url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | intro d' hd' | case h.left
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes... | case h.left
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | simp only [coeff_smul, smul_eq_mul] at hd' | case h.left
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes... | case h.left
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | obtain ⟨d, hdd', hd0⟩ := coeff_rename_ne_zero _ _ _ (right_ne_zero_of_mul hd') | case h.left
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes... | case h.left.intro.intro
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __s... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | rw [← hdd'] | case h.left.intro.intro
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __s... | case h.left.intro.intro
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __s... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.intro.intro
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | simp only [hp', hf, coeff_baseChange_apply, coe_mk] at hd0 | case h.left.intro.intro
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __s... | case h.left.intro.intro
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __s... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.intro.intro
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | have hd0' : coeff d p ≠ 0 := by
intro h
simp only [h, map_zero, ne_eq, not_true_eq_false] at hd0 | case h.left.intro.intro
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __s... | case h.left.intro.intro
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __s... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.intro.intro
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | convert hpn hd0' | case h.left.intro.intro
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __s... | case h.e'_2
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.intro.intro
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | simp only [weightedDegree, LinearMap.toAddMonoidHom_coe, Finsupp.total_mapDomain] | case h.e'_2
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes... | case h.e'_2
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | rfl | case h.e'_2
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | intro h | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes' := ⋯ }
hf ... | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes' := ⋯ }
hf ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src :=... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | simp only [h, map_zero, ne_eq, not_true_eq_false] at hd0 | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes' := ⋯ }
hf ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src :=... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | have h_eq : (algebraMap S (MvPolynomial (ℕ × M) S)).comp (algebraMap R S) =
(algebraMap R (MvPolynomial (ℕ × M) S)) := rfl | case h.right
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commute... | case h.right
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commute... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | have h_eq' : (algebraMap S (DividedPowerAlgebra S (S ⊗[R] M))).comp (algebraMap R S) =
(algebraMap R (DividedPowerAlgebra S (S ⊗[R] M))) := rfl | case h.right
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commute... | case h.right
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commute... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | rw [LinearMapClass.map_smul, dpScalarExtension_tmul] | case h.right
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commute... | case h.right
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commute... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | congr 1 | case h.right
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commute... | case h.right.e_a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, com... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | simp only [SetLike.mem_coe, mem_weightedHomogeneousSubmodule, IsWeightedHomogeneous] at hpn | case h.right.e_a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, com... | case h.right.e_a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, com... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.e_a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | simp only [hp', baseChange, hf, coe_ringHom_mk, coe_mk, coe_eval₂RingHom, ← hpa,
MvPolynomial.rename, ← algebraMap_eq, h_eq, dpScalarExtension, AlgHom.baseChange,
toRingHom_eq_coe, coe_mk, RingHom.coe_coe, productMap_apply_tmul,
_root_.map_one, one_mul, lift', RingQuot.liftAlgHom_mkAlgHom_apply, coe_eval₂AlgHom,
... | case h.right.e_a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, com... | case h.right.e_a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, com... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.e_a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | rw [← AlgHom.comp_apply, MvPolynomial.comp_aeval, aeval_def] | case h.right.e_a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, com... | case h.right.e_a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, com... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.e_a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | simp only [eval₂_eq, MvPolynomial.algebraMap_apply, prod_X_pow_eq_monomial, RingHom.coe_comp,
Function.comp_apply] | case h.right.e_a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, com... | case h.right.e_a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, com... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.e_a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | apply Finset.sum_congr | case h.right.e_a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, com... | case h.right.e_a.h
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, c... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.e_a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | intro d | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes' := ⋯ }
