mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: case a
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
this✝ : ⨅ s, ⨅ (_ : span R s = p), s.encard = (⇑toENat ∘ ofENat) (⨅ s, ⨅ (_ : span R s = p), s.encard)
s : { s // span R s = p }
this : ↑(⨅ s', ⨅ (_ : span R s' = p), s'.encard) ≤ ↑(↑s).encard
⊢ ↑(toENat #↑↑s) ≤ #↑↑s

Tactic:
]

Goal state after:
case a
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
this✝ : ⨅ s, ⨅ (_ : span R s = p), s.encard = (⇑toENat ∘ ofENat) (⨅ s, ⨅ (_ : span R s = p), s.encard)
s : { s // span R s = p }
this : ↑(⨅ s', ⨅ (_ : span R s' = p), s'.encard) ≤ ↑(↑s).encard
⊢ ↑(toENat #↑↑s) ≤ #↑↑s
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap

Goal state before:
pp: ι : Type u_1
inst✝⁴ : Fintype ι
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
b : ι → M
inst✝ : DecidableEq ι
hb : Submodule.span R (Set.range b) = ⊤
f : Module.End R M
I : Ideal R
hI : LinearMap.range f ≤ I • ⊤
bM' : M → ι →₀ ↥I
hbM' : ∀ (x : M), (Ideal.finsuppTotal ι M I b) (bM' x) = f x
A : Matrix ι ι R := fun i j ↦ ↑((bM' (b j)) i)
⊢ ∀ (j : ι), ∑ i, A i j • b i = f (b j)

Tactic:
]

Goal state after:
ι : Type u_1
inst✝⁴ : Fintype ι
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
b : ι → M
inst✝ : DecidableEq ι
hb : Submodule.span R (Set.range b) = ⊤
f : Module.End R M
I : Ideal R
hI : LinearMap.range f ≤ I • ⊤
bM' : M → ι →₀ ↥I
hbM' : ∀ (x : M), (Ideal.finsuppTotal ι M I b) (bM' x) = f x
A : Matrix ι ι R := fun i j ↦ ↑((bM' (b j)) i)
⊢ ∀ (j : ι), ∑ i, A i j • b i = f (b j)
Module: Mathlib.RingTheory.Polynomial.Nilpotent

Goal state before:
pp: R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : R[X]
a b : S
hb : IsUnit ((aeval b) P)
hab : IsNilpotent (a - b)
⊢ IsUnit ((aeval b) P + ((aeval a) P - (aeval b) P))

Tactic:
]

Goal state after:
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : R[X]
a b : S
hb : IsUnit ((aeval b) P)
hab : IsNilpotent (a - b)
⊢ IsUnit ((aeval b) P + ((aeval a) P - (aeval b) P))
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
⊢ ⨅ s, ⨅ (_ : span R s = p), s.encard = ⨅ s, ↑(↑s).card

Tactic:
]

Goal state after:
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
⊢ ⨅ s, ⨅ (_ : span R s = p), s.encard = ⨅ s, ↑(↑s).card
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap

Goal state before:
pp: ι : Type u_1
inst✝⁴ : Fintype ι
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
b : ι → M
inst✝ : DecidableEq ι
hb : Submodule.span R (Set.range b) = ⊤
f : Module.End R M
I : Ideal R
hI : LinearMap.range f ≤ I • ⊤
bM' : M → ι →₀ ↥I
A : Matrix ι ι R := fun i j ↦ ↑((bM' (b j)) i)
j : ι
hbM' : ∑ i, ↑((bM' (b j)) i) • b i = f (b j)
⊢ ∑ i, ↑((bM' (b j)) i) • b i = f (b j)

Tactic:
]

Goal state after:
ι : Type u_1
inst✝⁴ : Fintype ι
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
b : ι → M
inst✝ : DecidableEq ι
hb : Submodule.span R (Set.range b) = ⊤
f : Module.End R M
I : Ideal R
hI : LinearMap.range f ≤ I • ⊤
bM' : M → ι →₀ ↥I
A : Matrix ι ι R := fun i j ↦ ↑((bM' (b j)) i)
j : ι
hbM' : ∑ i, ↑((bM' (b j)) i) • b i = f (b j)
⊢ ∑ i, ↑((bM' (b j)) i) • b i = f (b j)
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: case inl
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
h1 : ⨅ s, ⨅ (_ : span R s = p), s.encard = ⊤
⊢ ⨅ s, ↑(↑s).card = ⊤

