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/PolynomialMap/Homogeneous.lean | Polynomial.rTensor'_sum | [697, 1] | [734, 34] | simp only [coe_restrictScalars, lcoeff_apply, coeff_C_mul, coeff_X_pow,
↓reduceIte, mul_one, rTensor_tmul, Polynomial.lsum_apply,
lsmul_apply, smul_eq_mul] | case tmul.h_monomial
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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
n : N
p : ℕ
s : S
a✝ :
(((Polynomial.rTensor R N S) ((C s * X ^ ... | case tmul.h_monomial
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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
n : N
p : ℕ
s : S
a✝ :
(((Polynomial.rTensor R N S) ((C s * X ^ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case tmul.h_monomial
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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
n : N... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | Polynomial.rTensor'_sum | [697, 1] | [734, 34] | rw [C_mul_X_pow_eq_monomial, sum_monomial_index, smul_tmul', smul_eq_mul] | case tmul.h_monomial
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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
n : N
p : ℕ
s : S
a✝ :
(((Polynomial.rTensor R N S) ((C s * X ^ ... | case tmul.h_monomial.hf
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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
n : N
p : ℕ
s : S
a✝ :
(((Polynomial.rTensor R N S) ((C s * X... | Please generate a tactic in lean4 to solve the state.
STATE:
case tmul.h_monomial
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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
n : N... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | Polynomial.rTensor'_sum | [697, 1] | [734, 34] | rw [mul_zero] | case tmul.h_monomial.hf
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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
n : N
p : ℕ
s : S
a✝ :
(((Polynomial.rTensor R N S) ((C s * X... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case tmul.h_monomial.hf
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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
n ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | Polynomial.rTensor'_sum | [697, 1] | [734, 34] | intro b _ hb | case tmul.h_monomial.h₀
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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
n : N
p : ℕ
s : S
a✝ :
(((Polynomial.rTensor R N S) ((C s * X... | case tmul.h_monomial.h₀
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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
n : N
p : ℕ
s : S
a✝¹ :
(((Polynomial.rTensor R N S) ((C s * ... | Please generate a tactic in lean4 to solve the state.
STATE:
case tmul.h_monomial.h₀
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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
n ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | Polynomial.rTensor'_sum | [697, 1] | [734, 34] | simp only [rTensor_apply, rTensor_tmul, coe_restrictScalars, lcoeff_apply, coeff_C_mul,
coeff_X_pow, mul_ite, mul_one, mul_zero, if_neg hb, zero_tmul, smul_zero] | case tmul.h_monomial.h₀
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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
n : N
p : ℕ
s : S
a✝¹ :
(((Polynomial.rTensor R N S) ((C s * ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case tmul.h_monomial.h₀
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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
n ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | Polynomial.rTensor'_sum | [697, 1] | [734, 34] | exact fun _ ↦ smul_zero _ | case tmul.h_monomial.h₁
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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
n : N
p : ℕ
s : S
a✝ :
(((Polynomial.rTensor R N S) ((C s * X... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case tmul.h_monomial.h₁
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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
n ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | Polynomial.rTensor'_sum | [697, 1] | [734, 34] | simp only [add_tmul, LinearEquiv.map_add] | case 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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
p q : S[X] ⊗[R] N
hp :
(((Polynomial.rTensor R N S) p).sum fun p sn => φ p •... | case 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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
p q : S[X] ⊗[R] N
hp :
(((Polynomial.rTensor R N S) p).sum fun p sn => φ p •... | Please generate a tactic in lean4 to solve the state.
STATE:
case 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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
p q : S[X] ⊗[R] N... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | Polynomial.rTensor'_sum | [697, 1] | [734, 34] | rw [Finsupp.sum_add_index, hp, hq, LinearMap.map_add] | case 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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
p q : S[X] ⊗[R] N
hp :
(((Polynomial.rTensor R N S) p).sum fun p sn => φ p •... | case add.h_zero
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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
p q : S[X] ⊗[R] N
hp :
(((Polynomial.rTensor R N S) p).sum fun p sn =... | Please generate a tactic in lean4 to solve the state.
STATE:
case 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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
p q : S[X] ⊗[R] N... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | Polynomial.rTensor'_sum | [697, 1] | [734, 34] | intro x _ | case add.h_zero
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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
p q : S[X] ⊗[R] N
hp :
(((Polynomial.rTensor R N S) p).sum fun p sn =... | case add.h_zero
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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
p q : S[X] ⊗[R] N
hp :
(((Polynomial.rTensor R N S) p).sum fun p sn =... | Please generate a tactic in lean4 to solve the state.
STATE:
case add.h_zero
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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
p q : S[X]... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | Polynomial.rTensor'_sum | [697, 1] | [734, 34] | exact smul_zero _ | case add.h_zero
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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
p q : S[X] ⊗[R] N
hp :
(((Polynomial.rTensor R N S) p).sum fun p sn =... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case add.h_zero
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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
p q : S[X]... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | Polynomial.rTensor'_sum | [697, 1] | [734, 34] | intro x _ | case add.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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
p q : S[X] ⊗[R] N
hp :
(((Polynomial.rTensor R N S) p).sum fun p sn =>... | case add.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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
p q : S[X] ⊗[R] N
hp :
(((Polynomial.rTensor R N S) p).sum fun p sn =>... | Please generate a tactic in lean4 to solve the state.
STATE:
case add.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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
p q : S[X] ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | Polynomial.rTensor'_sum | [697, 1] | [734, 34] | exact smul_add _ | case add.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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
p q : S[X] ⊗[R] N
hp :
(((Polynomial.rTensor R N S) p).sum fun p sn =>... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case add.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 : M →ₚ[R] N
p✝ : ℕ
S : Type u_3
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
φ : ℕ → S
p q : S[X] ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | PolynomialMap.recompose_component | [740, 1] | [760, 31] | ext S _ _ sm | R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
⊢ (lfsum fun p => component p f) = f | case toFun'.h.h.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
⊢ (lfsum fun p => component p f).toFun' S sm = f.toFun' S sm | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
⊢ (lfsum fun p => component p f) = f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | PolynomialMap.recompose_component | [740, 1] | [760, 31] | rw [lfsum_eq (LocFinsupp_component f), LocFinsupp_component_eq] | case toFun'.h.h.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
⊢ (lfsum fun p => component p f).toFun' S sm = f.toFun' S sm | case toFun'.h.h.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
⊢ (((Polynomial.rTensor R N S) (f.toFun' S[X] ((LinearMap.rTen... | Please generate a tactic in lean4 to solve the state.
