Primcodable α\ninst✝³ : Primcodable β\ninst✝² : Primcodable γ\ninst✝¹ : Primcodable δ\ninst✝ : Primcodable σ\nf : α → ℕ\nh₂ : PrimrecRel fun a b => f a = b\ng : α → ℕ\npg : Primrec g\nhg : ∀ (x : α), f x ≤ g x\n⊢ Primrec f",
"tactic": "refine (nat_findGreatest pg h₂).of_eq fun n => ?_"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.161236\nγ : Type ?u.161239\nδ : Type ?u.161242\nσ : Type ?u.161245\ninst✝⁴ : Primcodable α\ninst✝³ : Primcodable β\ninst✝² : Primcodable γ\ninst✝¹ : Primcodable δ\ninst✝ : Primcodable σ\nf : α → ℕ\nh₂ : PrimrecRel fun a b => f a = b\ng : α → ℕ\npg : Primrec g\nhg : ∀ (x : α), f x ≤ g x\nn : α\n⊢ Nat.findGreatest (fun b => f n = b) (g n) = f n",
"tactic": "exact (Nat.findGreatest_spec (P := fun b => f n = b) (hg n) rfl).symm"
}
] |
[
812,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
808,
1
] |
Mathlib/Order/SymmDiff.lean
|
sdiff_symmDiff_eq_sup
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.32704\nα : Type u_1\nβ : Type ?u.32710\nπ : ι → Type ?u.32715\ninst✝ : GeneralizedCoheytingAlgebra α\na b c d : α\n⊢ (a \\ b) ∆ b = a ⊔ b",
"tactic": "rw [symmDiff_comm, symmDiff_sdiff_eq_sup, sup_comm]"
}
] |
[
189,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/Init/CcLemmas.lean
|
imp_eq_of_eq_false_right
|
[] |
[
61,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.unique_ext
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.26168\nγ : Type ?u.26171\nι : Type ?u.26174\nM : Type u_2\nM' : Type ?u.26180\nN : Type ?u.26183\nP : Type ?u.26186\nG : Type ?u.26189\nH : Type ?u.26192\nR : Type ?u.26195\nS : Type ?u.26198\ninst✝¹ : Zero M\ninst✝ : Unique α\nf g : α →₀ M\nh : ↑f default = ↑g default\na : α\n⊢ ↑f a = ↑g a",
"tactic": "rwa [Unique.eq_default a]"
}
] |
[
275,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
274,
1
] |
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.integrableOn_Ioi_comp_mul_right_iff
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nc a : ℝ\nha : 0 < a\n⊢ IntegrableOn (fun x => f (x * a)) (Ioi c) ↔ IntegrableOn f (Ioi (c * a))",
"tactic": "simpa only [mul_comm, MulZeroClass.mul_zero] using integrableOn_Ioi_comp_mul_left_iff f c ha"
}
] |
[
950,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
948,
1
] |
Mathlib/Algebra/EuclideanDomain/Basic.lean
|
EuclideanDomain.dvd_lcm_left
|
[
{
"state_after": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y : R\nhxy : gcd x y = 0\n⊢ x ∣ 0",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y : R\nhxy : gcd x y = 0\n⊢ x ∣ lcm x y",
"tactic": "rw [lcm, hxy, div_zero]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y : R\nhxy : gcd x y = 0\n⊢ x ∣ 0",
"tactic": "exact dvd_zero _"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y : R\nhxy : ¬gcd x y = 0\nz : R\nhz : y = gcd x y * z\n⊢ x * z * gcd x y = x * y",
"tactic": "rw [mul_right_comm, mul_assoc, ← hz]"
}
] |
[
258,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
251,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Basic.lean
|
Pmf.apply_eq_zero_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.6289\nγ : Type ?u.6292\np : Pmf α\na : α\n⊢ ↑p a = 0 ↔ ¬a ∈ support p",
"tactic": "rw [mem_support_iff, Classical.not_not]"
}
] |
[
106,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Data/Real/GoldenRatio.lean
|
gold_irrational
|
[
{
"state_after": "this : Irrational (sqrt ↑5)\n⊢ Irrational φ",
"state_before": "⊢ Irrational φ",
"tactic": "have := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)"
},
{
"state_after": "this✝ : Irrational (sqrt ↑5)\nthis : Irrational (↑1 + sqrt ↑5)\n⊢ Irrational φ",
"state_before": "this : Irrational (sqrt ↑5)\n⊢ Irrational φ",
"tactic": "have := this.rat_add 1"
},
{
"state_after": "this✝¹ : Irrational (sqrt ↑5)\nthis✝ : Irrational (↑1 + sqrt ↑5)\nthis : Irrational (↑0.5 * (↑1 + sqrt ↑5))\n⊢ Irrational φ",
"state_before": "this✝ : Irrational (sqrt ↑5)\nthis : Irrational (↑1 + sqrt ↑5)\n⊢ Irrational φ",
"tactic": "have := this.rat_mul (show (0.5 : ℚ) ≠ 0 by norm_num)"
},
{
"state_after": "case h.e'_1\nthis✝¹ : Irrational (sqrt ↑5)\nthis✝ : Irrational (↑1 + sqrt ↑5)\nthis : Irrational (↑0.5 * (↑1 + sqrt ↑5))\n⊢ φ = ↑0.5 * (↑1 + sqrt ↑5)",
"state_before": "this✝¹ : Irrational (sqrt ↑5)\nthis✝ : Irrational (↑1 + sqrt ↑5)\nthis : Irrational (↑0.5 * (↑1 + sqrt ↑5))\n⊢ Irrational φ",
"tactic": "convert this"
},
{
"state_after": "case h.e'_1\nthis✝¹ : Irrational (sqrt ↑5)\nthis✝ : Irrational (↑1 + sqrt ↑5)\nthis : Irrational (↑0.5 * (↑1 + sqrt ↑5))\n⊢ φ = 1 / 2 * (1 + sqrt 5)",
"state_before": "case h.e'_1\nthis✝¹ : Irrational (sqrt ↑5)\nthis✝ : Irrational (↑1 + sqrt ↑5)\nthis : Irrational (↑0.5 * (↑1 + sqrt ↑5))\n⊢ φ = ↑0.5 * (↑1 + sqrt ↑5)",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "case h.e'_1\nthis✝¹ : Irrational (sqrt ↑5)\nthis✝ : Irrational (↑1 + sqrt ↑5)\nthis : Irrational (↑0.5 * (↑1 + sqrt ↑5))\n⊢ φ = 1 / 2 * (1 + sqrt 5)",
"tactic": "field_simp"
},
{
"state_after": "no goals",
"state_before": "⊢ Nat.Prime 5",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "this✝ : Irrational (sqrt ↑5)\nthis : Irrational (↑1 + sqrt ↑5)\n⊢ 0.5 ≠ 0",
"tactic": "norm_num"
}
] |
[
147,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/ModelTheory/Semantics.lean
|
FirstOrder.Language.Relations.realize_antisymmetric
|
[] |
[
1017,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1014,
1
] |
Mathlib/Combinatorics/SetFamily/Intersecting.lean
|
Set.Intersecting.mono
|
[] |
[
50,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean
|
BoxIntegral.TaggedPrepartition.iUnion_toPrepartition
|
[] |
[
77,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Topology/MetricSpace/PiNat.lean
|
PiNat.firstDiff_le_longestPrefix
|
[
{
"state_after": "E : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nx y : (n : ℕ) → E n\nhx : ¬x ∈ s\nhy : y ∈ s\n⊢ firstDiff x y + 1 ≤ shortestPrefixDiff x s\n\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nx y : (n : ℕ) → E n\nhx : ¬x ∈ s\nhy : y ∈ s\n⊢ 1 ≤ shortestPrefixDiff x s",
"state_before": "E : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nx y : (n : ℕ) → E n\nhx : ¬x ∈ s\nhy : y ∈ s\n⊢ firstDiff x y ≤ longestPrefix x s",
"tactic": "rw [longestPrefix, le_tsub_iff_right]"
},
{
"state_after": "no goals",
"state_before": "E : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nx y : (n : ℕ) → E n\nhx : ¬x ∈ s\nhy : y ∈ s\n⊢ firstDiff x y + 1 ≤ shortestPrefixDiff x s",
"tactic": "exact firstDiff_lt_shortestPrefixDiff hs hx hy"
},
{
"state_after": "no goals",
"state_before": "E : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nx y : (n : ℕ) → E n\nhx : ¬x ∈ s\nhy : y ∈ s\n⊢ 1 ≤ shortestPrefixDiff x s",
"tactic": "exact shortestPrefixDiff_pos hs ⟨y, hy⟩ hx"
}
] |
[
542,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
538,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.comapDomain_single
|
[
{
"state_after": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\nκ : Type u_1\ninst✝¹ : DecidableEq κ\ninst✝ : (i : ι) → Zero (β i)\nh : κ → ι\nhh : Function.Injective h\nk : κ\nx : β (h k)\ni : κ\n⊢ ↑(comapDomain h hh (single (h k) x)) i = ↑(single k x) i",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\nκ : Type u_1\ninst✝¹ : DecidableEq κ\ninst✝ : (i : ι) → Zero (β i)\nh : κ → ι\nhh : Function.Injective h\nk : κ\nx : β (h k)\n⊢ comapDomain h hh (single (h k) x) = single k x",
"tactic": "ext i"
},
{
"state_after": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\nκ : Type u_1\ninst✝¹ : DecidableEq κ\ninst✝ : (i : ι) → Zero (β i)\nh : κ → ι\nhh : Function.Injective h\nk : κ\nx : β (h k)\ni : κ\n⊢ ↑(single (h k) x) (h i) = ↑(single k x) i",
"state_before": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\nκ : Type u_1\ninst✝¹ : DecidableEq κ\ninst✝ : (i : ι) → Zero (β i)\nh : κ → ι\nhh : Function.Injective h\nk : κ\nx : β (h k)\ni : κ\n⊢ ↑(comapDomain h hh (single (h k) x)) i = ↑(single k x) i",
"tactic": "rw [comapDomain_apply]"
},
{
"state_after": "case h.inl\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\nκ : Type u_1\ninst✝¹ : DecidableEq κ\ninst✝ : (i : ι) → Zero (β i)\nh : κ → ι\nhh : Function.Injective h\ni : κ\nx : β (h i)\n⊢ ↑(single (h i) x) (h i) = ↑(single i x) i\n\ncase h.inr\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\nκ : Type u_1\ninst✝¹ : DecidableEq κ\ninst✝ : (i : ι) → Zero (β i)\nh : κ → ι\nhh : Function.Injective h\nk : κ\nx : β (h k)\ni : κ\nhik : i ≠ k\n⊢ ↑(single (h k) x) (h i) = ↑(single k x) i",
"state_before": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\nκ : Type u_1\ninst✝¹ : DecidableEq κ\ninst✝ : (i : ι) → Zero (β i)\nh : κ → ι\nhh : Function.Injective h\nk : κ\nx : β (h k)\ni : κ\n⊢ ↑(single (h k) x) (h i) = ↑(single k x) i",
"tactic": "obtain rfl | hik := Decidable.eq_or_ne i k"
},
{
"state_after": "no goals",
"state_before": "case h.inl\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\nκ : Type u_1\ninst✝¹ : DecidableEq κ\ninst✝ : (i : ι) → Zero (β i)\nh : κ → ι\nhh : Function.Injective h\ni : κ\nx : β (h i)\n⊢ ↑(single (h i) x) (h i) = ↑(single i x) i",
"tactic": "rw [single_eq_same, single_eq_same]"
},
{
"state_after": "no goals",
"state_before": "case h.inr\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\nκ : Type u_1\ninst✝¹ : DecidableEq κ\ninst✝ : (i : ι) → Zero (β i)\nh : κ → ι\nhh : Function.Injective h\nk : κ\nx : β (h k)\ni : κ\nhik : i ≠ k\n⊢ ↑(single (h k) x) (h i) = ↑(single k x) i",
"tactic": "rw [single_eq_of_ne hik.symm, single_eq_of_ne (hh.ne hik.symm)]"
}
] |
[
1350,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1344,
1
] |
Std/Data/Option/Init/Lemmas.lean
|
Option.getD_some
|
[] |
[
17,
55
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
17,
9
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
PowerSeries.inv_mul_cancel
|
[] |
[
2164,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2163,
11
] |
Mathlib/Algebra/Star/StarAlgHom.lean
|
StarAlgEquiv.symm_bijective
|
[] |
[
842,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
841,
1
] |
Mathlib/GroupTheory/Perm/Sign.lean
|
Equiv.Perm.sign_symm
|
[] |
[
575,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
574,
1
] |
Mathlib/Probability/Kernel/Basic.lean
|
ProbabilityTheory.kernel.integral_deterministic
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_3\nι : Type ?u.401033\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf : β → E\ng : α → β\na : α\nhg : Measurable g\ninst✝ : MeasurableSingletonClass β\n⊢ (∫ (x : β), f x ∂↑(deterministic g hg) a) = f (g a)",
"tactic": "rw [kernel.deterministic_apply, integral_dirac _ (g a)]"
}
] |
[
405,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
402,
1
] |
Mathlib/Data/Vector/Basic.lean
|
Vector.ne_cons_iff
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\nα : Type u_1\na : α\nv : Vector α (Nat.succ n)\nv' : Vector α n\n⊢ v ≠ a ::ᵥ v' ↔ head v ≠ a ∨ tail v ≠ v'",
"tactic": "rw [Ne.def, eq_cons_iff a v v', not_and_or]"
}
] |
[
71,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/CategoryTheory/ConcreteCategory/Basic.lean
|
CategoryTheory.ConcreteCategory.epi_iff_surjective_of_preservesPushout
|
[] |
[
186,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
1
] |
Mathlib/Computability/Halting.lean
|
ComputablePred.halting_problem_not_re
|
[] |
[
269,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
268,
1
] |
Mathlib/MeasureTheory/Group/FundamentalDomain.lean
|
MeasureTheory.sdiff_fundamentalFrontier
|
[] |
[
611,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
610,
1
] |
Mathlib/Order/Basic.lean
|
lt_iff_le_and_ne
|
[] |
[
378,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
377,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Complex.cos_two_pi
|
[
{
"state_after": "no goals",
"state_before": "⊢ cos (2 * ↑π) = 1",
"tactic": "simp [two_mul, cos_add]"
}
] |
[
1134,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1134,
1
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
WeakBilin.embedding
|
[] |
[
136,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
Codisjoint.himp_inf_cancel_left
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.74711\nα : Type u_1\nβ : Type ?u.74717\ninst✝ : GeneralizedHeytingAlgebra α\na b c d : α\nh : Codisjoint a b\n⊢ b ⇨ a ⊓ b = a",
"tactic": "rw [himp_inf_distrib, himp_self, inf_top_eq, h.himp_eq_right]"
}
] |
[
450,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
449,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
Num.of_to_nat
|
[] |
[
508,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
507,
1
] |
Mathlib/Algebra/Hom/Equiv/Basic.lean
|
MulEquiv.comp_symm_eq
|
[] |
[
432,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
430,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.nextCoeff_X_sub_C
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c✝ d : R\nn m : ℕ\ninst✝² : Ring R\ninst✝¹ : Nontrivial R\ninst✝ : Ring S\nc : S\n⊢ nextCoeff (X - ↑C c) = -c",
"tactic": "rw [sub_eq_add_neg, ← map_neg C c, nextCoeff_X_add_C]"
}
] |
[
1478,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1477,
1
] |
Mathlib/Data/Finset/MulAntidiagonal.lean
|
Finset.isPwo_support_mulAntidiagonal
|
[] |
[
113,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.int_cast_eq_int_cast_iff'
|
[] |
[
463,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
462,
1
] |
Mathlib/Order/FixedPoints.lean
|
OrderHom.map_lfp
|
[] |
[
82,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.sumElim_apply
|
[] |
[
1319,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1317,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.right_mem_Ico
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.5009\ninst✝ : Preorder α\na a₁ a₂ b b₁ b₂ c x : α\n⊢ b ∈ Ico a b ↔ False",
"tactic": "simp [lt_irrefl]"
}
] |
[
213,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean
|
CategoryTheory.Limits.MultispanIndex.multispan_obj_right
|
[] |
[
276,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
275,
1
] |
Mathlib/Algebra/Algebra/Unitization.lean
|
Unitization.inr_zero
|
[] |
[
286,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
285,
1
] |
Mathlib/CategoryTheory/Monad/Algebra.lean
|
CategoryTheory.Comonad.Coalgebra.id_f
|
[] |
[
411,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
410,
1
] |
Mathlib/Order/GaloisConnection.lean
|
GaloisConnection.l_unique
|
[] |
[
218,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/GroupTheory/Finiteness.lean
|
Subgroup.rank_closure_finite_le_nat_card
|
[
{
"state_after": "M : Type ?u.118917\nN : Type ?u.118920\ninst✝⁴ : Monoid M\ninst✝³ : AddMonoid N\nG : Type u_1\nH : Type ?u.118932\ninst✝² : Group G\ninst✝¹ : AddGroup H\ns : Set G\ninst✝ : Finite ↑s\nthis : Fintype ↑s\n⊢ Group.rank { x // x ∈ closure s } ≤ Nat.card ↑s",
"state_before": "M : Type ?u.118917\nN : Type ?u.118920\ninst✝⁴ : Monoid M\ninst✝³ : AddMonoid N\nG : Type u_1\nH : Type ?u.118932\ninst✝² : Group G\ninst✝¹ : AddGroup H\ns : Set G\ninst✝ : Finite ↑s\n⊢ Group.rank { x // x ∈ closure s } ≤ Nat.card ↑s",
"tactic": "haveI := Fintype.ofFinite s"
},
{
"state_after": "M : Type ?u.118917\nN : Type ?u.118920\ninst✝⁴ : Monoid M\ninst✝³ : AddMonoid N\nG : Type u_1\nH : Type ?u.118932\ninst✝² : Group G\ninst✝¹ : AddGroup H\ns : Set G\ninst✝ : Finite ↑s\nthis : Fintype ↑s\n⊢ Group.rank { x // x ∈ closure ↑(Set.toFinset s) } ≤ Finset.card (Set.toFinset s)",
"state_before": "M : Type ?u.118917\nN : Type ?u.118920\ninst✝⁴ : Monoid M\ninst✝³ : AddMonoid N\nG : Type u_1\nH : Type ?u.118932\ninst✝² : Group G\ninst✝¹ : AddGroup H\ns : Set G\ninst✝ : Finite ↑s\nthis : Fintype ↑s\n⊢ Group.rank { x // x ∈ closure s } ≤ Nat.card ↑s",
"tactic": "rw [Nat.card_eq_fintype_card, ← s.toFinset_card, ← rank_congr (congr_arg _ s.coe_toFinset)]"
},
{
"state_after": "no goals",
"state_before": "M : Type ?u.118917\nN : Type ?u.118920\ninst✝⁴ : Monoid M\ninst✝³ : AddMonoid N\nG : Type u_1\nH : Type ?u.118932\ninst✝² : Group G\ninst✝¹ : AddGroup H\ns : Set G\ninst✝ : Finite ↑s\nthis : Fintype ↑s\n⊢ Group.rank { x // x ∈ closure ↑(Set.toFinset s) } ≤ Finset.card (Set.toFinset s)",
"tactic": "exact rank_closure_finset_le_card s.toFinset"
}
] |
[
460,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
456,
1
] |
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