tactic stringlengths 1 5.59k | name stringlengths 1 85 | haveDraft stringlengths 1 44.5k | goal stringlengths 7 64.3k |
|---|---|---|---|
of_symm_smul, | this | (aeval a : R[X] → A) X • ((of R M a).symm : AEval R M a → M) m = a • ((of R M a).symm : AEval R M a → M) m | R : Type u_1
A : Type u_3
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
a : A
inst✝⁴ : Algebra R A
inst✝³ : AddCommMonoid M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower R A M
m : AEval R M a
⊢ ((of R M a).symm : AEval R M a → M) (X • m) = a • ((of R M a).symm : AEval R M a → M) m |
aeval_X | aeval_X | a • ((of R M a).symm : AEval R M a → M) m = a • ((of R M a).symm : AEval R M a → M) m | R : Type u_1
A : Type u_3
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
a : A
inst✝⁴ : Algebra R A
inst✝³ : AddCommMonoid M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower R A M
m : AEval R M a
⊢ (aeval a : R[X] → A) X • ((of R M a).symm : AEval R M a → M) m = a • ((of R M a).symm : AEval R M a → M) m |
apply (of R M a).symm.injective | a | ((of R M a).symm : AEval R M a → M) ((r • f) • m) = ((of R M a).symm : AEval R M a → M) (r • f • m) | R : Type u_1
A : Type u_3
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
a : A
inst✝⁴ : Algebra R A
inst✝³ : AddCommMonoid M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower R A M
r : R
f : R[X]
m : AEval R M a
⊢ (r • f) • m = r • f • m |
of_symm_smul, | a | (aeval a : R[X] → A) (r • f) • ((of R M a).symm : AEval R M a → M) m = ((of R M a).symm : AEval R M a → M) (r • f • m) | R : Type u_1
A : Type u_3
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
a : A
inst✝⁴ : Algebra R A
inst✝³ : AddCommMonoid M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower R A M
r : R
f : R[X]
m : AEval R M a
⊢ ((of R M a).symm : AEval R M a → M) ((r • f) • m) = ((of R M a).symm : AEval R M a → M) (r • f • m) |
map_smul, | a | (r • (aeval a : R[X] → A) f) • ((of R M a).symm : AEval R M a → M) m = ((of R M a).symm : AEval R M a → M) (r • f • m) | R : Type u_1
A : Type u_3
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
a : A
inst✝⁴ : Algebra R A
inst✝³ : AddCommMonoid M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower R A M
r : R
f : R[X]
m : AEval R M a
⊢ (aeval a : R[X] → A) (r • f) • ((of R M a).symm : AEval R M a → M) m = ((of R M a).symm : AEval R M a → M) (r • f • m) |
smul_assoc, | a | r • (aeval a : R[X] → A) f • ((of R M a).symm : AEval R M a → M) m = ((of R M a).symm : AEval R M a → M) (r • f • m) | R : Type u_1
A : Type u_3
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
a : A
inst✝⁴ : Algebra R A
inst✝³ : AddCommMonoid M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower R A M
r : R
f : R[X]
m : AEval R M a
⊢ (r • (aeval a : R[X] → A) f) • ((of R M a).symm : AEval R M a → M) m = ((of R M a).symm : AEval R M a → M) (r • f • m) |
map_smul, | a | r • (aeval a : R[X] → A) f • ((of R M a).symm : AEval R M a → M) m = r • ((of R M a).symm : AEval R M a → M) (f • m) | R : Type u_1
A : Type u_3
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
a : A
inst✝⁴ : Algebra R A
inst✝³ : AddCommMonoid M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower R A M
r : R
f : R[X]
m : AEval R M a
⊢ r • (aeval a : R[X] → A) f • ((of R M a).symm : AEval R M a → M) m = ((of R M a).symm : AEval R M a → M) (r • f • m) |
of_symm_smul | a | r • (aeval a : R[X] → A) f • ((of R M a).symm : AEval R M a → M) m =
r • (aeval a : R[X] → A) f • ((of R M a).symm : AEval R M a → M) m | R : Type u_1
A : Type u_3
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
a : A
inst✝⁴ : Algebra R A
inst✝³ : AddCommMonoid M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower R A M
r : R
f : R[X]
m : AEval R M a
⊢ r • (aeval a : R[X] → A) f • ((of R M a).symm : AEval R M a → M) m = r • ((of R M a).symm : AEval R M a → M) (f • m) |
simp_rw [RingHom.id_apply, AddHom.toFun_eq_coe, LinearMap.coe_toAddHom,
LinearMap.comp_apply, LinearEquiv.coe_toLinearMap] at h ⊢ | simp_rw | (f : M → N) (((of R M a).symm : AEval R M a → M) (((C : R → R[X]) k * X ^ (n + 1)) • m)) =
((C : R → R[X]) k * X ^ (n + 1)) • (f : M → N) (((of R M a).symm : AEval R M a → M) m) | R : Type u_1
A : Type u_2
M : Type u_3
inst✝¹⁰ : CommSemiring R
inst✝⁹ : Semiring A
a : A
inst✝⁸ : Algebra R A
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module A M
inst✝⁵ : Module R M
inst✝⁴ : IsScalarTower R A M
N : Type u_4
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : Module R[X] N
inst✝ : IsScalarTower R R[X] N
f : M →ₗ[R] N
hf : ∀ (m : M), (f : M → N) (a • m) = X • (f : M → N) m
p : R[X]
n : ℕ
k : R
h :
∀ (x : AEval R M a),
(f ∘ₗ (↑(of R M a).symm : AEval R M a →ₗ[R] M)).toFun (((C : R → R[X]) k * X ^ n) • x) =
(RingHom.id R[X] : R[X] → R[X]) ((C : R → R[X]) k * X ^ n) •
(f ∘ₗ (↑(of R M a).symm : AEval R M a →ₗ[R] M)).toFun x
m : AEval R M a
⊢ (f ∘ₗ (↑(of R M a).symm : AEval R M a →ₗ[R] M)).toFun (((C : R → R[X]) k * X ^ (n + 1)) • m) =
(RingHom.id R[X] : R[X] → R[X]) ((C : R → R[X]) k * X ^ (n + 1)) •