STATE:
case toFun'.h.h.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
⊢... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | PolynomialMap.recompose_component | [740, 1] | [760, 31] | have hsm : sm = ((aeval 1).restrictScalars R).toLinearMap.rTensor M
(((monomial 1).restrictScalars R).rTensor M sm) := by
rw [← LinearMap.rTensor_comp_apply, LinearMap.rTensor, eq_comm]
convert DFunLike.congr_fun TensorProduct.map_id sm
ext s
simp only [coe_comp, coe_restrictScalars, Function.comp_apply, AlgH... | case toFun'.h.h.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
⊢ (((Polynomial.rTensor R N S) (f.toFun' S[X] ((LinearMap.rTen... | case toFun'.h.h.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
hsm :
sm =
(LinearMap.rTensor M (AlgHom.restrictScalars ... | Please generate a tactic in lean4 to solve the state.
STATE:
case toFun'.h.h.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
⊢... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | PolynomialMap.recompose_component | [740, 1] | [760, 31] | conv_rhs => rw [hsm, ← f.isCompat_apply'] | case toFun'.h.h.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
hsm :
sm =
(LinearMap.rTensor M (AlgHom.restrictScalars ... | case toFun'.h.h.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
hsm :
sm =
(LinearMap.rTensor M (AlgHom.restrictScalars ... | Please generate a tactic in lean4 to solve the state.
STATE:
case toFun'.h.h.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | PolynomialMap.recompose_component | [740, 1] | [760, 31] | generalize f.toFun' S[X] (((monomial 1).restrictScalars R).rTensor M sm) = sn | case toFun'.h.h.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
hsm :
sm =
(LinearMap.rTensor M (AlgHom.restrictScalars ... | case toFun'.h.h.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
hsm :
sm =
(LinearMap.rTensor M (AlgHom.restrictScalars ... | Please generate a tactic in lean4 to solve the state.
STATE:
case toFun'.h.h.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | PolynomialMap.recompose_component | [740, 1] | [760, 31] | convert rTensor'_sum (R := R) (fun _ ↦ 1) sn | case toFun'.h.h.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
hsm :
sm =
(LinearMap.rTensor M (AlgHom.restrictScalars ... | case h.e'_2.h.e'_7.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝² : CommRing S
x✝¹ : Algebra R S
sm : S ⊗[R] M
hsm :
sm =
(LinearMap.rTensor M (AlgHom.restrictScal... | Please generate a tactic in lean4 to solve the state.
STATE:
case toFun'.h.h.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | PolynomialMap.recompose_component | [740, 1] | [760, 31] | simp only [_root_.one_smul] | case h.e'_2.h.e'_7.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝² : CommRing S
x✝¹ : Algebra R S
sm : S ⊗[R] M
hsm :
sm =
(LinearMap.rTensor M (AlgHom.restrictScal... | case h.e'_3.h.e'_5.h.e'_12.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
hsm :
sm =
(LinearMap.rTensor M (AlgHom.restric... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_7.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝² : CommRing S
x✝¹ : Algebra R S
sm : S ⊗[R]... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | PolynomialMap.recompose_component | [740, 1] | [760, 31] | ext p | case h.e'_3.h.e'_5.h.e'_12.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
hsm :
sm =
(LinearMap.rTensor M (AlgHom.restric... | case h.e'_3.h.e'_5.h.e'_12.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p✝ : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
hsm :
sm =
(LinearMap.rTensor M (AlgHom.rest... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_5.h.e'_12.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | PolynomialMap.recompose_component | [740, 1] | [760, 31] | simp only [AlgHom.toLinearMap_apply, AlgHom.coe_restrictScalars', coe_aeval_eq_eval,
Polynomial.lsum_apply, coe_restrictScalars, lsmul_apply, smul_eq_mul, one_mul, eval_eq_sum] | case h.e'_3.h.e'_5.h.e'_12.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p✝ : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
hsm :
sm =
(LinearMap.rTensor M (AlgHom.rest... | case h.e'_3.h.e'_5.h.e'_12.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p✝ : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
hsm :
sm =
(LinearMap.rTensor M (AlgHom.rest... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_5.h.e'_12.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p✝ : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | PolynomialMap.recompose_component | [740, 1] | [760, 31] | apply congr_arg₂ _ rfl | case h.e'_3.h.e'_5.h.e'_12.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p✝ : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
hsm :
sm =
(LinearMap.rTensor M (AlgHom.rest... | case h.e'_3.h.e'_5.h.e'_12.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p✝ : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
hsm :
sm =
(LinearMap.rTensor M (AlgHom.rest... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_5.h.e'_12.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p✝ : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | PolynomialMap.recompose_component | [740, 1] | [760, 31] | simp only [one_pow, mul_one] | case h.e'_3.h.e'_5.h.e'_12.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p✝ : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
hsm :
sm =
(LinearMap.rTensor M (AlgHom.rest... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_5.h.e'_12.h.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p✝ : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | PolynomialMap.recompose_component | [740, 1] | [760, 31] | rw [← LinearMap.rTensor_comp_apply, LinearMap.rTensor, eq_comm] | R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
⊢ sm =
(LinearMap.rTensor M (AlgHom.restrictScalars R (aeval 1)).toLinearMap) ... | R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
⊢ (TensorProduct.map ((AlgHom.restrictScalars R (aeval 1)).toLinearMap ∘ₗ ↑R (mono... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
⊢ sm =
(LinearMap... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | PolynomialMap.recompose_component | [740, 1] | [760, 31] | convert DFunLike.congr_fun TensorProduct.map_id sm | R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
⊢ (TensorProduct.map ((AlgHom.restrictScalars R (aeval 1)).toLinearMap ∘ₗ ↑R (mono... | case h.e'_2.h.e'_5.h.e'_15
