mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: M : Type u_4
N : Type u_5
R₂ : Type u
inst✝⁵ : CommSemiring R₂
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R₂ M
inst✝² : PartialOrder N
inst✝¹ : AddCommMonoid N
inst✝ : Module R₂ N
Q : QuadraticMap R₂ M N
hQ : Q.PosDef
x : M
hQx : Q x = 0
hx : ¬x = 0
⊢ 0 < 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.DirectSum.Internal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝⁵ : DecidableEq ι
inst✝⁴ : Semiring R
inst✝³ : SetLike σ R
inst✝² : AddSubmonoidClass σ R
A : ι → σ
inst✝¹ : AddMonoid ι
inst✝ : SetLike.GradedMonoid A
r : ⨁ (i : ι), ↥(A i)
i : ι
r' : ↥(A i)
j n : ι
H : ∀ (x : ι), x + i = n ↔ x = j
⊢ ↑((r * (of (fun i ↦ ↥(A i)) i) r') n) = ↑(r j) * ↑r'

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: M : Type u_4
N : Type u_5
R₂ : Type u
inst✝⁵ : CommSemiring R₂
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R₂ M
inst✝² : PartialOrder N
inst✝¹ : AddCommMonoid N
inst✝ : Module R₂ N
Q : QuadraticMap R₂ M N
hQ : Q.PosDef
x : M
h : x = 0
⊢ Q x ≤ 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.DirectSum.Internal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝⁵ : DecidableEq ι
inst✝⁴ : Semiring R
inst✝³ : SetLike σ R
inst✝² : AddSubmonoidClass σ R
A : ι → σ
inst✝¹ : AddMonoid ι
inst✝ : SetLike.GradedMonoid A
r : ↥(A 0)
r' : ⨁ (i : ι), ↥(A i)
j _x : ι
⊢ 0 + _x = j ↔ _x = j

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ (-ι Q) m * (-ι Q) m = (algebraMap R (CliffordAlgebra Q)) (Q m)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
⊢ involute.comp involute = AlgHom.id R (CliffordAlgebra Q)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ unop ((↑(opLinearEquiv R) ∘ₗ ι Q) m * (↑(opLinearEquiv R) ∘ₗ ι Q) m) =
    unop ((algebraMap R (CliffordAlgebra Q)ᵐᵒᵖ) (Q m))

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x✝ : M
⊢ ((AlgHom.unop (reverseOp.comp (AlgHom.opComm reverseOp))).toLinearMap ∘ₗ ι Q) x✝ =
    ((AlgHom.unop (AlgHom.id R (CliffordAlgebra Q)ᵐᵒᵖ)).toLinearMap ∘ₗ ι Q) x✝

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x✝ : M
⊢ (((AlgHom.opComm reverseOp).comp reverseOp).toLinearMap ∘ₗ ι Q) x✝ =
    ((AlgHom.id R (CliffordAlgebra Q)).toLinearMap ∘ₗ ι Q) x✝

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ reverse ((ι Q) m) = (ι Q) m

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: M : Type u_4
N : Type u_5
R₂ : Type u
inst✝⁵ : CommSemiring R₂
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R₂ M
inst✝² : PartialOrder N
inst✝¹ : AddCommMonoid N
inst✝ : Module R₂ N
Q : QuadraticMap R₂ M N
hQ : Q.PosDef
x : M
⊢ Q x ≤ 0 ↔ x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.DirectSum.Internal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝⁵ : DecidableEq ι
inst✝⁴ : Semiring R
inst✝³ : SetLike σ R
inst✝² : AddSubmonoidClass σ R
A : ι → σ
inst✝¹ : AddMonoid ι
inst✝ : SetLike.GradedMonoid A
r : ⨁ (i : ι), ↥(A i)
r' : ↥(A 0)
j _x : ι
⊢ _x + 0 = j ↔ _x = j

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: R : Type u_3
n : Type w
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : CommRing R
inst✝ : Invertible 2
a : R
Q : QuadraticForm R (n → R)
⊢ (a • Q).toMatrix' = a • Q.toMatrix'