(f ∘ₗ (↑(of R M a).symm : AEval R M a →ₗ[R] M)).toFun m |
ext p | h | p ∈ annihilator R[X] (AEval R M a) ↔ p ∈ RingHom.ker (aeval a) | R : Type u_3
A : Type u_1
M : Type u_2
inst✝⁷ : CommSemiring R
inst✝⁶ : Semiring A
a : A
inst✝⁵ : Algebra R A
inst✝⁴ : AddCommMonoid M
inst✝³ : Module A M
inst✝² : Module R M
inst✝¹ : IsScalarTower R A M
inst✝ : FaithfulSMul A M
⊢ annihilator R[X] (AEval R M a) = RingHom.ker (aeval a) |
simp_rw [mem_annihilator, RingHom.mem_ker] | h | (∀ (m : AEval R M a), p • m = 0) ↔ (aeval a : R[X] → A) p = 0 | R : Type u_3
A : Type u_1
M : Type u_2
inst✝⁷ : CommSemiring R
inst✝⁶ : Semiring A
a : A
inst✝⁵ : Algebra R A
inst✝⁴ : AddCommMonoid M
inst✝³ : Module A M
inst✝² : Module R M
inst✝¹ : IsScalarTower R A M
inst✝ : FaithfulSMul A M
p : R[X]
⊢ p ∈ annihilator R[X] (AEval R M a) ↔ p ∈ RingHom.ker (aeval a) |
change (∀ m : M, aeval a p • m = 0) ↔ _ | h | (∀ (m : M), (aeval a : R[X] → A) p • m = 0) ↔ (aeval a : R[X] → A) p = 0 | R : Type u_3
A : Type u_1
M : Type u_2
inst✝⁷ : CommSemiring R
inst✝⁶ : Semiring A
a : A
inst✝⁵ : Algebra R A
inst✝⁴ : AddCommMonoid M
inst✝³ : Module A M
inst✝² : Module R M
inst✝¹ : IsScalarTower R A M
inst✝ : FaithfulSMul A M
p : R[X]
⊢ (∀ (m : AEval R M a), p • m = 0) ↔ (aeval a : R[X] → A) p = 0 |
ext p | h | p ∈ ⊤.annihilator ↔ p ∈ RingHom.ker (aeval a) | R : Type u_3
A : Type u_1
M : Type u_2
inst✝⁷ : CommSemiring R
inst✝⁶ : Semiring A
a : A
inst✝⁵ : Algebra R A
inst✝⁴ : AddCommMonoid M
inst✝³ : Module A M
inst✝² : Module R M
inst✝¹ : IsScalarTower R A M
inst✝ : FaithfulSMul A M
⊢ ⊤.annihilator = RingHom.ker (aeval a) |
simp only [Submodule.mem_annihilator, Submodule.mem_top, forall_true_left, RingHom.mem_ker] | h | (∀ (n : AEval R M a), p • n = 0) ↔ (aeval a : R[X] → A) p = 0 | R : Type u_3
A : Type u_1
M : Type u_2
inst✝⁷ : CommSemiring R
inst✝⁶ : Semiring A
a : A
inst✝⁵ : Algebra R A
inst✝⁴ : AddCommMonoid M
inst✝³ : Module A M
inst✝² : Module R M
inst✝¹ : IsScalarTower R A M
inst✝ : FaithfulSMul A M
p : R[X]
⊢ p ∈ ⊤.annihilator ↔ p ∈ RingHom.ker (aeval a) |
change (∀ m : M, aeval a p • m = 0) ↔ _ | h | (∀ (m : M), (aeval a : R[X] → A) p • m = 0) ↔ (aeval a : R[X] → A) p = 0 | R : Type u_3
A : Type u_1
M : Type u_2
inst✝⁷ : CommSemiring R
inst✝⁶ : Semiring A
a : A
inst✝⁵ : Algebra R A
inst✝⁴ : AddCommMonoid M
inst✝³ : Module A M
inst✝² : Module R M
inst✝¹ : IsScalarTower R A M
inst✝ : FaithfulSMul A M
p : R[X]
⊢ (∀ (n : AEval R M a), p • n = 0) ↔ (aeval a : R[X] → A) p = 0 |
ext | a | x✝ ∈
(↑((fun q ↦
⟨((Submodule.orderIsoMapComap (of R M a)).symm : Submodule R (AEval R M a) → Submodule R M)
(Submodule.restrictScalars R q),
sorry⟩)
((fun p ↦
let __AddSubmonoid := AddSubmonoid.map (of R M a) (↑p : Submodule R M).toAddSubmonoid;
{ toAddSubmonoid := __AddSubmonoid, smul_mem' := sorry })
p)) :
Submodule R M) ↔
x✝ ∈ (↑p : Submodule R M) | R : Type u_1
A : Type u_2
M : Type u_3
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
a : A
inst✝⁴ : Algebra R A
inst✝³ : AddCommMonoid M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower R A M
p : ↥((Algebra.lsmul R R M : A → End R M) a).invtSubmodule
⊢ (fun q ↦
⟨((Submodule.orderIsoMapComap (of R M a)).symm : Submodule R (AEval R M a) → Submodule R M)
(Submodule.restrictScalars R q),
⋯⟩)
((fun p ↦
let __AddSubmonoid := AddSubmonoid.map (of R M a) (↑p : Submodule R M).toAddSubmonoid;
{ toAddSubmonoid := __AddSubmonoid, smul_mem' := ⋯ })
p) =
p |
ext | h | x✝ ∈
(fun p ↦
let __AddSubmonoid := AddSubmonoid.map (of R M a) (↑p : Submodule R M).toAddSubmonoid;
{ toAddSubmonoid := __AddSubmonoid, smul_mem' := sorry })
((fun q ↦
⟨((Submodule.orderIsoMapComap (of R M a)).symm : Submodule R (AEval R M a) → Submodule R M)
(Submodule.restrictScalars R q),
sorry⟩)
q) ↔
x✝ ∈ q | R : Type u_1
A : Type u_2
M : Type u_3
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
a : A
inst✝⁴ : Algebra R A
inst✝³ : AddCommMonoid M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower R A M
q : Submodule R[X] (AEval R M a)
⊢ (fun p ↦
let __AddSubmonoid := AddSubmonoid.map (of R M a) (↑p : Submodule R M).toAddSubmonoid;
{ toAddSubmonoid := __AddSubmonoid, smul_mem' := ⋯ })
((fun q ↦
⟨((Submodule.orderIsoMapComap (of R M a)).symm : Submodule R (AEval R M a) → Submodule R M)
(Submodule.restrictScalars R q),
⋯⟩)
q) =
q |
obtain ⟨x, hx⟩ := x | mk | ((of R M a).symm : AEval R M a → M) (↑⟨x, sorry⟩ : AEval R M a) ∈ p | R : Type u_1
A : Type u_2
M : Type u_3
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
a : A
inst✝⁴ : Algebra R A
inst✝³ : AddCommMonoid M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower R A M
p : Submodule R M
hp : p ∈ ((Algebra.lsmul R R M : A → End R M) a).invtSubmodule
x :
↥((mapSubmodule R M a : ↥((Algebra.lsmul R R M : A → End R M) a).invtSubmodule → Submodule R[X] (AEval R M a))
⟨p, hp⟩)
⊢ ((of R M a).symm : AEval R M a → M) (↑x : AEval R M a) ∈ p |
refine (Fintype.truncEquivFinOfCardEq <| Fintype.card_coe s).lift
(fun e ↦ (finAntidiagonal s.card n).map ⟨fun f i ↦ if hi : i ∈ s then f (e ⟨i, hi⟩) else 0, ?_⟩)