hf ... | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes' := ⋯ }
hf ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src :=... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | conv_lhs => rw [MvPolynomial.as_sum (p := p)] | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes' := ⋯ }
hf ... | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes' := ⋯ }
hf ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src :=... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | simp only [coeff_sum, coeff_C_mul, _root_.map_sum, coeff_monomial, mul_ite, mul_one, mul_zero] | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes' := ⋯ }
hf ... | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes' := ⋯ }
hf ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src :=... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | refine Finset.sum_congr rfl (fun x _hx => ?_) | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes' := ⋯ }
hf ... | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes' := ⋯ }
hf ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src :=... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | split_ifs | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes' := ⋯ }
hf ... | case pos
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes' :... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src :=... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | rfl | case pos
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes' :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | rw [map_zero] | case neg
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, commutes' :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | ext d | case h.right.e_a.h
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, c... | case h.right.e_a.h.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src,... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.e_a.h
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | simp only [mem_support_iff, ne_eq, not_iff_not] | case h.right.e_a.h.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src,... | case h.right.e_a.h.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src,... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.e_a.h.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | conv_rhs => rw [MvPolynomial.as_sum (p := p)] | case h.right.e_a.h.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src,... | case h.right.e_a.h.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src,... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.e_a.h.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | simp only [coeff_sum, coeff_C_mul, coeff_monomial, mul_ite, mul_one, mul_zero,
Finset.sum_ite_eq', mem_support_iff, ne_eq, ite_not, ite_eq_left_iff,
support_sum_monomial_coeff] | case h.right.e_a.h.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src,... | case h.right.e_a.h.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src,... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.e_a.h.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | refine ⟨fun hd => ?_, fun hd hd' => absurd hd hd'⟩ | case h.right.e_a.h.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src,... | case h.right.e_a.h.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src,... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.e_a.h.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | by_contra h0 | case h.right.e_a.h.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src,... | case h.right.e_a.h.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src,... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.e_a.h.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | exact h0 (hinj (map_zero (f := algebraMap R S) ▸ hd h0)) | case h.right.e_a.h.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src,... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.e_a.h.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | intro d _hd | case h.right.e_a.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, c... | case h.right.e_a.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, c... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.e_a.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | rw [← algebraMap_eq, coeff_sum] | case h.right.e_a.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, c... | case h.right.e_a.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, c... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.e_a.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | simp only [MvPolynomial.algebraMap_apply, coeff_C_mul, map_sum, dp_def'] | case h.right.e_a.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, c... | case h.right.e_a.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, c... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.e_a.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | rw [← h_eq', RingHom.coe_comp, Function.comp_apply, h_sum d, coeff_sum, map_sum] | case h.right.e_a.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, c... | case h.right.e_a.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, c... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.e_a.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/BaseChange.lean | DividedPowerAlgebra.dpScalarExtension_mem_grade | [356, 1] | [420, 98] | simp only [Algebra.id.map_eq_id, RingHomCompTriple.comp_apply, _root_.map_mul, coeff_C_mul] | case h.right.e_a.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] S :=
let __src := algebraMap R S;
{ toRingHom := __src, c... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.e_a.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hinj : Function.Injective ⇑(algebraMap R S)
a : DividedPowerAlgebra R M
n : ℕ
s : S