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
⊢ (AlgHom.restrictScalars R (aeval 1)).toLinearMap ∘ₗ ↑... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
⊢ (TensorProduct.map ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | PolynomialMap.recompose_component | [740, 1] | [760, 31] | ext s | case h.e'_2.h.e'_5.h.e'_15
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
⊢ (AlgHom.restrictScalars R (aeval 1)).toLinearMap ∘ₗ ↑... | case h.e'_2.h.e'_5.h.e'_15.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
s : S
⊢ ((AlgHom.restrictScalars R (aeval 1)).toLinea... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5.h.e'_15
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/PolynomialMap/Homogeneous.lean | PolynomialMap.recompose_component | [740, 1] | [760, 31] | simp only [coe_comp, coe_restrictScalars, Function.comp_apply, AlgHom.toLinearMap_apply,
AlgHom.coe_restrictScalars', aeval_monomial, Algebra.id.map_eq_id, RingHom.id_apply, pow_one,
mul_one, id_coe, id_eq] | case h.e'_2.h.e'_5.h.e'_15.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S ⊗[R] M
s : S
⊢ ((AlgHom.restrictScalars R (aeval 1)).toLinea... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5.h.e'_15.h
R : Type u
inst✝⁴ : CommRing R
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f✝ : M →ₚ[R] N
p : ℕ
f : M →ₚ[R] N
S : Type u
x✝¹ : CommRing S
x✝ : Algebra R S
sm : S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Exponential.lean | DividedPowers.isExponential_dpowExp | [14, 1] | [20, 54] | rw [isExponential_iff] | R : Type u_1
inst✝ : CommRing R
I : Ideal R
hI : DividedPowers I
a : R
ha : a ∈ I
⊢ (hI.dpowExp a).IsExponential | R : Type u_1
inst✝ : CommRing R
I : Ideal R
hI : DividedPowers I
a : R
ha : a ∈ I
⊢ (constantCoeff R) (hI.dpowExp a) = 1 ∧
∀ (p q : ℕ),
↑((p + q).choose p) * (coeff R (p + q)) (hI.dpowExp a) = (coeff R p) (hI.dpowExp a) * (coeff R q) (hI.dpowExp a) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
I : Ideal R
hI : DividedPowers I
a : R
ha : a ∈ I
⊢ (hI.dpowExp a).IsExponential
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Exponential.lean | DividedPowers.isExponential_dpowExp | [14, 1] | [20, 54] | constructor | R : Type u_1
inst✝ : CommRing R
I : Ideal R
hI : DividedPowers I
a : R
ha : a ∈ I
⊢ (constantCoeff R) (hI.dpowExp a) = 1 ∧
∀ (p q : ℕ),
↑((p + q).choose p) * (coeff R (p + q)) (hI.dpowExp a) = (coeff R p) (hI.dpowExp a) * (coeff R q) (hI.dpowExp a) | case left
R : Type u_1
inst✝ : CommRing R
I : Ideal R
hI : DividedPowers I
a : R
ha : a ∈ I
⊢ (constantCoeff R) (hI.dpowExp a) = 1
case right
R : Type u_1
inst✝ : CommRing R
I : Ideal R
hI : DividedPowers I
a : R
ha : a ∈ I
⊢ ∀ (p q : ℕ),
↑((p + q).choose p) * (coeff R (p + q)) (hI.dpowExp a) = (coeff R p) (hI.dpo... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
I : Ideal R
hI : DividedPowers I
a : R
ha : a ∈ I
⊢ (constantCoeff R) (hI.dpowExp a) = 1 ∧
∀ (p q : ℕ),
↑((p + q).choose p) * (coeff R (p + q)) (hI.dpowExp a) = (coeff R p) (hI.dpowExp a) * (coeff R q) (hI.dpowExp a)
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Exponential.lean | DividedPowers.isExponential_dpowExp | [14, 1] | [20, 54] | simp only [dpowExp, ← coeff_zero_eq_constantCoeff_apply, coeff_mk, dpow_zero _ ha] | case left
R : Type u_1
inst✝ : CommRing R
I : Ideal R
hI : DividedPowers I
a : R
ha : a ∈ I
⊢ (constantCoeff R) (hI.dpowExp a) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
R : Type u_1
inst✝ : CommRing R
I : Ideal R
hI : DividedPowers I
a : R
ha : a ∈ I
⊢ (constantCoeff R) (hI.dpowExp a) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Exponential.lean | DividedPowers.isExponential_dpowExp | [14, 1] | [20, 54] | intro p q | case right
R : Type u_1
inst✝ : CommRing R
I : Ideal R
hI : DividedPowers I
a : R
ha : a ∈ I
⊢ ∀ (p q : ℕ),
↑((p + q).choose p) * (coeff R (p + q)) (hI.dpowExp a) = (coeff R p) (hI.dpowExp a) * (coeff R q) (hI.dpowExp a) | case right
R : Type u_1
inst✝ : CommRing R
I : Ideal R
hI : DividedPowers I
a : R
ha : a ∈ I
p q : ℕ
⊢ ↑((p + q).choose p) * (coeff R (p + q)) (hI.dpowExp a) = (coeff R p) (hI.dpowExp a) * (coeff R q) (hI.dpowExp a) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
R : Type u_1
inst✝ : CommRing R
I : Ideal R
hI : DividedPowers I
a : R
ha : a ∈ I
⊢ ∀ (p q : ℕ),
↑((p + q).choose p) * (coeff R (p + q)) (hI.dpowExp a) = (coeff R p) (hI.dpowExp a) * (coeff R q) (hI.dpowExp a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/Exponential.lean | DividedPowers.isExponential_dpowExp | [14, 1] | [20, 54] | simp only [dpowExp, coeff_mk, hI.dpow_mul p q ha] | case right
R : Type u_1
inst✝ : CommRing R
I : Ideal R
hI : DividedPowers I
a : R
ha : a ∈ I
p q : ℕ
⊢ ↑((p + q).choose p) * (coeff R (p + q)) (hI.dpowExp a) = (coeff R p) (hI.dpowExp a) * (coeff R q) (hI.dpowExp a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
R : Type u_1
inst✝ : CommRing R
I : Ideal R
hI : DividedPowers I
a : R
ha : a ∈ I
p q : ℕ
⊢ ↑((p + q).choose p) * (coeff R (p + q)) (hI.dpowExp a) = (coeff R p) (hI.dpowExp a) * (coeff R q) (hI.dpowExp a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | Ideal.sub_mem_ofRel_of_rel | [199, 1] | [202, 59] | rw [sub_add_cancel] | R✝ : Type u_1
M : Type u_2
inst✝³ : CommSemiring R✝
inst✝² : AddCommMonoid M
inst✝¹ : Module R✝ M
R : Type u_3
inst✝ : Ring R
r : R → R → Prop
a b : R
hr : r a b
⊢ a - b + b = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u_1
M : Type u_2
inst✝³ : CommSemiring R✝
inst✝² : AddCommMonoid M
inst✝¹ : Module R✝ M
R : Type u_3
inst✝ : Ring R
r : R → R → Prop
a b : R
hr : r a b
⊢ a - b + b = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.mk_surjective | [221, 1] | [222, 37] | apply RingQuot.mkAlgHom_surjective | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ Function.Surjective ⇑mk | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ Function.Surjective ⇑mk
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.mk_C | [224, 1] | [225, 53] | rw [← MvPolynomial.algebraMap_eq, AlgHom.commutes] | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
a : R
⊢ mk (C a) = (algebraMap R (DividedPowerAlgebra R M)) a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
a : R
⊢ mk (C a) = (algebraMap R (DividedPowerAlgebra R M)) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_eq_mkRingHom | [241, 1] | [244, 6] | rw [dp_def, ← mkAlgHom_coe R] | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