fun e₁ e₂ ↦ ?_ | refine_1 | Injective fun f i ↦ if hi : i ∈ s then f ((e : { x // x ∈ s } → Fin #s) ⟨i, sorry⟩) else 0 | ι : Type u_1
μ : Type u_2
μ' : Type u_3
inst✝³ : DecidableEq ι
inst✝² : AddCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
n✝ : μ
s : Finset ι
n : μ
⊢ Finset (ι → μ) |
refine (Fintype.truncEquivFinOfCardEq <| Fintype.card_coe s).lift
(fun e ↦ (finAntidiagonal s.card n).map ⟨fun f i ↦ if hi : i ∈ s then f (e ⟨i, hi⟩) else 0, ?_⟩)
fun e₁ e₂ ↦ ?_ | refine_2 | (fun e ↦
map { toFun := fun f i ↦ if hi : i ∈ s then f ((e : { x // x ∈ s } → Fin #s) ⟨i, sorry⟩) else 0, inj' := sorry }
(finAntidiagonal (#s) n))
e₁ =
(fun e ↦
map { toFun := fun f i ↦ if hi : i ∈ s then f ((e : { x // x ∈ s } → Fin #s) ⟨i, sorry⟩) else 0, inj' := sorry }
(finAntidiagonal (#s) n))
e₂ | ι : Type u_1
μ : Type u_2
μ' : Type u_3
inst✝³ : DecidableEq ι
inst✝² : AddCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
n✝ : μ
s : Finset ι
n : μ
refine_1 : Injective fun f i ↦ if hi : i ∈ s then f ((_fvar.12352 : { x // x ∈ s } → Fin #s) ⟨i, hi⟩) else 0
⊢ Finset (ι → μ) |
ext i | refine_1 | f i = g i | ι : Type u_1
μ : Type u_2
μ' : Type u_3
inst✝³ : DecidableEq ι
inst✝² : AddCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
n✝ : μ
s : Finset ι
n : μ
e : { x // x ∈ s } ≃ Fin #s
f g : Fin #s → μ
hfg :
(fun f i ↦ if hi : i ∈ s then f ((e : { x // x ∈ s } → Fin #s) ⟨i, hi⟩) else 0) f =
(fun f i ↦ if hi : i ∈ s then f ((e : { x // x ∈ s } → Fin #s) ⟨i, hi⟩) else 0) g
⊢ f = g |
ext f | refine_2 | f ∈
(fun e ↦
map { toFun := fun f i ↦ if hi : i ∈ s then f ((e : { x // x ∈ s } → Fin #s) ⟨i, sorry⟩) else 0, inj' := sorry }
(finAntidiagonal (#s) n))
e₁ ↔
f ∈
(fun e ↦
map { toFun := fun f i ↦ if hi : i ∈ s then f ((e : { x // x ∈ s } → Fin #s) ⟨i, sorry⟩) else 0, inj' := sorry }
(finAntidiagonal (#s) n))
e₂ | ι : Type u_1
μ : Type u_2
μ' : Type u_3
inst✝³ : DecidableEq ι
inst✝² : AddCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
n✝ : μ
s : Finset ι
n : μ
e₁ e₂ : { x // x ∈ s } ≃ Fin #s
⊢ (fun e ↦
map { toFun := fun f i ↦ if hi : i ∈ s then f ((e : { x // x ∈ s } → Fin #s) ⟨i, hi⟩) else 0, inj' := ⋯ }
(finAntidiagonal (#s) n))
e₁ =
(fun e ↦
map { toFun := fun f i ↦ if hi : i ∈ s then f ((e : { x // x ∈ s } → Fin #s) ⟨i, hi⟩) else 0, inj' := ⋯ }
(finAntidiagonal (#s) n))
e₂ |
simp only [mem_map, mem_finAntidiagonal] | refine_2 | (∃ a,
∑ i, a i = n ∧
({ toFun := fun f i ↦ if hi : i ∈ s then f ((e₁ : { x // x ∈ s } → Fin #s) ⟨i, sorry⟩) else 0, inj' := sorry } :
(Fin #s → μ) → ι → μ)
a =
f) ↔
∃ a,
∑ i, a i = n ∧
({ toFun := fun f i ↦ if hi : i ∈ s then f ((e₂ : { x // x ∈ s } → Fin #s) ⟨i, sorry⟩) else 0, inj' := sorry } :
(Fin #s → μ) → ι → μ)
a =
f | ι : Type u_1
μ : Type u_2
μ' : Type u_3
inst✝³ : DecidableEq ι
inst✝² : AddCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
n✝ : μ
s : Finset ι
n : μ
e₁ e₂ : { x // x ∈ s } ≃ Fin #s
f : ι → μ
⊢ f ∈
(fun e ↦
map { toFun := fun f i ↦ if hi : i ∈ s then f ((e : { x // x ∈ s } → Fin #s) ⟨i, hi⟩) else 0, inj' := ⋯ }
(finAntidiagonal (#s) n))
e₁ ↔
f ∈
(fun e ↦
map { toFun := fun f i ↦ if hi : i ∈ s then f ((e : { x // x ∈ s } → Fin #s) ⟨i, hi⟩) else 0, inj' := ⋯ }
(finAntidiagonal (#s) n))
e₂ |
refine Equiv.exists_congr ((e₁.symm.trans e₂).arrowCongr <| .refl _) fun g ↦ ?_ | refine_2 | ∑ i, g i = n ∧
({ toFun := fun f i ↦ if hi : i ∈ s then f ((e₁ : { x // x ∈ s } → Fin #s) ⟨i, sorry⟩) else 0, inj' := sorry } :
(Fin #s → μ) → ι → μ)
g =
f ↔
∑ i, ((e₁.symm.trans e₂).arrowCongr (Equiv.refl μ) : (Fin #s → μ) → Fin #s → μ) g i = n ∧
({ toFun := fun f i ↦ if hi : i ∈ s then f ((e₂ : { x // x ∈ s } → Fin #s) ⟨i, sorry⟩) else 0, inj' := sorry } :
(Fin #s → μ) → ι → μ)
(((e₁.symm.trans e₂).arrowCongr (Equiv.refl μ) : (Fin #s → μ) → Fin #s → μ) g) =
f | ι : Type u_1
μ : Type u_2
μ' : Type u_3
inst✝³ : DecidableEq ι
inst✝² : AddCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
n✝ : μ
s : Finset ι
n : μ
e₁ e₂ : { x // x ∈ s } ≃ Fin #s
f : ι → μ
⊢ (∃ a,
∑ i, a i = n ∧
({ toFun := fun f i ↦ if hi : i ∈ s then f ((e₁ : { x // x ∈ s } → Fin #s) ⟨i, hi⟩) else 0, inj' := ⋯ } :
(Fin #s → μ) → ι → μ)
a =
f) ↔
∃ a,
∑ i, a i = n ∧
({ toFun := fun f i ↦ if hi : i ∈ s then f ((e₂ : { x // x ∈ s } → Fin #s) ⟨i, hi⟩) else 0, inj' := ⋯ } :
(Fin #s → μ) → ι → μ)
a =
f |
have := Fintype.sum_equiv (e₂.symm.trans e₁) _ g fun _ ↦ rfl | refine_2 | ∑ i, g i = n ∧
({ toFun := fun f i ↦ if hi : i ∈ s then f ((e₁ : { x // x ∈ s } → Fin #s) ⟨i, sorry⟩) else 0, inj' := sorry } :
(Fin #s → μ) → ι → μ)
g =
f ↔
∑ i, ((e₁.symm.trans e₂).arrowCongr (Equiv.refl μ) : (Fin #s → μ) → Fin #s → μ) g i = n ∧
({ toFun := fun f i ↦ if hi : i ∈ s then f ((e₂ : { x // x ∈ s } → Fin #s) ⟨i, sorry⟩) else 0, inj' := sorry } :
(Fin #s → μ) → ι → μ)
(((e₁.symm.trans e₂).arrowCongr (Equiv.refl μ) : (Fin #s → μ) → Fin #s → μ) g) =
f | ι : Type u_1
μ : Type u_2
μ' : Type u_3
inst✝³ : DecidableEq ι
inst✝² : AddCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
n✝ : μ
s : Finset ι
n : μ
e₁ e₂ : { x // x ∈ s } ≃ Fin #s
f : ι → μ
g : Fin #s → μ