f : R →ₐ[R] ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | Subalgebra.fg_sup | [86, 1] | [92, 21] | classical
obtain ⟨s, hs⟩ := hA
obtain ⟨t, ht⟩ := hB
rw [← hs, ← ht, ← Algebra.adjoin_union, ← Finset.coe_union]
exact ⟨s ∪ t, rfl⟩ | R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
A B : Subalgebra R S
hA : A.FG
hB : B.FG
⊢ (A ⊔ B).FG | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
A B : Subalgebra R S
hA : A.FG
hB : B.FG
⊢ (A ⊔ B).FG
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | Subalgebra.fg_sup | [86, 1] | [92, 21] | obtain ⟨s, hs⟩ := hA | R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
A B : Subalgebra R S
hA : A.FG
hB : B.FG
⊢ (A ⊔ B).FG | case intro
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
A B : Subalgebra R S
hB : B.FG
s : Finset S
hs : Algebra.adjoin R ↑s = A
⊢ (A ⊔ B).FG | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
A B : Subalgebra R S
hA : A.FG
hB : B.FG
⊢ (A ⊔ B).FG
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | Subalgebra.fg_sup | [86, 1] | [92, 21] | obtain ⟨t, ht⟩ := hB | case intro
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
A B : Subalgebra R S
hB : B.FG
s : Finset S
hs : Algebra.adjoin R ↑s = A
⊢ (A ⊔ B).FG | case intro.intro
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
A B : Subalgebra R S
s : Finset S
hs : Algebra.adjoin R ↑s = A
t : Finset S
ht : Algebra.adjoin R ↑t = B
⊢ (A ⊔ B).FG | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
A B : Subalgebra R S
hB : B.FG
s : Finset S
hs : Algebra.adjoin R ↑s = A
⊢ (A ⊔ B).FG
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | Subalgebra.fg_sup | [86, 1] | [92, 21] | rw [← hs, ← ht, ← Algebra.adjoin_union, ← Finset.coe_union] | case intro.intro
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
A B : Subalgebra R S
s : Finset S
hs : Algebra.adjoin R ↑s = A
t : Finset S
ht : Algebra.adjoin R ↑t = B
⊢ (A ⊔ B).FG | case intro.intro
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
A B : Subalgebra R S
s : Finset S
hs : Algebra.adjoin R ↑s = A
t : Finset S
ht : Algebra.adjoin R ↑t = B
⊢ (Algebra.adjoin R ↑(s ∪ t)).FG | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
A B : Subalgebra R S
s : Finset S
hs : Algebra.adjoin R ↑s = A
t : Finset S
ht : Algebra.adjoin R ↑t = B
⊢ (A ⊔ B).FG
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | Subalgebra.fg_sup | [86, 1] | [92, 21] | exact ⟨s ∪ t, rfl⟩ | case intro.intro
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
A B : Subalgebra R S
s : Finset S
hs : Algebra.adjoin R ↑s = A
t : Finset S
ht : Algebra.adjoin R ↑t = B
⊢ (Algebra.adjoin R ↑(s ∪ t)).FG | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
A B : Subalgebra R S
s : Finset S
hs : Algebra.adjoin R ↑s = A
t : Finset S
ht : Algebra.adjoin R ↑t = B
⊢ (Algebra.adjoin R ↑(s ∪ t)).FG
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | AlgHom.factor | [94, 1] | [97, 63] | ext | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
⊢ φ = φ.range.val.comp ((↑(Ideal.quotientKerEquivRange φ)).comp (Ideal.Quotient.mkₐ R (RingHom.ker φ))) | case H
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
x✝ : S
⊢ φ x✝ = (φ.range.val.comp ((↑(Ideal.quotientKerEquivRange φ)).comp (Ideal.Quotient.mkₐ R (RingHom.ker φ)))) x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
⊢ φ = φ.range.val.comp ((↑(Ideal.quotientKerEquivRange φ)).comp (Ideal.Quotient.mkₐ R (RingHom.ker φ)))
T... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | AlgHom.factor | [94, 1] | [97, 63] | rfl | case H
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
x✝ : S
⊢ φ x✝ = (φ.range.val.comp ((↑(Ideal.quotientKerEquivRange φ)).comp (Ideal.Quotient.mkₐ R (RingHom.ker φ)))) x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
x✝ : S
⊢ φ x✝ = (φ.range.val.comp ((↑(Ideal.quotientKerEquivRange φ)).comp (Ideal.Quotient.mkₐ R (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | AlgHom.comp_rangeRestrict | [99, 1] | [102, 11] | ext | R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : Semiring S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
⊢ φ.range.val.comp φ.rangeRestrict = φ | case H
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : Semiring S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
x✝ : S
⊢ (φ.range.val.comp φ.rangeRestrict) x✝ = φ x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : Semiring S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
⊢ φ.range.val.comp φ.rangeRestrict = φ
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | AlgHom.comp_rangeRestrict | [99, 1] | [102, 11] | rfl | case H
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : Semiring S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
x✝ : S
⊢ (φ.range.val.comp φ.rangeRestrict) x✝ = φ x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : Semiring S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
x✝ : S
⊢ (φ.range.val.comp φ.rangeRestrict) x✝ = φ x✝
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | AlgHom.quotientKerEquivRange_mk | [104, 1] | [111, 6] | ext s | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
⊢ (↑(Ideal.quotientKerEquivRange φ)).comp (Ideal.Quotient.mkₐ R (RingHom.ker φ)) = φ.rangeRestrict | case H.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
s : S
⊢ ↑(((↑(Ideal.quotientKerEquivRange φ)).comp (Ideal.Quotient.mkₐ R (RingHom.ker φ))) s) = ↑(φ.rangeRestrict s) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
⊢ (↑(Ideal.quotientKerEquivRange φ)).comp (Ideal.Quotient.mkₐ R (RingHom.ker φ)) = φ.rangeRestrict
TACTIC... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | AlgHom.quotientKerEquivRange_mk | [104, 1] | [111, 6] | simp only [AlgEquiv.toAlgHom_eq_coe, coe_comp, AlgHom.coe_coe,
Ideal.Quotient.mkₐ_eq_mk, Function.comp_apply, coe_codRestrict] | case H.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
s : S
⊢ ↑(((↑(Ideal.quotientKerEquivRange φ)).comp (Ideal.Quotient.mkₐ R (RingHom.ker φ))) s) = ↑(φ.rangeRestrict s) | case H.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
s : S
⊢ ↑((Ideal.quotientKerEquivRange φ) ((Ideal.Quotient.mk (RingHom.ker φ)) s)) = φ s | Please generate a tactic in lean4 to solve the state.
STATE:
case H.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
s : S
⊢ ↑(((↑(Ideal.quotientKerEquivRange φ)).comp (Ideal.Quotient.mkₐ R (RingHom.ker φ))) s) = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | AlgHom.quotientKerEquivRange_mk | [104, 1] | [111, 6] | rfl | case H.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
s : S
⊢ ↑((Ideal.quotientKerEquivRange φ) ((Ideal.Quotient.mk (RingHom.ker φ)) s)) = φ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H.a
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
s : S
⊢ ↑((Ideal.quotientKerEquivRange φ) ((Ideal.Quotient.mk (RingHom.ker φ)) s)) = φ s
TACTIC:... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | Ideal.kerLiftAlg_eq_val_comp_Equiv | [113, 1] | [120, 6] | apply Ideal.Quotient.algHom_ext | R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
⊢ kerLiftAlg φ = φ.range.val.comp ↑(quotientKerEquivRange φ) | case h
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
⊢ (kerLiftAlg φ).comp (Quotient.mkₐ R (RingHom.ker φ)) =
(φ.range.val.comp ↑(quotientKerEquivRange φ)).comp (Quotient.mkₐ R (RingHom.ker φ)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
⊢ kerLiftAlg φ = φ.range.val.comp ↑(quotientKerEquivRange φ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | Ideal.kerLiftAlg_eq_val_comp_Equiv | [113, 1] | [120, 6] | ext s | case h
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
⊢ (kerLiftAlg φ).comp (Quotient.mkₐ R (RingHom.ker φ)) =
(φ.range.val.comp ↑(quotientKerEquivRange φ)).comp (Quotient.mkₐ R (RingHom.ker φ)) | case h.H
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
s : S
⊢ ((kerLiftAlg φ).comp (Quotient.mkₐ R (RingHom.ker φ))) s =
((φ.range.val.comp ↑(quotientKerEquivRange φ)).comp (Quotient.mkₐ R (RingHom.ker φ))) ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
⊢ (kerLiftAlg φ).comp (Quotient.mkₐ R (RingHom.ker φ)) =
(φ.range.val.comp ↑(quotientKerEquivR... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | Ideal.kerLiftAlg_eq_val_comp_Equiv | [113, 1] | [120, 6] | simp only [AlgHom.coe_comp, Quotient.mkₐ_eq_mk, Function.comp_apply, kerLiftAlg_mk,
AlgEquiv.toAlgHom_eq_coe, Subalgebra.coe_val, AlgHom.coe_coe] | case h.H
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
s : S
⊢ ((kerLiftAlg φ).comp (Quotient.mkₐ R (RingHom.ker φ))) s =
((φ.range.val.comp ↑(quotientKerEquivRange φ)).comp (Quotient.mkₐ R (RingHom.ker φ))) ... | case h.H
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
s : S
⊢ φ s = ↑((quotientKerEquivRange φ) ((Quotient.mk (RingHom.ker φ)) s)) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
s : S
⊢ ((kerLiftAlg φ).comp (Quotient.mkₐ R (RingHom.ker φ))) s =
((φ.range.val.comp ↑(quot... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | Ideal.kerLiftAlg_eq_val_comp_Equiv | [113, 1] | [120, 6] | rfl | case h.H
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
s : S
⊢ φ s = ↑((quotientKerEquivRange φ) ((Quotient.mk (RingHom.ker φ)) s)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : Semiring T
inst✝ : Algebra R T
φ : S →ₐ[R] T
s : S
⊢ φ s = ↑((quotientKerEquivRange φ) ((Quotient.mk (RingHom.ker φ)) s))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | MvPolynomial.aeval_range | [122, 1] | [143, 66] | apply le_antisymm | R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
⊢ (aeval s).range = Algebra.adjoin R (Set.range s) | case a
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
⊢ (aeval s).range ≤ Algebra.adjoin R (Set.range s)
case a
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
⊢ Algebra.adjoin R (... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
⊢ (aeval s).range = Algebra.adjoin R (Set.range s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | MvPolynomial.aeval_range | [122, 1] | [143, 66] | rintro x ⟨p, rfl⟩ | case a
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
⊢ (aeval s).range ≤ Algebra.adjoin R (Set.range s) | case a.intro
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
p : MvPolynomial σ R
⊢ (aeval s).toRingHom p ∈ Algebra.adjoin R (Set.range s) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
⊢ (aeval s).range ≤ Algebra.adjoin R (Set.range s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | MvPolynomial.aeval_range | [122, 1] | [143, 66] | simp only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe] | case a.intro
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
p : MvPolynomial σ R
⊢ (aeval s).toRingHom p ∈ Algebra.adjoin R (Set.range s) | case a.intro
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
p : MvPolynomial σ R
⊢ (aeval s) p ∈ Algebra.adjoin R (Set.range s) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
p : MvPolynomial σ R
⊢ (aeval s).toRingHom p ∈ Algebra.adjoin R (Set.range s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | MvPolynomial.aeval_range | [122, 1] | [143, 66] | induction p using induction_on with
| h_C a =>
simp only [aeval_C, Algebra.mem_adjoin_iff]
apply Subsemiring.subset_closure
left
use a
| h_add p q hp hq => simp only [map_add]; exact Subalgebra.add_mem _ hp hq
| h_X p n h =>
simp only [map_mul, aeval_X]
apply Subalgebra.mul_mem _ h
apply Algebra.subset_ad... | case a.intro
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
p : MvPolynomial σ R
⊢ (aeval s) p ∈ Algebra.adjoin R (Set.range s) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
p : MvPolynomial σ R
⊢ (aeval s) p ∈ Algebra.adjoin R (Set.range s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | MvPolynomial.aeval_range | [122, 1] | [143, 66] | simp only [aeval_C, Algebra.mem_adjoin_iff] | case a.intro.h_C
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
