declaration
stringlengths 27
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stringlengths 52
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| context
dict | tactic_states
listlengths 1
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|
|---|---|---|---|
theorem map_zsmul_const [AddGroup G] [AddGroup H] [AddConstMapClass F G H a b]
(f : F) (n : ℤ) : f (n • a) = f 0 + n • b := by
simpa using map_add_zsmul f 0 n
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/AddConstMap/Basic.lean
|
{
"open": [
"Function Set"
],
"variables": [
"{F G H : Type*} [FunLike F G H] {a : G} {b : H}"
]
}
|
[
{
"line": "simpa using map_add_zsmul f 0 n",
"before_state": "F : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝³ : FunLike F G H\na : G\nb : H\ninst✝² : AddGroup G\ninst✝¹ : AddGroup H\ninst✝ : AddConstMapClass F G H a b\nf : F\nn : ℤ\n⊢ f (n • a) = f 0 + n • b",
"after_state": "No Goals!"
}
] |
theorem map_sub_zsmul [AddGroup G] [AddGroup H] [AddConstMapClass F G H a b]
(f : F) (x : G) (n : ℤ) : f (x - n • a) = f x - n • b := by
simpa [sub_eq_add_neg] using map_add_zsmul f x (-n)
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/AddConstMap/Basic.lean
|
{
"open": [
"Function Set"
],
"variables": [
"{F G H : Type*} [FunLike F G H] {a : G} {b : H}"
]
}
|
[
{
"line": "simpa [sub_eq_add_neg] using map_add_zsmul f x (-n)",
"before_state": "F : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝³ : FunLike F G H\na : G\nb : H\ninst✝² : AddGroup G\ninst✝¹ : AddGroup H\ninst✝ : AddConstMapClass F G H a b\nf : F\nx : G\nn : ℤ\n⊢ f (x - n • a) = f x - n • b",
"after_state": "No Goals!"
}
] |
theorem singleton_vsub_self (p : P) : ({p} : Set P) -ᵥ {p} = {(0 : G)} := by
rw [Set.singleton_vsub_singleton]
rw [vsub_self]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/AddTorsor/Basic.lean
|
{
"open": [
"scoped Pointwise"
],
"variables": [
"{G : Type*} {P : Type*} [AddGroup G] [T : AddTorsor G P]"
]
}
|
[
{
"line": "rw [Set.singleton_vsub_singleton]",
"before_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {p} -ᵥ {p} = {0}",
"after_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {p -ᵥ p} = {0}"
},
{
"line": "rewrite [Set.singleton_vsub_singleton]",
"before_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {p} -ᵥ {p} = {0}",
"after_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {p -ᵥ p} = {0}"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {p -ᵥ p} = {0}",
"after_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {p -ᵥ p} = {0}"
},
{
"line": "try (with_reducible rfl)",
"before_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {p -ᵥ p} = {0}",
"after_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {p -ᵥ p} = {0}"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {p -ᵥ p} = {0}",
"after_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {p -ᵥ p} = {0}"
},
{
"line": "with_reducible rfl",
"before_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {p -ᵥ p} = {0}",
"after_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {p -ᵥ p} = {0}"
},
{
"line": "rfl",
"before_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {p -ᵥ p} = {0}",
"after_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {p -ᵥ p} = {0}"
},
{
"line": "apply_rfl",
"before_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {p -ᵥ p} = {0}",
"after_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {p -ᵥ p} = {0}"
},
{
"line": "skip",
"before_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {p -ᵥ p} = {0}",
"after_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {p -ᵥ p} = {0}"
},
{
"line": "rw [vsub_self]",
"before_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {p -ᵥ p} = {0}",
"after_state": "No Goals!"
},
{
"line": "rewrite [vsub_self]",
"before_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {p -ᵥ p} = {0}",
"after_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {0} = {0}"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {0} = {0}",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {0} = {0}",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {0} = {0}",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {0} = {0}",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {0} = {0}",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {0} = {0}",
"after_state": "No Goals!"
}
] |
theorem Algebra.ext {S : Type u} {A : Type v} [CommSemiring S] [Semiring A] (h1 h2 : Algebra S A)
(h : ∀ (r : S) (x : A), (by have I := h1; exact r • x) = r • x) : h1 = h2 :=
Algebra.algebra_ext _ _ fun r => by
simpa only [@Algebra.smul_def _ _ _ _ h1,@Algebra.smul_def _ _ _ _ h2,mul_one] using h r 1
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Tower.lean
|
{
"open": [
"Pointwise"
],
"variables": [
"(R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)",
"[CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B]",
"[AddCommMonoid M] [Module R M] [Module A M] [Module B M]",
"[IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M]",
"{A}",
"[CommSemiring R] [Semiring A] [Algebra R A]",
"[MulAction A M]",
"{R} {M}",
"{A} in",
"(R M) in",
"[CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]",
"[Algebra R S] [Algebra S A] [Algebra S B]",
"{R S A}",
"(R S A)",
"[Algebra R A] [Algebra R B]",
"[IsScalarTower R S A] [IsScalarTower R S B]"
]
}
|
[
{
"line": "have I := h1",
"before_state": "R : Type u\nS✝ : Type v\nA✝ : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁹ : CommSemiring R\ninst✝¹⁸ : Semiring A✝\ninst✝¹⁷ : Semiring B\ninst✝¹⁶ : Algebra R A✝\ninst✝¹⁵ : Algebra R B\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Module R M\ninst✝¹² : Module A✝ M\ninst✝¹¹ : Module B M\ninst✝¹⁰ : IsScalarTower R A✝ M\ninst✝⁹ : IsScalarTower R B M\ninst✝⁸ : SMulCommClass A✝ B M\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R A✝\ninst✝⁴ : MulAction A✝ M\ninst✝³ : Algebra R A✝\ninst✝² : Algebra R B\nS : Type u\nA : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Semiring A\nh1 h2 : Algebra S A\nr : S\nx : A\n⊢ A",
"after_state": "R : Type u\nS✝ : Type v\nA✝ : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁹ : CommSemiring R\ninst✝¹⁸ : Semiring A✝\ninst✝¹⁷ : Semiring B\ninst✝¹⁶ : Algebra R A✝\ninst✝¹⁵ : Algebra R B\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Module R M\ninst✝¹² : Module A✝ M\ninst✝¹¹ : Module B M\ninst✝¹⁰ : IsScalarTower R A✝ M\ninst✝⁹ : IsScalarTower R B M\ninst✝⁸ : SMulCommClass A✝ B M\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R A✝\ninst✝⁴ : MulAction A✝ M\ninst✝³ : Algebra R A✝\ninst✝² : Algebra R B\nS : Type u\nA : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Semiring A\nh1 h2 : Algebra S A\nr : S\nx : A\nI : Algebra S A\n⊢ A"
},
{
"line": "refine_lift\n have I := h1;\n ?_",
"before_state": "R : Type u\nS✝ : Type v\nA✝ : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁹ : CommSemiring R\ninst✝¹⁸ : Semiring A✝\ninst✝¹⁷ : Semiring B\ninst✝¹⁶ : Algebra R A✝\ninst✝¹⁵ : Algebra R B\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Module R M\ninst✝¹² : Module A✝ M\ninst✝¹¹ : Module B M\ninst✝¹⁰ : IsScalarTower R A✝ M\ninst✝⁹ : IsScalarTower R B M\ninst✝⁸ : SMulCommClass A✝ B M\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R A✝\ninst✝⁴ : MulAction A✝ M\ninst✝³ : Algebra R A✝\ninst✝² : Algebra R B\nS : Type u\nA : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Semiring A\nh1 h2 : Algebra S A\nr : S\nx : A\n⊢ A",
"after_state": "R : Type u\nS✝ : Type v\nA✝ : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁹ : CommSemiring R\ninst✝¹⁸ : Semiring A✝\ninst✝¹⁷ : Semiring B\ninst✝¹⁶ : Algebra R A✝\ninst✝¹⁵ : Algebra R B\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Module R M\ninst✝¹² : Module A✝ M\ninst✝¹¹ : Module B M\ninst✝¹⁰ : IsScalarTower R A✝ M\ninst✝⁹ : IsScalarTower R B M\ninst✝⁸ : SMulCommClass A✝ B M\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R A✝\ninst✝⁴ : MulAction A✝ M\ninst✝³ : Algebra R A✝\ninst✝² : Algebra R B\nS : Type u\nA : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Semiring A\nh1 h2 : Algebra S A\nr : S\nx : A\nI : Algebra S A\n⊢ A"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have I := h1;\n ?_);\n rotate_right)",
"before_state": "R : Type u\nS✝ : Type v\nA✝ : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁹ : CommSemiring R\ninst✝¹⁸ : Semiring A✝\ninst✝¹⁷ : Semiring B\ninst✝¹⁶ : Algebra R A✝\ninst✝¹⁵ : Algebra R B\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Module R M\ninst✝¹² : Module A✝ M\ninst✝¹¹ : Module B M\ninst✝¹⁰ : IsScalarTower R A✝ M\ninst✝⁹ : IsScalarTower R B M\ninst✝⁸ : SMulCommClass A✝ B M\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R A✝\ninst✝⁴ : MulAction A✝ M\ninst✝³ : Algebra R A✝\ninst✝² : Algebra R B\nS : Type u\nA : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Semiring A\nh1 h2 : Algebra S A\nr : S\nx : A\n⊢ A",
"after_state": "R : Type u\nS✝ : Type v\nA✝ : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁹ : CommSemiring R\ninst✝¹⁸ : Semiring A✝\ninst✝¹⁷ : Semiring B\ninst✝¹⁶ : Algebra R A✝\ninst✝¹⁵ : Algebra R B\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Module R M\ninst✝¹² : Module A✝ M\ninst✝¹¹ : Module B M\ninst✝¹⁰ : IsScalarTower R A✝ M\ninst✝⁹ : IsScalarTower R B M\ninst✝⁸ : SMulCommClass A✝ B M\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R A✝\ninst✝⁴ : MulAction A✝ M\ninst✝³ : Algebra R A✝\ninst✝² : Algebra R B\nS : Type u\nA : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Semiring A\nh1 h2 : Algebra S A\nr : S\nx : A\nI : Algebra S A\n⊢ A"
},
{
"line": "refine\n no_implicit_lambda%\n (have I := h1;\n ?_)",
"before_state": "R : Type u\nS✝ : Type v\nA✝ : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁹ : CommSemiring R\ninst✝¹⁸ : Semiring A✝\ninst✝¹⁷ : Semiring B\ninst✝¹⁶ : Algebra R A✝\ninst✝¹⁵ : Algebra R B\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Module R M\ninst✝¹² : Module A✝ M\ninst✝¹¹ : Module B M\ninst✝¹⁰ : IsScalarTower R A✝ M\ninst✝⁹ : IsScalarTower R B M\ninst✝⁸ : SMulCommClass A✝ B M\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R A✝\ninst✝⁴ : MulAction A✝ M\ninst✝³ : Algebra R A✝\ninst✝² : Algebra R B\nS : Type u\nA : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Semiring A\nh1 h2 : Algebra S A\nr : S\nx : A\n⊢ A",
"after_state": "R : Type u\nS✝ : Type v\nA✝ : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁹ : CommSemiring R\ninst✝¹⁸ : Semiring A✝\ninst✝¹⁷ : Semiring B\ninst✝¹⁶ : Algebra R A✝\ninst✝¹⁵ : Algebra R B\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Module R M\ninst✝¹² : Module A✝ M\ninst✝¹¹ : Module B M\ninst✝¹⁰ : IsScalarTower R A✝ M\ninst✝⁹ : IsScalarTower R B M\ninst✝⁸ : SMulCommClass A✝ B M\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R A✝\ninst✝⁴ : MulAction A✝ M\ninst✝³ : Algebra R A✝\ninst✝² : Algebra R B\nS : Type u\nA : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Semiring A\nh1 h2 : Algebra S A\nr : S\nx : A\nI : Algebra S A\n⊢ A"
},
{
"line": "rotate_right",
"before_state": "R : Type u\nS✝ : Type v\nA✝ : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁹ : CommSemiring R\ninst✝¹⁸ : Semiring A✝\ninst✝¹⁷ : Semiring B\ninst✝¹⁶ : Algebra R A✝\ninst✝¹⁵ : Algebra R B\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Module R M\ninst✝¹² : Module A✝ M\ninst✝¹¹ : Module B M\ninst✝¹⁰ : IsScalarTower R A✝ M\ninst✝⁹ : IsScalarTower R B M\ninst✝⁸ : SMulCommClass A✝ B M\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R A✝\ninst✝⁴ : MulAction A✝ M\ninst✝³ : Algebra R A✝\ninst✝² : Algebra R B\nS : Type u\nA : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Semiring A\nh1 h2 : Algebra S A\nr : S\nx : A\nI : Algebra S A\n⊢ A",
"after_state": "R : Type u\nS✝ : Type v\nA✝ : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁹ : CommSemiring R\ninst✝¹⁸ : Semiring A✝\ninst✝¹⁷ : Semiring B\ninst✝¹⁶ : Algebra R A✝\ninst✝¹⁵ : Algebra R B\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Module R M\ninst✝¹² : Module A✝ M\ninst✝¹¹ : Module B M\ninst✝¹⁰ : IsScalarTower R A✝ M\ninst✝⁹ : IsScalarTower R B M\ninst✝⁸ : SMulCommClass A✝ B M\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R A✝\ninst✝⁴ : MulAction A✝ M\ninst✝³ : Algebra R A✝\ninst✝² : Algebra R B\nS : Type u\nA : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Semiring A\nh1 h2 : Algebra S A\nr : S\nx : A\nI : Algebra S A\n⊢ A"
},
{
"line": "exact r • x",
"before_state": "R : Type u\nS✝ : Type v\nA✝ : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁹ : CommSemiring R\ninst✝¹⁸ : Semiring A✝\ninst✝¹⁷ : Semiring B\ninst✝¹⁶ : Algebra R A✝\ninst✝¹⁵ : Algebra R B\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Module R M\ninst✝¹² : Module A✝ M\ninst✝¹¹ : Module B M\ninst✝¹⁰ : IsScalarTower R A✝ M\ninst✝⁹ : IsScalarTower R B M\ninst✝⁸ : SMulCommClass A✝ B M\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R A✝\ninst✝⁴ : MulAction A✝ M\ninst✝³ : Algebra R A✝\ninst✝² : Algebra R B\nS : Type u\nA : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Semiring A\nh1 h2 : Algebra S A\nr : S\nx : A\nI : Algebra S A\n⊢ A",
"after_state": "No Goals!"
},
{
"line": "simpa only [@Algebra.smul_def _ _ _ _ h1, @Algebra.smul_def _ _ _ _ h2, mul_one] using h r 1",
"before_state": "S : Type u\nA : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Semiring A\nh1 h2 : Algebra S A\nh :\n ∀ (r : S) (x : A),\n (let_fun I := h1;\n r • x) =\n r • x\nr : S\n⊢ (algebraMap S A) r = (algebraMap S A) r",
"after_state": "No Goals!"
}
] |
theorem algebra_ext {R : Type*} [CommSemiring R] {A : Type*} [Semiring A] (P Q : Algebra R A)
(h : ∀ r : R, (haveI := P; algebraMap R A r) = haveI := Q; algebraMap R A r) :
P = Q := by
replace h : P.algebraMap = Q.algebraMap := DFunLike.ext _ _ h
have h' : (haveI := P; (· • ·) : R → A → A) = (haveI := Q; (· • ·) : R → A → A) := by
funext r a
rw [P.smul_def']
rw [Q.smul_def']
rw [h]
rcases P with @⟨⟨P⟩⟩
rcases Q with @⟨⟨Q⟩⟩
congr
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Defs.lean
|
{
"open": [],
"variables": [
"{R A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]",
"{R A : Type*} [CommRing R] [Ring A] [Algebra R A]",
"{R : Type u} {S : Type v} {A : Type w} {B : Type*}",
"[CommSemiring R] [CommSemiring S]",
"[Semiring A] [Algebra R A] [Semiring B] [Algebra R B]"
]
}
|
[
{
"line": "replace h : P.algebraMap = Q.algebraMap := DFunLike.ext _ _ h",
"before_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : ∀ (r : R), (algebraMap R A) r = (algebraMap R A) r\n⊢ P = Q",
"after_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\n⊢ P = Q"
},
{
"line": "have h : P.algebraMap = Q.algebraMap := DFunLike.ext _ _ h",
"before_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : ∀ (r : R), (algebraMap R A) r = (algebraMap R A) r\n⊢ P = Q",
"after_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh✝ : ∀ (r : R), (algebraMap R A) r = (algebraMap R A) r\nh : Algebra.algebraMap = Algebra.algebraMap\n⊢ P = Q"
},
{
"line": "refine_lift\n have h : P.algebraMap = Q.algebraMap := DFunLike.ext _ _ h;\n ?_",
"before_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : ∀ (r : R), (algebraMap R A) r = (algebraMap R A) r\n⊢ P = Q",
"after_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh✝ : ∀ (r : R), (algebraMap R A) r = (algebraMap R A) r\nh : Algebra.algebraMap = Algebra.algebraMap\n⊢ P = Q"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h : P.algebraMap = Q.algebraMap := DFunLike.ext _ _ h;\n ?_);\n rotate_right)",
"before_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : ∀ (r : R), (algebraMap R A) r = (algebraMap R A) r\n⊢ P = Q",
"after_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh✝ : ∀ (r : R), (algebraMap R A) r = (algebraMap R A) r\nh : Algebra.algebraMap = Algebra.algebraMap\n⊢ P = Q"
},
{
"line": "refine\n no_implicit_lambda%\n (have h : P.algebraMap = Q.algebraMap := DFunLike.ext _ _ h;\n ?_)",
"before_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : ∀ (r : R), (algebraMap R A) r = (algebraMap R A) r\n⊢ P = Q",
"after_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh✝ : ∀ (r : R), (algebraMap R A) r = (algebraMap R A) r\nh : Algebra.algebraMap = Algebra.algebraMap\n⊢ P = Q"
},
{
"line": "rotate_right",
"before_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh✝ : ∀ (r : R), (algebraMap R A) r = (algebraMap R A) r\nh : Algebra.algebraMap = Algebra.algebraMap\n⊢ P = Q",
"after_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh✝ : ∀ (r : R), (algebraMap R A) r = (algebraMap R A) r\nh : Algebra.algebraMap = Algebra.algebraMap\n⊢ P = Q"
},
{
"line": "have h' :\n (haveI := P;\n (· • ·) :\n R → A → A) =\n (haveI := Q;\n (· • ·) :\n R → A → A) :=\n by\n funext r a\n rw [P.smul_def']\n rw [Q.smul_def']\n rw [h]",
"before_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\n⊢ P = Q",
"after_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nh' : (fun x1 x2 => x1 • x2) = fun x1 x2 => x1 • x2\n⊢ P = Q"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h' :\n (haveI := P;\n (· • ·) :\n R → A → A) =\n (haveI := Q;\n (· • ·) :\n R → A → A) :=\n ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( funext r a\n rw [P.smul_def']\n rw [Q.smul_def']\n rw [h])",
"before_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\n⊢ P = Q",
"after_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nh' : (fun x1 x2 => x1 • x2) = fun x1 x2 => x1 • x2\n⊢ P = Q"
},
{
"line": "refine\n no_implicit_lambda%\n (have h' :\n (haveI := P;\n (· • ·) :\n R → A → A) =\n (haveI := Q;\n (· • ·) :\n R → A → A) :=\n ?body✝;\n ?_)",
"before_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\n⊢ P = Q",
"after_state": "case body\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\n⊢ (fun x1 x2 => x1 • x2) = fun x1 x2 => x1 • x2\n---\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nh' : (fun x1 x2 => x1 • x2) = fun x1 x2 => x1 • x2\n⊢ P = Q"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( funext r a\n rw [P.smul_def']\n rw [Q.smul_def']\n rw [h])",
"before_state": "case body\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\n⊢ (fun x1 x2 => x1 • x2) = fun x1 x2 => x1 • x2\n---\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nh' : (fun x1 x2 => x1 • x2) = fun x1 x2 => x1 • x2\n⊢ P = Q",
"after_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nh' : (fun x1 x2 => x1 • x2) = fun x1 x2 => x1 • x2\n⊢ P = Q"
},
{
"line": "with_annotate_state\"by\"\n ( funext r a\n rw [P.smul_def']\n rw [Q.smul_def']\n rw [h])",
"before_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\n⊢ (fun x1 x2 => x1 • x2) = fun x1 x2 => x1 • x2",
"after_state": "No Goals!"
},
{
"line": "funext r a",
"before_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\n⊢ (fun x1 x2 => x1 • x2) = fun x1 x2 => x1 • x2",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ r • a = r • a"
},
{
"line": "apply funext✝;\n intro r;\n funext a",
"before_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\n⊢ (fun x1 x2 => x1 • x2) = fun x1 x2 => x1 • x2",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ r • a = r • a"
},
{
"line": "apply funext✝",
"before_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\n⊢ (fun x1 x2 => x1 • x2) = fun x1 x2 => x1 • x2",
"after_state": "case h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\n⊢ ∀ (x : R), (fun x2 => x • x2) = fun x2 => x • x2"
},
{
"line": "intro r",
"before_state": "case h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\n⊢ ∀ (x : R), (fun x2 => x • x2) = fun x2 => x • x2",
"after_state": "case h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\n⊢ (fun x2 => r • x2) = fun x2 => r • x2"
},
{
"line": "funext a",
"before_state": "case h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\n⊢ (fun x2 => r • x2) = fun x2 => r • x2",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ r • a = r • a"
},
{
"line": "apply funext✝;\n intro a",
"before_state": "case h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\n⊢ (fun x2 => r • x2) = fun x2 => r • x2",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ r • a = r • a"
},
{
"line": "apply funext✝",
"before_state": "case h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\n⊢ (fun x2 => r • x2) = fun x2 => r • x2",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\n⊢ ∀ (x : A), r • x = r • x"
},
{
"line": "intro a",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\n⊢ ∀ (x : A), r • x = r • x",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ r • a = r • a"
},
{
"line": "rw [P.smul_def']",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ r • a = r • a",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = r • a"
},
{
"line": "rewrite [P.smul_def']",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ r • a = r • a",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = r • a"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = r • a",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = r • a"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = r • a",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = r • a"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = r • a",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = r • a"
},
{
"line": "with_reducible rfl",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = r • a",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = r • a"
},
{
"line": "rfl",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = r • a",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = r • a"
},
{
"line": "apply_rfl",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = r • a",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = r • a"
},
{
"line": "skip",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = r • a",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = r • a"
},
{
"line": "rw [Q.smul_def']",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = r • a",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a"
},
{
"line": "rewrite [Q.smul_def']",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = r • a",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a"
},
{
"line": "with_reducible rfl",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a"
},
{
"line": "rfl",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a"
},
{
"line": "apply_rfl",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a"
},
{
"line": "skip",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a"
},
{
"line": "rw [h]",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a",
"after_state": "No Goals!"
},
{
"line": "rewrite [h]",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a",
"after_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "case h.h\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nr : R\na : A\n⊢ Algebra.algebraMap r * a = Algebra.algebraMap r * a",
"after_state": "No Goals!"
},
{
"line": "rcases P with @⟨⟨P⟩⟩",
"before_state": "R : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nP Q : Algebra R A\nh : Algebra.algebraMap = Algebra.algebraMap\nh' : (fun x1 x2 => x1 • x2) = fun x1 x2 => x1 • x2\n⊢ P = Q",
"after_state": "case mk.mk\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nQ : Algebra R A\nalgebraMap✝ : R →+* A\ncommutes'✝ : ∀ (r : R) (x : A), algebraMap✝ r * x = x * algebraMap✝ r\nP : R → A → A\nsmul_def'✝ : ∀ (r : R) (x : A), r • x = algebraMap✝ r * x\nh : Algebra.algebraMap = Algebra.algebraMap\nh' : (fun x1 x2 => x1 • x2) = fun x1 x2 => x1 • x2\n⊢ { smul := P, algebraMap := algebraMap✝, commutes' := commutes'✝, smul_def' := smul_def'✝ } = Q"
},
{
"line": "rcases Q with @⟨⟨Q⟩⟩",
"before_state": "case mk.mk\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nQ : Algebra R A\nalgebraMap✝ : R →+* A\ncommutes'✝ : ∀ (r : R) (x : A), algebraMap✝ r * x = x * algebraMap✝ r\nP : R → A → A\nsmul_def'✝ : ∀ (r : R) (x : A), r • x = algebraMap✝ r * x\nh : Algebra.algebraMap = Algebra.algebraMap\nh' : (fun x1 x2 => x1 • x2) = fun x1 x2 => x1 • x2\n⊢ { smul := P, algebraMap := algebraMap✝, commutes' := commutes'✝, smul_def' := smul_def'✝ } = Q",
"after_state": "case mk.mk.mk.mk\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nalgebraMap✝¹ : R →+* A\ncommutes'✝¹ : ∀ (r : R) (x : A), algebraMap✝¹ r * x = x * algebraMap✝¹ r\nP : R → A → A\nsmul_def'✝¹ : ∀ (r : R) (x : A), r • x = algebraMap✝¹ r * x\nalgebraMap✝ : R →+* A\ncommutes'✝ : ∀ (r : R) (x : A), algebraMap✝ r * x = x * algebraMap✝ r\nQ : R → A → A\nsmul_def'✝ : ∀ (r : R) (x : A), r • x = algebraMap✝ r * x\nh : Algebra.algebraMap = Algebra.algebraMap\nh' : (fun x1 x2 => x1 • x2) = fun x1 x2 => x1 • x2\n⊢ { smul := P, algebraMap := algebraMap✝¹, commutes' := commutes'✝¹, smul_def' := smul_def'✝¹ } =\n { smul := Q, algebraMap := algebraMap✝, commutes' := commutes'✝, smul_def' := smul_def'✝ }"
},
{
"line": "congr",
"before_state": "case mk.mk.mk.mk\nR : Type u_6\ninst✝¹ : CommSemiring R\nA : Type u_7\ninst✝ : Semiring A\nalgebraMap✝¹ : R →+* A\ncommutes'✝¹ : ∀ (r : R) (x : A), algebraMap✝¹ r * x = x * algebraMap✝¹ r\nP : R → A → A\nsmul_def'✝¹ : ∀ (r : R) (x : A), r • x = algebraMap✝¹ r * x\nalgebraMap✝ : R →+* A\ncommutes'✝ : ∀ (r : R) (x : A), algebraMap✝ r * x = x * algebraMap✝ r\nQ : R → A → A\nsmul_def'✝ : ∀ (r : R) (x : A), r • x = algebraMap✝ r * x\nh : Algebra.algebraMap = Algebra.algebraMap\nh' : (fun x1 x2 => x1 • x2) = fun x1 x2 => x1 • x2\n⊢ { smul := P, algebraMap := algebraMap✝¹, commutes' := commutes'✝¹, smul_def' := smul_def'✝¹ } =\n { smul := Q, algebraMap := algebraMap✝, commutes' := commutes'✝, smul_def' := smul_def'✝ }",
"after_state": "No Goals!"
}
] |
theorem comp_symm (e : A₁ ≃ₐ[R] A₂) : AlgHom.comp (e : A₁ →ₐ[R] A₂) ↑e.symm = AlgHom.id R A₂ := by
ext
simp
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Equiv.lean
|
{
"open": [],
"variables": [
"{R : Type uR}",
"{A₁ : Type uA₁} {A₂ : Type uA₂} {A₃ : Type uA₃}",
"{A₁' : Type uA₁'} {A₂' : Type uA₂'} {A₃' : Type uA₃'}",
"[CommSemiring R] [Semiring A₁] [Semiring A₂] [Semiring A₃]",
"[Semiring A₁'] [Semiring A₂'] [Semiring A₃']",
"[Algebra R A₁] [Algebra R A₂] [Algebra R A₃]",
"[Algebra R A₁'] [Algebra R A₂'] [Algebra R A₃']",
"(e : A₁ ≃ₐ[R] A₂)"
]
}
|
[
{
"line": "ext",
"before_state": "R : Type uR\nA₁ : Type uA₁\nA₂ : Type uA₂\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A₁\ninst✝² : Semiring A₂\ninst✝¹ : Algebra R A₁\ninst✝ : Algebra R A₂\ne : A₁ ≃ₐ[R] A₂\n⊢ (↑e).comp ↑e.symm = AlgHom.id R A₂",
"after_state": "case H\nR : Type uR\nA₁ : Type uA₁\nA₂ : Type uA₂\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A₁\ninst✝² : Semiring A₂\ninst✝¹ : Algebra R A₁\ninst✝ : Algebra R A₂\ne : A₁ ≃ₐ[R] A₂\nx✝ : A₂\n⊢ ((↑e).comp ↑e.symm) x✝ = (AlgHom.id R A₂) x✝"
},
{
"line": "simp",
"before_state": "case H\nR : Type uR\nA₁ : Type uA₁\nA₂ : Type uA₂\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A₁\ninst✝² : Semiring A₂\ninst✝¹ : Algebra R A₁\ninst✝ : Algebra R A₂\ne : A₁ ≃ₐ[R] A₂\nx✝ : A₂\n⊢ ((↑e).comp ↑e.symm) x✝ = (AlgHom.id R A₂) x✝",
"after_state": "No Goals!"
}
] |
theorem symm_comp (e : A₁ ≃ₐ[R] A₂) : AlgHom.comp ↑e.symm (e : A₁ →ₐ[R] A₂) = AlgHom.id R A₁ := by
ext
simp
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Equiv.lean
|
{
"open": [],
"variables": [
"{R : Type uR}",
"{A₁ : Type uA₁} {A₂ : Type uA₂} {A₃ : Type uA₃}",
"{A₁' : Type uA₁'} {A₂' : Type uA₂'} {A₃' : Type uA₃'}",
"[CommSemiring R] [Semiring A₁] [Semiring A₂] [Semiring A₃]",
"[Semiring A₁'] [Semiring A₂'] [Semiring A₃']",
"[Algebra R A₁] [Algebra R A₂] [Algebra R A₃]",
"[Algebra R A₁'] [Algebra R A₂'] [Algebra R A₃']",
"(e : A₁ ≃ₐ[R] A₂)"
]
}
|
[
{
"line": "ext",
"before_state": "R : Type uR\nA₁ : Type uA₁\nA₂ : Type uA₂\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A₁\ninst✝² : Semiring A₂\ninst✝¹ : Algebra R A₁\ninst✝ : Algebra R A₂\ne : A₁ ≃ₐ[R] A₂\n⊢ (↑e.symm).comp ↑e = AlgHom.id R A₁",
"after_state": "case H\nR : Type uR\nA₁ : Type uA₁\nA₂ : Type uA₂\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A₁\ninst✝² : Semiring A₂\ninst✝¹ : Algebra R A₁\ninst✝ : Algebra R A₂\ne : A₁ ≃ₐ[R] A₂\nx✝ : A₁\n⊢ ((↑e.symm).comp ↑e) x✝ = (AlgHom.id R A₁) x✝"
},
{
"line": "simp",
"before_state": "case H\nR : Type uR\nA₁ : Type uA₁\nA₂ : Type uA₂\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A₁\ninst✝² : Semiring A₂\ninst✝¹ : Algebra R A₁\ninst✝ : Algebra R A₂\ne : A₁ ≃ₐ[R] A₂\nx✝ : A₁\n⊢ ((↑e.symm).comp ↑e) x✝ = (AlgHom.id R A₁) x✝",
"after_state": "No Goals!"
}
] |
theorem algebraMap_eq_apply (e : A₁ ≃ₐ[R] A₂) {y : R} {x : A₁} :
algebraMap R A₂ y = e x ↔ algebraMap R A₁ y = x :=
⟨fun h => by simpa using e.symm.toAlgHom.algebraMap_eq_apply h, fun h =>
e.toAlgHom.algebraMap_eq_apply h⟩
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Equiv.lean
|
{
"open": [],
"variables": [
"{R : Type uR}",
"{A₁ : Type uA₁} {A₂ : Type uA₂} {A₃ : Type uA₃}",
"{A₁' : Type uA₁'} {A₂' : Type uA₂'} {A₃' : Type uA₃'}",
"[CommSemiring R] [Semiring A₁] [Semiring A₂] [Semiring A₃]",
"[Semiring A₁'] [Semiring A₂'] [Semiring A₃']",
"[Algebra R A₁] [Algebra R A₂] [Algebra R A₃]",
"[Algebra R A₁'] [Algebra R A₂'] [Algebra R A₃']",
"(e : A₁ ≃ₐ[R] A₂)",
"(l : A₁ ≃ₗ[R] A₂) (map_one : l 1 = 1) (map_mul : ∀ x y : A₁, l (x * y) = l x * l y)"
]
}
|
[
{
"line": "simpa using e.symm.toAlgHom.algebraMap_eq_apply h",
"before_state": "R : Type uR\nA₁ : Type uA₁\nA₂ : Type uA₂\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A₁\ninst✝² : Semiring A₂\ninst✝¹ : Algebra R A₁\ninst✝ : Algebra R A₂\ne : A₁ ≃ₐ[R] A₂\ny : R\nx : A₁\nh : (algebraMap R A₂) y = e x\n⊢ (algebraMap R A₁) y = x",
"after_state": "No Goals!"
}
] |
theorem comp_ofId (φ : A →ₐ[R] B) : φ.comp (Algebra.ofId R A) = Algebra.ofId R B := by ext
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Hom.lean
|
{
"open": [],
"variables": [
"{R A B F : Type*} [CommSemiring R] [Semiring A] [Semiring B]",
"{R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}",
"[CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]",
"[Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]",
"(φ : A →ₐ[R] B)",
"(R A)",
"{R S : Type*}",
"(R : Type u) (A : Type v) (B : Type w)",
"[CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]",
"{R}"
]
}
|
[
{
"line": "ext",
"before_state": "R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nφ : A →ₐ[R] B\n⊢ φ.comp (Algebra.ofId R A) = Algebra.ofId R B",
"after_state": "No Goals!"
}
] |
lemma span_eq_toSubmodule (s : NonUnitalSubalgebra R A) :
Submodule.span R (s : Set A) = s.toSubmodule := by
simp [SetLike.ext'_iff, Submodule.coe_span_eq_self]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean
|
{
"open": [],
"variables": [
"{S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"[SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)",
"{s} in",
"{F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]",
"[Module R A] [Module R B] [Module R C]",
"[CommRing R]",
"[NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]",
"[Module R A] [Module R B] [Module R C]",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]",
"[Module R A] [Module R B] [Module R C]",
"{S : NonUnitalSubalgebra R A}",
"[FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"{R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"{F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]",
"[NonUnitalNonAssocSemiring C] [Module R C] [FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"{F : Type*} (R : Type u) {A : Type v} {B : Type w}",
"[CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]"
]
}
|
[
{
"line": "simp [SetLike.ext'_iff, Submodule.coe_span_eq_self]",
"before_state": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : NonUnitalNonAssocSemiring A\ninst✝ : Module R A\ns : NonUnitalSubalgebra R A\n⊢ Submodule.span R ↑s = s.toSubmodule",
"after_state": "No Goals!"
}
] |
theorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by
rw [← SetLike.le_def]
exact le_sup_left
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean
|
{
"open": [
"Submodule in",
"NonUnitalSubalgebra in",
"NonUnitalSubalgebra in"
],
"variables": [
"{S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"[SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)",
"{s} in",
"{F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]",
"[Module R A] [Module R B] [Module R C]",
"[CommRing R]",
"[NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]",
"[Module R A] [Module R B] [Module R C]",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]",
"[Module R A] [Module R B] [Module R C]",
"{S : NonUnitalSubalgebra R A}",
"[FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"{R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"{F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]",
"[NonUnitalNonAssocSemiring C] [Module R C] [FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"{F : Type*} (R : Type u) {A : Type v} {B : Type w}",
"[CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"[NonUnitalNonAssocSemiring B] [Module R B]",
"[FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"[IsScalarTower R A A] [SMulCommClass R A A]",
"{R}",
"(R A)",
"{R A}"
]
}
|
[
{
"line": "rw [← SetLike.le_def]",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ ∀ {x : A}, x ∈ S → x ∈ S ⊔ T",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ S ≤ S ⊔ T"
},
{
"line": "rewrite [← SetLike.le_def]",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ ∀ {x : A}, x ∈ S → x ∈ S ⊔ T",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ S ≤ S ⊔ T"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ S ≤ S ⊔ T",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ S ≤ S ⊔ T"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ S ≤ S ⊔ T",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ S ≤ S ⊔ T"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ S ≤ S ⊔ T",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ S ≤ S ⊔ T"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ S ≤ S ⊔ T",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ S ≤ S ⊔ T"
},
{
"line": "rfl",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ S ≤ S ⊔ T",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ S ≤ S ⊔ T"
},
{
"line": "apply_rfl",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ S ≤ S ⊔ T",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ S ≤ S ⊔ T"
},
{
"line": "skip",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ S ≤ S ⊔ T",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ S ≤ S ⊔ T"
},
{
"line": "exact le_sup_left",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ S ≤ S ⊔ T",
"after_state": "No Goals!"
}
] |
theorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by
rw [← SetLike.le_def]
exact le_sup_right
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean
|
{
"open": [
"Submodule in",
"NonUnitalSubalgebra in",
"NonUnitalSubalgebra in"
],
"variables": [
"{S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"[SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)",
"{s} in",
"{F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]",
"[Module R A] [Module R B] [Module R C]",
"[CommRing R]",
"[NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]",
"[Module R A] [Module R B] [Module R C]",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]",
"[Module R A] [Module R B] [Module R C]",
"{S : NonUnitalSubalgebra R A}",
"[FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"{R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"{F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]",
"[NonUnitalNonAssocSemiring C] [Module R C] [FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"{F : Type*} (R : Type u) {A : Type v} {B : Type w}",
"[CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"[NonUnitalNonAssocSemiring B] [Module R B]",
"[FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"[IsScalarTower R A A] [SMulCommClass R A A]",
"{R}",
"(R A)",
"{R A}"
]
}
|
[
{
"line": "rw [← SetLike.le_def]",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ ∀ {x : A}, x ∈ T → x ∈ S ⊔ T",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ T ≤ S ⊔ T"
},
{
"line": "rewrite [← SetLike.le_def]",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ ∀ {x : A}, x ∈ T → x ∈ S ⊔ T",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ T ≤ S ⊔ T"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ T ≤ S ⊔ T",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ T ≤ S ⊔ T"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ T ≤ S ⊔ T",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ T ≤ S ⊔ T"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ T ≤ S ⊔ T",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ T ≤ S ⊔ T"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ T ≤ S ⊔ T",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ T ≤ S ⊔ T"
},
{
"line": "rfl",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ T ≤ S ⊔ T",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ T ≤ S ⊔ T"
},
{
"line": "apply_rfl",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ T ≤ S ⊔ T",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ T ≤ S ⊔ T"
},
{
"line": "skip",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ T ≤ S ⊔ T",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ T ≤ S ⊔ T"
},
{
"line": "exact le_sup_right",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS T : NonUnitalSubalgebra R A\n⊢ T ≤ S ⊔ T",
"after_state": "No Goals!"
}
] |
theorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :
(sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=
SetLike.coe_injective <| by simp
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean
|
{
"open": [
"Submodule in",
"NonUnitalSubalgebra in",
"NonUnitalSubalgebra in"
],
"variables": [
"{S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"[SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)",
"{s} in",
"{F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]",
"[Module R A] [Module R B] [Module R C]",
"[CommRing R]",
"[NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]",
"[Module R A] [Module R B] [Module R C]",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]",
"[Module R A] [Module R B] [Module R C]",
"{S : NonUnitalSubalgebra R A}",
"[FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"{R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"{F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]",
"[NonUnitalNonAssocSemiring C] [Module R C] [FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"{F : Type*} (R : Type u) {A : Type v} {B : Type w}",
"[CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"[NonUnitalNonAssocSemiring B] [Module R B]",
"[FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"[IsScalarTower R A A] [SMulCommClass R A A]",
"{R}",
"(R A)",
"{R A}"
]
}
|
[
{
"line": "simp",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS : Set (NonUnitalSubalgebra R A)\n⊢ ↑(sInf S).toSubmodule = ↑(sInf (NonUnitalSubalgebra.toSubmodule '' S))",
"after_state": "No Goals!"
}
] |
theorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :
(sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=
SetLike.coe_injective <| by simp
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean
|
{
"open": [
"Submodule in",
"NonUnitalSubalgebra in",
"NonUnitalSubalgebra in"
],
"variables": [
"{S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"[SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)",
"{s} in",
"{F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]",
"[Module R A] [Module R B] [Module R C]",
"[CommRing R]",
"[NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]",
"[Module R A] [Module R B] [Module R C]",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]",
"[Module R A] [Module R B] [Module R C]",
"{S : NonUnitalSubalgebra R A}",
"[FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"{R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"{F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]",
"[NonUnitalNonAssocSemiring C] [Module R C] [FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"{F : Type*} (R : Type u) {A : Type v} {B : Type w}",
"[CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"[NonUnitalNonAssocSemiring B] [Module R B]",
"[FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"[IsScalarTower R A A] [SMulCommClass R A A]",
"{R}",
"(R A)",
"{R A}"
]
}
|
[
{
"line": "simp",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS : Set (NonUnitalSubalgebra R A)\n⊢ ↑(sInf S).toNonUnitalSubsemiring = ↑(sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S))",
"after_state": "No Goals!"
}
] |
theorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :
(↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean
|
{
"open": [
"Submodule in",
"NonUnitalSubalgebra in",
"NonUnitalSubalgebra in"
],
"variables": [
"{S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"[SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)",
"{s} in",
"{F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]",
"[Module R A] [Module R B] [Module R C]",
"[CommRing R]",
"[NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]",
"[Module R A] [Module R B] [Module R C]",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]",
"[Module R A] [Module R B] [Module R C]",
"{S : NonUnitalSubalgebra R A}",
"[FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"{R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"{F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]",
"[NonUnitalNonAssocSemiring C] [Module R C] [FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"{F : Type*} (R : Type u) {A : Type v} {B : Type w}",
"[CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"[NonUnitalNonAssocSemiring B] [Module R B]",
"[FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"[IsScalarTower R A A] [SMulCommClass R A A]",
"{R}",
"(R A)",
"{R A}"
]
}
|
[
{
"line": "simp [iInf]",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nι : Sort u_4\nS : ι → NonUnitalSubalgebra R A\n⊢ ↑(⨅ i, S i) = ⋂ i, ↑(S i)",
"after_state": "No Goals!"
}
] |
theorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :
(⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=
SetLike.coe_injective <| by simp
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean
|
{
"open": [
"Submodule in",
"NonUnitalSubalgebra in",
"NonUnitalSubalgebra in"
],
"variables": [
"{S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"[SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)",
"{s} in",
"{F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]",
"[Module R A] [Module R B] [Module R C]",
"[CommRing R]",
"[NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]",
"[Module R A] [Module R B] [Module R C]",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]",
"[Module R A] [Module R B] [Module R C]",
"{S : NonUnitalSubalgebra R A}",
"[FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"{R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"{F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]",
"[NonUnitalNonAssocSemiring C] [Module R C] [FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"{F : Type*} (R : Type u) {A : Type v} {B : Type w}",
"[CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"[NonUnitalNonAssocSemiring B] [Module R B]",
"[FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"[IsScalarTower R A A] [SMulCommClass R A A]",
"{R}",
"(R A)",
"{R A}"
]
}
|
[
{
"line": "simp",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nι : Sort u_4\nS : ι → NonUnitalSubalgebra R A\n⊢ ↑(⨅ i, S i).toSubmodule = ↑(⨅ i, (S i).toSubmodule)",
"after_state": "No Goals!"
}
] |
theorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=
show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by
rw [NonUnitalSubsemiring.closure_empty]
rw [NonUnitalSubsemiring.coe_bot]
rw [Submodule.span_zero_singleton]
rw [Submodule.mem_bot]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean
|
{
"open": [
"Submodule in",
"NonUnitalSubalgebra in",
"NonUnitalSubalgebra in"
],
"variables": [
"{S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"[SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)",
"{s} in",
"{F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]",
"[Module R A] [Module R B] [Module R C]",
"[CommRing R]",
"[NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]",
"[Module R A] [Module R B] [Module R C]",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]",
"[Module R A] [Module R B] [Module R C]",
"{S : NonUnitalSubalgebra R A}",
"[FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"{R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"{F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]",
"[NonUnitalNonAssocSemiring C] [Module R C] [FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"{F : Type*} (R : Type u) {A : Type v} {B : Type w}",
"[CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"[NonUnitalNonAssocSemiring B] [Module R B]",
"[FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"[IsScalarTower R A A] [SMulCommClass R A A]",
"{R}",
"(R A)",
"{R A}"
]
}
|
[
{
"line": "rw [NonUnitalSubsemiring.closure_empty]",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure ∅) ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R ↑⊥ ↔ x = 0"
},
{
"line": "rewrite [NonUnitalSubsemiring.closure_empty]",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure ∅) ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R ↑⊥ ↔ x = 0"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R ↑⊥ ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R ↑⊥ ↔ x = 0"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R ↑⊥ ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R ↑⊥ ↔ x = 0"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R ↑⊥ ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R ↑⊥ ↔ x = 0"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R ↑⊥ ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R ↑⊥ ↔ x = 0"
},
{
"line": "rfl",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R ↑⊥ ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R ↑⊥ ↔ x = 0"
},
{
"line": "apply_rfl",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R ↑⊥ ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R ↑⊥ ↔ x = 0"
},
{
"line": "skip",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R ↑⊥ ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R ↑⊥ ↔ x = 0"
},
{
"line": "rw [NonUnitalSubsemiring.coe_bot]",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R ↑⊥ ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R {0} ↔ x = 0"
},
{
"line": "rewrite [NonUnitalSubsemiring.coe_bot]",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R ↑⊥ ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R {0} ↔ x = 0"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R {0} ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R {0} ↔ x = 0"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R {0} ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R {0} ↔ x = 0"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R {0} ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R {0} ↔ x = 0"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R {0} ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R {0} ↔ x = 0"
},
{
"line": "rfl",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R {0} ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R {0} ↔ x = 0"
},
{
"line": "apply_rfl",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R {0} ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R {0} ↔ x = 0"
},
{
"line": "skip",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R {0} ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R {0} ↔ x = 0"
},
{
"line": "rw [Submodule.span_zero_singleton]",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R {0} ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ ⊥ ↔ x = 0"
},
{
"line": "rewrite [Submodule.span_zero_singleton]",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ Submodule.span R {0} ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ ⊥ ↔ x = 0"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ ⊥ ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ ⊥ ↔ x = 0"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ ⊥ ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ ⊥ ↔ x = 0"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ ⊥ ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ ⊥ ↔ x = 0"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ ⊥ ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ ⊥ ↔ x = 0"
},
{
"line": "rfl",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ ⊥ ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ ⊥ ↔ x = 0"
},
{
"line": "apply_rfl",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ ⊥ ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ ⊥ ↔ x = 0"
},
{
"line": "skip",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ ⊥ ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ ⊥ ↔ x = 0"
},
{
"line": "rw [Submodule.mem_bot]",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ ⊥ ↔ x = 0",
"after_state": "No Goals!"
},
{
"line": "rewrite [Submodule.mem_bot]",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x ∈ ⊥ ↔ x = 0",
"after_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x = 0 ↔ x = 0"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x = 0 ↔ x = 0",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x = 0 ↔ x = 0",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x = 0 ↔ x = 0",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x = 0 ↔ x = 0",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x = 0 ↔ x = 0",
"after_state": "No Goals!"