⊢ ∑ i, g i = n ∧
({ toFun := fun f i ↦ if hi : i ∈ s then f ((e₁ : { x // x ∈ s } → Fin #s) ⟨i, hi⟩) else 0, inj' := ⋯ } :
(Fin #s → μ) → ι → μ)
g =
f ↔
∑ i, ((e₁.symm.trans e₂).arrowCongr (Equiv.refl μ) : (Fin #s → μ) → Fin #s → μ) g i = n ∧
({ toFun := fun f i ↦ if hi : i ∈ s then f ((e₂ : { x // x ∈ s } → Fin #s) ⟨i, hi⟩) else 0, inj' := ⋯ } :
(Fin #s → μ) → ι → μ)
(((e₁.symm.trans e₂).arrowCongr (Equiv.refl μ) : (Fin #s → μ) → Fin #s → μ) g) =
f |
induction' Fintype.truncEquivFinOfCardEq (Fintype.card_coe s) using Trunc.ind with e | a | f ∈
Trunc.lift
(fun e ↦
map { toFun := fun f i ↦ if hi : i ∈ s then f ((e : { x // x ∈ s } → Fin #s) ⟨i, sorry⟩) else 0, inj' := sorry }
(finAntidiagonal (#s) n))
sorry (Trunc.mk e) ↔
s.sum f = n ∧ ∀ (i : ι), f i ≠ 0 → i ∈ s | ι : Type u_1
μ : Type u_2
inst✝³ : DecidableEq ι
inst✝² : AddCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
s : Finset ι
n : μ
f : ι → μ
⊢ f ∈
Trunc.lift
(fun e ↦
map { toFun := fun f i ↦ if hi : i ∈ s then f ((e : { x // x ∈ s } → Fin #s) ⟨i, hi⟩) else 0, inj' := ⋯ }
(finAntidiagonal (#s) n))
⋯ (Fintype.truncEquivFinOfCardEq ⋯) ↔
s.sum f = n ∧ ∀ (i : ι), f i ≠ 0 → i ∈ s |
simp only [Trunc.lift_mk, mem_map, mem_finAntidiagonal, Embedding.coeFn_mk] | a | (∃ a, ∑ i, a i = n ∧ (fun i ↦ if hi : i ∈ s then a ((e : { x // x ∈ s } → Fin #s) ⟨i, sorry⟩) else 0) = f) ↔
s.sum f = n ∧ ∀ (i : ι), f i ≠ 0 → i ∈ s | ι : Type u_1
μ : Type u_2
inst✝³ : DecidableEq ι
inst✝² : AddCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
s : Finset ι
n : μ
f : ι → μ
e : { x // x ∈ s } ≃ Fin #s
⊢ f ∈
Trunc.lift
(fun e ↦
map { toFun := fun f i ↦ if hi : i ∈ s then f ((e : { x // x ∈ s } → Fin #s) ⟨i, hi⟩) else 0, inj' := ⋯ }
(finAntidiagonal (#s) n))
⋯ (Trunc.mk e) ↔
s.sum f = n ∧ ∀ (i : ι), f i ≠ 0 → i ∈ s |
constructor | a | (∃ a, ∑ i, a i = n ∧ (fun i ↦ if hi : i ∈ s then a ((e : { x // x ∈ s } → Fin #s) ⟨i, sorry⟩) else 0) = f) →
s.sum f = n ∧ ∀ (i : ι), f i ≠ 0 → i ∈ s | ι : Type u_1
μ : Type u_2
inst✝³ : DecidableEq ι
inst✝² : AddCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
s : Finset ι
n : μ
f : ι → μ
e : { x // x ∈ s } ≃ Fin #s
⊢ (∃ a, ∑ i, a i = n ∧ (fun i ↦ if hi : i ∈ s then a ((e : { x // x ∈ s } → Fin #s) ⟨i, hi⟩) else 0) = f) ↔
s.sum f = n ∧ ∀ (i : ι), f i ≠ 0 → i ∈ s |
constructor | a | (s.sum f = n ∧ ∀ (i : ι), f i ≠ 0 → i ∈ s) →
∃ a, ∑ i, a i = n ∧ (fun i ↦ if hi : i ∈ s then a ((e : { x // x ∈ s } → Fin #s) ⟨i, sorry⟩) else 0) = f | ι : Type u_1
μ : Type u_2
inst✝³ : DecidableEq ι
inst✝² : AddCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
s : Finset ι
n : μ
f : ι → μ
e : { x // x ∈ s } ≃ Fin #s
a : (∃ a, ∑ i, a i = n ∧ (fun i ↦ if hi : i ∈ s then a ((e : { x // x ∈ s } → Fin #s) ⟨i, hi⟩) else 0) = f) →
s.sum f = n ∧ ∀ (i : ι), f i ≠ 0 → i ∈ s
⊢ (∃ a, ∑ i, a i = n ∧ (fun i ↦ if hi : i ∈ s then a ((e : { x // x ∈ s } → Fin #s) ⟨i, hi⟩) else 0) = f) ↔
s.sum f = n ∧ ∀ (i : ι), f i ≠ 0 → i ∈ s |
refine ⟨f ∘ (↑) ∘ e.symm, ?_, by ext i; have := not_imp_comm.1 (hf i); aesop⟩ | a | ∑ i, (f ∘ Subtype.val ∘ (⇑e.symm : Fin #s → { x // x ∈ s })) i = s.sum f | ι : Type u_1
μ : Type u_2
inst✝³ : DecidableEq ι
inst✝² : AddCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
s : Finset ι
f : ι → μ
e : { x // x ∈ s } ≃ Fin #s
hf : ∀ (i : ι), f i ≠ 0 → i ∈ s
⊢ ∃ a, ∑ i, a i = s.sum f ∧ (fun i ↦ if hi : i ∈ s then a ((e : { x // x ∈ s } → Fin #s) ⟨i, hi⟩) else 0) = f |
ext i | h | (if hi : i ∈ s then (f ∘ Subtype.val ∘ (⇑e.symm : Fin #s → { x // x ∈ s })) ((e : { x // x ∈ s } → Fin #s) ⟨i, sorry⟩)
else 0) =
f i | ι : Type u_1
μ : Type u_2
inst✝³ : DecidableEq ι
inst✝² : AddCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
s : Finset ι
f : ι → μ
e : { x // x ∈ s } ≃ Fin #s
hf : ∀ (i : ι), f i ≠ 0 → i ∈ s
⊢ (fun i ↦
if hi : i ∈ s then (f ∘ Subtype.val ∘ (⇑e.symm : Fin #s → { x // x ∈ s })) ((e : { x // x ∈ s } → Fin #s) ⟨i, hi⟩)
else 0) =
f |
have := not_imp_comm.1 (hf i) | h | (if hi : i ∈ s then (f ∘ Subtype.val ∘ (⇑e.symm : Fin #s → { x // x ∈ s })) ((e : { x // x ∈ s } → Fin #s) ⟨i, sorry⟩)
else 0) =
f i | ι : Type u_1
μ : Type u_2
inst✝³ : DecidableEq ι
inst✝² : AddCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
s : Finset ι
f : ι → μ
e : { x // x ∈ s } ≃ Fin #s
hf : ∀ (i : ι), f i ≠ 0 → i ∈ s
i : ι
⊢ (if hi : i ∈ s then (f ∘ Subtype.val ∘ (⇑e.symm : Fin #s → { x // x ∈ s })) ((e : { x // x ∈ s } → Fin #s) ⟨i, hi⟩)
else 0) =
f i |
rw [← sum_attach s] | a | ∑ i, (f ∘ Subtype.val ∘ (⇑e.symm : Fin #s → { x // x ∈ s })) i = ∑ x ∈ s.attach, f (↑x : ι) | ι : Type u_1
μ : Type u_2
inst✝³ : DecidableEq ι
inst✝² : AddCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
s : Finset ι
f : ι → μ
e : { x // x ∈ s } ≃ Fin #s
hf : ∀ (i : ι), f i ≠ 0 → i ∈ s
⊢ ∑ i, (f ∘ Subtype.val ∘ (⇑e.symm : Fin #s → { x // x ∈ s })) i = s.sum f |
← sum_attach s | a | ∑ i, (f ∘ Subtype.val ∘ (⇑e.symm : Fin #s → { x // x ∈ s })) i = ∑ x ∈ s.attach, f (↑x : ι) | ι : Type u_1
μ : Type u_2
inst✝³ : DecidableEq ι
inst✝² : AddCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