a : R
⊢ (aeval s) (C a) ∈ Algebra.adjoin R (Set.range s) | case a.intro.h_C
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
a : R
⊢ (algebraMap R S) a ∈ Algebra.adjoin R (Set.range s) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.h_C
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
a : R
⊢ (aeval s) (C a) ∈ Algebra.adjoin R (Set.range s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | MvPolynomial.aeval_range | [122, 1] | [143, 66] | apply Subsemiring.subset_closure | case a.intro.h_C
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
a : R
⊢ (algebraMap R S) a ∈ Algebra.adjoin R (Set.range s) | case a.intro.h_C.a
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
a : R
⊢ (algebraMap R S) a ∈ Set.range ⇑(algebraMap R S) ∪ Set.range s | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.h_C
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
a : R
⊢ (algebraMap R S) a ∈ Algebra.adjoin R (Set.range s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | MvPolynomial.aeval_range | [122, 1] | [143, 66] | left | case a.intro.h_C.a
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
a : R
⊢ (algebraMap R S) a ∈ Set.range ⇑(algebraMap R S) ∪ Set.range s | case a.intro.h_C.a.h
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
a : R
⊢ (algebraMap R S) a ∈ Set.range ⇑(algebraMap R S) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.h_C.a
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
a : R
⊢ (algebraMap R S) a ∈ Set.range ⇑(algebraMap R S) ∪ Set.range s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | MvPolynomial.aeval_range | [122, 1] | [143, 66] | use a | case a.intro.h_C.a.h
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
a : R
⊢ (algebraMap R S) a ∈ Set.range ⇑(algebraMap R S) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.h_C.a.h
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
a : R
⊢ (algebraMap R S) a ∈ Set.range ⇑(algebraMap R S)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | MvPolynomial.aeval_range | [122, 1] | [143, 66] | simp only [map_add] | case a.intro.h_add
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
p q : MvPolynomial σ R
hp : (aeval s) p ∈ Algebra.adjoin R (Set.range s)
hq : (aeval s) q ∈ Algebra.adjoin R (Set.range s)
⊢ (aeval s) (p + q) ∈ Algebra.adjoin R (Set.range s) | case a.intro.h_add
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
p q : MvPolynomial σ R
hp : (aeval s) p ∈ Algebra.adjoin R (Set.range s)
hq : (aeval s) q ∈ Algebra.adjoin R (Set.range s)
⊢ (aeval s) p + (aeval s) q ∈ Algebra.adjoin R (Set.range s) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.h_add
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
p q : MvPolynomial σ R
hp : (aeval s) p ∈ Algebra.adjoin R (Set.range s)
hq : (aeval s) q ∈ Algebra.adjoin R (Set.range s)
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | MvPolynomial.aeval_range | [122, 1] | [143, 66] | exact Subalgebra.add_mem _ hp hq | case a.intro.h_add
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
p q : MvPolynomial σ R
hp : (aeval s) p ∈ Algebra.adjoin R (Set.range s)
hq : (aeval s) q ∈ Algebra.adjoin R (Set.range s)
⊢ (aeval s) p + (aeval s) q ∈ Algebra.adjoin R (Set.range s) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.h_add
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
p q : MvPolynomial σ R
hp : (aeval s) p ∈ Algebra.adjoin R (Set.range s)
hq : (aeval s) q ∈ Algebra.adjoin R (Set.range s)
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | MvPolynomial.aeval_range | [122, 1] | [143, 66] | simp only [map_mul, aeval_X] | case a.intro.h_X
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
p : MvPolynomial σ R
n : σ
h : (aeval s) p ∈ Algebra.adjoin R (Set.range s)
⊢ (aeval s) (p * X n) ∈ Algebra.adjoin R (Set.range s) | case a.intro.h_X
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
p : MvPolynomial σ R
n : σ
h : (aeval s) p ∈ Algebra.adjoin R (Set.range s)
⊢ (aeval s) p * s n ∈ Algebra.adjoin R (Set.range s) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.h_X
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
p : MvPolynomial σ R
n : σ
h : (aeval s) p ∈ Algebra.adjoin R (Set.range s)
⊢ (aeval s) (p * X n) ∈ Algebra.adjoin R (Set.ran... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | MvPolynomial.aeval_range | [122, 1] | [143, 66] | apply Subalgebra.mul_mem _ h | case a.intro.h_X
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
p : MvPolynomial σ R
n : σ
h : (aeval s) p ∈ Algebra.adjoin R (Set.range s)
⊢ (aeval s) p * s n ∈ Algebra.adjoin R (Set.range s) | case a.intro.h_X
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
p : MvPolynomial σ R
n : σ
h : (aeval s) p ∈ Algebra.adjoin R (Set.range s)
⊢ s n ∈ Algebra.adjoin R (Set.range s) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.h_X
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
p : MvPolynomial σ R
n : σ
h : (aeval s) p ∈ Algebra.adjoin R (Set.range s)
⊢ (aeval s) p * s n ∈ Algebra.adjoin R (Set.range... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | MvPolynomial.aeval_range | [122, 1] | [143, 66] | apply Algebra.subset_adjoin | case a.intro.h_X
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
p : MvPolynomial σ R
n : σ
h : (aeval s) p ∈ Algebra.adjoin R (Set.range s)
⊢ s n ∈ Algebra.adjoin R (Set.range s) | case a.intro.h_X.a
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
p : MvPolynomial σ R
n : σ
h : (aeval s) p ∈ Algebra.adjoin R (Set.range s)
⊢ s n ∈ Set.range s | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.h_X
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
p : MvPolynomial σ R
n : σ
h : (aeval s) p ∈ Algebra.adjoin R (Set.range s)
⊢ s n ∈ Algebra.adjoin R (Set.range s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | MvPolynomial.aeval_range | [122, 1] | [143, 66] | use n | case a.intro.h_X.a
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