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
⊢ reverse ∘ₗ involute.toLinearMap = involute.toLinearMap ∘ₗ reverse

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: R : Type u_3
n : Type w
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : CommRing R
inst✝ : Invertible 2
Q : QuadraticForm R (n → R)
⊢ Q.toMatrix'.IsSymm

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
⊢ reverse (List.map ⇑(ι Q) []).prod = (List.map ⇑(ι Q) []).reverse.prod

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: R : Type u_3
n : Type w
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq n
inst✝³ : CommRing R
inst✝² : Invertible 2
m : Type w
inst✝¹ : DecidableEq m
inst✝ : Fintype m
Q : QuadraticForm R (m → R)
f : (n → R) →ₗ[R] m → R
⊢ toMatrix' (QuadraticMap.comp Q f) = (LinearMap.toMatrix' f)ᵀ * Q.toMatrix' * LinearMap.toMatrix' f

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x : M
xs : List M
⊢ reverse (List.map (⇑(ι Q)) (x :: xs)).prod = (List.map (⇑(ι Q)) (x :: xs)).reverse.prod

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: R : Type u_3
n : Type w
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : CommRing R
inst✝ : Invertible 2
Q : QuadraticForm R (n → R)
⊢ toMatrix (Pi.basisFun R n) Q = Q.toMatrix'

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
⊢ involute (List.map ⇑(ι Q) []).prod = (-1) ^ [].length • (List.map ⇑(ι Q) []).prod

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: R : Type u_3
N : Type u_5
n : Type w
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq n
inst✝³ : CommRing R
inst✝² : Invertible 2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
b : Basis n R N
a : R
Q : QuadraticForm R N
⊢ toMatrix b (a • Q) = a • toMatrix b Q

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x : M
xs : List M
⊢ involute (List.map (⇑(ι Q)) (x :: xs)).prod = (-1) ^ (x :: xs).length • (List.map (⇑(ι Q)) (x :: xs)).prod

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
⊢ Submodule.comap involute.toLinearMap (ι Q).range = (ι Q).range

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
n : ZMod 2
⊢ Submodule.map involute.toLinearMap (evenOdd Q n) = evenOdd Q n

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: R : Type u_3
N : Type u_5
n : Type w
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq n
inst✝³ : CommRing R
inst✝² : Invertible 2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
b : Basis n R N
Q : QuadraticForm R N
⊢ (toMatrix b Q).IsSymm

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
n : ZMod 2
⊢ Submodule.comap involute.toLinearMap (evenOdd Q n) = evenOdd Q n

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: R : Type u_3
N : Type u_5
P : Type u_6
n : Type w
inst✝⁹ : Fintype n
inst✝⁸ : DecidableEq n
inst✝⁷ : CommRing R
inst✝⁶ : Invertible 2
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
b : Basis n R N
m : Type w
inst✝³ : DecidableEq m
inst✝² : Fintype m
inst✝¹ : AddCommGroup P
inst✝ : Module R P
b' : Basis m R P
Q : QuadraticForm R P
f : N →ₗ[R] P
⊢ toMatrix b (QuadraticMap.comp Q f) = ((LinearMap.toMatrix b b') f)ᵀ * toMatrix b' Q * (LinearMap.toMatrix b b') f

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: R : Type u_3
n : Type w
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : CommRing R
inst✝ : Invertible 2
Q : QuadraticForm R (n → R)
a : R
⊢ (a • Q).discr' = a ^ Fintype.card n * Q.discr'

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.DirectSum.Internal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝⁷ : DecidableEq ι
inst✝⁶ : Semiring R
inst✝⁵ : SetLike σ R
inst✝⁴ : AddSubmonoidClass σ R
A : ι → σ
inst✝³ : AddCommMonoid ι
inst✝² : PartialOrder ι
inst✝¹ : CanonicallyOrderedAdd ι
inst✝ : SetLike.GradedMonoid A
i : ι
r : ↥(A i)
r' : ⨁ (i : ι), ↥(A i)
n : ι
h : ¬i ≤ n
⊢ ↑(((of (fun i ↦ ↥(A i)) i) r * r') n) = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: R : Type u_3
n : Type w
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : CommRing R
inst✝ : Invertible 2
Q : QuadraticForm R (n → R)
f : (n → R) →ₗ[R] n → R
⊢ discr' (QuadraticMap.comp Q f) = (LinearMap.toMatrix' f).det * (LinearMap.toMatrix' f).det * Q.discr'