m : M
⊢ dp R n m = (mkRingHom (Rel R M)) (X (n, m)) | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
m : M
⊢ (mkAlgHom R (Rel R M)) (X (n, m)) = ↑(mkAlgHom R (Rel R M)) (X (n, m)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
m : M
⊢ dp R n m = (mkRingHom (Rel R M)) (X (n, m))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_eq_mkRingHom | [241, 1] | [244, 6] | rfl | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
m : M
⊢ (mkAlgHom R (Rel R M)) (X (n, m)) = ↑(mkAlgHom R (Rel R M)) (X (n, m)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
m : M
⊢ (mkAlgHom R (Rel R M)) (X (n, m)) = ↑(mkAlgHom R (Rel R M)) (X (n, m))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_zero | [247, 1] | [249, 41] | rw [dp_def, ← map_one (mkAlgHom R (Rel R M))] | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
m : M
⊢ dp R 0 m = 1 | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
m : M
⊢ (mkAlgHom R (Rel R M)) (X (0, m)) = (mkAlgHom R (Rel R M)) 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
m : M
⊢ dp R 0 m = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_zero | [247, 1] | [249, 41] | exact RingQuot.mkAlgHom_rel R Rel.zero | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
m : M
⊢ (mkAlgHom R (Rel R M)) (X (0, m)) = (mkAlgHom R (Rel R M)) 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
m : M
⊢ (mkAlgHom R (Rel R M)) (X (0, m)) = (mkAlgHom R (Rel R M)) 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_smul | [252, 1] | [255, 32] | rw [dp_def, dp_def, ← map_smul] | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
r : R
n : ℕ
m : M
⊢ dp R n (r • m) = r ^ n • dp R n m | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
r : R
n : ℕ
m : M
⊢ (mkAlgHom R (Rel R M)) (X (n, r • m)) = (mkAlgHom R (Rel R M)) (r ^ n • X (n, m)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
r : R
n : ℕ
m : M
⊢ dp R n (r • m) = r ^ n • dp R n m
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_smul | [252, 1] | [255, 32] | exact mkAlgHom_rel R Rel.smul | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
r : R
n : ℕ
m : M
⊢ (mkAlgHom R (Rel R M)) (X (n, r • m)) = (mkAlgHom R (Rel R M)) (r ^ n • X (n, m)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
r : R
n : ℕ
m : M
⊢ (mkAlgHom R (Rel R M)) (X (n, r • m)) = (mkAlgHom R (Rel R M)) (r ^ n • X (n, m))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_null | [258, 1] | [263, 57] | cases' Nat.eq_zero_or_pos n with hn hn | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
⊢ dp R n 0 = if n = 0 then 1 else 0 | case inl
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
hn : n = 0
⊢ dp R n 0 = if n = 0 then 1 else 0
case inr
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
hn : n > 0
⊢ dp R n 0 = if n = 0 then 1 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
⊢ dp R n 0 = if n = 0 then 1 else 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_null | [258, 1] | [263, 57] | rw [if_pos hn, hn, dp_zero] | case inl
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
hn : n = 0
⊢ dp R n 0 = if n = 0 then 1 else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
hn : n = 0
⊢ dp R n 0 = if n = 0 then 1 else 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_null | [258, 1] | [263, 57] | rw [if_neg (ne_of_gt hn), ← zero_smul R (0 : M), dp_smul] | case inr
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
hn : n > 0
⊢ dp R n 0 = if n = 0 then 1 else 0 | case inr
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
hn : n > 0
⊢ 0 ^ n • dp R n 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
hn : n > 0
⊢ dp R n 0 = if n = 0 then 1 else 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_null | [258, 1] | [263, 57] | rw [zero_pow (Nat.pos_iff_ne_zero.mp hn), zero_smul] | case inr
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
hn : n > 0
⊢ 0 ^ n • dp R n 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
hn : n > 0
⊢ 0 ^ n • dp R n 0 = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_mul | [266, 1] | [269, 31] | simp only [dp_def, ← _root_.map_mul, ← map_nsmul] | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n p : ℕ
m : M
⊢ dp R n m * dp R p m = (n + p).choose n • dp R (n + p) m | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n p : ℕ
m : M
⊢ (mkAlgHom R (Rel R M)) (X (n, m) * X (p, m)) = (mkAlgHom R (Rel R M)) ((n + p).choose n • X (n + p, m)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n p : ℕ
m : M
⊢ dp R n m * dp R p m = (n + p).choose n • dp R (n + p) m
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_mul | [266, 1] | [269, 31] | exact mkAlgHom_rel R Rel.mul | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n p : ℕ
m : M
⊢ (mkAlgHom R (Rel R M)) (X (n, m) * X (p, m)) = (mkAlgHom R (Rel R M)) ((n + p).choose n • X (n + p, m)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n p : ℕ
m : M
⊢ (mkAlgHom R (Rel R M)) (X (n, m) * X (p, m)) = (mkAlgHom R (Rel R M)) ((n + p).choose n • X (n + p, m))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_add | [272, 1] | [280, 22] | simp only [dp_def] | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x y : M
⊢ dp R n (x + y) = ∑ k ∈ antidiagonal n, dp R k.1 x * dp R k.2 y | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x y : M
⊢ (mkAlgHom R (Rel R M)) (X (n, x + y)) =
∑ x_1 ∈ antidiagonal n, (mkAlgHom R (Rel R M)) (X (x_1.1, x)) * (mkAlgHom R (Rel R M)) (X (x_1.2, y)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x y : M
⊢ dp R n (x + y) = ∑ k ∈ antidiagonal n, dp R k.1 x * dp R k.2 y
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_add | [272, 1] | [280, 22] | rw [mkAlgHom_rel (A := MvPolynomial (ℕ × M) R) R Rel.add] | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x y : M
⊢ (mkAlgHom R (Rel R M)) (X (n, x + y)) =
∑ x_1 ∈ antidiagonal n, (mkAlgHom R (Rel R M)) (X (x_1.1, x)) * (mkAlgHom R (Rel R M)) (X (x_1.2, y)) | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x y : M
⊢ (mkAlgHom R (Rel R M)) (∑ k ∈ antidiagonal n, X (k.1, x) * X (k.2, y)) =
∑ x_1 ∈ antidiagonal n, (mkAlgHom R (Rel R M)) (X (x_1.1, x)) * (mkAlgHom R (Rel R M)) (X (x_1.2, y)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x y : M
⊢ (mkAlgHom R (Rel R M)) (X (n, x + y)) =
∑ x_1 ∈ antidiagonal n, (mkAlgHom R (Rel R M)) (X (x_1.1, x)) * (mkAlgHom R (Rel R M)) (X (x_1.2, y))
TAC... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_add | [272, 1] | [280, 22] | rw [AlgHom.map_sum] | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x y : M
⊢ (mkAlgHom R (Rel R M)) (∑ k ∈ antidiagonal n, X (k.1, x) * X (k.2, y)) =
∑ x_1 ∈ antidiagonal n, (mkAlgHom R (Rel R M)) (X (x_1.1, x)) * (mkAlgHom R (Rel R M)) (X (x_1.2, y)) | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x y : M
⊢ ∑ x_1 ∈ antidiagonal n, (mkAlgHom R (Rel R M)) (X (x_1.1, x) * X (x_1.2, y)) =
∑ x_1 ∈ antidiagonal n, (mkAlgHom R (Rel R M)) (X (x_1.1, x)) * (mkAlgHom R (Rel R M)) (X (x_1.2, y)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x y : M
⊢ (mkAlgHom R (Rel R M)) (∑ k ∈ antidiagonal n, X (k.1, x) * X (k.2, y)) =
∑ x_1 ∈ antidiagonal n, (mkAlgHom R (Rel R M)) (X (x_1.1, x)) * (mkAlgHo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_add | [272, 1] | [280, 22] | apply Finset.sum_congr rfl | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x y : M
⊢ ∑ x_1 ∈ antidiagonal n, (mkAlgHom R (Rel R M)) (X (x_1.1, x) * X (x_1.2, y)) =
∑ x_1 ∈ antidiagonal n, (mkAlgHom R (Rel R M)) (X (x_1.1, x)) * (mkAlgHom R (Rel R M)) (X (x_1.2, y)) | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x y : M
⊢ ∀ x_1 ∈ antidiagonal n,
(mkAlgHom R (Rel R M)) (X (x_1.1, x) * X (x_1.2, y)) =
(mkAlgHom R (Rel R M)) (X (x_1.1, x)) * (mkAlgHom R (Rel R M)) (X (x_1.2, y)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x y : M
⊢ ∑ x_1 ∈ antidiagonal n, (mkAlgHom R (Rel R M)) (X (x_1.1, x) * X (x_1.2, y)) =
∑ x_1 ∈ antidiagonal n, (mkAlgHom R (Rel R M)) (X (x_1.1, x)) * (m... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_add | [272, 1] | [280, 22] | intro k _ | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x y : M
⊢ ∀ x_1 ∈ antidiagonal n,
(mkAlgHom R (Rel R M)) (X (x_1.1, x) * X (x_1.2, y)) =
(mkAlgHom R (Rel R M)) (X (x_1.1, x)) * (mkAlgHom R (Rel R M)) (X (x_1.2, y)) | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x y : M
k : ℕ × ℕ
a✝ : k ∈ antidiagonal n
⊢ (mkAlgHom R (Rel R M)) (X (k.1, x) * X (k.2, y)) =
(mkAlgHom R (Rel R M)) (X (k.1, x)) * (mkAlgHom R (Rel R M)) (X (k.2, y)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x y : M
⊢ ∀ x_1 ∈ antidiagonal n,
(mkAlgHom R (Rel R M)) (X (x_1.1, x) * X (x_1.2, y)) =
(mkAlgHom R (Rel R M)) (X (x_1.1, x)) * (mkAlgHom R (Rel R M... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_add | [272, 1] | [280, 22] | rw [AlgHom.map_mul] | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x y : M
k : ℕ × ℕ
a✝ : k ∈ antidiagonal n
⊢ (mkAlgHom R (Rel R M)) (X (k.1, x) * X (k.2, y)) =
(mkAlgHom R (Rel R M)) (X (k.1, x)) * (mkAlgHom R (Rel R M)) (X (k.2, y)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x y : M
k : ℕ × ℕ
a✝ : k ∈ antidiagonal n
⊢ (mkAlgHom R (Rel R M)) (X (k.1, x) * X (k.2, y)) =
(mkAlgHom R (Rel R M)) (X (k.1, x)) * (mkAlgHom R (Rel R M))... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_sum | [284, 1] | [291, 44] | apply DividedPowers.dpow_sum_aux' | R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : DecidableEq ι
s : Finset ι
q : ℕ
x : ι → M
⊢ dp R q (s.sum x) = ∑ k ∈ s.sym q, ∏ i ∈ s, dp R (Multiset.count i ↑k) (x i) | case dpow_zero
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : DecidableEq ι
s : Finset ι
q : ℕ
x : ι → M
⊢ ∀ (x : M), dp R 0 x = 1
case dpow_add
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : DecidableEq ι
s : Finset ι
q : ℕ
x : ι → M
⊢ dp R q (s.sum x) = ∑ k ∈ s.sym q, ∏ i ∈ s, dp R (Multiset.count i ↑k) (x i)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_sum | [284, 1] | [291, 44] | intro x | case dpow_zero
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : DecidableEq ι
s : Finset ι
q : ℕ
x : ι → M
⊢ ∀ (x : M), dp R 0 x = 1 | case dpow_zero
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : DecidableEq ι
s : Finset ι
q : ℕ
x✝ : ι → M
x : M
⊢ dp R 0 x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_zero
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : DecidableEq ι
s : Finset ι
q : ℕ
x : ι → M
⊢ ∀ (x : M), dp R 0 x = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_sum | [284, 1] | [291, 44] | rw [dp_zero] | case dpow_zero
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : DecidableEq ι
s : Finset ι
q : ℕ
x✝ : ι → M
x : M
⊢ dp R 0 x = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_zero
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : DecidableEq ι
s : Finset ι
q : ℕ
x✝ : ι → M
x : M
⊢ dp R 0 x = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_sum | [284, 1] | [291, 44] | intro n x y | case dpow_add
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : DecidableEq ι
s : Finset ι
q : ℕ
x : ι → M
⊢ ∀ (n : ℕ) (x y : M),
dp R n (x + y) =
∑ x_1 ∈ antidiagonal n,
match x_1 with
| (k, l) => dp R k x * dp R l y | case dpow_add
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : DecidableEq ι
s : Finset ι
q : ℕ
x✝ : ι → M
n : ℕ
x y : M
⊢ dp R n (x + y) =
∑ x_1 ∈ antidiagonal n,
match x_1 with
| (k, l) => dp R k x * dp R l y | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_add
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : DecidableEq ι
s : Finset ι
q : ℕ
x : ι → M
⊢ ∀ (n : ℕ) (x y : M),
dp R n (x + y) =
∑ x_1 ∈ antidiagonal n,
matc... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_sum | [284, 1] | [291, 44] | rw [dp_add] | case dpow_add
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : DecidableEq ι
s : Finset ι
q : ℕ
x✝ : ι → M
n : ℕ
x y : M
⊢ dp R n (x + y) =
∑ x_1 ∈ antidiagonal n,
match x_1 with
| (k, l) => dp R k x * dp R l y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_add
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : DecidableEq ι
s : Finset ι
q : ℕ
x✝ : ι → M
n : ℕ
x y : M
⊢ dp R n (x + y) =
∑ x_1 ∈ antidiagonal n,
match x_1 with
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_sum | [284, 1] | [291, 44] | intro n hn | case dpow_eval_zero
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : DecidableEq ι
s : Finset ι
q : ℕ
x : ι → M
⊢ ∀ {n : ℕ}, n ≠ 0 → dp R n 0 = 0 | case dpow_eval_zero
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : DecidableEq ι
s : Finset ι
q : ℕ
x : ι → M
n : ℕ
hn : n ≠ 0
⊢ dp R n 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_eval_zero
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : DecidableEq ι
s : Finset ι
q : ℕ
x : ι → M
⊢ ∀ {n : ℕ}, n ≠ 0 → dp R n 0 = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_sum | [284, 1] | [291, 44] | rw [dp_null R n, if_neg hn] | case dpow_eval_zero
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : DecidableEq ι
s : Finset ι
q : ℕ
x : ι → M
n : ℕ
hn : n ≠ 0
⊢ dp R n 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_eval_zero
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : DecidableEq ι
s : Finset ι
q : ℕ
x : ι → M
n : ℕ
hn : n ≠ 0
⊢ dp R n 0 = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.dp_sum_smul | [294, 1] | [300, 86] | simp_rw [dp_sum, dp_smul, Algebra.smul_def, map_prod, ← Finset.prod_mul_distrib] | R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : DecidableEq ι
s : Finset ι
q : ℕ
a : ι → R
x : ι → M
⊢ dp R q (∑ i ∈ s, a i • x i) =
∑ k ∈ s.sym q, (∏ i ∈ s, a i ^ Multiset.count i ↑k) • ∏ i ∈ s, dp R (Multiset.count i ↑k) (x i) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : DecidableEq ι
s : Finset ι
q : ℕ
a : ι → R
x : ι → M
⊢ dp R q (∑ i ∈ s, a i • x i) =
∑ k ∈ s.sym q, (∏ i ∈ s, a i ^ Multiset.count i ↑k) • ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.ext_iff | [304, 1] | [315, 54] | constructor | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ f = g ↔ ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m) | case mp
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ f = g → ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ f = g ↔ ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.ext_iff | [304, 1] | [315, 54] | . intro h n m
rw [h] | case mp
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ f = g → ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝... | case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ (∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)) → f = g | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ f = g → ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
cas... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.ext_iff | [304, 1] | [315, 54] | . intro h
rw [DFunLike.ext'_iff]
apply Function.Surjective.injective_comp_right (mkAlgHom_surjective R (Rel R M))
dsimp only
rw [← AlgHom.coe_comp, ← AlgHom.coe_comp, ← DFunLike.ext'_iff]
exact MvPolynomial.algHom_ext fun ⟨n, m⟩ => h n m | case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ (∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)) → f = g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ (∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)) → f = g
T... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.ext_iff | [304, 1] | [315, 54] | intro h n m | case mp
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ f = g → ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m) | case mp
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
h : f = g
n : ℕ
m : M
⊢ f (dp R n m) = g (dp R n m) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ f = g → ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
TACT... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.ext_iff | [304, 1] | [315, 54] | rw [h] | case mp
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
h : f = g
n : ℕ
m : M
⊢ f (dp R n m) = g (dp R n m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
h : f = g
n : ℕ
m : M
⊢ f (dp R n m) = g (dp R n m)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.ext_iff | [304, 1] | [315, 54] | intro h | case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ (∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)) → f = g | case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
h : ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
⊢ f = g | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
⊢ (∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)) → f = g
T... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.ext_iff | [304, 1] | [315, 54] | rw [DFunLike.ext'_iff] | case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
h : ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
⊢ f = g | case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
h : ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
⊢ ⇑f = ⇑g | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
h : ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
⊢ f = g
T... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.ext_iff | [304, 1] | [315, 54] | apply Function.Surjective.injective_comp_right (mkAlgHom_surjective R (Rel R M)) | case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
h : ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
⊢ ⇑f = ⇑g | case mpr.a
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
h : ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
⊢ (fun g => g ∘ ⇑(mkAlgHom R (Rel R M))) ⇑f = (fun g => g ∘ ⇑(mkAlgH... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
h : ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
⊢ ⇑f = ⇑g... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.ext_iff | [304, 1] | [315, 54] | dsimp only | case mpr.a
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
h : ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
⊢ (fun g => g ∘ ⇑(mkAlgHom R (Rel R M))) ⇑f = (fun g => g ∘ ⇑(mkAlgH... | case mpr.a
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
h : ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
⊢ ⇑f ∘ ⇑(mkAlgHom R (Rel R M)) = ⇑g ∘ ⇑(mkAlgHom R (Rel R M)) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.a
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
h : ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
⊢ (fun ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.ext_iff | [304, 1] | [315, 54] | rw [← AlgHom.coe_comp, ← AlgHom.coe_comp, ← DFunLike.ext'_iff] | case mpr.a
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
h : ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
⊢ ⇑f ∘ ⇑(mkAlgHom R (Rel R M)) = ⇑g ∘ ⇑(mkAlgHom R (Rel R M)) | case mpr.a
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
h : ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
⊢ f.comp (mkAlgHom R (Rel R M)) = g.comp (mkAlgHom R (Rel R M)) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.a
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
h : ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
⊢ ⇑f ∘ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.ext_iff | [304, 1] | [315, 54] | exact MvPolynomial.algHom_ext fun ⟨n, m⟩ => h n m | case mpr.a
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
h : ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
⊢ f.comp (mkAlgHom R (Rel R M)) = g.comp (mkAlgHom R (Rel R M)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.a
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f g : DividedPowerAlgebra R M →ₐ[R] A
h : ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)
⊢ f.com... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.induction_on | [324, 11] | [341, 55] | obtain ⟨F, hf⟩ := RingQuot.mkRingHom_surjective (DividedPowerAlgebra.Rel R M) f | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
f : DividedPowerAlgebra R M
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f *... | case intro
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
f : DividedPowerAlgebra R M
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
f : DividedPowerAlgebra R M
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.induction_on | [324, 11] | [341, 55] | rw [← hf] | case intro
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
f : DividedPowerAlgebra R M
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P... | case intro
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
f : DividedPowerAlgebra R M
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
f : DividedPowerAlgebra R M
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f +... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.induction_on | [324, 11] | [341, 55] | induction F using MvPolynomial.induction_on generalizing f with
| h_C a =>
convert h_C a using 1;
rw [mk, mkAlgHom, AlgHom.coe_mk]
| h_add g1 g2 hg1 hg2 =>
rw [map_add]
exact h_add _ _ (hg1 ((mkRingHom (Rel R M)) g1) rfl) (hg2 ((mkRingHom (Rel R M)) g2) rfl)
| h_X g nm h =>
have h' : (mkRingHom (Rel... | case intro
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
f : DividedPowerAlgebra R M
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
f : DividedPowerAlgebra R M
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f +... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.induction_on | [324, 11] | [341, 55] | convert h_C a using 1 | case intro.h_C
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)
a ... | case h.e'_1
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)
a : R... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.h_C
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : Divide... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.induction_on | [324, 11] | [341, 55] | rw [mk, mkAlgHom, AlgHom.coe_mk] | case h.e'_1
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)
a : R... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : DividedPo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.induction_on | [324, 11] | [341, 55] | rw [map_add] | case intro.h_add
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)
... | case intro.h_add
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)
... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.h_add
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : Divi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.induction_on | [324, 11] | [341, 55] | exact h_add _ _ (hg1 ((mkRingHom (Rel R M)) g1) rfl) (hg2 ((mkRingHom (Rel R M)) g2) rfl) | case intro.h_add
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.h_add
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : Divi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.induction_on | [324, 11] | [341, 55] | have h' : (mkRingHom (Rel R M)) (X nm) = dp R nm.1 nm.2 := by
simp only [dp_def, Prod.mk.eta, mkAlgHom, AlgHom.coe_mk] | case intro.h_X
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)
g ... | case intro.h_X
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)
g ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.h_X
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : Divide... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.induction_on | [324, 11] | [341, 55] | rw [_root_.map_mul, h'] | case intro.h_X
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)
g ... | case intro.h_X
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)
g ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.h_X
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : Divide... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.induction_on | [324, 11] | [341, 55] | exact h_dp _ _ _ (h (mkRingHom (Rel R M) g) rfl) | case intro.h_X
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)
g ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.h_X
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : Divide... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.induction_on | [324, 11] | [341, 55] | simp only [dp_def, Prod.mk.eta, mkAlgHom, AlgHom.coe_mk] | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)
g : MvPolynomial ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : DividedPowerAlgebra R M → Prop
h_C : ∀ (a : R), P (mk (C a))
h_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)
h_dp : ∀ (f : DividedPowerAlgebra R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.lift'_imp | [369, 1] | [381, 17] | cases' h with a r n a m n a n a b <;>
simp only [eval₂AlgHom_X', map_one, map_zero, map_smul, AlgHom.map_mul, map_nsmul, AlgHom.map_sum] | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m)
hf_mul : ∀ (n p : ℕ) (m : M), f (n, m) * f (p, m) =... | case zero
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m)
hf_mul : ∀ (n p : ℕ) (m : M), f (n, m) * ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n •... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.lift'_imp | [369, 1] | [381, 17] | . apply hf_zero | case zero
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m)
hf_mul : ∀ (n p : ℕ) (m : M), f (n, m) * ... | case smul
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m)
hf_mul : ∀ (n p : ℕ) (m : M), f (n, m) * ... | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.lift'_imp | [369, 1] | [381, 17] | . apply hf_smul | case smul
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m)
hf_mul : ∀ (n p : ℕ) (m : M), f (n, m) * ... | case mul
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m)
hf_mul : ∀ (n p : ℕ) (m : M), f (n, m) * f... | Please generate a tactic in lean4 to solve the state.