p : MvPolynomial σ R
n : σ
h : (aeval s) p ∈ Algebra.adjoin R (Set.range s)
⊢ s n ∈ Set.range s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.h_X.a
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
p : MvPolynomial σ R
n : σ
h : (aeval s) p ∈ Algebra.adjoin R (Set.range s)
⊢ s n ∈ Set.range s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | MvPolynomial.aeval_range | [122, 1] | [143, 66] | rw [Algebra.adjoin_le_iff] | case a
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
⊢ Algebra.adjoin R (Set.range s) ≤ (aeval s).range | case a
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
⊢ Set.range s ⊆ ↑(aeval s).range | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
⊢ Algebra.adjoin R (Set.range s) ≤ (aeval s).range
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | MvPolynomial.aeval_range | [122, 1] | [143, 66] | rintro x ⟨i, rfl⟩ | case a
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
⊢ Set.range s ⊆ ↑(aeval s).range | case a.intro
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
i : σ
⊢ s i ∈ ↑(aeval s).range | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
⊢ Set.range s ⊆ ↑(aeval s).range
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | MvPolynomial.aeval_range | [122, 1] | [143, 66] | use X i | case a.intro
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
i : σ
⊢ s i ∈ ↑(aeval s).range | case h
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
i : σ
⊢ (aeval s).toRingHom (X i) = s i | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
i : σ
⊢ s i ∈ ↑(aeval s).range
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | MvPolynomial.aeval_range | [122, 1] | [143, 66] | simp only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe, aeval_X] | case h
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
i : σ
⊢ (aeval s).toRingHom (X i) = s i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
σ : Type u_3
s : σ → S
i : σ
⊢ (aeval s).toRingHom (X i) = s i
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | TensorProduct.includeRight_lid | [154, 1] | [163, 46] | suffices ∀ m, (rTensor M (algebraMap' R S).toLinearMap).comp
(TensorProduct.lid R M).symm.toLinearMap m = 1 ⊗ₜ[R] m by
simp only [← this, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
LinearEquiv.symm_apply_apply] | R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
m : R ⊗[R] M
⊢ 1 ⊗ₜ[R] (TensorProduct.lid R M) m = (rTensor M (algebraMap' R S).toLinearMap) m | R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
m : R ⊗[R] M
⊢ ∀ (m : M), (rTensor M (algebraMap' R S).toLinearMap ∘ₗ ↑(TensorProduct.lid R M).symm) m = 1 ⊗ₜ[R] m | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
m : R ⊗[R] M
⊢ 1 ⊗ₜ[R] (TensorProduct.lid R M) m = (rTensor M (algebraMap' R S).toLinearMap) m
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | TensorProduct.includeRight_lid | [154, 1] | [163, 46] | intro z | R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
m : R ⊗[R] M
⊢ ∀ (m : M), (rTensor M (algebraMap' R S).toLinearMap ∘ₗ ↑(TensorProduct.lid R M).symm) m = 1 ⊗ₜ[R] m | R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
m : R ⊗[R] M
z : M
⊢ (rTensor M (algebraMap' R S).toLinearMap ∘ₗ ↑(TensorProduct.lid R M).symm) z = 1 ⊗ₜ[R] z | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
m : R ⊗[R] M
⊢ ∀ (m : M), (rTensor M (algebraMap' R S).toLinearMap ∘ₗ ↑(TensorProduct.lid R M).symm) m = 1 ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | TensorProduct.includeRight_lid | [154, 1] | [163, 46] | simp only [LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, lid_symm_apply, rTensor_tmul,
AlgHom.toLinearMap_apply, _root_.map_one] | R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
m : R ⊗[R] M
z : M
⊢ (rTensor M (algebraMap' R S).toLinearMap ∘ₗ ↑(TensorProduct.lid R M).symm) z = 1 ⊗ₜ[R] z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
m : R ⊗[R] M
z : M
⊢ (rTensor M (algebraMap' R S).toLinearMap ∘ₗ ↑(TensorProduct.lid R M).symm) z = 1 ⊗ₜ[R]... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | TensorProduct.includeRight_lid | [154, 1] | [163, 46] | simp only [← this, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
LinearEquiv.symm_apply_apply] | R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
m : R ⊗[R] M
this : ∀ (m : M), (rTensor M (algebraMap' R S).toLinearMap ∘ₗ ↑(TensorProduct.lid R M).symm) m = 1 ⊗ₜ[R] m
⊢ 1 ⊗ₜ[R] (TensorProduct.lid R M) m = (rTensor M... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
m : R ⊗[R] M
this : ∀ (m : M), (rTensor M (algebraMap' R S).toLinearMap ∘ₗ ↑(TensorProduct.lid R M).symm) m... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.isCompat_apply' | [191, 1] | [194, 54] | simpa only using _root_.congr_fun (f.isCompat' φ) x | R : Type u
inst✝⁸ : CommRing R
M : Type u_1
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
N : Type u_2
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
f : M →ₚ[R] N
S : Type u
inst✝³ : CommRing S
inst✝² : Algebra R S
S' : Type u
inst✝¹ : CommRing S'
inst✝ : Algebra R S'
φ : S →ₐ[R] S'
x : S ⊗[R] M
⊢ (LinearMap.rTensor N φ.to... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁸ : CommRing R
M : Type u_1
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
N : Type u_2
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
f : M →ₚ[R] N
S : Type u
inst✝³ : CommRing S
inst✝² : Algebra R S
S' : Type u
inst✝¹ : CommRing S'
inst✝ : Algebra... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.φ_range | [214, 1] | [231, 48] | apply le_antisymm | R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
⊢ (PolynomialMap.φ R s).range = Algebra.adjoin R ↑s | case a
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
⊢ (PolynomialMap.φ R s).range ≤ Algebra.adjoin R ↑s
case a
R : Type u
inst✝⁶ : CommRin... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
⊢ (PolynomialMap.φ R s).range = A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.φ_range | [214, 1] | [231, 48] | rintro _ ⟨p, rfl⟩ | case a
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
⊢ (PolynomialMap.φ R s).range ≤ Algebra.adjoin R ↑s | case a.intro
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
p : MvPolynomial (Fin s.card) R
⊢ (PolynomialMap.φ R s).toRingHom p ∈ Algebra.adj... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
⊢ (PolynomialMap.φ R s).ra... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.φ_range | [214, 1] | [231, 48] | induction p using MvPolynomial.induction_on with
| h_C r => simp only [toRingHom_eq_coe, ← algebraMap_eq, RingHom.coe_coe, commutes,
algebraMap_mem]
| h_add p q hp hq => simp only [map_add, add_mem hp hq]
| h_X p n hp =>
rw [_root_.map_mul]
apply mul_mem hp
apply Algebra.subset_adjoin
simp [φ] | case a.intro
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
p : MvPolynomial (Fin s.card) R
⊢ (PolynomialMap.φ R s).toRingHom p ∈ Algebra.adj... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
p : MvPolynomial (Fi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.φ_range | [214, 1] | [231, 48] | simp only [toRingHom_eq_coe, ← algebraMap_eq, RingHom.coe_coe, commutes,
algebraMap_mem] | case a.intro.h_C
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
r : R
⊢ (PolynomialMap.φ R s).toRingHom (C r) ∈ Algebra.adjoin R ↑s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.h_C
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
r : R
⊢ (Polynom... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.φ_range | [214, 1] | [231, 48] | simp only [map_add, add_mem hp hq] | case a.intro.h_add
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
p q : MvPolynomial (Fin s.card) R
hp : (PolynomialMap.φ R s).toRingHom p ∈ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.h_add
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
p q : MvPolyno... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.φ_range | [214, 1] | [231, 48] | rw [_root_.map_mul] | case a.intro.h_X
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
p : MvPolynomial (Fin s.card) R
n : Fin s.card
hp : (PolynomialMap.φ R s).toR... | case a.intro.h_X
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
p : MvPolynomial (Fin s.card) R
n : Fin s.card
hp : (PolynomialMap.φ R s).toR... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.h_X
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
p : MvPolynomial... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.φ_range | [214, 1] | [231, 48] | apply mul_mem hp | case a.intro.h_X
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
p : MvPolynomial (Fin s.card) R
n : Fin s.card
hp : (PolynomialMap.φ R s).toR... | case a.intro.h_X
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
p : MvPolynomial (Fin s.card) R
n : Fin s.card
hp : (PolynomialMap.φ R s).toR... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.h_X
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
p : MvPolynomial... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.φ_range | [214, 1] | [231, 48] | apply Algebra.subset_adjoin | case a.intro.h_X
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
p : MvPolynomial (Fin s.card) R
n : Fin s.card
hp : (PolynomialMap.φ R s).toR... | case a.intro.h_X.a
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
p : MvPolynomial (Fin s.card) R
n : Fin s.card
hp : (PolynomialMap.φ R s).t... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.h_X
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
p : MvPolynomial... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.φ_range | [214, 1] | [231, 48] | simp [φ] | case a.intro.h_X.a
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
p : MvPolynomial (Fin s.card) R
n : Fin s.card
hp : (PolynomialMap.φ R s).t... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.h_X.a
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
p : MvPolynomi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.φ_range | [214, 1] | [231, 48] | rw [Algebra.adjoin_le_iff] | case a
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
⊢ Algebra.adjoin R ↑s ≤ (PolynomialMap.φ R s).range | case a
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
⊢ ↑s ⊆ ↑(PolynomialMap.φ R s).range | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
⊢ Algebra.adjoin R ↑s ≤ (P... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.φ_range | [214, 1] | [231, 48] | intro x | case a
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
⊢ ↑s ⊆ ↑(PolynomialMap.φ R s).range | case a
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
x : S
⊢ x ∈ ↑s → x ∈ ↑(PolynomialMap.φ R s).range | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
⊢ ↑s ⊆ ↑(PolynomialMap.φ R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.φ_range | [214, 1] | [231, 48] | simp only [Finset.mem_coe, φ, coe_range, Set.mem_range] | case a
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
x : S
⊢ x ∈ ↑s → x ∈ ↑(PolynomialMap.φ R s).range | case a
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
x : S
⊢ x ∈ s → ∃ y, (aeval fun n => ↑(s.equivFin.symm n)) y = x | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
x : S
⊢ x ∈ ↑s → x ∈ ↑(Pol... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.φ_range | [214, 1] | [231, 48] | intro hx | case a
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
x : S
⊢ x ∈ s → ∃ y, (aeval fun n => ↑(s.equivFin.symm n)) y = x | case a
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
x : S
hx : x ∈ s
⊢ ∃ y, (aeval fun n => ↑(s.equivFin.symm n)) y = x | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
x : S
⊢ x ∈ s → ∃ y, (aeva... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.φ_range | [214, 1] | [231, 48] | use X (s.equivFin ⟨x, hx⟩) | case a
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
x : S
hx : x ∈ s
⊢ ∃ y, (aeval fun n => ↑(s.equivFin.symm n)) y = x | case h