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.DirectSum.Internal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝⁷ : DecidableEq ι
inst✝⁶ : Semiring R
inst✝⁵ : SetLike σ R
inst✝⁴ : AddSubmonoidClass σ R
A : ι → σ
inst✝³ : AddCommMonoid ι
inst✝² : PartialOrder ι
inst✝¹ : CanonicallyOrderedAdd ι
inst✝ : SetLike.GradedMonoid A
r : ⨁ (i : ι), ↥(A i)
i : ι
r' : ↥(A i)
n : ι
h : ¬i ≤ n
⊢ ↑((r * (of (fun i ↦ ↥(A i)) i) r') n) = 0

Tactic:
coe_mul_apply_eq_dfinsuppSum,

Goal state after:
ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝⁷ : DecidableEq ι
inst✝⁶ : Semiring R
inst✝⁵ : SetLike σ R
inst✝⁴ : AddSubmonoidClass σ R
A : ι → σ
inst✝³ : AddCommMonoid ι
inst✝² : PartialOrder ι
inst✝¹ : CanonicallyOrderedAdd ι
inst✝ : SetLike.GradedMonoid A
r : ⨁ (i : ι), ↥(A i)
i : ι
r' : ↥(A i)
n : ι
h : ¬i ≤ n
⊢ (DFinsupp.sum r fun i_1 ri ↦
      DFinsupp.sum ((of (fun i ↦ ↥(A i)) i) r') fun j rj ↦ if i_1 + j = n then ↑ri * ↑rj else 0) =
    0
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: R : Type u_3
N : Type u_5
n : Type w
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq n
inst✝³ : CommRing R
inst✝² : Invertible 2
inst✝¹ : AddCommGroup N
inst✝ : Module R N
b : Basis n R N
Q : QuadraticForm R N
a : R
⊢ discr b (a • Q) = a ^ Fintype.card n * discr b Q

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
⊢ Submodule.map reverse (ι Q).range = (ι Q).range

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: R : Type u_3
N : Type u_5
P : Type u_6
n : Type w
inst✝⁷ : Fintype n
inst✝⁶ : DecidableEq n
inst✝⁵ : CommRing R
inst✝⁴ : Invertible 2
inst✝³ : AddCommGroup N
inst✝² : Module R N
b : Basis n R N
inst✝¹ : AddCommGroup P
inst✝ : Module R P
b' : Basis n R P
Q : QuadraticForm R P
f : N →ₗ[R] P
⊢ discr b (QuadraticMap.comp Q f) = ((LinearMap.toMatrix b b') f).det * ((LinearMap.toMatrix b b') f).det * discr b' Q

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
⊢ Submodule.comap reverse (ι Q).range = (ι Q).range

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: R : Type u_3
n : Type w
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : CommRing R
inst✝ : Invertible 2
Q : QuadraticForm R (n → R)
⊢ discr (Pi.basisFun R n) Q = Q.discr'

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: R : Type u_3
M : Type u_4
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
htwo : Invertible 2
B : BilinForm R M
hB₁ : B ≠ 0
hB₂ : IsSymm B
⊢ ∃ x, (B x) x ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: V : Type u
K : Type v
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
hK : Invertible 2
B : BilinForm K V
hB₂ : IsSymm B
this : ∀ (d : ℕ), finrank K V = d → ∃ v, IsOrthoᵢ B ⇑v
⊢ ∃ v, IsOrthoᵢ B ⇑v

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
p q : Submodule R (CliffordAlgebra Q)
⊢ Submodule.map reverse (p * q) = Submodule.map reverse q * Submodule.map reverse p

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
p q : Submodule R (CliffordAlgebra Q)
⊢ Submodule.comap reverse (p * q) = Submodule.comap reverse q * Submodule.comap reverse p

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
p : Submodule R (CliffordAlgebra Q)
n : ℕ
⊢ Submodule.map reverse (p ^ n) = Submodule.map reverse p ^ n

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
p : Submodule R (CliffordAlgebra Q)
n : ℕ
⊢ Submodule.map reverse (p ^ n) = Submodule.map reverse p ^ n

Tactic:
submodule_map_pow_reverse

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
p : Submodule R (CliffordAlgebra Q)
n : ℕ
⊢ Submodule.comap reverse (p ^ n) = Submodule.comap reverse p ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.DirectSum.Internal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝⁷ : DecidableEq ι
inst✝⁶ : Semiring R
inst✝⁵ : SetLike σ R
inst✝⁴ : AddSubmonoidClass σ R
A : ι → σ
inst✝³ : AddCommMonoid ι
inst✝² : PartialOrder ι
inst✝¹ : CanonicallyOrderedAdd ι
inst✝ : SetLike.GradedMonoid A
r : ⨁ (i : ι), ↥(A i)
i : ι
r' : ↥(A i)
n : ι
h : ¬i ≤ n
⊢ ↑((r * (of (fun i ↦ ↥(A i)) i) r') n) = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.DirectSum.Internal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝¹⁰ : DecidableEq ι
inst✝⁹ : Semiring R
inst✝⁸ : SetLike σ R
inst✝⁷ : AddSubmonoidClass σ R
A : ι → σ
inst✝⁶ : AddCommMonoid ι
inst✝⁵ : PartialOrder ι
inst✝⁴ : CanonicallyOrderedAdd ι
inst✝³ : SetLike.GradedMonoid A
inst✝² : Sub ι
inst✝¹ : OrderedSub ι
inst✝ : AddLeftReflectLE ι
i : ι
r : ↥(A i)
r' : ⨁ (i : ι), ↥(A i)
n : ι
h : i ≤ n
x : ι
⊢ i + x = n ↔ x = n - i

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
n : ZMod 2
⊢ Submodule.map reverse (evenOdd Q n) = evenOdd Q n

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.DirectSum.Internal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝¹¹ : DecidableEq ι
inst✝¹⁰ : Semiring R
inst✝⁹ : SetLike σ R
inst✝⁸ : AddSubmonoidClass σ R
A : ι → σ
inst✝⁷ : AddCommMonoid ι
inst✝⁶ : PartialOrder ι
inst✝⁵ : CanonicallyOrderedAdd ι
inst✝⁴ : SetLike.GradedMonoid A
inst✝³ : Sub ι
inst✝² : OrderedSub ι
inst✝¹ : AddLeftReflectLE ι
r : ⨁ (i : ι), ↥(A i)
i : ι
r' : ↥(A i)
n : ι
inst✝ : Decidable (i ≤ n)
⊢ ↑((r * (of (fun i ↦ ↥(A i)) i) r') n) = if i ≤ n then ↑(r (n - i)) * ↑r' else 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.DirectSum.Internal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝¹¹ : DecidableEq ι
inst✝¹⁰ : Semiring R
inst✝⁹ : SetLike σ R
inst✝⁸ : AddSubmonoidClass σ R
A : ι → σ
inst✝⁷ : AddCommMonoid ι
inst✝⁶ : PartialOrder ι
inst✝⁵ : CanonicallyOrderedAdd ι
inst✝⁴ : SetLike.GradedMonoid A
inst✝³ : Sub ι
inst✝² : OrderedSub ι
inst✝¹ : AddLeftReflectLE ι
i : ι
r : ↥(A i)
r' : ⨁ (i : ι), ↥(A i)
n : ι
inst✝ : Decidable (i ≤ n)
⊢ ↑(((of (fun i ↦ ↥(A i)) i) r * r') n) = if i ≤ n then ↑r * ↑(r' (n - i)) else 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
n : ZMod 2
⊢ Submodule.comap reverse (evenOdd Q n) = evenOdd Q n