s : Finset ι
f : ι → μ
e : { x // x ∈ s } ≃ Fin #s
hf : ∀ (i : ι), f i ≠ 0 → i ∈ s
⊢ ∑ i, (f ∘ Subtype.val ∘ (⇑e.symm : Fin #s → { x // x ∈ s })) i = s.sum f |
ext | h | a✝ ∈ ∅.piAntidiag 0 ↔ a✝ ∈ {0} | ι : Type u_1
μ : Type u_2
inst✝³ : DecidableEq ι
inst✝² : AddCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
⊢ ∅.piAntidiag 0 = {0} |
ext f | h | f ∈ (cons i s sorry).piAntidiag n ↔
f ∈
(antidiagonal n).disjiUnion (fun p ↦ map (addRightEmbedding fun t ↦ if t = i then p.1 else 0) (s.piAntidiag p.2))
sorry | ι : Type u_1
μ : Type u_2
inst✝³ : DecidableEq ι
inst✝² : AddCancelCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
i : ι
s : Finset ι
hi : i ∉ s
n : μ
⊢ (cons i s hi).piAntidiag n =
(antidiagonal n).disjiUnion (fun p ↦ map (addRightEmbedding fun t ↦ if t = i then p.1 else 0) (s.piAntidiag p.2)) ⋯ |
simp only [mem_piAntidiag, sum_cons, ne_eq, mem_cons, mem_disjiUnion, mem_antidiagonal, mem_map,
addLeftEmbedding_apply, Prod.exists] | h | (f i + ∑ x ∈ s, f x = n ∧ ∀ (i_1 : ι), ¬f i_1 = 0 → i_1 = i ∨ i_1 ∈ s) ↔
∃ a b,
a + b = n ∧
∃ a_1,
(s.sum a_1 = b ∧ ∀ (i : ι), ¬a_1 i = 0 → i ∈ s) ∧
(addRightEmbedding fun t ↦ if t = i then a else 0 : (ι → μ) → ι → μ) a_1 = f | ι : Type u_1
μ : Type u_2
inst✝³ : DecidableEq ι
inst✝² : AddCancelCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
i : ι
s : Finset ι
hi : i ∉ s
n : μ
f : ι → μ
⊢ f ∈ (cons i s hi).piAntidiag n ↔
f ∈
(antidiagonal n).disjiUnion (fun p ↦ map (addRightEmbedding fun t ↦ if t = i then p.1 else 0) (s.piAntidiag p.2))
⋯ |
constructor | h | (f i + ∑ x ∈ s, f x = n ∧ ∀ (i_1 : ι), ¬f i_1 = 0 → i_1 = i ∨ i_1 ∈ s) →
∃ a b,
a + b = n ∧
∃ a_1,
(s.sum a_1 = b ∧ ∀ (i : ι), ¬a_1 i = 0 → i ∈ s) ∧
(addRightEmbedding fun t ↦ if t = i then a else 0 : (ι → μ) → ι → μ) a_1 = f | ι : Type u_1
μ : Type u_2
inst✝³ : DecidableEq ι
inst✝² : AddCancelCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
i : ι
s : Finset ι
hi : i ∉ s
n : μ
f : ι → μ
⊢ (f i + ∑ x ∈ s, f x = n ∧ ∀ (i_1 : ι), ¬f i_1 = 0 → i_1 = i ∨ i_1 ∈ s) ↔
∃ a b,
a + b = n ∧
∃ a_1,
(s.sum a_1 = b ∧ ∀ (i : ι), ¬a_1 i = 0 → i ∈ s) ∧
(addRightEmbedding fun t ↦ if t = i then a else 0 : (ι → μ) → ι → μ) a_1 = f |
constructor | h | (∃ a b,
a + b = n ∧
∃ a_1,
(s.sum a_1 = b ∧ ∀ (i : ι), ¬a_1 i = 0 → i ∈ s) ∧
(addRightEmbedding fun t ↦ if t = i then a else 0 : (ι → μ) → ι → μ) a_1 = f) →
f i + ∑ x ∈ s, f x = n ∧ ∀ (i_1 : ι), ¬f i_1 = 0 → i_1 = i ∨ i_1 ∈ s | ι : Type u_1
μ : Type u_2
inst✝³ : DecidableEq ι
inst✝² : AddCancelCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
i : ι
s : Finset ι
hi : i ∉ s
n : μ
f : ι → μ
h : (f i + ∑ x ∈ s, f x = n ∧ ∀ (i_1 : ι), ¬f i_1 = 0 → i_1 = i ∨ i_1 ∈ s) →
∃ a b,
a + b = n ∧
∃ a_1,
(s.sum a_1 = b ∧ ∀ (i : ι), ¬a_1 i = 0 → i ∈ s) ∧
(addRightEmbedding fun t ↦ if t = i then a else 0 : (ι → μ) → ι → μ) a_1 = f
⊢ (f i + ∑ x ∈ s, f x = n ∧ ∀ (i_1 : ι), ¬f i_1 = 0 → i_1 = i ∨ i_1 ∈ s) ↔
∃ a b,
a + b = n ∧
∃ a_1,
(s.sum a_1 = b ∧ ∀ (i : ι), ¬a_1 i = 0 → i ∈ s) ∧
(addRightEmbedding fun t ↦ if t = i then a else 0 : (ι → μ) → ι → μ) a_1 = f |
refine ⟨_, _, hn, update f i 0, ⟨sum_update_of_notMem hi _ _, fun j ↦ ?_⟩, by aesop⟩ | h | ¬update f i 0 j = 0 → j ∈ s | ι : Type u_1
μ : Type u_2
inst✝³ : DecidableEq ι
inst✝² : AddCancelCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
i : ι
s : Finset ι
hi : i ∉ s
n : μ
f : ι → μ
hn : f i + ∑ x ∈ s, f x = n
hf : ∀ (i_1 : ι), ¬f i_1 = 0 → i_1 = i ∨ i_1 ∈ s
⊢ ∃ a b,
a + b = n ∧
∃ a_1,
(s.sum a_1 = b ∧ ∀ (i : ι), ¬a_1 i = 0 → i ∈ s) ∧
(addRightEmbedding fun t ↦ if t = i then a else 0 : (ι → μ) → ι → μ) a_1 = f |
have := fun h₁ h₂ ↦ (hf j h₁).resolve_left h₂ | h | ¬update f i 0 j = 0 → j ∈ s | ι : Type u_1
μ : Type u_2
inst✝³ : DecidableEq ι
inst✝² : AddCancelCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
i : ι
s : Finset ι
hi : i ∉ s
n : μ
f : ι → μ
hn : f i + ∑ x ∈ s, f x = n
hf : ∀ (i_1 : ι), ¬f i_1 = 0 → i_1 = i ∨ i_1 ∈ s
j : ι
⊢ ¬update f i 0 j = 0 → j ∈ s |
ext | h | a✝ ∈ s.piAntidiag 0 ↔ a✝ ∈ {0} | ι : Type u_1
μ : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid μ
inst✝³ : PartialOrder μ
inst✝² : CanonicallyOrderedAdd μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
s : Finset ι
⊢ s.piAntidiag 0 = {0} |
ext | h | a✝ ∈ univ.piAntidiag n ↔ a✝ ∈ Nat.antidiagonalTuple k n | n k : ℕ
⊢ univ.piAntidiag n = Nat.antidiagonalTuple k n |
ext f | h | f ∈ SMul.smul n (s.piAntidiag m) ↔ f ∈ {f ∈ s.piAntidiag (n * m) | ∀ i ∈ s, n ∣ f i} | ι : Type u_1
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq (ι → ℕ)
s : Finset ι
m n : ℕ
hn : n ≠ 0
⊢ SMul.smul n (s.piAntidiag m) = {f ∈ s.piAntidiag (n * m) | ∀ i ∈ s, n ∣ f i} |
refine mem_smul_finset.trans ?_ | h | (∃ y ∈ s.piAntidiag m, n • y = f) ↔ f ∈ {f ∈ s.piAntidiag (n * m) | ∀ i ∈ s, n ∣ f i} | ι : Type u_1
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq (ι → ℕ)
s : Finset ι
m n : ℕ
hn : n ≠ 0
f : ι → ℕ
⊢ f ∈ SMul.smul n (s.piAntidiag m) ↔ f ∈ {f ∈ s.piAntidiag (n * m) | ∀ i ∈ s, n ∣ f i} |
simp only [mem_smul_finset, mem_filter, mem_piAntidiag, Function.Embedding.coeFn_mk, exists_prop,
and_assoc] | h | (∃ y, s.sum y = m ∧ (∀ (i : ι), y i ≠ 0 → i ∈ s) ∧ n • y = f) ↔
s.sum f = n * m ∧ (∀ (i : ι), f i ≠ 0 → i ∈ s) ∧ ∀ i ∈ s, n ∣ f i | ι : Type u_1
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq (ι → ℕ)
s : Finset ι
m n : ℕ
hn : n ≠ 0
f : ι → ℕ
⊢ (∃ y ∈ s.piAntidiag m, n • y = f) ↔ f ∈ {f ∈ s.piAntidiag (n * m) | ∀ i ∈ s, n ∣ f i} |
constructor | h | (∃ y, s.sum y = m ∧ (∀ (i : ι), y i ≠ 0 → i ∈ s) ∧ n • y = f) →
s.sum f = n * m ∧ (∀ (i : ι), f i ≠ 0 → i ∈ s) ∧ ∀ i ∈ s, n ∣ f i | ι : Type u_1
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq (ι → ℕ)
s : Finset ι
m n : ℕ
hn : n ≠ 0
f : ι → ℕ
⊢ (∃ y, s.sum y = m ∧ (∀ (i : ι), y i ≠ 0 → i ∈ s) ∧ n • y = f) ↔
s.sum f = n * m ∧ (∀ (i : ι), f i ≠ 0 → i ∈ s) ∧ ∀ i ∈ s, n ∣ f i |
constructor | h | (s.sum f = n * m ∧ (∀ (i : ι), f i ≠ 0 → i ∈ s) ∧ ∀ i ∈ s, n ∣ f i) →
∃ y, s.sum y = m ∧ (∀ (i : ι), y i ≠ 0 → i ∈ s) ∧ n • y = f | ι : Type u_1
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq (ι → ℕ)
s : Finset ι
m n : ℕ
hn : n ≠ 0
f : ι → ℕ
h : (∃ y, s.sum y = m ∧ (∀ (i : ι), y i ≠ 0 → i ∈ s) ∧ n • y = f) →
s.sum f = n * m ∧ (∀ (i : ι), f i ≠ 0 → i ∈ s) ∧ ∀ i ∈ s, n ∣ f i
⊢ (∃ y, s.sum y = m ∧ (∀ (i : ι), y i ≠ 0 → i ∈ s) ∧ n • y = f) ↔
s.sum f = n * m ∧ (∀ (i : ι), f i ≠ 0 → i ∈ s) ∧ ∀ i ∈ s, n ∣ f i |
have (i) : n ∣ f i := by
by_cases hi : i ∈ s
· exact hfdvd _ hi
· rw [not_imp_comm.1 (hfsup _) hi]
exact dvd_zero _ | h | ∃ y, s.sum y = m ∧ (∀ (i : ι), y i ≠ 0 → i ∈ s) ∧ n • y = f | ι : Type u_1
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq (ι → ℕ)
s : Finset ι
m n : ℕ
hn : n ≠ 0
f : ι → ℕ
hfsum : s.sum f = n * m
hfsup : ∀ (i : ι), f i ≠ 0 → i ∈ s
hfdvd : ∀ i ∈ s, n ∣ f i
⊢ ∃ y, s.sum y = m ∧ (∀ (i : ι), y i ≠ 0 → i ∈ s) ∧ n • y = f |
by_cases hi : i ∈ s | pos | n ∣ f i | ι : Type u_1
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq (ι → ℕ)
s : Finset ι
m n : ℕ
hn : n ≠ 0
f : ι → ℕ
hfsum : s.sum f = n * m
hfsup : ∀ (i : ι), f i ≠ 0 → i ∈ s
hfdvd : ∀ i ∈ s, n ∣ f i
i : ι
⊢ n ∣ f i |
by_cases hi : i ∈ s | neg | n ∣ f i | ι : Type u_1
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq (ι → ℕ)
s : Finset ι
m n : ℕ
hn : n ≠ 0
f : ι → ℕ
hfsum : s.sum f = n * m
hfsup : ∀ (i : ι), f i ≠ 0 → i ∈ s
hfdvd : ∀ i ∈ s, n ∣ f i
i : ι
pos : n ∣ f i
⊢ n ∣ f i |
rw [not_imp_comm.1 (hfsup _) hi] | neg | n ∣ 0 | ι : Type u_1
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq (ι → ℕ)
s : Finset ι
m n : ℕ
hn : n ≠ 0
f : ι → ℕ
hfsum : s.sum f = n * m
hfsup : ∀ (i : ι), f i ≠ 0 → i ∈ s
hfdvd : ∀ i ∈ s, n ∣ f i
i : ι
hi : i ∉ s
⊢ n ∣ f i |
not_imp_comm.1 (hfsup _) hi | neg | n ∣ 0 | ι : Type u_1
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq (ι → ℕ)
s : Finset ι
m n : ℕ
hn : n ≠ 0
f : ι → ℕ
hfsum : s.sum f = n * m
hfsup : ∀ (i : ι), f i ≠ 0 → i ∈ s
hfdvd : ∀ i ∈ s, n ∣ f i
i : ι
hi : i ∉ s
⊢ n ∣ f i |
refine ⟨fun i ↦ f i / n, ?_⟩ | h | ∑ i ∈ s, f i / n = m ∧ (∀ (i : ι), (fun i ↦ f i / n) i ≠ 0 → i ∈ s) ∧ (n • fun i ↦ f i / n) = f | ι : Type u_1
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq (ι → ℕ)
s : Finset ι
m n : ℕ
hn : n ≠ 0
f : ι → ℕ
hfsum : s.sum f = n * m
hfsup : ∀ (i : ι), f i ≠ 0 → i ∈ s
hfdvd : ∀ i ∈ s, n ∣ f i
this : ∀ (i : ι), n ∣ f i
⊢ ∃ y, s.sum y = m ∧ (∀ (i : ι), y i ≠ 0 → i ∈ s) ∧ n • y = f |
simp [funext_iff, Nat.mul_div_cancel', ← Nat.sum_div, *] | h | ∀ (i : ι), n ≤ f i → i ∈ s | ι : Type u_1
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq (ι → ℕ)
s : Finset ι
m n : ℕ
hn : n ≠ 0
f : ι → ℕ
hfsum : s.sum f = n * m
hfsup : ∀ (i : ι), f i ≠ 0 → i ∈ s
hfdvd : ∀ i ∈ s, n ∣ f i
this : ∀ (i : ι), n ∣ f i
⊢ ∑ i ∈ s, f i / n = m ∧ (∀ (i : ι), (fun i ↦ f i / n) i ≠ 0 → i ∈ s) ∧ (n • fun i ↦ f i / n) = f |
rw [map_eq_image] | rw | image (⇑{ toFun := fun x ↦ n • x, inj' := sorry } : (ι → ℕ) → ι → ℕ) (s.piAntidiag m) =
{f ∈ s.piAntidiag (n * m) | ∀ i ∈ s, n ∣ f i} | ι : Type u_1
inst✝ : DecidableEq ι
s : Finset ι
m n : ℕ
hn : n ≠ 0
⊢ map { toFun := fun x ↦ n • x, inj' := ⋯ } (s.piAntidiag m) = {f ∈ s.piAntidiag (n * m) | ∀ i ∈ s, n ∣ f i} |
map_eq_image | map_eq_image | image (⇑{ toFun := fun x ↦ n • x, inj' := sorry } : (ι → ℕ) → ι → ℕ) (s.piAntidiag m) =
{f ∈ s.piAntidiag (n * m) | ∀ i ∈ s, n ∣ f i} | ι : Type u_1
inst✝ : DecidableEq ι
s : Finset ι
m n : ℕ
hn : n ≠ 0
⊢ map { toFun := fun x ↦ n • x, inj' := ⋯ } (s.piAntidiag m) = {f ∈ s.piAntidiag (n * m) | ∀ i ∈ s, n ∣ f i} |
ext f | h | f ∈ map { toFun := fun m a ↦ Multiset.count a (↑m : Multiset ι), inj' := sorry } (s.sym n) ↔ f ∈ s.piAntidiag n | ι : Type u_1
inst✝ : DecidableEq ι
s : Finset ι
n : ℕ
⊢ map { toFun := fun m a ↦ Multiset.count a (↑m : Multiset ι), inj' := ⋯ } (s.sym n) = s.piAntidiag n |
simp only [Sym.val_eq_coe, mem_map, mem_sym_iff, Embedding.coeFn_mk, funext_iff, Sym.exists,
Sym.mem_mk, Sym.coe_mk, exists_and_left, exists_prop, mem_piAntidiag, ne_eq] | h | (∃ s_1, (∀ a ∈ s_1, a ∈ s) ∧ s_1.card = n ∧ ∀ (x : ι), Multiset.count x s_1 = f x) ↔
s.sum f = n ∧ ∀ (i : ι), ¬f i = 0 → i ∈ s | ι : Type u_1