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
x : S
hx : x ∈ s
⊢ (aeval fun n => ↑(s.equivFin.symm n)) (X (s.equivFin ⟨x, hx⟩)) = x | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
x : S
hx : x ∈ s
⊢ ∃ y, (a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.φ_range | [214, 1] | [231, 48] | simp only [aeval_X, Equiv.symm_apply_apply] | case h
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
x : S
hx : x ∈ s
⊢ (aeval fun n => ↑(s.equivFin.symm n)) (X (s.equivFin ⟨x, hx⟩)) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
s : Finset S
x : S
hx : x ∈ s
⊢ (aeval ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.Subalgebra.FG.exists_range_eq | [254, 1] | [258, 26] | obtain ⟨s, hs⟩ := hB | R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
B : Subalgebra R S
hB : B.FG
⊢ ∃ s, (PolynomialMap.φ R s).range = B | case intro
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
B : Subalgebra R S
s : Finset S
hs : Algebra.adjoin R ↑s = B
⊢ ∃ s, (PolynomialMap.φ R s).range ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
B : Subalgebra R S
hB : B.FG
⊢ ∃ s, (Polynomia... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.Subalgebra.FG.exists_range_eq | [254, 1] | [258, 26] | use s | case intro
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
B : Subalgebra R S
s : Finset S
hs : Algebra.adjoin R ↑s = B
⊢ ∃ s, (PolynomialMap.φ R s).range ... | case h
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
B : Subalgebra R S
s : Finset S
hs : Algebra.adjoin R ↑s = B
⊢ (PolynomialMap.φ R s).range = B | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
B : Subalgebra R S
s : Finset S
hs ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.Subalgebra.FG.exists_range_eq | [254, 1] | [258, 26] | simp only [φ_range, hs] | case h
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
B : Subalgebra R S
s : Finset S
hs : Algebra.adjoin R ↑s = B
⊢ (PolynomialMap.φ R s).range = B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u
inst✝⁶ : CommRing R
M : Type u_1
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommGroup N
inst✝² : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
B : Subalgebra R S
s : Finset S
hs : Al... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.toFun'_eq_of_diagram | [261, 1] | [295, 6] | let θ := (Ideal.quotientKerEquivRange (R := R) ψ).symm.toAlgHom.comp
(h'.comp (Ideal.quotientKerEquivRange φ).toAlgHom) | R : Type u
inst✝¹² : CommRing R
M : Type u_1
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
N : Type u_2
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
φ : A →ₐ[R] S
p : A ⊗[R] M
T : Type w
inst✝³ : Comm... | R : Type u
inst✝¹² : CommRing R
M : Type u_1
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
N : Type u_2
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
φ : A →ₐ[R] S
p : A ⊗[R] M
T : Type w
inst✝³ : Comm... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝¹² : CommRing R
M : Type u_1
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
N : Type u_2
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Al... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.toFun'_eq_of_diagram | [261, 1] | [295, 6] | have ht : h.comp (φ.range.val.comp (Ideal.quotientKerEquivRange φ).toAlgHom) =
ψ.range.val.comp ((Ideal.quotientKerEquivRange ψ).toAlgHom.comp θ) := by
simp only [θ, ← AlgHom.comp_assoc, ← hh']
apply congr_arg₂ _ _ rfl
apply congr_arg₂ _ _ rfl
simp only [AlgEquiv.toAlgHom_eq_coe, AlgHom.comp_assoc, AlgEquiv... | R : Type u
inst✝¹² : CommRing R
M : Type u_1
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
N : Type u_2
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
φ : A →ₐ[R] S
p : A ⊗[R] M
T : Type w
inst✝³ : Comm... | R : Type u
inst✝¹² : CommRing R
M : Type u_1
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
N : Type u_2
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
φ : A →ₐ[R] S
p : A ⊗[R] M
T : Type w
inst✝³ : Comm... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝¹² : CommRing R
M : Type u_1
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
N : Type u_2
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Al... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.toFun'_eq_of_diagram | [261, 1] | [295, 6] | simp only [φ.factor, ψ.factor, ← AlgHom.comp_assoc] | R : Type u
inst✝¹² : CommRing R
M : Type u_1
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
N : Type u_2
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
φ : A →ₐ[R] S
p : A ⊗[R] M
T : Type w
inst✝³ : Comm... | R : Type u
inst✝¹² : CommRing R
M : Type u_1
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
N : Type u_2
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
φ : A →ₐ[R] S
p : A ⊗[R] M
T : Type w
inst✝³ : Comm... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝¹² : CommRing R
M : Type u_1
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
N : Type u_2
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Al... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Basic.lean | PolynomialMap.toFun'_eq_of_diagram | [261, 1] | [295, 6] | nth_rewrite 2 [AlgHom.comp_assoc] | R : Type u
inst✝¹² : CommRing R
M : Type u_1
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
N : Type u_2
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
φ : A →ₐ[R] S
p : A ⊗[R] M
T : Type w
inst✝³ : Comm... | R : Type u
inst✝¹² : CommRing R
M : Type u_1
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
N : Type u_2
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
φ : A →ₐ[R] S
p : A ⊗[R] M
T : Type w
inst✝³ : Comm... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝¹² : CommRing R
M : Type u_1
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
N : Type u_2
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
f✝ f : M →ₚ[R] N
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Al... |
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