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.DirectSum.Internal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
S : Type u_3
R : Type u_4
inst✝³ : DecidableEq ι
inst✝² : CommSemiring S
inst✝¹ : Semiring R
inst✝ : Algebra S R
p : Submodule S R
⊢ 1 ∈ p ^ 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.DirectSum.Internal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
S : Type u_3
R : Type u_4
inst✝³ : DecidableEq ι
inst✝² : CommSemiring S
inst✝¹ : Semiring R
inst✝ : Algebra S R
p✝ : Submodule S R
i j : ℕ
p q : R
hp : p ∈ p✝ ^ i
hq : q ∈ p✝ ^ j
⊢ p * q ∈ p✝ ^ (i + j)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.DirectSum.Internal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝⁷ : AddCommMonoid ι
inst✝⁶ : LinearOrder ι
inst✝⁵ : IsOrderedAddMonoid ι
inst✝⁴ : DecidableEq ι
inst✝³ : Semiring R
inst✝² : SetLike σ R
inst✝¹ : AddSubmonoidClass σ R
A : ι → σ
inst✝ : SetLike.GradedMonoid A
r r' : ⨁ (i : ι), ↥(A i)
m n : ι
hr : ∀ i < m, r i = 0
hr' : ∀ i < n, r' i = 0
k : ι
hk : k < m + n
x : ι × ι
hx : x ∈ {ij ∈ DFinsupp.support r ×ˢ DFinsupp.support r' | ij.1 + ij.2 = k}
hm : m ≤ x.1
hn : n ≤ x.2
⊢ x.1 + x.2 = k

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.DirectSum.Internal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝⁷ : AddCommMonoid ι
inst✝⁶ : LinearOrder ι
inst✝⁵ : IsOrderedAddMonoid ι
inst✝⁴ : DecidableEq ι
inst✝³ : Semiring R
inst✝² : SetLike σ R
inst✝¹ : AddSubmonoidClass σ R
A : ι → σ
inst✝ : SetLike.GradedMonoid A
r r' : ⨁ (i : ι), ↥(A i)
m n : ι
hr : ∀ i < m, r i = 0
hr' : ∀ i < n, r' i = 0
x : ι × ι
hm : m ≤ x.1
hn : n ≤ x.2
hk : x.1 + x.2 < m + n
hx : x ∈ {ij ∈ DFinsupp.support r ×ˢ DFinsupp.support r' | ij.1 + ij.2 = x.1 + x.2}
⊢ m + n ≤ x.1 + x.2

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.DirectSum.Internal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝⁷ : AddCommMonoid ι
inst✝⁶ : LinearOrder ι
inst✝⁵ : IsOrderedAddMonoid ι
inst✝⁴ : DecidableEq ι
inst✝³ : Semiring R
inst✝² : SetLike σ R
inst✝¹ : AddSubmonoidClass σ R
A : ι → σ
inst✝ : SetLike.GradedMonoid A
r r' : ⨁ (i : ι), ↥(A i)
m n : ι
hr : ∀ i < m, r i = 0
hr' : ∀ i < n, r' i = 0
k : ι
hk : k < m + n
x : ι × ι
hx : x ∈ {ij ∈ DFinsupp.support r ×ˢ DFinsupp.support r' | ij.1 + ij.2 = k}
⊢ x.1 < m ∨ x.2 < n

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.DirectSum.Internal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝⁷ : AddCommMonoid ι
inst✝⁶ : LinearOrder ι
inst✝⁵ : IsOrderedAddMonoid ι
inst✝⁴ : DecidableEq ι
inst✝³ : Semiring R
inst✝² : SetLike σ R
inst✝¹ : AddSubmonoidClass σ R
A : ι → σ
inst✝ : SetLike.GradedMonoid A
r r' : ⨁ (i : ι), ↥(A i)
m n : ι
hr : ∀ i < m, r i = 0
hr' : ∀ i < n, r' i = 0
k : ι
hk : k < m + n
⊢ (r * r') k = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x : CliffordAlgebra Q
h : x ∈ evenOdd Q 0
⊢ involute x = x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.DirectSum.Internal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝⁸ : AddCommMonoid ι
inst✝⁷ : LinearOrder ι
inst✝⁶ : IsOrderedAddMonoid ι
inst✝⁵ : DecidableEq ι
inst✝⁴ : Semiring R
inst✝³ : SetLike σ R
inst✝² : AddSubmonoidClass σ R
A : ι → σ
inst✝¹ : SetLike.GradedMonoid A
inst✝ : CanonicallyOrderedAdd ι
l : List ((⨁ (i : ι), ↥(A i)) × ι)
hl : ∀ xn ∈ l, ∀ k < xn.2, xn.1 k = 0
n : ι
hn : n < (List.map Prod.snd l).sum
⊢ (List.map Prod.fst l).prod n = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.DirectSum.Internal