Tactic:
]

Goal state after:
case inl
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
h1 : ⨅ s, ⨅ (_ : span R s = p), s.encard = ⊤
⊢ ⨅ s, ↑(↑s).card = ⊤
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: case neg
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
h2 : ⨅ s, ⨅ (_ : span R s = p), s.encard ≠ ⊤
s : Set M
h : span R s = p
hs : ¬s.Finite
⊢ ⨅ s, (↑↑s).encard ≤ ⊤

Tactic:
]

Goal state after:
case neg
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
h2 : ⨅ s, ⨅ (_ : span R s = p), s.encard ≠ ⊤
s : Set M
h : span R s = p
hs : ¬s.Finite
⊢ ⨅ s, (↑↑s).encard ≤ ⊤
Module: Mathlib.RingTheory.Finiteness.Subalgebra

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
I : (Submodule R A)ˣ
T T' : Finset A
hT : ↑T ⊆ ↑↑I
hT' : ↑T' ⊆ ↑↑I⁻¹
one_mem : 1 ∈ span R (↑T * ↑T')
⊢ ↑(1 * I) ≤ span R ↑T * 1

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
I : (Submodule R A)ˣ
T T' : Finset A
hT : ↑T ⊆ ↑↑I
hT' : ↑T' ⊆ ↑↑I⁻¹
one_mem : 1 ∈ span R (↑T * ↑T')
⊢ ↑(1 * I) ≤ span R ↑T * 1
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: case neg
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
h2 : ⨅ s, ⨅ (_ : span R s = p), s.encard ≠ ⊤
s : Set M
h : span R s = p
hs : ¬s.Finite
⊢ ⨅ s, (↑↑s).encard ≤ s.encard

Tactic:
rw [Set.Infinite.encard_eq hs]

Goal state after:
case neg
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
h2 : ⨅ s, ⨅ (_ : span R s = p), s.encard ≠ ⊤
s : Set M
h : span R s = p
hs : ¬s.Finite
⊢ ⨅ s, (↑↑s).encard ≤ ⊤
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
⊢ ⨅ s, #↑↑s < ℵ₀ ↔ ∃ S, S.Finite ∧ span R S = p

Tactic:
]

Goal state after:
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
⊢ ⨅ s, #↑↑s < ℵ₀ ↔ ∃ S, S.Finite ∧ span R S = p
Module: Mathlib.RingTheory.Finiteness.Subalgebra

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
I : (Submodule R A)ˣ
T T' : Finset A
hT : ↑T ⊆ ↑↑I
hT' : ↑T' ⊆ ↑↑I⁻¹
one_mem : 1 ∈ span R (↑T * ↑T')
⊢ span R (↑T * ↑T') * ↑I ≤ span R (↑T * ↑T') * ↑I

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Finiteness.Subalgebra

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
M N : Submodule R A
hm : M.FG
hn : N.FG
⊢ (Submodule.map₂ (LinearMap.mul R A) M N).FG

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
M N : Submodule R A
hm : M.FG
hn : N.FG
⊢ (Submodule.map₂ (LinearMap.mul R A) M N).FG
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
⊢ ↑(toNat p.spanRank) = p.spanRank ↔ p.spanRank < ℵ₀

Tactic:
]

Goal state after:
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
⊢ ↑(toNat p.spanRank) = p.spanRank ↔ p.spanRank < ℵ₀
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
⊢ ⨅ s_1, #↑↑s_1 ≤ #↑s

Tactic:
]

Goal state after:
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
⊢ ⨅ s_1, #↑↑s_1 ≤ #↑s
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: R : Type u_1
M : Type u
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : RankCondition R
ι : Type u
v : ι → M
hv : Function.Injective v
hs : LinearIndependent R v
x : { s // span R s = span R (Set.range v) }
this : #↑↑x = #↑(Subtype.val ⁻¹' ↑x)
⊢ #ι ≤ #↑(Subtype.val ⁻¹' ↑x)

Tactic:
]

Goal state after:
R : Type u_1
M : Type u
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : RankCondition R
ι : Type u
v : ι → M
hv : Function.Injective v
hs : LinearIndependent R v
x : { s // span R s = span R (Set.range v) }
this : #↑↑x = #↑(Subtype.val ⁻¹' ↑x)
⊢ #ι ≤ #↑(Subtype.val ⁻¹' ↑x)
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: case refine_1
R : Type u_1
M : Type u
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : RankCondition R
ι : Type u
v : ι → M
hv : Function.Injective v
hs : LinearIndependent R v
x : { s // span R s = span R (Set.range v) }
this : #↑↑x = #↑(Subtype.val ⁻¹' ↑x)
⊢ #ι ≤ #ι

pp: case refine_1.h
R : Type u_1
M : Type u
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : RankCondition R
ι : Type u
v : ι → M
hv : Function.Injective v
hs : LinearIndependent R v
x : { s // span R s = span R (Set.range v) }
this : #↑↑x = #↑(Subtype.val ⁻¹' ↑x)
⊢ Function.Injective ⇑(Module.Basis.span hs)

Tactic:
]

Goal state after:
case refine_1.h
R : Type u_1
M : Type u
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : RankCondition R
ι : Type u
v : ι → M
hv : Function.Injective v
hs : LinearIndependent R v
x : { s // span R s = span R (Set.range v) }
this : #↑↑x = #↑(Subtype.val ⁻¹' ↑x)
⊢ Function.Injective ⇑(Module.Basis.span hs)
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap

Goal state before:
pp: case inr
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
inst✝ : Module.Finite R M
I : Ideal R
h✝ : Nontrivial R
s : Finset M
hs_card : s.card = ⊤.spanFinrank
hs_span : Submodule.span R ↑s = ⊤
this : Submodule.span R (Set.range Subtype.val) = ⊤
A : ↥(isRepresentation R Subtype.val)
h : ∀ (i j : ↥s), ↑A i j ∈ I
hI : range ((isRepresentation.toEnd R Subtype.val this) A) ≤ I • ⊤
⊢ (isRepresentation.toEnd R Subtype.val this) ((aeval A) (↑A).charpoly) = (isRepresentation.toEnd R Subtype.val this) 0

Tactic:
]

Goal state after:
case inr
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
inst✝ : Module.Finite R M
I : Ideal R
h✝ : Nontrivial R
s : Finset M
hs_card : s.card = ⊤.spanFinrank
hs_span : Submodule.span R ↑s = ⊤
this : Submodule.span R (Set.range Subtype.val) = ⊤
A : ↥(isRepresentation R Subtype.val)
h : ∀ (i j : ↥s), ↑A i j ∈ I
hI : range ((isRepresentation.toEnd R Subtype.val this) A) ≤ I • ⊤
⊢ (isRepresentation.toEnd R Subtype.val this) ((aeval A) (↑A).charpoly) = (isRepresentation.toEnd R Subtype.val this) 0
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
⊢ ↑(toNat (span R s).spanRank) ≤ toENat #↑s

Tactic:
]

Goal state after:
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
⊢ ↑(toNat (span R s).spanRank) ≤ toENat #↑s
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
hs : s.Finite
⊢ ↑(span R s).spanFinrank ≤ ↑s.ncard

Tactic:
]

Goal state after:
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
hs : s.Finite
⊢ ↑(span R s).spanFinrank ≤ ↑s.ncard
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
⊢ ∃ s, #↑s = ⨅ s, #↑↑s ∧ span R s = p

Tactic:
]

Goal state after:
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
⊢ ∃ s, #↑s = ⨅ s, #↑↑s ∧ span R s = p
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
⊢ (∑ x ∈ univ.erase (Equiv.refl n), ↑↑(Equiv.Perm.sign x) * ∏ i, M.charmatrix (x i) i +
          ↑↑(Equiv.Perm.sign (Equiv.refl n)) * ∏ i, M.charmatrix ((Equiv.refl n) i) i -
        ∏ i, (X - C (M i i))).degree <
    ↑(Fintype.card n - 1)

Tactic:
]

Goal state after:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
⊢ (∑ x ∈ univ.erase (Equiv.refl n), ↑↑(Equiv.Perm.sign x) * ∏ i, M.charmatrix (x i) i +
          ↑↑(Equiv.Perm.sign (Equiv.refl n)) * ∏ i, M.charmatrix ((Equiv.refl n) i) i -
        ∏ i, (X - C (M i i))).degree <
    ↑(Fintype.card n - 1)
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap

Goal state before:
pp: case inr.e_6
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
inst✝ : Module.Finite R M
I : Ideal R
h✝ : Nontrivial R
s : Finset M
hs_card : s.card = ⊤.spanFinrank
hs_span : Submodule.span R ↑s = ⊤
this : Submodule.span R (Set.range Subtype.val) = ⊤
A : ↥(isRepresentation R Subtype.val)
h : ∀ (i j : ↥s), ↑A i j ∈ I
hI : range ((isRepresentation.toEnd R Subtype.val this) A) ≤ I • ⊤
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
⊢ (∑ x ∈ univ.erase (Equiv.refl n), ↑↑(Equiv.Perm.sign x) * ∏ i, M.charmatrix (x i) i).degree < ↑(Fintype.card n - 1)

Tactic:
← mem_degreeLT

Goal state after:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
⊢ ∑ x ∈ univ.erase (Equiv.refl n), ↑↑(Equiv.Perm.sign x) * ∏ i, M.charmatrix (x i) i ∈ degreeLT R (Fintype.card n - 1)
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
⊢ ∑ x ∈ univ.erase (Equiv.refl n), ↑↑(Equiv.Perm.sign x) * ∏ i, M.charmatrix (x i) i ∈ degreeLT R (Fintype.card n - 1)

Tactic:
]

Goal state after:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
⊢ ∑ x ∈ univ.erase (Equiv.refl n), ↑↑(Equiv.Perm.sign x) * ∏ i, M.charmatrix (x i) i ∈ degreeLT R (Fintype.card n - 1)
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
h : p.FG
s : Set M
hs₁ : #↑s = p.spanRank
hs₂ : span R s = p
this : p.spanRank = ↑p.spanFinrank
⊢ toENat ↑p.spanFinrank = ↑(toNat ↑p.spanFinrank)

Tactic:
]

Goal state after:
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
h : p.FG
s : Set M
hs₁ : #↑s = p.spanRank
hs₂ : span R s = p
this : p.spanRank = ↑p.spanFinrank
⊢ toENat ↑p.spanFinrank = ↑(toNat ↑p.spanFinrank)
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
⊢ ↑↑(Equiv.Perm.sign c) * ∏ i, M.charmatrix (c i) i ∈ degreeLT R (Fintype.card n - 1)

Tactic:
← C_eq_intCast,

Goal state after:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
⊢ C ↑↑(Equiv.Perm.sign c) * ∏ i, M.charmatrix (c i) i ∈ degreeLT R (Fintype.card n - 1)
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
⊢ ↑↑(Equiv.Perm.sign c) • ∏ i, M.charmatrix (c i) i ∈ degreeLT R (Fintype.card n - 1)

Tactic:
]

Goal state after:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
⊢ ↑↑(Equiv.Perm.sign c) • ∏ i, M.charmatrix (c i) i ∈ degreeLT R (Fintype.card n - 1)
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
⊢ (∏ i, M.charmatrix (c i) i).degree < ↑(Fintype.card n - 1)

Tactic:
]

Goal state after:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
⊢ (∏ i, M.charmatrix (c i) i).degree < ↑(Fintype.card n - 1)
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
⊢ (∏ i, M.charmatrix (c i) i).natDegree < Fintype.card n - 1

Tactic:
]

Goal state after:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
⊢ (∏ i, M.charmatrix (c i) i).natDegree < Fintype.card n - 1
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
⊢ ∑ i, (M.charmatrix (c i) i).natDegree ≤ ∑ x with c x = x, 1

Tactic:
]

Goal state after:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
⊢ ∑ i, (M.charmatrix (c i) i).natDegree ≤ ∑ x with c x = x, 1
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
⊢ ∑ i, (M.charmatrix (c i) i).natDegree ≤ ∑ a, if c a = a then 1 else 0

Tactic:
]

Goal state after:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
⊢ ∑ i, (M.charmatrix (c i) i).natDegree ≤ ∑ a, if c a = a then 1 else 0
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
k : ℕ
h : Fintype.card n - 1 ≤ k
⊢ (M.charpoly - ∏ i, (X - C (M i i))).coeff k = 0

Tactic:
]

Goal state after:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
k : ℕ
h : Fintype.card n - 1 ≤ k
⊢ (M.charpoly - ∏ i, (X - C (M i i))).coeff k = 0
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
k : ℕ
h : Fintype.card n - 1 ≤ k
⊢ Fintype.card n - 1 ≤ k

Tactic:
]

Goal state after:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
k : ℕ
h : Fintype.card n - 1 ≤ k
⊢ Fintype.card n - 1 ≤ k
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I : Submodule R M
h : I.spanRank = 0
⊢ 0 ≤ 0

Tactic:
]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: case pos
R : Type u
inst✝³ : CommRing R
n : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : Nontrivial R
M : Matrix n n R
h : Fintype.card n = 0
⊢ M.charpoly.degree = ↑0

Tactic:
]

Goal state after:
case pos
R : Type u
inst✝³ : CommRing R
n : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : Nontrivial R
M : Matrix n n R
h : Fintype.card n = 0
⊢ M.charpoly.degree = ↑0
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: case mpr
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ ⨅ s, #↑↑s = 0

Tactic:
]

Goal state after:
case mpr
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ ⨅ s, #↑↑s = 0
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: case pos
R : Type u
inst✝³ : CommRing R
n : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : Nontrivial R
M : Matrix n n R
h : Fintype.card n = 0
⊢ degree 1 = ↑0

pp: case pos
R : Type u
inst✝³ : CommRing R
n : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : Nontrivial R
M : Matrix n n R
h : Fintype.card n = 0
⊢ Fintype.card n = 0

Tactic:
]

Goal state after:
case pos
R : Type u
inst✝³ : CommRing R
n : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : Nontrivial R
M : Matrix n n R
h : Fintype.card n = 0
⊢ degree 1 = ↑0

case pos
R : Type u
inst✝³ : CommRing R
n : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : Nontrivial R
M : Matrix n n R
h : Fintype.card n = 0
⊢ Fintype.card n = 0
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
h : p.FG
⊢ ↑p.spanFinrank = p.spanRank

Tactic:
]

Goal state after:
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
h : p.FG
⊢ ↑p.spanFinrank = p.spanRank
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
hp : p.FG
⊢ p.spanRank < ℵ₀

Tactic:
]

Goal state after:
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
hp : p.FG
⊢ p.spanRank < ℵ₀
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: case neg
R : Type u
inst✝³ : CommRing R
n : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : Nontrivial R
M : Matrix n n R
h : ¬Fintype.card n = 0
⊢ (M.charpoly - ∏ i, (X - C (M i i)) + ∏ i, (X - C (M i i))).degree = ↑(Fintype.card n)

Tactic:
]

Goal state after:
case neg
R : Type u
inst✝³ : CommRing R
n : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : Nontrivial R
M : Matrix n n R
h : ¬Fintype.card n = 0
⊢ (M.charpoly - ∏ i, (X - C (M i i)) + ∏ i, (X - C (M i i))).degree = ↑(Fintype.card n)
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
⊢ p.generators ⊆ ↑(span R p.generators)

Tactic:
]

Goal state after:
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
⊢ p.generators ⊆ ↑(span R p.generators)
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
h : p.FG
heq : p.generators = ∅
⊢ p = ⊥

Tactic:
]

Goal state after:
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
h : p.FG
heq : p.generators = ∅
⊢ p = ⊥
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
n : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : Nontrivial R
M : Matrix n n R
h : ¬Fintype.card n = 0
⊢ ∑ i, (X - C (M i i)).natDegree = Fintype.card n

pp: case h
R : Type u
inst✝³ : CommRing R
n : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : Nontrivial R
M : Matrix n n R
h : ¬Fintype.card n = 0
⊢ ∏ i, (X - C (M i i)).leadingCoeff ≠ 0

Tactic:
]

Goal state after:
R : Type u
inst✝³ : CommRing R
n : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : Nontrivial R
M : Matrix n n R
h : ¬Fintype.card n = 0
⊢ ∑ i, (X - C (M i i)).natDegree = Fintype.card n

case h
R : Type u
inst✝³ : CommRing R
n : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : Nontrivial R
M : Matrix n n R
h : ¬Fintype.card n = 0
⊢ ∏ i, (X - C (M i i)).leadingCoeff ≠ 0
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
h : p.FG
heq : p.generators = ∅
⊢ ⊥ = ⊥

Tactic:
]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
n : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : Nontrivial R
M : Matrix n n R
h : ¬Fintype.card n = 0
⊢ #univ = #univ

Tactic:
]

Goal state after: