file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/Polynomial/Degree/TrailingDegree.lean
|
Polynomial.le_trailingDegree_C_mul_X_pow
|
[
{
"state_after": "R : Type u\nS : Type v\na✝ b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\na : R\n⊢ ↑n ≤ trailingDegree (↑(monomial n) a)",
"state_before": "R : Type u\nS : Type v\na✝ b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\na : R\n⊢ ↑n ≤ trailingDegree (↑C a * X ^ n)",
"tactic": "rw [C_mul_X_pow_eq_monomial]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na✝ b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\na : R\n⊢ ↑n ≤ trailingDegree (↑(monomial n) a)",
"tactic": "exact le_trailingDegree_monomial"
}
] |
[
261,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
1
] |
Mathlib/NumberTheory/Padics/PadicNorm.lean
|
padicNorm.of_nat
|
[] |
[
303,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
302,
1
] |
Mathlib/Data/Polynomial/FieldDivision.lean
|
Polynomial.rootMultiplicity_sub_one_le_derivative_rootMultiplicity
|
[
{
"state_after": "case pos\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\np : R[X]\nt : R\nh : IsRoot p t\n⊢ rootMultiplicity t p - 1 ≤ rootMultiplicity t (↑derivative p)\n\ncase neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\np : R[X]\nt : R\nh : ¬IsRoot p t\n⊢ rootMultiplicity t p - 1 ≤ rootMultiplicity t (↑derivative p)",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\np : R[X]\nt : R\n⊢ rootMultiplicity t p - 1 ≤ rootMultiplicity t (↑derivative p)",
"tactic": "by_cases p.IsRoot t"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\np : R[X]\nt : R\nh : IsRoot p t\n⊢ rootMultiplicity t p - 1 ≤ rootMultiplicity t (↑derivative p)",
"tactic": "exact (derivative_rootMultiplicity_of_root h).symm.le"
},
{
"state_after": "case neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\np : R[X]\nt : R\nh : ¬IsRoot p t\n⊢ 0 ≤ rootMultiplicity t (↑derivative p)",
"state_before": "case neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\np : R[X]\nt : R\nh : ¬IsRoot p t\n⊢ rootMultiplicity t p - 1 ≤ rootMultiplicity t (↑derivative p)",
"tactic": "rw [rootMultiplicity_eq_zero h, zero_tsub]"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\np : R[X]\nt : R\nh : ¬IsRoot p t\n⊢ 0 ≤ rootMultiplicity t (↑derivative p)",
"tactic": "exact zero_le _"
}
] |
[
68,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/Algebra/Order/WithZero.lean
|
le_of_le_mul_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\na b c d x y z : α\ninst✝ : LinearOrderedCommGroupWithZero α\nh : c ≠ 0\nhab : a * c ≤ b * c\n⊢ a ≤ b",
"tactic": "simpa only [mul_inv_cancel_right₀ h] using mul_le_mul_right' hab c⁻¹"
}
] |
[
137,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/Logic/Equiv/Defs.lean
|
Equiv.symm_comp_eq
|
[] |
[
602,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
601,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
contDiff_prod_mk_left
|
[] |
[
1639,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1638,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Concrete.lean
|
Equiv.Perm.length_toList_pos_of_mem_support
|
[] |
[
246,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
1
] |
Mathlib/Data/Set/Ncard.lean
|
Set.ncard_insert_eq_ite
|
[
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.11523\ns t : Set α\na b x y : α\nf : α → β\ninst✝ : Decidable (a ∈ s)\nhs : autoParam (Set.Finite s) _auto✝\nh : a ∈ s\n⊢ ncard (insert a s) = if a ∈ s then ncard s else ncard s + 1\n\ncase neg\nα : Type u_1\nβ : Type ?u.11523\ns t : Set α\na b x y : α\nf : α → β\ninst✝ : Decidable (a ∈ s)\nhs : autoParam (Set.Finite s) _auto✝\nh : ¬a ∈ s\n⊢ ncard (insert a s) = if a ∈ s then ncard s else ncard s + 1",
"state_before": "α : Type u_1\nβ : Type ?u.11523\ns t : Set α\na b x y : α\nf : α → β\ninst✝ : Decidable (a ∈ s)\nhs : autoParam (Set.Finite s) _auto✝\n⊢ ncard (insert a s) = if a ∈ s then ncard s else ncard s + 1",
"tactic": "by_cases h : a ∈ s"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.11523\ns t : Set α\na b x y : α\nf : α → β\ninst✝ : Decidable (a ∈ s)\nhs : autoParam (Set.Finite s) _auto✝\nh : a ∈ s\n⊢ ncard (insert a s) = if a ∈ s then ncard s else ncard s + 1",
"tactic": "rw [ncard_insert_of_mem h, if_pos h]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.11523\ns t : Set α\na b x y : α\nf : α → β\ninst✝ : Decidable (a ∈ s)\nhs : autoParam (Set.Finite s) _auto✝\nh : ¬a ∈ s\n⊢ ncard (insert a s) = if a ∈ s then ncard s else ncard s + 1",
"tactic": "rw [ncard_insert_of_not_mem h hs, if_neg h]"
}
] |
[
170,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/Topology/Sets/Opens.lean
|
TopologicalSpace.Opens.comap_comap
|
[] |
[
387,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
385,
11
] |
Mathlib/Topology/Algebra/InfiniteSum/Ring.lean
|
hasSum_mul_left_iff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nκ : Type ?u.8721\nR : Type ?u.8724\nα : Type u_1\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSemiring α\nf g : ι → α\na a₁ a₂ : α\nh : a₂ ≠ 0\nH : HasSum (fun i => a₂ * f i) (a₂ * a₁)\n⊢ HasSum f a₁",
"tactic": "simpa only [inv_mul_cancel_left₀ h] using H.mul_left a₂⁻¹"
}
] |
[
93,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.ball_bot
|
[] |
[
829,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
828,
1
] |
Std/Data/Nat/Gcd.lean
|
Nat.gcd_dvd_gcd_of_dvd_left
|
[] |
[
126,
67
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
125,
1
] |
Mathlib/Data/List/Basic.lean
|
List.subset_append_of_subset_right'
|
[] |
[
321,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
320,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.C_eq_zero
|
[] |
[
764,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
763,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
PosNum.add_to_nat
|
[
{
"state_after": "α : Type ?u.30847\nb : PosNum\n⊢ 1 + ↑b = ↑1 + ↑b",
"state_before": "α : Type ?u.30847\nb : PosNum\n⊢ ↑(1 + b) = ↑1 + ↑b",
"tactic": "rw [one_add b, succ_to_nat, add_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.30847\nb : PosNum\n⊢ 1 + ↑b = ↑1 + ↑b",
"tactic": "rfl"
},
{
"state_after": "α : Type ?u.30847\na : PosNum\n⊢ ↑a + 1 = ↑a + ↑1",
"state_before": "α : Type ?u.30847\na : PosNum\n⊢ ↑(a + 1) = ↑a + ↑1",
"tactic": "rw [add_one a, succ_to_nat]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.30847\na : PosNum\n⊢ ↑a + 1 = ↑a + ↑1",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.30847\na b : PosNum\n⊢ ↑a + ↑b + (↑a + ↑b) + 1 = ↑a + ↑a + (↑b + ↑b + 1)",
"tactic": "simp [add_left_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.30847\na b : PosNum\n⊢ ↑a + ↑b + (↑a + ↑b) + 1 = ↑a + ↑a + 1 + (↑b + ↑b)",
"tactic": "simp [add_comm, add_left_comm]"
},
{
"state_after": "α : Type ?u.30847\na b : PosNum\n⊢ ↑a + ↑b + 1 + (↑a + ↑b + 1) = ↑a + ↑a + 1 + (↑b + ↑b + 1)",
"state_before": "α : Type ?u.30847\na b : PosNum\n⊢ ↑(succ (a + b)) + ↑(succ (a + b)) = ↑a + ↑a + 1 + (↑b + ↑b + 1)",
"tactic": "rw [succ_to_nat, add_to_nat a b]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.30847\na b : PosNum\n⊢ ↑a + ↑b + 1 + (↑a + ↑b + 1) = ↑a + ↑a + 1 + (↑b + ↑b + 1)",
"tactic": "simp [add_left_comm]"
}
] |
[
102,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.eq_zero_or_pos
|
[] |
[
413,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
412,
1
] |
Mathlib/Topology/VectorBundle/Basic.lean
|
Pretrivialization.linearMapAt_apply
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_4\nB : Type u_1\nF : Type u_2\nE : B → Type u_3\ninst✝⁷ : Semiring R\ninst✝⁶ : TopologicalSpace F\ninst✝⁵ : TopologicalSpace B\ne✝ : Pretrivialization F TotalSpace.proj\nx : TotalSpace E\nb✝ : B\ny✝ : E b✝\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module R F\ninst✝² : (x : B) → AddCommMonoid (E x)\ninst✝¹ : (x : B) → Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ninst✝ : Pretrivialization.IsLinear R e\nb : B\ny : E b\n⊢ ↑(Pretrivialization.linearMapAt R e b) y = if b ∈ e.baseSet then (↑e (totalSpaceMk b y)).snd else 0",
"tactic": "rw [coe_linearMapAt]"
}
] |
[
136,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.union_congr_right
|
[] |
[
859,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
858,
1
] |
Mathlib/CategoryTheory/Limits/IsLimit.lean
|
CategoryTheory.Limits.IsColimit.hom_isIso
|
[] |
[
654,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
653,
1
] |
Mathlib/CategoryTheory/Subobject/Lattice.lean
|
CategoryTheory.Subobject.inf_factors
|
[
{
"state_after": "C : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nA B : C\nf : A ⟶ B\n⊢ ∀ {X Y : Subobject B}, Factors X f ∧ Factors Y f → Factors (X ⊓ Y) f",
"state_before": "C : Type u₁\ninst✝² : Category C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nA B : C\nX Y : Subobject B\nf : A ⟶ B\n⊢ Factors X f ∧ Factors Y f → Factors (X ⊓ Y) f",
"tactic": "revert X Y"
},
{
"state_after": "case h\nC : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nA B : C\nf : A ⟶ B\n⊢ ∀ (a₁ a₂ : MonoOver B),\n Factors (Quotient.mk'' a₁) f ∧ Factors (Quotient.mk'' a₂) f → Factors (Quotient.mk'' a₁ ⊓ Quotient.mk'' a₂) f",
"state_before": "C : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nA B : C\nf : A ⟶ B\n⊢ ∀ {X Y : Subobject B}, Factors X f ∧ Factors Y f → Factors (X ⊓ Y) f",
"tactic": "apply Quotient.ind₂'"
},
{
"state_after": "case h.intro.intro.intro\nC : Type u₁\ninst✝² : Category C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nA B : C\nX Y : MonoOver B\ng₁ : A ⟶ X.obj.left\ng₂ : A ⟶ Y.obj.left\nhg₂ : g₂ ≫ MonoOver.arrow Y = g₁ ≫ MonoOver.arrow X\n⊢ Factors (Quotient.mk'' X ⊓ Quotient.mk'' Y) (g₁ ≫ MonoOver.arrow X)",
"state_before": "case h\nC : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nA B : C\nf : A ⟶ B\n⊢ ∀ (a₁ a₂ : MonoOver B),\n Factors (Quotient.mk'' a₁) f ∧ Factors (Quotient.mk'' a₂) f → Factors (Quotient.mk'' a₁ ⊓ Quotient.mk'' a₂) f",
"tactic": "rintro X Y ⟨⟨g₁, rfl⟩, ⟨g₂, hg₂⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case h.intro.intro.intro\nC : Type u₁\ninst✝² : Category C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nA B : C\nX Y : MonoOver B\ng₁ : A ⟶ X.obj.left\ng₂ : A ⟶ Y.obj.left\nhg₂ : g₂ ≫ MonoOver.arrow Y = g₁ ≫ MonoOver.arrow X\n⊢ Factors (Quotient.mk'' X ⊓ Quotient.mk'' Y) (g₁ ≫ MonoOver.arrow X)",
"tactic": "exact ⟨_, pullback.lift_snd_assoc _ _ hg₂ _⟩"
}
] |
[
425,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
419,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.dvd_add_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.249217\nβ : Type ?u.249220\nγ : Type ?u.249223\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na c b : Ordinal\nx✝ : a ∣ a * b + c\nd : Ordinal\ne : a * b + c = a * d\n⊢ c = a * (d - b)",
"tactic": "rw [mul_sub, ← e, add_sub_cancel]"
},
{
"state_after": "α : Type ?u.249217\nβ : Type ?u.249220\nγ : Type ?u.249223\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na c b : Ordinal\nx✝ : a ∣ c\nd : Ordinal\ne : c = a * d\n⊢ a ∣ a * (b + d)",
"state_before": "α : Type ?u.249217\nβ : Type ?u.249220\nγ : Type ?u.249223\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na c b : Ordinal\nx✝ : a ∣ c\nd : Ordinal\ne : c = a * d\n⊢ a ∣ a * b + c",
"tactic": "rw [e, ← mul_add]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.249217\nβ : Type ?u.249220\nγ : Type ?u.249223\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na c b : Ordinal\nx✝ : a ∣ c\nd : Ordinal\ne : c = a * d\n⊢ a ∣ a * (b + d)",
"tactic": "apply dvd_mul_right"
}
] |
[
995,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
991,
1
] |
Mathlib/MeasureTheory/Measure/VectorMeasure.lean
|
MeasureTheory.VectorMeasure.AbsolutelyContinuous.eq
|
[] |
[
1096,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1095,
1
] |
Mathlib/Analysis/SpecialFunctions/Exp.lean
|
ContinuousAt.exp
|
[] |
[
154,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.mem_image_elim
|
[] |
[
244,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.sq_abs
|
[] |
[
973,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
972,
1
] |
Mathlib/Algebra/Group/TypeTags.lean
|
ofMul_toMul
|
[] |
[
118,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec₂.of_eq
|
[
{
"state_after": "case h.h\nα : Type u_1\nβ : Type u_2\nσ : Type u_3\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\nf g : α → β → σ\nhg : Primrec₂ f\nH : ∀ (a : α) (b : β), f a b = g a b\na : α\nb : β\n⊢ f a b = g a b",
"state_before": "α : Type u_1\nβ : Type u_2\nσ : Type u_3\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\nf g : α → β → σ\nhg : Primrec₂ f\nH : ∀ (a : α) (b : β), f a b = g a b\n⊢ f = g",
"tactic": "funext a b"
},
{
"state_after": "no goals",
"state_before": "case h.h\nα : Type u_1\nβ : Type u_2\nσ : Type u_3\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\nf g : α → β → σ\nhg : Primrec₂ f\nH : ∀ (a : α) (b : β), f a b = g a b\na : α\nb : β\n⊢ f a b = g a b",
"tactic": "apply H"
}
] |
[
418,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
417,
1
] |
Mathlib/Probability/Independence/Basic.lean
|
ProbabilityTheory.iIndepSets.iIndep
|
[
{
"state_after": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\n⊢ (∀ (i : ι), i ∈ s → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ s), f i) = ∏ i in s, ↑↑μ (f i)",
"state_before": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\n⊢ ProbabilityTheory.iIndep m",
"tactic": "intro s f"
},
{
"state_after": "case refine_1\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\n⊢ (∀ (i : ι), i ∈ ∅ → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ ∅), f i) = ∏ i in ∅, ↑↑μ (f i)\n\ncase refine_2\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\n⊢ ∀ ⦃a : ι⦄ {s : Finset ι},\n ¬a ∈ s →\n ((∀ (i : ι), i ∈ s → f i ∈ (fun x => {s | MeasurableSet s}) i) →\n ↑↑μ (⋂ (i : ι) (_ : i ∈ s), f i) = ∏ i in s, ↑↑μ (f i)) →\n (∀ (i : ι), i ∈ insert a s → f i ∈ (fun x => {s | MeasurableSet s}) i) →\n ↑↑μ (⋂ (i : ι) (_ : i ∈ insert a s), f i) = ∏ i in insert a s, ↑↑μ (f i)",
"state_before": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\n⊢ (∀ (i : ι), i ∈ s → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ s), f i) = ∏ i in s, ↑↑μ (f i)",
"tactic": "refine Finset.induction ?_ ?_ s"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\n⊢ (∀ (i : ι), i ∈ ∅ → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ ∅), f i) = ∏ i in ∅, ↑↑μ (f i)",
"tactic": "simp only [Finset.not_mem_empty, Set.mem_setOf_eq, IsEmpty.forall_iff, implies_true,\nSet.iInter_of_empty, Set.iInter_univ, measure_univ, Finset.prod_empty, forall_true_left]"
},
{
"state_after": "case refine_2\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\n⊢ ↑↑μ (⋂ (i : ι) (_ : i ∈ insert a S), f i) = ∏ i in insert a S, ↑↑μ (f i)",
"state_before": "case refine_2\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\n⊢ ∀ ⦃a : ι⦄ {s : Finset ι},\n ¬a ∈ s →\n ((∀ (i : ι), i ∈ s → f i ∈ (fun x => {s | MeasurableSet s}) i) →\n ↑↑μ (⋂ (i : ι) (_ : i ∈ s), f i) = ∏ i in s, ↑↑μ (f i)) →\n (∀ (i : ι), i ∈ insert a s → f i ∈ (fun x => {s | MeasurableSet s}) i) →\n ↑↑μ (⋂ (i : ι) (_ : i ∈ insert a s), f i) = ∏ i in insert a s, ↑↑μ (f i)",
"tactic": "intro a S ha_notin_S h_rec hf_m"
},
{
"state_after": "case refine_2\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\n⊢ ↑↑μ (⋂ (i : ι) (_ : i ∈ insert a S), f i) = ∏ i in insert a S, ↑↑μ (f i)",
"state_before": "case refine_2\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\n⊢ ↑↑μ (⋂ (i : ι) (_ : i ∈ insert a S), f i) = ∏ i in insert a S, ↑↑μ (f i)",
"tactic": "have hf_m_S : ∀ x ∈ S, MeasurableSet[m x] (f x) := fun x hx => hf_m x (by simp [hx])"
},
{
"state_after": "case refine_2\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\n⊢ ↑↑μ (f a ∩ ⋂ (x : ι) (_ : x ∈ S), f x) = ↑↑μ (f a) * ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i)",
"state_before": "case refine_2\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\n⊢ ↑↑μ (⋂ (i : ι) (_ : i ∈ insert a S), f i) = ∏ i in insert a S, ↑↑μ (f i)",
"tactic": "rw [Finset.set_biInter_insert, Finset.prod_insert ha_notin_S, ← h_rec hf_m_S]"
},
{
"state_after": "case refine_2\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\n⊢ ↑↑μ (f a ∩ ⋂ (x : ι) (_ : x ∈ S), f x) = ↑↑μ (f a) * ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i)",
"state_before": "case refine_2\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\n⊢ ↑↑μ (f a ∩ ⋂ (x : ι) (_ : x ∈ S), f x) = ↑↑μ (f a) * ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i)",
"tactic": "let p := piiUnionInter π S"
},
{
"state_after": "case refine_2\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\nm_p : MeasurableSpace Ω := generateFrom p\nhS_eq_generate : m_p = generateFrom p\n⊢ ↑↑μ (f a ∩ ⋂ (x : ι) (_ : x ∈ S), f x) = ↑↑μ (f a) * ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i)",
"state_before": "case refine_2\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\n⊢ ↑↑μ (f a ∩ ⋂ (x : ι) (_ : x ∈ S), f x) = ↑↑μ (f a) * ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i)",
"tactic": "set m_p := generateFrom p with hS_eq_generate"
},
{
"state_after": "case refine_2\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\nm_p : MeasurableSpace Ω := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a)\n⊢ ↑↑μ (f a ∩ ⋂ (x : ι) (_ : x ∈ S), f x) = ↑↑μ (f a) * ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i)",
"state_before": "case refine_2\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\nm_p : MeasurableSpace Ω := generateFrom p\nhS_eq_generate : m_p = generateFrom p\n⊢ ↑↑μ (f a ∩ ⋂ (x : ι) (_ : x ∈ S), f x) = ↑↑μ (f a) * ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i)",
"tactic": "have h_indep : @Indep Ω m_p (m a) m0 μ := by\n have hp : IsPiSystem p := isPiSystem_piiUnionInter π h_pi S\n have h_le' : ∀ i, generateFrom (π i) ≤ m0 := fun i => (h_generate i).symm.trans_le (h_le i)\n have hm_p : m_p ≤ m0 := generateFrom_piiUnionInter_le π h_le' S\n exact IndepSets.indep hm_p (h_le a) hp (h_pi a) hS_eq_generate (h_generate a)\n (iIndepSets.piiUnionInter_of_not_mem h_ind ha_notin_S)"
},
{
"state_after": "case refine_2\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\nm_p : MeasurableSpace Ω := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a)\n⊢ (⋂ (n : ι) (_ : n ∈ S), f n) ∈ {s | MeasurableSet s}",
"state_before": "case refine_2\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\nm_p : MeasurableSpace Ω := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a)\n⊢ ↑↑μ (f a ∩ ⋂ (x : ι) (_ : x ∈ S), f x) = ↑↑μ (f a) * ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i)",
"tactic": "refine @Indep.symm Ω _ _ m0 μ h_indep (f a) (⋂ n ∈ S, f n)\n (hf_m a (Finset.mem_insert_self a S)) ?_"
},
{
"state_after": "case refine_2\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\nm_p : MeasurableSpace Ω := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a)\nh_le_p : ∀ (i : ι), i ∈ S → m i ≤ m_p\n⊢ (⋂ (n : ι) (_ : n ∈ S), f n) ∈ {s | MeasurableSet s}",
"state_before": "case refine_2\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\nm_p : MeasurableSpace Ω := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a)\n⊢ (⋂ (n : ι) (_ : n ∈ S), f n) ∈ {s | MeasurableSet s}",
"tactic": "have h_le_p : ∀ i ∈ S, m i ≤ m_p := by\n intro n hn\n rw [hS_eq_generate, h_generate n]\n exact le_generateFrom_piiUnionInter _ hn"
},
{
"state_after": "case refine_2\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\nm_p : MeasurableSpace Ω := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a)\nh_le_p : ∀ (i : ι), i ∈ S → m i ≤ m_p\nh_S_f : ∀ (i : ι), i ∈ S → MeasurableSet (f i)\n⊢ (⋂ (n : ι) (_ : n ∈ S), f n) ∈ {s | MeasurableSet s}",
"state_before": "case refine_2\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\nm_p : MeasurableSpace Ω := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a)\nh_le_p : ∀ (i : ι), i ∈ S → m i ≤ m_p\n⊢ (⋂ (n : ι) (_ : n ∈ S), f n) ∈ {s | MeasurableSet s}",
"tactic": "have h_S_f : ∀ i ∈ S, MeasurableSet[m_p] (f i) :=\n fun i hi => (h_le_p i hi) (f i) (hf_m_S i hi)"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\nm_p : MeasurableSpace Ω := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a)\nh_le_p : ∀ (i : ι), i ∈ S → m i ≤ m_p\nh_S_f : ∀ (i : ι), i ∈ S → MeasurableSet (f i)\n⊢ (⋂ (n : ι) (_ : n ∈ S), f n) ∈ {s | MeasurableSet s}",
"tactic": "exact S.measurableSet_biInter h_S_f"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nx : ι\nhx : x ∈ S\n⊢ x ∈ insert a S",
"tactic": "simp [hx]"
},
{
"state_after": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\nm_p : MeasurableSpace Ω := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nhp : IsPiSystem p\n⊢ Indep m_p (m a)",
"state_before": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\nm_p : MeasurableSpace Ω := generateFrom p\nhS_eq_generate : m_p = generateFrom p\n⊢ Indep m_p (m a)",
"tactic": "have hp : IsPiSystem p := isPiSystem_piiUnionInter π h_pi S"
},
{
"state_after": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\nm_p : MeasurableSpace Ω := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nhp : IsPiSystem p\nh_le' : ∀ (i : ι), generateFrom (π i) ≤ m0\n⊢ Indep m_p (m a)",
"state_before": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\nm_p : MeasurableSpace Ω := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nhp : IsPiSystem p\n⊢ Indep m_p (m a)",
"tactic": "have h_le' : ∀ i, generateFrom (π i) ≤ m0 := fun i => (h_generate i).symm.trans_le (h_le i)"
},
{
"state_after": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\nm_p : MeasurableSpace Ω := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nhp : IsPiSystem p\nh_le' : ∀ (i : ι), generateFrom (π i) ≤ m0\nhm_p : m_p ≤ m0\n⊢ Indep m_p (m a)",
"state_before": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\nm_p : MeasurableSpace Ω := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nhp : IsPiSystem p\nh_le' : ∀ (i : ι), generateFrom (π i) ≤ m0\n⊢ Indep m_p (m a)",
"tactic": "have hm_p : m_p ≤ m0 := generateFrom_piiUnionInter_le π h_le' S"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\nm_p : MeasurableSpace Ω := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nhp : IsPiSystem p\nh_le' : ∀ (i : ι), generateFrom (π i) ≤ m0\nhm_p : m_p ≤ m0\n⊢ Indep m_p (m a)",
"tactic": "exact IndepSets.indep hm_p (h_le a) hp (h_pi a) hS_eq_generate (h_generate a)\n (iIndepSets.piiUnionInter_of_not_mem h_ind ha_notin_S)"
},
{
"state_after": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\nm_p : MeasurableSpace Ω := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a)\nn : ι\nhn : n ∈ S\n⊢ m n ≤ m_p",
"state_before": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\nm_p : MeasurableSpace Ω := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a)\n⊢ ∀ (i : ι), i ∈ S → m i ≤ m_p",
"tactic": "intro n hn"
},
{
"state_after": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\nm_p : MeasurableSpace Ω := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a)\nn : ι\nhn : n ∈ S\n⊢ generateFrom (π n) ≤ generateFrom p",
"state_before": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\nm_p : MeasurableSpace Ω := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a)\nn : ι\nhn : n ∈ S\n⊢ m n ≤ m_p",
"tactic": "rw [hS_eq_generate, h_generate n]"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ m0\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateFrom (π i)\nh_ind : iIndepSets π\ns : Finset ι\nf : ι → Set Ω\na : ι\nS : Finset ι\nha_notin_S : ¬a ∈ S\nh_rec :\n (∀ (i : ι), i ∈ S → f i ∈ (fun x => {s | MeasurableSet s}) i) → ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i) = ∏ i in S, ↑↑μ (f i)\nhf_m : ∀ (i : ι), i ∈ insert a S → f i ∈ (fun x => {s | MeasurableSet s}) i\nhf_m_S : ∀ (x : ι), x ∈ S → MeasurableSet (f x)\np : Set (Set Ω) := piiUnionInter π ↑S\nm_p : MeasurableSpace Ω := generateFrom p\nhS_eq_generate : m_p = generateFrom p\nh_indep : Indep m_p (m a)\nn : ι\nhn : n ∈ S\n⊢ generateFrom (π n) ≤ generateFrom p",
"tactic": "exact le_generateFrom_piiUnionInter _ hn"
}
] |
[
581,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
554,
1
] |
Mathlib/Algebra/Field/Defs.lean
|
uniq_inv_of_isField
|
[
{
"state_after": "α : Type ?u.15083\nβ : Type ?u.15086\nK : Type ?u.15089\nR : Type u\ninst✝ : Ring R\nhf : IsField R\nx : R\nhx : x ≠ 0\n⊢ ∃! y, x * y = 1",
"state_before": "α : Type ?u.15083\nβ : Type ?u.15086\nK : Type ?u.15089\nR : Type u\ninst✝ : Ring R\nhf : IsField R\n⊢ ∀ (x : R), x ≠ 0 → ∃! y, x * y = 1",
"tactic": "intro x hx"
},
{
"state_after": "case hex\nα : Type ?u.15083\nβ : Type ?u.15086\nK : Type ?u.15089\nR : Type u\ninst✝ : Ring R\nhf : IsField R\nx : R\nhx : x ≠ 0\n⊢ ∃ x_1, x * x_1 = 1\n\ncase hunique\nα : Type ?u.15083\nβ : Type ?u.15086\nK : Type ?u.15089\nR : Type u\ninst✝ : Ring R\nhf : IsField R\nx : R\nhx : x ≠ 0\n⊢ ∀ (y₁ y₂ : R), x * y₁ = 1 → x * y₂ = 1 → y₁ = y₂",
"state_before": "α : Type ?u.15083\nβ : Type ?u.15086\nK : Type ?u.15089\nR : Type u\ninst✝ : Ring R\nhf : IsField R\nx : R\nhx : x ≠ 0\n⊢ ∃! y, x * y = 1",
"tactic": "apply exists_unique_of_exists_of_unique"
},
{
"state_after": "no goals",
"state_before": "case hex\nα : Type ?u.15083\nβ : Type ?u.15086\nK : Type ?u.15089\nR : Type u\ninst✝ : Ring R\nhf : IsField R\nx : R\nhx : x ≠ 0\n⊢ ∃ x_1, x * x_1 = 1",
"tactic": "exact hf.mul_inv_cancel hx"
},
{
"state_after": "case hunique\nα : Type ?u.15083\nβ : Type ?u.15086\nK : Type ?u.15089\nR : Type u\ninst✝ : Ring R\nhf : IsField R\nx : R\nhx : x ≠ 0\ny z : R\nhxy : x * y = 1\nhxz : x * z = 1\n⊢ y = z",
"state_before": "case hunique\nα : Type ?u.15083\nβ : Type ?u.15086\nK : Type ?u.15089\nR : Type u\ninst✝ : Ring R\nhf : IsField R\nx : R\nhx : x ≠ 0\n⊢ ∀ (y₁ y₂ : R), x * y₁ = 1 → x * y₂ = 1 → y₁ = y₂",
"tactic": "intro y z hxy hxz"
},
{
"state_after": "no goals",
"state_before": "case hunique\nα : Type ?u.15083\nβ : Type ?u.15086\nK : Type ?u.15089\nR : Type u\ninst✝ : Ring R\nhf : IsField R\nx : R\nhx : x ≠ 0\ny z : R\nhxy : x * y = 1\nhxz : x * z = 1\n⊢ y = z",
"tactic": "calc\n y = y * (x * z) := by rw [hxz, mul_one]\n _ = x * y * z := by rw [← mul_assoc, hf.mul_comm y x]\n _ = z := by rw [hxy, one_mul]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.15083\nβ : Type ?u.15086\nK : Type ?u.15089\nR : Type u\ninst✝ : Ring R\nhf : IsField R\nx : R\nhx : x ≠ 0\ny z : R\nhxy : x * y = 1\nhxz : x * z = 1\n⊢ y = y * (x * z)",
"tactic": "rw [hxz, mul_one]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.15083\nβ : Type ?u.15086\nK : Type ?u.15089\nR : Type u\ninst✝ : Ring R\nhf : IsField R\nx : R\nhx : x ≠ 0\ny z : R\nhxy : x * y = 1\nhxz : x * z = 1\n⊢ y * (x * z) = x * y * z",
"tactic": "rw [← mul_assoc, hf.mul_comm y x]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.15083\nβ : Type ?u.15086\nK : Type ?u.15089\nR : Type u\ninst✝ : Ring R\nhf : IsField R\nx : R\nhx : x ≠ 0\ny z : R\nhxy : x * y = 1\nhxz : x * z = 1\n⊢ x * y * z = z",
"tactic": "rw [hxy, one_mul]"
}
] |
[
251,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
1
] |
Mathlib/Order/Filter/Ultrafilter.lean
|
Filter.iSup_ultrafilter_le_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.42657\nf✝ : Filter α\ns : Set α\na : α\nf f' : Filter α\n⊢ (⨆ (g : Ultrafilter α) (_ : ↑g ≤ f), ↑g) ≤ f' ↔ f ≤ f'",
"tactic": "simp only [iSup_le_iff, ← le_iff_ultrafilter]"
}
] |
[
448,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
446,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean
|
CategoryTheory.IsPullback.of_has_biproduct
|
[] |
[
568,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
566,
1
] |
Mathlib/Analysis/NormedSpace/LpEquiv.lean
|
coe_addEquiv_lpPiLp
|
[] |
[
92,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/MeasureTheory/Measure/VectorMeasure.lean
|
MeasureTheory.VectorMeasure.ext
|
[] |
[
143,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean
|
LinearMap.mkContinuous_apply
|
[] |
[
109,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Localization.mulEquivOfQuotient_mk'
|
[] |
[
1697,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1696,
1
] |
Mathlib/Data/PNat/Basic.lean
|
PNat.recOn_one
|
[] |
[
335,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
1
] |
Mathlib/Order/CompleteLattice.lean
|
Equiv.iSup_congr
|
[] |
[
656,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
654,
11
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.toNat_lift
|
[
{
"state_after": "case a\nα β : Type u\nc : Cardinal\n⊢ ↑(↑toNat (lift c)) = ↑(↑toNat c)",
"state_before": "α β : Type u\nc : Cardinal\n⊢ ↑toNat (lift c) = ↑toNat c",
"tactic": "apply natCast_injective"
},
{
"state_after": "case a.inl\nα β : Type u\nc : Cardinal\nhc : c < ℵ₀\n⊢ ↑(↑toNat (lift c)) = ↑(↑toNat c)\n\ncase a.inr\nα β : Type u\nc : Cardinal\nhc : c ≥ ℵ₀\n⊢ ↑(↑toNat (lift c)) = ↑(↑toNat c)",
"state_before": "case a\nα β : Type u\nc : Cardinal\n⊢ ↑(↑toNat (lift c)) = ↑(↑toNat c)",
"tactic": "cases' lt_or_ge c ℵ₀ with hc hc"
},
{
"state_after": "case a.inl\nα β : Type u\nc : Cardinal\nhc : c < ℵ₀\n⊢ lift c < ℵ₀",
"state_before": "case a.inl\nα β : Type u\nc : Cardinal\nhc : c < ℵ₀\n⊢ ↑(↑toNat (lift c)) = ↑(↑toNat c)",
"tactic": "rw [cast_toNat_of_lt_aleph0, ← lift_natCast.{u,v}, cast_toNat_of_lt_aleph0 hc]"
},
{
"state_after": "no goals",
"state_before": "case a.inl\nα β : Type u\nc : Cardinal\nhc : c < ℵ₀\n⊢ lift c < ℵ₀",
"tactic": "rwa [lift_lt_aleph0]"
},
{
"state_after": "case a.inr\nα β : Type u\nc : Cardinal\nhc : c ≥ ℵ₀\n⊢ ℵ₀ ≤ lift c",
"state_before": "case a.inr\nα β : Type u\nc : Cardinal\nhc : c ≥ ℵ₀\n⊢ ↑(↑toNat (lift c)) = ↑(↑toNat c)",
"tactic": "rw [cast_toNat_of_aleph0_le, ← lift_natCast.{u,v}, cast_toNat_of_aleph0_le hc, lift_zero]"
},
{
"state_after": "no goals",
"state_before": "case a.inr\nα β : Type u\nc : Cardinal\nhc : c ≥ ℵ₀\n⊢ ℵ₀ ≤ lift c",
"tactic": "rwa [aleph0_le_lift]"
}
] |
[
1783,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1777,
1
] |
Mathlib/Data/Pi/Algebra.lean
|
Pi.mulSingle_comm
|
[
{
"state_after": "no goals",
"state_before": "I : Type u\nα : Type ?u.12756\nβ : Type u_1\nγ : Type ?u.12762\nf : I → Type v₁\ng : I → Type v₂\nh : I → Type v₃\nx✝ y : (i : I) → f i\ni✝ : I\ninst✝⁴ : DecidableEq I\ninst✝³ : (i : I) → One (f i)\ninst✝² : (i : I) → One (g i)\ninst✝¹ : (i : I) → One (h i)\ninst✝ : One β\ni : I\nx : β\ni' : I\n⊢ mulSingle i x i' = mulSingle i' x i",
"tactic": "simp [mulSingle_apply, eq_comm]"
}
] |
[
283,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
281,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.coe_piecewise
|
[
{
"state_after": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝³ : (i : ι) → Zero (β i)\ninst✝² : (i : ι) → Zero (β₁ i)\ninst✝¹ : (i : ι) → Zero (β₂ i)\nx y : Π₀ (i : ι), β i\ns : Set ι\ninst✝ : (i : ι) → Decidable (i ∈ s)\nx✝ : ι\n⊢ ↑(piecewise x y s) x✝ = Set.piecewise s (↑x) (↑y) x✝",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝³ : (i : ι) → Zero (β i)\ninst✝² : (i : ι) → Zero (β₁ i)\ninst✝¹ : (i : ι) → Zero (β₂ i)\nx y : Π₀ (i : ι), β i\ns : Set ι\ninst✝ : (i : ι) → Decidable (i ∈ s)\n⊢ ↑(piecewise x y s) = Set.piecewise s ↑x ↑y",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝³ : (i : ι) → Zero (β i)\ninst✝² : (i : ι) → Zero (β₁ i)\ninst✝¹ : (i : ι) → Zero (β₂ i)\nx y : Π₀ (i : ι), β i\ns : Set ι\ninst✝ : (i : ι) → Decidable (i ∈ s)\nx✝ : ι\n⊢ ↑(piecewise x y s) x✝ = Set.piecewise s (↑x) (↑y) x✝",
"tactic": "apply piecewise_apply"
}
] |
[
228,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
226,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.Equiv.trans
|
[] |
[
997,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
996,
1
] |
Mathlib/Algebra/Order/Sub/Defs.lean
|
tsub_eq_of_eq_add
|
[] |
[
346,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
345,
1
] |
Mathlib/Data/Sum/Order.lean
|
OrderIso.sumLexDualAntidistrib_symm_inl
|
[] |
[
724,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
722,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.addNat_one
|
[
{
"state_after": "case h\nn m : ℕ\ni : Fin n\n⊢ ↑(↑(addNat 1) i) = ↑(succ i)",
"state_before": "n m : ℕ\ni : Fin n\n⊢ ↑(addNat 1) i = succ i",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nn m : ℕ\ni : Fin n\n⊢ ↑(↑(addNat 1) i) = ↑(succ i)",
"tactic": "rw [coe_addNat, val_succ]"
}
] |
[
1354,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1352,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.EqOnSource.symm_eqOn
|
[] |
[
860,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
857,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
LocalHomeomorph.isOpen_extend_preimage
|
[
{
"state_after": "𝕜 : Type u_4\nE : Type u_1\nM : Type u_2\nH : Type u_3\nE' : Type ?u.149123\nM' : Type ?u.149126\nH' : Type ?u.149129\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx : M\ns✝ t : Set M\ns : Set E\nhs : IsOpen s\n⊢ IsOpen ((extend f I).source ∩ ↑(extend f I) ⁻¹' s)",
"state_before": "𝕜 : Type u_4\nE : Type u_1\nM : Type u_2\nH : Type u_3\nE' : Type ?u.149123\nM' : Type ?u.149126\nH' : Type ?u.149129\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx : M\ns✝ t : Set M\ns : Set E\nhs : IsOpen s\n⊢ IsOpen (f.source ∩ ↑(extend f I) ⁻¹' s)",
"tactic": "rw [← extend_source f I]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_4\nE : Type u_1\nM : Type u_2\nH : Type u_3\nE' : Type ?u.149123\nM' : Type ?u.149126\nH' : Type ?u.149129\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx : M\ns✝ t : Set M\ns : Set E\nhs : IsOpen s\n⊢ IsOpen ((extend f I).source ∩ ↑(extend f I) ⁻¹' s)",
"tactic": "exact isOpen_extend_preimage' f I hs"
}
] |
[
883,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
882,
1
] |
Mathlib/Algebra/Order/Group/Abs.lean
|
abs_le_max_abs_abs
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\na b c d : α\nhab : a ≤ b\nhbc : b ≤ c\n⊢ b ≤ max (abs a) (abs c)",
"tactic": "simp [hbc.trans (le_abs_self c)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\na b c d : α\nhab : a ≤ b\nhbc : b ≤ c\n⊢ -b ≤ max (abs a) (abs c)",
"tactic": "simp [((@neg_le_neg_iff α ..).mpr hab).trans (neg_le_abs_self a)]"
}
] |
[
328,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
325,
1
] |
Mathlib/Analysis/Calculus/MeanValue.lean
|
Monotone.convexOn_univ_of_deriv
|
[] |
[
1140,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1137,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.spanSingleton_pow
|
[
{
"state_after": "case zero\nR : 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.1336678\ninst✝³ : CommRing R₁\nK : Type ?u.1336684\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx : P\n⊢ spanSingleton S x ^ Nat.zero = spanSingleton S (x ^ Nat.zero)\n\ncase succ\nR : 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.1336678\ninst✝³ : CommRing R₁\nK : Type ?u.1336684\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx : P\nn : ℕ\nhn : spanSingleton S x ^ n = spanSingleton S (x ^ n)\n⊢ spanSingleton S x ^ Nat.succ n = spanSingleton S (x ^ Nat.succ n)",
"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.1336678\ninst✝³ : CommRing R₁\nK : Type ?u.1336684\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx : P\nn : ℕ\n⊢ spanSingleton S x ^ n = spanSingleton S (x ^ n)",
"tactic": "induction' n with n hn"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : 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.1336678\ninst✝³ : CommRing R₁\nK : Type ?u.1336684\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx : P\n⊢ spanSingleton S x ^ Nat.zero = spanSingleton S (x ^ Nat.zero)",
"tactic": "rw [pow_zero, pow_zero, spanSingleton_one]"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : 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.1336678\ninst✝³ : CommRing R₁\nK : Type ?u.1336684\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx : P\nn : ℕ\nhn : spanSingleton S x ^ n = spanSingleton S (x ^ n)\n⊢ spanSingleton S x ^ Nat.succ n = spanSingleton S (x ^ Nat.succ n)",
"tactic": "rw [pow_succ, hn, spanSingleton_mul_spanSingleton, pow_succ]"
}
] |
[
1377,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1374,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.enum_succ_eq_top
|
[
{
"state_after": "α : Type ?u.99408\nβ : Type ?u.99411\nγ : Type ?u.99414\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\n⊢ o < succ o",
"state_before": "α : Type ?u.99408\nβ : Type ?u.99411\nγ : Type ?u.99414\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\n⊢ o < type fun x x_1 => x < x_1",
"tactic": "rw [type_lt]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.99408\nβ : Type ?u.99411\nγ : Type ?u.99414\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\n⊢ o < succ o",
"tactic": "exact lt_succ o"
}
] |
[
354,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
348,
1
] |
Mathlib/CategoryTheory/Monoidal/Category.lean
|
CategoryTheory.MonoidalCategory.leftAssocTensor_map
|
[] |
[
428,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
427,
1
] |
Mathlib/Algebra/Lie/IdealOperations.lean
|
LieSubmodule.comap_bracket_eq
|
[
{
"state_after": "R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nf : M →ₗ⁅R,L⁆ M₂\nhf₁ : LieModuleHom.ker f = ⊥\nhf₂ : N₂ ≤ LieModuleHom.range f\n⊢ comap f ⁅I, map f (comap f N₂)⁆ = ⁅I, comap f N₂⁆",
"state_before": "R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nf : M →ₗ⁅R,L⁆ M₂\nhf₁ : LieModuleHom.ker f = ⊥\nhf₂ : N₂ ≤ LieModuleHom.range f\n⊢ comap f ⁅I, N₂⁆ = ⁅I, comap f N₂⁆",
"tactic": "conv_lhs => rw [← map_comap_eq N₂ f hf₂]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nf : M →ₗ⁅R,L⁆ M₂\nhf₁ : LieModuleHom.ker f = ⊥\nhf₂ : N₂ ≤ LieModuleHom.range f\n⊢ comap f ⁅I, map f (comap f N₂)⁆ = ⁅I, comap f N₂⁆",
"tactic": "rw [← map_bracket_eq, comap_map_eq _ f hf₁]"
}
] |
[
245,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
1
] |
Mathlib/Data/PNat/Xgcd.lean
|
PNat.gcd_props
|
[
{
"state_after": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\n⊢ w * z = succPNat (x * y) ∧\n a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "a b : ℕ+\n⊢ let d := gcdD a b;\n let w := gcdW a b;\n let x := gcdX a b;\n let y := gcdY a b;\n let z := gcdZ a b;\n let a' := gcdA' a b;\n let b' := gcdB' a b;\n w * z = succPNat (x * y) ∧\n a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "intros d w x y z a' b'"
},
{
"state_after": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\n⊢ w * z = succPNat (x * y) ∧\n a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\n⊢ w * z = succPNat (x * y) ∧\n a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "let u := XgcdType.start a b"
},
{
"state_after": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\n⊢ w * z = succPNat (x * y) ∧\n a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\n⊢ w * z = succPNat (x * y) ∧\n a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "let ur := u.reduce"
},
{
"state_after": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝ : d = XgcdType.a ur\n⊢ w * z = succPNat (x * y) ∧\n a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\n⊢ w * z = succPNat (x * y) ∧\n a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "have _ : d = ur.a := rfl"
},
{
"state_after": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\n⊢ w * z = succPNat (x * y) ∧\n a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝ : d = XgcdType.a ur\n⊢ w * z = succPNat (x * y) ∧\n a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "have hb : d = ur.b := u.reduce_isReduced'"
},
{
"state_after": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\n⊢ w * z = succPNat (x * y) ∧\n a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\n⊢ w * z = succPNat (x * y) ∧\n a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "have ha' : (a' : ℕ) = w + x := gcdA'_coe a b"
},
{
"state_after": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\n⊢ w * z = succPNat (x * y) ∧\n a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\n⊢ w * z = succPNat (x * y) ∧\n a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "have hb' : (b' : ℕ) = y + z := gcdB'_coe a b"
},
{
"state_after": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\n⊢ w * z = succPNat (x * y) ∧\n a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\n⊢ w * z = succPNat (x * y) ∧\n a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "have hdet : w * z = succPNat (x * y) := u.reduce_isSpecial' rfl"
},
{
"state_after": "case left\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\n⊢ w * z = succPNat (x * y)\n\ncase right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\n⊢ a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\n⊢ w * z = succPNat (x * y) ∧\n a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "constructor"
},
{
"state_after": "case right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\n⊢ a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "case left\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\n⊢ w * z = succPNat (x * y)\n\ncase right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\n⊢ a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "exact hdet"
},
{
"state_after": "case right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\n⊢ a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "case right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\n⊢ a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "have hdet' : (w * z : ℕ) = x * y + 1 := by rw [← mul_coe, hdet, succPNat_coe]"
},
{
"state_after": "case right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\n⊢ a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "case right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\n⊢ a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "have _ : u.v = ⟨a, b⟩ := XgcdType.start_v a b"
},
{
"state_after": "case right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑w * ↑d + x * ↑(XgcdType.b ur), y * ↑d + ↑z * ↑(XgcdType.b ur)) = (↑a, ↑b) :=\n Eq.trans (XgcdType.reduce_v u) (XgcdType.start_v a b)\n⊢ a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "case right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\n⊢ a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "let hv : Prod.mk (w * d + x * ur.b : ℕ) (y * d + z * ur.b : ℕ) = ⟨a, b⟩ :=\n u.reduce_v.trans (XgcdType.start_v a b)"
},
{
"state_after": "case right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\n⊢ a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "case right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑w * ↑d + x * ↑(XgcdType.b ur), y * ↑d + ↑z * ↑(XgcdType.b ur)) = (↑a, ↑b) :=\n Eq.trans (XgcdType.reduce_v u) (XgcdType.start_v a b)\n⊢ a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "rw [← hb, ← add_mul, ← add_mul, ← ha', ← hb'] at hv"
},
{
"state_after": "case right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\n⊢ a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "case right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\n⊢ a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "have ha'' : (a : ℕ) = a' * d := (congr_arg Prod.fst hv).symm"
},
{
"state_after": "case right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\n⊢ a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "case right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\n⊢ a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "have hb'' : (b : ℕ) = b' * d := (congr_arg Prod.snd hv).symm"
},
{
"state_after": "case right.left\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\n⊢ a = a' * d\n\ncase right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\n⊢ b = b' * d ∧ z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "case right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\n⊢ a = a' * d ∧\n b = b' * d ∧\n z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "constructor"
},
{
"state_after": "case right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\n⊢ b = b' * d ∧ z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "case right.left\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\n⊢ a = a' * d\n\ncase right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\n⊢ b = b' * d ∧ z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "exact eq ha''"
},
{
"state_after": "case right.right.left\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\n⊢ b = b' * d\n\ncase right.right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\n⊢ z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "case right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\n⊢ b = b' * d ∧ z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "constructor"
},
{
"state_after": "case right.right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\n⊢ z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "case right.right.left\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\n⊢ b = b' * d\n\ncase right.right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\n⊢ z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "exact eq hb''"
},
{
"state_after": "case right.right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\n⊢ z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "case right.right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\n⊢ z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "have hza' : (z * a' : ℕ) = x * b' + 1 := by\n rw [ha', hb', mul_add, mul_add, mul_comm (z : ℕ), hdet']\n ring"
},
{
"state_after": "case right.right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\nhwb' : ↑w * ↑b' = y * ↑a' + 1\n⊢ z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "case right.right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\n⊢ z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "have hwb' : (w * b' : ℕ) = y * a' + 1 := by\n rw [ha', hb', mul_add, mul_add, hdet']\n ring"
},
{
"state_after": "case right.right.right.left\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\nhwb' : ↑w * ↑b' = y * ↑a' + 1\n⊢ z * a' = succPNat (x * ↑b')\n\ncase right.right.right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\nhwb' : ↑w * ↑b' = y * ↑a' + 1\n⊢ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "case right.right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\nhwb' : ↑w * ↑b' = y * ↑a' + 1\n⊢ z * a' = succPNat (x * ↑b') ∧ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "constructor"
},
{
"state_after": "case right.right.right.right.left\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\nhwb' : ↑w * ↑b' = y * ↑a' + 1\n⊢ w * b' = succPNat (y * ↑a')\n\ncase right.right.right.right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\nhwb' : ↑w * ↑b' = y * ↑a' + 1\n⊢ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"state_before": "case right.right.right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\nhwb' : ↑w * ↑b' = y * ↑a' + 1\n⊢ w * b' = succPNat (y * ↑a') ∧ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "constructor"
},
{
"state_after": "case right.right.right.right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\nhwb' : ↑w * ↑b' = y * ↑a' + 1\n⊢ ↑z * (↑a' * ↑d) = x * (↑b' * ↑d) + ↑d ∧ ↑w * (↑b' * ↑d) = y * (↑a' * ↑d) + ↑d",
"state_before": "case right.right.right.right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\nhwb' : ↑w * ↑b' = y * ↑a' + 1\n⊢ ↑z * ↑a = x * ↑b + ↑d ∧ ↑w * ↑b = y * ↑a + ↑d",
"tactic": "rw [ha'', hb'']"
},
{
"state_after": "case right.right.right.right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\nhwb' : ↑w * ↑b' = y * ↑a' + 1\n⊢ ↑z * ↑a' * ↑d = x * ↑b' * ↑d + ↑d ∧ ↑w * ↑b' * ↑d = y * ↑a' * ↑d + ↑d",
"state_before": "case right.right.right.right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\nhwb' : ↑w * ↑b' = y * ↑a' + 1\n⊢ ↑z * (↑a' * ↑d) = x * (↑b' * ↑d) + ↑d ∧ ↑w * (↑b' * ↑d) = y * (↑a' * ↑d) + ↑d",
"tactic": "repeat' rw [← @mul_assoc]"
},
{
"state_after": "case right.right.right.right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\nhwb' : ↑w * ↑b' = y * ↑a' + 1\n⊢ (x * ↑b' + 1) * ↑d = x * ↑b' * ↑d + ↑d ∧ (y * ↑a' + 1) * ↑d = y * ↑a' * ↑d + ↑d",
"state_before": "case right.right.right.right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\nhwb' : ↑w * ↑b' = y * ↑a' + 1\n⊢ ↑z * ↑a' * ↑d = x * ↑b' * ↑d + ↑d ∧ ↑w * ↑b' * ↑d = y * ↑a' * ↑d + ↑d",
"tactic": "rw [hza', hwb']"
},
{
"state_after": "no goals",
"state_before": "case right.right.right.right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\nhwb' : ↑w * ↑b' = y * ↑a' + 1\n⊢ (x * ↑b' + 1) * ↑d = x * ↑b' * ↑d + ↑d ∧ (y * ↑a' + 1) * ↑d = y * ↑a' * ↑d + ↑d",
"tactic": "constructor <;> ring"
},
{
"state_after": "no goals",
"state_before": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\n⊢ ↑w * ↑z = x * y + 1",
"tactic": "rw [← mul_coe, hdet, succPNat_coe]"
},
{
"state_after": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\n⊢ x * y + 1 + ↑z * x = x * y + x * ↑z + 1",
"state_before": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\n⊢ ↑z * ↑a' = x * ↑b' + 1",
"tactic": "rw [ha', hb', mul_add, mul_add, mul_comm (z : ℕ), hdet']"
},
{
"state_after": "no goals",
"state_before": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\n⊢ x * y + 1 + ↑z * x = x * y + x * ↑z + 1",
"tactic": "ring"
},
{
"state_after": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\n⊢ ↑w * y + (x * y + 1) = y * ↑w + y * x + 1",
"state_before": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\n⊢ ↑w * ↑b' = y * ↑a' + 1",
"tactic": "rw [ha', hb', mul_add, mul_add, hdet']"
},
{
"state_after": "no goals",
"state_before": "a b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\n⊢ ↑w * y + (x * y + 1) = y * ↑w + y * x + 1",
"tactic": "ring"
},
{
"state_after": "case right.right.right.left.a\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\nhwb' : ↑w * ↑b' = y * ↑a' + 1\n⊢ ↑(z * a') = ↑(succPNat (x * ↑b'))",
"state_before": "case right.right.right.left\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\nhwb' : ↑w * ↑b' = y * ↑a' + 1\n⊢ z * a' = succPNat (x * ↑b')",
"tactic": "apply eq"
},
{
"state_after": "no goals",
"state_before": "case right.right.right.left.a\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\nhwb' : ↑w * ↑b' = y * ↑a' + 1\n⊢ ↑(z * a') = ↑(succPNat (x * ↑b'))",
"tactic": "rw [succPNat_coe, Nat.succ_eq_add_one, mul_coe, hza']"
},
{
"state_after": "case right.right.right.right.left.a\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\nhwb' : ↑w * ↑b' = y * ↑a' + 1\n⊢ ↑(w * b') = ↑(succPNat (y * ↑a'))",
"state_before": "case right.right.right.right.left\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\nhwb' : ↑w * ↑b' = y * ↑a' + 1\n⊢ w * b' = succPNat (y * ↑a')",
"tactic": "apply eq"
},
{
"state_after": "no goals",
"state_before": "case right.right.right.right.left.a\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\nhwb' : ↑w * ↑b' = y * ↑a' + 1\n⊢ ↑(w * b') = ↑(succPNat (y * ↑a'))",
"tactic": "rw [succPNat_coe, Nat.succ_eq_add_one, mul_coe, hwb']"
},
{
"state_after": "case right.right.right.right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\nhwb' : ↑w * ↑b' = y * ↑a' + 1\n⊢ ↑z * ↑a' * ↑d = x * ↑b' * ↑d + ↑d ∧ ↑w * ↑b' * ↑d = y * ↑a' * ↑d + ↑d",
"state_before": "case right.right.right.right.right\na b : ℕ+\nd : ℕ+ := gcdD a b\nw : ℕ+ := gcdW a b\nx : ℕ := gcdX a b\ny : ℕ := gcdY a b\nz : ℕ+ := gcdZ a b\na' : ℕ+ := gcdA' a b\nb' : ℕ+ := gcdB' a b\nu : XgcdType := XgcdType.start a b\nur : XgcdType := XgcdType.reduce u\nx✝¹ : d = XgcdType.a ur\nhb : d = XgcdType.b ur\nha' : ↑a' = ↑w + x\nhb' : ↑b' = y + ↑z\nhdet : w * z = succPNat (x * y)\nhdet' : ↑w * ↑z = x * y + 1\nx✝ : XgcdType.v u = (↑a, ↑b)\nhv : (↑a' * ↑d, ↑b' * ↑d) = (↑a, ↑b)\nha'' : ↑a = ↑a' * ↑d\nhb'' : ↑b = ↑b' * ↑d\nhza' : ↑z * ↑a' = x * ↑b' + 1\nhwb' : ↑w * ↑b' = y * ↑a' + 1\n⊢ ↑z * ↑a' * ↑d = x * ↑b' * ↑d + ↑d ∧ ↑w * ↑b' * ↑d = y * ↑a' * ↑d + ↑d",
"tactic": "rw [← @mul_assoc]"
}
] |
[
506,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
456,
1
] |
Mathlib/Order/Filter/Partial.lean
|
Filter.mem_rcomap'
|
[] |
[
159,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.blockDiagonal_pow
|
[] |
[
484,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
482,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.range_nonempty
|
[] |
[
744,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
743,
1
] |
Mathlib/Analysis/SpecificLimits/Normed.lean
|
summable_norm_pow_mul_geometric_of_norm_lt_1
|
[
{
"state_after": "case intro.intro\nα : Type ?u.150661\nβ : Type ?u.150664\nι : Type ?u.150667\nR : Type u_1\ninst✝ : NormedRing R\nk : ℕ\nr : R\nhr : ‖r‖ < 1\nr' : ℝ\nhrr' : ‖r‖ < r'\nh : r' < 1\n⊢ Summable fun n => ‖↑n ^ k * r ^ n‖",
"state_before": "α : Type ?u.150661\nβ : Type ?u.150664\nι : Type ?u.150667\nR : Type u_1\ninst✝ : NormedRing R\nk : ℕ\nr : R\nhr : ‖r‖ < 1\n⊢ Summable fun n => ‖↑n ^ k * r ^ n‖",
"tactic": "rcases exists_between hr with ⟨r', hrr', h⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type ?u.150661\nβ : Type ?u.150664\nι : Type ?u.150667\nR : Type u_1\ninst✝ : NormedRing R\nk : ℕ\nr : R\nhr : ‖r‖ < 1\nr' : ℝ\nhrr' : ‖r‖ < r'\nh : r' < 1\n⊢ Summable fun n => ‖↑n ^ k * r ^ n‖",
"tactic": "exact summable_of_isBigO_nat (summable_geometric_of_lt_1 ((norm_nonneg _).trans hrr'.le) h)\n (isLittleO_pow_const_mul_const_pow_const_pow_of_norm_lt _ hrr').isBigO.norm_left"
}
] |
[
337,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
333,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
Summable.tendsto_cofinite_zero
|
[
{
"state_after": "α : Type u_2\nβ : Type ?u.775371\nγ : Type ?u.775374\nδ : Type ?u.775377\nG : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : AddCommGroup G\ninst✝ : TopologicalAddGroup G\nf : α → G\nhf : Summable f\ne : Set G\nhe : e ∈ 𝓝 0\n⊢ e ∈ Filter.map f cofinite",
"state_before": "α : Type u_2\nβ : Type ?u.775371\nγ : Type ?u.775374\nδ : Type ?u.775377\nG : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : AddCommGroup G\ninst✝ : TopologicalAddGroup G\nf : α → G\nhf : Summable f\n⊢ Tendsto f cofinite (𝓝 0)",
"tactic": "intro e he"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.775371\nγ : Type ?u.775374\nδ : Type ?u.775377\nG : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : AddCommGroup G\ninst✝ : TopologicalAddGroup G\nf : α → G\nhf : Summable f\ne : Set G\nhe : e ∈ 𝓝 0\n⊢ f ⁻¹' e ∈ cofinite",
"state_before": "α : Type u_2\nβ : Type ?u.775371\nγ : Type ?u.775374\nδ : Type ?u.775377\nG : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : AddCommGroup G\ninst✝ : TopologicalAddGroup G\nf : α → G\nhf : Summable f\ne : Set G\nhe : e ∈ 𝓝 0\n⊢ e ∈ Filter.map f cofinite",
"tactic": "rw [Filter.mem_map]"
},
{
"state_after": "case intro\nα : Type u_2\nβ : Type ?u.775371\nγ : Type ?u.775374\nδ : Type ?u.775377\nG : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : AddCommGroup G\ninst✝ : TopologicalAddGroup G\nf : α → G\nhf : Summable f\ne : Set G\nhe : e ∈ 𝓝 0\ns : Finset α\nhs : ∀ (t : Finset α), Disjoint t s → ∑ k in t, f k ∈ e\n⊢ f ⁻¹' e ∈ cofinite",
"state_before": "α : Type u_2\nβ : Type ?u.775371\nγ : Type ?u.775374\nδ : Type ?u.775377\nG : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : AddCommGroup G\ninst✝ : TopologicalAddGroup G\nf : α → G\nhf : Summable f\ne : Set G\nhe : e ∈ 𝓝 0\n⊢ f ⁻¹' e ∈ cofinite",
"tactic": "rcases hf.vanishing he with ⟨s, hs⟩"
},
{
"state_after": "case intro\nα : Type u_2\nβ : Type ?u.775371\nγ : Type ?u.775374\nδ : Type ?u.775377\nG : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : AddCommGroup G\ninst✝ : TopologicalAddGroup G\nf : α → G\nhf : Summable f\ne : Set G\nhe : e ∈ 𝓝 0\ns : Finset α\nhs : ∀ (t : Finset α), Disjoint t s → ∑ k in t, f k ∈ e\nx : α\nhx : ¬x ∈ s\n⊢ f x ∈ e",
"state_before": "case intro\nα : Type u_2\nβ : Type ?u.775371\nγ : Type ?u.775374\nδ : Type ?u.775377\nG : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : AddCommGroup G\ninst✝ : TopologicalAddGroup G\nf : α → G\nhf : Summable f\ne : Set G\nhe : e ∈ 𝓝 0\ns : Finset α\nhs : ∀ (t : Finset α), Disjoint t s → ∑ k in t, f k ∈ e\n⊢ f ⁻¹' e ∈ cofinite",
"tactic": "refine' s.eventually_cofinite_nmem.mono fun x hx => _"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_2\nβ : Type ?u.775371\nγ : Type ?u.775374\nδ : Type ?u.775377\nG : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : AddCommGroup G\ninst✝ : TopologicalAddGroup G\nf : α → G\nhf : Summable f\ne : Set G\nhe : e ∈ 𝓝 0\ns : Finset α\nhs : ∀ (t : Finset α), Disjoint t s → ∑ k in t, f k ∈ e\nx : α\nhx : ¬x ∈ s\n⊢ f x ∈ e",
"tactic": "simpa using hs {x} (disjoint_singleton_left.2 hx)"
}
] |
[
1265,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1260,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
padicNormE.eq_padic_norm'
|
[] |
[
639,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
638,
1
] |
Mathlib/Order/WithBot.lean
|
WithBot.ofDual_map
|
[] |
[
969,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
967,
1
] |
Mathlib/Data/List/Basic.lean
|
List.zipLeft'_nil_right
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.463625\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nas : List α\nb : β\nbs : List β\n⊢ zipLeft' as [] = (map (fun a => (a, none)) as, [])",
"tactic": "cases as <;> rfl"
}
] |
[
4040,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4039,
1
] |
Mathlib/Data/Fin/Tuple/Basic.lean
|
Fin.cons_zero
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\n⊢ cons x p 0 = x",
"tactic": "simp [cons]"
}
] |
[
82,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/Data/QPF/Univariate/Basic.lean
|
Qpf.Cofix.bisim'
|
[
{
"state_after": "case intro.intro.intro.intro.intro\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u_1\nQ : α → Prop\nu v : α → Cofix F\nh :\n ∀ (x : α),\n Q x →\n ∃ a f f',\n dest (u x) = abs { fst := a, snd := f } ∧\n dest (v x) = abs { fst := a, snd := f' } ∧ ∀ (i : PFunctor.B (P F) a), ∃ x', Q x' ∧ f i = u x' ∧ f' i = v x'\nx✝¹ : α\nQx : Q x✝¹\nR : Cofix F → Cofix F → Prop := fun w z => ∃ x', Q x' ∧ w = u x' ∧ z = v x'\nx y : Cofix F\nx✝ : R x y\nx' : α\nQx' : Q x'\nxeq : x = u x'\nyeq : y = v x'\na : (P F).A\nf f' : PFunctor.B (P F) a → Cofix F\nux'eq : dest (u x') = abs { fst := a, snd := f }\nvx'eq : dest (v x') = abs { fst := a, snd := f' }\nh' : ∀ (i : PFunctor.B (P F) a), ∃ x', Q x' ∧ f i = u x' ∧ f' i = v x'\n⊢ Liftr R (dest x) (dest y)",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u_1\nQ : α → Prop\nu v : α → Cofix F\nh :\n ∀ (x : α),\n Q x →\n ∃ a f f',\n dest (u x) = abs { fst := a, snd := f } ∧\n dest (v x) = abs { fst := a, snd := f' } ∧ ∀ (i : PFunctor.B (P F) a), ∃ x', Q x' ∧ f i = u x' ∧ f' i = v x'\nx✝¹ : α\nQx : Q x✝¹\nR : Cofix F → Cofix F → Prop := fun w z => ∃ x', Q x' ∧ w = u x' ∧ z = v x'\nx y : Cofix F\nx✝ : R x y\nx' : α\nQx' : Q x'\nxeq : x = u x'\nyeq : y = v x'\n⊢ Liftr R (dest x) (dest y)",
"tactic": "rcases h x' Qx' with ⟨a, f, f', ux'eq, vx'eq, h'⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u_1\nQ : α → Prop\nu v : α → Cofix F\nh :\n ∀ (x : α),\n Q x →\n ∃ a f f',\n dest (u x) = abs { fst := a, snd := f } ∧\n dest (v x) = abs { fst := a, snd := f' } ∧ ∀ (i : PFunctor.B (P F) a), ∃ x', Q x' ∧ f i = u x' ∧ f' i = v x'\nx✝¹ : α\nQx : Q x✝¹\nR : Cofix F → Cofix F → Prop := fun w z => ∃ x', Q x' ∧ w = u x' ∧ z = v x'\nx y : Cofix F\nx✝ : R x y\nx' : α\nQx' : Q x'\nxeq : x = u x'\nyeq : y = v x'\na : (P F).A\nf f' : PFunctor.B (P F) a → Cofix F\nux'eq : dest (u x') = abs { fst := a, snd := f }\nvx'eq : dest (v x') = abs { fst := a, snd := f' }\nh' : ∀ (i : PFunctor.B (P F) a), ∃ x', Q x' ∧ f i = u x' ∧ f' i = v x'\n⊢ ∃ a f₀ f₁,\n dest x = abs { fst := a, snd := f₀ } ∧\n dest y = abs { fst := a, snd := f₁ } ∧ ∀ (i : PFunctor.B (P F) a), R (f₀ i) (f₁ i)",
"state_before": "case intro.intro.intro.intro.intro\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u_1\nQ : α → Prop\nu v : α → Cofix F\nh :\n ∀ (x : α),\n Q x →\n ∃ a f f',\n dest (u x) = abs { fst := a, snd := f } ∧\n dest (v x) = abs { fst := a, snd := f' } ∧ ∀ (i : PFunctor.B (P F) a), ∃ x', Q x' ∧ f i = u x' ∧ f' i = v x'\nx✝¹ : α\nQx : Q x✝¹\nR : Cofix F → Cofix F → Prop := fun w z => ∃ x', Q x' ∧ w = u x' ∧ z = v x'\nx y : Cofix F\nx✝ : R x y\nx' : α\nQx' : Q x'\nxeq : x = u x'\nyeq : y = v x'\na : (P F).A\nf f' : PFunctor.B (P F) a → Cofix F\nux'eq : dest (u x') = abs { fst := a, snd := f }\nvx'eq : dest (v x') = abs { fst := a, snd := f' }\nh' : ∀ (i : PFunctor.B (P F) a), ∃ x', Q x' ∧ f i = u x' ∧ f' i = v x'\n⊢ Liftr R (dest x) (dest y)",
"tactic": "rw [liftr_iff]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u_1\nQ : α → Prop\nu v : α → Cofix F\nh :\n ∀ (x : α),\n Q x →\n ∃ a f f',\n dest (u x) = abs { fst := a, snd := f } ∧\n dest (v x) = abs { fst := a, snd := f' } ∧ ∀ (i : PFunctor.B (P F) a), ∃ x', Q x' ∧ f i = u x' ∧ f' i = v x'\nx✝¹ : α\nQx : Q x✝¹\nR : Cofix F → Cofix F → Prop := fun w z => ∃ x', Q x' ∧ w = u x' ∧ z = v x'\nx y : Cofix F\nx✝ : R x y\nx' : α\nQx' : Q x'\nxeq : x = u x'\nyeq : y = v x'\na : (P F).A\nf f' : PFunctor.B (P F) a → Cofix F\nux'eq : dest (u x') = abs { fst := a, snd := f }\nvx'eq : dest (v x') = abs { fst := a, snd := f' }\nh' : ∀ (i : PFunctor.B (P F) a), ∃ x', Q x' ∧ f i = u x' ∧ f' i = v x'\n⊢ ∃ a f₀ f₁,\n dest x = abs { fst := a, snd := f₀ } ∧\n dest y = abs { fst := a, snd := f₁ } ∧ ∀ (i : PFunctor.B (P F) a), R (f₀ i) (f₁ i)",
"tactic": "refine' ⟨a, f, f', xeq.symm ▸ ux'eq, yeq.symm ▸ vx'eq, h'⟩"
}
] |
[
512,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
502,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.Nonempty.subset_one_iff
|
[] |
[
140,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/Data/Matrix/Kronecker.lean
|
Matrix.kronecker_zero
|
[] |
[
302,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieIdeal.map_le_iff_le_comap
|
[
{
"state_after": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI I₂ : LieIdeal R L\nJ : LieIdeal R L'\n⊢ ↑f '' ↑I ⊆ ↑J ↔ I ≤ comap f J",
"state_before": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI I₂ : LieIdeal R L\nJ : LieIdeal R L'\n⊢ map f I ≤ J ↔ I ≤ comap f J",
"tactic": "rw [map_le]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI I₂ : LieIdeal R L\nJ : LieIdeal R L'\n⊢ ↑f '' ↑I ⊆ ↑J ↔ I ≤ comap f J",
"tactic": "exact Set.image_subset_iff"
}
] |
[
837,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
835,
1
] |
Mathlib/Tactic/Ring/Basic.lean
|
Mathlib.Tactic.Ring.neg_zero
|
[
{
"state_after": "no goals",
"state_before": "u : Lean.Level\nR✝ : Type ?u.213173\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝¹ : CommSemiring R✝\nR : Type u_1\ninst✝ : Ring R\n⊢ -0 = 0",
"tactic": "simp"
}
] |
[
553,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
553,
1
] |
Mathlib/NumberTheory/ArithmeticFunction.lean
|
Nat.ArithmeticFunction.pow_zero_eq_zeta
|
[
{
"state_after": "case h\nR : Type ?u.511482\nn : ℕ\n⊢ ↑(pow 0) n = ↑ζ n",
"state_before": "R : Type ?u.511482\n⊢ pow 0 = ζ",
"tactic": "ext n"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type ?u.511482\nn : ℕ\n⊢ ↑(pow 0) n = ↑ζ n",
"tactic": "simp"
}
] |
[
771,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
769,
1
] |
Mathlib/Order/Hom/Lattice.lean
|
BoundedLatticeHom.comp_assoc
|
[] |
[
1339,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1337,
1
] |
Mathlib/FieldTheory/Separable.lean
|
Polynomial.card_rootSet_eq_natDegree
|
[
{
"state_after": "F : Type u\ninst✝² : Field F\nK : Type v\ninst✝¹ : Field K\ninst✝ : Algebra F K\np : F[X]\nhsep : Separable p\nhsplit : Splits (algebraMap F K) p\n⊢ card (Multiset.toFinset (roots (map (algebraMap F K) p))) = natDegree p",
"state_before": "F : Type u\ninst✝² : Field F\nK : Type v\ninst✝¹ : Field K\ninst✝ : Algebra F K\np : F[X]\nhsep : Separable p\nhsplit : Splits (algebraMap F K) p\n⊢ Fintype.card ↑(rootSet p K) = natDegree p",
"tactic": "simp_rw [rootSet_def, Finset.coe_sort_coe, Fintype.card_coe]"
},
{
"state_after": "F : Type u\ninst✝² : Field F\nK : Type v\ninst✝¹ : Field K\ninst✝ : Algebra F K\np : F[X]\nhsep : Separable p\nhsplit : Splits (algebraMap F K) p\n⊢ Multiset.Nodup (roots (map (algebraMap F K) p))",
"state_before": "F : Type u\ninst✝² : Field F\nK : Type v\ninst✝¹ : Field K\ninst✝ : Algebra F K\np : F[X]\nhsep : Separable p\nhsplit : Splits (algebraMap F K) p\n⊢ card (Multiset.toFinset (roots (map (algebraMap F K) p))) = natDegree p",
"tactic": "rw [Multiset.toFinset_card_of_nodup, ← natDegree_eq_card_roots hsplit]"
},
{
"state_after": "no goals",
"state_before": "F : Type u\ninst✝² : Field F\nK : Type v\ninst✝¹ : Field K\ninst✝ : Algebra F K\np : F[X]\nhsep : Separable p\nhsplit : Splits (algebraMap F K) p\n⊢ Multiset.Nodup (roots (map (algebraMap F K) p))",
"tactic": "exact nodup_roots hsep.map"
}
] |
[
427,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
423,
1
] |
Mathlib/Data/Set/Sups.lean
|
Set.mem_infs
|
[] |
[
233,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/Algebra/Order/LatticeGroup.lean
|
inv_sup_eq_inv_inf_inv
|
[] |
[
103,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.FinMeasSupp.add
|
[
{
"state_after": "α : Type u_2\nβ✝ : Type ?u.913715\nγ : Type ?u.913718\nδ : Type ?u.913721\nm : MeasurableSpace α\ninst✝² : Zero β✝\ninst✝¹ : Zero γ\nμ : Measure α\nf✝ : α →ₛ β✝\nβ : Type u_1\ninst✝ : AddMonoid β\nf g : α →ₛ β\nhf : SimpleFunc.FinMeasSupp f μ\nhg : SimpleFunc.FinMeasSupp g μ\n⊢ SimpleFunc.FinMeasSupp (map (fun p => p.fst + p.snd) (pair f g)) μ",
"state_before": "α : Type u_2\nβ✝ : Type ?u.913715\nγ : Type ?u.913718\nδ : Type ?u.913721\nm : MeasurableSpace α\ninst✝² : Zero β✝\ninst✝¹ : Zero γ\nμ : Measure α\nf✝ : α →ₛ β✝\nβ : Type u_1\ninst✝ : AddMonoid β\nf g : α →ₛ β\nhf : SimpleFunc.FinMeasSupp f μ\nhg : SimpleFunc.FinMeasSupp g μ\n⊢ SimpleFunc.FinMeasSupp (f + g) μ",
"tactic": "rw [add_eq_map₂]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ✝ : Type ?u.913715\nγ : Type ?u.913718\nδ : Type ?u.913721\nm : MeasurableSpace α\ninst✝² : Zero β✝\ninst✝¹ : Zero γ\nμ : Measure α\nf✝ : α →ₛ β✝\nβ : Type u_1\ninst✝ : AddMonoid β\nf g : α →ₛ β\nhf : SimpleFunc.FinMeasSupp f μ\nhg : SimpleFunc.FinMeasSupp g μ\n⊢ SimpleFunc.FinMeasSupp (map (fun p => p.fst + p.snd) (pair f g)) μ",
"tactic": "exact hf.map₂ hg (zero_add 0)"
}
] |
[
1239,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1236,
11
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.smul_apply
|
[] |
[
307,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
306,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isLittleO_pow_sub_sub
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.660809\nβ : Type ?u.660812\nE : Type ?u.660815\nF : Type ?u.660818\nG : Type ?u.660821\nE' : Type u_1\nF' : Type ?u.660827\nG' : Type ?u.660830\nE'' : Type ?u.660833\nF'' : Type ?u.660836\nG'' : Type ?u.660839\nR : Type ?u.660842\nR' : Type ?u.660845\n𝕜 : Type ?u.660848\n𝕜' : Type ?u.660851\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nx₀ : E'\nm : ℕ\nh : 1 < m\n⊢ (fun x => ‖x - x₀‖ ^ m) =o[𝓝 x₀] fun x => x - x₀",
"tactic": "simpa only [isLittleO_norm_right, pow_one] using isLittleO_pow_sub_pow_sub x₀ h"
}
] |
[
2068,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2066,
1
] |
Mathlib/Data/Nat/PartENat.lean
|
PartENat.withTopEquiv_symm_zero
|
[] |
[
731,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
730,
1
] |
Mathlib/Data/Int/ModEq.lean
|
Int.neg_modEq_neg
|
[
{
"state_after": "no goals",
"state_before": "m n a b c d : ℤ\n⊢ -a ≡ -b [ZMOD n] ↔ a ≡ b [ZMOD n]",
"tactic": "simp [-sub_neg_eq_add, neg_sub_neg, modEq_iff_dvd, dvd_sub_comm]"
}
] |
[
118,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Analysis/Convex/Integral.lean
|
ConvexOn.map_integral_le
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.1581214\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : IsProbabilityMeasure μ\nhg : ConvexOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed s\nhfs : ∀ᵐ (x : α) ∂μ, f x ∈ s\nhfi : Integrable f\nhgi : Integrable (g ∘ f)\n⊢ g (∫ (x : α), f x ∂μ) ≤ ∫ (x : α), g (f x) ∂μ",
"tactic": "simpa only [average_eq_integral] using\n hg.map_average_le hgc hsc (IsProbabilityMeasure.ne_zero μ) hfs hfi hgi"
}
] |
[
216,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
LinearMap.inl_injective
|
[
{
"state_after": "no goals",
"state_before": "R : 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.51858\nM₆ : Type ?u.51861\nS : Type ?u.51864\ninst✝¹³ : Semiring R\ninst✝¹² : Semiring S\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M₄\ninst✝⁷ : AddCommMonoid M₅\ninst✝⁶ : AddCommMonoid M₆\ninst✝⁵ : Module R M\ninst✝⁴ : Module R M₂\ninst✝³ : Module R M₃\ninst✝² : Module R M₄\ninst✝¹ : Module R M₅\ninst✝ : Module R M₆\nf : M →ₗ[R] M₂\nx✝ : M\n⊢ ∀ ⦃a₂ : M⦄, ↑(inl R M M₂) x✝ = ↑(inl R M M₂) a₂ → x✝ = a₂",
"tactic": "simp"
}
] |
[
206,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/Order/Max.lean
|
Prod.isTop_iff
|
[] |
[
452,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
451,
1
] |
Mathlib/Data/MvPolynomial/Division.lean
|
MvPolynomial.monomial_one_dvd_monomial_one
|
[
{
"state_after": "σ : Type u_2\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\ni j : σ →₀ ℕ\n⊢ (1 = 0 ∨ i ≤ j) ∧ 1 ∣ 1 ↔ i ≤ j",
"state_before": "σ : Type u_2\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\ni j : σ →₀ ℕ\n⊢ ↑(monomial i) 1 ∣ ↑(monomial j) 1 ↔ i ≤ j",
"tactic": "rw [monomial_dvd_monomial]"
},
{
"state_after": "no goals",
"state_before": "σ : Type u_2\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\ni j : σ →₀ ℕ\n⊢ (1 = 0 ∨ i ≤ j) ∧ 1 ∣ 1 ↔ i ≤ j",
"tactic": "simp_rw [one_ne_zero, false_or_iff, dvd_rfl, and_true_iff]"
}
] |
[
250,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.rightInverse_surjInv
|
[] |
[
495,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
494,
1
] |
Mathlib/Analysis/LocallyConvex/Polar.lean
|
LinearMap.subset_bipolar
|
[
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type u_3\ninst✝⁴ : NormedCommRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\ns : Set E\nx : E\nhx : x ∈ s\ny : F\nhy : y ∈ polar B s\n⊢ ‖↑(↑B x) y‖ ≤ 1",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type u_3\ninst✝⁴ : NormedCommRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\ns : Set E\nx : E\nhx : x ∈ s\ny : F\nhy : y ∈ polar B s\n⊢ ‖↑(↑(flip B) y) x‖ ≤ 1",
"tactic": "rw [B.flip_apply]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type u_3\ninst✝⁴ : NormedCommRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\ns : Set E\nx : E\nhx : x ∈ s\ny : F\nhy : y ∈ polar B s\n⊢ ‖↑(↑B x) y‖ ≤ 1",
"tactic": "exact hy x hx"
}
] |
[
118,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Data/List/Cycle.lean
|
Cycle.chain_coe_cons
|
[] |
[
941,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
939,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.card_le_of_le
|
[] |
[
762,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
761,
1
] |
Mathlib/Algebra/Order/Hom/Ring.lean
|
OrderRingIso.symm_symm
|
[] |
[
472,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
471,
1
] |
Mathlib/GroupTheory/Subsemigroup/Membership.lean
|
Subsemigroup.mem_sup_right
|
[
{
"state_after": "ι : Sort ?u.7990\nM : Type u_1\nA : Type ?u.7996\nB : Type ?u.7999\ninst✝ : Mul M\nS T : Subsemigroup M\nthis : T ≤ S ⊔ T\n⊢ ∀ {x : M}, x ∈ T → x ∈ S ⊔ T",
"state_before": "ι : Sort ?u.7990\nM : Type u_1\nA : Type ?u.7996\nB : Type ?u.7999\ninst✝ : Mul M\nS T : Subsemigroup M\n⊢ ∀ {x : M}, x ∈ T → x ∈ S ⊔ T",
"tactic": "have : T ≤ S ⊔ T := le_sup_right"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.7990\nM : Type u_1\nA : Type ?u.7996\nB : Type ?u.7999\ninst✝ : Mul M\nS T : Subsemigroup M\nthis : T ≤ S ⊔ T\n⊢ ∀ {x : M}, x ∈ T → x ∈ S ⊔ T",
"tactic": "tauto"
}
] |
[
92,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Order/WithBot.lean
|
WithBot.unbot'_lt_iff
|
[
{
"state_after": "case none\nα : Type u_1\nβ : Type ?u.40647\nγ : Type ?u.40650\nδ : Type ?u.40653\na b✝ : α\ninst✝ : LT α\nb c : α\nha : none ≠ ⊥\n⊢ unbot' b none < c ↔ none < ↑c\n\ncase some\nα : Type u_1\nβ : Type ?u.40647\nγ : Type ?u.40650\nδ : Type ?u.40653\na b✝ : α\ninst✝ : LT α\nb c val✝ : α\nha : Option.some val✝ ≠ ⊥\n⊢ unbot' b (Option.some val✝) < c ↔ Option.some val✝ < ↑c",
"state_before": "α : Type u_1\nβ : Type ?u.40647\nγ : Type ?u.40650\nδ : Type ?u.40653\na✝ b✝ : α\ninst✝ : LT α\na : WithBot α\nb c : α\nha : a ≠ ⊥\n⊢ unbot' b a < c ↔ a < ↑c",
"tactic": "cases a"
},
{
"state_after": "no goals",
"state_before": "case some\nα : Type u_1\nβ : Type ?u.40647\nγ : Type ?u.40650\nδ : Type ?u.40653\na b✝ : α\ninst✝ : LT α\nb c val✝ : α\nha : Option.some val✝ ≠ ⊥\n⊢ unbot' b (Option.some val✝) < c ↔ Option.some val✝ < ↑c",
"tactic": ". rw [some_eq_coe, unbot'_coe, coe_lt_coe]"
},
{
"state_after": "no goals",
"state_before": "case none\nα : Type u_1\nβ : Type ?u.40647\nγ : Type ?u.40650\nδ : Type ?u.40653\na b✝ : α\ninst✝ : LT α\nb c : α\nha : none ≠ ⊥\n⊢ unbot' b none < c ↔ none < ↑c",
"tactic": "exact (ha rfl).elim"
},
{
"state_after": "no goals",
"state_before": "case some\nα : Type u_1\nβ : Type ?u.40647\nγ : Type ?u.40650\nδ : Type ?u.40653\na b✝ : α\ninst✝ : LT α\nb c val✝ : α\nha : Option.some val✝ ≠ ⊥\n⊢ unbot' b (Option.some val✝) < c ↔ Option.some val✝ < ↑c",
"tactic": "rw [some_eq_coe, unbot'_coe, coe_lt_coe]"
}
] |
[
396,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
393,
1
] |
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
|
WithSeminorms.image_isVonNBounded_iff_finset_seminorm_bounded
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\n𝕜₂ : Type ?u.473913\n𝕝 : Type ?u.473916\n𝕝₂ : Type ?u.473919\nE : Type u_3\nF : Type ?u.473925\nG : Type u_1\nι : Type u_4\nι' : Type ?u.473934\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\nf : G → E\ns : Set G\nhp : WithSeminorms p\n⊢ Bornology.IsVonNBounded 𝕜 (f '' s) ↔ ∀ (I : Finset ι), ∃ r, r > 0 ∧ ∀ (x : G), x ∈ s → ↑(Finset.sup I p) (f x) < r",
"tactic": "simp_rw [hp.isVonNBounded_iff_finset_seminorm_bounded, Set.ball_image_iff]"
}
] |
[
551,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
547,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.mapRange_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type ?u.178673\nγ : Type ?u.178676\nι : Type ?u.178679\nM : Type u_2\nM' : Type ?u.178685\nN : Type u_1\nP : Type ?u.178691\nG : Type ?u.178694\nH : Type ?u.178697\nR : Type ?u.178700\nS : Type ?u.178703\ninst✝² : Zero M\ninst✝¹ : Zero N\ninst✝ : Zero P\nf : M → N\nhf : f 0 = 0\na : α\n⊢ ↑(mapRange f hf 0) a = ↑0 a",
"tactic": "simp only [hf, zero_apply, mapRange_apply]"
}
] |
[
779,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
778,
1
] |
Mathlib/RingTheory/Ideal/Quotient.lean
|
Ideal.quotEquivOfEq_symm
|
[
{
"state_after": "case h\nR✝ : Type u\ninst✝¹ : CommRing R✝\nI✝ : Ideal R✝\na b : R✝\nS : Type v\nR : Type u_1\ninst✝ : CommRing R\nI J : Ideal R\nh : I = J\nx✝ : R ⧸ J\n⊢ ↑(RingEquiv.symm (quotEquivOfEq h)) x✝ = ↑(quotEquivOfEq (_ : J = I)) x✝",
"state_before": "R✝ : Type u\ninst✝¹ : CommRing R✝\nI✝ : Ideal R✝\na b : R✝\nS : Type v\nR : Type u_1\ninst✝ : CommRing R\nI J : Ideal R\nh : I = J\n⊢ RingEquiv.symm (quotEquivOfEq h) = quotEquivOfEq (_ : J = I)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR✝ : Type u\ninst✝¹ : CommRing R✝\nI✝ : Ideal R✝\na b : R✝\nS : Type v\nR : Type u_1\ninst✝ : CommRing R\nI J : Ideal R\nh : I = J\nx✝ : R ⧸ J\n⊢ ↑(RingEquiv.symm (quotEquivOfEq h)) x✝ = ↑(quotEquivOfEq (_ : J = I)) x✝",
"tactic": "rfl"
}
] |
[
328,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
327,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
MonoidHom.apply_mnat
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.341364\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : Monoid M\nf : Multiplicative ℕ →* M\nn : Multiplicative ℕ\n⊢ ↑f n = ↑f (↑Multiplicative.ofAdd 1) ^ ↑Multiplicative.toAdd n",
"tactic": "rw [← powersHom_symm_apply, ← powersHom_apply, Equiv.apply_symm_apply]"
}
] |
[
896,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
894,
1
] |
Mathlib/Data/Set/Intervals/Group.lean
|
Set.sub_mem_Ioc_iff_left
|
[] |
[
112,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/RingTheory/IntegralClosure.lean
|
IsIntegralClosure.mk'_algebraMap
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : IsIntegralClosure A R B\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R A B\nx : R\nh : optParam (IsIntegral R (↑(algebraMap R B) x)) (_ : IsIntegral R (↑(algebraMap R ((fun x => B) x)) x))\n⊢ ↑(algebraMap A B) (mk' A (↑(algebraMap R B) x) h) = ↑(algebraMap A B) (↑(algebraMap R A) x)",
"tactic": "rw [algebraMap_mk', ← IsScalarTower.algebraMap_apply]"
}
] |
[
897,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
894,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.disjoint_principal_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.56617\nγ : Type ?u.56620\nι : Sort ?u.56623\nι' : Sort ?u.56626\nl l' : Filter α\np : ι → Prop\ns✝ : ι → Set α\nt : Set α\ni : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nf : Filter α\ns : Set α\n⊢ Disjoint (𝓟 s) f ↔ sᶜ ∈ f",
"tactic": "rw [disjoint_comm, disjoint_principal_right]"
}
] |
[
704,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
703,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Equiv.lean
|
ContinuousLinearEquiv.comp_right_differentiableAt_iff
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_3\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.169117\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx✝ : E\ns t : Set E\nL L₁ L₂ : Filter E\niso : E ≃L[𝕜] F\nf : F → G\nx : E\n⊢ DifferentiableAt 𝕜 (f ∘ ↑iso) x ↔ DifferentiableAt 𝕜 f (↑iso x)",
"tactic": "simp only [← differentiableWithinAt_univ, ← iso.comp_right_differentiableWithinAt_iff,\n preimage_univ]"
}
] |
[
188,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
185,
1
] |
Mathlib/RingTheory/DedekindDomain/PID.lean
|
IsLocalization.OverPrime.mem_normalizedFactors_of_isPrime
|
[
{
"state_after": "R : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\n⊢ P ∈ normalizedFactors (Ideal.map (algebraMap R Sₚ) p)",
"state_before": "R : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\n⊢ P ∈ normalizedFactors (Ideal.map (algebraMap R Sₚ) p)",
"tactic": "have non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰ :=\n map_le_nonZeroDivisors_of_injective _ (NoZeroSMulDivisors.algebraMap_injective _ _)\n p.primeCompl_le_nonZeroDivisors"
},
{
"state_after": "R : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\n⊢ P ∈ normalizedFactors (Ideal.map (algebraMap R Sₚ) p)",
"state_before": "R : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\n⊢ P ∈ normalizedFactors (Ideal.map (algebraMap R Sₚ) p)",
"tactic": "letI : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S"
},
{
"state_after": "R : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis : IsScalarTower R (Localization.AtPrime p) Sₚ\n⊢ P ∈ normalizedFactors (Ideal.map (algebraMap R Sₚ) p)",
"state_before": "R : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\n⊢ P ∈ normalizedFactors (Ideal.map (algebraMap R Sₚ) p)",
"tactic": "haveI : IsScalarTower R (Localization.AtPrime p) Sₚ :=\n IsScalarTower.of_algebraMap_eq fun x => by\n rw [IsScalarTower.algebraMap_apply R S]\n exact (IsLocalization.map_eq (T := Algebra.algebraMapSubmonoid S (primeCompl p))\n (Submonoid.le_comap_map _) x).symm"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\n⊢ P ∈ normalizedFactors (Ideal.map (algebraMap R Sₚ) p)",
"state_before": "R : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis : IsScalarTower R (Localization.AtPrime p) Sₚ\n⊢ P ∈ normalizedFactors (Ideal.map (algebraMap R Sₚ) p)",
"tactic": "obtain ⟨pid, p', ⟨hp'0, hp'p⟩, hpu⟩ :=\n (DiscreteValuationRing.iff_pid_with_one_nonzero_prime (Localization.AtPrime p)).mp\n (IsLocalization.AtPrime.discreteValuationRing_of_dedekind_domain R hp0 _)"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis✝ : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\nthis : LocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥\n⊢ P ∈ normalizedFactors (Ideal.map (algebraMap R Sₚ) p)",
"state_before": "case intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\n⊢ P ∈ normalizedFactors (Ideal.map (algebraMap R Sₚ) p)",
"tactic": "have : LocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥ := by\n rw [Submodule.ne_bot_iff] at hp0 ⊢\n obtain ⟨x, x_mem, x_ne⟩ := hp0\n exact\n ⟨algebraMap _ _ x, (IsLocalization.AtPrime.to_map_mem_maximal_iff _ _ _).mpr x_mem,\n IsLocalization.to_map_ne_zero_of_mem_nonZeroDivisors _ p.primeCompl_le_nonZeroDivisors\n (mem_nonZeroDivisors_of_ne_zero x_ne)⟩"
},
{
"state_after": "R : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis✝ : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\nthis : LocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥\n⊢ comap (algebraMap (Localization.AtPrime p) Sₚ) P ≠ ⊥\n\nR : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis✝ : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\nthis : LocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥\n⊢ IsPrime (comap (algebraMap (Localization.AtPrime p) Sₚ) P)\n\nR : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis✝ : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\nthis : LocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥\n⊢ IsPrime (LocalRing.maximalIdeal (Localization.AtPrime p))\n\ncase intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis✝ : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\nthis : LocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥\n⊢ Ideal.map (algebraMap R Sₚ) p ≠ 0",
"state_before": "case intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis✝ : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\nthis : LocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥\n⊢ P ∈ normalizedFactors (Ideal.map (algebraMap R Sₚ) p)",
"tactic": "rw [← Multiset.singleton_le, ← normalize_eq P, ←\n normalizedFactors_irreducible (Ideal.prime_of_isPrime hP0 hP).irreducible, ←\n dvd_iff_normalizedFactors_le_normalizedFactors hP0, dvd_iff_le,\n IsScalarTower.algebraMap_eq R (Localization.AtPrime p) Sₚ, ← Ideal.map_map,\n Localization.AtPrime.map_eq_maximalIdeal, Ideal.map_le_iff_le_comap,\n hpu (LocalRing.maximalIdeal _) ⟨this, _⟩, hpu (comap _ _) ⟨_, _⟩]"
},
{
"state_after": "R : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nx : R\n⊢ ↑(algebraMap S Sₚ) (↑(algebraMap R S) x) =\n ↑(algebraMap (Localization.AtPrime p) Sₚ) (↑(algebraMap R (Localization.AtPrime p)) x)",
"state_before": "R : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nx : R\n⊢ ↑(algebraMap R Sₚ) x = ↑(algebraMap (Localization.AtPrime p) Sₚ) (↑(algebraMap R (Localization.AtPrime p)) x)",
"tactic": "rw [IsScalarTower.algebraMap_apply R S]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nx : R\n⊢ ↑(algebraMap S Sₚ) (↑(algebraMap R S) x) =\n ↑(algebraMap (Localization.AtPrime p) Sₚ) (↑(algebraMap R (Localization.AtPrime p)) x)",
"tactic": "exact (IsLocalization.map_eq (T := Algebra.algebraMapSubmonoid S (primeCompl p))\n (Submonoid.le_comap_map _) x).symm"
},
{
"state_after": "R : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : ∃ x, x ∈ p ∧ x ≠ 0\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\n⊢ ∃ x, x ∈ LocalRing.maximalIdeal (Localization.AtPrime p) ∧ x ≠ 0",
"state_before": "R : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\n⊢ LocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥",
"tactic": "rw [Submodule.ne_bot_iff] at hp0 ⊢"
},
{
"state_after": "case intro.intro\nR : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\nx : R\nx_mem : x ∈ p\nx_ne : x ≠ 0\n⊢ ∃ x, x ∈ LocalRing.maximalIdeal (Localization.AtPrime p) ∧ x ≠ 0",
"state_before": "R : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : ∃ x, x ∈ p ∧ x ≠ 0\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\n⊢ ∃ x, x ∈ LocalRing.maximalIdeal (Localization.AtPrime p) ∧ x ≠ 0",
"tactic": "obtain ⟨x, x_mem, x_ne⟩ := hp0"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\nx : R\nx_mem : x ∈ p\nx_ne : x ≠ 0\n⊢ ∃ x, x ∈ LocalRing.maximalIdeal (Localization.AtPrime p) ∧ x ≠ 0",
"tactic": "exact\n ⟨algebraMap _ _ x, (IsLocalization.AtPrime.to_map_mem_maximal_iff _ _ _).mpr x_mem,\n IsLocalization.to_map_ne_zero_of_mem_nonZeroDivisors _ p.primeCompl_le_nonZeroDivisors\n (mem_nonZeroDivisors_of_ne_zero x_ne)⟩"
},
{
"state_after": "R : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis✝ : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\nthis : LocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥\nhRS : Algebra.IsIntegral R S\n⊢ comap (algebraMap (Localization.AtPrime p) Sₚ) P ≠ ⊥",
"state_before": "R : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis✝ : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\nthis : LocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥\n⊢ comap (algebraMap (Localization.AtPrime p) Sₚ) P ≠ ⊥",
"tactic": "have hRS : Algebra.IsIntegral R S :=\n isIntegral_of_noetherian (isNoetherian_of_fg_of_noetherian' Module.Finite.out)"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis✝ : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\nthis : LocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥\nhRS : Algebra.IsIntegral R S\n⊢ comap (algebraMap (Localization.AtPrime p) Sₚ) P ≠ ⊥",
"tactic": "exact mt (Ideal.eq_bot_of_comap_eq_bot (isIntegral_localization hRS)) hP0"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis✝ : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\nthis : LocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥\n⊢ IsPrime (comap (algebraMap (Localization.AtPrime p) Sₚ) P)",
"tactic": "exact Ideal.comap_isPrime (algebraMap (Localization.AtPrime p) Sₚ) P"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis✝ : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\nthis : LocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥\n⊢ IsPrime (LocalRing.maximalIdeal (Localization.AtPrime p))",
"tactic": "exact (LocalRing.maximalIdeal.isMaximal _).isPrime"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis✝ : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\nthis : LocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥\n⊢ ¬p = ⊥\n\ncase intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis✝ : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\nthis : LocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥\n⊢ Function.Injective ↑(algebraMap R Sₚ)",
"state_before": "case intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis✝ : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\nthis : LocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥\n⊢ Ideal.map (algebraMap R Sₚ) p ≠ 0",
"tactic": "rw [Ne.def, zero_eq_bot, Ideal.map_eq_bot_iff_of_injective]"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis✝ : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\nthis : LocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥\n⊢ Function.Injective ↑(RingHom.comp (algebraMap S Sₚ) (algebraMap R S))",
"state_before": "case intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis✝ : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\nthis : LocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥\n⊢ Function.Injective ↑(algebraMap R Sₚ)",
"tactic": "rw [IsScalarTower.algebraMap_eq R S Sₚ]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis✝ : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\nthis : LocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥\n⊢ Function.Injective ↑(RingHom.comp (algebraMap S Sₚ) (algebraMap R S))",
"tactic": "exact\n (IsLocalization.injective Sₚ non_zero_div).comp (NoZeroSMulDivisors.algebraMap_injective _ _)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : IsDomain R\ninst✝¹⁴ : IsDedekindDomain R\nS : Type ?u.233823\ninst✝¹³ : CommRing S\ninst✝¹² : IsDomain S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Free R S\ninst✝⁹ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁸ : IsPrime p\nSₚ : Type u_1\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Algebra S Sₚ\ninst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ\ninst✝⁴ : Algebra R Sₚ\ninst✝³ : IsScalarTower R S Sₚ\ninst✝² : IsDomain Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : DecidableEq (Ideal Sₚ)\nP : Ideal Sₚ\nhP : IsPrime P\nhP0 : P ≠ ⊥\nnon_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰\nthis✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) S\nthis✝ : IsScalarTower R (Localization.AtPrime p) Sₚ\npid : IsPrincipalIdealRing (Localization.AtPrime p)\np' : Ideal (Localization.AtPrime p)\nhpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = p'\nhp'0 : p' ≠ ⊥\nhp'p : IsPrime p'\nthis : LocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥\n⊢ ¬p = ⊥",
"tactic": "assumption"
}
] |
[
254,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/RingTheory/Ideal/MinimalPrime.lean
|
Ideal.exists_minimalPrimes_comap_eq
|
[
{
"state_after": "case intro.intro.intro\nR : Type u_2\nS : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI✝ J : Ideal R\nI : Ideal S\nf : R →+* S\np : Ideal R\nH : p ∈ Ideal.minimalPrimes (comap f I)\np' : Ideal S\nh₁ : IsPrime p'\nh₂ : I ≤ p'\nh₃ : comap f p' = p\n⊢ ∃ p', p' ∈ Ideal.minimalPrimes I ∧ comap f p' = p",
"state_before": "R : Type u_2\nS : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI✝ J : Ideal R\nI : Ideal S\nf : R →+* S\np : Ideal R\nH : p ∈ Ideal.minimalPrimes (comap f I)\n⊢ ∃ p', p' ∈ Ideal.minimalPrimes I ∧ comap f p' = p",
"tactic": "obtain ⟨p', h₁, h₂, h₃⟩ := Ideal.exists_comap_eq_of_mem_minimalPrimes f p H"
},
{
"state_after": "case intro.intro.intro.intro.intro\nR : Type u_2\nS : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI✝ J : Ideal R\nI : Ideal S\nf : R →+* S\np : Ideal R\nH : p ∈ Ideal.minimalPrimes (comap f I)\np' : Ideal S\nh₁ : IsPrime p'\nh₂ : I ≤ p'\nh₃ : comap f p' = p\nq : Ideal S\nhq : q ∈ Ideal.minimalPrimes I\nhq' : q ≤ p'\n⊢ ∃ p', p' ∈ Ideal.minimalPrimes I ∧ comap f p' = p",
"state_before": "case intro.intro.intro\nR : Type u_2\nS : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI✝ J : Ideal R\nI : Ideal S\nf : R →+* S\np : Ideal R\nH : p ∈ Ideal.minimalPrimes (comap f I)\np' : Ideal S\nh₁ : IsPrime p'\nh₂ : I ≤ p'\nh₃ : comap f p' = p\n⊢ ∃ p', p' ∈ Ideal.minimalPrimes I ∧ comap f p' = p",
"tactic": "obtain ⟨q, hq, hq'⟩ := Ideal.exists_minimalPrimes_le h₂"
},
{
"state_after": "case intro.intro.intro.intro.intro\nR : Type u_2\nS : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI✝ J : Ideal R\nI : Ideal S\nf : R →+* S\np : Ideal R\nH : p ∈ Ideal.minimalPrimes (comap f I)\np' : Ideal S\nh₁ : IsPrime p'\nh₂ : I ≤ p'\nh₃ : comap f p' = p\nq : Ideal S\nhq : q ∈ Ideal.minimalPrimes I\nhq' : q ≤ p'\n⊢ p = comap f q",
"state_before": "case intro.intro.intro.intro.intro\nR : Type u_2\nS : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI✝ J : Ideal R\nI : Ideal S\nf : R →+* S\np : Ideal R\nH : p ∈ Ideal.minimalPrimes (comap f I)\np' : Ideal S\nh₁ : IsPrime p'\nh₂ : I ≤ p'\nh₃ : comap f p' = p\nq : Ideal S\nhq : q ∈ Ideal.minimalPrimes I\nhq' : q ≤ p'\n⊢ ∃ p', p' ∈ Ideal.minimalPrimes I ∧ comap f p' = p",
"tactic": "refine' ⟨q, hq, Eq.symm _⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro\nR : Type u_2\nS : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI✝ J : Ideal R\nI : Ideal S\nf : R →+* S\np : Ideal R\nH : p ∈ Ideal.minimalPrimes (comap f I)\np' : Ideal S\nh₁ : IsPrime p'\nh₂ : I ≤ p'\nh₃ : comap f p' = p\nq : Ideal S\nhq : q ∈ Ideal.minimalPrimes I\nhq' : q ≤ p'\nthis : IsPrime q\n⊢ p = comap f q",
"state_before": "case intro.intro.intro.intro.intro\nR : Type u_2\nS : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI✝ J : Ideal R\nI : Ideal S\nf : R →+* S\np : Ideal R\nH : p ∈ Ideal.minimalPrimes (comap f I)\np' : Ideal S\nh₁ : IsPrime p'\nh₂ : I ≤ p'\nh₃ : comap f p' = p\nq : Ideal S\nhq : q ∈ Ideal.minimalPrimes I\nhq' : q ≤ p'\n⊢ p = comap f q",
"tactic": "have := hq.1.1"
},
{
"state_after": "case intro.intro.intro.intro.intro\nR : Type u_2\nS : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI✝ J : Ideal R\nI : Ideal S\nf : R →+* S\np : Ideal R\nH : p ∈ Ideal.minimalPrimes (comap f I)\np' : Ideal S\nh₁ : IsPrime p'\nh₂ : I ≤ p'\nh₃ : comap f p' = p\nq : Ideal S\nhq : q ∈ Ideal.minimalPrimes I\nhq' : q ≤ p'\nthis✝ : IsPrime q\nthis : comap f q ≤ p\n⊢ p = comap f q",
"state_before": "case intro.intro.intro.intro.intro\nR : Type u_2\nS : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI✝ J : Ideal R\nI : Ideal S\nf : R →+* S\np : Ideal R\nH : p ∈ Ideal.minimalPrimes (comap f I)\np' : Ideal S\nh₁ : IsPrime p'\nh₂ : I ≤ p'\nh₃ : comap f p' = p\nq : Ideal S\nhq : q ∈ Ideal.minimalPrimes I\nhq' : q ≤ p'\nthis : IsPrime q\n⊢ p = comap f q",
"tactic": "have := (Ideal.comap_mono hq').trans_eq h₃"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\nR : Type u_2\nS : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI✝ J : Ideal R\nI : Ideal S\nf : R →+* S\np : Ideal R\nH : p ∈ Ideal.minimalPrimes (comap f I)\np' : Ideal S\nh₁ : IsPrime p'\nh₂ : I ≤ p'\nh₃ : comap f p' = p\nq : Ideal S\nhq : q ∈ Ideal.minimalPrimes I\nhq' : q ≤ p'\nthis✝ : IsPrime q\nthis : comap f q ≤ p\n⊢ p = comap f q",
"tactic": "exact (H.2 ⟨inferInstance, Ideal.comap_mono hq.1.2⟩ this).antisymm this"
}
] |
[
166,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/Analysis/Calculus/Deriv/Mul.lean
|
HasStrictDerivAt.const_mul
|
[
{
"state_after": "case h.e'_7\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\n𝕜' : Type ?u.270691\n𝔸 : Type u_1\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕜 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝔸\nc✝ d : 𝕜 → 𝔸\nc' d' : 𝔸\nu v : 𝕜 → 𝕜'\nc : 𝔸\nhd : HasStrictDerivAt d d' x\n⊢ c * d' = 0 * d x + c * d'",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\n𝕜' : Type ?u.270691\n𝔸 : Type u_1\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕜 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝔸\nc✝ d : 𝕜 → 𝔸\nc' d' : 𝔸\nu v : 𝕜 → 𝕜'\nc : 𝔸\nhd : HasStrictDerivAt d d' x\n⊢ HasStrictDerivAt (fun y => c * d y) (c * d') x",
"tactic": "convert (hasStrictDerivAt_const _ _).mul hd using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_7\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\n𝕜' : Type ?u.270691\n𝔸 : Type u_1\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕜 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝔸\nc✝ d : 𝕜 → 𝔸\nc' d' : 𝔸\nu v : 𝕜 → 𝕜'\nc : 𝔸\nhd : HasStrictDerivAt d d' x\n⊢ c * d' = 0 * d x + c * d'",
"tactic": "rw [zero_mul, zero_add]"
}
] |
[
264,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.isUnit_iff_singleton
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.120690\nα : Type u_1\nβ : Type ?u.120696\nγ : Type ?u.120699\ninst✝ : Group α\ns t : Set α\na b : α\n⊢ IsUnit s ↔ ∃ a, s = {a}",
"tactic": "simp only [isUnit_iff, Group.isUnit, and_true_iff]"
}
] |
[
1194,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1193,
1
] |
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