STATE:
case smul
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.lift'_imp | [369, 1] | [381, 17] | . apply hf_mul | case mul
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m)
hf_mul : ∀ (n p : ℕ) (m : M), f (n, m) * f... | case add
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m)
hf_mul : ∀ (n p : ℕ) (m : M), f (n, m) * f... | Please generate a tactic in lean4 to solve the state.
STATE:
case mul
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.lift'_imp | [369, 1] | [381, 17] | . apply hf_add | case add
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m)
hf_mul : ∀ (n p : ℕ) (m : M), f (n, m) * f... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case add
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.lift'_imp | [369, 1] | [381, 17] | apply hf_zero | case zero
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m)
hf_mul : ∀ (n p : ℕ) (m : M), f (n, m) * ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.lift'_imp | [369, 1] | [381, 17] | apply hf_smul | case smul
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m)
hf_mul : ∀ (n p : ℕ) (m : M), f (n, m) * ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case smul
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.lift'_imp | [369, 1] | [381, 17] | apply hf_mul | case mul
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m)
hf_mul : ∀ (n p : ℕ) (m : M), f (n, m) * f... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mul
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.lift'_imp | [369, 1] | [381, 17] | apply hf_add | case add
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m)
hf_mul : ∀ (n p : ℕ) (m : M), f (n, m) * f... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case add
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.lift'AlgHom_apply | [394, 1] | [402, 6] | rw [mk, lift', RingQuot.liftAlgHom_mkAlgHom_apply, coe_eval₂AlgHom] | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m)
hf_mul : ∀ (n p : ℕ) (m : M), f (n, m) * f (p, m) =... | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m)
hf_mul : ∀ (n p : ℕ) (m : M), f (n, m) * f (p, m) =... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n •... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.lift'AlgHom_apply | [394, 1] | [402, 6] | rfl | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m)
hf_mul : ∀ (n p : ℕ) (m : M), f (n, m) * f (p, m) =... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n •... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.lift'AlgHom_apply_dp | [406, 1] | [412, 80] | rw [dp_def, ← mk, lift'AlgHom_apply f hf_zero hf_smul hf_mul hf_add, aeval_X] | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n • f (n, m)
hf_mul : ∀ (n p : ℕ) (m : M), f (n, m) * f (p, m) =... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : ℕ × M → A
hf_zero : ∀ (m : M), f (0, m) = 1
hf_smul : ∀ (n : ℕ) (r : R) (m : M), f (n, r • m) = r ^ n •... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.liftAlgHom_apply | [437, 1] | [440, 34] | rw [lift, lift'AlgHom_apply] | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
p : MvPolynomial (ℕ × M) R
⊢ (lift hI φ hφ) (mk p) = (aeval fun nm => hI.dpow nm.1 (φ nm.2)) p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
p : MvPolynomial (ℕ × M) R
⊢ (lift h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.liftAlgHom_apply_dp | [450, 1] | [452, 34] | rw [lift, lift'AlgHom_apply_dp] | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
n : ℕ
m : M
⊢ (lift hI φ hφ) (dp R n m) = hI.dpow n (φ m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
n : ℕ
m : M
⊢ (lift hI φ hφ) (dp R n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.liftAlgHom_unique | [457, 1] | [462, 31] | apply DividedPowerAlgebra.ext | R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
f : DividedPowerAlgebra R M →ₐ[R] A
hf : ∀ (n : ℕ) (m : M), f (dp R n m) = hI.dpow n (φ m)
⊢ f = ... | case h
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
f : DividedPowerAlgebra R M →ₐ[R] A
hf : ∀ (n : ℕ) (m : M), f (dp R n m) = hI.dpow n (φ m)... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
f : DividedPowerAlgebra R M →ₐ[R] A
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.liftAlgHom_unique | [457, 1] | [462, 31] | intro n m | case h
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
f : DividedPowerAlgebra R M →ₐ[R] A
hf : ∀ (n : ℕ) (m : M), f (dp R n m) = hI.dpow n (φ m)... | case h
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
f : DividedPowerAlgebra R M →ₐ[R] A
hf : ∀ (n : ℕ) (m : M), f (dp R n m) = hI.dpow n (φ m)... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
f : DividedPowerAlgebra R M →... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/DPAlgebra/Init.lean | DividedPowerAlgebra.liftAlgHom_unique | [457, 1] | [462, 31] | rw [liftAlgHom_apply_dp, hf] | case h
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
f : DividedPowerAlgebra R M →ₐ[R] A
hf : ∀ (n : ℕ) (m : M), f (dp R n m) = hI.dpow n (φ m)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
φ : M →ₗ[R] A
hφ : ∀ (m : M), φ m ∈ I
f : DividedPowerAlgebra R M →... |
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