},
{
"line": "exact Iff.rfl✝",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : A\n⊢ x = 0 ↔ x = 0",
"after_state": "No Goals!"
}
] |
theorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by
simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean
|
{
"open": [
"Submodule in",
"NonUnitalSubalgebra in",
"NonUnitalSubalgebra in"
],
"variables": [
"{S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"[SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)",
"{s} in",
"{F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]",
"[Module R A] [Module R B] [Module R C]",
"[CommRing R]",
"[NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]",
"[Module R A] [Module R B] [Module R C]",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]",
"[Module R A] [Module R B] [Module R C]",
"{S : NonUnitalSubalgebra R A}",
"[FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"{R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"{F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]",
"[NonUnitalNonAssocSemiring C] [Module R C] [FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"{F : Type*} (R : Type u) {A : Type v} {B : Type w}",
"[CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"[NonUnitalNonAssocSemiring B] [Module R B]",
"[FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"[IsScalarTower R A A] [SMulCommClass R A A]",
"{R}",
"(R A)",
"{R A}"
]
}
|
[
{
"line": "simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]",
"before_state": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\n⊢ ↑⊥ = {0}",
"after_state": "No Goals!"
}
] |
theorem map_bot [IsScalarTower R B B] [SMulCommClass R B B]
(f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=
SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean
|
{
"open": [
"Submodule in",
"NonUnitalSubalgebra in",
"NonUnitalSubalgebra in"
],
"variables": [
"{S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"[SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)",
"{s} in",
"{F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]",
"[Module R A] [Module R B] [Module R C]",
"[CommRing R]",
"[NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]",
"[Module R A] [Module R B] [Module R C]",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]",
"[Module R A] [Module R B] [Module R C]",
"{S : NonUnitalSubalgebra R A}",
"[FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"{R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"{F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}",
"[CommSemiring R]",
"[NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]",
"[NonUnitalNonAssocSemiring C] [Module R C] [FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"{F : Type*} (R : Type u) {A : Type v} {B : Type w}",
"[CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]",
"[NonUnitalNonAssocSemiring B] [Module R B]",
"[FunLike F A B] [NonUnitalAlgHomClass F R A B]",
"[IsScalarTower R A A] [SMulCommClass R A A]",
"{R}",
"(R A)",
"{R A}"
]
}
|
[
{
"line": "simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]",
"before_state": "R : Type u\nA : Type v\nB : Type w\ninst✝⁸ : CommSemiring R\ninst✝⁷ : NonUnitalNonAssocSemiring A\ninst✝⁶ : Module R A\ninst✝⁵ : NonUnitalNonAssocSemiring B\ninst✝⁴ : Module R B\ninst✝³ : IsScalarTower R A A\ninst✝² : SMulCommClass R A A\ninst✝¹ : IsScalarTower R B B\ninst✝ : SMulCommClass R B B\nf : A →ₙₐ[R] B\n⊢ ↑(NonUnitalSubalgebra.map f ⊥) = ↑⊥",
"after_state": "No Goals!"
}
] |
theorem mem_one' {x : A} : x ∈ (1 : SubMulAction R A) ↔ ∃ y, algebraMap R A y = x :=
exists_congr fun r => by rw [algebraMap_eq_smul_one]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Operations.lean
|
{
"open": [
"Algebra Set MulOpposite",
"Pointwise"
],
"variables": [
"{R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]"
]
}
|
[
{
"line": "rw [algebraMap_eq_smul_one]",
"before_state": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A\nr : R\n⊢ (fun r => r • 1) r = x ↔ (algebraMap R A) r = x",
"after_state": "No Goals!"
},
{
"line": "rewrite [algebraMap_eq_smul_one]",
"before_state": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A\nr : R\n⊢ (fun r => r • 1) r = x ↔ (algebraMap R A) r = x",
"after_state": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A\nr : R\n⊢ (fun r => r • 1) r = x ↔ r • 1 = x"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A\nr : R\n⊢ (fun r => r • 1) r = x ↔ r • 1 = x",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A\nr : R\n⊢ (fun r => r • 1) r = x ↔ r • 1 = x",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A\nr : R\n⊢ (fun r => r • 1) r = x ↔ r • 1 = x",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A\nr : R\n⊢ (fun r => r • 1) r = x ↔ r • 1 = x",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A\nr : R\n⊢ (fun r => r • 1) r = x ↔ r • 1 = x",
"after_state": "No Goals!"
},
{
"line": "exact Iff.rfl✝",
"before_state": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A\nr : R\n⊢ (fun r => r • 1) r = x ↔ r • 1 = x",
"after_state": "No Goals!"
}
] |
theorem le_one_toAddSubmonoid : 1 ≤ (1 : Submodule R A).toAddSubmonoid := by
rintro x ⟨n, rfl⟩
exact ⟨n, show (n : R) • (1 : A) = n by rw [Nat.cast_smul_eq_nsmul, nsmul_one]⟩
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Operations.lean
|
{
"open": [
"Algebra Set MulOpposite",
"Pointwise"
],
"variables": [
"{R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]",
"{R : Type u} [Semiring R] {A : Type v} [Semiring A] [Module R A]"
]
}
|
[
{
"line": "rintro x ⟨n, rfl⟩",
"before_state": "R : Type u\ninst✝² : Semiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Module R A\n⊢ 1 ≤ Submodule.toAddSubmonoid 1",
"after_state": "case intro\nR : Type u\ninst✝² : Semiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Module R A\nn : ℕ\n⊢ (Nat.castAddMonoidHom A) n ∈ Submodule.toAddSubmonoid 1"
},
{
"line": "exact ⟨n, show (n : R) • (1 : A) = n by rw [Nat.cast_smul_eq_nsmul, nsmul_one]⟩",
"before_state": "case intro\nR : Type u\ninst✝² : Semiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Module R A\nn : ℕ\n⊢ (Nat.castAddMonoidHom A) n ∈ Submodule.toAddSubmonoid 1",
"after_state": "No Goals!"
},
{
"line": "rw [Nat.cast_smul_eq_nsmul, nsmul_one]",
"before_state": "R : Type u\ninst✝² : Semiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Module R A\nn : ℕ\n⊢ ↑n • 1 = ↑n",
"after_state": "No Goals!"
},
{
"line": "rewrite [Nat.cast_smul_eq_nsmul, nsmul_one]",
"before_state": "R : Type u\ninst✝² : Semiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Module R A\nn : ℕ\n⊢ ↑n • 1 = ↑n",
"after_state": "R : Type u\ninst✝² : Semiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Module R A\nn : ℕ\n⊢ ↑n = ↑n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u\ninst✝² : Semiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Module R A\nn : ℕ\n⊢ ↑n = ↑n",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u\ninst✝² : Semiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Module R A\nn : ℕ\n⊢ ↑n = ↑n",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u\ninst✝² : Semiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Module R A\nn : ℕ\n⊢ ↑n = ↑n",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u\ninst✝² : Semiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Module R A\nn : ℕ\n⊢ ↑n = ↑n",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "R : Type u\ninst✝² : Semiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Module R A\nn : ℕ\n⊢ ↑n = ↑n",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "R : Type u\ninst✝² : Semiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Module R A\nn : ℕ\n⊢ ↑n = ↑n",
"after_state": "No Goals!"
}
] |
theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop}
(smul : ∀ (r : A) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn))
(add : ∀ x hx y hy, p x hx → p y hy → p (x + y) (add_mem ‹_› ‹_›)) : p x hx := by
refine Exists.elim ?_ fun (h : x ∈ I • N) (H : p x h) ↦ H
exact smul_induction_on hx (fun a ha x hx ↦ ⟨_, smul _ ha _ hx⟩)
fun x y ⟨_, hx⟩ ⟨_, hy⟩ ↦ ⟨_, add _ _ _ _ hx hy⟩
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Operations.lean
|
{
"open": [
"Algebra Set MulOpposite",
"Pointwise"
],
"variables": [
"{R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]",
"{R : Type u} [Semiring R] {A : Type v} [Semiring A] [Module R A]",
"{M : Type*} [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]",
"{I J : Submodule R A} {N P : Submodule R M}"
]
}
|
[
{
"line": "assumption",
"before_state": "R✝ : Type u\nA✝ : Type v\ninst✝⁹ : CommSemiring R✝\ninst✝⁸ : Semiring A✝\ninst✝⁷ : Algebra R✝ A✝\nR : Type u\ninst✝⁶ : Semiring R\nA : Type v\ninst✝⁵ : Semiring A\ninst✝⁴ : Module R A\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nI J : Submodule R A\nN P : Submodule R M\nx✝ : M\nhx✝ : x✝ ∈ I • N\np : (x : M) → x ∈ I • N → Prop\nsmul : ∀ r ∈ I, ∀ n ∈ N, p (r • n) ⋯\nx : M\nhx : x ∈ I • N\ny : M\nhy : y ∈ I • N\n⊢ x ∈ I • N",
"after_state": "No Goals!"
},
{
"line": "assumption",
"before_state": "R✝ : Type u\nA✝ : Type v\ninst✝⁹ : CommSemiring R✝\ninst✝⁸ : Semiring A✝\ninst✝⁷ : Algebra R✝ A✝\nR : Type u\ninst✝⁶ : Semiring R\nA : Type v\ninst✝⁵ : Semiring A\ninst✝⁴ : Module R A\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nI J : Submodule R A\nN P : Submodule R M\nx✝ : M\nhx✝ : x✝ ∈ I • N\np : (x : M) → x ∈ I • N → Prop\nsmul : ∀ r ∈ I, ∀ n ∈ N, p (r • n) ⋯\nx : M\nhx : x ∈ I • N\ny : M\nhy : y ∈ I • N\n⊢ y ∈ I • N",
"after_state": "No Goals!"
},
{
"line": "refine Exists.elim ?_ fun (h : x ∈ I • N) (H : p x h) ↦ H",
"before_state": "R : Type u\ninst✝⁶ : Semiring R\nA : Type v\ninst✝⁵ : Semiring A\ninst✝⁴ : Module R A\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nI : Submodule R A\nN : Submodule R M\nx : M\nhx : x ∈ I • N\np : (x : M) → x ∈ I • N → Prop\nsmul : ∀ r ∈ I, ∀ n ∈ N, p (r • n) ⋯\nadd : ∀ (x : M) (hx : x ∈ I • N) (y : M) (hy : y ∈ I • N), p x hx → p y hy → p (x + y) ⋯\n⊢ p x hx",
"after_state": "R : Type u\ninst✝⁶ : Semiring R\nA : Type v\ninst✝⁵ : Semiring A\ninst✝⁴ : Module R A\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nI : Submodule R A\nN : Submodule R M\nx : M\nhx : x ∈ I • N\np : (x : M) → x ∈ I • N → Prop\nsmul : ∀ r ∈ I, ∀ n ∈ N, p (r • n) ⋯\nadd : ∀ (x : M) (hx : x ∈ I • N) (y : M) (hy : y ∈ I • N), p x hx → p y hy → p (x + y) ⋯\n⊢ ∃ (x_1 : x ∈ I • N), p x x_1"
},
{
"line": "exact smul_induction_on hx (fun a ha x hx ↦ ⟨_, smul _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ ↦ ⟨_, add _ _ _ _ hx hy⟩",
"before_state": "R : Type u\ninst✝⁶ : Semiring R\nA : Type v\ninst✝⁵ : Semiring A\ninst✝⁴ : Module R A\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nI : Submodule R A\nN : Submodule R M\nx : M\nhx : x ∈ I • N\np : (x : M) → x ∈ I • N → Prop\nsmul : ∀ r ∈ I, ∀ n ∈ N, p (r • n) ⋯\nadd : ∀ (x : M) (hx : x ∈ I • N) (y : M) (hy : y ∈ I • N), p x hx → p y hy → p (x + y) ⋯\n⊢ ∃ (x_1 : x ∈ I • N), p x x_1",
"after_state": "No Goals!"
}
] |
theorem le_pow_toAddSubmonoid {n : ℕ} : M.toAddSubmonoid ^ n ≤ (M ^ n).toAddSubmonoid := by
obtain rfl | hn := Decidable.eq_or_ne n 0
· rw [Submodule.pow_zero, pow_zero]
exact le_one_toAddSubmonoid
· exact (pow_toAddSubmonoid M hn).ge
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Operations.lean
|
{
"open": [
"Algebra Set MulOpposite",
"Pointwise"
],
"variables": [
"{R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]",
"{R : Type u} [Semiring R] {A : Type v} [Semiring A] [Module R A]",
"{M : Type*} [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]",
"{I J : Submodule R A} {N P : Submodule R M}",
"(I J N P)",
"[IsScalarTower R A A]",
"(S T : Set A) {M N P Q : Submodule R A} {m n : A}",
"(M)",
"{M}",
"(M N P)",
"{ι : Sort uι}"
]
}
|
[
{
"line": "obtain rfl | hn := Decidable.eq_or_ne n 0",
"before_state": "R : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\nn : ℕ\n⊢ M.toAddSubmonoid ^ n ≤ (M ^ n).toAddSubmonoid",
"after_state": "case inl\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ M.toAddSubmonoid ^ 0 ≤ (M ^ 0).toAddSubmonoid\n---\ncase inr\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\nn : ℕ\nhn : n ≠ 0\n⊢ M.toAddSubmonoid ^ n ≤ (M ^ n).toAddSubmonoid"
},
{
"line": "rw [Submodule.pow_zero, pow_zero]",
"before_state": "case inl\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ M.toAddSubmonoid ^ 0 ≤ (M ^ 0).toAddSubmonoid",
"after_state": "case inl\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ 1 ≤ Submodule.toAddSubmonoid 1"
},
{
"line": "rewrite [Submodule.pow_zero, pow_zero]",
"before_state": "case inl\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ M.toAddSubmonoid ^ 0 ≤ (M ^ 0).toAddSubmonoid",
"after_state": "case inl\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ 1 ≤ Submodule.toAddSubmonoid 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case inl\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ 1 ≤ Submodule.toAddSubmonoid 1",
"after_state": "case inl\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ 1 ≤ Submodule.toAddSubmonoid 1"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case inl\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ 1 ≤ Submodule.toAddSubmonoid 1",
"after_state": "case inl\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ 1 ≤ Submodule.toAddSubmonoid 1"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case inl\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ 1 ≤ Submodule.toAddSubmonoid 1",
"after_state": "case inl\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ 1 ≤ Submodule.toAddSubmonoid 1"
},
{
"line": "with_reducible rfl",
"before_state": "case inl\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ 1 ≤ Submodule.toAddSubmonoid 1",
"after_state": "case inl\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ 1 ≤ Submodule.toAddSubmonoid 1"
},
{
"line": "rfl",
"before_state": "case inl\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ 1 ≤ Submodule.toAddSubmonoid 1",
"after_state": "case inl\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ 1 ≤ Submodule.toAddSubmonoid 1"
},
{
"line": "apply_rfl",
"before_state": "case inl\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ 1 ≤ Submodule.toAddSubmonoid 1",
"after_state": "case inl\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ 1 ≤ Submodule.toAddSubmonoid 1"
},
{
"line": "skip",
"before_state": "case inl\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ 1 ≤ Submodule.toAddSubmonoid 1",
"after_state": "case inl\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ 1 ≤ Submodule.toAddSubmonoid 1"
},
{
"line": "exact le_one_toAddSubmonoid",
"before_state": "case inl\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ 1 ≤ Submodule.toAddSubmonoid 1",
"after_state": "No Goals!"
},
{
"line": "exact (pow_toAddSubmonoid M hn).ge",
"before_state": "case inr\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\nn : ℕ\nhn : n ≠ 0\n⊢ M.toAddSubmonoid ^ n ≤ (M ^ n).toAddSubmonoid",
"after_state": "No Goals!"
}
] |
theorem mem_one {x : A} : x ∈ (1 : Submodule R A) ↔ ∃ y, algebraMap R A y = x := by
simp [one_eq_range]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Operations.lean
|
{
"open": [
"Algebra Set MulOpposite",
"Pointwise"
],
"variables": [
"{R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]",
"{R : Type u} [Semiring R] {A : Type v} [Semiring A] [Module R A]",
"{M : Type*} [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]",
"{I J : Submodule R A} {N P : Submodule R M}",
"(I J N P)",
"[IsScalarTower R A A]",
"(S T : Set A) {M N P Q : Submodule R A} {m n : A}",
"(M)",
"{M}",
"(M N P)",
"{ι : Sort uι}",
"{ι : Sort uι}",
"{R : Type u} [CommSemiring R]",
"{A : Type v} [Semiring A] [Algebra R A]",
"(S T : Set A) {M N P Q : Submodule R A} {m n : A}"
]
}
|
[
{
"line": "simp [one_eq_range]",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A\n⊢ x ∈ 1 ↔ ∃ y, (algebraMap R A) y = x",
"after_state": "No Goals!"
}
] |
theorem mem_span_mul_finite_of_mem_mul {P Q : Submodule R A} {x : A} (hx : x ∈ P * Q) :
∃ T T' : Finset A, (T : Set A) ⊆ P ∧ (T' : Set A) ⊆ Q ∧ x ∈ span R (T * T' : Set A) :=
Submodule.mem_span_mul_finite_of_mem_span_mul
(by rwa [← Submodule.span_eq P, ← Submodule.span_eq Q, Submodule.span_mul_span] at hx)
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Operations.lean
|
{
"open": [
"Algebra Set MulOpposite",
"Pointwise",
"Pointwise"
],
"variables": [
"{R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]",
"{R : Type u} [Semiring R] {A : Type v} [Semiring A] [Module R A]",
"{M : Type*} [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]",
"{I J : Submodule R A} {N P : Submodule R M}",
"(I J N P)",
"[IsScalarTower R A A]",
"(S T : Set A) {M N P Q : Submodule R A} {m n : A}",
"(M)",
"{M}",
"(M N P)",
"{ι : Sort uι}",
"{ι : Sort uι}",
"{R : Type u} [CommSemiring R]",
"{A : Type v} [Semiring A] [Algebra R A]",
"(S T : Set A) {M N P Q : Submodule R A} {m n : A}",
"(R M N)",
"{R} (P Q)",
"{α : Type*} [Monoid α] [DistribMulAction α A] [SMulCommClass α R A]"
]
}
|
[
{
"line": "rwa [← Submodule.span_eq P, ← Submodule.span_eq Q, Submodule.span_mul_span] at hx",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx✝ : Sort u_3\nspan : x✝\nP Q : Submodule R A\nx : A\nhx : x ∈ P * Q\n⊢ ?m.89073 ∈ Submodule.span ?m.89065 (?m.89071 * ?m.89072)",
"after_state": "No Goals!"
},
{
"line": "rw [← Submodule.span_eq P, ← Submodule.span_eq Q, Submodule.span_mul_span] at hx",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx✝ : Sort u_3\nspan : x✝\nP Q : Submodule R A\nx : A\nhx : x ∈ P * Q\n⊢ ?m.89073 ∈ Submodule.span ?m.89065 (?m.89071 * ?m.89072)",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx✝ : Sort u_3\nspan : x✝\nP Q : Submodule R A\nx : A\nhx✝ : x ∈ P * Q\nhx : x ∈ Submodule.span R (↑P * ↑Q)\n⊢ ?m.89073 ∈ Submodule.span ?m.89065 (?m.89071 * ?m.89072)"
},
{
"line": "rewrite [← Submodule.span_eq P, ← Submodule.span_eq Q, Submodule.span_mul_span] at hx",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx✝ : Sort u_3\nspan : x✝\nP Q : Submodule R A\nx : A\nhx : x ∈ P * Q\n⊢ ?m.89073 ∈ Submodule.span ?m.89065 (?m.89071 * ?m.89072)",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx✝ : Sort u_3\nspan : x✝\nP Q : Submodule R A\nx : A\nhx✝ : x ∈ P * Q\nhx : x ∈ Submodule.span R (↑P * ↑Q)\n⊢ ?m.89073 ∈ Submodule.span ?m.89065 (?m.89071 * ?m.89072)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx✝ : Sort u_3\nspan : x✝\nP Q : Submodule R A\nx : A\nhx✝ : x ∈ P * Q\nhx : x ∈ Submodule.span R (↑P * ↑Q)\n⊢ ?m.89073 ∈ Submodule.span ?m.89065 (?m.89071 * ?m.89072)",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx✝ : Sort u_3\nspan : x✝\nP Q : Submodule R A\nx : A\nhx✝ : x ∈ P * Q\nhx : x ∈ Submodule.span R (↑P * ↑Q)\n⊢ ?m.89073 ∈ Submodule.span ?m.89065 (?m.89071 * ?m.89072)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx✝ : Sort u_3\nspan : x✝\nP Q : Submodule R A\nx : A\nhx✝ : x ∈ P * Q\nhx : x ∈ Submodule.span R (↑P * ↑Q)\n⊢ ?m.89073 ∈ Submodule.span ?m.89065 (?m.89071 * ?m.89072)",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx✝ : Sort u_3\nspan : x✝\nP Q : Submodule R A\nx : A\nhx✝ : x ∈ P * Q\nhx : x ∈ Submodule.span R (↑P * ↑Q)\n⊢ ?m.89073 ∈ Submodule.span ?m.89065 (?m.89071 * ?m.89072)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx✝ : Sort u_3\nspan : x✝\nP Q : Submodule R A\nx : A\nhx✝ : x ∈ P * Q\nhx : x ∈ Submodule.span R (↑P * ↑Q)\n⊢ ?m.89073 ∈ Submodule.span ?m.89065 (?m.89071 * ?m.89072)",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx✝ : Sort u_3\nspan : x✝\nP Q : Submodule R A\nx : A\nhx✝ : x ∈ P * Q\nhx : x ∈ Submodule.span R (↑P * ↑Q)\n⊢ ?m.89073 ∈ Submodule.span ?m.89065 (?m.89071 * ?m.89072)"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx✝ : Sort u_3\nspan : x✝\nP Q : Submodule R A\nx : A\nhx✝ : x ∈ P * Q\nhx : x ∈ Submodule.span R (↑P * ↑Q)\n⊢ ?m.89073 ∈ Submodule.span ?m.89065 (?m.89071 * ?m.89072)",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx✝ : Sort u_3\nspan : x✝\nP Q : Submodule R A\nx : A\nhx✝ : x ∈ P * Q\nhx : x ∈ Submodule.span R (↑P * ↑Q)\n⊢ ?m.89073 ∈ Submodule.span ?m.89065 (?m.89071 * ?m.89072)"
},
{
"line": "rfl",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx✝ : Sort u_3\nspan : x✝\nP Q : Submodule R A\nx : A\nhx✝ : x ∈ P * Q\nhx : x ∈ Submodule.span R (↑P * ↑Q)\n⊢ ?m.89073 ∈ Submodule.span ?m.89065 (?m.89071 * ?m.89072)",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx✝ : Sort u_3\nspan : x✝\nP Q : Submodule R A\nx : A\nhx✝ : x ∈ P * Q\nhx : x ∈ Submodule.span R (↑P * ↑Q)\n⊢ ?m.89073 ∈ Submodule.span ?m.89065 (?m.89071 * ?m.89072)"
},
{
"line": "apply_rfl",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx✝ : Sort u_3\nspan : x✝\nP Q : Submodule R A\nx : A\nhx✝ : x ∈ P * Q\nhx : x ∈ Submodule.span R (↑P * ↑Q)\n⊢ ?m.89073 ∈ Submodule.span ?m.89065 (?m.89071 * ?m.89072)",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx✝ : Sort u_3\nspan : x✝\nP Q : Submodule R A\nx : A\nhx✝ : x ∈ P * Q\nhx : x ∈ Submodule.span R (↑P * ↑Q)\n⊢ ?m.89073 ∈ Submodule.span ?m.89065 (?m.89071 * ?m.89072)"
},
{
"line": "skip",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx✝ : Sort u_3\nspan : x✝\nP Q : Submodule R A\nx : A\nhx✝ : x ∈ P * Q\nhx : x ∈ Submodule.span R (↑P * ↑Q)\n⊢ ?m.89073 ∈ Submodule.span ?m.89065 (?m.89071 * ?m.89072)",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx✝ : Sort u_3\nspan : x✝\nP Q : Submodule R A\nx : A\nhx✝ : x ∈ P * Q\nhx : x ∈ Submodule.span R (↑P * ↑Q)\n⊢ ?m.89073 ∈ Submodule.span ?m.89065 (?m.89071 * ?m.89072)"
},
{
"line": "assumption",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx✝ : Sort u_3\nspan : x✝\nP Q : Submodule R A\nx : A\nhx✝ : x ∈ P * Q\nhx : x ∈ Submodule.span R (↑P * ↑Q)\n⊢ ?m.89073 ∈ Submodule.span ?m.89065 (?m.89071 * ?m.89072)",
"after_state": "No Goals!"
}
] |
lemma mem_smul_iff_inv_mul_mem {S} [DivisionSemiring S] [Algebra R S] {x : S} {p : Submodule R S}
{y : S} (hx : x ≠ 0) : y ∈ x • p ↔ x⁻¹ * y ∈ p := by
constructor
· rintro ⟨a, ha : a ∈ p, rfl⟩; simpa [inv_mul_cancel_left₀ hx]
· exact fun h ↦ ⟨_, h, by simp [mul_inv_cancel_left₀ hx]⟩
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Operations.lean
|
{
"open": [
"Algebra Set MulOpposite",
"Pointwise",
"Pointwise"
],
"variables": [
"{R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]",
"{R : Type u} [Semiring R] {A : Type v} [Semiring A] [Module R A]",
"{M : Type*} [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]",
"{I J : Submodule R A} {N P : Submodule R M}",
"(I J N P)",
"[IsScalarTower R A A]",
"(S T : Set A) {M N P Q : Submodule R A} {m n : A}",
"(M)",
"{M}",
"(M N P)",
"{ι : Sort uι}",
"{ι : Sort uι}",
"{R : Type u} [CommSemiring R]",
"{A : Type v} [Semiring A] [Algebra R A]",
"(S T : Set A) {M N P Q : Submodule R A} {m n : A}",
"(R M N)",
"{R} (P Q)",
"{α : Type*} [Monoid α] [DistribMulAction α A] [SMulCommClass α R A]",
"{M N P}"
]
}
|
[
{
"line": "constructor",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nS : Type u_3\ninst✝¹ : DivisionSemiring S\ninst✝ : Algebra R S\nx : S\np : Submodule R S\ny : S\nhx : x ≠ 0\n⊢ y ∈ x • p ↔ x⁻¹ * y ∈ p",
"after_state": "case mp\nR : Type u\ninst✝² : CommSemiring R\nS : Type u_3\ninst✝¹ : DivisionSemiring S\ninst✝ : Algebra R S\nx : S\np : Submodule R S\ny : S\nhx : x ≠ 0\n⊢ y ∈ x • p → x⁻¹ * y ∈ p\n---\ncase mpr\nR : Type u\ninst✝² : CommSemiring R\nS : Type u_3\ninst✝¹ : DivisionSemiring S\ninst✝ : Algebra R S\nx : S\np : Submodule R S\ny : S\nhx : x ≠ 0\n⊢ x⁻¹ * y ∈ p → y ∈ x • p"
},
{
"line": "rintro ⟨a, ha : a ∈ p, rfl⟩",
"before_state": "case mp\nR : Type u\ninst✝² : CommSemiring R\nS : Type u_3\ninst✝¹ : DivisionSemiring S\ninst✝ : Algebra R S\nx : S\np : Submodule R S\ny : S\nhx : x ≠ 0\n⊢ y ∈ x • p → x⁻¹ * y ∈ p",
"after_state": "case mp.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nS : Type u_3\ninst✝¹ : DivisionSemiring S\ninst✝ : Algebra R S\nx : S\np : Submodule R S\nhx : x ≠ 0\na : S\nha : a ∈ p\n⊢ x⁻¹ * (DistribMulAction.toLinearMap R S x) a ∈ p"
},
{
"line": "simpa [inv_mul_cancel_left₀ hx]",
"before_state": "case mp.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nS : Type u_3\ninst✝¹ : DivisionSemiring S\ninst✝ : Algebra R S\nx : S\np : Submodule R S\nhx : x ≠ 0\na : S\nha : a ∈ p\n⊢ x⁻¹ * (DistribMulAction.toLinearMap R S x) a ∈ p",
"after_state": "No Goals!"
},
{
"line": "exact fun h ↦ ⟨_, h, by simp [mul_inv_cancel_left₀ hx]⟩",
"before_state": "case mpr\nR : Type u\ninst✝² : CommSemiring R\nS : Type u_3\ninst✝¹ : DivisionSemiring S\ninst✝ : Algebra R S\nx : S\np : Submodule R S\ny : S\nhx : x ≠ 0\n⊢ x⁻¹ * y ∈ p → y ∈ x • p",
"after_state": "No Goals!"
},
{
"line": "simp [mul_inv_cancel_left₀ hx]",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nS : Type u_3\ninst✝¹ : DivisionSemiring S\ninst✝ : Algebra R S\nx : S\np : Submodule R S\ny : S\nhx : x ≠ 0\nh : x⁻¹ * y ∈ p\n⊢ (DistribMulAction.toLinearMap R S x) (x⁻¹ * y) = y",
"after_state": "No Goals!"
}
] |
theorem mul_smul_mul_eq_smul_mul_smul (x y : R) : (x * y) • (M * N) = (x • M) * (y • N) := by
ext
refine ⟨?_, fun hx ↦ Submodule.mul_induction_on hx ?_ fun _ _ hx hy ↦ Submodule.add_mem _ hx hy⟩
· rintro ⟨_, hx, rfl⟩
rw [DistribMulAction.toLinearMap_apply]
refine Submodule.mul_induction_on hx (fun m hm n hn ↦ ?_) (fun _ _ hn hm ↦ ?_)
· rw [mul_smul_mul_comm]
exact mul_mem_mul (smul_mem_pointwise_smul m x M hm) (smul_mem_pointwise_smul n y N hn)
· rw [smul_add]
exact Submodule.add_mem _ hn hm
· rintro _ ⟨m, hm, rfl⟩ _ ⟨n, hn, rfl⟩
simp_rw [DistribMulAction.toLinearMap_apply, smul_mul_smul_comm]
exact smul_mem_pointwise_smul _ _ _ (mul_mem_mul hm hn)
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Operations.lean
|
{
"open": [
"Algebra Set MulOpposite",
"Pointwise",
"Pointwise"
],
"variables": [
"{R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]",
"{R : Type u} [Semiring R] {A : Type v} [Semiring A] [Module R A]",
"{M : Type*} [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]",
"{I J : Submodule R A} {N P : Submodule R M}",
"(I J N P)",
"[IsScalarTower R A A]",
"(S T : Set A) {M N P Q : Submodule R A} {m n : A}",
"(M)",
"{M}",
"(M N P)",
"{ι : Sort uι}",
"{ι : Sort uι}",
"{R : Type u} [CommSemiring R]",
"{A : Type v} [Semiring A] [Algebra R A]",
"(S T : Set A) {M N P Q : Submodule R A} {m n : A}",
"(R M N)",
"{R} (P Q)",
"{α : Type*} [Monoid α] [DistribMulAction α A] [SMulCommClass α R A]",
"{M N P}",
"(M N) in"
]
}
|
[
{
"line": "ext",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\n⊢ (x * y) • (M * N) = x • M * y • N",
"after_state": "case h\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nx✝ : A\n⊢ x✝ ∈ (x * y) • (M * N) ↔ x✝ ∈ x • M * y • N"
},
{
"line": "refine ⟨?_, fun hx ↦ Submodule.mul_induction_on hx ?_ fun _ _ hx hy ↦ Submodule.add_mem _ hx hy⟩",
"before_state": "case h\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nx✝ : A\n⊢ x✝ ∈ (x * y) • (M * N) ↔ x✝ ∈ x • M * y • N",
"after_state": "case h.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nx✝ : A\n⊢ x✝ ∈ (x * y) • (M * N) → x✝ ∈ x • M * y • N\n---\ncase h.refine_2\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nx✝ : A\nhx : x✝ ∈ x • M * y • N\n⊢ ∀ m ∈ x • M, ∀ n ∈ y • N, m * n ∈ (x * y) • (M * N)"
},
{
"line": "rintro ⟨_, hx, rfl⟩",
"before_state": "case h.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nx✝ : A\n⊢ x✝ ∈ (x * y) • (M * N) → x✝ ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\n⊢ (DistribMulAction.toLinearMap R A (x * y)) w✝ ∈ x • M * y • N"
},
{
"line": "rw [DistribMulAction.toLinearMap_apply]",
"before_state": "case h.refine_1.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\n⊢ (DistribMulAction.toLinearMap R A (x * y)) w✝ ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\n⊢ (x * y) • w✝ ∈ x • M * y • N"
},
{
"line": "rewrite [DistribMulAction.toLinearMap_apply]",
"before_state": "case h.refine_1.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\n⊢ (DistribMulAction.toLinearMap R A (x * y)) w✝ ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\n⊢ (x * y) • w✝ ∈ x • M * y • N"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.refine_1.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\n⊢ (x * y) • w✝ ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\n⊢ (x * y) • w✝ ∈ x • M * y • N"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.refine_1.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\n⊢ (x * y) • w✝ ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\n⊢ (x * y) • w✝ ∈ x • M * y • N"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.refine_1.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\n⊢ (x * y) • w✝ ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\n⊢ (x * y) • w✝ ∈ x • M * y • N"
},
{
"line": "with_reducible rfl",
"before_state": "case h.refine_1.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\n⊢ (x * y) • w✝ ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\n⊢ (x * y) • w✝ ∈ x • M * y • N"
},
{
"line": "rfl",
"before_state": "case h.refine_1.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\n⊢ (x * y) • w✝ ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\n⊢ (x * y) • w✝ ∈ x • M * y • N"
},
{
"line": "apply_rfl",
"before_state": "case h.refine_1.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\n⊢ (x * y) • w✝ ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\n⊢ (x * y) • w✝ ∈ x • M * y • N"
},
{
"line": "skip",
"before_state": "case h.refine_1.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\n⊢ (x * y) • w✝ ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\n⊢ (x * y) • w✝ ∈ x • M * y • N"
},
{
"line": "refine Submodule.mul_induction_on hx (fun m hm n hn ↦ ?_) (fun _ _ hn hm ↦ ?_)",
"before_state": "case h.refine_1.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\n⊢ (x * y) • w✝ ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nm : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ (x * y) • (m * n) ∈ x • M * y • N\n---\ncase h.refine_1.intro.intro.refine_2\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nx✝¹ x✝ : A\nhn : (x * y) • x✝¹ ∈ x • M * y • N\nhm : (x * y) • x✝ ∈ x • M * y • N\n⊢ (x * y) • (x✝¹ + x✝) ∈ x • M * y • N"
},
{
"line": "rw [mul_smul_mul_comm]",
"before_state": "case h.refine_1.intro.intro.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nm : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ (x * y) • (m * n) ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nm : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ x • m * y • n ∈ x • M * y • N"
},
{
"line": "rewrite [mul_smul_mul_comm]",
"before_state": "case h.refine_1.intro.intro.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nm : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ (x * y) • (m * n) ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nm : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ x • m * y • n ∈ x • M * y • N"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.refine_1.intro.intro.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nm : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ x • m * y • n ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nm : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ x • m * y • n ∈ x • M * y • N"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.refine_1.intro.intro.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nm : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ x • m * y • n ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nm : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ x • m * y • n ∈ x • M * y • N"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.refine_1.intro.intro.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nm : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ x • m * y • n ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nm : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ x • m * y • n ∈ x • M * y • N"
},
{
"line": "with_reducible rfl",
"before_state": "case h.refine_1.intro.intro.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nm : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ x • m * y • n ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nm : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ x • m * y • n ∈ x • M * y • N"
},
{
"line": "rfl",
"before_state": "case h.refine_1.intro.intro.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nm : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ x • m * y • n ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nm : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ x • m * y • n ∈ x • M * y • N"
},
{
"line": "apply_rfl",
"before_state": "case h.refine_1.intro.intro.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nm : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ x • m * y • n ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nm : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ x • m * y • n ∈ x • M * y • N"
},
{
"line": "skip",
"before_state": "case h.refine_1.intro.intro.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nm : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ x • m * y • n ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nm : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ x • m * y • n ∈ x • M * y • N"
},
{
"line": "exact mul_mem_mul (smul_mem_pointwise_smul m x M hm) (smul_mem_pointwise_smul n y N hn)",
"before_state": "case h.refine_1.intro.intro.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nm : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ x • m * y • n ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro.refine_1\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nm : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ x • m * y • n ∈ x • M * y • N"
},
{
"line": "rw [smul_add]",
"before_state": "case h.refine_1.intro.intro.refine_2\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nx✝¹ x✝ : A\nhn : (x * y) • x✝¹ ∈ x • M * y • N\nhm : (x * y) • x✝ ∈ x • M * y • N\n⊢ (x * y) • (x✝¹ + x✝) ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro.refine_2\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nx✝¹ x✝ : A\nhn : (x * y) • x✝¹ ∈ x • M * y • N\nhm : (x * y) • x✝ ∈ x • M * y • N\n⊢ (x * y) • x✝¹ + (x * y) • x✝ ∈ x • M * y • N"
},
{
"line": "rewrite [smul_add]",
"before_state": "case h.refine_1.intro.intro.refine_2\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nx✝¹ x✝ : A\nhn : (x * y) • x✝¹ ∈ x • M * y • N\nhm : (x * y) • x✝ ∈ x • M * y • N\n⊢ (x * y) • (x✝¹ + x✝) ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro.refine_2\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nx✝¹ x✝ : A\nhn : (x * y) • x✝¹ ∈ x • M * y • N\nhm : (x * y) • x✝ ∈ x • M * y • N\n⊢ (x * y) • x✝¹ + (x * y) • x✝ ∈ x • M * y • N"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.refine_1.intro.intro.refine_2\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nx✝¹ x✝ : A\nhn : (x * y) • x✝¹ ∈ x • M * y • N\nhm : (x * y) • x✝ ∈ x • M * y • N\n⊢ (x * y) • x✝¹ + (x * y) • x✝ ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro.refine_2\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nx✝¹ x✝ : A\nhn : (x * y) • x✝¹ ∈ x • M * y • N\nhm : (x * y) • x✝ ∈ x • M * y • N\n⊢ (x * y) • x✝¹ + (x * y) • x✝ ∈ x • M * y • N"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.refine_1.intro.intro.refine_2\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nx✝¹ x✝ : A\nhn : (x * y) • x✝¹ ∈ x • M * y • N\nhm : (x * y) • x✝ ∈ x • M * y • N\n⊢ (x * y) • x✝¹ + (x * y) • x✝ ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro.refine_2\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nx✝¹ x✝ : A\nhn : (x * y) • x✝¹ ∈ x • M * y • N\nhm : (x * y) • x✝ ∈ x • M * y • N\n⊢ (x * y) • x✝¹ + (x * y) • x✝ ∈ x • M * y • N"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.refine_1.intro.intro.refine_2\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nx✝¹ x✝ : A\nhn : (x * y) • x✝¹ ∈ x • M * y • N\nhm : (x * y) • x✝ ∈ x • M * y • N\n⊢ (x * y) • x✝¹ + (x * y) • x✝ ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro.refine_2\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nx✝¹ x✝ : A\nhn : (x * y) • x✝¹ ∈ x • M * y • N\nhm : (x * y) • x✝ ∈ x • M * y • N\n⊢ (x * y) • x✝¹ + (x * y) • x✝ ∈ x • M * y • N"
},
{
"line": "with_reducible rfl",
"before_state": "case h.refine_1.intro.intro.refine_2\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nx✝¹ x✝ : A\nhn : (x * y) • x✝¹ ∈ x • M * y • N\nhm : (x * y) • x✝ ∈ x • M * y • N\n⊢ (x * y) • x✝¹ + (x * y) • x✝ ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro.refine_2\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nx✝¹ x✝ : A\nhn : (x * y) • x✝¹ ∈ x • M * y • N\nhm : (x * y) • x✝ ∈ x • M * y • N\n⊢ (x * y) • x✝¹ + (x * y) • x✝ ∈ x • M * y • N"
},
{
"line": "rfl",
"before_state": "case h.refine_1.intro.intro.refine_2\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nx✝¹ x✝ : A\nhn : (x * y) • x✝¹ ∈ x • M * y • N\nhm : (x * y) • x✝ ∈ x • M * y • N\n⊢ (x * y) • x✝¹ + (x * y) • x✝ ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro.refine_2\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nx✝¹ x✝ : A\nhn : (x * y) • x✝¹ ∈ x • M * y • N\nhm : (x * y) • x✝ ∈ x • M * y • N\n⊢ (x * y) • x✝¹ + (x * y) • x✝ ∈ x • M * y • N"
},
{
"line": "apply_rfl",
"before_state": "case h.refine_1.intro.intro.refine_2\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nx✝¹ x✝ : A\nhn : (x * y) • x✝¹ ∈ x • M * y • N\nhm : (x * y) • x✝ ∈ x • M * y • N\n⊢ (x * y) • x✝¹ + (x * y) • x✝ ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro.refine_2\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nx✝¹ x✝ : A\nhn : (x * y) • x✝¹ ∈ x • M * y • N\nhm : (x * y) • x✝ ∈ x • M * y • N\n⊢ (x * y) • x✝¹ + (x * y) • x✝ ∈ x • M * y • N"
},
{
"line": "skip",
"before_state": "case h.refine_1.intro.intro.refine_2\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nx✝¹ x✝ : A\nhn : (x * y) • x✝¹ ∈ x • M * y • N\nhm : (x * y) • x✝ ∈ x • M * y • N\n⊢ (x * y) • x✝¹ + (x * y) • x✝ ∈ x • M * y • N",
"after_state": "case h.refine_1.intro.intro.refine_2\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nx✝¹ x✝ : A\nhn : (x * y) • x✝¹ ∈ x • M * y • N\nhm : (x * y) • x✝ ∈ x • M * y • N\n⊢ (x * y) • x✝¹ + (x * y) • x✝ ∈ x • M * y • N"
},
{
"line": "exact Submodule.add_mem _ hn hm",
"before_state": "case h.refine_1.intro.intro.refine_2\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nw✝ : A\nhx : w✝ ∈ ↑(M * N)\nx✝¹ x✝ : A\nhn : (x * y) • x✝¹ ∈ x • M * y • N\nhm : (x * y) • x✝ ∈ x • M * y • N\n⊢ (x * y) • x✝¹ + (x * y) • x✝ ∈ x • M * y • N",
"after_state": "No Goals!"
},
{
"line": "rintro _ ⟨m, hm, rfl⟩ _ ⟨n, hn, rfl⟩",
"before_state": "case h.refine_2\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nx✝ : A\nhx : x✝ ∈ x • M * y • N\n⊢ ∀ m ∈ x • M, ∀ n ∈ y • N, m * n ∈ (x * y) • (M * N)",
"after_state": "case h.refine_2.intro.intro.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nx✝ : A\nhx : x✝ ∈ x • M * y • N\nm : A\nhm : m ∈ ↑M\nn : A\nhn : n ∈ ↑N\n⊢ (DistribMulAction.toLinearMap R A x) m * (DistribMulAction.toLinearMap R A y) n ∈ (x * y) • (M * N)"
},
{
"line": "simp_rw [DistribMulAction.toLinearMap_apply, smul_mul_smul_comm]",
"before_state": "case h.refine_2.intro.intro.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nx✝ : A\nhx : x✝ ∈ x • M * y • N\nm : A\nhm : m ∈ ↑M\nn : A\nhn : n ∈ ↑N\n⊢ (DistribMulAction.toLinearMap R A x) m * (DistribMulAction.toLinearMap R A y) n ∈ (x * y) • (M * N)",
"after_state": "case h.refine_2.intro.intro.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nx✝ : A\nhx : x✝ ∈ x • M * y • N\nm : A\nhm : m ∈ ↑M\nn : A\nhn : n ∈ ↑N\n⊢ (x * y) • (m * n) ∈ (x * y) • (M * N)"
},
{
"line": "simp (failIfUnchanged✝ := false✝) only",
"before_state": "case h.refine_2.intro.intro.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nx✝ : A\nhx : x✝ ∈ x • M * y • N\nm : A\nhm : m ∈ ↑M\nn : A\nhn : n ∈ ↑N\n⊢ (DistribMulAction.toLinearMap R A x) m * (DistribMulAction.toLinearMap R A y) n ∈ (x * y) • (M * N)",
"after_state": "case h.refine_2.intro.intro.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nx✝ : A\nhx : x✝ ∈ x • M * y • N\nm : A\nhm : m ∈ ↑M\nn : A\nhn : n ∈ ↑N\n⊢ (DistribMulAction.toLinearMap R A x) m * (DistribMulAction.toLinearMap R A y) n ∈ (x * y) • (M * N)"
},
{
"line": "simp only [DistribMulAction.toLinearMap_apply]",
"before_state": "case h.refine_2.intro.intro.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nx✝ : A\nhx : x✝ ∈ x • M * y • N\nm : A\nhm : m ∈ ↑M\nn : A\nhn : n ∈ ↑N\n⊢ (DistribMulAction.toLinearMap R A x) m * (DistribMulAction.toLinearMap R A y) n ∈ (x * y) • (M * N)",
"after_state": "case h.refine_2.intro.intro.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nx✝ : A\nhx : x✝ ∈ x • M * y • N\nm : A\nhm : m ∈ ↑M\nn : A\nhn : n ∈ ↑N\n⊢ x • m * y • n ∈ (x * y) • (M * N)"
},
{
"line": "simp only [smul_mul_smul_comm]",
"before_state": "case h.refine_2.intro.intro.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nx✝ : A\nhx : x✝ ∈ x • M * y • N\nm : A\nhm : m ∈ ↑M\nn : A\nhn : n ∈ ↑N\n⊢ x • m * y • n ∈ (x * y) • (M * N)",
"after_state": "case h.refine_2.intro.intro.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nx✝ : A\nhx : x✝ ∈ x • M * y • N\nm : A\nhm : m ∈ ↑M\nn : A\nhn : n ∈ ↑N\n⊢ (x * y) • (m * n) ∈ (x * y) • (M * N)"
},
{
"line": "exact smul_mem_pointwise_smul _ _ _ (mul_mem_mul hm hn)",
"before_state": "case h.refine_2.intro.intro.intro.intro\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nx y : R\nx✝ : A\nhx : x✝ ∈ x • M * y • N\nm : A\nhm : m ∈ ↑M\nn : A\nhn : n ∈ ↑N\n⊢ (x * y) • (m * n) ∈ (x * y) • (M * N)",
"after_state": "No Goals!"
}
] |
theorem mem_div_iff_smul_subset {x : A} {I J : Submodule R A} : x ∈ I / J ↔ x • (J : Set A) ⊆ I :=
⟨fun h y ⟨y', hy', xy'_eq_y⟩ => by rw [← xy'_eq_y]; exact h _ hy',
fun h _ hy => h (Set.smul_mem_smul_set hy)⟩
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Operations.lean
|
{
"open": [
"Algebra Set MulOpposite",
"Pointwise",
"Pointwise"
],
"variables": [
"{R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]",
"{R : Type u} [Semiring R] {A : Type v} [Semiring A] [Module R A]",
"{M : Type*} [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]",
"{I J : Submodule R A} {N P : Submodule R M}",
"(I J N P)",
"[IsScalarTower R A A]",
"(S T : Set A) {M N P Q : Submodule R A} {m n : A}",
"(M)",
"{M}",
"(M N P)",
"{ι : Sort uι}",
"{ι : Sort uι}",
"{R : Type u} [CommSemiring R]",
"{A : Type v} [Semiring A] [Algebra R A]",
"(S T : Set A) {M N P Q : Submodule R A} {m n : A}",
"(R M N)",
"{R} (P Q)",
"{α : Type*} [Monoid α] [DistribMulAction α A] [SMulCommClass α R A]",
"{M N P}",
"(M N) in",
"(M)",
"{α : Type*} [Monoid α] [MulSemiringAction α A] [SMulCommClass α R A]",
"{A : Type v} [CommSemiring A] [Algebra R A]",
"{M N : Submodule R A} {m n : A}",
"(M N)",
"(R A)",
"{R A}"
]
}
|
[
{
"line": "rw [← xy'_eq_y]",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\nI J : Submodule R A\nh : x ∈ I / J\ny : A\nx✝ : y ∈ x • ↑J\ny' : A\nhy' : y' ∈ ↑J\nxy'_eq_y : (fun x_1 => x • x_1) y' = y\n⊢ y ∈ ↑I",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\nI J : Submodule R A\nh : x ∈ I / J\ny : A\nx✝ : y ∈ x • ↑J\ny' : A\nhy' : y' ∈ ↑J\nxy'_eq_y : (fun x_1 => x • x_1) y' = y\n⊢ (fun x_1 => x • x_1) y' ∈ ↑I"
},
{
"line": "rewrite [← xy'_eq_y]",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\nI J : Submodule R A\nh : x ∈ I / J\ny : A\nx✝ : y ∈ x • ↑J\ny' : A\nhy' : y' ∈ ↑J\nxy'_eq_y : (fun x_1 => x • x_1) y' = y\n⊢ y ∈ ↑I",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\nI J : Submodule R A\nh : x ∈ I / J\ny : A\nx✝ : y ∈ x • ↑J\ny' : A\nhy' : y' ∈ ↑J\nxy'_eq_y : (fun x_1 => x • x_1) y' = y\n⊢ (fun x_1 => x • x_1) y' ∈ ↑I"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\nI J : Submodule R A\nh : x ∈ I / J\ny : A\nx✝ : y ∈ x • ↑J\ny' : A\nhy' : y' ∈ ↑J\nxy'_eq_y : (fun x_1 => x • x_1) y' = y\n⊢ (fun x_1 => x • x_1) y' ∈ ↑I",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\nI J : Submodule R A\nh : x ∈ I / J\ny : A\nx✝ : y ∈ x • ↑J\ny' : A\nhy' : y' ∈ ↑J\nxy'_eq_y : (fun x_1 => x • x_1) y' = y\n⊢ (fun x_1 => x • x_1) y' ∈ ↑I"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\nI J : Submodule R A\nh : x ∈ I / J\ny : A\nx✝ : y ∈ x • ↑J\ny' : A\nhy' : y' ∈ ↑J\nxy'_eq_y : (fun x_1 => x • x_1) y' = y\n⊢ (fun x_1 => x • x_1) y' ∈ ↑I",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\nI J : Submodule R A\nh : x ∈ I / J\ny : A\nx✝ : y ∈ x • ↑J\ny' : A\nhy' : y' ∈ ↑J\nxy'_eq_y : (fun x_1 => x • x_1) y' = y\n⊢ (fun x_1 => x • x_1) y' ∈ ↑I"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\nI J : Submodule R A\nh : x ∈ I / J\ny : A\nx✝ : y ∈ x • ↑J\ny' : A\nhy' : y' ∈ ↑J\nxy'_eq_y : (fun x_1 => x • x_1) y' = y\n⊢ (fun x_1 => x • x_1) y' ∈ ↑I",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\nI J : Submodule R A\nh : x ∈ I / J\ny : A\nx✝ : y ∈ x • ↑J\ny' : A\nhy' : y' ∈ ↑J\nxy'_eq_y : (fun x_1 => x • x_1) y' = y\n⊢ (fun x_1 => x • x_1) y' ∈ ↑I"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\nI J : Submodule R A\nh : x ∈ I / J\ny : A\nx✝ : y ∈ x • ↑J\ny' : A\nhy' : y' ∈ ↑J\nxy'_eq_y : (fun x_1 => x • x_1) y' = y\n⊢ (fun x_1 => x • x_1) y' ∈ ↑I",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\nI J : Submodule R A\nh : x ∈ I / J\ny : A\nx✝ : y ∈ x • ↑J\ny' : A\nhy' : y' ∈ ↑J\nxy'_eq_y : (fun x_1 => x • x_1) y' = y\n⊢ (fun x_1 => x • x_1) y' ∈ ↑I"
},
{
"line": "rfl",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\nI J : Submodule R A\nh : x ∈ I / J\ny : A\nx✝ : y ∈ x • ↑J\ny' : A\nhy' : y' ∈ ↑J\nxy'_eq_y : (fun x_1 => x • x_1) y' = y\n⊢ (fun x_1 => x • x_1) y' ∈ ↑I",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\nI J : Submodule R A\nh : x ∈ I / J\ny : A\nx✝ : y ∈ x • ↑J\ny' : A\nhy' : y' ∈ ↑J\nxy'_eq_y : (fun x_1 => x • x_1) y' = y\n⊢ (fun x_1 => x • x_1) y' ∈ ↑I"
},
{
"line": "apply_rfl",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\nI J : Submodule R A\nh : x ∈ I / J\ny : A\nx✝ : y ∈ x • ↑J\ny' : A\nhy' : y' ∈ ↑J\nxy'_eq_y : (fun x_1 => x • x_1) y' = y\n⊢ (fun x_1 => x • x_1) y' ∈ ↑I",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\nI J : Submodule R A\nh : x ∈ I / J\ny : A\nx✝ : y ∈ x • ↑J\ny' : A\nhy' : y' ∈ ↑J\nxy'_eq_y : (fun x_1 => x • x_1) y' = y\n⊢ (fun x_1 => x • x_1) y' ∈ ↑I"
},
{
"line": "skip",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\nI J : Submodule R A\nh : x ∈ I / J\ny : A\nx✝ : y ∈ x • ↑J\ny' : A\nhy' : y' ∈ ↑J\nxy'_eq_y : (fun x_1 => x • x_1) y' = y\n⊢ (fun x_1 => x • x_1) y' ∈ ↑I",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\nI J : Submodule R A\nh : x ∈ I / J\ny : A\nx✝ : y ∈ x • ↑J\ny' : A\nhy' : y' ∈ ↑J\nxy'_eq_y : (fun x_1 => x • x_1) y' = y\n⊢ (fun x_1 => x • x_1) y' ∈ ↑I"
},
{
"line": "exact h _ hy'",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\nI J : Submodule R A\nh : x ∈ I / J\ny : A\nx✝ : y ∈ x • ↑J\ny' : A\nhy' : y' ∈ ↑J\nxy'_eq_y : (fun x_1 => x • x_1) y' = y\n⊢ (fun x_1 => x • x_1) y' ∈ ↑I",
"after_state": "No Goals!"
}
] |
theorem mul_one_div_le_one {I : Submodule R A} : I * (1 / I) ≤ 1 := by
rw [Submodule.mul_le]
intro m hm n hn
rw [Submodule.mem_div_iff_forall_mul_mem] at hn
rw [mul_comm]
exact hn m hm
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Operations.lean
|
{
"open": [
"Algebra Set MulOpposite",
"Pointwise",
"Pointwise"
],
"variables": [
"{R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]",
"{R : Type u} [Semiring R] {A : Type v} [Semiring A] [Module R A]",
"{M : Type*} [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]",
"{I J : Submodule R A} {N P : Submodule R M}",
"(I J N P)",
"[IsScalarTower R A A]",
"(S T : Set A) {M N P Q : Submodule R A} {m n : A}",
"(M)",
"{M}",
"(M N P)",
"{ι : Sort uι}",
"{ι : Sort uι}",
"{R : Type u} [CommSemiring R]",
"{A : Type v} [Semiring A] [Algebra R A]",
"(S T : Set A) {M N P Q : Submodule R A} {m n : A}",
"(R M N)",
"{R} (P Q)",
"{α : Type*} [Monoid α] [DistribMulAction α A] [SMulCommClass α R A]",
"{M N P}",
"(M N) in",
"(M)",
"{α : Type*} [Monoid α] [MulSemiringAction α A] [SMulCommClass α R A]",
"{A : Type v} [CommSemiring A] [Algebra R A]",
"{M N : Submodule R A} {m n : A}",
"(M N)",
"(R A)",
"{R A}"
]
}
|
[
{
"line": "rw [Submodule.mul_le]",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\n⊢ I * (1 / I) ≤ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\n⊢ ∀ m ∈ I, ∀ n ∈ 1 / I, m * n ∈ 1"
},
{
"line": "rewrite [Submodule.mul_le]",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\n⊢ I * (1 / I) ≤ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\n⊢ ∀ m ∈ I, ∀ n ∈ 1 / I, m * n ∈ 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\n⊢ ∀ m ∈ I, ∀ n ∈ 1 / I, m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\n⊢ ∀ m ∈ I, ∀ n ∈ 1 / I, m * n ∈ 1"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\n⊢ ∀ m ∈ I, ∀ n ∈ 1 / I, m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\n⊢ ∀ m ∈ I, ∀ n ∈ 1 / I, m * n ∈ 1"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\n⊢ ∀ m ∈ I, ∀ n ∈ 1 / I, m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\n⊢ ∀ m ∈ I, ∀ n ∈ 1 / I, m * n ∈ 1"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\n⊢ ∀ m ∈ I, ∀ n ∈ 1 / I, m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\n⊢ ∀ m ∈ I, ∀ n ∈ 1 / I, m * n ∈ 1"
},
{
"line": "rfl",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\n⊢ ∀ m ∈ I, ∀ n ∈ 1 / I, m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\n⊢ ∀ m ∈ I, ∀ n ∈ 1 / I, m * n ∈ 1"
},
{
"line": "apply_rfl",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\n⊢ ∀ m ∈ I, ∀ n ∈ 1 / I, m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\n⊢ ∀ m ∈ I, ∀ n ∈ 1 / I, m * n ∈ 1"
},
{
"line": "skip",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\n⊢ ∀ m ∈ I, ∀ n ∈ 1 / I, m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\n⊢ ∀ m ∈ I, ∀ n ∈ 1 / I, m * n ∈ 1"
},
{
"line": "intro m hm n hn",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\n⊢ ∀ m ∈ I, ∀ n ∈ 1 / I, m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : n ∈ 1 / I\n⊢ m * n ∈ 1"
},
{
"line": "intro m;\n intro hm n hn",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\n⊢ ∀ m ∈ I, ∀ n ∈ 1 / I, m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : n ∈ 1 / I\n⊢ m * n ∈ 1"
},
{
"line": "intro m",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\n⊢ ∀ m ∈ I, ∀ n ∈ 1 / I, m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\n⊢ m ∈ I → ∀ n ∈ 1 / I, m * n ∈ 1"
},
{
"line": "intro hm n hn",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\n⊢ m ∈ I → ∀ n ∈ 1 / I, m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : n ∈ 1 / I\n⊢ m * n ∈ 1"
},
{
"line": "intro hm;\n intro n hn",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\n⊢ m ∈ I → ∀ n ∈ 1 / I, m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : n ∈ 1 / I\n⊢ m * n ∈ 1"
},
{
"line": "intro hm",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\n⊢ m ∈ I → ∀ n ∈ 1 / I, m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\n⊢ ∀ n ∈ 1 / I, m * n ∈ 1"
},
{
"line": "intro n hn",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\n⊢ ∀ n ∈ 1 / I, m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : n ∈ 1 / I\n⊢ m * n ∈ 1"
},
{
"line": "intro n;\n intro hn",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\n⊢ ∀ n ∈ 1 / I, m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : n ∈ 1 / I\n⊢ m * n ∈ 1"
},
{
"line": "intro n",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\n⊢ ∀ n ∈ 1 / I, m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\n⊢ n ∈ 1 / I → m * n ∈ 1"
},
{
"line": "intro hn",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\n⊢ n ∈ 1 / I → m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : n ∈ 1 / I\n⊢ m * n ∈ 1"
},
{
"line": "rw [Submodule.mem_div_iff_forall_mul_mem] at hn",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : n ∈ 1 / I\n⊢ m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ m * n ∈ 1"
},
{
"line": "rewrite [Submodule.mem_div_iff_forall_mul_mem] at hn",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : n ∈ 1 / I\n⊢ m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ m * n ∈ 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ m * n ∈ 1"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ m * n ∈ 1"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ m * n ∈ 1"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ m * n ∈ 1"
},
{
"line": "rfl",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ m * n ∈ 1"
},
{
"line": "apply_rfl",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ m * n ∈ 1"
},
{
"line": "skip",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ m * n ∈ 1"
},
{
"line": "rw [mul_comm]",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ n * m ∈ 1"
},
{
"line": "rewrite [mul_comm]",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ m * n ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ n * m ∈ 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ n * m ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ n * m ∈ 1"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ n * m ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ n * m ∈ 1"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ n * m ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ n * m ∈ 1"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ n * m ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ n * m ∈ 1"
},
{
"line": "rfl",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ n * m ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ n * m ∈ 1"
},
{
"line": "apply_rfl",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ n * m ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ n * m ∈ 1"
},
{
"line": "skip",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ n * m ∈ 1",
"after_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ n * m ∈ 1"
},
{
"line": "exact hn m hm",
"before_state": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nm : A\nhm : m ∈ I\nn : A\nhn : ∀ y ∈ I, n * y ∈ 1\n⊢ n * m ∈ 1",
"after_state": "No Goals!"
}
] |
theorem algebraMap_mem_iff (S : Type*) {R A : Type*} [CommSemiring R] [CommSemiring S]
[Ring A] [Algebra R S] [Algebra R A] [Algebra S A] [IsScalarTower R S A] {a : A} {r : R} :
algebraMap R S r ∈ spectrum S a ↔ r ∈ spectrum R a := by
simp only [spectrum.mem_iff]
simp only [Algebra.algebraMap_eq_smul_one]
simp only [smul_assoc]
simp only [one_smul]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Spectrum/Basic.lean
|
{
"open": [
"Set",
"scoped Pointwise"
],
"variables": [
"(R : Type u) {A : Type v}",
"[CommSemiring R] [Ring A] [Algebra R A]",
"{R}",
"{R : Type u} {A : Type v}",
"[CommSemiring R] [Ring A] [Algebra R A]",
"(R)",
"{R}"
]
}
|
[
{
"line": "simp only [spectrum.mem_iff]",
"before_state": "S : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\na : A\nr : R\n⊢ (algebraMap R S) r ∈ spectrum S a ↔ r ∈ spectrum R a",
"after_state": "S : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\na : A\nr : R\n⊢ ¬IsUnit ((algebraMap S A) ((algebraMap R S) r) - a) ↔ ¬IsUnit ((algebraMap R A) r - a)"
},
{
"line": "simp only [Algebra.algebraMap_eq_smul_one]",
"before_state": "S : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\na : A\nr : R\n⊢ ¬IsUnit ((algebraMap S A) ((algebraMap R S) r) - a) ↔ ¬IsUnit ((algebraMap R A) r - a)",
"after_state": "S : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\na : A\nr : R\n⊢ ¬IsUnit ((r • 1) • 1 - a) ↔ ¬IsUnit (r • 1 - a)"
},
{
"line": "simp only [smul_assoc]",
"before_state": "S : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\na : A\nr : R\n⊢ ¬IsUnit ((r • 1) • 1 - a) ↔ ¬IsUnit (r • 1 - a)",
"after_state": "S : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\na : A\nr : R\n⊢ ¬IsUnit (r • 1 • 1 - a) ↔ ¬IsUnit (r • 1 - a)"
},
{
"line": "simp only [one_smul]",
"before_state": "S : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\na : A\nr : R\n⊢ ¬IsUnit (r • 1 • 1 - a) ↔ ¬IsUnit (r • 1 - a)",
"after_state": "No Goals!"
}
] |
theorem resolventSet_of_subsingleton [Subsingleton A] (a : A) : resolventSet R a = Set.univ := by
simp_rw [resolventSet, Subsingleton.elim (algebraMap R A _ - a) 1, isUnit_one, Set.setOf_true]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Spectrum/Basic.lean
|
{
"open": [
"Set",
"scoped Pointwise"
],
"variables": [
"(R : Type u) {A : Type v}",
"[CommSemiring R] [Ring A] [Algebra R A]",
"{R}",
"{R : Type u} {A : Type v}",
"[CommSemiring R] [Ring A] [Algebra R A]",
"(R)",
"{R}"
]
}
|
[
{
"line": "simp_rw [resolventSet, Subsingleton.elim (algebraMap R A _ - a) 1, isUnit_one, Set.setOf_true]",
"before_state": "R : Type u\nA : Type v\ninst✝³ : CommSemiring R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : Subsingleton A\na : A\n⊢ resolventSet R a = univ",
"after_state": "No Goals!"
},
{
"line": "simp (failIfUnchanged✝ := false✝) only",
"before_state": "R : Type u\nA : Type v\ninst✝³ : CommSemiring R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : Subsingleton A\na : A\n⊢ resolventSet R a = univ",
"after_state": "R : Type u\nA : Type v\ninst✝³ : CommSemiring R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : Subsingleton A\na : A\n⊢ resolventSet R a = univ"
},
{
"line": "simp only [resolventSet]",
"before_state": "R : Type u\nA : Type v\ninst✝³ : CommSemiring R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : Subsingleton A\na : A\n⊢ resolventSet R a = univ",
"after_state": "R : Type u\nA : Type v\ninst✝³ : CommSemiring R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : Subsingleton A\na : A\n⊢ {r | IsUnit ((algebraMap R A) r - a)} = univ"
},
{
"line": "simp only [Subsingleton.elim (algebraMap R A _ - a) 1]",
"before_state": "R : Type u\nA : Type v\ninst✝³ : CommSemiring R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : Subsingleton A\na : A\n⊢ {r | IsUnit ((algebraMap R A) r - a)} = univ",
"after_state": "R : Type u\nA : Type v\ninst✝³ : CommSemiring R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : Subsingleton A\na : A\n⊢ {r | IsUnit 1} = univ"
},
{
"line": "simp only [isUnit_one]",
"before_state": "R : Type u\nA : Type v\ninst✝³ : CommSemiring R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : Subsingleton A\na : A\n⊢ {r | IsUnit 1} = univ",
"after_state": "R : Type u\nA : Type v\ninst✝³ : CommSemiring R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : Subsingleton A\na : A\n⊢ {r | True} = univ"
},
{
"line": "simp only [Set.setOf_true]",
"before_state": "R : Type u\nA : Type v\ninst✝³ : CommSemiring R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : Subsingleton A\na : A\n⊢ {r | True} = univ",
"after_state": "No Goals!"
}
] |
theorem units_smul_resolvent_self {r : Rˣ} {a : A} :
r • resolvent a (r : R) = resolvent (r⁻¹ • a) (1 : R) := by
simpa only [Units.smul_def,Algebra.id.smul_eq_mul,Units.inv_mul] using
@units_smul_resolvent _ _ _ _ _ r r a
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Spectrum/Basic.lean
|
{
"open": [
"Set",
"scoped Pointwise"
],
"variables": [
"(R : Type u) {A : Type v}",
"[CommSemiring R] [Ring A] [Algebra R A]",
"{R}",
"{R : Type u} {A : Type v}",
"[CommSemiring R] [Ring A] [Algebra R A]",
"(R)",
"{R}"
]
}
|
[
{
"line": "simpa only [Units.smul_def, Algebra.id.smul_eq_mul, Units.inv_mul] using @units_smul_resolvent _ _ _ _ _ r r a",
"before_state": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : Rˣ\na : A\n⊢ r • resolvent a ↑r = resolvent (r⁻¹ • a) 1",
"after_state": "No Goals!"
}
] |
lemma inv₀_mem_iff {r : R} {a : Aˣ} :
r⁻¹ ∈ spectrum R (a : A) ↔ r ∈ spectrum R (↑a⁻¹ : A) := by
obtain (rfl | hr) := eq_or_ne r 0
· simp [zero_mem_iff]
· lift r to Rˣ using hr.isUnit
simp [inv_mem_iff]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Spectrum/Basic.lean
|
{
"open": [
"Set",
"scoped Pointwise"
],
"variables": [
"(R : Type u) {A : Type v}",
"[CommSemiring R] [Ring A] [Algebra R A]",
"{R}",
"{R : Type u} {A : Type v}",
"[CommSemiring R] [Ring A] [Algebra R A]",
"(R)",
"{R}",
"[InvolutiveStar R] [StarRing A] [StarModule R A]",
"{R : Type u} {A : Type v}",
"[CommRing R] [Ring A] [Algebra R A]",
"{R : Type u} {A : Type v} [Semifield R] [Ring A] [Algebra R A]"
]
}
|
[
{
"line": "obtain (rfl | hr) := eq_or_ne r 0",
"before_state": "R : Type u\nA : Type v\ninst✝² : Semifield R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : R\na : Aˣ\n⊢ r⁻¹ ∈ spectrum R ↑a ↔ r ∈ spectrum R ↑a⁻¹",
"after_state": "case inl\nR : Type u\nA : Type v\ninst✝² : Semifield R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : Aˣ\n⊢ 0⁻¹ ∈ spectrum R ↑a ↔ 0 ∈ spectrum R ↑a⁻¹\n---\ncase inr\nR : Type u\nA : Type v\ninst✝² : Semifield R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : R\na : Aˣ\nhr : r ≠ 0\n⊢ r⁻¹ ∈ spectrum R ↑a ↔ r ∈ spectrum R ↑a⁻¹"
},
{
"line": "simp [zero_mem_iff]",
"before_state": "case inl\nR : Type u\nA : Type v\ninst✝² : Semifield R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : Aˣ\n⊢ 0⁻¹ ∈ spectrum R ↑a ↔ 0 ∈ spectrum R ↑a⁻¹",
"after_state": "No Goals!"
},
{
"line": "lift r to Rˣ using hr.isUnit",
"before_state": "case inr\nR : Type u\nA : Type v\ninst✝² : Semifield R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : R\na : Aˣ\nhr : r ≠ 0\n⊢ r⁻¹ ∈ spectrum R ↑a ↔ r ∈ spectrum R ↑a⁻¹",
"after_state": "case inr.intro\nR : Type u\nA : Type v\ninst✝² : Semifield R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : Aˣ\nr : Rˣ\nhr : ↑r ≠ 0\n⊢ (↑r)⁻¹ ∈ spectrum R ↑a ↔ ↑r ∈ spectrum R ↑a⁻¹"
},
{
"line": "simp [inv_mem_iff]",
"before_state": "case inr.intro\nR : Type u\nA : Type v\ninst✝² : Semifield R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : Aˣ\nr : Rˣ\nhr : ↑r ≠ 0\n⊢ (↑r)⁻¹ ∈ spectrum R ↑a ↔ ↑r ∈ spectrum R ↑a⁻¹",
"after_state": "No Goals!"
}
] |
lemma inv₀_mem_inv_iff {r : R} {a : Aˣ} :
r⁻¹ ∈ spectrum R (↑a⁻¹ : A) ↔ r ∈ spectrum R (a : A) := by
simp
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Spectrum/Basic.lean
|
{
"open": [
"Set",
"scoped Pointwise"
],
"variables": [
"(R : Type u) {A : Type v}",
"[CommSemiring R] [Ring A] [Algebra R A]",
"{R}",
"{R : Type u} {A : Type v}",
"[CommSemiring R] [Ring A] [Algebra R A]",
"(R)",
"{R}",
"[InvolutiveStar R] [StarRing A] [StarModule R A]",
"{R : Type u} {A : Type v}",
"[CommRing R] [Ring A] [Algebra R A]",
"{R : Type u} {A : Type v} [Semifield R] [Ring A] [Algebra R A]"
]
}
|
[
{
"line": "simp",
"before_state": "R : Type u\nA : Type v\ninst✝² : Semifield R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : R\na : Aˣ\n⊢ r⁻¹ ∈ spectrum R ↑a⁻¹ ↔ r ∈ spectrum R ↑a",
"after_state": "No Goals!"
}
] |
theorem mem_resolventSet_apply (φ : F) {a : A} {r : R} (h : r ∈ resolventSet R a) :
r ∈ resolventSet R ((φ : A → B) a) := by
simpa only [map_sub,AlgHomClass.commutes] using h.map φ
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Spectrum/Basic.lean
|
{
"open": [
"Set",
"scoped Pointwise"
],
"variables": [
"(R : Type u) {A : Type v}",
"[CommSemiring R] [Ring A] [Algebra R A]",
"{R}",
"{R : Type u} {A : Type v}",
"[CommSemiring R] [Ring A] [Algebra R A]",
"(R)",
"{R}",
"[InvolutiveStar R] [StarRing A] [StarModule R A]",
"{R : Type u} {A : Type v}",
"[CommRing R] [Ring A] [Algebra R A]",
"{R : Type u} {A : Type v} [Semifield R] [Ring A] [Algebra R A]",
"{𝕜 : Type u} {A : Type v}",
"[Field 𝕜] [Ring A] [Algebra 𝕜 A]",
"{F R A B : Type*} [CommSemiring R] [Ring A] [Algebra R A] [Ring B] [Algebra R B]",
"[FunLike F A B] [AlgHomClass F R A B]"
]
}
|
[
{
"line": "simpa only [map_sub, AlgHomClass.commutes] using h.map φ",
"before_state": "F : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra R A\ninst✝³ : Ring B\ninst✝² : Algebra R B\ninst✝¹ : FunLike F A B\ninst✝ : AlgHomClass F R A B\nφ : F\na : A\nr : R\nh : r ∈ resolventSet R a\n⊢ r ∈ resolventSet R (φ a)",
"after_state": "No Goals!"
}
] |
lemma inv_add_add_mul_eq_zero (u : (PreQuasiregular R)ˣ) :
u⁻¹.val.val + u.val.val + u.val.val * u⁻¹.val.val = 0 := by
simpa [-Units.mul_inv] using congr($(u.mul_inv).val)
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Spectrum/Quasispectrum.lean
|
{
"open": [],
"variables": [
"{R : Type*} [NonUnitalSemiring R]"
]
}
|
[
{
"line": "<failed to pretty print>",
"before_state": "R : Type u_1\ninst✝ : NonUnitalSemiring R\nu : (PreQuasiregular R)ˣ\n⊢ (↑u⁻¹).val + (↑u).val + (↑u).val * (↑u⁻¹).val = 0",
"after_state": "No Goals!"
}
] |
lemma add_inv_add_mul_eq_zero (u : (PreQuasiregular R)ˣ) :
u.val.val + u⁻¹.val.val + u⁻¹.val.val * u.val.val = 0 := by
simpa [-Units.inv_mul] using congr($(u.inv_mul).val)
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Spectrum/Quasispectrum.lean
|
{
"open": [],
"variables": [
"{R : Type*} [NonUnitalSemiring R]"
]
}
|
[
{
"line": "<failed to pretty print>",
"before_state": "R : Type u_1\ninst✝ : NonUnitalSemiring R\nu : (PreQuasiregular R)ˣ\n⊢ (↑u).val + (↑u⁻¹).val + (↑u⁻¹).val * (↑u).val = 0",
"after_state": "No Goals!"
}
] |
lemma zero_mem_spectrum_inr (R S : Type*) {A : Type*} [CommSemiring R]
[CommRing S] [Nontrivial S] [NonUnitalRing A] [Algebra R S] [Module S A] [IsScalarTower S A A]
[SMulCommClass S A A] [Module R A] [IsScalarTower R S A] (a : A) :
0 ∈ spectrum R (a : Unitization S A) := by
rw [spectrum.zero_mem_iff]
rintro ⟨u, hu⟩
simpa [-Units.mul_inv, hu] using congr($(u.mul_inv).fst)
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Spectrum/Quasispectrum.lean
|
{
"open": [
"PreQuasiregular",
"PreQuasiregular"
],
"variables": [
"{R : Type*} [NonUnitalSemiring R]",
"{R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A]",
"(R A) in",
"(R) in",
"{R : Type*} [NonUnitalSemiring R]",
"(R : Type*) {A : Type*} [CommSemiring R] [NonUnitalRing A]",
"{R} in",
"{R}",
"[IsScalarTower R A A] [SMulCommClass R A A]"
]
}
|
[
{
"line": "rw [spectrum.zero_mem_iff]",
"before_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\n⊢ 0 ∈ spectrum R ↑a",
"after_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\n⊢ ¬IsUnit ↑a"
},
{
"line": "rewrite [spectrum.zero_mem_iff]",
"before_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\n⊢ 0 ∈ spectrum R ↑a",
"after_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\n⊢ ¬IsUnit ↑a"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\n⊢ ¬IsUnit ↑a",
"after_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\n⊢ ¬IsUnit ↑a"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\n⊢ ¬IsUnit ↑a",
"after_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\n⊢ ¬IsUnit ↑a"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\n⊢ ¬IsUnit ↑a",
"after_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\n⊢ ¬IsUnit ↑a"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\n⊢ ¬IsUnit ↑a",
"after_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\n⊢ ¬IsUnit ↑a"
},
{
"line": "rfl",
"before_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\n⊢ ¬IsUnit ↑a",
"after_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\n⊢ ¬IsUnit ↑a"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\n⊢ ¬IsUnit ↑a",
"after_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\n⊢ ¬IsUnit ↑a"
},
{
"line": "skip",
"before_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\n⊢ ¬IsUnit ↑a",
"after_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\n⊢ ¬IsUnit ↑a"
},
{
"line": "rintro ⟨u, hu⟩",
"before_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\n⊢ ¬IsUnit ↑a",
"after_state": "case intro\nR : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\nu : (Unitization S A)ˣ\nhu : ↑u = ↑a\n⊢ False"
},
{
"line": "<failed to pretty print>",
"before_state": "case intro\nR : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\nu : (Unitization S A)ˣ\nhu : ↑u = ↑a\n⊢ False",
"after_state": "No Goals!"
}
] |
lemma iff_spectrum_nonneg {𝕜 A : Type*} [Semifield 𝕜] [LinearOrder 𝕜] [Ring A] [PartialOrder A]
[Algebra 𝕜 A] : NonnegSpectrumClass 𝕜 A ↔ ∀ a : A, 0 ≤ a → ∀ x ∈ spectrum 𝕜 a, 0 ≤ x := by
simp [show NonnegSpectrumClass 𝕜 A ↔ _ from ⟨fun ⟨h⟩ ↦ h, (⟨·⟩)⟩,
quasispectrum_eq_spectrum_union_zero]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Spectrum/Quasispectrum.lean
|
{
"open": [
"PreQuasiregular",
"PreQuasiregular"
],
"variables": [
"{R : Type*} [NonUnitalSemiring R]",
"{R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A]",
"(R A) in",
"(R) in",
"{R : Type*} [NonUnitalSemiring R]",
"(R : Type*) {A : Type*} [CommSemiring R] [NonUnitalRing A]",
"{R} in",
"{R}",
"[IsScalarTower R A A] [SMulCommClass R A A]"
]
}
|
[
{
"line": "simp [show NonnegSpectrumClass 𝕜 A ↔ _ from ⟨fun ⟨h⟩ ↦ h, (⟨·⟩)⟩, quasispectrum_eq_spectrum_union_zero]",
"before_state": "𝕜 : Type u_6\nA : Type u_7\ninst✝⁴ : Semifield 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : Ring A\ninst✝¹ : PartialOrder A\ninst✝ : Algebra 𝕜 A\n⊢ NonnegSpectrumClass 𝕜 A ↔ ∀ (a : A), 0 ≤ a → ∀ x ∈ spectrum 𝕜 a, 0 ≤ x",
"after_state": "No Goals!"
}
] |
theorem of_subset_range_algebraMap (hf : f.LeftInverse (algebraMap R S))
(h : quasispectrum S a ⊆ Set.range (algebraMap R S)) : QuasispectrumRestricts a f where
rightInvOn := fun s hs => by obtain ⟨r, rfl⟩ := h hs; rw [hf r]
left_inv := hf
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Spectrum/Quasispectrum.lean
|
{
"open": [
"PreQuasiregular",
"PreQuasiregular"
],
"variables": [
"{R : Type*} [NonUnitalSemiring R]",
"{R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A]",
"(R A) in",
"(R) in",
"{R : Type*} [NonUnitalSemiring R]",
"(R : Type*) {A : Type*} [CommSemiring R] [NonUnitalRing A]",
"{R} in",
"{R}",
"[IsScalarTower R A A] [SMulCommClass R A A]",
"{R S A : Type*} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A]",
"[Algebra R S] {a : A} {f : S → R}"
]
}
|
[
{
"line": "obtain ⟨r, rfl⟩ := h hs",
"before_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁵ : Semifield R\ninst✝⁴ : Field S\ninst✝³ : NonUnitalRing A\ninst✝² : Module R A\ninst✝¹ : Module S A\ninst✝ : Algebra R S\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : quasispectrum S a ⊆ Set.range ⇑(algebraMap R S)\ns : S\nhs : s ∈ quasispectrum S a\n⊢ (algebraMap R S) (f s) = s",
"after_state": "case intro\nR : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁵ : Semifield R\ninst✝⁴ : Field S\ninst✝³ : NonUnitalRing A\ninst✝² : Module R A\ninst✝¹ : Module S A\ninst✝ : Algebra R S\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : quasispectrum S a ⊆ Set.range ⇑(algebraMap R S)\nr : R\nhs : (algebraMap R S) r ∈ quasispectrum S a\n⊢ (algebraMap R S) (f ((algebraMap R S) r)) = (algebraMap R S) r"
},
{
"line": "rw [hf r]",
"before_state": "case intro\nR : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁵ : Semifield R\ninst✝⁴ : Field S\ninst✝³ : NonUnitalRing A\ninst✝² : Module R A\ninst✝¹ : Module S A\ninst✝ : Algebra R S\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : quasispectrum S a ⊆ Set.range ⇑(algebraMap R S)\nr : R\nhs : (algebraMap R S) r ∈ quasispectrum S a\n⊢ (algebraMap R S) (f ((algebraMap R S) r)) = (algebraMap R S) r",
"after_state": "No Goals!"
},
{
"line": "rewrite [hf r]",
"before_state": "case intro\nR : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁵ : Semifield R\ninst✝⁴ : Field S\ninst✝³ : NonUnitalRing A\ninst✝² : Module R A\ninst✝¹ : Module S A\ninst✝ : Algebra R S\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : quasispectrum S a ⊆ Set.range ⇑(algebraMap R S)\nr : R\nhs : (algebraMap R S) r ∈ quasispectrum S a\n⊢ (algebraMap R S) (f ((algebraMap R S) r)) = (algebraMap R S) r",
"after_state": "case intro\nR : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁵ : Semifield R\ninst✝⁴ : Field S\ninst✝³ : NonUnitalRing A\ninst✝² : Module R A\ninst✝¹ : Module S A\ninst✝ : Algebra R S\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : quasispectrum S a ⊆ Set.range ⇑(algebraMap R S)\nr : R\nhs : (algebraMap R S) r ∈ quasispectrum S a\n⊢ (algebraMap R S) r = (algebraMap R S) r"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case intro\nR : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁵ : Semifield R\ninst✝⁴ : Field S\ninst✝³ : NonUnitalRing A\ninst✝² : Module R A\ninst✝¹ : Module S A\ninst✝ : Algebra R S\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : quasispectrum S a ⊆ Set.range ⇑(algebraMap R S)\nr : R\nhs : (algebraMap R S) r ∈ quasispectrum S a\n⊢ (algebraMap R S) r = (algebraMap R S) r",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case intro\nR : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁵ : Semifield R\ninst✝⁴ : Field S\ninst✝³ : NonUnitalRing A\ninst✝² : Module R A\ninst✝¹ : Module S A\ninst✝ : Algebra R S\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : quasispectrum S a ⊆ Set.range ⇑(algebraMap R S)\nr : R\nhs : (algebraMap R S) r ∈ quasispectrum S a\n⊢ (algebraMap R S) r = (algebraMap R S) r",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case intro\nR : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁵ : Semifield R\ninst✝⁴ : Field S\ninst✝³ : NonUnitalRing A\ninst✝² : Module R A\ninst✝¹ : Module S A\ninst✝ : Algebra R S\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : quasispectrum S a ⊆ Set.range ⇑(algebraMap R S)\nr : R\nhs : (algebraMap R S) r ∈ quasispectrum S a\n⊢ (algebraMap R S) r = (algebraMap R S) r",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "case intro\nR : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁵ : Semifield R\ninst✝⁴ : Field S\ninst✝³ : NonUnitalRing A\ninst✝² : Module R A\ninst✝¹ : Module S A\ninst✝ : Algebra R S\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : quasispectrum S a ⊆ Set.range ⇑(algebraMap R S)\nr : R\nhs : (algebraMap R S) r ∈ quasispectrum S a\n⊢ (algebraMap R S) r = (algebraMap R S) r",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "case intro\nR : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁵ : Semifield R\ninst✝⁴ : Field S\ninst✝³ : NonUnitalRing A\ninst✝² : Module R A\ninst✝¹ : Module S A\ninst✝ : Algebra R S\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : quasispectrum S a ⊆ Set.range ⇑(algebraMap R S)\nr : R\nhs : (algebraMap R S) r ∈ quasispectrum S a\n⊢ (algebraMap R S) r = (algebraMap R S) r",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "case intro\nR : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁵ : Semifield R\ninst✝⁴ : Field S\ninst✝³ : NonUnitalRing A\ninst✝² : Module R A\ninst✝¹ : Module S A\ninst✝ : Algebra R S\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : quasispectrum S a ⊆ Set.range ⇑(algebraMap R S)\nr : R\nhs : (algebraMap R S) r ∈ quasispectrum S a\n⊢ (algebraMap R S) r = (algebraMap R S) r",
"after_state": "No Goals!"
}
] |
lemma mul_comm_iff {f : S → R} {a b : A} :
QuasispectrumRestricts (a * b) f ↔ QuasispectrumRestricts (b * a) f := by
simp only [quasispectrumRestricts_iff]
simp only [quasispectrum.mul_comm]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Spectrum/Quasispectrum.lean
|
{
"open": [
"PreQuasiregular",
"PreQuasiregular"
],
"variables": [
"{R : Type*} [NonUnitalSemiring R]",
"{R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A]",
"(R A) in",
"(R) in",
"{R : Type*} [NonUnitalSemiring R]",
"(R : Type*) {A : Type*} [CommSemiring R] [NonUnitalRing A]",
"{R} in",
"{R}",
"[IsScalarTower R A A] [SMulCommClass R A A]",
"{R S A : Type*} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A]",
"[Algebra R S] {a : A} {f : S → R}",
"[IsScalarTower S A A] [SMulCommClass S A A]"
]
}
|
[
{
"line": "simp only [quasispectrumRestricts_iff]",
"before_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁷ : Semifield R\ninst✝⁶ : Field S\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : Module R A\ninst✝³ : Module S A\ninst✝² : Algebra R S\ninst✝¹ : IsScalarTower S A A\ninst✝ : SMulCommClass S A A\nf : S → R\na b : A\n⊢ QuasispectrumRestricts (a * b) f ↔ QuasispectrumRestricts (b * a) f",
"after_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁷ : Semifield R\ninst✝⁶ : Field S\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : Module R A\ninst✝³ : Module S A\ninst✝² : Algebra R S\ninst✝¹ : IsScalarTower S A A\ninst✝ : SMulCommClass S A A\nf : S → R\na b : A\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S (a * b)) ∧ Function.LeftInverse f ⇑(algebraMap R S) ↔\n Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S (b * a)) ∧ Function.LeftInverse f ⇑(algebraMap R S)"
},
{
"line": "simp only [quasispectrum.mul_comm]",
"before_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁷ : Semifield R\ninst✝⁶ : Field S\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : Module R A\ninst✝³ : Module S A\ninst✝² : Algebra R S\ninst✝¹ : IsScalarTower S A A\ninst✝ : SMulCommClass S A A\nf : S → R\na b : A\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S (a * b)) ∧ Function.LeftInverse f ⇑(algebraMap R S) ↔\n Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S (b * a)) ∧ Function.LeftInverse f ⇑(algebraMap R S)",
"after_state": "No Goals!"
}
] |
theorem algebraMap_image (h : QuasispectrumRestricts a f) :
algebraMap R S '' quasispectrum R a = quasispectrum S a := by
refine Set.eq_of_subset_of_subset ?_ fun s hs => ⟨f s, ?_⟩
· simpa only [quasispectrum.preimage_algebraMap] using
(quasispectrum S a).image_preimage_subset (algebraMap R S)
exact ⟨quasispectrum.of_algebraMap_mem S ((h.rightInvOn hs).symm ▸ hs), h.rightInvOn hs⟩
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Spectrum/Quasispectrum.lean
|
{
"open": [
"PreQuasiregular",
"PreQuasiregular"
],
"variables": [
"{R : Type*} [NonUnitalSemiring R]",
"{R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A]",
"(R A) in",
"(R) in",
"{R : Type*} [NonUnitalSemiring R]",
"(R : Type*) {A : Type*} [CommSemiring R] [NonUnitalRing A]",
"{R} in",
"{R}",
"[IsScalarTower R A A] [SMulCommClass R A A]",
"{R S A : Type*} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A]",
"[Algebra R S] {a : A} {f : S → R}",
"[IsScalarTower S A A] [SMulCommClass S A A]",
"[IsScalarTower R S A]"
]
}
|
[
{
"line": "refine Set.eq_of_subset_of_subset ?_ fun s hs => ⟨f s, ?_⟩",
"before_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁸ : Semifield R\ninst✝⁷ : Field S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module R A\ninst✝⁴ : Module S A\ninst✝³ : Algebra R S\na : A\nf : S → R\ninst✝² : IsScalarTower S A A\ninst✝¹ : SMulCommClass S A A\ninst✝ : IsScalarTower R S A\nh : QuasispectrumRestricts a f\n⊢ ⇑(algebraMap R S) '' quasispectrum R a = quasispectrum S a",
"after_state": "case refine_1\nR : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁸ : Semifield R\ninst✝⁷ : Field S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module R A\ninst✝⁴ : Module S A\ninst✝³ : Algebra R S\na : A\nf : S → R\ninst✝² : IsScalarTower S A A\ninst✝¹ : SMulCommClass S A A\ninst✝ : IsScalarTower R S A\nh : QuasispectrumRestricts a f\n⊢ ⇑(algebraMap R S) '' quasispectrum R a ⊆ quasispectrum S a\n---\ncase refine_2\nR : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁸ : Semifield R\ninst✝⁷ : Field S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module R A\ninst✝⁴ : Module S A\ninst✝³ : Algebra R S\na : A\nf : S → R\ninst✝² : IsScalarTower S A A\ninst✝¹ : SMulCommClass S A A\ninst✝ : IsScalarTower R S A\nh : QuasispectrumRestricts a f\ns : S\nhs : s ∈ quasispectrum S a\n⊢ f s ∈ quasispectrum R a ∧ (algebraMap R S) (f s) = s"
},
{
"line": "simpa only [quasispectrum.preimage_algebraMap] using (quasispectrum S a).image_preimage_subset (algebraMap R S)",
"before_state": "case refine_1\nR : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁸ : Semifield R\ninst✝⁷ : Field S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module R A\ninst✝⁴ : Module S A\ninst✝³ : Algebra R S\na : A\nf : S → R\ninst✝² : IsScalarTower S A A\ninst✝¹ : SMulCommClass S A A\ninst✝ : IsScalarTower R S A\nh : QuasispectrumRestricts a f\n⊢ ⇑(algebraMap R S) '' quasispectrum R a ⊆ quasispectrum S a",
"after_state": "No Goals!"
},
{
"line": "exact ⟨quasispectrum.of_algebraMap_mem S ((h.rightInvOn hs).symm ▸ hs), h.rightInvOn hs⟩",
"before_state": "case refine_2\nR : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁸ : Semifield R\ninst✝⁷ : Field S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module R A\ninst✝⁴ : Module S A\ninst✝³ : Algebra R S\na : A\nf : S → R\ninst✝² : IsScalarTower S A A\ninst✝¹ : SMulCommClass S A A\ninst✝ : IsScalarTower R S A\nh : QuasispectrumRestricts a f\ns : S\nhs : s ∈ quasispectrum S a\n⊢ f s ∈ quasispectrum R a ∧ (algebraMap R S) (f s) = s",
"after_state": "No Goals!"
}
] |
theorem image (h : QuasispectrumRestricts a f) : f '' quasispectrum S a = quasispectrum R a := by
simp only [← h.algebraMap_image]
simp only [Set.image_image]
simp only [h.left_inv _]
simp only [Set.image_id']
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Spectrum/Quasispectrum.lean
|
{
"open": [
"PreQuasiregular",
"PreQuasiregular"
],
"variables": [
"{R : Type*} [NonUnitalSemiring R]",
"{R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A]",
"(R A) in",
"(R) in",
"{R : Type*} [NonUnitalSemiring R]",
"(R : Type*) {A : Type*} [CommSemiring R] [NonUnitalRing A]",
"{R} in",
"{R}",
"[IsScalarTower R A A] [SMulCommClass R A A]",
"{R S A : Type*} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A]",
"[Algebra R S] {a : A} {f : S → R}",
"[IsScalarTower S A A] [SMulCommClass S A A]",
"[IsScalarTower R S A]"
]
}
|
[
{
"line": "simp only [← h.algebraMap_image]",
"before_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁸ : Semifield R\ninst✝⁷ : Field S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module R A\ninst✝⁴ : Module S A\ninst✝³ : Algebra R S\na : A\nf : S → R\ninst✝² : IsScalarTower S A A\ninst✝¹ : SMulCommClass S A A\ninst✝ : IsScalarTower R S A\nh : QuasispectrumRestricts a f\n⊢ f '' quasispectrum S a = quasispectrum R a",
"after_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁸ : Semifield R\ninst✝⁷ : Field S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module R A\ninst✝⁴ : Module S A\ninst✝³ : Algebra R S\na : A\nf : S → R\ninst✝² : IsScalarTower S A A\ninst✝¹ : SMulCommClass S A A\ninst✝ : IsScalarTower R S A\nh : QuasispectrumRestricts a f\n⊢ f '' (⇑(algebraMap R S) '' quasispectrum R a) = quasispectrum R a"
},
{
"line": "simp only [Set.image_image]",
"before_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁸ : Semifield R\ninst✝⁷ : Field S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module R A\ninst✝⁴ : Module S A\ninst✝³ : Algebra R S\na : A\nf : S → R\ninst✝² : IsScalarTower S A A\ninst✝¹ : SMulCommClass S A A\ninst✝ : IsScalarTower R S A\nh : QuasispectrumRestricts a f\n⊢ f '' (⇑(algebraMap R S) '' quasispectrum R a) = quasispectrum R a",
"after_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁸ : Semifield R\ninst✝⁷ : Field S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module R A\ninst✝⁴ : Module S A\ninst✝³ : Algebra R S\na : A\nf : S → R\ninst✝² : IsScalarTower S A A\ninst✝¹ : SMulCommClass S A A\ninst✝ : IsScalarTower R S A\nh : QuasispectrumRestricts a f\n⊢ (fun x => f ((algebraMap R S) x)) '' quasispectrum R a = quasispectrum R a"
},
{
"line": "simp only [h.left_inv _]",
"before_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁸ : Semifield R\ninst✝⁷ : Field S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module R A\ninst✝⁴ : Module S A\ninst✝³ : Algebra R S\na : A\nf : S → R\ninst✝² : IsScalarTower S A A\ninst✝¹ : SMulCommClass S A A\ninst✝ : IsScalarTower R S A\nh : QuasispectrumRestricts a f\n⊢ (fun x => f ((algebraMap R S) x)) '' quasispectrum R a = quasispectrum R a",
"after_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁸ : Semifield R\ninst✝⁷ : Field S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module R A\ninst✝⁴ : Module S A\ninst✝³ : Algebra R S\na : A\nf : S → R\ninst✝² : IsScalarTower S A A\ninst✝¹ : SMulCommClass S A A\ninst✝ : IsScalarTower R S A\nh : QuasispectrumRestricts a f\n⊢ (fun a => a) '' quasispectrum R a = quasispectrum R a"
},
{
"line": "simp only [Set.image_id']",
"before_state": "R : Type u_6\nS : Type u_7\nA : Type u_8\ninst✝⁸ : Semifield R\ninst✝⁷ : Field S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module R A\ninst✝⁴ : Module S A\ninst✝³ : Algebra R S\na : A\nf : S → R\ninst✝² : IsScalarTower S A A\ninst✝¹ : SMulCommClass S A A\ninst✝ : IsScalarTower R S A\nh : QuasispectrumRestricts a f\n⊢ (fun a => a) '' quasispectrum R a = quasispectrum R a",
"after_state": "No Goals!"
}
] |
theorem of_rightInvOn (h₁ : Function.LeftInverse f (algebraMap R S))
(h₂ : (spectrum S a).RightInvOn f (algebraMap R S)) : SpectrumRestricts a f where
rightInvOn x hx := by
obtain (rfl | hx) := mem_quasispectrum_iff.mp hx
· simpa using h₁ 0
· exact h₂ hx
left_inv := h₁
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Spectrum/Quasispectrum.lean
|
{
"open": [
"PreQuasiregular",
"PreQuasiregular"
],
"variables": [
"{R : Type*} [NonUnitalSemiring R]",
"{R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A]",
"(R A) in",
"(R) in",
"{R : Type*} [NonUnitalSemiring R]",
"(R : Type*) {A : Type*} [CommSemiring R] [NonUnitalRing A]",
"{R} in",
"{R}",
"[IsScalarTower R A A] [SMulCommClass R A A]",
"{R S A : Type*} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A]",
"[Algebra R S] {a : A} {f : S → R}",
"[IsScalarTower S A A] [SMulCommClass S A A]",
"[IsScalarTower R S A]",
"{R S A : Type*} [Semifield R] [Semifield S] [Ring A]",
"[Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R}"
]
}
|
[
{
"line": "obtain (rfl | hx) := mem_quasispectrum_iff.mp hx",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nh₁ : Function.LeftInverse f ⇑(algebraMap R S)\nh₂ : Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a)\nx : S\nhx : x ∈ quasispectrum S a\n⊢ (algebraMap R S) (f x) = x",
"after_state": "case inl\nR : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nh₁ : Function.LeftInverse f ⇑(algebraMap R S)\nh₂ : Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a)\nhx : 0 ∈ quasispectrum S a\n⊢ (algebraMap R S) (f 0) = 0\n---\ncase inr\nR : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nh₁ : Function.LeftInverse f ⇑(algebraMap R S)\nh₂ : Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a)\nx : S\nhx✝ : x ∈ quasispectrum S a\nhx : x ∈ spectrum S a\n⊢ (algebraMap R S) (f x) = x"
},
{
"line": "simpa using h₁ 0",
"before_state": "case inl\nR : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nh₁ : Function.LeftInverse f ⇑(algebraMap R S)\nh₂ : Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a)\nhx : 0 ∈ quasispectrum S a\n⊢ (algebraMap R S) (f 0) = 0",
"after_state": "No Goals!"
},
{
"line": "exact h₂ hx",
"before_state": "case inr\nR : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nh₁ : Function.LeftInverse f ⇑(algebraMap R S)\nh₂ : Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a)\nx : S\nhx✝ : x ∈ quasispectrum S a\nhx : x ∈ spectrum S a\n⊢ (algebraMap R S) (f x) = x",
"after_state": "No Goals!"
}
] |
theorem of_subset_range_algebraMap (hf : f.LeftInverse (algebraMap R S))
(h : spectrum S a ⊆ Set.range (algebraMap R S)) : SpectrumRestricts a f where
rightInvOn := fun s hs => by
rw [mem_quasispectrum_iff] at hs
obtain (rfl | hs) := hs
· simpa using hf 0
· obtain ⟨r, rfl⟩ := h hs
rw [hf r]
left_inv := hf
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Spectrum/Quasispectrum.lean
|
{
"open": [
"PreQuasiregular",
"PreQuasiregular"
],
"variables": [
"{R : Type*} [NonUnitalSemiring R]",
"{R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A]",
"(R A) in",
"(R) in",
"{R : Type*} [NonUnitalSemiring R]",
"(R : Type*) {A : Type*} [CommSemiring R] [NonUnitalRing A]",
"{R} in",
"{R}",
"[IsScalarTower R A A] [SMulCommClass R A A]",
"{R S A : Type*} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A]",
"[Algebra R S] {a : A} {f : S → R}",
"[IsScalarTower S A A] [SMulCommClass S A A]",
"[IsScalarTower R S A]",
"{R S A : Type*} [Semifield R] [Semifield S] [Ring A]",
"[Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R}"
]
}
|
[
{
"line": "rw [mem_quasispectrum_iff] at hs",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\ns : S\nhs : s ∈ quasispectrum S a\n⊢ (algebraMap R S) (f s) = s",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\ns : S\nhs : s = 0 ∨ s ∈ spectrum S a\n⊢ (algebraMap R S) (f s) = s"
},
{
"line": "rewrite [mem_quasispectrum_iff] at hs",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\ns : S\nhs : s ∈ quasispectrum S a\n⊢ (algebraMap R S) (f s) = s",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\ns : S\nhs : s = 0 ∨ s ∈ spectrum S a\n⊢ (algebraMap R S) (f s) = s"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\ns : S\nhs : s = 0 ∨ s ∈ spectrum S a\n⊢ (algebraMap R S) (f s) = s",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\ns : S\nhs : s = 0 ∨ s ∈ spectrum S a\n⊢ (algebraMap R S) (f s) = s"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\ns : S\nhs : s = 0 ∨ s ∈ spectrum S a\n⊢ (algebraMap R S) (f s) = s",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\ns : S\nhs : s = 0 ∨ s ∈ spectrum S a\n⊢ (algebraMap R S) (f s) = s"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\ns : S\nhs : s = 0 ∨ s ∈ spectrum S a\n⊢ (algebraMap R S) (f s) = s",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\ns : S\nhs : s = 0 ∨ s ∈ spectrum S a\n⊢ (algebraMap R S) (f s) = s"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\ns : S\nhs : s = 0 ∨ s ∈ spectrum S a\n⊢ (algebraMap R S) (f s) = s",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\ns : S\nhs : s = 0 ∨ s ∈ spectrum S a\n⊢ (algebraMap R S) (f s) = s"
},
{
"line": "rfl",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\ns : S\nhs : s = 0 ∨ s ∈ spectrum S a\n⊢ (algebraMap R S) (f s) = s",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\ns : S\nhs : s = 0 ∨ s ∈ spectrum S a\n⊢ (algebraMap R S) (f s) = s"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\ns : S\nhs : s = 0 ∨ s ∈ spectrum S a\n⊢ (algebraMap R S) (f s) = s",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\ns : S\nhs : s = 0 ∨ s ∈ spectrum S a\n⊢ (algebraMap R S) (f s) = s"
},
{
"line": "skip",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\ns : S\nhs : s = 0 ∨ s ∈ spectrum S a\n⊢ (algebraMap R S) (f s) = s",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\ns : S\nhs : s = 0 ∨ s ∈ spectrum S a\n⊢ (algebraMap R S) (f s) = s"
},
{
"line": "obtain (rfl | hs) := hs",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\ns : S\nhs : s = 0 ∨ s ∈ spectrum S a\n⊢ (algebraMap R S) (f s) = s",
"after_state": "case inl\nR : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\n⊢ (algebraMap R S) (f 0) = 0\n---\ncase inr\nR : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\ns : S\nhs : s ∈ spectrum S a\n⊢ (algebraMap R S) (f s) = s"
},
{
"line": "simpa using hf 0",
"before_state": "case inl\nR : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\n⊢ (algebraMap R S) (f 0) = 0",
"after_state": "No Goals!"
},
{
"line": "obtain ⟨r, rfl⟩ := h hs",
"before_state": "case inr\nR : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\ns : S\nhs : s ∈ spectrum S a\n⊢ (algebraMap R S) (f s) = s",
"after_state": "case inr.intro\nR : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\nr : R\nhs : (algebraMap R S) r ∈ spectrum S a\n⊢ (algebraMap R S) (f ((algebraMap R S) r)) = (algebraMap R S) r"
},
{
"line": "rw [hf r]",
"before_state": "case inr.intro\nR : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\nr : R\nhs : (algebraMap R S) r ∈ spectrum S a\n⊢ (algebraMap R S) (f ((algebraMap R S) r)) = (algebraMap R S) r",
"after_state": "No Goals!"
},
{
"line": "rewrite [hf r]",
"before_state": "case inr.intro\nR : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\nr : R\nhs : (algebraMap R S) r ∈ spectrum S a\n⊢ (algebraMap R S) (f ((algebraMap R S) r)) = (algebraMap R S) r",
"after_state": "case inr.intro\nR : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\nr : R\nhs : (algebraMap R S) r ∈ spectrum S a\n⊢ (algebraMap R S) r = (algebraMap R S) r"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case inr.intro\nR : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\nr : R\nhs : (algebraMap R S) r ∈ spectrum S a\n⊢ (algebraMap R S) r = (algebraMap R S) r",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case inr.intro\nR : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\nr : R\nhs : (algebraMap R S) r ∈ spectrum S a\n⊢ (algebraMap R S) r = (algebraMap R S) r",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case inr.intro\nR : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\nr : R\nhs : (algebraMap R S) r ∈ spectrum S a\n⊢ (algebraMap R S) r = (algebraMap R S) r",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "case inr.intro\nR : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\nr : R\nhs : (algebraMap R S) r ∈ spectrum S a\n⊢ (algebraMap R S) r = (algebraMap R S) r",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "case inr.intro\nR : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\nr : R\nhs : (algebraMap R S) r ∈ spectrum S a\n⊢ (algebraMap R S) r = (algebraMap R S) r",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "case inr.intro\nR : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\nr : R\nhs : (algebraMap R S) r ∈ spectrum S a\n⊢ (algebraMap R S) r = (algebraMap R S) r",
"after_state": "No Goals!"
}
] |
lemma of_spectrum_eq {a b : A} {f : S → R} (ha : SpectrumRestricts a f)
(h : spectrum S a = spectrum S b) : SpectrumRestricts b f where
rightInvOn := by
rw [quasispectrum_eq_spectrum_union_zero]
rw [← h]
rw [← quasispectrum_eq_spectrum_union_zero]
exact QuasispectrumRestricts.rightInvOn ha
left_inv := ha.left_inv
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Spectrum/Quasispectrum.lean
|
{
"open": [
"PreQuasiregular",
"PreQuasiregular"
],
"variables": [
"{R : Type*} [NonUnitalSemiring R]",
"{R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A]",
"(R A) in",
"(R) in",
"{R : Type*} [NonUnitalSemiring R]",
"(R : Type*) {A : Type*} [CommSemiring R] [NonUnitalRing A]",
"{R} in",
"{R}",
"[IsScalarTower R A A] [SMulCommClass R A A]",
"{R S A : Type*} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A]",
"[Algebra R S] {a : A} {f : S → R}",
"[IsScalarTower S A A] [SMulCommClass S A A]",
"[IsScalarTower R S A]",
"{R S A : Type*} [Semifield R] [Semifield S] [Ring A]",
"[Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R}"
]
}
|
[
{
"line": "rw [quasispectrum_eq_spectrum_union_zero]",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S b)",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S b ∪ {0})"
},
{
"line": "rewrite [quasispectrum_eq_spectrum_union_zero]",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S b)",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S b ∪ {0})"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S b ∪ {0})",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S b ∪ {0})"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S b ∪ {0})",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S b ∪ {0})"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S b ∪ {0})",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S b ∪ {0})"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S b ∪ {0})",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S b ∪ {0})"
},
{
"line": "rfl",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S b ∪ {0})",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S b ∪ {0})"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S b ∪ {0})",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S b ∪ {0})"
},
{
"line": "skip",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S b ∪ {0})",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S b ∪ {0})"
},
{
"line": "rw [← h]",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S b ∪ {0})",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a ∪ {0})"
},
{
"line": "rewrite [← h]",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S b ∪ {0})",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a ∪ {0})"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a ∪ {0})",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a ∪ {0})"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a ∪ {0})",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a ∪ {0})"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a ∪ {0})",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a ∪ {0})"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a ∪ {0})",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a ∪ {0})"
},
{
"line": "rfl",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a ∪ {0})",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a ∪ {0})"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a ∪ {0})",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a ∪ {0})"
},
{
"line": "skip",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a ∪ {0})",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a ∪ {0})"
},
{
"line": "rw [← quasispectrum_eq_spectrum_union_zero]",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a ∪ {0})",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S a)"
},
{
"line": "rewrite [← quasispectrum_eq_spectrum_union_zero]",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a ∪ {0})",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S a)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S a)",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S a)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S a)",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S a)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S a)",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S a)"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S a)",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S a)"
},
{
"line": "rfl",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S a)",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S a)"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S a)",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S a)"
},
{
"line": "skip",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S a)",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S a)"
},
{
"line": "exact QuasispectrumRestricts.rightInvOn ha",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na b : A\nf : S → R\nha : SpectrumRestricts a f\nh : spectrum S a = spectrum S b\n⊢ Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S a)",
"after_state": "No Goals!"
}
] |
theorem algebraMap_image (h : SpectrumRestricts a f) :
algebraMap R S '' spectrum R a = spectrum S a := by
refine Set.eq_of_subset_of_subset ?_ fun s hs => ⟨f s, ?_⟩
· simpa only [spectrum.preimage_algebraMap] using
(spectrum S a).image_preimage_subset (algebraMap R S)
exact ⟨spectrum.of_algebraMap_mem S ((h.rightInvOn hs).symm ▸ hs), h.rightInvOn hs⟩
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Spectrum/Quasispectrum.lean
|
{
"open": [
"PreQuasiregular",
"PreQuasiregular"
],
"variables": [
"{R : Type*} [NonUnitalSemiring R]",
"{R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A]",
"(R A) in",
"(R) in",
"{R : Type*} [NonUnitalSemiring R]",
"(R : Type*) {A : Type*} [CommSemiring R] [NonUnitalRing A]",
"{R} in",
"{R}",
"[IsScalarTower R A A] [SMulCommClass R A A]",
"{R S A : Type*} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A]",
"[Algebra R S] {a : A} {f : S → R}",
"[IsScalarTower S A A] [SMulCommClass S A A]",
"[IsScalarTower R S A]",
"{R S A : Type*} [Semifield R] [Semifield S] [Ring A]",
"[Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R}",
"[IsScalarTower R S A]"
]
}
|
[
{
"line": "refine Set.eq_of_subset_of_subset ?_ fun s hs => ⟨f s, ?_⟩",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁶ : Semifield R\ninst✝⁵ : Semifield S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\na : A\nf : S → R\ninst✝ : IsScalarTower R S A\nh : SpectrumRestricts a f\n⊢ ⇑(algebraMap R S) '' spectrum R a = spectrum S a",
"after_state": "case refine_1\nR : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁶ : Semifield R\ninst✝⁵ : Semifield S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\na : A\nf : S → R\ninst✝ : IsScalarTower R S A\nh : SpectrumRestricts a f\n⊢ ⇑(algebraMap R S) '' spectrum R a ⊆ spectrum S a\n---\ncase refine_2\nR : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁶ : Semifield R\ninst✝⁵ : Semifield S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\na : A\nf : S → R\ninst✝ : IsScalarTower R S A\nh : SpectrumRestricts a f\ns : S\nhs : s ∈ spectrum S a\n⊢ f s ∈ spectrum R a ∧ (algebraMap R S) (f s) = s"
},
{
"line": "simpa only [spectrum.preimage_algebraMap] using (spectrum S a).image_preimage_subset (algebraMap R S)",
"before_state": "case refine_1\nR : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁶ : Semifield R\ninst✝⁵ : Semifield S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\na : A\nf : S → R\ninst✝ : IsScalarTower R S A\nh : SpectrumRestricts a f\n⊢ ⇑(algebraMap R S) '' spectrum R a ⊆ spectrum S a",
"after_state": "No Goals!"
},
{
"line": "exact ⟨spectrum.of_algebraMap_mem S ((h.rightInvOn hs).symm ▸ hs), h.rightInvOn hs⟩",
"before_state": "case refine_2\nR : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁶ : Semifield R\ninst✝⁵ : Semifield S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\na : A\nf : S → R\ninst✝ : IsScalarTower R S A\nh : SpectrumRestricts a f\ns : S\nhs : s ∈ spectrum S a\n⊢ f s ∈ spectrum R a ∧ (algebraMap R S) (f s) = s",
"after_state": "No Goals!"
}
] |
theorem image (h : SpectrumRestricts a f) : f '' spectrum S a = spectrum R a := by
simp only [← h.algebraMap_image]
simp only [Set.image_image]
simp only [h.left_inv _]
simp only [Set.image_id']
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Spectrum/Quasispectrum.lean
|
{
"open": [
"PreQuasiregular",
"PreQuasiregular"
],
"variables": [
"{R : Type*} [NonUnitalSemiring R]",
"{R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A]",
"(R A) in",
"(R) in",
"{R : Type*} [NonUnitalSemiring R]",
"(R : Type*) {A : Type*} [CommSemiring R] [NonUnitalRing A]",
"{R} in",
"{R}",
"[IsScalarTower R A A] [SMulCommClass R A A]",
"{R S A : Type*} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A]",
"[Algebra R S] {a : A} {f : S → R}",
"[IsScalarTower S A A] [SMulCommClass S A A]",
"[IsScalarTower R S A]",
"{R S A : Type*} [Semifield R] [Semifield S] [Ring A]",
"[Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R}",
"[IsScalarTower R S A]"
]
}
|
[
{
"line": "simp only [← h.algebraMap_image]",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁶ : Semifield R\ninst✝⁵ : Semifield S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\na : A\nf : S → R\ninst✝ : IsScalarTower R S A\nh : SpectrumRestricts a f\n⊢ f '' spectrum S a = spectrum R a",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁶ : Semifield R\ninst✝⁵ : Semifield S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\na : A\nf : S → R\ninst✝ : IsScalarTower R S A\nh : SpectrumRestricts a f\n⊢ f '' (⇑(algebraMap R S) '' spectrum R a) = spectrum R a"
},
{
"line": "simp only [Set.image_image]",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁶ : Semifield R\ninst✝⁵ : Semifield S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\na : A\nf : S → R\ninst✝ : IsScalarTower R S A\nh : SpectrumRestricts a f\n⊢ f '' (⇑(algebraMap R S) '' spectrum R a) = spectrum R a",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁶ : Semifield R\ninst✝⁵ : Semifield S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\na : A\nf : S → R\ninst✝ : IsScalarTower R S A\nh : SpectrumRestricts a f\n⊢ (fun x => f ((algebraMap R S) x)) '' spectrum R a = spectrum R a"
},
{
"line": "simp only [h.left_inv _]",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁶ : Semifield R\ninst✝⁵ : Semifield S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\na : A\nf : S → R\ninst✝ : IsScalarTower R S A\nh : SpectrumRestricts a f\n⊢ (fun x => f ((algebraMap R S) x)) '' spectrum R a = spectrum R a",
"after_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁶ : Semifield R\ninst✝⁵ : Semifield S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\na : A\nf : S → R\ninst✝ : IsScalarTower R S A\nh : SpectrumRestricts a f\n⊢ (fun a => a) '' spectrum R a = spectrum R a"
},
{
"line": "simp only [Set.image_id']",
"before_state": "R : Type u_9\nS : Type u_10\nA : Type u_11\ninst✝⁶ : Semifield R\ninst✝⁵ : Semifield S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\na : A\nf : S → R\ninst✝ : IsScalarTower R S A\nh : SpectrumRestricts a f\n⊢ (fun a => a) '' spectrum R a = spectrum R a",
"after_state": "No Goals!"
}
] |
theorem op_adjoin (s : Set A) :
(Algebra.adjoin R s).op = Algebra.adjoin R (MulOpposite.unop ⁻¹' s) := by
apply toSubsemiring_injective
simp_rw [Algebra.adjoin, op_toSubsemiring, Subsemiring.op_closure, Set.preimage_union]
congr with x
simp_rw [Set.mem_preimage, Set.mem_range, MulOpposite.algebraMap_apply]
congr!
rw [← MulOpposite.op_injective.eq_iff (b := x.unop)]
rw [MulOpposite.op_unop]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean
|
{
"open": [],
"variables": [
"{ι : Sort*} {R A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]"
]
}
|
[
{
"line": "apply toSubsemiring_injective",
"before_state": "R : Type u_2\nA : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\ns : Set A\n⊢ (Algebra.adjoin R s).op = Algebra.adjoin R (MulOpposite.unop ⁻¹' s)",
"after_state": "No Goals!"
}
] |
theorem unop_adjoin (s : Set Aᵐᵒᵖ) :
(Algebra.adjoin R s).unop = Algebra.adjoin R (MulOpposite.op ⁻¹' s) := by
apply toSubsemiring_injective
simp_rw [Algebra.adjoin, unop_toSubsemiring, Subsemiring.unop_closure, Set.preimage_union]
congr with x
simp
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean
|
{
"open": [],
"variables": [
"{ι : Sort*} {R A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]"
]
}
|
[
{
"line": "apply toSubsemiring_injective",
"before_state": "R : Type u_2\nA : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\ns : Set Aᵐᵒᵖ\n⊢ (Algebra.adjoin R s).unop = Algebra.adjoin R (MulOpposite.op ⁻¹' s)",
"after_state": "No Goals!"
}
] |
theorem toSubalgebra_mk (s : Submodule R A) (h1 hmul) :
s.toSubalgebra h1 hmul =
Subalgebra.mk ⟨⟨⟨s, @hmul⟩, h1⟩, s.add_mem, s.zero_mem⟩
(by intro r; rw [Algebra.algebraMap_eq_smul_one]; apply s.smul_mem _ h1) :=
rfl
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
{
"open": [],
"variables": [
"{R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}",
"[CommSemiring R]",
"[Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C]",
"(S : Subalgebra R A)",
"{S R A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]",
"[SetLike S A] [SubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)",
"{R A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]",
"(p : Submodule R A)"
]
}
|
[
{
"line": "intro r",
"before_state": "R' : Type u'\nR✝¹ : Type u\nA✝¹ : Type v\nB : Type w\nC : Type w'\ninst✝¹⁴ : CommSemiring R✝¹\ninst✝¹³ : Semiring A✝¹\ninst✝¹² : Algebra R✝¹ A✝¹\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R✝¹ B\ninst✝⁹ : Semiring C\ninst✝⁸ : Algebra R✝¹ C\nS✝ : Subalgebra R✝¹ A✝¹\nS : Type u_1\nR✝ : Type u_2\nA✝ : Type u_3\ninst✝⁷ : CommSemiring R✝\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : SetLike S A✝\ninst✝³ : SubsemiringClass S A✝\nhSR : SMulMemClass S R✝ A✝\ns✝ : S\nR : Type u_4\nA : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\n⊢ ∀ (r : R),\n (algebraMap R A) r ∈ { carrier := ↑s, mul_mem' := hmul, one_mem' := h1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier",
"after_state": "R' : Type u'\nR✝¹ : Type u\nA✝¹ : Type v\nB : Type w\nC : Type w'\ninst✝¹⁴ : CommSemiring R✝¹\ninst✝¹³ : Semiring A✝¹\ninst✝¹² : Algebra R✝¹ A✝¹\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R✝¹ B\ninst✝⁹ : Semiring C\ninst✝⁸ : Algebra R✝¹ C\nS✝ : Subalgebra R✝¹ A✝¹\nS : Type u_1\nR✝ : Type u_2\nA✝ : Type u_3\ninst✝⁷ : CommSemiring R✝\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : SetLike S A✝\ninst✝³ : SubsemiringClass S A✝\nhSR : SMulMemClass S R✝ A✝\ns✝ : S\nR : Type u_4\nA : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ (algebraMap R A) r ∈ { carrier := ↑s, mul_mem' := hmul, one_mem' := h1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier"
},
{
"line": "rw [Algebra.algebraMap_eq_smul_one]",
"before_state": "R' : Type u'\nR✝¹ : Type u\nA✝¹ : Type v\nB : Type w\nC : Type w'\ninst✝¹⁴ : CommSemiring R✝¹\ninst✝¹³ : Semiring A✝¹\ninst✝¹² : Algebra R✝¹ A✝¹\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R✝¹ B\ninst✝⁹ : Semiring C\ninst✝⁸ : Algebra R✝¹ C\nS✝ : Subalgebra R✝¹ A✝¹\nS : Type u_1\nR✝ : Type u_2\nA✝ : Type u_3\ninst✝⁷ : CommSemiring R✝\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : SetLike S A✝\ninst✝³ : SubsemiringClass S A✝\nhSR : SMulMemClass S R✝ A✝\ns✝ : S\nR : Type u_4\nA : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ (algebraMap R A) r ∈ { carrier := ↑s, mul_mem' := hmul, one_mem' := h1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier",
"after_state": "R' : Type u'\nR✝¹ : Type u\nA✝¹ : Type v\nB : Type w\nC : Type w'\ninst✝¹⁴ : CommSemiring R✝¹\ninst✝¹³ : Semiring A✝¹\ninst✝¹² : Algebra R✝¹ A✝¹\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R✝¹ B\ninst✝⁹ : Semiring C\ninst✝⁸ : Algebra R✝¹ C\nS✝ : Subalgebra R✝¹ A✝¹\nS : Type u_1\nR✝ : Type u_2\nA✝ : Type u_3\ninst✝⁷ : CommSemiring R✝\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : SetLike S A✝\ninst✝³ : SubsemiringClass S A✝\nhSR : SMulMemClass S R✝ A✝\ns✝ : S\nR : Type u_4\nA : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ r • 1 ∈ { carrier := ↑s, mul_mem' := hmul, one_mem' := h1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier"
},
{
"line": "rewrite [Algebra.algebraMap_eq_smul_one]",
"before_state": "R' : Type u'\nR✝¹ : Type u\nA✝¹ : Type v\nB : Type w\nC : Type w'\ninst✝¹⁴ : CommSemiring R✝¹\ninst✝¹³ : Semiring A✝¹\ninst✝¹² : Algebra R✝¹ A✝¹\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R✝¹ B\ninst✝⁹ : Semiring C\ninst✝⁸ : Algebra R✝¹ C\nS✝ : Subalgebra R✝¹ A✝¹\nS : Type u_1\nR✝ : Type u_2\nA✝ : Type u_3\ninst✝⁷ : CommSemiring R✝\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : SetLike S A✝\ninst✝³ : SubsemiringClass S A✝\nhSR : SMulMemClass S R✝ A✝\ns✝ : S\nR : Type u_4\nA : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ (algebraMap R A) r ∈ { carrier := ↑s, mul_mem' := hmul, one_mem' := h1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier",
"after_state": "R' : Type u'\nR✝¹ : Type u\nA✝¹ : Type v\nB : Type w\nC : Type w'\ninst✝¹⁴ : CommSemiring R✝¹\ninst✝¹³ : Semiring A✝¹\ninst✝¹² : Algebra R✝¹ A✝¹\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R✝¹ B\ninst✝⁹ : Semiring C\ninst✝⁸ : Algebra R✝¹ C\nS✝ : Subalgebra R✝¹ A✝¹\nS : Type u_1\nR✝ : Type u_2\nA✝ : Type u_3\ninst✝⁷ : CommSemiring R✝\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : SetLike S A✝\ninst✝³ : SubsemiringClass S A✝\nhSR : SMulMemClass S R✝ A✝\ns✝ : S\nR : Type u_4\nA : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ r • 1 ∈ { carrier := ↑s, mul_mem' := hmul, one_mem' := h1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R' : Type u'\nR✝¹ : Type u\nA✝¹ : Type v\nB : Type w\nC : Type w'\ninst✝¹⁴ : CommSemiring R✝¹\ninst✝¹³ : Semiring A✝¹\ninst✝¹² : Algebra R✝¹ A✝¹\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R✝¹ B\ninst✝⁹ : Semiring C\ninst✝⁸ : Algebra R✝¹ C\nS✝ : Subalgebra R✝¹ A✝¹\nS : Type u_1\nR✝ : Type u_2\nA✝ : Type u_3\ninst✝⁷ : CommSemiring R✝\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : SetLike S A✝\ninst✝³ : SubsemiringClass S A✝\nhSR : SMulMemClass S R✝ A✝\ns✝ : S\nR : Type u_4\nA : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ r • 1 ∈ { carrier := ↑s, mul_mem' := hmul, one_mem' := h1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier",
"after_state": "R' : Type u'\nR✝¹ : Type u\nA✝¹ : Type v\nB : Type w\nC : Type w'\ninst✝¹⁴ : CommSemiring R✝¹\ninst✝¹³ : Semiring A✝¹\ninst✝¹² : Algebra R✝¹ A✝¹\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R✝¹ B\ninst✝⁹ : Semiring C\ninst✝⁸ : Algebra R✝¹ C\nS✝ : Subalgebra R✝¹ A✝¹\nS : Type u_1\nR✝ : Type u_2\nA✝ : Type u_3\ninst✝⁷ : CommSemiring R✝\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : SetLike S A✝\ninst✝³ : SubsemiringClass S A✝\nhSR : SMulMemClass S R✝ A✝\ns✝ : S\nR : Type u_4\nA : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ r • 1 ∈ { carrier := ↑s, mul_mem' := hmul, one_mem' := h1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R' : Type u'\nR✝¹ : Type u\nA✝¹ : Type v\nB : Type w\nC : Type w'\ninst✝¹⁴ : CommSemiring R✝¹\ninst✝¹³ : Semiring A✝¹\ninst✝¹² : Algebra R✝¹ A✝¹\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R✝¹ B\ninst✝⁹ : Semiring C\ninst✝⁸ : Algebra R✝¹ C\nS✝ : Subalgebra R✝¹ A✝¹\nS : Type u_1\nR✝ : Type u_2\nA✝ : Type u_3\ninst✝⁷ : CommSemiring R✝\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : SetLike S A✝\ninst✝³ : SubsemiringClass S A✝\nhSR : SMulMemClass S R✝ A✝\ns✝ : S\nR : Type u_4\nA : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ r • 1 ∈ { carrier := ↑s, mul_mem' := hmul, one_mem' := h1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier",
"after_state": "R' : Type u'\nR✝¹ : Type u\nA✝¹ : Type v\nB : Type w\nC : Type w'\ninst✝¹⁴ : CommSemiring R✝¹\ninst✝¹³ : Semiring A✝¹\ninst✝¹² : Algebra R✝¹ A✝¹\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R✝¹ B\ninst✝⁹ : Semiring C\ninst✝⁸ : Algebra R✝¹ C\nS✝ : Subalgebra R✝¹ A✝¹\nS : Type u_1\nR✝ : Type u_2\nA✝ : Type u_3\ninst✝⁷ : CommSemiring R✝\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : SetLike S A✝\ninst✝³ : SubsemiringClass S A✝\nhSR : SMulMemClass S R✝ A✝\ns✝ : S\nR : Type u_4\nA : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ r • 1 ∈ { carrier := ↑s, mul_mem' := hmul, one_mem' := h1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R' : Type u'\nR✝¹ : Type u\nA✝¹ : Type v\nB : Type w\nC : Type w'\ninst✝¹⁴ : CommSemiring R✝¹\ninst✝¹³ : Semiring A✝¹\ninst✝¹² : Algebra R✝¹ A✝¹\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R✝¹ B\ninst✝⁹ : Semiring C\ninst✝⁸ : Algebra R✝¹ C\nS✝ : Subalgebra R✝¹ A✝¹\nS : Type u_1\nR✝ : Type u_2\nA✝ : Type u_3\ninst✝⁷ : CommSemiring R✝\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : SetLike S A✝\ninst✝³ : SubsemiringClass S A✝\nhSR : SMulMemClass S R✝ A✝\ns✝ : S\nR : Type u_4\nA : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ r • 1 ∈ { carrier := ↑s, mul_mem' := hmul, one_mem' := h1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier",
"after_state": "R' : Type u'\nR✝¹ : Type u\nA✝¹ : Type v\nB : Type w\nC : Type w'\ninst✝¹⁴ : CommSemiring R✝¹\ninst✝¹³ : Semiring A✝¹\ninst✝¹² : Algebra R✝¹ A✝¹\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R✝¹ B\ninst✝⁹ : Semiring C\ninst✝⁸ : Algebra R✝¹ C\nS✝ : Subalgebra R✝¹ A✝¹\nS : Type u_1\nR✝ : Type u_2\nA✝ : Type u_3\ninst✝⁷ : CommSemiring R✝\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : SetLike S A✝\ninst✝³ : SubsemiringClass S A✝\nhSR : SMulMemClass S R✝ A✝\ns✝ : S\nR : Type u_4\nA : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ r • 1 ∈ { carrier := ↑s, mul_mem' := hmul, one_mem' := h1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier"
},
{
"line": "with_reducible rfl",
"before_state": "R' : Type u'\nR✝¹ : Type u\nA✝¹ : Type v\nB : Type w\nC : Type w'\ninst✝¹⁴ : CommSemiring R✝¹\ninst✝¹³ : Semiring A✝¹\ninst✝¹² : Algebra R✝¹ A✝¹\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R✝¹ B\ninst✝⁹ : Semiring C\ninst✝⁸ : Algebra R✝¹ C\nS✝ : Subalgebra R✝¹ A✝¹\nS : Type u_1\nR✝ : Type u_2\nA✝ : Type u_3\ninst✝⁷ : CommSemiring R✝\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : SetLike S A✝\ninst✝³ : SubsemiringClass S A✝\nhSR : SMulMemClass S R✝ A✝\ns✝ : S\nR : Type u_4\nA : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ r • 1 ∈ { carrier := ↑s, mul_mem' := hmul, one_mem' := h1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier",
"after_state": "R' : Type u'\nR✝¹ : Type u\nA✝¹ : Type v\nB : Type w\nC : Type w'\ninst✝¹⁴ : CommSemiring R✝¹\ninst✝¹³ : Semiring A✝¹\ninst✝¹² : Algebra R✝¹ A✝¹\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R✝¹ B\ninst✝⁹ : Semiring C\ninst✝⁸ : Algebra R✝¹ C\nS✝ : Subalgebra R✝¹ A✝¹\nS : Type u_1\nR✝ : Type u_2\nA✝ : Type u_3\ninst✝⁷ : CommSemiring R✝\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : SetLike S A✝\ninst✝³ : SubsemiringClass S A✝\nhSR : SMulMemClass S R✝ A✝\ns✝ : S\nR : Type u_4\nA : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ r • 1 ∈ { carrier := ↑s, mul_mem' := hmul, one_mem' := h1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier"
},
{
"line": "rfl",
"before_state": "R' : Type u'\nR✝¹ : Type u\nA✝¹ : Type v\nB : Type w\nC : Type w'\ninst✝¹⁴ : CommSemiring R✝¹\ninst✝¹³ : Semiring A✝¹\ninst✝¹² : Algebra R✝¹ A✝¹\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R✝¹ B\ninst✝⁹ : Semiring C\ninst✝⁸ : Algebra R✝¹ C\nS✝ : Subalgebra R✝¹ A✝¹\nS : Type u_1\nR✝ : Type u_2\nA✝ : Type u_3\ninst✝⁷ : CommSemiring R✝\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : SetLike S A✝\ninst✝³ : SubsemiringClass S A✝\nhSR : SMulMemClass S R✝ A✝\ns✝ : S\nR : Type u_4\nA : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ r • 1 ∈ { carrier := ↑s, mul_mem' := hmul, one_mem' := h1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier",
"after_state": "R' : Type u'\nR✝¹ : Type u\nA✝¹ : Type v\nB : Type w\nC : Type w'\ninst✝¹⁴ : CommSemiring R✝¹\ninst✝¹³ : Semiring A✝¹\ninst✝¹² : Algebra R✝¹ A✝¹\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R✝¹ B\ninst✝⁹ : Semiring C\ninst✝⁸ : Algebra R✝¹ C\nS✝ : Subalgebra R✝¹ A✝¹\nS : Type u_1\nR✝ : Type u_2\nA✝ : Type u_3\ninst✝⁷ : CommSemiring R✝\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : SetLike S A✝\ninst✝³ : SubsemiringClass S A✝\nhSR : SMulMemClass S R✝ A✝\ns✝ : S\nR : Type u_4\nA : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ r • 1 ∈ { carrier := ↑s, mul_mem' := hmul, one_mem' := h1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier"
},
{
"line": "apply_rfl",
"before_state": "R' : Type u'\nR✝¹ : Type u\nA✝¹ : Type v\nB : Type w\nC : Type w'\ninst✝¹⁴ : CommSemiring R✝¹\ninst✝¹³ : Semiring A✝¹\ninst✝¹² : Algebra R✝¹ A✝¹\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R✝¹ B\ninst✝⁹ : Semiring C\ninst✝⁸ : Algebra R✝¹ C\nS✝ : Subalgebra R✝¹ A✝¹\nS : Type u_1\nR✝ : Type u_2\nA✝ : Type u_3\ninst✝⁷ : CommSemiring R✝\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : SetLike S A✝\ninst✝³ : SubsemiringClass S A✝\nhSR : SMulMemClass S R✝ A✝\ns✝ : S\nR : Type u_4\nA : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ r • 1 ∈ { carrier := ↑s, mul_mem' := hmul, one_mem' := h1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier",
"after_state": "R' : Type u'\nR✝¹ : Type u\nA✝¹ : Type v\nB : Type w\nC : Type w'\ninst✝¹⁴ : CommSemiring R✝¹\ninst✝¹³ : Semiring A✝¹\ninst✝¹² : Algebra R✝¹ A✝¹\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R✝¹ B\ninst✝⁹ : Semiring C\ninst✝⁸ : Algebra R✝¹ C\nS✝ : Subalgebra R✝¹ A✝¹\nS : Type u_1\nR✝ : Type u_2\nA✝ : Type u_3\ninst✝⁷ : CommSemiring R✝\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : SetLike S A✝\ninst✝³ : SubsemiringClass S A✝\nhSR : SMulMemClass S R✝ A✝\ns✝ : S\nR : Type u_4\nA : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ r • 1 ∈ { carrier := ↑s, mul_mem' := hmul, one_mem' := h1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier"
},
{
"line": "skip",
"before_state": "R' : Type u'\nR✝¹ : Type u\nA✝¹ : Type v\nB : Type w\nC : Type w'\ninst✝¹⁴ : CommSemiring R✝¹\ninst✝¹³ : Semiring A✝¹\ninst✝¹² : Algebra R✝¹ A✝¹\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R✝¹ B\ninst✝⁹ : Semiring C\ninst✝⁸ : Algebra R✝¹ C\nS✝ : Subalgebra R✝¹ A✝¹\nS : Type u_1\nR✝ : Type u_2\nA✝ : Type u_3\ninst✝⁷ : CommSemiring R✝\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : SetLike S A✝\ninst✝³ : SubsemiringClass S A✝\nhSR : SMulMemClass S R✝ A✝\ns✝ : S\nR : Type u_4\nA : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ r • 1 ∈ { carrier := ↑s, mul_mem' := hmul, one_mem' := h1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier",
"after_state": "R' : Type u'\nR✝¹ : Type u\nA✝¹ : Type v\nB : Type w\nC : Type w'\ninst✝¹⁴ : CommSemiring R✝¹\ninst✝¹³ : Semiring A✝¹\ninst✝¹² : Algebra R✝¹ A✝¹\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R✝¹ B\ninst✝⁹ : Semiring C\ninst✝⁸ : Algebra R✝¹ C\nS✝ : Subalgebra R✝¹ A✝¹\nS : Type u_1\nR✝ : Type u_2\nA✝ : Type u_3\ninst✝⁷ : CommSemiring R✝\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : SetLike S A✝\ninst✝³ : SubsemiringClass S A✝\nhSR : SMulMemClass S R✝ A✝\ns✝ : S\nR : Type u_4\nA : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ r • 1 ∈ { carrier := ↑s, mul_mem' := hmul, one_mem' := h1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier"
},
{
"line": "apply s.smul_mem _ h1",
"before_state": "R' : Type u'\nR✝¹ : Type u\nA✝¹ : Type v\nB : Type w\nC : Type w'\ninst✝¹⁴ : CommSemiring R✝¹\ninst✝¹³ : Semiring A✝¹\ninst✝¹² : Algebra R✝¹ A✝¹\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R✝¹ B\ninst✝⁹ : Semiring C\ninst✝⁸ : Algebra R✝¹ C\nS✝ : Subalgebra R✝¹ A✝¹\nS : Type u_1\nR✝ : Type u_2\nA✝ : Type u_3\ninst✝⁷ : CommSemiring R✝\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : SetLike S A✝\ninst✝³ : SubsemiringClass S A✝\nhSR : SMulMemClass S R✝ A✝\ns✝ : S\nR : Type u_4\nA : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ r • 1 ∈ { carrier := ↑s, mul_mem' := hmul, one_mem' := h1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier",
"after_state": "No Goals!"
}
] |
lemma centralizer_coe_image_includeLeft_eq_center_tensorProduct
(S : Set A) [Module.Free R B] :
Subalgebra.centralizer R
(Algebra.TensorProduct.includeLeft (S := R) '' S) =
(Algebra.TensorProduct.map (Subalgebra.centralizer R (S : Set A)).val
(AlgHom.id R B)).range := by
classical
ext w
constructor
· intro hw
rw [mem_centralizer_iff] at hw
let ℬ := Module.Free.chooseBasis R B
obtain ⟨b, rfl⟩ := TensorProduct.eq_repr_basis_right ℬ w
refine Subalgebra.sum_mem _ fun j hj => ⟨⟨b j, ?_⟩ ⊗ₜ[R] ℬ j, by simp⟩
rw [Subalgebra.mem_centralizer_iff]
intro x hx
suffices x • b = b.mapRange (· * x) (by simp) from Finsupp.ext_iff.1 this j
specialize hw (x ⊗ₜ[R] 1) ⟨x, hx, rfl⟩
simp only [Finsupp.sum] at hw
simp only [Finset.mul_sum] at hw
simp only [Algebra.TensorProduct.tmul_mul_tmul] at hw
simp only [one_mul] at hw
simp only [Finset.sum_mul] at hw
simp only [mul_one] at hw
refine TensorProduct.sum_tmul_basis_right_injective ℬ ?_
simp only [Finsupp.coe_lsum]
rw [sum_of_support_subset (s := b.support) (hs := Finsupp.support_smul) (h := by aesop)]
rw [sum_of_support_subset (s := b.support) (hs := support_mapRange) (h := by aesop)]
simpa only [Finsupp.coe_smul,Pi.smul_apply,smul_eq_mul,LinearMap.flip_apply,TensorProduct.mk_apply,Finsupp.mapRange_apply] using hw
· rintro ⟨w, rfl⟩
rw [Subalgebra.mem_centralizer_iff]
rintro _ ⟨x, hx, rfl⟩
induction w using TensorProduct.induction_on with
| zero => simp
| tmul b c =>
simp [Subalgebra.mem_centralizer_iff _ |>.1 b.2 x hx]
| add y z hy hz => rw [map_add, mul_add, hy, hz, add_mul]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Subalgebra/Centralizer.lean
|
{
"open": [
"Algebra.TensorProduct",
"Finsupp TensorProduct"
],
"variables": [
"{R : Type*} [CommSemiring R]",
"{A : Type*} [Semiring A] [Algebra R A]",
"(R : Type*) [CommSemiring R]",
"(A : Type*) [Semiring A] [Algebra R A]",
"(B : Type*) [Semiring B] [Algebra R B]"
]
}
|
[
{
"line": "classical\next w\nconstructor\n· intro hw\n rw [mem_centralizer_iff] at hw\n let ℬ := Module.Free.chooseBasis R B\n obtain ⟨b, rfl⟩ := TensorProduct.eq_repr_basis_right ℬ w\n refine Subalgebra.sum_mem _ fun j hj => ⟨⟨b j, ?_⟩ ⊗ₜ[R] ℬ j, by simp⟩\n rw [Subalgebra.mem_centralizer_iff]\n intro x hx\n suffices x • b = b.mapRange (· * x) (by simp) from Finsupp.ext_iff.1 this j\n specialize hw (x ⊗ₜ[R] 1) ⟨x, hx, rfl⟩\n simp only [Finsupp.sum] at hw\n simp only [Finset.mul_sum] at hw\n simp only [Algebra.TensorProduct.tmul_mul_tmul] at hw\n simp only [one_mul] at hw\n simp only [Finset.sum_mul] at hw\n simp only [mul_one] at hw\n refine TensorProduct.sum_tmul_basis_right_injective ℬ ?_\n simp only [Finsupp.coe_lsum]\n rw [sum_of_support_subset (s := b.support) (hs := Finsupp.support_smul) (h := by aesop)]\n rw [sum_of_support_subset (s := b.support) (hs := support_mapRange) (h := by aesop)]\n simpa only [Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul, LinearMap.flip_apply, TensorProduct.mk_apply,\n Finsupp.mapRange_apply] using hw\n· rintro ⟨w, rfl⟩\n rw [Subalgebra.mem_centralizer_iff]\n rintro _ ⟨x, hx, rfl⟩\n induction w using TensorProduct.induction_on with\n | zero => simp\n | tmul b c => simp [Subalgebra.mem_centralizer_iff _ |>.1 b.2 x hx]\n | add y z hy hz => rw [map_add, mul_add, hy, hz, add_mul]",
"before_state": "R : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\n⊢ Subalgebra.centralizer R (⇑includeLeft '' S) =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).range",
"after_state": "No Goals!"
},
{
"line": "ext w",
"before_state": "R : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\n⊢ Subalgebra.centralizer R (⇑includeLeft '' S) =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).range",
"after_state": "case h\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : A ⊗[R] B\n⊢ w ∈ Subalgebra.centralizer R (⇑includeLeft '' S) ↔\n w ∈ (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).range"
},
{
"line": "constructor",
"before_state": "case h\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : A ⊗[R] B\n⊢ w ∈ Subalgebra.centralizer R (⇑includeLeft '' S) ↔\n w ∈ (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).range",
"after_state": "case h.mp\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : A ⊗[R] B\n⊢ w ∈ Subalgebra.centralizer R (⇑includeLeft '' S) →\n w ∈ (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).range\n---\ncase h.mpr\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : A ⊗[R] B\n⊢ w ∈ (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).range →\n w ∈ Subalgebra.centralizer R (⇑includeLeft '' S)"
},
{
"line": "intro hw",
"before_state": "case h.mp\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : A ⊗[R] B\n⊢ w ∈ Subalgebra.centralizer R (⇑includeLeft '' S) →\n w ∈ (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).range",
"after_state": "case h.mp\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : A ⊗[R] B\nhw : w ∈ Subalgebra.centralizer R (⇑includeLeft '' S)\n⊢ w ∈ (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).range"
},
{
"line": "rw [mem_centralizer_iff] at hw",
"before_state": "case h.mp\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : A ⊗[R] B\nhw : w ∈ Subalgebra.centralizer R (⇑includeLeft '' S)\n⊢ w ∈ (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).range",
"after_state": "case h.mp\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : A ⊗[R] B\nhw : w ∈ Subalgebra.centralizer R (⇑includeLeft '' S)\n⊢ w ∈ (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).range"
},
{
"line": "rewrite [mem_centralizer_iff] at hw",
"before_state": "case h.mp\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : A ⊗[R] B\nhw : w ∈ Subalgebra.centralizer R (⇑includeLeft '' S)\n⊢ w ∈ (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).range",
"after_state": "case h.mp\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : A ⊗[R] B\nhw : w ∈ Subalgebra.centralizer R (⇑includeLeft '' S)\n⊢ w ∈ (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).range"
},
{
"line": "rintro ⟨w, rfl⟩",
"before_state": "case h.mpr\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : A ⊗[R] B\n⊢ w ∈ (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).range →\n w ∈ Subalgebra.centralizer R (⇑includeLeft '' S)",
"after_state": "case h.mpr.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\n⊢ (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w ∈\n Subalgebra.centralizer R (⇑includeLeft '' S)"
},
{
"line": "rw [Subalgebra.mem_centralizer_iff]",
"before_state": "case h.mpr.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\n⊢ (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w ∈\n Subalgebra.centralizer R (⇑includeLeft '' S)",
"after_state": "case h.mpr.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\n⊢ ∀ g ∈ ⇑includeLeft '' S,\n g * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w * g"
},
{
"line": "rewrite [Subalgebra.mem_centralizer_iff]",
"before_state": "case h.mpr.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\n⊢ (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w ∈\n Subalgebra.centralizer R (⇑includeLeft '' S)",
"after_state": "case h.mpr.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\n⊢ ∀ g ∈ ⇑includeLeft '' S,\n g * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w * g"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.mpr.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\n⊢ ∀ g ∈ ⇑includeLeft '' S,\n g * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w * g",
"after_state": "case h.mpr.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\n⊢ ∀ g ∈ ⇑includeLeft '' S,\n g * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w * g"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.mpr.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\n⊢ ∀ g ∈ ⇑includeLeft '' S,\n g * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w * g",
"after_state": "case h.mpr.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\n⊢ ∀ g ∈ ⇑includeLeft '' S,\n g * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w * g"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.mpr.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\n⊢ ∀ g ∈ ⇑includeLeft '' S,\n g * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w * g",
"after_state": "case h.mpr.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\n⊢ ∀ g ∈ ⇑includeLeft '' S,\n g * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w * g"
},
{
"line": "with_reducible rfl",
"before_state": "case h.mpr.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\n⊢ ∀ g ∈ ⇑includeLeft '' S,\n g * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w * g",
"after_state": "case h.mpr.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\n⊢ ∀ g ∈ ⇑includeLeft '' S,\n g * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w * g"
},
{
"line": "rfl",
"before_state": "case h.mpr.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\n⊢ ∀ g ∈ ⇑includeLeft '' S,\n g * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w * g",
"after_state": "case h.mpr.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\n⊢ ∀ g ∈ ⇑includeLeft '' S,\n g * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w * g"
},
{
"line": "apply_rfl",
"before_state": "case h.mpr.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\n⊢ ∀ g ∈ ⇑includeLeft '' S,\n g * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w * g",
"after_state": "case h.mpr.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\n⊢ ∀ g ∈ ⇑includeLeft '' S,\n g * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w * g"
},
{
"line": "skip",
"before_state": "case h.mpr.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\n⊢ ∀ g ∈ ⇑includeLeft '' S,\n g * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w * g",
"after_state": "case h.mpr.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\n⊢ ∀ g ∈ ⇑includeLeft '' S,\n g * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w * g"
},
{
"line": "rintro _ ⟨x, hx, rfl⟩",
"before_state": "case h.mpr.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\n⊢ ∀ g ∈ ⇑includeLeft '' S,\n g * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w * g",
"after_state": "case h.mpr.intro.intro.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\nx : A\nhx : x ∈ S\n⊢ includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w * includeLeft x"
},
{
"line": "induction w using TensorProduct.induction_on with\n| zero => simp\n| tmul b c => simp [Subalgebra.mem_centralizer_iff _ |>.1 b.2 x hx]\n| add y z hy hz => rw [map_add, mul_add, hy, hz, add_mul]",
"before_state": "case h.mpr.intro.intro.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\nx : A\nhx : x ∈ S\n⊢ includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w * includeLeft x",
"after_state": "No Goals!"
},
{
"line": "induction w using TensorProduct.induction_on with\n| zero => simp\n| tmul b c => simp [Subalgebra.mem_centralizer_iff _ |>.1 b.2 x hx]\n| add y z hy hz => rw [map_add, mul_add, hy, hz, add_mul]",
"before_state": "case h.mpr.intro.intro.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\nx : A\nhx : x ∈ S\n⊢ includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w * includeLeft x",
"after_state": "case h.mpr.intro.intro.intro\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nw : ↥(Subalgebra.centralizer R S) ⊗[R] B\nx : A\nhx : x ∈ S\n⊢ includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom w * includeLeft x"
},
{
"line": "simp",
"before_state": "case h.mpr.intro.intro.intro.zero\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nx : A\nhx : x ∈ S\n⊢ includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom 0 =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom 0 * includeLeft x",
"after_state": "No Goals!"
},
{
"line": "simp [Subalgebra.mem_centralizer_iff _ |>.1 b.2 x hx]",
"before_state": "case h.mpr.intro.intro.intro.tmul\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nx : A\nhx : x ∈ S\nb : ↥(Subalgebra.centralizer R S)\nc : B\n⊢ includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom (b ⊗ₜ[R] c) =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom (b ⊗ₜ[R] c) * includeLeft x",
"after_state": "No Goals!"
},
{
"line": "rw [map_add, mul_add, hy, hz, add_mul]",
"before_state": "case h.mpr.intro.intro.intro.add\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nx : A\nhx : x ∈ S\ny z : ↥(Subalgebra.centralizer R S) ⊗[R] B\nhy :\n includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x\nhz :\n includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x\n⊢ includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom (y + z) =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom (y + z) * includeLeft x",
"after_state": "No Goals!"
},
{
"line": "rewrite [map_add, mul_add, hy, hz, add_mul]",
"before_state": "case h.mpr.intro.intro.intro.add\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nx : A\nhx : x ∈ S\ny z : ↥(Subalgebra.centralizer R S) ⊗[R] B\nhy :\n includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x\nhz :\n includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x\n⊢ includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom (y + z) =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom (y + z) * includeLeft x",
"after_state": "case h.mpr.intro.intro.intro.add\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nx : A\nhx : x ∈ S\ny z : ↥(Subalgebra.centralizer R S) ⊗[R] B\nhy :\n includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x\nhz :\n includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x\n⊢ (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x +\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x +\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.mpr.intro.intro.intro.add\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nx : A\nhx : x ∈ S\ny z : ↥(Subalgebra.centralizer R S) ⊗[R] B\nhy :\n includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x\nhz :\n includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x\n⊢ (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x +\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x +\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.mpr.intro.intro.intro.add\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nx : A\nhx : x ∈ S\ny z : ↥(Subalgebra.centralizer R S) ⊗[R] B\nhy :\n includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x\nhz :\n includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x\n⊢ (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x +\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x +\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.mpr.intro.intro.intro.add\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nx : A\nhx : x ∈ S\ny z : ↥(Subalgebra.centralizer R S) ⊗[R] B\nhy :\n includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x\nhz :\n includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x\n⊢ (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x +\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x +\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "case h.mpr.intro.intro.intro.add\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nx : A\nhx : x ∈ S\ny z : ↥(Subalgebra.centralizer R S) ⊗[R] B\nhy :\n includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x\nhz :\n includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x\n⊢ (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x +\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x +\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "case h.mpr.intro.intro.intro.add\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nx : A\nhx : x ∈ S\ny z : ↥(Subalgebra.centralizer R S) ⊗[R] B\nhy :\n includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x\nhz :\n includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x\n⊢ (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x +\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x +\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "case h.mpr.intro.intro.intro.add\nR : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_4\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_5\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nx : A\nhx : x ∈ S\ny z : ↥(Subalgebra.centralizer R S) ⊗[R] B\nhy :\n includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x\nhz :\n includeLeft x * (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x\n⊢ (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x +\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x =\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom y * includeLeft x +\n (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).toRingHom z * includeLeft x",
"after_state": "No Goals!"
}
] |
theorem map_comap_eq_self
{f : A →ₐ[R] B} {S : Subalgebra R B} (h : S ≤ f.range) : (S.comap f).map f = S := by
simpa only [inf_of_le_left h] using map_comap_eq f S
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Subalgebra/Lattice.lean
|
{
"open": [
"Subalgebra in",
"Subalgebra in",
"Subalgebra in",
"Algebra"
],
"variables": [
"(R : Type u) {A : Type v} {B : Type w}",
"[CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]",
"{R}",
"{R : Type u} {A : Type v} {B : Type w}",
"[CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]",
"(S : Subalgebra R A)",
"(R A)",
"(R)",
"{R A B : Type*} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]",
"{F : Type*}",
"[FunLike F A B] [AlgHomClass F R A B]",
"{R A B : Type*} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]"
]
}
|
[
{
"line": "simpa only [inf_of_le_left h] using map_comap_eq f S",
"before_state": "R : Type u_5\nA : Type u_6\nB : Type u_7\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R B\nh : S ≤ f.range\n⊢ Subalgebra.map f (Subalgebra.comap f S) = S",
"after_state": "No Goals!"
}
] |
theorem ext_of_eq_adjoin {S : Subalgebra R A} {s : Set A} (hS : S = adjoin R s) ⦃φ₁ φ₂ : S →ₐ[R] B⦄
(h : ∀ x hx, φ₁ ⟨x, hS.ge (subset_adjoin hx)⟩ = φ₂ ⟨x, hS.ge (subset_adjoin hx)⟩) :
φ₁ = φ₂ := by
subst hS; exact adjoin_ext h
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Subalgebra/Lattice.lean
|
{
"open": [
"Subalgebra in",
"Subalgebra in",
"Subalgebra in",
"Algebra",
"Submodule Subsemiring",
"Algebra Subalgebra"
],
"variables": [
"(R : Type u) {A : Type v} {B : Type w}",
"[CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]",
"{R}",
"{R : Type u} {A : Type v} {B : Type w}",
"[CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]",
"(S : Subalgebra R A)",
"(R A)",
"(R)",
"{R A B : Type*} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]",
"{F : Type*}",
"[FunLike F A B] [AlgHomClass F R A B]",
"{R A B : Type*} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]",
"{R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB}",
"[CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]",
"[Algebra R S] [Algebra R A] [Algebra S A] [Algebra R B] [IsScalarTower R S A]",
"{s t : Set A}",
"(R A)",
"{R} in",
"{A} (s)",
"{R} in",
"{R}",
"(R)",
"[CommSemiring R] [CommSemiring A]",
"[Algebra R A] {s t : Set A}",
"(R s t)",
"[CommRing R] [Ring A]",
"[Algebra R A] {s t : Set A}",
"(R)",
"[CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B]"
]
}
|
[
{
"line": "subst hS",
"before_state": "R : Type uR\nA : Type uA\nB : Type uB\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : Semiring A\ninst✝¹³ : Semiring B\ninst✝¹² : Algebra R A\ninst✝¹¹ : Algebra R B\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring A\ninst✝⁸ : Algebra R A\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nS : Subalgebra R A\ns : Set A\nhS : S = adjoin R s\nφ₁ φ₂ : ↥S →ₐ[R] B\nh : ∀ (x : A) (hx : x ∈ s), φ₁ ⟨x, ⋯⟩ = φ₂ ⟨x, ⋯⟩\n⊢ φ₁ = φ₂",
"after_state": "R : Type uR\nA : Type uA\nB : Type uB\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : Semiring A\ninst✝¹³ : Semiring B\ninst✝¹² : Algebra R A\ninst✝¹¹ : Algebra R B\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring A\ninst✝⁸ : Algebra R A\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns : Set A\nφ₁ φ₂ : ↥(adjoin R s) →ₐ[R] B\nh : ∀ (x : A) (hx : x ∈ s), φ₁ ⟨x, ⋯⟩ = φ₂ ⟨x, ⋯⟩\n⊢ φ₁ = φ₂"
},
{
"line": "exact adjoin_ext h",
"before_state": "R : Type uR\nA : Type uA\nB : Type uB\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : Semiring A\ninst✝¹³ : Semiring B\ninst✝¹² : Algebra R A\ninst✝¹¹ : Algebra R B\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring A\ninst✝⁸ : Algebra R A\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns : Set A\nφ₁ φ₂ : ↥(adjoin R s) →ₐ[R] B\nh : ∀ (x : A) (hx : x ∈ s), φ₁ ⟨x, ⋯⟩ = φ₂ ⟨x, ⋯⟩\n⊢ φ₁ = φ₂",
"after_state": "No Goals!"
}
] |
theorem comap_map_eq (f : A →ₐ[R] B) (S : Subalgebra R A) :
(S.map f).comap f = S ⊔ Algebra.adjoin R (f ⁻¹' {0}) := by
apply le_antisymm
· intro x hx
rw [mem_comap] at hx
rw [mem_map] at hx
obtain ⟨y, hy, hxy⟩ := hx
replace hxy : x - y ∈ f ⁻¹' {0} := by simp [hxy]
rw [← Algebra.adjoin_eq S]
rw [← Algebra.adjoin_union]
rw [← add_sub_cancel y x]
exact Subalgebra.add_mem _
(Algebra.subset_adjoin <| Or.inl hy) (Algebra.subset_adjoin <| Or.inr hxy)
· rw [← map_le, Algebra.map_sup, f.map_adjoin]
apply le_of_eq
rw [sup_eq_left]
rw [Algebra.adjoin_le_iff]
exact (Set.image_preimage_subset f {0}).trans (Set.singleton_subset_iff.2 (S.map f).zero_mem)
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Subalgebra/Lattice.lean
|
{
"open": [
"Subalgebra in",
"Subalgebra in",
"Subalgebra in",
"Algebra",
"Submodule Subsemiring",
"Algebra Subalgebra"
],
"variables": [
"(R : Type u) {A : Type v} {B : Type w}",
"[CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]",
"{R}",
"{R : Type u} {A : Type v} {B : Type w}",
"[CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]",
"(S : Subalgebra R A)",
"(R A)",
"(R)",
"{R A B : Type*} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]",
"{F : Type*}",
"[FunLike F A B] [AlgHomClass F R A B]",
"{R A B : Type*} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]",
"{R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB}",
"[CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]",
"[Algebra R S] [Algebra R A] [Algebra S A] [Algebra R B] [IsScalarTower R S A]",
"{s t : Set A}",
"(R A)",
"{R} in",
"{A} (s)",
"{R} in",
"{R}",
"(R)",
"[CommSemiring R] [CommSemiring A]",
"[Algebra R A] {s t : Set A}",
"(R s t)",
"[CommRing R] [Ring A]",
"[Algebra R A] {s t : Set A}",
"(R)",
"[CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B]",
"(F E : Type*) {K : Type*} [CommSemiring E] [Semiring K] [SMul F E] [Algebra E K]",
"[CommSemiring F] [Algebra F K] [IsScalarTower F E K] (L : Subalgebra F K) {F}",
"(R) [CommSemiring R] [Ring A] [Algebra R A] [Ring B] [Algebra R B]",
"[CommSemiring R] [Ring A] [Algebra R A] [Ring B] [Algebra R B]"
]
}
|
[
{
"line": "apply le_antisymm",
"before_state": "R : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.comap f (Subalgebra.map f S) = S ⊔ adjoin R (⇑f ⁻¹' {0})",
"after_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.comap f (Subalgebra.map f S) ≤ S ⊔ adjoin R (⇑f ⁻¹' {0})\n---\ncase a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ S ⊔ adjoin R (⇑f ⁻¹' {0}) ≤ Subalgebra.comap f (Subalgebra.map f S)"
},
{
"line": "intro x hx",
"before_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.comap f (Subalgebra.map f S) ≤ S ⊔ adjoin R (⇑f ⁻¹' {0})",
"after_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\nx : A\nhx : x ∈ Subalgebra.comap f (Subalgebra.map f S)\n⊢ x ∈ S ⊔ adjoin R (⇑f ⁻¹' {0})"
},
{
"line": "intro x;\n intro hx",
"before_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.comap f (Subalgebra.map f S) ≤ S ⊔ adjoin R (⇑f ⁻¹' {0})",
"after_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\nx : A\nhx : x ∈ Subalgebra.comap f (Subalgebra.map f S)\n⊢ x ∈ S ⊔ adjoin R (⇑f ⁻¹' {0})"
},
{
"line": "intro x",
"before_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.comap f (Subalgebra.map f S) ≤ S ⊔ adjoin R (⇑f ⁻¹' {0})",
"after_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\nx : A\n⊢ x ∈ Subalgebra.comap f (Subalgebra.map f S) → x ∈ S ⊔ adjoin R (⇑f ⁻¹' {0})"
},
{
"line": "intro hx",
"before_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\nx : A\n⊢ x ∈ Subalgebra.comap f (Subalgebra.map f S) → x ∈ S ⊔ adjoin R (⇑f ⁻¹' {0})",
"after_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\nx : A\nhx : x ∈ Subalgebra.comap f (Subalgebra.map f S)\n⊢ x ∈ S ⊔ adjoin R (⇑f ⁻¹' {0})"
},
{
"line": "rw [mem_comap] at hx",
"before_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\nx : A\nhx : x ∈ Subalgebra.comap f (Subalgebra.map f S)\n⊢ x ∈ S ⊔ adjoin R (⇑f ⁻¹' {0})",
"after_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\nx : A\nhx : x ∈ Subalgebra.comap f (Subalgebra.map f S)\n⊢ x ∈ S ⊔ adjoin R (⇑f ⁻¹' {0})"
},
{
"line": "rewrite [mem_comap] at hx",
"before_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\nx : A\nhx : x ∈ Subalgebra.comap f (Subalgebra.map f S)\n⊢ x ∈ S ⊔ adjoin R (⇑f ⁻¹' {0})",
"after_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\nx : A\nhx : x ∈ Subalgebra.comap f (Subalgebra.map f S)\n⊢ x ∈ S ⊔ adjoin R (⇑f ⁻¹' {0})"
},
{
"line": "rw [← map_le, Algebra.map_sup, f.map_adjoin]",
"before_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ S ⊔ adjoin R (⇑f ⁻¹' {0}) ≤ Subalgebra.comap f (Subalgebra.map f S)",
"after_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.map f S ⊔ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S"
},
{
"line": "rewrite [← map_le, Algebra.map_sup, f.map_adjoin]",
"before_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ S ⊔ adjoin R (⇑f ⁻¹' {0}) ≤ Subalgebra.comap f (Subalgebra.map f S)",
"after_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.map f S ⊔ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.map f S ⊔ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S",
"after_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.map f S ⊔ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.map f S ⊔ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S",
"after_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.map f S ⊔ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.map f S ⊔ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S",
"after_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.map f S ⊔ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S"
},
{
"line": "with_reducible rfl",
"before_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.map f S ⊔ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S",
"after_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.map f S ⊔ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S"
},
{
"line": "rfl",
"before_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.map f S ⊔ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S",
"after_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.map f S ⊔ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S"
},
{
"line": "apply_rfl",
"before_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.map f S ⊔ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S",
"after_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.map f S ⊔ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S"
},
{
"line": "skip",
"before_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.map f S ⊔ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S",
"after_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.map f S ⊔ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S"
},
{
"line": "apply le_of_eq",
"before_state": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.map f S ⊔ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S",
"after_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.map f S ⊔ adjoin R (⇑f '' (⇑f ⁻¹' {0})) = Subalgebra.map f S"
},
{
"line": "rw [sup_eq_left]",
"before_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.map f S ⊔ adjoin R (⇑f '' (⇑f ⁻¹' {0})) = Subalgebra.map f S",
"after_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S"
},
{
"line": "rewrite [sup_eq_left]",
"before_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ Subalgebra.map f S ⊔ adjoin R (⇑f '' (⇑f ⁻¹' {0})) = Subalgebra.map f S",
"after_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S",
"after_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S",
"after_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S",
"after_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S"
},
{
"line": "with_reducible rfl",
"before_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S",
"after_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S"
},
{
"line": "rfl",
"before_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S",
"after_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S"
},
{
"line": "apply_rfl",
"before_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S",
"after_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S"
},
{
"line": "skip",
"before_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S",
"after_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S"
},
{
"line": "rw [Algebra.adjoin_le_iff]",
"before_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S",
"after_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ ⇑f '' (⇑f ⁻¹' {0}) ⊆ ↑(Subalgebra.map f S)"
},
{
"line": "rewrite [Algebra.adjoin_le_iff]",
"before_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ adjoin R (⇑f '' (⇑f ⁻¹' {0})) ≤ Subalgebra.map f S",
"after_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ ⇑f '' (⇑f ⁻¹' {0}) ⊆ ↑(Subalgebra.map f S)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ ⇑f '' (⇑f ⁻¹' {0}) ⊆ ↑(Subalgebra.map f S)",
"after_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ ⇑f '' (⇑f ⁻¹' {0}) ⊆ ↑(Subalgebra.map f S)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ ⇑f '' (⇑f ⁻¹' {0}) ⊆ ↑(Subalgebra.map f S)",
"after_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ ⇑f '' (⇑f ⁻¹' {0}) ⊆ ↑(Subalgebra.map f S)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ ⇑f '' (⇑f ⁻¹' {0}) ⊆ ↑(Subalgebra.map f S)",
"after_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ ⇑f '' (⇑f ⁻¹' {0}) ⊆ ↑(Subalgebra.map f S)"
},
{
"line": "with_reducible rfl",
"before_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ ⇑f '' (⇑f ⁻¹' {0}) ⊆ ↑(Subalgebra.map f S)",
"after_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ ⇑f '' (⇑f ⁻¹' {0}) ⊆ ↑(Subalgebra.map f S)"
},
{
"line": "rfl",
"before_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ ⇑f '' (⇑f ⁻¹' {0}) ⊆ ↑(Subalgebra.map f S)",
"after_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ ⇑f '' (⇑f ⁻¹' {0}) ⊆ ↑(Subalgebra.map f S)"
},
{
"line": "apply_rfl",
"before_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ ⇑f '' (⇑f ⁻¹' {0}) ⊆ ↑(Subalgebra.map f S)",
"after_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ ⇑f '' (⇑f ⁻¹' {0}) ⊆ ↑(Subalgebra.map f S)"
},
{
"line": "skip",
"before_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ ⇑f '' (⇑f ⁻¹' {0}) ⊆ ↑(Subalgebra.map f S)",
"after_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ ⇑f '' (⇑f ⁻¹' {0}) ⊆ ↑(Subalgebra.map f S)"
},
{
"line": "exact (Set.image_preimage_subset f {0}).trans (Set.singleton_subset_iff.2 (S.map f).zero_mem)",
"before_state": "case a.hab\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝²⁰ : CommSemiring R\ninst✝¹⁹ : Semiring A\ninst✝¹⁸ : Semiring B\ninst✝¹⁷ : Algebra R A\ninst✝¹⁶ : Algebra R B\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : CommSemiring A\ninst✝¹³ : Algebra R A\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ ⇑f '' (⇑f ⁻¹' {0}) ⊆ ↑(Subalgebra.map f S)",
"after_state": "No Goals!"
}
] |
theorem mem_of_finset_sum_eq_one_of_pow_smul_mem
{ι : Type*} (ι' : Finset ι) (s : ι → S) (l : ι → S)
(e : ∑ i ∈ ι', l i * s i = 1) (hs : ∀ i, s i ∈ S') (hl : ∀ i, l i ∈ S') (x : S)
(H : ∀ i, ∃ n : ℕ, (s i ^ n : S) • x ∈ S') : x ∈ S' := by
suffices x ∈ Subalgebra.toSubmodule (Algebra.ofId S' S).range by
obtain ⟨x, rfl⟩ := this
exact x.2
choose n hn using H
let s' : ι → S' := fun x => ⟨s x, hs x⟩
let l' : ι → S' := fun x => ⟨l x, hl x⟩
have e' : ∑ i ∈ ι', l' i * s' i = 1 := by
ext
show S'.subtype (∑ i ∈ ι', l' i * s' i) = 1
simpa only [map_sum,map_mul] using e
have : Ideal.span (s' '' ι') = ⊤ := by
rw [Ideal.eq_top_iff_one]
rw [← e']
apply sum_mem
intros i hi
exact Ideal.mul_mem_left _ _ <| Ideal.subset_span <| Set.mem_image_of_mem s' hi
let N := ι'.sup n
have hN := Ideal.span_pow_eq_top _ this N
apply (Algebra.ofId S' S).range.toSubmodule.mem_of_span_top_of_smul_mem _ hN
rintro ⟨_, _, ⟨i, hi, rfl⟩, rfl⟩
change s' i ^ N • x ∈ _
rw [← tsub_add_cancel_of_le (show n i ≤ N from Finset.le_sup hi)]
rw [pow_add]
rw [mul_smul]
refine Submodule.smul_mem _ (⟨_, pow_mem (hs i) _⟩ : S') ?_
exact ⟨⟨_, hn i⟩, rfl⟩
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Subalgebra/Operations.lean
|
{
"open": [
"Algebra"
],
"variables": [
"{R A B : Type*} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]",
"{R S : Type*} [CommSemiring R] [CommRing S] [Algebra R S]",
"(S' : Subalgebra R S)"
]
}
|
[
{
"line": "suffices x ∈ Subalgebra.toSubmodule (Algebra.ofId S' S).range\n by\n obtain ⟨x, rfl⟩ := this\n exact x.2",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nH : ∀ (i : ι), ∃ n, s i ^ n • x ∈ S'\n⊢ x ∈ S'",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nH : ∀ (i : ι), ∃ n, s i ^ n • x ∈ S'\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "refine_lift\n suffices x ∈ Subalgebra.toSubmodule (Algebra.ofId S' S).range\n by\n obtain ⟨x, rfl⟩ := this\n exact x.2;\n ?_",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nH : ∀ (i : ι), ∃ n, s i ^ n • x ∈ S'\n⊢ x ∈ S'",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nH : ∀ (i : ι), ∃ n, s i ^ n • x ∈ S'\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (suffices x ∈ Subalgebra.toSubmodule (Algebra.ofId S' S).range\n by\n obtain ⟨x, rfl⟩ := this\n exact x.2;\n ?_);\n rotate_right)",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nH : ∀ (i : ι), ∃ n, s i ^ n • x ∈ S'\n⊢ x ∈ S'",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nH : ∀ (i : ι), ∃ n, s i ^ n • x ∈ S'\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "refine\n no_implicit_lambda%\n (suffices x ∈ Subalgebra.toSubmodule (Algebra.ofId S' S).range\n by\n obtain ⟨x, rfl⟩ := this\n exact x.2;\n ?_)",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nH : ∀ (i : ι), ∃ n, s i ^ n • x ∈ S'\n⊢ x ∈ S'",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nH : ∀ (i : ι), ∃ n, s i ^ n • x ∈ S'\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "obtain ⟨x, rfl⟩ := this",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nH : ∀ (i : ι), ∃ n, s i ^ n • x ∈ S'\nthis : x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range\n⊢ x ∈ S'",
"after_state": "case intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : ↥S'\nH : ∀ (i : ι), ∃ n, s i ^ n • (ofId (↥S') S).toRingHom x ∈ S'\n⊢ (ofId (↥S') S).toRingHom x ∈ S'"
},
{
"line": "exact x.2",
"before_state": "case intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : ↥S'\nH : ∀ (i : ι), ∃ n, s i ^ n • (ofId (↥S') S).toRingHom x ∈ S'\n⊢ (ofId (↥S') S).toRingHom x ∈ S'",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nH : ∀ (i : ι), ∃ n, s i ^ n • x ∈ S'\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nH : ∀ (i : ι), ∃ n, s i ^ n • x ∈ S'\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "choose n hn using H",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nH : ∀ (i : ι), ∃ n, s i ^ n • x ∈ S'\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "let s' : ι → S' := fun x => ⟨s x, hs x⟩",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "refine_lift\n let s' : ι → S' := fun x => ⟨s x, hs x⟩;\n ?_",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (let s' : ι → S' := fun x => ⟨s x, hs x⟩;\n ?_);\n rotate_right)",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "refine\n no_implicit_lambda%\n (let s' : ι → S' := fun x => ⟨s x, hs x⟩;\n ?_)",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "rotate_right",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "let l' : ι → S' := fun x => ⟨l x, hl x⟩",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "refine_lift\n let l' : ι → S' := fun x => ⟨l x, hl x⟩;\n ?_",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (let l' : ι → S' := fun x => ⟨l x, hl x⟩;\n ?_);\n rotate_right)",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "refine\n no_implicit_lambda%\n (let l' : ι → S' := fun x => ⟨l x, hl x⟩;\n ?_)",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "rotate_right",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "have e' : ∑ i ∈ ι', l' i * s' i = 1 := by\n ext\n show S'.subtype (∑ i ∈ ι', l' i * s' i) = 1\n simpa only [map_sum, map_mul] using e",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have e' : ∑ i ∈ ι', l' i * s' i = 1 := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( ext\n show S'.subtype (∑ i ∈ ι', l' i * s' i) = 1\n simpa only [map_sum, map_mul] using e)",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "refine\n no_implicit_lambda%\n (have e' : ∑ i ∈ ι', l' i * s' i = 1 := ?body✝;\n ?_)",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case body\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ ∑ i ∈ ι', l' i * s' i = 1\n---\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( ext\n show S'.subtype (∑ i ∈ ι', l' i * s' i) = 1\n simpa only [map_sum, map_mul] using e)",
"before_state": "case body\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ ∑ i ∈ ι', l' i * s' i = 1\n---\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "with_annotate_state\"by\"\n ( ext\n show S'.subtype (∑ i ∈ ι', l' i * s' i) = 1\n simpa only [map_sum, map_mul] using e)",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ ∑ i ∈ ι', l' i * s' i = 1",
"after_state": "No Goals!"
},
{
"line": "ext",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ ∑ i ∈ ι', l' i * s' i = 1",
"after_state": "case a\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ ↑(∑ i ∈ ι', l' i * s' i) = ↑1"
},
{
"line": "show S'.subtype (∑ i ∈ ι', l' i * s' i) = 1",
"before_state": "case a\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ ↑(∑ i ∈ ι', l' i * s' i) = ↑1",
"after_state": "case a\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ S'.subtype (∑ i ∈ ι', l' i * s' i) = 1"
},
{
"line": "refine_lift show S'.subtype (∑ i ∈ ι', l' i * s' i) = 1 from ?_",
"before_state": "case a\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ ↑(∑ i ∈ ι', l' i * s' i) = ↑1",
"after_state": "case a\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ S'.subtype (∑ i ∈ ι', l' i * s' i) = 1"
},
{
"line": "focus (refine no_implicit_lambda% (show S'.subtype (∑ i ∈ ι', l' i * s' i) = 1 from ?_); rotate_right)",
"before_state": "case a\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ ↑(∑ i ∈ ι', l' i * s' i) = ↑1",
"after_state": "case a\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ S'.subtype (∑ i ∈ ι', l' i * s' i) = 1"
},
{
"line": "refine no_implicit_lambda% (show S'.subtype (∑ i ∈ ι', l' i * s' i) = 1 from ?_)",
"before_state": "case a\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ ↑(∑ i ∈ ι', l' i * s' i) = ↑1",
"after_state": "case a\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ S'.subtype (∑ i ∈ ι', l' i * s' i) = 1"
},
{
"line": "rotate_right",
"before_state": "case a\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ S'.subtype (∑ i ∈ ι', l' i * s' i) = 1",
"after_state": "case a\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ S'.subtype (∑ i ∈ ι', l' i * s' i) = 1"
},
{
"line": "simpa only [map_sum, map_mul] using e",
"before_state": "case a\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\n⊢ S'.subtype (∑ i ∈ ι', l' i * s' i) = 1",
"after_state": "No Goals!"
},
{
"line": "have : Ideal.span (s' '' ι') = ⊤ := by\n rw [Ideal.eq_top_iff_one]\n rw [← e']\n apply sum_mem\n intros i hi\n exact Ideal.mul_mem_left _ _ <| Ideal.subset_span <| Set.mem_image_of_mem s' hi",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have : Ideal.span (s' '' ι') = ⊤ := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( rw [Ideal.eq_top_iff_one]\n rw [← e']\n apply sum_mem\n intros i hi\n exact Ideal.mul_mem_left _ _ <| Ideal.subset_span <| Set.mem_image_of_mem s' hi)",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "refine\n no_implicit_lambda%\n (have : Ideal.span (s' '' ι') = ⊤ := ?body✝;\n ?_)",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case body\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ Ideal.span (s' '' ↑ι') = ⊤\n---\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( rw [Ideal.eq_top_iff_one]\n rw [← e']\n apply sum_mem\n intros i hi\n exact Ideal.mul_mem_left _ _ <| Ideal.subset_span <| Set.mem_image_of_mem s' hi)",
"before_state": "case body\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ Ideal.span (s' '' ↑ι') = ⊤\n---\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "with_annotate_state\"by\"\n ( rw [Ideal.eq_top_iff_one]\n rw [← e']\n apply sum_mem\n intros i hi\n exact Ideal.mul_mem_left _ _ <| Ideal.subset_span <| Set.mem_image_of_mem s' hi)",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ Ideal.span (s' '' ↑ι') = ⊤",
"after_state": "No Goals!"
},
{
"line": "rw [Ideal.eq_top_iff_one]",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ Ideal.span (s' '' ↑ι') = ⊤",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ 1 ∈ Ideal.span (s' '' ↑ι')"
},
{
"line": "rewrite [Ideal.eq_top_iff_one]",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ Ideal.span (s' '' ↑ι') = ⊤",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ 1 ∈ Ideal.span (s' '' ↑ι')"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ 1 ∈ Ideal.span (s' '' ↑ι')",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ 1 ∈ Ideal.span (s' '' ↑ι')"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ 1 ∈ Ideal.span (s' '' ↑ι')",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ 1 ∈ Ideal.span (s' '' ↑ι')"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ 1 ∈ Ideal.span (s' '' ↑ι')",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ 1 ∈ Ideal.span (s' '' ↑ι')"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ 1 ∈ Ideal.span (s' '' ↑ι')",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ 1 ∈ Ideal.span (s' '' ↑ι')"
},
{
"line": "rfl",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ 1 ∈ Ideal.span (s' '' ↑ι')",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ 1 ∈ Ideal.span (s' '' ↑ι')"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ 1 ∈ Ideal.span (s' '' ↑ι')",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ 1 ∈ Ideal.span (s' '' ↑ι')"
},
{
"line": "skip",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ 1 ∈ Ideal.span (s' '' ↑ι')",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ 1 ∈ Ideal.span (s' '' ↑ι')"
},
{
"line": "rw [← e']",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ 1 ∈ Ideal.span (s' '' ↑ι')",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ ∑ i ∈ ι', l' i * s' i ∈ Ideal.span (s' '' ↑ι')"
},
{
"line": "rewrite [← e']",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ 1 ∈ Ideal.span (s' '' ↑ι')",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ ∑ i ∈ ι', l' i * s' i ∈ Ideal.span (s' '' ↑ι')"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ ∑ i ∈ ι', l' i * s' i ∈ Ideal.span (s' '' ↑ι')",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ ∑ i ∈ ι', l' i * s' i ∈ Ideal.span (s' '' ↑ι')"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ ∑ i ∈ ι', l' i * s' i ∈ Ideal.span (s' '' ↑ι')",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ ∑ i ∈ ι', l' i * s' i ∈ Ideal.span (s' '' ↑ι')"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ ∑ i ∈ ι', l' i * s' i ∈ Ideal.span (s' '' ↑ι')",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ ∑ i ∈ ι', l' i * s' i ∈ Ideal.span (s' '' ↑ι')"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ ∑ i ∈ ι', l' i * s' i ∈ Ideal.span (s' '' ↑ι')",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ ∑ i ∈ ι', l' i * s' i ∈ Ideal.span (s' '' ↑ι')"
},
{
"line": "rfl",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ ∑ i ∈ ι', l' i * s' i ∈ Ideal.span (s' '' ↑ι')",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ ∑ i ∈ ι', l' i * s' i ∈ Ideal.span (s' '' ↑ι')"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ ∑ i ∈ ι', l' i * s' i ∈ Ideal.span (s' '' ↑ι')",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ ∑ i ∈ ι', l' i * s' i ∈ Ideal.span (s' '' ↑ι')"
},
{
"line": "skip",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ ∑ i ∈ ι', l' i * s' i ∈ Ideal.span (s' '' ↑ι')",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ ∑ i ∈ ι', l' i * s' i ∈ Ideal.span (s' '' ↑ι')"
},
{
"line": "apply sum_mem",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ ∑ i ∈ ι', l' i * s' i ∈ Ideal.span (s' '' ↑ι')",
"after_state": "case h\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ ∀ c ∈ ι', l' c * s' c ∈ Ideal.span (s' '' ↑ι')"
},
{
"line": "intros i hi",
"before_state": "case h\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\n⊢ ∀ c ∈ ι', l' c * s' c ∈ Ideal.span (s' '' ↑ι')",
"after_state": "case h\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\ni : ι\nhi : i ∈ ι'\n⊢ l' i * s' i ∈ Ideal.span (s' '' ↑ι')"
},
{
"line": "exact Ideal.mul_mem_left _ _ <| Ideal.subset_span <| Set.mem_image_of_mem s' hi",
"before_state": "case h\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\ni : ι\nhi : i ∈ ι'\n⊢ l' i * s' i ∈ Ideal.span (s' '' ↑ι')",
"after_state": "No Goals!"
},
{
"line": "let N := ι'.sup n",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "refine_lift\n let N := ι'.sup n;\n ?_",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (let N := ι'.sup n;\n ?_);\n rotate_right)",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "refine\n no_implicit_lambda%\n (let N := ι'.sup n;\n ?_)",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "rotate_right",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "have hN := Ideal.span_pow_eq_top _ this N",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "refine_lift\n have hN := Ideal.span_pow_eq_top _ this N;\n ?_",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hN := Ideal.span_pow_eq_top _ this N;\n ?_);\n rotate_right)",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "refine\n no_implicit_lambda%\n (have hN := Ideal.span_pow_eq_top _ this N;\n ?_)",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "rotate_right",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "apply (Algebra.ofId S' S).range.toSubmodule.mem_of_span_top_of_smul_mem _ hN",
"before_state": "R : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\n⊢ x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\n⊢ ∀ (r : ↑((fun x => x ^ N) '' (s' '' ↑ι'))), ↑r • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "rintro ⟨_, _, ⟨i, hi, rfl⟩, rfl⟩",
"before_state": "case H\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\n⊢ ∀ (r : ↑((fun x => x ^ N) '' (s' '' ↑ι'))), ↑r • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ ↑⟨(fun x => x ^ N) (s' i), ⋯⟩ • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "change s' i ^ N • x ∈ _",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ ↑⟨(fun x => x ^ N) (s' i), ⋯⟩ • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ N • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "rw [← tsub_add_cancel_of_le (show n i ≤ N from Finset.le_sup hi)]",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ N • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i + n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "rewrite [← tsub_add_cancel_of_le (show n i ≤ N from Finset.le_sup hi)]",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ N • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i + n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i + n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i + n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i + n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i + n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i + n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i + n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "with_reducible rfl",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i + n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i + n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "rfl",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i + n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i + n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "apply_rfl",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i + n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i + n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "skip",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i + n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i + n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "rw [pow_add]",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i + n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ (s' i ^ (N - n i) * s' i ^ n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "rewrite [pow_add]",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i + n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ (s' i ^ (N - n i) * s' i ^ n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ (s' i ^ (N - n i) * s' i ^ n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ (s' i ^ (N - n i) * s' i ^ n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ (s' i ^ (N - n i) * s' i ^ n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ (s' i ^ (N - n i) * s' i ^ n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ (s' i ^ (N - n i) * s' i ^ n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ (s' i ^ (N - n i) * s' i ^ n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "with_reducible rfl",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ (s' i ^ (N - n i) * s' i ^ n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ (s' i ^ (N - n i) * s' i ^ n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "rfl",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ (s' i ^ (N - n i) * s' i ^ n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ (s' i ^ (N - n i) * s' i ^ n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "apply_rfl",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ (s' i ^ (N - n i) * s' i ^ n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ (s' i ^ (N - n i) * s' i ^ n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "skip",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ (s' i ^ (N - n i) * s' i ^ n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ (s' i ^ (N - n i) * s' i ^ n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "rw [mul_smul]",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ (s' i ^ (N - n i) * s' i ^ n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i) • s' i ^ n i • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "rewrite [mul_smul]",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ (s' i ^ (N - n i) * s' i ^ n i) • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i) • s' i ^ n i • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i) • s' i ^ n i • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i) • s' i ^ n i • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i) • s' i ^ n i • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i) • s' i ^ n i • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i) • s' i ^ n i • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i) • s' i ^ n i • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "with_reducible rfl",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i) • s' i ^ n i • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i) • s' i ^ n i • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "rfl",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i) • s' i ^ n i • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i) • s' i ^ n i • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "apply_rfl",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i) • s' i ^ n i • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i) • s' i ^ n i • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "skip",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i) • s' i ^ n i • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i) • s' i ^ n i • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "refine Submodule.smul_mem _ (⟨_, pow_mem (hs i) _⟩ : S') ?_",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ (N - n i) • s' i ^ n i • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ n i • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range"
},
{
"line": "exact ⟨⟨_, hn i⟩, rfl⟩",
"before_state": "case H.mk.intro.intro.intro.intro\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nS' : Subalgebra R S\nι : Type u_6\nι' : Finset ι\ns l : ι → S\ne : ∑ i ∈ ι', l i * s i = 1\nhs : ∀ (i : ι), s i ∈ S'\nhl : ∀ (i : ι), l i ∈ S'\nx : S\nn : ι → ℕ\nhn : ∀ (i : ι), s i ^ n i • x ∈ S'\ns' : ι → ↥S' := fun x => ⟨s x, ⋯⟩\nl' : ι → ↥S' := fun x => ⟨l x, ⋯⟩\ne' : ∑ i ∈ ι', l' i * s' i = 1\nthis : Ideal.span (s' '' ↑ι') = ⊤\nN : ℕ := ι'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' ↑ι')) = ⊤\ni : ι\nhi : i ∈ ↑ι'\n⊢ s' i ^ n i • x ∈ Subalgebra.toSubmodule (ofId (↥S') S).range",
"after_state": "No Goals!"
}
] |
theorem mul_toSubmodule_le (S T : Subalgebra R A) :
(Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by
rw [Submodule.mul_le]
intro y hy z hz
show y * z ∈ S ⊔ T
exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean
|
{
"open": [],
"variables": [
"{R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]"
]
}
|
[
{
"line": "rw [Submodule.mul_le]",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ Subalgebra.toSubmodule S * Subalgebra.toSubmodule T ≤ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ m ∈ Subalgebra.toSubmodule S, ∀ n ∈ Subalgebra.toSubmodule T, m * n ∈ Subalgebra.toSubmodule (S ⊔ T)"
},
{
"line": "rewrite [Submodule.mul_le]",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ Subalgebra.toSubmodule S * Subalgebra.toSubmodule T ≤ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ m ∈ Subalgebra.toSubmodule S, ∀ n ∈ Subalgebra.toSubmodule T, m * n ∈ Subalgebra.toSubmodule (S ⊔ T)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ m ∈ Subalgebra.toSubmodule S, ∀ n ∈ Subalgebra.toSubmodule T, m * n ∈ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ m ∈ Subalgebra.toSubmodule S, ∀ n ∈ Subalgebra.toSubmodule T, m * n ∈ Subalgebra.toSubmodule (S ⊔ T)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ m ∈ Subalgebra.toSubmodule S, ∀ n ∈ Subalgebra.toSubmodule T, m * n ∈ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ m ∈ Subalgebra.toSubmodule S, ∀ n ∈ Subalgebra.toSubmodule T, m * n ∈ Subalgebra.toSubmodule (S ⊔ T)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ m ∈ Subalgebra.toSubmodule S, ∀ n ∈ Subalgebra.toSubmodule T, m * n ∈ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ m ∈ Subalgebra.toSubmodule S, ∀ n ∈ Subalgebra.toSubmodule T, m * n ∈ Subalgebra.toSubmodule (S ⊔ T)"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ m ∈ Subalgebra.toSubmodule S, ∀ n ∈ Subalgebra.toSubmodule T, m * n ∈ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ m ∈ Subalgebra.toSubmodule S, ∀ n ∈ Subalgebra.toSubmodule T, m * n ∈ Subalgebra.toSubmodule (S ⊔ T)"
},
{
"line": "rfl",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ m ∈ Subalgebra.toSubmodule S, ∀ n ∈ Subalgebra.toSubmodule T, m * n ∈ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ m ∈ Subalgebra.toSubmodule S, ∀ n ∈ Subalgebra.toSubmodule T, m * n ∈ Subalgebra.toSubmodule (S ⊔ T)"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ m ∈ Subalgebra.toSubmodule S, ∀ n ∈ Subalgebra.toSubmodule T, m * n ∈ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ m ∈ Subalgebra.toSubmodule S, ∀ n ∈ Subalgebra.toSubmodule T, m * n ∈ Subalgebra.toSubmodule (S ⊔ T)"
},
{
"line": "skip",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ m ∈ Subalgebra.toSubmodule S, ∀ n ∈ Subalgebra.toSubmodule T, m * n ∈ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ m ∈ Subalgebra.toSubmodule S, ∀ n ∈ Subalgebra.toSubmodule T, m * n ∈ Subalgebra.toSubmodule (S ⊔ T)"
},
{
"line": "intro y hy z hz",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ m ∈ Subalgebra.toSubmodule S, ∀ n ∈ Subalgebra.toSubmodule T, m * n ∈ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\nz : A\nhz : z ∈ Subalgebra.toSubmodule T\n⊢ y * z ∈ Subalgebra.toSubmodule (S ⊔ T)"
},
{
"line": "intro y;\n intro hy z hz",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ m ∈ Subalgebra.toSubmodule S, ∀ n ∈ Subalgebra.toSubmodule T, m * n ∈ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\nz : A\nhz : z ∈ Subalgebra.toSubmodule T\n⊢ y * z ∈ Subalgebra.toSubmodule (S ⊔ T)"
},
{
"line": "intro y",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ m ∈ Subalgebra.toSubmodule S, ∀ n ∈ Subalgebra.toSubmodule T, m * n ∈ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\n⊢ y ∈ Subalgebra.toSubmodule S → ∀ n ∈ Subalgebra.toSubmodule T, y * n ∈ Subalgebra.toSubmodule (S ⊔ T)"
},
{
"line": "intro hy z hz",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\n⊢ y ∈ Subalgebra.toSubmodule S → ∀ n ∈ Subalgebra.toSubmodule T, y * n ∈ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\nz : A\nhz : z ∈ Subalgebra.toSubmodule T\n⊢ y * z ∈ Subalgebra.toSubmodule (S ⊔ T)"
},
{
"line": "intro hy;\n intro z hz",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\n⊢ y ∈ Subalgebra.toSubmodule S → ∀ n ∈ Subalgebra.toSubmodule T, y * n ∈ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\nz : A\nhz : z ∈ Subalgebra.toSubmodule T\n⊢ y * z ∈ Subalgebra.toSubmodule (S ⊔ T)"
},
{
"line": "intro hy",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\n⊢ y ∈ Subalgebra.toSubmodule S → ∀ n ∈ Subalgebra.toSubmodule T, y * n ∈ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\n⊢ ∀ n ∈ Subalgebra.toSubmodule T, y * n ∈ Subalgebra.toSubmodule (S ⊔ T)"
},
{
"line": "intro z hz",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\n⊢ ∀ n ∈ Subalgebra.toSubmodule T, y * n ∈ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\nz : A\nhz : z ∈ Subalgebra.toSubmodule T\n⊢ y * z ∈ Subalgebra.toSubmodule (S ⊔ T)"
},
{
"line": "intro z;\n intro hz",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\n⊢ ∀ n ∈ Subalgebra.toSubmodule T, y * n ∈ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\nz : A\nhz : z ∈ Subalgebra.toSubmodule T\n⊢ y * z ∈ Subalgebra.toSubmodule (S ⊔ T)"
},
{
"line": "intro z",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\n⊢ ∀ n ∈ Subalgebra.toSubmodule T, y * n ∈ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\nz : A\n⊢ z ∈ Subalgebra.toSubmodule T → y * z ∈ Subalgebra.toSubmodule (S ⊔ T)"
},
{
"line": "intro hz",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\nz : A\n⊢ z ∈ Subalgebra.toSubmodule T → y * z ∈ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\nz : A\nhz : z ∈ Subalgebra.toSubmodule T\n⊢ y * z ∈ Subalgebra.toSubmodule (S ⊔ T)"
},
{
"line": "show y * z ∈ S ⊔ T",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\nz : A\nhz : z ∈ Subalgebra.toSubmodule T\n⊢ y * z ∈ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\nz : A\nhz : z ∈ Subalgebra.toSubmodule T\n⊢ y * z ∈ S ⊔ T"
},
{
"line": "refine_lift show y * z ∈ S ⊔ T from ?_",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\nz : A\nhz : z ∈ Subalgebra.toSubmodule T\n⊢ y * z ∈ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\nz : A\nhz : z ∈ Subalgebra.toSubmodule T\n⊢ y * z ∈ S ⊔ T"
},
{
"line": "focus (refine no_implicit_lambda% (show y * z ∈ S ⊔ T from ?_); rotate_right)",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\nz : A\nhz : z ∈ Subalgebra.toSubmodule T\n⊢ y * z ∈ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\nz : A\nhz : z ∈ Subalgebra.toSubmodule T\n⊢ y * z ∈ S ⊔ T"
},
{
"line": "refine no_implicit_lambda% (show y * z ∈ S ⊔ T from ?_)",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\nz : A\nhz : z ∈ Subalgebra.toSubmodule T\n⊢ y * z ∈ Subalgebra.toSubmodule (S ⊔ T)",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\nz : A\nhz : z ∈ Subalgebra.toSubmodule T\n⊢ y * z ∈ S ⊔ T"
},
{
"line": "rotate_right",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\nz : A\nhz : z ∈ Subalgebra.toSubmodule T\n⊢ y * z ∈ S ⊔ T",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\nz : A\nhz : z ∈ Subalgebra.toSubmodule T\n⊢ y * z ∈ S ⊔ T"
},
{
"line": "exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ Subalgebra.toSubmodule S\nz : A\nhz : z ∈ Subalgebra.toSubmodule T\n⊢ y * z ∈ S ⊔ T",
"after_state": "No Goals!"
}
] |
theorem isIdempotentElem_toSubmodule (S : Subalgebra R A) :
IsIdempotentElem S.toSubmodule := by
apply le_antisymm
· refine (mul_toSubmodule_le _ _).trans_eq ?_
rw [sup_idem]
· intro x hx1
rw [← mul_one x]
exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean
|
{
"open": [],
"variables": [
"{R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]"
]
}
|
[
{
"line": "apply le_antisymm",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ IsIdempotentElem (Subalgebra.toSubmodule S)",
"after_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S ≤ Subalgebra.toSubmodule S\n---\ncase a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ Subalgebra.toSubmodule S ≤ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S"
},
{
"line": "refine (mul_toSubmodule_le _ _).trans_eq ?_",
"before_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S ≤ Subalgebra.toSubmodule S",
"after_state": "No Goals!"
},
{
"line": "intro x hx1",
"before_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ Subalgebra.toSubmodule S ≤ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S",
"after_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ Subalgebra.toSubmodule S\n⊢ x ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S"
},
{
"line": "intro x;\n intro hx1",
"before_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ Subalgebra.toSubmodule S ≤ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S",
"after_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ Subalgebra.toSubmodule S\n⊢ x ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S"
},
{
"line": "intro x",
"before_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ Subalgebra.toSubmodule S ≤ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S",
"after_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\n⊢ x ∈ Subalgebra.toSubmodule S → x ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S"
},
{
"line": "intro hx1",
"before_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\n⊢ x ∈ Subalgebra.toSubmodule S → x ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S",
"after_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ Subalgebra.toSubmodule S\n⊢ x ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S"
},
{
"line": "rw [← mul_one x]",
"before_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ Subalgebra.toSubmodule S\n⊢ x ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S",
"after_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ Subalgebra.toSubmodule S\n⊢ x * 1 ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S"
},
{
"line": "rewrite [← mul_one x]",
"before_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ Subalgebra.toSubmodule S\n⊢ x ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S",
"after_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ Subalgebra.toSubmodule S\n⊢ x * 1 ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ Subalgebra.toSubmodule S\n⊢ x * 1 ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S",
"after_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ Subalgebra.toSubmodule S\n⊢ x * 1 ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ Subalgebra.toSubmodule S\n⊢ x * 1 ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S",
"after_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ Subalgebra.toSubmodule S\n⊢ x * 1 ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ Subalgebra.toSubmodule S\n⊢ x * 1 ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S",
"after_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ Subalgebra.toSubmodule S\n⊢ x * 1 ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S"
},
{
"line": "with_reducible rfl",
"before_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ Subalgebra.toSubmodule S\n⊢ x * 1 ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S",
"after_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ Subalgebra.toSubmodule S\n⊢ x * 1 ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S"
},
{
"line": "rfl",
"before_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ Subalgebra.toSubmodule S\n⊢ x * 1 ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S",
"after_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ Subalgebra.toSubmodule S\n⊢ x * 1 ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S"
},
{
"line": "apply_rfl",
"before_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ Subalgebra.toSubmodule S\n⊢ x * 1 ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S",
"after_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ Subalgebra.toSubmodule S\n⊢ x * 1 ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S"
},
{
"line": "skip",
"before_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ Subalgebra.toSubmodule S\n⊢ x * 1 ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S",
"after_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ Subalgebra.toSubmodule S\n⊢ x * 1 ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S"
},
{
"line": "exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)",
"before_state": "case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ Subalgebra.toSubmodule S\n⊢ x * 1 ∈ Subalgebra.toSubmodule S * Subalgebra.toSubmodule S",
"after_state": "No Goals!"
}
] |
theorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ :=
SetLike.coe_injective <| by dsimp -- Porting note: why does `rfl` not work instead of `by dsimp`?
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Algebra/Subalgebra/Tower.lean
|
{
"open": [
"Pointwise",
"IsScalarTower"
],
"variables": [
"(R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)",
"[CommSemiring R] [Semiring A] [Algebra R A]",
"[AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]",
"{A}",
"[CommSemiring R] [CommSemiring S] [Semiring A]",
"[Algebra R S] [Algebra S A]",
"[Algebra R A] [IsScalarTower R S A]",
"{S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]",
"[Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]",
"[IsScalarTower R S A] [IsScalarTower R S B]"
]
}
|
[
{
"line": "dsimp\n -- Porting note: why does `rfl` not work instead of `by dsimp`?",
"before_state": "x✝ : Sort u_1\nrestrictScalars : x✝\n⊢ ↑?m.19805 = ↑?m.19806",
"after_state": "No Goals!"
}
] |
lemma mulLeftRight_comp_congr (e : A ≃ₐ[R] B) :
(AlgHom.mulLeftRight R B).comp (Algebra.TensorProduct.congr e e.op).toAlgHom =
(e.toLinearEquiv.algConj R).toAlgHom.comp (AlgHom.mulLeftRight R A) := by
apply AlgHom.ext
intro a
induction a using TensorProduct.induction_on with
| zero => simp
| tmul a a' =>
ext; simp [AlgHom.mulLeftRight_apply, LinearEquiv.algConj, LinearEquiv.conj]
| add _ _ _ _ => simp_all [map_add]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Azumaya/Basic.lean
|
{
"open": [
"scoped TensorProduct",
"MulOpposite"
],
"variables": [
"(R A B : Type*) [CommSemiring R] [Ring A] [Ring B] [Algebra R A] [Algebra R B]"
]
}
|
[
{
"line": "apply AlgHom.ext",
"before_state": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ne : A ≃ₐ[R] B\n⊢ (AlgHom.mulLeftRight R B).comp ↑(Algebra.TensorProduct.congr e (AlgEquiv.op e)) =\n (↑(LinearEquiv.algConj R e.toLinearEquiv)).comp (AlgHom.mulLeftRight R A)",
"after_state": "case H\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ne : A ≃ₐ[R] B\n⊢ ∀ (x : A ⊗[R] Aᵐᵒᵖ),\n ((AlgHom.mulLeftRight R B).comp ↑(Algebra.TensorProduct.congr e (AlgEquiv.op e))) x =\n ((↑(LinearEquiv.algConj R e.toLinearEquiv)).comp (AlgHom.mulLeftRight R A)) x"
},
{
"line": "intro a",
"before_state": "case H\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ne : A ≃ₐ[R] B\n⊢ ∀ (x : A ⊗[R] Aᵐᵒᵖ),\n ((AlgHom.mulLeftRight R B).comp ↑(Algebra.TensorProduct.congr e (AlgEquiv.op e))) x =\n ((↑(LinearEquiv.algConj R e.toLinearEquiv)).comp (AlgHom.mulLeftRight R A)) x",
"after_state": "case H\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ne : A ≃ₐ[R] B\na : A ⊗[R] Aᵐᵒᵖ\n⊢ ((AlgHom.mulLeftRight R B).comp ↑(Algebra.TensorProduct.congr e (AlgEquiv.op e))) a =\n ((↑(LinearEquiv.algConj R e.toLinearEquiv)).comp (AlgHom.mulLeftRight R A)) a"
},
{
"line": "induction a using TensorProduct.induction_on with\n| zero => simp\n| tmul a a' => ext; simp [AlgHom.mulLeftRight_apply, LinearEquiv.algConj, LinearEquiv.conj]\n| add _ _ _ _ => simp_all [map_add]",
"before_state": "case H\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ne : A ≃ₐ[R] B\na : A ⊗[R] Aᵐᵒᵖ\n⊢ ((AlgHom.mulLeftRight R B).comp ↑(Algebra.TensorProduct.congr e (AlgEquiv.op e))) a =\n ((↑(LinearEquiv.algConj R e.toLinearEquiv)).comp (AlgHom.mulLeftRight R A)) a",
"after_state": "No Goals!"
},
{
"line": "induction a using TensorProduct.induction_on with\n| zero => simp\n| tmul a a' => ext; simp [AlgHom.mulLeftRight_apply, LinearEquiv.algConj, LinearEquiv.conj]\n| add _ _ _ _ => simp_all [map_add]",
"before_state": "case H\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ne : A ≃ₐ[R] B\na : A ⊗[R] Aᵐᵒᵖ\n⊢ ((AlgHom.mulLeftRight R B).comp ↑(Algebra.TensorProduct.congr e (AlgEquiv.op e))) a =\n ((↑(LinearEquiv.algConj R e.toLinearEquiv)).comp (AlgHom.mulLeftRight R A)) a",
"after_state": "case H\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ne : A ≃ₐ[R] B\na : A ⊗[R] Aᵐᵒᵖ\n⊢ ((AlgHom.mulLeftRight R B).comp ↑(Algebra.TensorProduct.congr e (AlgEquiv.op e))) a =\n ((↑(LinearEquiv.algConj R e.toLinearEquiv)).comp (AlgHom.mulLeftRight R A)) a"
},
{
"line": "simp",
"before_state": "case H.zero\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ne : A ≃ₐ[R] B\n⊢ ((AlgHom.mulLeftRight R B).comp ↑(Algebra.TensorProduct.congr e (AlgEquiv.op e))) 0 =\n ((↑(LinearEquiv.algConj R e.toLinearEquiv)).comp (AlgHom.mulLeftRight R A)) 0",
"after_state": "No Goals!"
},
{
"line": "ext",
"before_state": "case H.tmul\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ne : A ≃ₐ[R] B\na : A\na' : Aᵐᵒᵖ\n⊢ ((AlgHom.mulLeftRight R B).comp ↑(Algebra.TensorProduct.congr e (AlgEquiv.op e))) (a ⊗ₜ[R] a') =\n ((↑(LinearEquiv.algConj R e.toLinearEquiv)).comp (AlgHom.mulLeftRight R A)) (a ⊗ₜ[R] a')",
"after_state": "case H.tmul.h\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ne : A ≃ₐ[R] B\na : A\na' : Aᵐᵒᵖ\nx✝ : B\n⊢ (((AlgHom.mulLeftRight R B).comp ↑(Algebra.TensorProduct.congr e (AlgEquiv.op e))) (a ⊗ₜ[R] a')) x✝ =\n (((↑(LinearEquiv.algConj R e.toLinearEquiv)).comp (AlgHom.mulLeftRight R A)) (a ⊗ₜ[R] a')) x✝"
},
{
"line": "simp [AlgHom.mulLeftRight_apply, LinearEquiv.algConj, LinearEquiv.conj]",
"before_state": "case H.tmul.h\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ne : A ≃ₐ[R] B\na : A\na' : Aᵐᵒᵖ\nx✝ : B\n⊢ (((AlgHom.mulLeftRight R B).comp ↑(Algebra.TensorProduct.congr e (AlgEquiv.op e))) (a ⊗ₜ[R] a')) x✝ =\n (((↑(LinearEquiv.algConj R e.toLinearEquiv)).comp (AlgHom.mulLeftRight R A)) (a ⊗ₜ[R] a')) x✝",
"after_state": "No Goals!"
},
{
"line": "simp_all [map_add]",
"before_state": "case H.add\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ne : A ≃ₐ[R] B\nx✝ y✝ : A ⊗[R] Aᵐᵒᵖ\na✝¹ :\n ((AlgHom.mulLeftRight R B).comp ↑(Algebra.TensorProduct.congr e (AlgEquiv.op e))) x✝ =\n ((↑(LinearEquiv.algConj R e.toLinearEquiv)).comp (AlgHom.mulLeftRight R A)) x✝\na✝ :\n ((AlgHom.mulLeftRight R B).comp ↑(Algebra.TensorProduct.congr e (AlgEquiv.op e))) y✝ =\n ((↑(LinearEquiv.algConj R e.toLinearEquiv)).comp (AlgHom.mulLeftRight R A)) y✝\n⊢ ((AlgHom.mulLeftRight R B).comp ↑(Algebra.TensorProduct.congr e (AlgEquiv.op e))) (x✝ + y✝) =\n ((↑(LinearEquiv.algConj R e.toLinearEquiv)).comp (AlgHom.mulLeftRight R A)) (x✝ + y✝)",
"after_state": "No Goals!"
}
] |
theorem matrix [Nonempty n] : IsAzumaya R (Matrix n n R) where
eq_of_smul_eq_smul := by nontriviality R; exact eq_of_smul_eq_smul
bij := Function.bijective_iff_has_inverse.mpr
⟨AlgHom.mulLeftRightMatrix_inv R n,
DFunLike.congr_fun (AlgHom.mulLeftRightMatrix.inv_comp R n),
DFunLike.congr_fun (AlgHom.mulLeftRightMatrix.comp_inv R n)⟩
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Azumaya/Matrix.lean
|
{
"open": [
"scoped TensorProduct",
"Matrix MulOpposite"
],
"variables": [
"(R n : Type*) [CommSemiring R] [Fintype n] [DecidableEq n]"
]
}
|
[
{
"line": "nontriviality R",
"before_state": "R : Type u_1\nn : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : Nonempty n\n⊢ ∀ {m₁ m₂ : R}, (∀ (a : Matrix n n R), m₁ • a = m₂ • a) → m₁ = m₂",
"after_state": "R : Type u_1\nn : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : Nonempty n\na✝ : Nontrivial R\n⊢ ∀ {m₁ m₂ : R}, (∀ (a : Matrix n n R), m₁ • a = m₂ • a) → m₁ = m₂"
},
{
"line": "simp [nontriviality✝]",
"before_state": "R : Type u_1\nn : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : Nonempty n\na✝ : Subsingleton R\n⊢ ∀ {m₁ m₂ : R}, (∀ (a : Matrix n n R), m₁ • a = m₂ • a) → m₁ = m₂",
"after_state": "No Goals!"
},
{
"line": "exact eq_of_smul_eq_smul",
"before_state": "R : Type u_1\nn : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : Nonempty n\na✝ : Nontrivial R\n⊢ ∀ {m₁ m₂ : R}, (∀ (a : Matrix n n R), m₁ • a = m₂ • a) → m₁ = m₂",
"after_state": "No Goals!"
}
] |
theorem exists_mem_multiset_dvd (hp : Prime p) {s : Multiset α} : p ∣ s.prod → ∃ a ∈ s, p ∣ a :=
Multiset.induction_on s (fun h => (hp.not_dvd_one h).elim) fun a s ih h =>
have : p ∣ a * s.prod := by simpa using h
match hp.dvd_or_dvd this with
| Or.inl h => ⟨a, Multiset.mem_cons_self a s, h⟩
| Or.inr h =>
let ⟨a, has, h⟩ := ih h
⟨a, Multiset.mem_cons_of_mem has, h⟩
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Associated.lean
|
{
"open": [],
"variables": [
"{α β γ δ : Type*}",
"[CommMonoidWithZero α] {p : α}"
]
}
|
[
{
"line": "simpa using h",
"before_state": "α : Type u_1\ninst✝ : CommMonoidWithZero α\np : α\nhp : Prime p\ns✝ : Multiset α\na : α\ns : Multiset α\nih : p ∣ s.prod → ∃ a ∈ s, p ∣ a\nh : p ∣ (a ::ₘ s).prod\n⊢ p ∣ a * s.prod",
"after_state": "No Goals!"
}
] |
theorem exists_mem_multiset_map_dvd (hp : Prime p) {s : Multiset β} {f : β → α} :
p ∣ (s.map f).prod → ∃ a ∈ s, p ∣ f a := fun h => by
simpa only [exists_prop,Multiset.mem_map,exists_exists_and_eq_and] using
hp.exists_mem_multiset_dvd h
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Associated.lean
|
{
"open": [],
"variables": [
"{α β γ δ : Type*}",
"[CommMonoidWithZero α] {p : α}"
]
}
|
[
{
"line": "simpa only [exists_prop, Multiset.mem_map, exists_exists_and_eq_and] using hp.exists_mem_multiset_dvd h",
"before_state": "α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoidWithZero α\np : α\nhp : Prime p\ns : Multiset β\nf : β → α\nh : p ∣ (Multiset.map f s).prod\n⊢ ∃ a ∈ s, p ∣ f a",
"after_state": "No Goals!"
}
] |
lemma expect_sum_comm (s : Finset ι) (t : Finset κ) (f : ι → κ → M) :
𝔼 i ∈ s, ∑ j ∈ t, f i j = ∑ j ∈ t, 𝔼 i ∈ s, f i j := by
simpa only [expect,smul_sum] using sum_comm
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Expect.lean
|
{
"open": [
"Finset Function",
"Fintype (card)",
"scoped Pointwise",
"Batteries.ExtendedBinder Lean Meta",
"Lean Meta Parser.Term PrettyPrinter.Delaborator SubExpr",
"Batteries.ExtendedBinder",
"scoped BigOperators"
],
"variables": [
"{ι κ M N : Type*}",
"[AddCommMonoid M] [Module ℚ≥0 M] [AddCommMonoid N] [Module ℚ≥0 N] {s t : Finset ι}"
]
}
|
[
{
"line": "simpa only [expect, smul_sum] using sum_comm",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ 𝔼 i ∈ s, ∑ j ∈ t, f i j = ∑ j ∈ t, 𝔼 i ∈ s, f i j",
"after_state": "No Goals!"
}
] |
lemma expect_comm (s : Finset ι) (t : Finset κ) (f : ι → κ → M) :
𝔼 i ∈ s, 𝔼 j ∈ t, f i j = 𝔼 j ∈ t, 𝔼 i ∈ s, f i j := by
rw [expect]
rw [expect]
rw [← expect_sum_comm]
rw [← expect_sum_comm]
rw [expect]
rw [expect]
rw [smul_comm]
rw [sum_comm]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Expect.lean
|
{
"open": [
"Finset Function",
"Fintype (card)",
"scoped Pointwise",
"Batteries.ExtendedBinder Lean Meta",
"Lean Meta Parser.Term PrettyPrinter.Delaborator SubExpr",
"Batteries.ExtendedBinder",
"scoped BigOperators"
],
"variables": [
"{ι κ M N : Type*}",
"[AddCommMonoid M] [Module ℚ≥0 M] [AddCommMonoid N] [Module ℚ≥0 N] {s t : Finset ι}"
]
}
|
[
{
"line": "rw [expect]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ 𝔼 i ∈ s, 𝔼 j ∈ t, f i j = 𝔼 j ∈ t, 𝔼 i ∈ s, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = 𝔼 j ∈ t, 𝔼 i ∈ s, f i j"
},
{
"line": "rewrite [expect]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ 𝔼 i ∈ s, 𝔼 j ∈ t, f i j = 𝔼 j ∈ t, 𝔼 i ∈ s, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = 𝔼 j ∈ t, 𝔼 i ∈ s, f i j"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = 𝔼 j ∈ t, 𝔼 i ∈ s, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = 𝔼 j ∈ t, 𝔼 i ∈ s, f i j"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = 𝔼 j ∈ t, 𝔼 i ∈ s, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = 𝔼 j ∈ t, 𝔼 i ∈ s, f i j"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = 𝔼 j ∈ t, 𝔼 i ∈ s, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = 𝔼 j ∈ t, 𝔼 i ∈ s, f i j"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = 𝔼 j ∈ t, 𝔼 i ∈ s, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = 𝔼 j ∈ t, 𝔼 i ∈ s, f i j"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = 𝔼 j ∈ t, 𝔼 i ∈ s, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = 𝔼 j ∈ t, 𝔼 i ∈ s, f i j"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = 𝔼 j ∈ t, 𝔼 i ∈ s, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = 𝔼 j ∈ t, 𝔼 i ∈ s, f i j"
},
{
"line": "skip",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = 𝔼 j ∈ t, 𝔼 i ∈ s, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = 𝔼 j ∈ t, 𝔼 i ∈ s, f i j"
},
{
"line": "rw [expect]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = 𝔼 j ∈ t, 𝔼 i ∈ s, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i"
},
{
"line": "rewrite [expect]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = 𝔼 j ∈ t, 𝔼 i ∈ s, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i"
},
{
"line": "skip",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i"
},
{
"line": "rw [← expect_sum_comm]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i"
},
{
"line": "rewrite [← expect_sum_comm]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, 𝔼 j ∈ t, f i j = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i"
},
{
"line": "skip",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i"
},
{
"line": "rw [← expect_sum_comm]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "rewrite [← expect_sum_comm]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • ∑ i ∈ t, 𝔼 i_1 ∈ s, f i_1 i",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "skip",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "rw [expect]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "rewrite [expect]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • 𝔼 i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "skip",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "rw [expect]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "rewrite [expect]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • 𝔼 i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "skip",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "rw [smul_comm]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "rewrite [smul_comm]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#s))⁻¹ • (↑(#t))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "skip",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "rw [sum_comm]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "No Goals!"
},
{
"line": "rewrite [sum_comm]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ t, ∑ j ∈ s, f j i = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ y ∈ s, ∑ x ∈ t, f y x = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ y ∈ s, ∑ x ∈ t, f y x = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ y ∈ s, ∑ x ∈ t, f y x = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ y ∈ s, ∑ x ∈ t, f y x = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ y ∈ s, ∑ x ∈ t, f y x = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ y ∈ s, ∑ x ∈ t, f y x = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ y ∈ s, ∑ x ∈ t, f y x = (↑(#t))⁻¹ • (↑(#s))⁻¹ • ∑ i ∈ s, ∑ j ∈ t, f i j",
"after_state": "No Goals!"
}
] |
lemma expect_add_distrib (s : Finset ι) (f g : ι → M) :
𝔼 i ∈ s, (f i + g i) = 𝔼 i ∈ s, f i + 𝔼 i ∈ s, g i := by
simp [expect, sum_add_distrib]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Expect.lean
|
{
"open": [
"Finset Function",
"Fintype (card)",
"scoped Pointwise",
"Batteries.ExtendedBinder Lean Meta",
"Lean Meta Parser.Term PrettyPrinter.Delaborator SubExpr",
"Batteries.ExtendedBinder",
"scoped BigOperators"
],
"variables": [
"{ι κ M N : Type*}",
"[AddCommMonoid M] [Module ℚ≥0 M] [AddCommMonoid N] [Module ℚ≥0 N] {s t : Finset ι}"
]
}
|
[
{
"line": "simp [expect, sum_add_distrib]",
"before_state": "ι : Type u_1\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nf g : ι → M\n⊢ 𝔼 i ∈ s, (f i + g i) = 𝔼 i ∈ s, f i + 𝔼 i ∈ s, g i",
"after_state": "No Goals!"
}
] |
lemma expect_add_expect_comm (f₁ f₂ g₁ g₂ : ι → M) :
𝔼 i ∈ s, (f₁ i + f₂ i) + 𝔼 i ∈ s, (g₁ i + g₂ i) =
𝔼 i ∈ s, (f₁ i + g₁ i) + 𝔼 i ∈ s, (f₂ i + g₂ i) := by
simp_rw [expect_add_distrib, add_add_add_comm]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Expect.lean
|
{
"open": [
"Finset Function",
"Fintype (card)",
"scoped Pointwise",
"Batteries.ExtendedBinder Lean Meta",
"Lean Meta Parser.Term PrettyPrinter.Delaborator SubExpr",
"Batteries.ExtendedBinder",
"scoped BigOperators"
],
"variables": [
"{ι κ M N : Type*}",
"[AddCommMonoid M] [Module ℚ≥0 M] [AddCommMonoid N] [Module ℚ≥0 N] {s t : Finset ι}"
]
}
|
[
{
"line": "simp_rw [expect_add_distrib, add_add_add_comm]",
"before_state": "ι : Type u_1\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nf₁ f₂ g₁ g₂ : ι → M\n⊢ 𝔼 i ∈ s, (f₁ i + f₂ i) + 𝔼 i ∈ s, (g₁ i + g₂ i) = 𝔼 i ∈ s, (f₁ i + g₁ i) + 𝔼 i ∈ s, (f₂ i + g₂ i)",
"after_state": "No Goals!"
},
{
"line": "simp (failIfUnchanged✝ := false✝) only",
"before_state": "ι : Type u_1\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nf₁ f₂ g₁ g₂ : ι → M\n⊢ 𝔼 i ∈ s, (f₁ i + f₂ i) + 𝔼 i ∈ s, (g₁ i + g₂ i) = 𝔼 i ∈ s, (f₁ i + g₁ i) + 𝔼 i ∈ s, (f₂ i + g₂ i)",
"after_state": "ι : Type u_1\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nf₁ f₂ g₁ g₂ : ι → M\n⊢ 𝔼 i ∈ s, (f₁ i + f₂ i) + 𝔼 i ∈ s, (g₁ i + g₂ i) = 𝔼 i ∈ s, (f₁ i + g₁ i) + 𝔼 i ∈ s, (f₂ i + g₂ i)"
},
{
"line": "simp only [expect_add_distrib]",
"before_state": "ι : Type u_1\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nf₁ f₂ g₁ g₂ : ι → M\n⊢ 𝔼 i ∈ s, (f₁ i + f₂ i) + 𝔼 i ∈ s, (g₁ i + g₂ i) = 𝔼 i ∈ s, (f₁ i + g₁ i) + 𝔼 i ∈ s, (f₂ i + g₂ i)",
"after_state": "ι : Type u_1\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nf₁ f₂ g₁ g₂ : ι → M\n⊢ 𝔼 i ∈ s, f₁ i + 𝔼 i ∈ s, f₂ i + (𝔼 i ∈ s, g₁ i + 𝔼 i ∈ s, g₂ i) =\n 𝔼 i ∈ s, f₁ i + 𝔼 i ∈ s, g₁ i + (𝔼 i ∈ s, f₂ i + 𝔼 i ∈ s, g₂ i)"
},
{
"line": "simp only [add_add_add_comm]",
"before_state": "ι : Type u_1\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ns : Finset ι\nf₁ f₂ g₁ g₂ : ι → M\n⊢ 𝔼 i ∈ s, f₁ i + 𝔼 i ∈ s, f₂ i + (𝔼 i ∈ s, g₁ i + 𝔼 i ∈ s, g₂ i) =\n 𝔼 i ∈ s, f₁ i + 𝔼 i ∈ s, g₁ i + (𝔼 i ∈ s, f₂ i + 𝔼 i ∈ s, g₂ i)",
"after_state": "No Goals!"
}
] |
lemma expect_ite_mem (s t : Finset ι) (f : ι → M) :
𝔼 i ∈ s, (if i ∈ t then f i else 0) = (#(s ∩ t) / #s : ℚ≥0) • 𝔼 i ∈ s ∩ t, f i := by
obtain hst | hst := (s ∩ t).eq_empty_or_nonempty
· simp [expect, hst]
· simp [expect, smul_smul, ← inv_mul_eq_div, hst.card_ne_zero]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Expect.lean
|
{
"open": [
"Finset Function",
"Fintype (card)",
"scoped Pointwise",
"Batteries.ExtendedBinder Lean Meta",
"Lean Meta Parser.Term PrettyPrinter.Delaborator SubExpr",
"Batteries.ExtendedBinder",
"scoped BigOperators"
],
"variables": [
"{ι κ M N : Type*}",
"[AddCommMonoid M] [Module ℚ≥0 M] [AddCommMonoid N] [Module ℚ≥0 N] {s t : Finset ι}",
"[DecidableEq ι]"
]
}
|
[
{
"line": "obtain hst | hst := (s ∩ t).eq_empty_or_nonempty",
"before_state": "ι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns t : Finset ι\nf : ι → M\n⊢ (𝔼 i ∈ s, if i ∈ t then f i else 0) = (↑(#(s ∩ t)) / ↑(#s)) • 𝔼 i ∈ s ∩ t, f i",
"after_state": "case inl\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns t : Finset ι\nf : ι → M\nhst : s ∩ t = ∅\n⊢ (𝔼 i ∈ s, if i ∈ t then f i else 0) = (↑(#(s ∩ t)) / ↑(#s)) • 𝔼 i ∈ s ∩ t, f i\n---\ncase inr\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns t : Finset ι\nf : ι → M\nhst : (s ∩ t).Nonempty\n⊢ (𝔼 i ∈ s, if i ∈ t then f i else 0) = (↑(#(s ∩ t)) / ↑(#s)) • 𝔼 i ∈ s ∩ t, f i"
},
{
"line": "simp [expect, hst]",
"before_state": "case inl\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns t : Finset ι\nf : ι → M\nhst : s ∩ t = ∅\n⊢ (𝔼 i ∈ s, if i ∈ t then f i else 0) = (↑(#(s ∩ t)) / ↑(#s)) • 𝔼 i ∈ s ∩ t, f i",
"after_state": "No Goals!"
},
{
"line": "simp [expect, smul_smul, ← inv_mul_eq_div, hst.card_ne_zero]",
"before_state": "case inr\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns t : Finset ι\nf : ι → M\nhst : (s ∩ t).Nonempty\n⊢ (𝔼 i ∈ s, if i ∈ t then f i else 0) = (↑(#(s ∩ t)) / ↑(#s)) • 𝔼 i ∈ s ∩ t, f i",
"after_state": "No Goals!"
}
] |
lemma card_smul_expect (s : Finset ι) (f : ι → M) : #s • 𝔼 i ∈ s, f i = ∑ i ∈ s, f i := by
obtain rfl | hs := s.eq_empty_or_nonempty
· simp
· rw [expect, ← Nat.cast_smul_eq_nsmul ℚ≥0, smul_inv_smul₀]
exact mod_cast hs.card_ne_zero
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Expect.lean
|
{
"open": [
"Finset Function",
"Fintype (card)",
"scoped Pointwise",
"Batteries.ExtendedBinder Lean Meta",
"Lean Meta Parser.Term PrettyPrinter.Delaborator SubExpr",
"Batteries.ExtendedBinder",
"scoped BigOperators"
],
"variables": [
"{ι κ M N : Type*}",
"[AddCommMonoid M] [Module ℚ≥0 M] [AddCommMonoid N] [Module ℚ≥0 N] {s t : Finset ι}",
"[DecidableEq ι]",
"{t : Finset κ} {g : κ → M}"
]
}
|
[
{
"line": "obtain rfl | hs := s.eq_empty_or_nonempty",
"before_state": "ι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\n⊢ #s • 𝔼 i ∈ s, f i = ∑ i ∈ s, f i",
"after_state": "case inl\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\nf : ι → M\n⊢ #∅ • 𝔼 i ∈ ∅, f i = ∑ i ∈ ∅, f i\n---\ncase inr\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ #s • 𝔼 i ∈ s, f i = ∑ i ∈ s, f i"
},
{
"line": "simp",
"before_state": "case inl\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\nf : ι → M\n⊢ #∅ • 𝔼 i ∈ ∅, f i = ∑ i ∈ ∅, f i",
"after_state": "No Goals!"
},
{
"line": "rw [expect, ← Nat.cast_smul_eq_nsmul ℚ≥0, smul_inv_smul₀]",
"before_state": "case inr\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ #s • 𝔼 i ∈ s, f i = ∑ i ∈ s, f i",
"after_state": "case inr.ha\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ↑(#s) ≠ 0"
},
{
"line": "rewrite [expect, ← Nat.cast_smul_eq_nsmul ℚ≥0, smul_inv_smul₀]",
"before_state": "case inr\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ #s • 𝔼 i ∈ s, f i = ∑ i ∈ s, f i",
"after_state": "case inr\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ∑ i ∈ s, f i = ∑ i ∈ s, f i\n---\ncase inr.ha\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ↑(#s) ≠ 0"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case inr\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ∑ i ∈ s, f i = ∑ i ∈ s, f i\n---\ncase inr.ha\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ↑(#s) ≠ 0",
"after_state": "case inr.ha\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ↑(#s) ≠ 0"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case inr\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ∑ i ∈ s, f i = ∑ i ∈ s, f i\n---\ncase inr.ha\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ↑(#s) ≠ 0",
"after_state": "case inr.ha\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ↑(#s) ≠ 0"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case inr\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ∑ i ∈ s, f i = ∑ i ∈ s, f i\n---\ncase inr.ha\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ↑(#s) ≠ 0",
"after_state": "case inr.ha\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ↑(#s) ≠ 0"
},
{
"line": "with_reducible rfl",
"before_state": "case inr\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ∑ i ∈ s, f i = ∑ i ∈ s, f i\n---\ncase inr.ha\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ↑(#s) ≠ 0",
"after_state": "case inr.ha\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ↑(#s) ≠ 0"
},
{
"line": "rfl",
"before_state": "case inr\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ∑ i ∈ s, f i = ∑ i ∈ s, f i\n---\ncase inr.ha\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ↑(#s) ≠ 0",
"after_state": "case inr.ha\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ↑(#s) ≠ 0"
},
{
"line": "eq_refl",
"before_state": "case inr\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ∑ i ∈ s, f i = ∑ i ∈ s, f i\n---\ncase inr.ha\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ↑(#s) ≠ 0",
"after_state": "case inr.ha\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ↑(#s) ≠ 0"
},
{
"line": "exact mod_cast hs.card_ne_zero",
"before_state": "case inr.ha\nι : Type u_1\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : _root_.Module ℚ≥0 M\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ↑(#s) ≠ 0",
"after_state": "No Goals!"
}
] |
lemma smul_expect {G : Type*} [DistribSMul G M] [SMulCommClass G ℚ≥0 M] (a : G)
(s : Finset ι) (f : ι → M) : a • 𝔼 i ∈ s, f i = 𝔼 i ∈ s, a • f i := by
simp only [expect]
simp only [smul_sum]
simp only [smul_comm]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Expect.lean
|
{
"open": [
"Finset Function",
"Fintype (card)",
"scoped Pointwise",
"Batteries.ExtendedBinder Lean Meta",
"Lean Meta Parser.Term PrettyPrinter.Delaborator SubExpr",
"Batteries.ExtendedBinder",
"scoped BigOperators"
],
"variables": [
"{ι κ M N : Type*}",
"[AddCommMonoid M] [Module ℚ≥0 M] [AddCommMonoid N] [Module ℚ≥0 N] {s t : Finset ι}",
"[DecidableEq ι]",
"{t : Finset κ} {g : κ → M}"
]
}
|
[
{
"line": "simp only [expect]",
"before_state": "ι : Type u_1\nM : Type u_3\ninst✝⁴ : AddCommMonoid M\ninst✝³ : _root_.Module ℚ≥0 M\ninst✝² : DecidableEq ι\nG : Type u_5\ninst✝¹ : DistribSMul G M\ninst✝ : SMulCommClass G ℚ≥0 M\na : G\ns : Finset ι\nf : ι → M\n⊢ a • 𝔼 i ∈ s, f i = 𝔼 i ∈ s, a • f i",
"after_state": "ι : Type u_1\nM : Type u_3\ninst✝⁴ : AddCommMonoid M\ninst✝³ : _root_.Module ℚ≥0 M\ninst✝² : DecidableEq ι\nG : Type u_5\ninst✝¹ : DistribSMul G M\ninst✝ : SMulCommClass G ℚ≥0 M\na : G\ns : Finset ι\nf : ι → M\n⊢ a • (↑(#s))⁻¹ • ∑ x ∈ s, f x = (↑(#s))⁻¹ • ∑ x ∈ s, a • f x"
},
{
"line": "simp only [smul_sum]",
"before_state": "ι : Type u_1\nM : Type u_3\ninst✝⁴ : AddCommMonoid M\ninst✝³ : _root_.Module ℚ≥0 M\ninst✝² : DecidableEq ι\nG : Type u_5\ninst✝¹ : DistribSMul G M\ninst✝ : SMulCommClass G ℚ≥0 M\na : G\ns : Finset ι\nf : ι → M\n⊢ a • (↑(#s))⁻¹ • ∑ x ∈ s, f x = (↑(#s))⁻¹ • ∑ x ∈ s, a • f x",
"after_state": "ι : Type u_1\nM : Type u_3\ninst✝⁴ : AddCommMonoid M\ninst✝³ : _root_.Module ℚ≥0 M\ninst✝² : DecidableEq ι\nG : Type u_5\ninst✝¹ : DistribSMul G M\ninst✝ : SMulCommClass G ℚ≥0 M\na : G\ns : Finset ι\nf : ι → M\n⊢ ∑ x ∈ s, a • (↑(#s))⁻¹ • f x = ∑ x ∈ s, (↑(#s))⁻¹ • a • f x"
},
{
"line": "simp only [smul_comm]",
"before_state": "ι : Type u_1\nM : Type u_3\ninst✝⁴ : AddCommMonoid M\ninst✝³ : _root_.Module ℚ≥0 M\ninst✝² : DecidableEq ι\nG : Type u_5\ninst✝¹ : DistribSMul G M\ninst✝ : SMulCommClass G ℚ≥0 M\na : G\ns : Finset ι\nf : ι → M\n⊢ ∑ x ∈ s, a • (↑(#s))⁻¹ • f x = ∑ x ∈ s, (↑(#s))⁻¹ • a • f x",
"after_state": "No Goals!"
}
] |
lemma expect_bijective (e : ι → κ) (he : Bijective e) (f : ι → M) (g : κ → M)
(h : ∀ i, f i = g (e i)) : 𝔼 i, f i = 𝔼 i, g i :=
expect_nbij e (fun _ _ ↦ mem_univ _) (fun i _ ↦ h i) he.injective.injOn <| by
simpa using he.surjective.surjOn _
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Expect.lean
|
{
"open": [
"Finset Function",
"Fintype (card)",
"scoped Pointwise",
"Batteries.ExtendedBinder Lean Meta",
"Lean Meta Parser.Term PrettyPrinter.Delaborator SubExpr",
"Batteries.ExtendedBinder",
"scoped BigOperators"
],
"variables": [
"{ι κ M N : Type*}",
"[AddCommMonoid M] [Module ℚ≥0 M] [AddCommMonoid N] [Module ℚ≥0 N] {s t : Finset ι}",
"[DecidableEq ι]",
"{t : Finset κ} {g : κ → M}",
"[AddCommGroup M] [Module ℚ≥0 M]",
"[Semiring M] [Module ℚ≥0 M]",
"[CommSemiring M] [Module ℚ≥0 M] [IsScalarTower ℚ≥0 M M] [SMulCommClass ℚ≥0 M M]",
"[Semifield M] [CharZero M]",
"[Semifield M] [CharZero M] [Semifield N] [CharZero N] [Algebra M N]",
"[Fintype ι] [Fintype κ]",
"[AddCommMonoid M] [Module ℚ≥0 M]"
]
}
|
[
{
"line": "simpa using he.surjective.surjOn _",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹⁸ : AddCommMonoid M\ninst✝¹⁷ : _root_.Module ℚ≥0 M\ninst✝¹⁶ : DecidableEq ι\ninst✝¹⁵ : AddCommGroup M\ninst✝¹⁴ : _root_.Module ℚ≥0 M\ninst✝¹³ : Semiring M\ninst✝¹² : _root_.Module ℚ≥0 M\ninst✝¹¹ : CommSemiring M\ninst✝¹⁰ : _root_.Module ℚ≥0 M\ninst✝⁹ : IsScalarTower ℚ≥0 M M\ninst✝⁸ : SMulCommClass ℚ≥0 M M\ninst✝⁷ : Semifield M\ninst✝⁶ : CharZero M\ninst✝⁵ : Semifield M\ninst✝⁴ : CharZero M\ninst✝³ : Fintype ι\ninst✝² : Fintype κ\ninst✝¹ : AddCommMonoid M\ninst✝ : _root_.Module ℚ≥0 M\ne : ι → κ\nhe : Bijective e\nf : ι → M\ng : κ → M\nh : ∀ (i : ι), f i = g (e i)\n⊢ Set.SurjOn e ↑univ ↑univ",
"after_state": "No Goals!"
}
] |
lemma dens_disjiUnion (s : Finset α) (t : α → Finset β) (h) :
(s.disjiUnion t h).dens = ∑ a ∈ s, (t a).dens := by simp [dens, sum_div]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Field.lean
|
{
"open": [
"Fintype"
],
"variables": [
"{ι K : Type*} [DivisionSemiring K]",
"{α β : Type*} [Fintype β]"
]
}
|
[
{
"line": "simp [dens, sum_div]",
"before_state": "α : Type u_3\nβ : Type u_4\ninst✝ : Fintype β\ns : Finset α\nt : α → Finset β\nh : (↑s).PairwiseDisjoint t\n⊢ (s.disjiUnion t h).dens = ∑ a ∈ s, (t a).dens",
"after_state": "No Goals!"
}
] |
theorem prod_ofFn (f : Fin n → M) : (List.ofFn f).prod = ∏ i, f i := by
simp [prod_eq_multiset_prod]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Fin.lean
|
{
"open": [
"Finset"
],
"variables": [
"{α M : Type*}",
"[CommMonoid M] {n : ℕ}"
]
}
|
[
{
"line": "simp [prod_eq_multiset_prod]",
"before_state": "M : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nf : Fin n → M\n⊢ (List.ofFn f).prod = ∏ i, f i",
"after_state": "No Goals!"
}
] |
theorem prod_univ_getElem (l : List M) : ∏ i : Fin l.length, l[i.1] = l.prod := by
simp [Finset.prod_eq_multiset_prod]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Fin.lean
|
{
"open": [
"Finset"
],
"variables": [
"{α M : Type*}",
"[CommMonoid M] {n : ℕ}"
]
}
|
[
{
"line": "get_elem_tactic",
"before_state": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List M\ni : Fin l.length\n⊢ ↑i < l.length",
"after_state": "No Goals!"
},
{
"line": "first\n| done\n| assumption\n| get_elem_tactic_trivial\n|\n fail \"failed to prove index is valid, possible solutions:\n - Use `have`-expressions to prove the index is valid\n - Use `a[i]!` notation instead, runtime check is performed, and 'Panic' error message is produced if index is not valid\n - Use `a[i]?` notation instead, result is an `Option` type\n - Use `a[i]'h` notation instead, where `h` is a proof that index is valid\"",
"before_state": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List M\ni : Fin l.length\n⊢ ↑i < l.length",
"after_state": "No Goals!"
},
{
"line": "done",
"before_state": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List M\ni : Fin l.length\n⊢ ↑i < l.length",
"after_state": "No Goals!"
},
{
"line": "assumption",
"before_state": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List M\ni : Fin l.length\n⊢ ↑i < l.length",
"after_state": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List M\ni : Fin l.length\n⊢ ↑i < l.length"
},
{
"line": "get_elem_tactic_trivial",
"before_state": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List M\ni : Fin l.length\n⊢ ↑i < l.length",
"after_state": "No Goals!"
},
{
"line": "(with_reducible apply Fin.val_lt_of_le✝);\n get_elem_tactic_trivial;\n done",
"before_state": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List M\ni : Fin l.length\n⊢ ↑i < l.length",
"after_state": "No Goals!"
},
{
"line": "with_reducible apply Fin.val_lt_of_le✝",
"before_state": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List M\ni : Fin l.length\n⊢ ↑i < l.length",
"after_state": "case h\nα : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List M\ni : Fin l.length\n⊢ l.length ≤ l.length"
},
{
"line": "apply Fin.val_lt_of_le✝",
"before_state": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List M\ni : Fin l.length\n⊢ ↑i < l.length",
"after_state": "case h\nα : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List M\ni : Fin l.length\n⊢ l.length ≤ l.length"
},
{
"line": "get_elem_tactic_trivial",
"before_state": "case h\nα : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List M\ni : Fin l.length\n⊢ l.length ≤ l.length",
"after_state": "No Goals!"
},
{
"line": "trivial",
"before_state": "case h\nα : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List M\ni : Fin l.length\n⊢ l.length ≤ l.length",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "case h\nα : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List M\ni : Fin l.length\n⊢ l.length ≤ l.length",
"after_state": "No Goals!"
},
{
"line": "apply_rfl",
"before_state": "case h\nα : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List M\ni : Fin l.length\n⊢ l.length ≤ l.length",
"after_state": "No Goals!"
},
{
"line": "simp [Finset.prod_eq_multiset_prod]",
"before_state": "M : Type u_2\ninst✝ : CommMonoid M\nl : List M\n⊢ ∏ i, l[↑i] = l.prod",
"after_state": "No Goals!"
}
] |
theorem prod_univ_fun_getElem (l : List α) (f : α → M) :
∏ i : Fin l.length, f l[i.1] = (l.map f).prod := by
simp [Finset.prod_eq_multiset_prod]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Fin.lean
|
{
"open": [
"Finset"
],
"variables": [
"{α M : Type*}",
"[CommMonoid M] {n : ℕ}"
]
}
|
[
{
"line": "get_elem_tactic",
"before_state": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List α\nf : α → M\ni : Fin l.length\n⊢ ↑i < l.length",
"after_state": "No Goals!"
},
{
"line": "first\n| done\n| assumption\n| get_elem_tactic_trivial\n|\n fail \"failed to prove index is valid, possible solutions:\n - Use `have`-expressions to prove the index is valid\n - Use `a[i]!` notation instead, runtime check is performed, and 'Panic' error message is produced if index is not valid\n - Use `a[i]?` notation instead, result is an `Option` type\n - Use `a[i]'h` notation instead, where `h` is a proof that index is valid\"",
"before_state": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List α\nf : α → M\ni : Fin l.length\n⊢ ↑i < l.length",
"after_state": "No Goals!"
},
{
"line": "done",
"before_state": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List α\nf : α → M\ni : Fin l.length\n⊢ ↑i < l.length",
"after_state": "No Goals!"
},
{
"line": "assumption",
"before_state": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List α\nf : α → M\ni : Fin l.length\n⊢ ↑i < l.length",
"after_state": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List α\nf : α → M\ni : Fin l.length\n⊢ ↑i < l.length"
},
{
"line": "get_elem_tactic_trivial",
"before_state": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List α\nf : α → M\ni : Fin l.length\n⊢ ↑i < l.length",
"after_state": "No Goals!"
},
{
"line": "(with_reducible apply Fin.val_lt_of_le✝);\n get_elem_tactic_trivial;\n done",
"before_state": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List α\nf : α → M\ni : Fin l.length\n⊢ ↑i < l.length",
"after_state": "No Goals!"
},
{
"line": "with_reducible apply Fin.val_lt_of_le✝",
"before_state": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List α\nf : α → M\ni : Fin l.length\n⊢ ↑i < l.length",
"after_state": "case h\nα : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List α\nf : α → M\ni : Fin l.length\n⊢ l.length ≤ l.length"
},
{
"line": "apply Fin.val_lt_of_le✝",
"before_state": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List α\nf : α → M\ni : Fin l.length\n⊢ ↑i < l.length",
"after_state": "case h\nα : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List α\nf : α → M\ni : Fin l.length\n⊢ l.length ≤ l.length"
},
{
"line": "get_elem_tactic_trivial",
"before_state": "case h\nα : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List α\nf : α → M\ni : Fin l.length\n⊢ l.length ≤ l.length",
"after_state": "No Goals!"
},
{
"line": "trivial",
"before_state": "case h\nα : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List α\nf : α → M\ni : Fin l.length\n⊢ l.length ≤ l.length",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "case h\nα : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List α\nf : α → M\ni : Fin l.length\n⊢ l.length ≤ l.length",
"after_state": "No Goals!"
},
{
"line": "apply_rfl",
"before_state": "case h\nα : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nl : List α\nf : α → M\ni : Fin l.length\n⊢ l.length ≤ l.length",
"after_state": "No Goals!"
},
{
"line": "simp [Finset.prod_eq_multiset_prod]",
"before_state": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nl : List α\nf : α → M\n⊢ ∏ i, f l[↑i] = (List.map f l).prod",
"after_state": "No Goals!"
}
] |
theorem prod_univ_one (f : Fin 1 → M) : ∏ i, f i = f 0 := by simp
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Fin.lean
|
{
"open": [
"Finset"
],
"variables": [
"{α M : Type*}",
"[CommMonoid M] {n : ℕ}"
]
}
|
[
{
"line": "simp",
"before_state": "M : Type u_2\ninst✝ : CommMonoid M\nf : Fin 1 → M\n⊢ ∏ i, f i = f 0",
"after_state": "No Goals!"
}
] |
theorem prod_univ_two (f : Fin 2 → M) : ∏ i, f i = f 0 * f 1 := by
simp [prod_univ_succ]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Fin.lean
|
{
"open": [
"Finset"
],
"variables": [
"{α M : Type*}",
"[CommMonoid M] {n : ℕ}"
]
}
|
[
{
"line": "simp [prod_univ_succ]",
"before_state": "M : Type u_2\ninst✝ : CommMonoid M\nf : Fin 2 → M\n⊢ ∏ i, f i = f 0 * f 1",
"after_state": "No Goals!"
}
] |
theorem prod_const (n : ℕ) (x : M) : ∏ _i : Fin n, x = x ^ n := by simp
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Fin.lean
|
{
"open": [
"Finset"
],
"variables": [
"{α M : Type*}",
"[CommMonoid M] {n : ℕ}"
]
}
|
[
{
"line": "simp",
"before_state": "M : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nx : M\n⊢ ∏ _i, x = x ^ n",
"after_state": "No Goals!"
}
] |
theorem prod_congr' {a b : ℕ} (f : Fin b → M) (h : a = b) :
(∏ i : Fin a, f (i.cast h)) = ∏ i : Fin b, f i := by
subst h
congr
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Fin.lean
|
{
"open": [
"Finset"
],
"variables": [
"{α M : Type*}",
"[CommMonoid M] {n : ℕ}"
]
}
|
[
{
"line": "subst h",
"before_state": "M : Type u_2\ninst✝ : CommMonoid M\na b : ℕ\nf : Fin b → M\nh : a = b\n⊢ ∏ i, f (Fin.cast h i) = ∏ i, f i",
"after_state": "M : Type u_2\ninst✝ : CommMonoid M\na : ℕ\nf : Fin a → M\n⊢ ∏ i, f (Fin.cast ⋯ i) = ∏ i, f i"
},
{
"line": "congr",
"before_state": "M : Type u_2\ninst✝ : CommMonoid M\na : ℕ\nf : Fin a → M\n⊢ ∏ i, f (Fin.cast ⋯ i) = ∏ i, f i",
"after_state": "No Goals!"
}
] |
theorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :
(∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by
simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Fin.lean
|
{
"open": [
"Finset"
],
"variables": [
"{α M : Type*}",
"[CommMonoid M] {n : ℕ}"
]
}
|
[
{
"line": "simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b",
"before_state": "n : ℕ\nR : Type u_3\ninst✝ : CommSemiring R\na b : R\n⊢ ∑ s, a ^ #s * b ^ (n - #s) = (a + b) ^ n",
"after_state": "No Goals!"
}
] |
theorem sum_Ico_Ico_comm {M : Type*} [AddCommMonoid M] (a b : ℕ) (f : ℕ → ℕ → M) :
(∑ i ∈ Finset.Ico a b, ∑ j ∈ Finset.Ico i b, f i j) =
∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j := by
rw [Finset.sum_sigma']
rw [Finset.sum_sigma']
refine sum_nbij' (fun x ↦ ⟨x.2, x.1⟩) (fun x ↦ ⟨x.2, x.1⟩) ?_ ?_ (fun _ _ ↦ rfl) (fun _ _ ↦ rfl)
(fun _ _ ↦ rfl) <;>
simp only [Finset.mem_Ico] <;>
simp only [Sigma.forall] <;>
simp only [Finset.mem_sigma] <;>
rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>
omega
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Intervals.lean
|
{
"open": [
"Nat"
],
"variables": [
"{α M : Type*}",
"[PartialOrder α] [CommMonoid M] {f : α → M} {a b : α}",
"[LocallyFiniteOrder α]",
"[LocallyFiniteOrderTop α]",
"[LocallyFiniteOrderBot α]",
"[Fintype α] [LinearOrder α] [LocallyFiniteOrderTop α] [LocallyFiniteOrderBot α]",
"[CommMonoid M] {s₂ s₁ s : Finset α} {a : α} {g f : α → M}"
]
}
|
[
{
"line": "rw [Finset.sum_sigma']",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ i ∈ Finset.Ico a b, ∑ j ∈ Finset.Ico i b, f i j = ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j"
},
{
"line": "rewrite [Finset.sum_sigma']",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ i ∈ Finset.Ico a b, ∑ j ∈ Finset.Ico i b, f i j = ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j"
},
{
"line": "try (with_reducible rfl)",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j"
},
{
"line": "with_reducible rfl",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j"
},
{
"line": "rfl",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j"
},
{
"line": "apply_rfl",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j"
},
{
"line": "skip",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j"
},
{
"line": "rw [Finset.sum_sigma']",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma fun j => Finset.Ico a (j + 1), f x.snd x.fst"
},
{
"line": "rewrite [Finset.sum_sigma']",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma fun j => Finset.Ico a (j + 1), f x.snd x.fst"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma fun j => Finset.Ico a (j + 1), f x.snd x.fst",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma fun j => Finset.Ico a (j + 1), f x.snd x.fst"
},
{
"line": "try (with_reducible rfl)",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma fun j => Finset.Ico a (j + 1), f x.snd x.fst",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma fun j => Finset.Ico a (j + 1), f x.snd x.fst"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma fun j => Finset.Ico a (j + 1), f x.snd x.fst",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma fun j => Finset.Ico a (j + 1), f x.snd x.fst"
},
{
"line": "with_reducible rfl",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma fun j => Finset.Ico a (j + 1), f x.snd x.fst",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma fun j => Finset.Ico a (j + 1), f x.snd x.fst"
},
{
"line": "rfl",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma fun j => Finset.Ico a (j + 1), f x.snd x.fst",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma fun j => Finset.Ico a (j + 1), f x.snd x.fst"
},
{
"line": "apply_rfl",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma fun j => Finset.Ico a (j + 1), f x.snd x.fst",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma fun j => Finset.Ico a (j + 1), f x.snd x.fst"
},
{
"line": "skip",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma fun j => Finset.Ico a (j + 1), f x.snd x.fst",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma fun j => Finset.Ico a (j + 1), f x.snd x.fst"
},
{
"line": "focus\n refine sum_nbij' (fun x ↦ ⟨x.2, x.1⟩) (fun x ↦ ⟨x.2, x.1⟩) ?_ ?_ (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) <;>\n simp only [Finset.mem_Ico] <;>\n simp only [Sigma.forall] <;>\n simp only [Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩\n with_annotate_state\"<;>\" skip\n all_goals omega",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma fun j => Finset.Ico a (j + 1), f x.snd x.fst",
"after_state": "No Goals!"
},
{
"line": "focus\n refine sum_nbij' (fun x ↦ ⟨x.2, x.1⟩) (fun x ↦ ⟨x.2, x.1⟩) ?_ ?_ (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) <;>\n simp only [Finset.mem_Ico] <;>\n simp only [Sigma.forall] <;>\n simp only [Finset.mem_sigma]\n with_annotate_state\"<;>\" skip\n all_goals rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma fun j => Finset.Ico a (j + 1), f x.snd x.fst",
"after_state": "No Goals!"
},
{
"line": "focus\n refine sum_nbij' (fun x ↦ ⟨x.2, x.1⟩) (fun x ↦ ⟨x.2, x.1⟩) ?_ ?_ (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) <;>\n simp only [Finset.mem_Ico] <;>\n simp only [Sigma.forall]\n with_annotate_state\"<;>\" skip\n all_goals simp only [Finset.mem_sigma]",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma fun j => Finset.Ico a (j + 1), f x.snd x.fst",
"after_state": "No Goals!"
},
{
"line": "focus\n refine sum_nbij' (fun x ↦ ⟨x.2, x.1⟩) (fun x ↦ ⟨x.2, x.1⟩) ?_ ?_ (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) <;>\n simp only [Finset.mem_Ico]\n with_annotate_state\"<;>\" skip\n all_goals simp only [Sigma.forall]",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma fun j => Finset.Ico a (j + 1), f x.snd x.fst",
"after_state": "No Goals!"
},
{
"line": "focus\n refine sum_nbij' (fun x ↦ ⟨x.2, x.1⟩) (fun x ↦ ⟨x.2, x.1⟩) ?_ ?_ (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl)\n with_annotate_state\"<;>\" skip\n all_goals simp only [Finset.mem_Ico]",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma fun j => Finset.Ico a (j + 1), f x.snd x.fst",
"after_state": "No Goals!"
},
{
"line": "refine sum_nbij' (fun x ↦ ⟨x.2, x.1⟩) (fun x ↦ ⟨x.2, x.1⟩) ?_ ?_ (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl)",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico i b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma fun j => Finset.Ico a (j + 1), f x.snd x.fst",
"after_state": "No Goals!"
}
] |
theorem sum_Ico_Ico_comm' {M : Type*} [AddCommMonoid M] (a b : ℕ) (f : ℕ → ℕ → M) :
(∑ i ∈ Finset.Ico a b, ∑ j ∈ Finset.Ico (i + 1) b, f i j) =
∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j := by
rw [Finset.sum_sigma']
rw [Finset.sum_sigma']
refine sum_nbij' (fun x ↦ ⟨x.2, x.1⟩) (fun x ↦ ⟨x.2, x.1⟩) ?_ ?_ (fun _ _ ↦ rfl) (fun _ _ ↦ rfl)
(fun _ _ ↦ rfl) <;>
simp only [Finset.mem_Ico] <;>
simp only [Sigma.forall] <;>
simp only [Finset.mem_sigma] <;>
rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>
omega
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Intervals.lean
|
{
"open": [
"Nat"
],
"variables": [
"{α M : Type*}",
"[PartialOrder α] [CommMonoid M] {f : α → M} {a b : α}",
"[LocallyFiniteOrder α]",
"[LocallyFiniteOrderTop α]",
"[LocallyFiniteOrderBot α]",
"[Fintype α] [LinearOrder α] [LocallyFiniteOrderTop α] [LocallyFiniteOrderBot α]",
"[CommMonoid M] {s₂ s₁ s : Finset α} {a : α} {g f : α → M}"
]
}
|
[
{
"line": "rw [Finset.sum_sigma']",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ i ∈ Finset.Ico a b, ∑ j ∈ Finset.Ico (i + 1) b, f i j = ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j"
},
{
"line": "rewrite [Finset.sum_sigma']",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ i ∈ Finset.Ico a b, ∑ j ∈ Finset.Ico (i + 1) b, f i j = ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j"
},
{
"line": "try (with_reducible rfl)",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j"
},
{
"line": "with_reducible rfl",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j"
},
{
"line": "rfl",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j"
},
{
"line": "apply_rfl",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j"
},
{
"line": "skip",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j"
},
{
"line": "rw [Finset.sum_sigma']",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma (Finset.Ico a), f x.snd x.fst"
},
{
"line": "rewrite [Finset.sum_sigma']",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma (Finset.Ico a), f x.snd x.fst"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma (Finset.Ico a), f x.snd x.fst",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma (Finset.Ico a), f x.snd x.fst"
},
{
"line": "try (with_reducible rfl)",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma (Finset.Ico a), f x.snd x.fst",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma (Finset.Ico a), f x.snd x.fst"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma (Finset.Ico a), f x.snd x.fst",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma (Finset.Ico a), f x.snd x.fst"
},
{
"line": "with_reducible rfl",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma (Finset.Ico a), f x.snd x.fst",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma (Finset.Ico a), f x.snd x.fst"
},
{
"line": "rfl",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma (Finset.Ico a), f x.snd x.fst",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma (Finset.Ico a), f x.snd x.fst"
},
{
"line": "apply_rfl",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma (Finset.Ico a), f x.snd x.fst",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma (Finset.Ico a), f x.snd x.fst"
},
{
"line": "skip",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma (Finset.Ico a), f x.snd x.fst",
"after_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma (Finset.Ico a), f x.snd x.fst"
},
{
"line": "focus\n refine sum_nbij' (fun x ↦ ⟨x.2, x.1⟩) (fun x ↦ ⟨x.2, x.1⟩) ?_ ?_ (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) <;>\n simp only [Finset.mem_Ico] <;>\n simp only [Sigma.forall] <;>\n simp only [Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩\n with_annotate_state\"<;>\" skip\n all_goals omega",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma (Finset.Ico a), f x.snd x.fst",
"after_state": "No Goals!"
},
{
"line": "focus\n refine sum_nbij' (fun x ↦ ⟨x.2, x.1⟩) (fun x ↦ ⟨x.2, x.1⟩) ?_ ?_ (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) <;>\n simp only [Finset.mem_Ico] <;>\n simp only [Sigma.forall] <;>\n simp only [Finset.mem_sigma]\n with_annotate_state\"<;>\" skip\n all_goals rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma (Finset.Ico a), f x.snd x.fst",
"after_state": "No Goals!"
},
{
"line": "focus\n refine sum_nbij' (fun x ↦ ⟨x.2, x.1⟩) (fun x ↦ ⟨x.2, x.1⟩) ?_ ?_ (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) <;>\n simp only [Finset.mem_Ico] <;>\n simp only [Sigma.forall]\n with_annotate_state\"<;>\" skip\n all_goals simp only [Finset.mem_sigma]",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma (Finset.Ico a), f x.snd x.fst",
"after_state": "No Goals!"
},
{
"line": "focus\n refine sum_nbij' (fun x ↦ ⟨x.2, x.1⟩) (fun x ↦ ⟨x.2, x.1⟩) ?_ ?_ (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) <;>\n simp only [Finset.mem_Ico]\n with_annotate_state\"<;>\" skip\n all_goals simp only [Sigma.forall]",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma (Finset.Ico a), f x.snd x.fst",
"after_state": "No Goals!"
},
{
"line": "focus\n refine sum_nbij' (fun x ↦ ⟨x.2, x.1⟩) (fun x ↦ ⟨x.2, x.1⟩) ?_ ?_ (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl)\n with_annotate_state\"<;>\" skip\n all_goals simp only [Finset.mem_Ico]",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma (Finset.Ico a), f x.snd x.fst",
"after_state": "No Goals!"
},
{
"line": "refine sum_nbij' (fun x ↦ ⟨x.2, x.1⟩) (fun x ↦ ⟨x.2, x.1⟩) ?_ ?_ (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl)",
"before_state": "M : Type u_3\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ x ∈ (Finset.Ico a b).sigma fun i => Finset.Ico (i + 1) b, f x.fst x.snd =\n ∑ x ∈ (Finset.Ico a b).sigma (Finset.Ico a), f x.snd x.fst",
"after_state": "No Goals!"
}
] |
theorem sum_sym2_filter_not_isDiag {ι α} [LinearOrder ι] [AddCommMonoid α]
(s : Finset ι) (p : Sym2 ι → α) :
∑ i ∈ s.sym2 with ¬ i.IsDiag, p i = ∑ i ∈ s.offDiag with i.1 < i.2, p s(i.1, i.2) := by
rw [Finset.offDiag_filter_lt_eq_filter_le]
conv_rhs => rw [← Finset.sum_subtype_eq_sum_filter]
refine (Finset.sum_equiv Sym2.sortEquiv.symm ?_ ?_).symm
all_goals aesop
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Sym.lean
|
{
"open": [
"Multiset"
],
"variables": []
}
|
[
{
"line": "rw [Finset.offDiag_filter_lt_eq_filter_le]",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 < i.2}, p s(i.1, i.2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)"
},
{
"line": "rewrite [Finset.offDiag_filter_lt_eq_filter_le]",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 < i.2}, p s(i.1, i.2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)"
},
{
"line": "skip",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)"
},
{
"line": "conv_rhs => rw [← Finset.sum_subtype_eq_sum_filter]",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ x ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑x).1, (↑x).2)"
},
{
"line": "conv => rhs; (rw [← Finset.sum_subtype_eq_sum_filter])",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ x ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑x).1, (↑x).2)"
},
{
"line": "rhs; (rw [← Finset.sum_subtype_eq_sum_filter])",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n| ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n| ∑ x ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑x).1, (↑x).2)"
},
{
"line": "rhs; (rw [← Finset.sum_subtype_eq_sum_filter])",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n| ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n| ∑ x ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑x).1, (↑x).2)"
},
{
"line": "rhs",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n| ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n| ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)"
},
{
"line": "(rw [← Finset.sum_subtype_eq_sum_filter])",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n| ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n| ∑ x ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑x).1, (↑x).2)"
},
{
"line": "rw [← Finset.sum_subtype_eq_sum_filter]",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n| ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n| ∑ x ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑x).1, (↑x).2)"
},
{
"line": "rw [← Finset.sum_subtype_eq_sum_filter]",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n| ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n| ∑ x ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑x).1, (↑x).2)"
},
{
"line": "rw [← Finset.sum_subtype_eq_sum_filter]",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n| ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n| ∑ x ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑x).1, (↑x).2)"
},
{
"line": "rewrite [← Finset.sum_subtype_eq_sum_filter]",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n| ∑ i ∈ {i ∈ s.offDiag | i.1 ≤ i.2}, p s(i.1, i.2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n| ∑ x ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑x).1, (↑x).2)"
},
{
"line": "try with_reducible rfl",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ x ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑x).1, (↑x).2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ x ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑x).1, (↑x).2)"
},
{
"line": "first\n| with_reducible rfl\n| skip",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ x ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑x).1, (↑x).2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ x ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑x).1, (↑x).2)"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ x ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑x).1, (↑x).2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ x ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑x).1, (↑x).2)"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ x ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑x).1, (↑x).2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ x ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑x).1, (↑x).2)"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ x ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑x).1, (↑x).2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ x ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑x).1, (↑x).2)"
},
{
"line": "skip",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ x ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑x).1, (↑x).2)",
"after_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ x ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑x).1, (↑x).2)"
},
{
"line": "refine (Finset.sum_equiv Sym2.sortEquiv.symm ?_ ?_).symm",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∑ i ∈ {i ∈ s.sym2 | ¬i.IsDiag}, p i = ∑ x ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑x).1, (↑x).2)",
"after_state": "case refine_1\nι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∀ (i : { p // p.1 ≤ p.2 }),\n i ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag ↔ Sym2.sortEquiv.symm i ∈ {i ∈ s.sym2 | ¬i.IsDiag}\n---\ncase refine_2\nι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∀ i ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑i).1, (↑i).2) = p (Sym2.sortEquiv.symm i)"
},
{
"line": "all_goals aesop",
"before_state": "case refine_1\nι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∀ (i : { p // p.1 ≤ p.2 }),\n i ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag ↔ Sym2.sortEquiv.symm i ∈ {i ∈ s.sym2 | ¬i.IsDiag}\n---\ncase refine_2\nι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∀ i ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑i).1, (↑i).2) = p (Sym2.sortEquiv.symm i)",
"after_state": "No Goals!"
},
{
"line": "aesop",
"before_state": "case refine_1\nι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∀ (i : { p // p.1 ≤ p.2 }),\n i ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag ↔ Sym2.sortEquiv.symm i ∈ {i ∈ s.sym2 | ¬i.IsDiag}",
"after_state": "No Goals!"
},
{
"line": "aesop",
"before_state": "case refine_2\nι : Type u_1\nα : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid α\ns : Finset ι\np : Sym2 ι → α\n⊢ ∀ i ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag, p s((↑i).1, (↑i).2) = p (Sym2.sortEquiv.symm i)",
"after_state": "No Goals!"
}
] |
theorem sum_count_of_mem_sym {α} [DecidableEq α] {m : ℕ} {k : Sym α m} {s : Finset α}
(hk : k ∈ s.sym m) : (∑ i ∈ s, count i k) = m := by
simp_rw [← k.prop, ← toFinset_sum_count_eq, eq_comm]
refine sum_subset_zero_on_sdiff (fun _ _ ↦ ?_) ?_ (fun _ _ ↦ rfl)
all_goals aesop
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Sym.lean
|
{
"open": [
"Multiset"
],
"variables": []
}
|
[
{
"line": "simp_rw [← k.prop, ← toFinset_sum_count_eq, eq_comm]",
"before_state": "α : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nk : Sym α m\ns : Finset α\nhk : k ∈ s.sym m\n⊢ ∑ i ∈ s, count i ↑k = m",
"after_state": "α : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nk : Sym α m\ns : Finset α\nhk : k ∈ s.sym m\n⊢ ∑ a ∈ (↑k).toFinset, count a ↑k = ∑ i ∈ s, count i ↑k"
},
{
"line": "simp (failIfUnchanged✝ := false✝) only",
"before_state": "α : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nk : Sym α m\ns : Finset α\nhk : k ∈ s.sym m\n⊢ ∑ i ∈ s, count i ↑k = m",
"after_state": "α : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nk : Sym α m\ns : Finset α\nhk : k ∈ s.sym m\n⊢ ∑ i ∈ s, count i ↑k = m"
},
{
"line": "simp only [← k.prop]",
"before_state": "α : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nk : Sym α m\ns : Finset α\nhk : k ∈ s.sym m\n⊢ ∑ i ∈ s, count i ↑k = m",
"after_state": "α : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nk : Sym α m\ns : Finset α\nhk : k ∈ s.sym m\n⊢ ∑ i ∈ s, count i ↑k = (↑k).card"
},
{
"line": "simp only [← toFinset_sum_count_eq]",
"before_state": "α : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nk : Sym α m\ns : Finset α\nhk : k ∈ s.sym m\n⊢ ∑ i ∈ s, count i ↑k = (↑k).card",
"after_state": "α : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nk : Sym α m\ns : Finset α\nhk : k ∈ s.sym m\n⊢ ∑ i ∈ s, count i ↑k = ∑ a ∈ (↑k).toFinset, count a ↑k"
},
{
"line": "simp only [eq_comm]",
"before_state": "α : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nk : Sym α m\ns : Finset α\nhk : k ∈ s.sym m\n⊢ ∑ i ∈ s, count i ↑k = ∑ a ∈ (↑k).toFinset, count a ↑k",
"after_state": "α : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nk : Sym α m\ns : Finset α\nhk : k ∈ s.sym m\n⊢ ∑ a ∈ (↑k).toFinset, count a ↑k = ∑ i ∈ s, count i ↑k"
},
{
"line": "refine sum_subset_zero_on_sdiff (fun _ _ ↦ ?_) ?_ (fun _ _ ↦ rfl)",
"before_state": "α : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nk : Sym α m\ns : Finset α\nhk : k ∈ s.sym m\n⊢ ∑ a ∈ (↑k).toFinset, count a ↑k = ∑ i ∈ s, count i ↑k",
"after_state": "No Goals!"
}
] |
lemma sum_ne_top : ∑ i ∈ s, f i ≠ ⊤ ↔ ∀ i ∈ s, f i ≠ ⊤ := by simp
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/WithTop.lean
|
{
"open": [
"Finset"
],
"variables": [
"{ι α : Type*}",
"[AddCommMonoid α] {s : Finset ι} {f : ι → WithTop α}"
]
}
|
[
{
"line": "simp",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝ : AddCommMonoid α\ns : Finset ι\nf : ι → WithTop α\n⊢ ∑ i ∈ s, f i ≠ ⊤ ↔ ∀ i ∈ s, f i ≠ ⊤",
"after_state": "No Goals!"
}
] |
lemma sum_eq_bot_iff : ∑ i ∈ s, f i = ⊥ ↔ ∃ i ∈ s, f i = ⊥ := by
induction s using Finset.cons_induction <;> simp [*]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/WithTop.lean
|
{
"open": [
"Finset"
],
"variables": [
"{ι α : Type*}",
"[AddCommMonoid α] {s : Finset ι} {f : ι → WithTop α}",
"[LT α]",
"[CommMonoidWithZero α] [NoZeroDivisors α] [Nontrivial α] [DecidableEq α]",
"[AddCommMonoid α] {s : Finset ι} {f : ι → WithBot α}"
]
}
|
[
{
"line": "focus\n induction s using Finset.cons_induction\n with_annotate_state\"<;>\" skip\n all_goals simp [*]",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝⁶ : AddCommMonoid α\ninst✝⁵ : LT α\ninst✝⁴ : CommMonoidWithZero α\ninst✝³ : NoZeroDivisors α\ninst✝² : Nontrivial α\ninst✝¹ : DecidableEq α\ninst✝ : AddCommMonoid α\ns : Finset ι\nf : ι → WithBot α\n⊢ ∑ i ∈ s, f i = ⊥ ↔ ∃ i ∈ s, f i = ⊥",
"after_state": "No Goals!"
},
{
"line": "induction s using Finset.cons_induction",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝⁶ : AddCommMonoid α\ninst✝⁵ : LT α\ninst✝⁴ : CommMonoidWithZero α\ninst✝³ : NoZeroDivisors α\ninst✝² : Nontrivial α\ninst✝¹ : DecidableEq α\ninst✝ : AddCommMonoid α\ns : Finset ι\nf : ι → WithBot α\n⊢ ∑ i ∈ s, f i = ⊥ ↔ ∃ i ∈ s, f i = ⊥",
"after_state": "case empty\nι : Type u_1\nα : Type u_2\ninst✝⁶ : AddCommMonoid α\ninst✝⁵ : LT α\ninst✝⁴ : CommMonoidWithZero α\ninst✝³ : NoZeroDivisors α\ninst✝² : Nontrivial α\ninst✝¹ : DecidableEq α\ninst✝ : AddCommMonoid α\ns : Finset ι\nf : ι → WithBot α\n⊢ ∑ i ∈ ∅, f i = ⊥ ↔ ∃ i ∈ ∅, f i = ⊥\n---\ncase cons\nι : Type u_1\nα : Type u_2\ninst✝⁶ : AddCommMonoid α\ninst✝⁵ : LT α\ninst✝⁴ : CommMonoidWithZero α\ninst✝³ : NoZeroDivisors α\ninst✝² : Nontrivial α\ninst✝¹ : DecidableEq α\ninst✝ : AddCommMonoid α\ns : Finset ι\nf : ι → WithBot α\na✝¹ : ι\ns✝ : Finset ι\nh✝ : a✝¹ ∉ s✝\na✝ : ∑ i ∈ s✝, f i = ⊥ ↔ ∃ i ∈ s✝, f i = ⊥\n⊢ ∑ i ∈ cons a✝¹ s✝ h✝, f i = ⊥ ↔ ∃ i ∈ cons a✝¹ s✝ h✝, f i = ⊥"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case empty\nι : Type u_1\nα : Type u_2\ninst✝⁶ : AddCommMonoid α\ninst✝⁵ : LT α\ninst✝⁴ : CommMonoidWithZero α\ninst✝³ : NoZeroDivisors α\ninst✝² : Nontrivial α\ninst✝¹ : DecidableEq α\ninst✝ : AddCommMonoid α\ns : Finset ι\nf : ι → WithBot α\n⊢ ∑ i ∈ ∅, f i = ⊥ ↔ ∃ i ∈ ∅, f i = ⊥\n---\ncase cons\nι : Type u_1\nα : Type u_2\ninst✝⁶ : AddCommMonoid α\ninst✝⁵ : LT α\ninst✝⁴ : CommMonoidWithZero α\ninst✝³ : NoZeroDivisors α\ninst✝² : Nontrivial α\ninst✝¹ : DecidableEq α\ninst✝ : AddCommMonoid α\ns : Finset ι\nf : ι → WithBot α\na✝¹ : ι\ns✝ : Finset ι\nh✝ : a✝¹ ∉ s✝\na✝ : ∑ i ∈ s✝, f i = ⊥ ↔ ∃ i ∈ s✝, f i = ⊥\n⊢ ∑ i ∈ cons a✝¹ s✝ h✝, f i = ⊥ ↔ ∃ i ∈ cons a✝¹ s✝ h✝, f i = ⊥",
"after_state": "case empty\nι : Type u_1\nα : Type u_2\ninst✝⁶ : AddCommMonoid α\ninst✝⁵ : LT α\ninst✝⁴ : CommMonoidWithZero α\ninst✝³ : NoZeroDivisors α\ninst✝² : Nontrivial α\ninst✝¹ : DecidableEq α\ninst✝ : AddCommMonoid α\ns : Finset ι\nf : ι → WithBot α\n⊢ ∑ i ∈ ∅, f i = ⊥ ↔ ∃ i ∈ ∅, f i = ⊥\n---\ncase cons\nι : Type u_1\nα : Type u_2\ninst✝⁶ : AddCommMonoid α\ninst✝⁵ : LT α\ninst✝⁴ : CommMonoidWithZero α\ninst✝³ : NoZeroDivisors α\ninst✝² : Nontrivial α\ninst✝¹ : DecidableEq α\ninst✝ : AddCommMonoid α\ns : Finset ι\nf : ι → WithBot α\na✝¹ : ι\ns✝ : Finset ι\nh✝ : a✝¹ ∉ s✝\na✝ : ∑ i ∈ s✝, f i = ⊥ ↔ ∃ i ∈ s✝, f i = ⊥\n⊢ ∑ i ∈ cons a✝¹ s✝ h✝, f i = ⊥ ↔ ∃ i ∈ cons a✝¹ s✝ h✝, f i = ⊥"
},
{
"line": "skip",
"before_state": "case empty\nι : Type u_1\nα : Type u_2\ninst✝⁶ : AddCommMonoid α\ninst✝⁵ : LT α\ninst✝⁴ : CommMonoidWithZero α\ninst✝³ : NoZeroDivisors α\ninst✝² : Nontrivial α\ninst✝¹ : DecidableEq α\ninst✝ : AddCommMonoid α\ns : Finset ι\nf : ι → WithBot α\n⊢ ∑ i ∈ ∅, f i = ⊥ ↔ ∃ i ∈ ∅, f i = ⊥\n---\ncase cons\nι : Type u_1\nα : Type u_2\ninst✝⁶ : AddCommMonoid α\ninst✝⁵ : LT α\ninst✝⁴ : CommMonoidWithZero α\ninst✝³ : NoZeroDivisors α\ninst✝² : Nontrivial α\ninst✝¹ : DecidableEq α\ninst✝ : AddCommMonoid α\ns : Finset ι\nf : ι → WithBot α\na✝¹ : ι\ns✝ : Finset ι\nh✝ : a✝¹ ∉ s✝\na✝ : ∑ i ∈ s✝, f i = ⊥ ↔ ∃ i ∈ s✝, f i = ⊥\n⊢ ∑ i ∈ cons a✝¹ s✝ h✝, f i = ⊥ ↔ ∃ i ∈ cons a✝¹ s✝ h✝, f i = ⊥",
"after_state": "case empty\nι : Type u_1\nα : Type u_2\ninst✝⁶ : AddCommMonoid α\ninst✝⁵ : LT α\ninst✝⁴ : CommMonoidWithZero α\ninst✝³ : NoZeroDivisors α\ninst✝² : Nontrivial α\ninst✝¹ : DecidableEq α\ninst✝ : AddCommMonoid α\ns : Finset ι\nf : ι → WithBot α\n⊢ ∑ i ∈ ∅, f i = ⊥ ↔ ∃ i ∈ ∅, f i = ⊥\n---\ncase cons\nι : Type u_1\nα : Type u_2\ninst✝⁶ : AddCommMonoid α\ninst✝⁵ : LT α\ninst✝⁴ : CommMonoidWithZero α\ninst✝³ : NoZeroDivisors α\ninst✝² : Nontrivial α\ninst✝¹ : DecidableEq α\ninst✝ : AddCommMonoid α\ns : Finset ι\nf : ι → WithBot α\na✝¹ : ι\ns✝ : Finset ι\nh✝ : a✝¹ ∉ s✝\na✝ : ∑ i ∈ s✝, f i = ⊥ ↔ ∃ i ∈ s✝, f i = ⊥\n⊢ ∑ i ∈ cons a✝¹ s✝ h✝, f i = ⊥ ↔ ∃ i ∈ cons a✝¹ s✝ h✝, f i = ⊥"
},
{
"line": "all_goals simp [*]",
"before_state": "case empty\nι : Type u_1\nα : Type u_2\ninst✝⁶ : AddCommMonoid α\ninst✝⁵ : LT α\ninst✝⁴ : CommMonoidWithZero α\ninst✝³ : NoZeroDivisors α\ninst✝² : Nontrivial α\ninst✝¹ : DecidableEq α\ninst✝ : AddCommMonoid α\ns : Finset ι\nf : ι → WithBot α\n⊢ ∑ i ∈ ∅, f i = ⊥ ↔ ∃ i ∈ ∅, f i = ⊥\n---\ncase cons\nι : Type u_1\nα : Type u_2\ninst✝⁶ : AddCommMonoid α\ninst✝⁵ : LT α\ninst✝⁴ : CommMonoidWithZero α\ninst✝³ : NoZeroDivisors α\ninst✝² : Nontrivial α\ninst✝¹ : DecidableEq α\ninst✝ : AddCommMonoid α\ns : Finset ι\nf : ι → WithBot α\na✝¹ : ι\ns✝ : Finset ι\nh✝ : a✝¹ ∉ s✝\na✝ : ∑ i ∈ s✝, f i = ⊥ ↔ ∃ i ∈ s✝, f i = ⊥\n⊢ ∑ i ∈ cons a✝¹ s✝ h✝, f i = ⊥ ↔ ∃ i ∈ cons a✝¹ s✝ h✝, f i = ⊥",
"after_state": "No Goals!"
},
{
"line": "simp [*]",
"before_state": "case empty\nι : Type u_1\nα : Type u_2\ninst✝⁶ : AddCommMonoid α\ninst✝⁵ : LT α\ninst✝⁴ : CommMonoidWithZero α\ninst✝³ : NoZeroDivisors α\ninst✝² : Nontrivial α\ninst✝¹ : DecidableEq α\ninst✝ : AddCommMonoid α\ns : Finset ι\nf : ι → WithBot α\n⊢ ∑ i ∈ ∅, f i = ⊥ ↔ ∃ i ∈ ∅, f i = ⊥",
"after_state": "No Goals!"
},
{
"line": "simp [*]",
"before_state": "case cons\nι : Type u_1\nα : Type u_2\ninst✝⁶ : AddCommMonoid α\ninst✝⁵ : LT α\ninst✝⁴ : CommMonoidWithZero α\ninst✝³ : NoZeroDivisors α\ninst✝² : Nontrivial α\ninst✝¹ : DecidableEq α\ninst✝ : AddCommMonoid α\ns : Finset ι\nf : ι → WithBot α\na✝¹ : ι\ns✝ : Finset ι\nh✝ : a✝¹ ∉ s✝\na✝ : ∑ i ∈ s✝, f i = ⊥ ↔ ∃ i ∈ s✝, f i = ⊥\n⊢ ∑ i ∈ cons a✝¹ s✝ h✝, f i = ⊥ ↔ ∃ i ∈ cons a✝¹ s✝ h✝, f i = ⊥",
"after_state": "No Goals!"
}
] |
theorem prod_of_support_subset (f : α →₀ M) {s : Finset α} (hs : f.support ⊆ s) (g : α → M → N)
(h : ∀ i ∈ s, g i 0 = 1) : f.prod g = ∏ x ∈ s, g x (f x) := by
refine Finset.prod_subset hs fun x hxs hx => h x hxs ▸ (congr_arg (g x) ?_)
exact not_mem_support_iff.1 hx
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Finsupp/Basic.lean
|
{
"open": [
"Finset Function"
],
"variables": [
"{α ι γ A B C : Type*} [AddCommMonoid A] [AddCommMonoid B] [AddCommMonoid C]",
"{t : ι → A → C}",
"{s : Finset α} {f : α → ι →₀ A} (i : ι)",
"(g : ι →₀ A) (k : ι → A → γ → B) (x : γ)",
"{β M M' N P G H R S : Type*}",
"[Zero M] [Zero M'] [CommMonoid N]"
]
}
|
[
{
"line": "refine Finset.prod_subset hs fun x hxs hx => h x hxs ▸ (congr_arg (g x) ?_)",
"before_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝¹ : Zero M\ninst✝ : CommMonoid N\nf : α →₀ M\ns : Finset α\nhs : f.support ⊆ s\ng : α → M → N\nh : ∀ i ∈ s, g i 0 = 1\n⊢ f.prod g = ∏ x ∈ s, g x (f x)",
"after_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝¹ : Zero M\ninst✝ : CommMonoid N\nf : α →₀ M\ns : Finset α\nhs : f.support ⊆ s\ng : α → M → N\nh : ∀ i ∈ s, g i 0 = 1\nx : α\nhxs : x ∈ s\nhx : x ∉ f.support\n⊢ f x = 0"
},
{
"line": "exact not_mem_support_iff.1 hx",
"before_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝¹ : Zero M\ninst✝ : CommMonoid N\nf : α →₀ M\ns : Finset α\nhs : f.support ⊆ s\ng : α → M → N\nh : ∀ i ∈ s, g i 0 = 1\nx : α\nhxs : x ∈ s\nhx : x ∉ f.support\n⊢ f x = 0",
"after_state": "No Goals!"
}
] |
theorem prod_ite_eq [DecidableEq α] (f : α →₀ M) (a : α) (b : α → M → N) :
(f.prod fun x v => ite (a = x) (b x v) 1) = ite (a ∈ f.support) (b a (f a)) 1 := by
dsimp [Finsupp.prod]
rw [f.support.prod_ite_eq]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Finsupp/Basic.lean
|
{
"open": [
"Finset Function"
],
"variables": [
"{α ι γ A B C : Type*} [AddCommMonoid A] [AddCommMonoid B] [AddCommMonoid C]",
"{t : ι → A → C}",
"{s : Finset α} {f : α → ι →₀ A} (i : ι)",
"(g : ι →₀ A) (k : ι → A → γ → B) (x : γ)",
"{β M M' N P G H R S : Type*}",
"[Zero M] [Zero M'] [CommMonoid N]"
]
}
|
[
{
"line": "dsimp [Finsupp.prod]",
"before_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝² : Zero M\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (f.prod fun x v => if a = x then b x v else 1) = if a ∈ f.support then b a (f a) else 1",
"after_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝² : Zero M\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (∏ a_1 ∈ f.support, if a = a_1 then b a_1 (f a_1) else 1) = if a ∈ f.support then b a (f a) else 1"
},
{
"line": "rw [f.support.prod_ite_eq]",
"before_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝² : Zero M\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (∏ a_1 ∈ f.support, if a = a_1 then b a_1 (f a_1) else 1) = if a ∈ f.support then b a (f a) else 1",
"after_state": "No Goals!"
},
{
"line": "rewrite [f.support.prod_ite_eq]",
"before_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝² : Zero M\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (∏ a_1 ∈ f.support, if a = a_1 then b a_1 (f a_1) else 1) = if a ∈ f.support then b a (f a) else 1",
"after_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝² : Zero M\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (if a ∈ f.support then b a (f a) else 1) = if a ∈ f.support then b a (f a) else 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝² : Zero M\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (if a ∈ f.support then b a (f a) else 1) = if a ∈ f.support then b a (f a) else 1",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝² : Zero M\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (if a ∈ f.support then b a (f a) else 1) = if a ∈ f.support then b a (f a) else 1",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝² : Zero M\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (if a ∈ f.support then b a (f a) else 1) = if a ∈ f.support then b a (f a) else 1",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝² : Zero M\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (if a ∈ f.support then b a (f a) else 1) = if a ∈ f.support then b a (f a) else 1",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝² : Zero M\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (if a ∈ f.support then b a (f a) else 1) = if a ∈ f.support then b a (f a) else 1",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝² : Zero M\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (if a ∈ f.support then b a (f a) else 1) = if a ∈ f.support then b a (f a) else 1",
"after_state": "No Goals!"
}
] |
theorem sum_ite_self_eq [DecidableEq α] {N : Type*} [AddCommMonoid N] (f : α →₀ N) (a : α) :
(f.sum fun x v => ite (a = x) v 0) = f a := by
classical
convert f.sum_ite_eq a fun _ => id
simp [ite_eq_right_iff.2 Eq.symm]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Finsupp/Basic.lean
|
{
"open": [
"Finset Function"
],
"variables": [
"{α ι γ A B C : Type*} [AddCommMonoid A] [AddCommMonoid B] [AddCommMonoid C]",
"{t : ι → A → C}",
"{s : Finset α} {f : α → ι →₀ A} (i : ι)",
"(g : ι →₀ A) (k : ι → A → γ → B) (x : γ)",
"{β M M' N P G H R S : Type*}",
"[Zero M] [Zero M'] [CommMonoid N]"
]
}
|
[
{
"line": "classical\nconvert f.sum_ite_eq a fun _ => id\nsimp [ite_eq_right_iff.2 Eq.symm]",
"before_state": "α : Type u_1\ninst✝¹ : DecidableEq α\nN : Type u_16\ninst✝ : AddCommMonoid N\nf : α →₀ N\na : α\n⊢ (f.sum fun x v => if a = x then v else 0) = f a",
"after_state": "No Goals!"
},
{
"line": "convert f.sum_ite_eq a fun _ => id",
"before_state": "α : Type u_1\ninst✝¹ : DecidableEq α\nN : Type u_16\ninst✝ : AddCommMonoid N\nf : α →₀ N\na : α\n⊢ (f.sum fun x v => if a = x then v else 0) = f a",
"after_state": "case h.e'_3\nα : Type u_1\ninst✝¹ : DecidableEq α\nN : Type u_16\ninst✝ : AddCommMonoid N\nf : α →₀ N\na : α\n⊢ f a = if a ∈ f.support then id (f a) else 0"
},
{
"line": "simp [ite_eq_right_iff.2 Eq.symm]",
"before_state": "case h.e'_3\nα : Type u_1\ninst✝¹ : DecidableEq α\nN : Type u_16\ninst✝ : AddCommMonoid N\nf : α →₀ N\na : α\n⊢ f a = if a ∈ f.support then id (f a) else 0",
"after_state": "No Goals!"
}
] |
theorem prod_ite_eq' [DecidableEq α] (f : α →₀ M) (a : α) (b : α → M → N) :
(f.prod fun x v => ite (x = a) (b x v) 1) = ite (a ∈ f.support) (b a (f a)) 1 := by
dsimp [Finsupp.prod]
rw [f.support.prod_ite_eq']
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Finsupp/Basic.lean
|
{
"open": [
"Finset Function"
],
"variables": [
"{α ι γ A B C : Type*} [AddCommMonoid A] [AddCommMonoid B] [AddCommMonoid C]",
"{t : ι → A → C}",
"{s : Finset α} {f : α → ι →₀ A} (i : ι)",
"(g : ι →₀ A) (k : ι → A → γ → B) (x : γ)",
"{β M M' N P G H R S : Type*}",
"[Zero M] [Zero M'] [CommMonoid N]"
]
}
|
[
{
"line": "dsimp [Finsupp.prod]",
"before_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝² : Zero M\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (f.prod fun x v => if x = a then b x v else 1) = if a ∈ f.support then b a (f a) else 1",
"after_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝² : Zero M\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (∏ a_1 ∈ f.support, if a_1 = a then b a_1 (f a_1) else 1) = if a ∈ f.support then b a (f a) else 1"
},
{
"line": "rw [f.support.prod_ite_eq']",
"before_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝² : Zero M\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (∏ a_1 ∈ f.support, if a_1 = a then b a_1 (f a_1) else 1) = if a ∈ f.support then b a (f a) else 1",
"after_state": "No Goals!"
},
{
"line": "rewrite [f.support.prod_ite_eq']",
"before_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝² : Zero M\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (∏ a_1 ∈ f.support, if a_1 = a then b a_1 (f a_1) else 1) = if a ∈ f.support then b a (f a) else 1",
"after_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝² : Zero M\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (if a ∈ f.support then b a (f a) else 1) = if a ∈ f.support then b a (f a) else 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝² : Zero M\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (if a ∈ f.support then b a (f a) else 1) = if a ∈ f.support then b a (f a) else 1",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝² : Zero M\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (if a ∈ f.support then b a (f a) else 1) = if a ∈ f.support then b a (f a) else 1",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝² : Zero M\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (if a ∈ f.support then b a (f a) else 1) = if a ∈ f.support then b a (f a) else 1",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝² : Zero M\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (if a ∈ f.support then b a (f a) else 1) = if a ∈ f.support then b a (f a) else 1",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝² : Zero M\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (if a ∈ f.support then b a (f a) else 1) = if a ∈ f.support then b a (f a) else 1",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝² : Zero M\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (if a ∈ f.support then b a (f a) else 1) = if a ∈ f.support then b a (f a) else 1",
"after_state": "No Goals!"
}
] |
theorem sum_ite_self_eq' [DecidableEq α] {N : Type*} [AddCommMonoid N] (f : α →₀ N) (a : α) :
(f.sum fun x v => ite (x = a) v 0) = f a := by
classical
convert f.sum_ite_eq' a fun _ => id
simp [ite_eq_right_iff.2 Eq.symm]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Finsupp/Basic.lean
|
{
"open": [
"Finset Function"
],
"variables": [
"{α ι γ A B C : Type*} [AddCommMonoid A] [AddCommMonoid B] [AddCommMonoid C]",
"{t : ι → A → C}",
"{s : Finset α} {f : α → ι →₀ A} (i : ι)",
"(g : ι →₀ A) (k : ι → A → γ → B) (x : γ)",
"{β M M' N P G H R S : Type*}",
"[Zero M] [Zero M'] [CommMonoid N]"
]
}
|
[
{
"line": "classical\nconvert f.sum_ite_eq' a fun _ => id\nsimp [ite_eq_right_iff.2 Eq.symm]",
"before_state": "α : Type u_1\ninst✝¹ : DecidableEq α\nN : Type u_16\ninst✝ : AddCommMonoid N\nf : α →₀ N\na : α\n⊢ (f.sum fun x v => if x = a then v else 0) = f a",
"after_state": "No Goals!"
},
{
"line": "convert f.sum_ite_eq' a fun _ => id",
"before_state": "α : Type u_1\ninst✝¹ : DecidableEq α\nN : Type u_16\ninst✝ : AddCommMonoid N\nf : α →₀ N\na : α\n⊢ (f.sum fun x v => if x = a then v else 0) = f a",
"after_state": "case h.e'_3\nα : Type u_1\ninst✝¹ : DecidableEq α\nN : Type u_16\ninst✝ : AddCommMonoid N\nf : α →₀ N\na : α\n⊢ f a = if a ∈ f.support then id (f a) else 0"
},
{
"line": "simp [ite_eq_right_iff.2 Eq.symm]",
"before_state": "case h.e'_3\nα : Type u_1\ninst✝¹ : DecidableEq α\nN : Type u_16\ninst✝ : AddCommMonoid N\nf : α →₀ N\na : α\n⊢ f a = if a ∈ f.support then id (f a) else 0",
"after_state": "No Goals!"
}
] |
theorem support_sum [DecidableEq β] [Zero M] [AddCommMonoid N] {f : α →₀ M} {g : α → M → β →₀ N} :
(f.sum g).support ⊆ f.support.biUnion fun a => (g a (f a)).support := by
have : ∀ c, (f.sum fun a b => g a b c) ≠ 0 → ∃ a, f a ≠ 0 ∧ ¬(g a (f a)) c = 0 := fun a₁ h =>
let ⟨a, ha, ne⟩ := Finset.exists_ne_zero_of_sum_ne_zero h
⟨a, mem_support_iff.mp ha, ne⟩
simpa only [Finset.subset_iff,mem_support_iff,Finset.mem_biUnion,sum_apply,exists_prop]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Finsupp/Basic.lean
|
{
"open": [
"Finset Function"
],
"variables": [
"{α ι γ A B C : Type*} [AddCommMonoid A] [AddCommMonoid B] [AddCommMonoid C]",
"{t : ι → A → C}",
"{s : Finset α} {f : α → ι →₀ A} (i : ι)",
"(g : ι →₀ A) (k : ι → A → γ → B) (x : γ)",
"{β M M' N P G H R S : Type*}",
"[Zero M] [Zero M'] [CommMonoid N]",
"[Zero α] [CommMonoidWithZero β] [Nontrivial β] [NoZeroDivisors β]"
]
}
|
[
{
"line": "have : ∀ c, (f.sum fun a b => g a b c) ≠ 0 → ∃ a, f a ≠ 0 ∧ ¬(g a (f a)) c = 0 := fun a₁ h =>\n let ⟨a, ha, ne⟩ := Finset.exists_ne_zero_of_sum_ne_zero h\n ⟨a, mem_support_iff.mp ha, ne⟩",
"before_state": "α : Type u_1\nβ : Type u_7\nM : Type u_8\nN : Type u_10\ninst✝⁸ : Zero M\ninst✝⁷ : CommMonoid N\ninst✝⁶ : Zero α\ninst✝⁵ : CommMonoidWithZero β\ninst✝⁴ : Nontrivial β\ninst✝³ : NoZeroDivisors β\ninst✝² : DecidableEq β\ninst✝¹ : Zero M\ninst✝ : AddCommMonoid N\nf : α →₀ M\ng : α → M → β →₀ N\n⊢ (f.sum g).support ⊆ f.support.biUnion fun a => (g a (f a)).support",
"after_state": "α : Type u_1\nβ : Type u_7\nM : Type u_8\nN : Type u_10\ninst✝⁸ : Zero M\ninst✝⁷ : CommMonoid N\ninst✝⁶ : Zero α\ninst✝⁵ : CommMonoidWithZero β\ninst✝⁴ : Nontrivial β\ninst✝³ : NoZeroDivisors β\ninst✝² : DecidableEq β\ninst✝¹ : Zero M\ninst✝ : AddCommMonoid N\nf : α →₀ M\ng : α → M → β →₀ N\nthis : ∀ (c : β), (f.sum fun a b => (g a b) c) ≠ 0 → ∃ a, f a ≠ 0 ∧ ¬(g a (f a)) c = 0\n⊢ (f.sum g).support ⊆ f.support.biUnion fun a => (g a (f a)).support"
},
{
"line": "refine_lift\n have : ∀ c, (f.sum fun a b => g a b c) ≠ 0 → ∃ a, f a ≠ 0 ∧ ¬(g a (f a)) c = 0 := fun a₁ h =>\n let ⟨a, ha, ne⟩ := Finset.exists_ne_zero_of_sum_ne_zero h\n ⟨a, mem_support_iff.mp ha, ne⟩;\n ?_",
"before_state": "α : Type u_1\nβ : Type u_7\nM : Type u_8\nN : Type u_10\ninst✝⁸ : Zero M\ninst✝⁷ : CommMonoid N\ninst✝⁶ : Zero α\ninst✝⁵ : CommMonoidWithZero β\ninst✝⁴ : Nontrivial β\ninst✝³ : NoZeroDivisors β\ninst✝² : DecidableEq β\ninst✝¹ : Zero M\ninst✝ : AddCommMonoid N\nf : α →₀ M\ng : α → M → β →₀ N\n⊢ (f.sum g).support ⊆ f.support.biUnion fun a => (g a (f a)).support",
"after_state": "α : Type u_1\nβ : Type u_7\nM : Type u_8\nN : Type u_10\ninst✝⁸ : Zero M\ninst✝⁷ : CommMonoid N\ninst✝⁶ : Zero α\ninst✝⁵ : CommMonoidWithZero β\ninst✝⁴ : Nontrivial β\ninst✝³ : NoZeroDivisors β\ninst✝² : DecidableEq β\ninst✝¹ : Zero M\ninst✝ : AddCommMonoid N\nf : α →₀ M\ng : α → M → β →₀ N\nthis : ∀ (c : β), (f.sum fun a b => (g a b) c) ≠ 0 → ∃ a, f a ≠ 0 ∧ ¬(g a (f a)) c = 0\n⊢ (f.sum g).support ⊆ f.support.biUnion fun a => (g a (f a)).support"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have : ∀ c, (f.sum fun a b => g a b c) ≠ 0 → ∃ a, f a ≠ 0 ∧ ¬(g a (f a)) c = 0 := fun a₁ h =>\n let ⟨a, ha, ne⟩ := Finset.exists_ne_zero_of_sum_ne_zero h\n ⟨a, mem_support_iff.mp ha, ne⟩;\n ?_);\n rotate_right)",
"before_state": "α : Type u_1\nβ : Type u_7\nM : Type u_8\nN : Type u_10\ninst✝⁸ : Zero M\ninst✝⁷ : CommMonoid N\ninst✝⁶ : Zero α\ninst✝⁵ : CommMonoidWithZero β\ninst✝⁴ : Nontrivial β\ninst✝³ : NoZeroDivisors β\ninst✝² : DecidableEq β\ninst✝¹ : Zero M\ninst✝ : AddCommMonoid N\nf : α →₀ M\ng : α → M → β →₀ N\n⊢ (f.sum g).support ⊆ f.support.biUnion fun a => (g a (f a)).support",
"after_state": "α : Type u_1\nβ : Type u_7\nM : Type u_8\nN : Type u_10\ninst✝⁸ : Zero M\ninst✝⁷ : CommMonoid N\ninst✝⁶ : Zero α\ninst✝⁵ : CommMonoidWithZero β\ninst✝⁴ : Nontrivial β\ninst✝³ : NoZeroDivisors β\ninst✝² : DecidableEq β\ninst✝¹ : Zero M\ninst✝ : AddCommMonoid N\nf : α →₀ M\ng : α → M → β →₀ N\nthis : ∀ (c : β), (f.sum fun a b => (g a b) c) ≠ 0 → ∃ a, f a ≠ 0 ∧ ¬(g a (f a)) c = 0\n⊢ (f.sum g).support ⊆ f.support.biUnion fun a => (g a (f a)).support"
},
{
"line": "refine\n no_implicit_lambda%\n (have : ∀ c, (f.sum fun a b => g a b c) ≠ 0 → ∃ a, f a ≠ 0 ∧ ¬(g a (f a)) c = 0 := fun a₁ h =>\n let ⟨a, ha, ne⟩ := Finset.exists_ne_zero_of_sum_ne_zero h\n ⟨a, mem_support_iff.mp ha, ne⟩;\n ?_)",
"before_state": "α : Type u_1\nβ : Type u_7\nM : Type u_8\nN : Type u_10\ninst✝⁸ : Zero M\ninst✝⁷ : CommMonoid N\ninst✝⁶ : Zero α\ninst✝⁵ : CommMonoidWithZero β\ninst✝⁴ : Nontrivial β\ninst✝³ : NoZeroDivisors β\ninst✝² : DecidableEq β\ninst✝¹ : Zero M\ninst✝ : AddCommMonoid N\nf : α →₀ M\ng : α → M → β →₀ N\n⊢ (f.sum g).support ⊆ f.support.biUnion fun a => (g a (f a)).support",
"after_state": "α : Type u_1\nβ : Type u_7\nM : Type u_8\nN : Type u_10\ninst✝⁸ : Zero M\ninst✝⁷ : CommMonoid N\ninst✝⁶ : Zero α\ninst✝⁵ : CommMonoidWithZero β\ninst✝⁴ : Nontrivial β\ninst✝³ : NoZeroDivisors β\ninst✝² : DecidableEq β\ninst✝¹ : Zero M\ninst✝ : AddCommMonoid N\nf : α →₀ M\ng : α → M → β →₀ N\nthis : ∀ (c : β), (f.sum fun a b => (g a b) c) ≠ 0 → ∃ a, f a ≠ 0 ∧ ¬(g a (f a)) c = 0\n⊢ (f.sum g).support ⊆ f.support.biUnion fun a => (g a (f a)).support"
},
{
"line": "rotate_right",
"before_state": "α : Type u_1\nβ : Type u_7\nM : Type u_8\nN : Type u_10\ninst✝⁸ : Zero M\ninst✝⁷ : CommMonoid N\ninst✝⁶ : Zero α\ninst✝⁵ : CommMonoidWithZero β\ninst✝⁴ : Nontrivial β\ninst✝³ : NoZeroDivisors β\ninst✝² : DecidableEq β\ninst✝¹ : Zero M\ninst✝ : AddCommMonoid N\nf : α →₀ M\ng : α → M → β →₀ N\nthis : ∀ (c : β), (f.sum fun a b => (g a b) c) ≠ 0 → ∃ a, f a ≠ 0 ∧ ¬(g a (f a)) c = 0\n⊢ (f.sum g).support ⊆ f.support.biUnion fun a => (g a (f a)).support",
"after_state": "α : Type u_1\nβ : Type u_7\nM : Type u_8\nN : Type u_10\ninst✝⁸ : Zero M\ninst✝⁷ : CommMonoid N\ninst✝⁶ : Zero α\ninst✝⁵ : CommMonoidWithZero β\ninst✝⁴ : Nontrivial β\ninst✝³ : NoZeroDivisors β\ninst✝² : DecidableEq β\ninst✝¹ : Zero M\ninst✝ : AddCommMonoid N\nf : α →₀ M\ng : α → M → β →₀ N\nthis : ∀ (c : β), (f.sum fun a b => (g a b) c) ≠ 0 → ∃ a, f a ≠ 0 ∧ ¬(g a (f a)) c = 0\n⊢ (f.sum g).support ⊆ f.support.biUnion fun a => (g a (f a)).support"
},
{
"line": "simpa only [Finset.subset_iff, mem_support_iff, Finset.mem_biUnion, sum_apply, exists_prop]",
"before_state": "α : Type u_1\nβ : Type u_7\nM : Type u_8\nN : Type u_10\ninst✝⁸ : Zero M\ninst✝⁷ : CommMonoid N\ninst✝⁶ : Zero α\ninst✝⁵ : CommMonoidWithZero β\ninst✝⁴ : Nontrivial β\ninst✝³ : NoZeroDivisors β\ninst✝² : DecidableEq β\ninst✝¹ : Zero M\ninst✝ : AddCommMonoid N\nf : α →₀ M\ng : α → M → β →₀ N\nthis : ∀ (c : β), (f.sum fun a b => (g a b) c) ≠ 0 → ∃ a, f a ≠ 0 ∧ ¬(g a (f a)) c = 0\n⊢ (f.sum g).support ⊆ f.support.biUnion fun a => (g a (f a)).support",
"after_state": "No Goals!"
}
] |
lemma prod_mul_eq_prod_mul_of_exists [Zero M] [CommMonoid N]
{f : α →₀ M} {g : α → M → N} {n₁ n₂ : N}
(a : α) (ha : a ∈ f.support)
(h : g a (f a) * n₁ = g a (f a) * n₂) :
f.prod g * n₁ = f.prod g * n₂ := by
classical
exact Finset.prod_mul_eq_prod_mul_of_exists a ha h
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Finsupp/Basic.lean
|
{
"open": [
"Finset Function"
],
"variables": [
"{α ι γ A B C : Type*} [AddCommMonoid A] [AddCommMonoid B] [AddCommMonoid C]",
"{t : ι → A → C}",
"{s : Finset α} {f : α → ι →₀ A} (i : ι)",
"(g : ι →₀ A) (k : ι → A → γ → B) (x : γ)",
"{β M M' N P G H R S : Type*}",
"[Zero M] [Zero M'] [CommMonoid N]",
"[Zero α] [CommMonoidWithZero β] [Nontrivial β] [NoZeroDivisors β]"
]
}
|
[
{
"line": "classical exact Finset.prod_mul_eq_prod_mul_of_exists a ha h",
"before_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝⁴ : Zero M\ninst✝³ : CommMonoid N\ninst✝² : Zero α\ninst✝¹ : Zero M\ninst✝ : CommMonoid N\nf : α →₀ M\ng : α → M → N\nn₁ n₂ : N\na : α\nha : a ∈ f.support\nh : g a (f a) * n₁ = g a (f a) * n₂\n⊢ f.prod g * n₁ = f.prod g * n₂",
"after_state": "No Goals!"
},
{
"line": "exact Finset.prod_mul_eq_prod_mul_of_exists a ha h",
"before_state": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝⁴ : Zero M\ninst✝³ : CommMonoid N\ninst✝² : Zero α\ninst✝¹ : Zero M\ninst✝ : CommMonoid N\nf : α →₀ M\ng : α → M → N\nn₁ n₂ : N\na : α\nha : a ∈ f.support\nh : g a (f a) * n₁ = g a (f a) * n₂\n⊢ f.prod g * n₁ = f.prod g * n₂",
"after_state": "No Goals!"
}
] |
theorem prod_pow_pos_of_zero_not_mem_support {f : ℕ →₀ ℕ} (nhf : 0 ∉ f.support) :
0 < f.prod (· ^ ·) :=
Nat.pos_iff_ne_zero.mpr <| Finset.prod_ne_zero_iff.mpr fun _ hf =>
pow_ne_zero _ fun H => by subst H; exact nhf hf
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Finsupp/Basic.lean
|
{
"open": [
"Finset Function"
],
"variables": [
"{α ι γ A B C : Type*} [AddCommMonoid A] [AddCommMonoid B] [AddCommMonoid C]",
"{t : ι → A → C}",
"{s : Finset α} {f : α → ι →₀ A} (i : ι)",
"(g : ι →₀ A) (k : ι → A → γ → B) (x : γ)",
"{β M M' N P G H R S : Type*}",
"[Zero M] [Zero M'] [CommMonoid N]",
"[Zero α] [CommMonoidWithZero β] [Nontrivial β] [NoZeroDivisors β]",
"[NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S]"
]
}
|
[
{
"line": "subst H",
"before_state": "f : ℕ →₀ ℕ\nnhf : 0 ∉ f.support\nx✝ : ℕ\nhf : x✝ ∈ f.support\nH : x✝ = 0\n⊢ False",
"after_state": "f : ℕ →₀ ℕ\nnhf : 0 ∉ f.support\nhf : 0 ∈ f.support\n⊢ False"
},
{
"line": "exact nhf hf",
"before_state": "f : ℕ →₀ ℕ\nnhf : 0 ∉ f.support\nhf : 0 ∈ f.support\n⊢ False",
"after_state": "No Goals!"
}
] |
lemma prod_univ_pi [DecidableEq ι] [Fintype ι] {κ : ι → Type*} (t : ∀ i, Finset (κ i))
(f : (∀ i ∈ (univ : Finset ι), κ i) → β) :
∏ x ∈ univ.pi t, f x = ∏ x ∈ Fintype.piFinset t, f fun a _ ↦ x a := by
apply prod_nbij' (fun x i ↦ x i <| mem_univ _) (fun x i _ ↦ x i) <;> simp
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Group/Finset/Pi.lean
|
{
"open": [
"Fin Function"
],
"variables": [
"{ι β : Type*}",
"[CommMonoid β]"
]
}
|
[
{
"line": "focus\n apply prod_nbij' (fun x i ↦ x i <| mem_univ _) (fun x i _ ↦ x i)\n with_annotate_state\"<;>\" skip\n all_goals simp",
"before_state": "ι : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid β\nuniv : Finset ι\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nκ : ι → Type u_3\nt : (i : ι) → Finset (κ i)\nf : ((i : ι) → i ∈ univ → κ i) → β\n⊢ ∏ x ∈ univ.pi t, f x = ∏ x ∈ Fintype.piFinset t, f fun a x_1 => x a",
"after_state": "No Goals!"
},
{
"line": "apply prod_nbij' (fun x i ↦ x i <| mem_univ _) (fun x i _ ↦ x i)",
"before_state": "ι : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid β\nuniv : Finset ι\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nκ : ι → Type u_3\nt : (i : ι) → Finset (κ i)\nf : ((i : ι) → i ∈ univ → κ i) → β\n⊢ ∏ x ∈ univ.pi t, f x = ∏ x ∈ Fintype.piFinset t, f fun a x_1 => x a",
"after_state": "No Goals!"
}
] |
theorem prod_comm' {s : Finset γ} {t : γ → Finset α} {t' : Finset α} {s' : α → Finset γ}
(h : ∀ x y, x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t') {f : γ → α → β} :
(∏ x ∈ s, ∏ y ∈ t x, f x y) = ∏ y ∈ t', ∏ x ∈ s' y, f x y := by
classical
have : ∀ z : γ × α, (z ∈ s.biUnion fun x => (t x).map <| Function.Embedding.sectR x _) ↔
z.1 ∈ s ∧ z.2 ∈ t z.1 := by
rintro ⟨x, y⟩
simp only [mem_biUnion]
simp only [mem_map]
simp only [Function.Embedding.sectR_apply]
simp only [Prod.mk.injEq]
simp only [exists_eq_right]
simp only [← and_assoc]
exact
(prod_finset_product' _ _ _ this).symm.trans
((prod_finset_product_right' _ _ _) fun ⟨x, y⟩ => (this _).trans ((h x y).trans and_comm))
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Group/Finset/Sigma.lean
|
{
"open": [
"Fin Function"
],
"variables": [
"{ι κ α β γ : Type*}",
"{s s₁ s₂ : Finset α} {a : α} {f g : α → β}",
"[CommMonoid β]"
]
}
|
[
{
"line": "classical\nhave : ∀ z : γ × α, (z ∈ s.biUnion fun x => (t x).map <| Function.Embedding.sectR x _) ↔ z.1 ∈ s ∧ z.2 ∈ t z.1 :=\n by\n rintro ⟨x, y⟩\n simp only [mem_biUnion]\n simp only [mem_map]\n simp only [Function.Embedding.sectR_apply]\n simp only [Prod.mk.injEq]\n simp only [exists_eq_right]\n simp only [← and_assoc]\nexact\n (prod_finset_product' _ _ _ this).symm.trans\n ((prod_finset_product_right' _ _ _) fun ⟨x, y⟩ => (this _).trans ((h x y).trans and_comm))",
"before_state": "α : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : CommMonoid β\ns : Finset γ\nt : γ → Finset α\nt' : Finset α\ns' : α → Finset γ\nh : ∀ (x : γ) (y : α), x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t'\nf : γ → α → β\n⊢ ∏ x ∈ s, ∏ y ∈ t x, f x y = ∏ y ∈ t', ∏ x ∈ s' y, f x y",
"after_state": "No Goals!"
},
{
"line": "have : ∀ z : γ × α, (z ∈ s.biUnion fun x => (t x).map <| Function.Embedding.sectR x _) ↔ z.1 ∈ s ∧ z.2 ∈ t z.1 :=\n by\n rintro ⟨x, y⟩\n simp only [mem_biUnion]\n simp only [mem_map]\n simp only [Function.Embedding.sectR_apply]\n simp only [Prod.mk.injEq]\n simp only [exists_eq_right]\n simp only [← and_assoc]",
"before_state": "α : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : CommMonoid β\ns : Finset γ\nt : γ → Finset α\nt' : Finset α\ns' : α → Finset γ\nh : ∀ (x : γ) (y : α), x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t'\nf : γ → α → β\n⊢ ∏ x ∈ s, ∏ y ∈ t x, f x y = ∏ y ∈ t', ∏ x ∈ s' y, f x y",
"after_state": "α : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : CommMonoid β\ns : Finset γ\nt : γ → Finset α\nt' : Finset α\ns' : α → Finset γ\nh : ∀ (x : γ) (y : α), x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t'\nf : γ → α → β\nthis : ∀ (z : γ × α), (z ∈ s.biUnion fun x => Finset.map (Embedding.sectR x α) (t x)) ↔ z.1 ∈ s ∧ z.2 ∈ t z.1\n⊢ ∏ x ∈ s, ∏ y ∈ t x, f x y = ∏ y ∈ t', ∏ x ∈ s' y, f x y"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have : ∀ z : γ × α, (z ∈ s.biUnion fun x => (t x).map <| Function.Embedding.sectR x _) ↔ z.1 ∈ s ∧ z.2 ∈ t z.1 :=\n ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( rintro ⟨x, y⟩\n simp only [mem_biUnion]\n simp only [mem_map]\n simp only [Function.Embedding.sectR_apply]\n simp only [Prod.mk.injEq]\n simp only [exists_eq_right]\n simp only [← and_assoc])",
"before_state": "α : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : CommMonoid β\ns : Finset γ\nt : γ → Finset α\nt' : Finset α\ns' : α → Finset γ\nh : ∀ (x : γ) (y : α), x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t'\nf : γ → α → β\n⊢ ∏ x ∈ s, ∏ y ∈ t x, f x y = ∏ y ∈ t', ∏ x ∈ s' y, f x y",
"after_state": "α : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : CommMonoid β\ns : Finset γ\nt : γ → Finset α\nt' : Finset α\ns' : α → Finset γ\nh : ∀ (x : γ) (y : α), x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t'\nf : γ → α → β\nthis : ∀ (z : γ × α), (z ∈ s.biUnion fun x => Finset.map (Embedding.sectR x α) (t x)) ↔ z.1 ∈ s ∧ z.2 ∈ t z.1\n⊢ ∏ x ∈ s, ∏ y ∈ t x, f x y = ∏ y ∈ t', ∏ x ∈ s' y, f x y"
},
{
"line": "refine\n no_implicit_lambda%\n (have : ∀ z : γ × α, (z ∈ s.biUnion fun x => (t x).map <| Function.Embedding.sectR x _) ↔ z.1 ∈ s ∧ z.2 ∈ t z.1 :=\n ?body✝;\n ?_)",
"before_state": "α : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : CommMonoid β\ns : Finset γ\nt : γ → Finset α\nt' : Finset α\ns' : α → Finset γ\nh : ∀ (x : γ) (y : α), x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t'\nf : γ → α → β\n⊢ ∏ x ∈ s, ∏ y ∈ t x, f x y = ∏ y ∈ t', ∏ x ∈ s' y, f x y",
"after_state": "case body\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : CommMonoid β\ns : Finset γ\nt : γ → Finset α\nt' : Finset α\ns' : α → Finset γ\nh : ∀ (x : γ) (y : α), x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t'\nf : γ → α → β\n⊢ ∀ (z : γ × α), (z ∈ s.biUnion fun x => Finset.map (Embedding.sectR x α) (t x)) ↔ z.1 ∈ s ∧ z.2 ∈ t z.1\n---\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : CommMonoid β\ns : Finset γ\nt : γ → Finset α\nt' : Finset α\ns' : α → Finset γ\nh : ∀ (x : γ) (y : α), x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t'\nf : γ → α → β\nthis : ∀ (z : γ × α), (z ∈ s.biUnion fun x => Finset.map (Embedding.sectR x α) (t x)) ↔ z.1 ∈ s ∧ z.2 ∈ t z.1\n⊢ ∏ x ∈ s, ∏ y ∈ t x, f x y = ∏ y ∈ t', ∏ x ∈ s' y, f x y"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( rintro ⟨x, y⟩\n simp only [mem_biUnion]\n simp only [mem_map]\n simp only [Function.Embedding.sectR_apply]\n simp only [Prod.mk.injEq]\n simp only [exists_eq_right]\n simp only [← and_assoc])",
"before_state": "case body\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : CommMonoid β\ns : Finset γ\nt : γ → Finset α\nt' : Finset α\ns' : α → Finset γ\nh : ∀ (x : γ) (y : α), x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t'\nf : γ → α → β\n⊢ ∀ (z : γ × α), (z ∈ s.biUnion fun x => Finset.map (Embedding.sectR x α) (t x)) ↔ z.1 ∈ s ∧ z.2 ∈ t z.1\n---\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : CommMonoid β\ns : Finset γ\nt : γ → Finset α\nt' : Finset α\ns' : α → Finset γ\nh : ∀ (x : γ) (y : α), x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t'\nf : γ → α → β\nthis : ∀ (z : γ × α), (z ∈ s.biUnion fun x => Finset.map (Embedding.sectR x α) (t x)) ↔ z.1 ∈ s ∧ z.2 ∈ t z.1\n⊢ ∏ x ∈ s, ∏ y ∈ t x, f x y = ∏ y ∈ t', ∏ x ∈ s' y, f x y",
"after_state": "α : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : CommMonoid β\ns : Finset γ\nt : γ → Finset α\nt' : Finset α\ns' : α → Finset γ\nh : ∀ (x : γ) (y : α), x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t'\nf : γ → α → β\nthis : ∀ (z : γ × α), (z ∈ s.biUnion fun x => Finset.map (Embedding.sectR x α) (t x)) ↔ z.1 ∈ s ∧ z.2 ∈ t z.1\n⊢ ∏ x ∈ s, ∏ y ∈ t x, f x y = ∏ y ∈ t', ∏ x ∈ s' y, f x y"
},
{
"line": "with_annotate_state\"by\"\n ( rintro ⟨x, y⟩\n simp only [mem_biUnion]\n simp only [mem_map]\n simp only [Function.Embedding.sectR_apply]\n simp only [Prod.mk.injEq]\n simp only [exists_eq_right]\n simp only [← and_assoc])",
"before_state": "α : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : CommMonoid β\ns : Finset γ\nt : γ → Finset α\nt' : Finset α\ns' : α → Finset γ\nh : ∀ (x : γ) (y : α), x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t'\nf : γ → α → β\n⊢ ∀ (z : γ × α), (z ∈ s.biUnion fun x => Finset.map (Embedding.sectR x α) (t x)) ↔ z.1 ∈ s ∧ z.2 ∈ t z.1",
"after_state": "case mk\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : CommMonoid β\ns : Finset γ\nt : γ → Finset α\nt' : Finset α\ns' : α → Finset γ\nh : ∀ (x : γ) (y : α), x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t'\nf : γ → α → β\nx : γ\ny : α\n⊢ ((x, y) ∈ s.biUnion fun x => Finset.map (Embedding.sectR x α) (t x)) ↔ x ∈ s ∧ y ∈ t x"
},
{
"line": "rintro ⟨x, y⟩",
"before_state": "α : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : CommMonoid β\ns : Finset γ\nt : γ → Finset α\nt' : Finset α\ns' : α → Finset γ\nh : ∀ (x : γ) (y : α), x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t'\nf : γ → α → β\n⊢ ∀ (z : γ × α), (z ∈ s.biUnion fun x => Finset.map (Embedding.sectR x α) (t x)) ↔ z.1 ∈ s ∧ z.2 ∈ t z.1",
"after_state": "case mk\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : CommMonoid β\ns : Finset γ\nt : γ → Finset α\nt' : Finset α\ns' : α → Finset γ\nh : ∀ (x : γ) (y : α), x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t'\nf : γ → α → β\nx : γ\ny : α\n⊢ ((x, y) ∈ s.biUnion fun x => Finset.map (Embedding.sectR x α) (t x)) ↔ (x, y).1 ∈ s ∧ (x, y).2 ∈ t (x, y).1"
},
{
"line": "simp only [mem_biUnion]",
"before_state": "case mk\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : CommMonoid β\ns : Finset γ\nt : γ → Finset α\nt' : Finset α\ns' : α → Finset γ\nh : ∀ (x : γ) (y : α), x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t'\nf : γ → α → β\nx : γ\ny : α\n⊢ ((x, y) ∈ s.biUnion fun x => Finset.map (Embedding.sectR x α) (t x)) ↔ (x, y).1 ∈ s ∧ (x, y).2 ∈ t (x, y).1",
"after_state": "case mk\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : CommMonoid β\ns : Finset γ\nt : γ → Finset α\nt' : Finset α\ns' : α → Finset γ\nh : ∀ (x : γ) (y : α), x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t'\nf : γ → α → β\nx : γ\ny : α\n⊢ ((x, y) ∈ s.biUnion fun x => Finset.map (Embedding.sectR x α) (t x)) ↔ x ∈ s ∧ y ∈ t x"
},
{
"line": "simp only [mem_map]",
"before_state": "case mk\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : CommMonoid β\ns : Finset γ\nt : γ → Finset α\nt' : Finset α\ns' : α → Finset γ\nh : ∀ (x : γ) (y : α), x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t'\nf : γ → α → β\nx : γ\ny : α\n⊢ ((x, y) ∈ s.biUnion fun x => Finset.map (Embedding.sectR x α) (t x)) ↔ x ∈ s ∧ y ∈ t x",
"after_state": "case mk\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : CommMonoid β\ns : Finset γ\nt : γ → Finset α\nt' : Finset α\ns' : α → Finset γ\nh : ∀ (x : γ) (y : α), x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t'\nf : γ → α → β\nx : γ\ny : α\n⊢ ((x, y) ∈ s.biUnion fun x => Finset.map (Embedding.sectR x α) (t x)) ↔ x ∈ s ∧ y ∈ t x"
},
{
"line": "exact\n (prod_finset_product' _ _ _ this).symm.trans\n ((prod_finset_product_right' _ _ _) fun ⟨x, y⟩ => (this _).trans ((h x y).trans and_comm))",
"before_state": "α : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : CommMonoid β\ns : Finset γ\nt : γ → Finset α\nt' : Finset α\ns' : α → Finset γ\nh : ∀ (x : γ) (y : α), x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t'\nf : γ → α → β\nthis : ∀ (z : γ × α), (z ∈ s.biUnion fun x => Finset.map (Embedding.sectR x α) (t x)) ↔ z.1 ∈ s ∧ z.2 ∈ t z.1\n⊢ ∏ x ∈ s, ∏ y ∈ t x, f x y = ∏ y ∈ t', ∏ x ∈ s' y, f x y",
"after_state": "No Goals!"
}
] |
theorem prod_congr (h : s₁ = s₂) : (∀ x ∈ s₂, f x = g x) → s₁.prod f = s₂.prod g := by
rw [h]; exact fold_congr
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Group/Finset/Basic.lean
|
{
"open": [
"Fin Function"
],
"variables": [
"{ι κ G M : Type*} {s s₁ s₂ : Finset ι} {a : ι}",
"[CommMonoid M] {f g : ι → M}"
]
}
|
[
{
"line": "rw [h]",
"before_state": "ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝ : CommMonoid M\nf g : ι → M\nh : s₁ = s₂\n⊢ (∀ x ∈ s₂, f x = g x) → s₁.prod f = s₂.prod g",
"after_state": "ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝ : CommMonoid M\nf g : ι → M\nh : s₁ = s₂\n⊢ (∀ x ∈ s₂, f x = g x) → s₂.prod f = s₂.prod g"
},
{
"line": "rewrite [h]",
"before_state": "ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝ : CommMonoid M\nf g : ι → M\nh : s₁ = s₂\n⊢ (∀ x ∈ s₂, f x = g x) → s₁.prod f = s₂.prod g",
"after_state": "ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝ : CommMonoid M\nf g : ι → M\nh : s₁ = s₂\n⊢ (∀ x ∈ s₂, f x = g x) → s₂.prod f = s₂.prod g"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝ : CommMonoid M\nf g : ι → M\nh : s₁ = s₂\n⊢ (∀ x ∈ s₂, f x = g x) → s₂.prod f = s₂.prod g",
"after_state": "ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝ : CommMonoid M\nf g : ι → M\nh : s₁ = s₂\n⊢ (∀ x ∈ s₂, f x = g x) → s₂.prod f = s₂.prod g"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝ : CommMonoid M\nf g : ι → M\nh : s₁ = s₂\n⊢ (∀ x ∈ s₂, f x = g x) → s₂.prod f = s₂.prod g",
"after_state": "ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝ : CommMonoid M\nf g : ι → M\nh : s₁ = s₂\n⊢ (∀ x ∈ s₂, f x = g x) → s₂.prod f = s₂.prod g"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝ : CommMonoid M\nf g : ι → M\nh : s₁ = s₂\n⊢ (∀ x ∈ s₂, f x = g x) → s₂.prod f = s₂.prod g",
"after_state": "ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝ : CommMonoid M\nf g : ι → M\nh : s₁ = s₂\n⊢ (∀ x ∈ s₂, f x = g x) → s₂.prod f = s₂.prod g"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝ : CommMonoid M\nf g : ι → M\nh : s₁ = s₂\n⊢ (∀ x ∈ s₂, f x = g x) → s₂.prod f = s₂.prod g",
"after_state": "ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝ : CommMonoid M\nf g : ι → M\nh : s₁ = s₂\n⊢ (∀ x ∈ s₂, f x = g x) → s₂.prod f = s₂.prod g"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝ : CommMonoid M\nf g : ι → M\nh : s₁ = s₂\n⊢ (∀ x ∈ s₂, f x = g x) → s₂.prod f = s₂.prod g",
"after_state": "ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝ : CommMonoid M\nf g : ι → M\nh : s₁ = s₂\n⊢ (∀ x ∈ s₂, f x = g x) → s₂.prod f = s₂.prod g"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝ : CommMonoid M\nf g : ι → M\nh : s₁ = s₂\n⊢ (∀ x ∈ s₂, f x = g x) → s₂.prod f = s₂.prod g",
"after_state": "ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝ : CommMonoid M\nf g : ι → M\nh : s₁ = s₂\n⊢ (∀ x ∈ s₂, f x = g x) → s₂.prod f = s₂.prod g"
},
{
"line": "skip",
"before_state": "ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝ : CommMonoid M\nf g : ι → M\nh : s₁ = s₂\n⊢ (∀ x ∈ s₂, f x = g x) → s₂.prod f = s₂.prod g",
"after_state": "ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝ : CommMonoid M\nf g : ι → M\nh : s₁ = s₂\n⊢ (∀ x ∈ s₂, f x = g x) → s₂.prod f = s₂.prod g"
},
{
"line": "exact fold_congr",
"before_state": "ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝ : CommMonoid M\nf g : ι → M\nh : s₁ = s₂\n⊢ (∀ x ∈ s₂, f x = g x) → s₂.prod f = s₂.prod g",
"after_state": "No Goals!"
}
] |
lemma prod_eq_one_iff [Subsingleton Mˣ] : ∏ i ∈ s, f i = 1 ↔ ∀ i ∈ s, f i = 1 := by
induction' s using Finset.cons_induction with i s hi ih <;> simp [*]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Group/Finset/Basic.lean
|
{
"open": [
"Fin Function"
],
"variables": [
"{ι κ G M : Type*} {s s₁ s₂ : Finset ι} {a : ι}",
"[CommMonoid M] {f g : ι → M}"
]
}
|
[
{
"line": "focus\n induction' s using Finset.cons_induction with i s hi ih\n with_annotate_state\"<;>\" skip\n all_goals simp [*]",
"before_state": "ι : Type u_1\nM : Type u_4\ns : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : Subsingleton Mˣ\n⊢ ∏ i ∈ s, f i = 1 ↔ ∀ i ∈ s, f i = 1",
"after_state": "No Goals!"
},
{
"line": "induction' s using Finset.cons_induction with i s hi ih",
"before_state": "ι : Type u_1\nM : Type u_4\ns : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : Subsingleton Mˣ\n⊢ ∏ i ∈ s, f i = 1 ↔ ∀ i ∈ s, f i = 1",
"after_state": "case empty\nι : Type u_1\nM : Type u_4\ns : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : Subsingleton Mˣ\n⊢ ∏ i ∈ ∅, f i = 1 ↔ ∀ i ∈ ∅, f i = 1\n---\ncase cons\nι : Type u_1\nM : Type u_4\ns✝ : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : Subsingleton Mˣ\ni : ι\ns : Finset ι\nhi : i ∉ s\nih : ∏ i ∈ s, f i = 1 ↔ ∀ i ∈ s, f i = 1\n⊢ ∏ i ∈ Finset.cons i s hi, f i = 1 ↔ ∀ i_1 ∈ Finset.cons i s hi, f i_1 = 1"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case empty\nι : Type u_1\nM : Type u_4\ns : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : Subsingleton Mˣ\n⊢ ∏ i ∈ ∅, f i = 1 ↔ ∀ i ∈ ∅, f i = 1\n---\ncase cons\nι : Type u_1\nM : Type u_4\ns✝ : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : Subsingleton Mˣ\ni : ι\ns : Finset ι\nhi : i ∉ s\nih : ∏ i ∈ s, f i = 1 ↔ ∀ i ∈ s, f i = 1\n⊢ ∏ i ∈ Finset.cons i s hi, f i = 1 ↔ ∀ i_1 ∈ Finset.cons i s hi, f i_1 = 1",
"after_state": "case empty\nι : Type u_1\nM : Type u_4\ns : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : Subsingleton Mˣ\n⊢ ∏ i ∈ ∅, f i = 1 ↔ ∀ i ∈ ∅, f i = 1\n---\ncase cons\nι : Type u_1\nM : Type u_4\ns✝ : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : Subsingleton Mˣ\ni : ι\ns : Finset ι\nhi : i ∉ s\nih : ∏ i ∈ s, f i = 1 ↔ ∀ i ∈ s, f i = 1\n⊢ ∏ i ∈ Finset.cons i s hi, f i = 1 ↔ ∀ i_1 ∈ Finset.cons i s hi, f i_1 = 1"
},
{
"line": "skip",
"before_state": "case empty\nι : Type u_1\nM : Type u_4\ns : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : Subsingleton Mˣ\n⊢ ∏ i ∈ ∅, f i = 1 ↔ ∀ i ∈ ∅, f i = 1\n---\ncase cons\nι : Type u_1\nM : Type u_4\ns✝ : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : Subsingleton Mˣ\ni : ι\ns : Finset ι\nhi : i ∉ s\nih : ∏ i ∈ s, f i = 1 ↔ ∀ i ∈ s, f i = 1\n⊢ ∏ i ∈ Finset.cons i s hi, f i = 1 ↔ ∀ i_1 ∈ Finset.cons i s hi, f i_1 = 1",
"after_state": "case empty\nι : Type u_1\nM : Type u_4\ns : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : Subsingleton Mˣ\n⊢ ∏ i ∈ ∅, f i = 1 ↔ ∀ i ∈ ∅, f i = 1\n---\ncase cons\nι : Type u_1\nM : Type u_4\ns✝ : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : Subsingleton Mˣ\ni : ι\ns : Finset ι\nhi : i ∉ s\nih : ∏ i ∈ s, f i = 1 ↔ ∀ i ∈ s, f i = 1\n⊢ ∏ i ∈ Finset.cons i s hi, f i = 1 ↔ ∀ i_1 ∈ Finset.cons i s hi, f i_1 = 1"
},
{
"line": "all_goals simp [*]",
"before_state": "case empty\nι : Type u_1\nM : Type u_4\ns : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : Subsingleton Mˣ\n⊢ ∏ i ∈ ∅, f i = 1 ↔ ∀ i ∈ ∅, f i = 1\n---\ncase cons\nι : Type u_1\nM : Type u_4\ns✝ : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : Subsingleton Mˣ\ni : ι\ns : Finset ι\nhi : i ∉ s\nih : ∏ i ∈ s, f i = 1 ↔ ∀ i ∈ s, f i = 1\n⊢ ∏ i ∈ Finset.cons i s hi, f i = 1 ↔ ∀ i_1 ∈ Finset.cons i s hi, f i_1 = 1",
"after_state": "No Goals!"
},
{
"line": "simp [*]",
"before_state": "case empty\nι : Type u_1\nM : Type u_4\ns : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : Subsingleton Mˣ\n⊢ ∏ i ∈ ∅, f i = 1 ↔ ∀ i ∈ ∅, f i = 1",
"after_state": "No Goals!"
},
{
"line": "simp [*]",
"before_state": "case cons\nι : Type u_1\nM : Type u_4\ns✝ : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : Subsingleton Mˣ\ni : ι\ns : Finset ι\nhi : i ∉ s\nih : ∏ i ∈ s, f i = 1 ↔ ∀ i ∈ s, f i = 1\n⊢ ∏ i ∈ Finset.cons i s hi, f i = 1 ↔ ∀ i_1 ∈ Finset.cons i s hi, f i_1 = 1",
"after_state": "No Goals!"
}
] |
lemma prod_sum_eq_prod_toLeft_mul_prod_toRight (s : Finset (ι ⊕ κ)) (f : ι ⊕ κ → M) :
∏ x ∈ s, f x = (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) := by
rw [← Finset.toLeft_disjSum_toRight (u := s)]
rw [Finset.prod_disj_sum]
rw [Finset.toLeft_disjSum]
rw [Finset.toRight_disjSum]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Group/Finset/Basic.lean
|
{
"open": [
"Fin Function"
],
"variables": [
"{ι κ G M : Type*} {s s₁ s₂ : Finset ι} {a : ι}",
"[CommMonoid M] {f g : ι → M}"
]
}
|
[
{
"line": "rw [← Finset.toLeft_disjSum_toRight (u := s)]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ ∏ x ∈ s, f x = (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ ∏ x ∈ s.toLeft.disjSum s.toRight, f x =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "rewrite [← Finset.toLeft_disjSum_toRight (u := s)]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ ∏ x ∈ s, f x = (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ ∏ x ∈ s.toLeft.disjSum s.toRight, f x =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ ∏ x ∈ s.toLeft.disjSum s.toRight, f x =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ ∏ x ∈ s.toLeft.disjSum s.toRight, f x =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ ∏ x ∈ s.toLeft.disjSum s.toRight, f x =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ ∏ x ∈ s.toLeft.disjSum s.toRight, f x =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ ∏ x ∈ s.toLeft.disjSum s.toRight, f x =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ ∏ x ∈ s.toLeft.disjSum s.toRight, f x =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ ∏ x ∈ s.toLeft.disjSum s.toRight, f x =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ ∏ x ∈ s.toLeft.disjSum s.toRight, f x =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ ∏ x ∈ s.toLeft.disjSum s.toRight, f x =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ ∏ x ∈ s.toLeft.disjSum s.toRight, f x =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ ∏ x ∈ s.toLeft.disjSum s.toRight, f x =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ ∏ x ∈ s.toLeft.disjSum s.toRight, f x =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "skip",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ ∏ x ∈ s.toLeft.disjSum s.toRight, f x =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ ∏ x ∈ s.toLeft.disjSum s.toRight, f x =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "rw [Finset.prod_disj_sum]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ ∏ x ∈ s.toLeft.disjSum s.toRight, f x =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "rewrite [Finset.prod_disj_sum]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ ∏ x ∈ s.toLeft.disjSum s.toRight, f x =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "skip",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "rw [Finset.toLeft_disjSum]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "rewrite [Finset.toLeft_disjSum]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ (s.toLeft.disjSum s.toRight).toLeft, f (Sum.inl x)) *\n ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "skip",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)"
},
{
"line": "rw [Finset.toRight_disjSum]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "No Goals!"
},
{
"line": "rewrite [Finset.toRight_disjSum]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ (s.toLeft.disjSum s.toRight).toRight, f (Sum.inr x)",
"after_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x)",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x)",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x)",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x)",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x)",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset (ι ⊕ κ)\nf : ι ⊕ κ → M\n⊢ (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x) =\n (∏ x ∈ s.toLeft, f (Sum.inl x)) * ∏ x ∈ s.toRight, f (Sum.inr x)",
"after_state": "No Goals!"
}
] |
theorem prod_sumElim (s : Finset ι) (t : Finset κ) (f : ι → M) (g : κ → M) :
∏ x ∈ s.disjSum t, Sum.elim f g x = (∏ x ∈ s, f x) * ∏ x ∈ t, g x := by simp
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/BigOperators/Group/Finset/Basic.lean
|
{
"open": [
"Fin Function"
],
"variables": [
"{ι κ G M : Type*} {s s₁ s₂ : Finset ι} {a : ι}",
"[CommMonoid M] {f g : ι → M}"
]
}
|
[
{
"line": "simp",
"before_state": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset ι\nt : Finset κ\nf : ι → M\ng : κ → M\n⊢ ∏ x ∈ s.disjSum t, Sum.elim f g x = (∏ x ∈ s, f x) * ∏ x ∈ t, g x",
"after_state": "No Goals!"
}
] |
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