inst✝ : DecidableEq ι
s : Finset ι
n : ℕ
f : ι → ℕ
⊢ f ∈ map { toFun := fun m a ↦ Multiset.count a (↑m : Multiset ι), inj' := ⋯ } (s.sym n) ↔ f ∈ s.piAntidiag n |
constructor | h | (∃ s_1, (∀ a ∈ s_1, a ∈ s) ∧ s_1.card = n ∧ ∀ (x : ι), Multiset.count x s_1 = f x) →
s.sum f = n ∧ ∀ (i : ι), ¬f i = 0 → i ∈ s | ι : Type u_1
inst✝ : DecidableEq ι
s : Finset ι
n : ℕ
f : ι → ℕ
⊢ (∃ s_1, (∀ a ∈ s_1, a ∈ s) ∧ s_1.card = n ∧ ∀ (x : ι), Multiset.count x s_1 = f x) ↔
s.sum f = n ∧ ∀ (i : ι), ¬f i = 0 → i ∈ s |
constructor | h | (s.sum f = n ∧ ∀ (i : ι), ¬f i = 0 → i ∈ s) →
∃ s_1, (∀ a ∈ s_1, a ∈ s) ∧ s_1.card = n ∧ ∀ (x : ι), Multiset.count x s_1 = f x | ι : Type u_1
inst✝ : DecidableEq ι
s : Finset ι
n : ℕ
f : ι → ℕ
h : (∃ s_1, (∀ a ∈ s_1, a ∈ s) ∧ s_1.card = n ∧ ∀ (x : ι), Multiset.count x s_1 = f x) →
s.sum f = n ∧ ∀ (i : ι), ¬f i = 0 → i ∈ s
⊢ (∃ s_1, (∀ a ∈ s_1, a ∈ s) ∧ s_1.card = n ∧ ∀ (x : ι), Multiset.count x s_1 = f x) ↔
s.sum f = n ∧ ∀ (i : ι), ¬f i = 0 → i ∈ s |
refine ⟨∑ a ∈ s, f a • {a}, ?_, ?_⟩ | h | ∀ a ∈ ∑ a ∈ s, f a • {a}, a ∈ s | ι : Type u_1
inst✝ : DecidableEq ι
s : Finset ι
f : ι → ℕ
hf : ∀ (i : ι), ¬f i = 0 → i ∈ s
⊢ ∃ s_1, (∀ a ∈ s_1, a ∈ s) ∧ s_1.card = s.sum f ∧ ∀ (x : ι), Multiset.count x s_1 = f x |
refine ⟨∑ a ∈ s, f a • {a}, ?_, ?_⟩ | h | (∑ a ∈ s, f a • {a}).card = s.sum f ∧ ∀ (x : ι), Multiset.count x (∑ a ∈ s, f a • {a}) = f x | ι : Type u_1
inst✝ : DecidableEq ι
s : Finset ι
f : ι → ℕ
hf : ∀ (i : ι), ¬f i = 0 → i ∈ s
h : ∀ a ∈ ∑ a ∈ s, f a • {a}, a ∈ s
⊢ ∃ s_1, (∀ a ∈ s_1, a ∈ s) ∧ s_1.card = s.sum f ∧ ∀ (x : ι), Multiset.count x s_1 = f x |
by_cases hp : p = 0 | pos | p.mirror.natDegree = p.natDegree | R : Type u_1
inst✝ : Semiring R
p : R[X]
⊢ p.mirror.natDegree = p.natDegree |
by_cases hp : p = 0 | neg | p.mirror.natDegree = p.natDegree | R : Type u_1
inst✝ : Semiring R
p : R[X]
pos : p.mirror.natDegree = p.natDegree
⊢ p.mirror.natDegree = p.natDegree |
hp, | pos | (mirror 0).natDegree = natDegree 0 | R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : p = 0
⊢ p.mirror.natDegree = p.natDegree |
mirror_zero | pos | natDegree 0 = natDegree 0 | R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : p = 0
⊢ (mirror 0).natDegree = natDegree 0 |
rw [mirror, natDegree_mul', reverse_natDegree, natDegree_X_pow,
tsub_add_cancel_of_le p.natTrailingDegree_le_natDegree] | rw | p.reverse.leadingCoeff * (X ^ p.natTrailingDegree).leadingCoeff ≠ 0 | R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : ¬p = 0
a✝ : Nontrivial R
⊢ p.mirror.natDegree = p.natDegree |
mirror, | this | (p.reverse * X ^ p.natTrailingDegree).natDegree = p.natDegree | R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : ¬p = 0
a✝ : Nontrivial R
⊢ p.mirror.natDegree = p.natDegree |
natDegree_mul', | this | p.reverse.natDegree + (X ^ p.natTrailingDegree).natDegree = p.natDegree | R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : ¬p = 0
a✝ : Nontrivial R
⊢ (p.reverse * X ^ p.natTrailingDegree).natDegree = p.natDegree |
natDegree_mul', | this | p.reverse.leadingCoeff * (X ^ p.natTrailingDegree).leadingCoeff ≠ 0 | R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : ¬p = 0
a✝ : Nontrivial R
this : p.reverse.natDegree + (X ^ p.natTrailingDegree).natDegree = p.natDegree
⊢ (p.reverse * X ^ p.natTrailingDegree).natDegree = p.natDegree |
leadingCoeff_X_pow, | this | p.reverse.leadingCoeff * 1 ≠ 0 | R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : ¬p = 0
a✝ : Nontrivial R
⊢ p.reverse.leadingCoeff * (X ^ p.natTrailingDegree).leadingCoeff ≠ 0 |
mul_one, | this | p.reverse.leadingCoeff ≠ 0 | R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : ¬p = 0
a✝ : Nontrivial R
⊢ p.reverse.leadingCoeff * 1 ≠ 0 |
reverse_leadingCoeff, | this | p.trailingCoeff ≠ 0 | R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : ¬p = 0
a✝ : Nontrivial R
⊢ p.reverse.leadingCoeff ≠ 0 |
Ne, | this | ¬p.trailingCoeff = 0 | R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : ¬p = 0
a✝ : Nontrivial R
⊢ p.trailingCoeff ≠ 0 |
trailingCoeff_eq_zero | trailingCoeff_eq_zero | ¬p = 0 | R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : ¬p = 0
a✝ : Nontrivial R
⊢ ¬p.trailingCoeff = 0 |
by_cases hp : p = 0 | pos | p.mirror.natTrailingDegree = p.natTrailingDegree | R : Type u_1
inst✝ : Semiring R
p : R[X]
⊢ p.mirror.natTrailingDegree = p.natTrailingDegree |
by_cases hp : p = 0 | neg | p.mirror.natTrailingDegree = p.natTrailingDegree | R : Type u_1
inst✝ : Semiring R
p : R[X]
pos : p.mirror.natTrailingDegree = p.natTrailingDegree
⊢ p.mirror.natTrailingDegree = p.natTrailingDegree |
hp, | pos | (mirror 0).natTrailingDegree = natTrailingDegree 0 | R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : p = 0
⊢ p.mirror.natTrailingDegree = p.natTrailingDegree |
mirror_zero | pos | natTrailingDegree 0 = natTrailingDegree 0 | R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : p = 0
⊢ (mirror 0).natTrailingDegree = natTrailingDegree 0 |
mirror, | neg | (p.reverse * X ^ p.natTrailingDegree).natTrailingDegree = p.natTrailingDegree | R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : ¬p = 0
⊢ p.mirror.natTrailingDegree = p.natTrailingDegree |
natTrailingDegree_mul_X_pow ((mt reverse_eq_zero.mp) hp), | neg | p.reverse.natTrailingDegree + p.natTrailingDegree = p.natTrailingDegree | R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : ¬p = 0
⊢ (p.reverse * X ^ p.natTrailingDegree).natTrailingDegree = p.natTrailingDegree |
natTrailingDegree_reverse, | neg | 0 + p.natTrailingDegree = p.natTrailingDegree | R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : ¬p = 0
⊢ p.reverse.natTrailingDegree + p.natTrailingDegree = p.natTrailingDegree |
zero_add | neg | p.natTrailingDegree = p.natTrailingDegree | R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : ¬p = 0
⊢ 0 + p.natTrailingDegree = p.natTrailingDegree |
by_cases h2 : p.natDegree < n | pos | p.mirror.coeff n = p.coeff ((revAt (p.natDegree + p.natTrailingDegree) : ℕ → ℕ) n) | R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
⊢ p.mirror.coeff n = p.coeff ((revAt (p.natDegree + p.natTrailingDegree) : ℕ → ℕ) n) |
by_cases h2 : p.natDegree < n | neg | p.mirror.coeff n = p.coeff ((revAt (p.natDegree + p.natTrailingDegree) : ℕ → ℕ) n) | R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
pos : p.mirror.coeff n = p.coeff ((revAt (p.natDegree + p.natTrailingDegree) : ℕ → ℕ) n)
⊢ p.mirror.coeff n = p.coeff ((revAt (p.natDegree + p.natTrailingDegree) : ℕ → ℕ) n) |
rw [coeff_eq_zero_of_natDegree_lt (by rwa [mirror_natDegree])] | pos | 0 = p.coeff ((revAt (p.natDegree + p.natTrailingDegree) : ℕ → ℕ) n) | R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
⊢ p.mirror.coeff n = p.coeff ((revAt (p.natDegree + p.natTrailingDegree) : ℕ → ℕ) n) |
coeff_eq_zero_of_natDegree_lt (by rwa [mirror_natDegree]) | pos | 0 = p.coeff ((revAt (p.natDegree + p.natTrailingDegree) : ℕ → ℕ) n) | R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
⊢ p.mirror.coeff n = p.coeff ((revAt (p.natDegree + p.natTrailingDegree) : ℕ → ℕ) n) |
mirror_natDegree | mirror_natDegree | p.natDegree < n | R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
⊢ p.mirror.natDegree < n |
by_cases h1 : n ≤ p.natDegree + p.natTrailingDegree | pos | 0 = p.coeff ((revAt (p.natDegree + p.natTrailingDegree) : ℕ → ℕ) n) | R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
⊢ 0 = p.coeff ((revAt (p.natDegree + p.natTrailingDegree) : ℕ → ℕ) n) |
by_cases h1 : n ≤ p.natDegree + p.natTrailingDegree | neg | 0 = p.coeff ((revAt (p.natDegree + p.natTrailingDegree) : ℕ → ℕ) n) | R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
pos : 0 = p.coeff ((revAt (p.natDegree + p.natTrailingDegree) : ℕ → ℕ) n)
⊢ 0 = p.coeff ((revAt (p.natDegree + p.natTrailingDegree) : ℕ → ℕ) n) |
rw [revAt_le h1, coeff_eq_zero_of_lt_natTrailingDegree] | pos | p.natDegree + p.natTrailingDegree - n < p.natTrailingDegree | R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
h1 : n ≤ p.natDegree + p.natTrailingDegree
⊢ 0 = p.coeff ((revAt (p.natDegree + p.natTrailingDegree) : ℕ → ℕ) n) |
revAt_le h1, | pos | 0 = p.coeff (p.natDegree + p.natTrailingDegree - n) | R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
h1 : n ≤ p.natDegree + p.natTrailingDegree
⊢ 0 = p.coeff ((revAt (p.natDegree + p.natTrailingDegree) : ℕ → ℕ) n) |
coeff_eq_zero_of_lt_natTrailingDegree | pos | 0 = 0 | R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
h1 : n ≤ p.natDegree + p.natTrailingDegree
⊢ 0 = p.coeff (p.natDegree + p.natTrailingDegree - n) |
coeff_eq_zero_of_lt_natTrailingDegree | pos | p.natDegree + p.natTrailingDegree - n < p.natTrailingDegree | R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
h1 : n ≤ p.natDegree + p.natTrailingDegree
pos : 0 = 0
⊢ 0 = p.coeff (p.natDegree + p.natTrailingDegree - n) |
← revAtFun_eq, | neg | 0 = p.coeff (revAtFun (p.natDegree + p.natTrailingDegree) n) | R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
h1 : ¬n ≤ p.natDegree + p.natTrailingDegree
⊢ 0 = p.coeff ((revAt (p.natDegree + p.natTrailingDegree) : ℕ → ℕ) n) |
revAtFun, | neg | 0 = p.coeff (if n ≤ p.natDegree + p.natTrailingDegree then p.natDegree + p.natTrailingDegree - n else n) | R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
h1 : ¬n ≤ p.natDegree + p.natTrailingDegree
⊢ 0 = p.coeff (revAtFun (p.natDegree + p.natTrailingDegree) n) |
if_neg h1, | neg | 0 = p.coeff n | R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
h1 : ¬n ≤ p.natDegree + p.natTrailingDegree
⊢ 0 = p.coeff (if n ≤ p.natDegree + p.natTrailingDegree then p.natDegree + p.natTrailingDegree - n else n) |
coeff_eq_zero_of_natDegree_lt h2 | neg | 0 = 0 | R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
h1 : ¬n ≤ p.natDegree + p.natTrailingDegree
⊢ 0 = p.coeff n |
rw [revAt_le (h2.trans (Nat.le_add_right _ _))] | neg | p.mirror.coeff n = p.coeff (p.natDegree + p.natTrailingDegree - n) | R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : n ≤ p.natDegree
⊢ p.mirror.coeff n = p.coeff ((revAt (p.natDegree + p.natTrailingDegree) : ℕ → ℕ) n) |
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