Goal state before:
pp: case cons
ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝⁸ : AddCommMonoid ι
inst✝⁷ : LinearOrder ι
inst✝⁶ : IsOrderedAddMonoid ι
inst✝⁵ : DecidableEq ι
inst✝⁴ : Semiring R
inst✝³ : SetLike σ R
inst✝² : AddSubmonoidClass σ R
A : ι → σ
inst✝¹ : SetLike.GradedMonoid A
inst✝ : CanonicallyOrderedAdd ι
m : ι
head : ⨁ (i : ι), ↥(A i)
tail : List (⨁ (i : ι), ↥(A i))
ih : (∀ x ∈ tail, ∀ k < m, x k = 0) → ∀ ⦃n : ι⦄, n < tail.length • m → tail.prod n = 0
hl : ∀ x ∈ head :: tail, ∀ k < m, x k = 0
n : ι
hn : n < (head :: tail).length • m
⊢ (head :: tail).prod n = 0

Tactic:
simp only [List.mem_cons, forall_eq_or_imp, List.length_cons, List.prod_cons] at hl hn ⊢
    refine mul_apply_eq_zero hl.1 (ih hl.2) ?_
    simpa [add_smul, add_comm m] using hn

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x : CliffordAlgebra Q
h : x ∈ evenOdd Q 1
⊢ involute x = -x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.DirectSum.Internal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝⁸ : AddCommMonoid ι
inst✝⁷ : LinearOrder ι
inst✝⁶ : IsOrderedAddMonoid ι
inst✝⁵ : DecidableEq ι
inst✝⁴ : Semiring R
inst✝³ : SetLike σ R
inst✝² : AddSubmonoidClass σ R
A : ι → σ
inst✝¹ : SetLike.GradedMonoid A
inst✝ : CanonicallyOrderedAdd ι
l : List (⨁ (i : ι), ↥(A i))
m : ι
hl : ∀ x ∈ l, ∀ k < m, x k = 0
n : ι
hn : n < l.length • m
⊢ l.prod n = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.DirectSum.Internal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝⁸ : AddCommMonoid ι
inst✝⁷ : LinearOrder ι
inst✝⁶ : IsOrderedAddMonoid ι
inst✝⁵ : DecidableEq ι
inst✝⁴ : CanonicallyOrderedAdd ι
inst✝³ : CommSemiring R
inst✝² : SetLike σ R
inst✝¹ : AddSubmonoidClass σ R
A : ι → σ
inst✝ : SetLike.GradedMonoid A
s : Multiset ((⨁ (i : ι), ↥(A i)) × ι)
hs : ∀ xn ∈ s, ∀ k < xn.2, xn.1 k = 0
n : ι
hn : n < (Multiset.map Prod.snd s).sum
⊢ ∀ xn ∈ s.toList, ∀ k < xn.2, xn.1 k = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.DirectSum.Internal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝⁸ : AddCommMonoid ι
inst✝⁷ : LinearOrder ι
inst✝⁶ : IsOrderedAddMonoid ι
inst✝⁵ : DecidableEq ι
inst✝⁴ : CanonicallyOrderedAdd ι
inst✝³ : CommSemiring R
inst✝² : SetLike σ R
inst✝¹ : AddSubmonoidClass σ R
A : ι → σ
inst✝ : SetLike.GradedMonoid A
s : Multiset ((⨁ (i : ι), ↥(A i)) × ι)
hs : ∀ xn ∈ s, ∀ k < xn.2, xn.1 k = 0
n : ι
hn : n < (Multiset.map Prod.snd s).sum
⊢ ?m.66 < (List.map Prod.snd s.toList).sum

Tactic:
by

Goal state after: