mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: case inr
α : Type u_1
M : Matroid α
C : Set α
aux : ∀ {α : Type u_1} (M : Matroid α), ∀ C ⊆ M.E, M ／ M.closure C = M ／ C ＼ (M.closure C \ C)
hCE : ¬C ⊆ M.E
⊢ M ／ (C ∩ M.E) ＼ (M.closure (C ∩ M.E) \ C) = M ／ (C ∩ M.E) ＼ (M.closure (C ∩ M.E) \ C)

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Nullstellensatz

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
σ : Type u_2
inst✝¹ : Finite σ
inst✝ : IsDomain R
P : MvPolynomial PEmpty.{u_2 + 1} R
S : PEmpty.{u_2 + 1} → Finset R
Hdeg : ∀ (i : PEmpty.{u_2 + 1}), degreeOf i P < #(S i)
Heval : constantCoeff P = 0
this : P = C (constantCoeff P)
⊢ P = 0

Tactic:
]

Goal state after:
R : Type u_1
inst✝² : CommRing R
σ : Type u_2
inst✝¹ : Finite σ
inst✝ : IsDomain R
P : MvPolynomial PEmpty.{u_2 + 1} R
S : PEmpty.{u_2 + 1} → Finset R
Hdeg : ∀ (i : PEmpty.{u_2 + 1}), degreeOf i P < #(S i)
Heval : constantCoeff P = 0
this : P = C (constantCoeff P)
⊢ P = 0
Module: Mathlib.Combinatorics.Nullstellensatz

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
σ : Type u_2
inst✝¹ : Finite σ
inst✝ : IsDomain R
P : MvPolynomial PEmpty.{u_2 + 1} R
S : PEmpty.{u_2 + 1} → Finset R
Hdeg : ∀ (i : PEmpty.{u_2 + 1}), degreeOf i P < #(S i)
Heval : constantCoeff P = 0
this : P = C (constantCoeff P)
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α✝ α : Type u_1
M : Matroid α
C : Set α
hCE : C ⊆ M.E
I : Set α
hI : M.IsBasis I C
⊢ M ／ I ＼ (M.closure C \ I) = M ／ I ＼ (M.closure C \ I)

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
M : Matroid α
C X : Set α
⊢ (M ／ C).closure X \ ((M ／ C).closure X \ X) ⊆ M.closure (C ∪ X) \ C ∩ X ∧
    M.closure (C ∪ X) \ C ∩ X ⊆ (M ／ C).closure X

Tactic:
]

Goal state after:
α : Type u_1
M : Matroid α
C X : Set α
⊢ (M ／ C).closure X \ ((M ／ C).closure X \ X) ⊆ M.closure (C ∪ X) \ C ∩ X ∧
    M.closure (C ∪ X) \ C ∩ X ⊆ (M ／ C).closure X
Module: Mathlib.Combinatorics.Matroid.Sum

Goal state before:
pp: case refine_1.inl
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
I X : Set ((i : ι) × α i)
hI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)
x✝ :
  I ⊆ X ∧
    (∀ ⦃t : Set ((i : ι) × α i)⦄, (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' t)) → t ⊆ X → I ⊆ t → I = t) ∧
      X ⊆ univ.sigma fun i ↦ (M i).E
hIX : I ⊆ X
h' : X ⊆ univ.sigma fun i ↦ (M i).E
i : ι
I₀ : Set (α i)
hI₀ : (M i).Indep I₀
hI₀X : I₀ ⊆ Sigma.mk i ⁻¹' X
hII₀ : Sigma.mk i ⁻¹' I ⊆ I₀
h : (∀ (i_1 : ι), (M i_1).Indep (Sigma.mk i_1 ⁻¹' I ∪ Sigma.mk i_1 ⁻¹' Sigma.mk i '' I₀)) → I = I ∪ Sigma.mk i '' I₀
⊢ (M i).Indep I₀

Tactic:
]

Goal state after:
case refine_1.inl
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
I X : Set ((i : ι) × α i)
hI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)
x✝ :
  I ⊆ X ∧
    (∀ ⦃t : Set ((i : ι) × α i)⦄, (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' t)) → t ⊆ X → I ⊆ t → I = t) ∧
      X ⊆ univ.sigma fun i ↦ (M i).E
hIX : I ⊆ X
h' : X ⊆ univ.sigma fun i ↦ (M i).E
i : ι
I₀ : Set (α i)
hI₀ : (M i).Indep I₀
hI₀X : I₀ ⊆ Sigma.mk i ⁻¹' X
hII₀ : Sigma.mk i ⁻¹' I ⊆ I₀
h : (∀ (i_1 : ι), (M i_1).Indep (Sigma.mk i_1 ⁻¹' I ∪ Sigma.mk i_1 ⁻¹' Sigma.mk i '' I₀)) → I = I ∪ Sigma.mk i '' I₀
⊢ (M i).Indep I₀
Module: Mathlib.RingTheory.MvPolynomial.Groebner

Goal state before:
pp: case h.right.left
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommRing R
ι : Type u_3
b : ι → MvPolynomial σ R
hb : ∀ (i : ι), IsUnit (m.leadingCoeff (b i))
f : MvPolynomial σ R
hb' : ∀ (i : ι), m.degree (b i) ≠ 0
hf0 : ¬f = 0
i : ι
hf : m.degree (b i) ≤ m.degree f
deg_reduce : m.toSyn (m.degree (m.reduce ⋯ f)) < m.toSyn (m.degree f)
g' : ι →₀ MvPolynomial σ R
r' : MvPolynomial σ R
H' :
  m.reduce ⋯ f = (Finsupp.linearCombination (MvPolynomial σ R) b) g' + r' ∧
    (∀ (i_1 : ι), m.toSyn (m.degree (b i_1 * g' i_1)) ≤ m.toSyn (m.degree (m.reduce ⋯ f))) ∧
      ∀ c ∈ r'.support, ∀ (i : ι), ¬m.degree (b i) ≤ c
j : ι
⊢ m.toSyn
      (m.degree
        (b j * g' j +
          b j * (Finsupp.single i ((monomial (m.degree f - m.degree (b i))) (↑⋯.unit⁻¹ * m.leadingCoeff f))) j)) ≤
    m.toSyn (m.degree f)

Tactic:
]

Goal state after:
case h.right.left
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommRing R
ι : Type u_3
b : ι → MvPolynomial σ R
hb : ∀ (i : ι), IsUnit (m.leadingCoeff (b i))
f : MvPolynomial σ R
hb' : ∀ (i : ι), m.degree (b i) ≠ 0
hf0 : ¬f = 0
i : ι
hf : m.degree (b i) ≤ m.degree f
deg_reduce : m.toSyn (m.degree (m.reduce ⋯ f)) < m.toSyn (m.degree f)
g' : ι →₀ MvPolynomial σ R
r' : MvPolynomial σ R
H' :
  m.reduce ⋯ f = (Finsupp.linearCombination (MvPolynomial σ R) b) g' + r' ∧
    (∀ (i_1 : ι), m.toSyn (m.degree (b i_1 * g' i_1)) ≤ m.toSyn (m.degree (m.reduce ⋯ f))) ∧
      ∀ c ∈ r'.support, ∀ (i : ι), ¬m.degree (b i) ≤ c
j : ι
⊢ m.toSyn
      (m.degree
        (b j * g' j +
          b j * (Finsupp.single i ((monomial (m.degree f - m.degree (b i))) (↑⋯.unit⁻¹ * m.leadingCoeff f))) j)) ≤
    m.toSyn (m.degree f)
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
M : Matroid α
X C : Set α
hC : C ⊆ M.E
⊢ Disjoint X C → X ⊆ M.E → (M.closure (X ∪ C) \ C = M.E \ C ↔ M.closure (X ∪ C) = M.E ∧ C ⊆ M.E)

Tactic:
]

Goal state after:
α : Type u_1
M : Matroid α
X C : Set α
hC : C ⊆ M.E
⊢ Disjoint X C → X ⊆ M.E → (M.closure (X ∪ C) \ C = M.E \ C ↔ M.closure (X ∪ C) = M.E ∧ C ⊆ M.E)
Module: Mathlib.Combinatorics.Matroid.Sum

Goal state before:
pp: case refine_1.inr
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
I X : Set ((i : ι) × α i)
hI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)
x✝ :
  I ⊆ X ∧
    (∀ ⦃t : Set ((i : ι) × α i)⦄, (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' t)) → t ⊆ X → I ⊆ t → I = t) ∧
      X ⊆ univ.sigma fun i ↦ (M i).E
hIX : I ⊆ X
h' : X ⊆ univ.sigma fun i ↦ (M i).E
i : ι
I₀ : Set (α i)
hI₀ : (M i).Indep I₀
hI₀X : I₀ ⊆ Sigma.mk i ⁻¹' X
hII₀ : Sigma.mk i ⁻¹' I ⊆ I₀
h : (∀ (i_1 : ι), (M i_1).Indep (Sigma.mk i_1 ⁻¹' I ∪ Sigma.mk i_1 ⁻¹' Sigma.mk i '' I₀)) → I = I ∪ Sigma.mk i '' I₀
j : ι
hij : i ≠ j
⊢ (M j).Indep (Sigma.mk j ⁻¹' I)

Tactic:
]

Goal state after:
case refine_1.inr
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
I X : Set ((i : ι) × α i)
hI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)
x✝ :
  I ⊆ X ∧
    (∀ ⦃t : Set ((i : ι) × α i)⦄, (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' t)) → t ⊆ X → I ⊆ t → I = t) ∧
      X ⊆ univ.sigma fun i ↦ (M i).E
hIX : I ⊆ X
h' : X ⊆ univ.sigma fun i ↦ (M i).E
i : ι
I₀ : Set (α i)
hI₀ : (M i).Indep I₀
hI₀X : I₀ ⊆ Sigma.mk i ⁻¹' X
hII₀ : Sigma.mk i ⁻¹' I ⊆ I₀
h : (∀ (i_1 : ι), (M i_1).Indep (Sigma.mk i_1 ⁻¹' I ∪ Sigma.mk i_1 ⁻¹' Sigma.mk i '' I₀)) → I = I ∪ Sigma.mk i '' I₀
j : ι
hij : i ≠ j
⊢ (M j).Indep (Sigma.mk j ⁻¹' I)
Module: Mathlib.Combinatorics.Nullstellensatz

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
σ✝ : Type u_2
inst✝² : Finite σ✝
inst✝¹ : IsDomain R
σ : Type u_2
inst✝ : Fintype σ
h :
  ∀ (P : MvPolynomial σ R) (S : σ → Finset R),
    (∀ (i : σ), degreeOf i P < #(S i)) → (∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) P = 0) → P = 0
P : MvPolynomial (Option σ) R
S : Option σ → Finset R
Hdeg : ∀ (i : Option σ), degreeOf i P < #(S i)
Heval : ∀ (x : Option σ → R), (∀ (i : Option σ), x i ∈ S i) → (eval x) P = 0
Q : Polynomial (MvPolynomial σ R) := (optionEquivLeft R σ) P
hQ : Q = (optionEquivLeft R σ) P
this : Q = 0
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
M : Matroid α
X C : Set α
hC : C ⊆ M.E
hdj : Disjoint X C
hX : X ⊆ M.E
h : M.closure (X ∪ C) \ C = M.E \ C
⊢ M.E = M.E

pp: α : Type u_1
M : Matroid α
X C : Set α
hC : C ⊆ M.E
hdj : Disjoint X C
hX : X ⊆ M.E
h : M.closure (X ∪ C) \ C = M.E \ C
⊢ C ⊆ M.E

Tactic:
]

Goal state after:
α : Type u_1
M : Matroid α
X C : Set α
hC : C ⊆ M.E
hdj : Disjoint X C
hX : X ⊆ M.E
h : M.closure (X ∪ C) \ C = M.E \ C
⊢ C ⊆ M.E
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
M : Matroid α
X C : Set α
⊢ M.Spanning (X ∪ C ∩ M.E) → (Disjoint X (C ∩ M.E) ↔ Disjoint X C)

Tactic:
]

Goal state after:
α : Type u_1
M : Matroid α
X C : Set α
⊢ M.Spanning (X ∪ C ∩ M.E) → (Disjoint X (C ∩ M.E) ↔ Disjoint X C)
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
M : Matroid α
X C : Set α
h : M.Spanning (X ∪ C ∩ M.E)
hdj : Disjoint X (C ∩ M.E)
⊢ Disjoint X (C \ M.E)

Tactic:
]

Goal state after:
α : Type u_1
M : Matroid α
X C : Set α
h : M.Spanning (X ∪ C ∩ M.E)
hdj : Disjoint X (C ∩ M.E)
⊢ Disjoint X (C \ M.E)
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
M : Matroid α
X : Set α
hX : M.Spanning X
C : Set α
hXE : X ⊆ M.E
⊢ M.Spanning (X \ C ∪ C ∩ M.E)

Tactic:
]

Goal state after:
α : Type u_1
M : Matroid α
X : Set α
hX : M.Spanning X
C : Set α
hXE : X ⊆ M.E
⊢ M.Spanning (X \ C ∪ C ∩ M.E)
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
M : Matroid α
X : Set α
hX : M.Spanning X
C : Set α
hXE : X ⊆ M.E
⊢ (M ／ C).Spanning (X \ C)

Tactic:
rw [contract_spanning_iff', and_iff_left disjoint_sdiff_left]

Goal state after:
α : Type u_1
M : Matroid α
X : Set α
hX : M.Spanning X
C : Set α
hXE : X ⊆ M.E
⊢ M.Spanning (X \ C ∪ C ∩ M.E)
Module: Mathlib.RingTheory.MvPolynomial.Groebner

Goal state before:
pp: case h.right.left.right
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommRing R
ι : Type u_3
b : ι → MvPolynomial σ R
hb : ∀ (i : ι), IsUnit (m.leadingCoeff (b i))
f : MvPolynomial σ R
hb' : ∀ (i : ι), m.degree (b i) ≠ 0
hf0 : ¬f = 0
i : ι
hf : m.degree (b i) ≤ m.degree f
deg_reduce : m.toSyn (m.degree (m.reduce ⋯ f)) < m.toSyn (m.degree f)
g' : ι →₀ MvPolynomial σ R
r' : MvPolynomial σ R
H' :
  m.reduce ⋯ f = (Finsupp.linearCombination (MvPolynomial σ R) b) g' + r' ∧
    (∀ (i_1 : ι), m.toSyn (m.degree (b i_1 * g' i_1)) ≤ m.toSyn (m.degree (m.reduce ⋯ f))) ∧
      ∀ c ∈ r'.support, ∀ (i : ι), ¬m.degree (b i) ≤ c
j : ι
⊢ m.toSyn
      (m.degree (b j * if i = j then (monomial (m.degree f - m.degree (b i))) (↑⋯.unit⁻¹ * m.leadingCoeff f) else 0)) ≤
    m.toSyn (m.degree f)

Tactic:
]

Goal state after:
case h.right.left.right
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommRing R
ι : Type u_3
b : ι → MvPolynomial σ R
hb : ∀ (i : ι), IsUnit (m.leadingCoeff (b i))
f : MvPolynomial σ R
hb' : ∀ (i : ι), m.degree (b i) ≠ 0
hf0 : ¬f = 0
i : ι
hf : m.degree (b i) ≤ m.degree f
deg_reduce : m.toSyn (m.degree (m.reduce ⋯ f)) < m.toSyn (m.degree f)
g' : ι →₀ MvPolynomial σ R
r' : MvPolynomial σ R
H' :
  m.reduce ⋯ f = (Finsupp.linearCombination (MvPolynomial σ R) b) g' + r' ∧
    (∀ (i_1 : ι), m.toSyn (m.degree (b i_1 * g' i_1)) ≤ m.toSyn (m.degree (m.reduce ⋯ f))) ∧
      ∀ c ∈ r'.support, ∀ (i : ι), ¬m.degree (b i) ≤ c
j : ι
⊢ m.toSyn
      (m.degree (b j * if i = j then (monomial (m.degree f - m.degree (b i))) (↑⋯.unit⁻¹ * m.leadingCoeff f) else 0)) ≤
    m.toSyn (m.degree f)
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
M : Matroid α
X : Set α
hX : M.Spanning X
⊢ (M ／ X).loops = M.E \ X ∧ (M ／ X).E = M.E \ X

Tactic:
]

Goal state after:
α : Type u_1
M : Matroid α
X : Set α
hX : M.Spanning X
⊢ (M ／ X).loops = M.E \ X ∧ (M ／ X).E = M.E \ X
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
M : Matroid α
K C : Set α
hK : M.IsCircuit K
hCK : Disjoint C K
I : Set α
hI : M.IsBasis I (C ∩ M.E)
⊢ ¬M.Indep (K ∪ I)

Tactic:
]

Goal state after:
α : Type u_1
M : Matroid α
K C : Set α
hK : M.IsCircuit K
hCK : Disjoint C K
I : Set α
hI : M.IsBasis I (C ∩ M.E)
⊢ ¬M.Indep (K ∪ I)
Module: Mathlib.Combinatorics.Nullstellensatz

Goal state before:
pp: case heval
R : Type u_1
inst✝³ : CommRing R
σ✝ : Type u_2
inst✝² : Finite σ✝
inst✝¹ : IsDomain R
σ : Type u_2
inst✝ : Fintype σ
h :
  ∀ (P : MvPolynomial σ R) (S : σ → Finset R),
    (∀ (i : σ), degreeOf i P < #(S i)) → (∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) P = 0) → P = 0
P : MvPolynomial (Option σ) R
S : Option σ → Finset R
Hdeg : ∀ (i : Option σ), degreeOf i P < #(S i)
Heval : ∀ (x : Option σ → R), (∀ (i : Option σ), x i ∈ S i) → (eval x) P = 0
Q : Polynomial (MvPolynomial σ R) := (optionEquivLeft R σ) P
hQ : Q = (optionEquivLeft R σ) P
x : σ → R
hx : ∀ (i : σ), x i ∈ S (Option.some i)
y : R
hy : y ∈ S none
⊢ (eval fun x_1 ↦ x_1.elim y x) P = 0

Tactic:
]

Goal state after:
case heval
R : Type u_1
inst✝³ : CommRing R
σ✝ : Type u_2
inst✝² : Finite σ✝
inst✝¹ : IsDomain R
σ : Type u_2
inst✝ : Fintype σ
h :
  ∀ (P : MvPolynomial σ R) (S : σ → Finset R),
    (∀ (i : σ), degreeOf i P < #(S i)) → (∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) P = 0) → P = 0
P : MvPolynomial (Option σ) R
S : Option σ → Finset R
Hdeg : ∀ (i : Option σ), degreeOf i P < #(S i)
Heval : ∀ (x : Option σ → R), (∀ (i : Option σ), x i ∈ S i) → (eval x) P = 0
Q : Polynomial (MvPolynomial σ R) := (optionEquivLeft R σ) P
hQ : Q = (optionEquivLeft R σ) P
x : σ → R
hx : ∀ (i : σ), x i ∈ S (Option.some i)
y : R
hy : y ∈ S none
⊢ (eval fun x_1 ↦ x_1.elim y x) P = 0
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
M : Matroid α
K : Set α
hK : M.IsCircuit K
C : Set α
hKC : ¬K ⊆ C
h' : (M ／ (C ∩ K)).IsCircuit (K \ C)
hwin : (M ／ C).Dep (K \ C)
⊢ (M ／ C).Dep (K \ C)

Tactic:
]

Goal state after:
α : Type u_1
M : Matroid α
K : Set α
hK : M.IsCircuit K
C : Set α
hKC : ¬K ⊆ C
h' : (M ／ (C ∩ K)).IsCircuit (K \ C)
hwin : (M ／ C).Dep (K \ C)
⊢ (M ／ C).Dep (K \ C)
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
M : Matroid α
K C : Set α
hC : M.IsCircuit C
hK : K.Nonempty
hKC : K ⊆ C
⊢ K.Nonempty

Tactic:
]

Goal state after:
α : Type u_1
M : Matroid α
K C : Set α
hC : M.IsCircuit C
hK : K.Nonempty
hKC : K ⊆ C
⊢ K.Nonempty
Module: Mathlib.RingTheory.MvPolynomial.Groebner

Goal state before:
pp: case pos
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommRing R
ι : Type u_3
b : ι → MvPolynomial σ R
hb : ∀ (i : ι), IsUnit (m.leadingCoeff (b i))
f : MvPolynomial σ R
hb' : ∀ (i : ι), m.degree (b i) ≠ 0
hf0 : ¬f = 0
i : ι
hf : m.degree (b i) ≤ m.degree f
deg_reduce : m.toSyn (m.degree (m.reduce ⋯ f)) < m.toSyn (m.degree f)
g' : ι →₀ MvPolynomial σ R
r' : MvPolynomial σ R
H' :
  m.reduce ⋯ f = (Finsupp.linearCombination (MvPolynomial σ R) b) g' + r' ∧
    (∀ (i_1 : ι), m.toSyn (m.degree (b i_1 * g' i_1)) ≤ m.toSyn (m.degree (m.reduce ⋯ f))) ∧
      ∀ c ∈ r'.support, ∀ (i : ι), ¬m.degree (b i) ≤ c
⊢ m.toSyn (m.degree f) ≤ m.toSyn (m.degree f)

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: case inr
α : Type u_1
M : Matroid α
K C : Set α
aux : ∀ {K : Set α}, (M ／ K).IsCircuit C → M.Indep K → ∃ C', M.IsCircuit C' ∧ C ⊆ C' ∧ C' ⊆ C ∪ K
hKi : ¬M.Indep K
I : Set α
hC : (M ／ I).IsCircuit C ∧ Disjoint C (K \ I)
hI : M.IsBasis' I K
⊢ ∃ C', M.IsCircuit C' ∧ C ⊆ C' ∧ C' ⊆ C ∪ K

Tactic:
]

Goal state after:
case inr
α : Type u_1
M : Matroid α
K C : Set α
aux : ∀ {K : Set α}, (M ／ K).IsCircuit C → M.Indep K → ∃ C', M.IsCircuit C' ∧ C ⊆ C' ∧ C' ⊆ C ∪ K
hKi : ¬M.Indep K
I : Set α
hC : (M ／ I).IsCircuit C ∧ Disjoint C (K \ I)
hI : M.IsBasis' I K
⊢ ∃ C', M.IsCircuit C' ∧ C ⊆ C' ∧ C' ⊆ C ∪ K
Module: Mathlib.Combinatorics.Nullstellensatz

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
σ✝ : Type u_2
inst✝² : Finite σ✝
inst✝¹ : IsDomain R
σ : Type u_2
inst✝ : Fintype σ
h :
  ∀ (P : MvPolynomial σ R) (S : σ → Finset R),
    (∀ (i : σ), degreeOf i P < #(S i)) → (∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) P = 0) → P = 0
P : MvPolynomial (Option σ) R
S : Option σ → Finset R
Hdeg : ∀ (i : Option σ), degreeOf i P < #(S i)
Heval : ∀ (x : Option σ → R), (∀ (i : Option σ), x i ∈ S i) → (eval x) P = 0
Q : Polynomial (MvPolynomial σ R) := (optionEquivLeft R σ) P
hQ : Q = (optionEquivLeft R σ) P
x : σ → R
hx : ∀ (i : σ), x i ∈ S (Option.some i)
⊢ ∀ (N : ℕ), degreeOf none P < N → (Polynomial.map (eval x) Q).coeff N = 0

Tactic:
]

Goal state after:
R : Type u_1
inst✝³ : CommRing R
σ✝ : Type u_2
inst✝² : Finite σ✝
inst✝¹ : IsDomain R
σ : Type u_2
inst✝ : Fintype σ
h :
  ∀ (P : MvPolynomial σ R) (S : σ → Finset R),
    (∀ (i : σ), degreeOf i P < #(S i)) → (∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) P = 0) → P = 0
P : MvPolynomial (Option σ) R
S : Option σ → Finset R
Hdeg : ∀ (i : Option σ), degreeOf i P < #(S i)
Heval : ∀ (x : Option σ → R), (∀ (i : Option σ), x i ∈ S i) → (eval x) P = 0
Q : Polynomial (MvPolynomial σ R) := (optionEquivLeft R σ) P
hQ : Q = (optionEquivLeft R σ) P
x : σ → R
hx : ∀ (i : σ), x i ∈ S (Option.some i)
⊢ ∀ (N : ℕ), degreeOf none P < N → (Polynomial.map (eval x) Q).coeff N = 0
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
M : Matroid α
K✝ C K : Set α
hC : (M ／ K).IsCircuit C
hKi : M.Indep K
hCE : C ⊆ M.E
hCK : Disjoint C K
C' : Set α
hC'ss : C' ⊆ C ∪ K
hC' : M.IsCircuit C'
⊢ M.Dep (C' \ K ∪ K)

Tactic:
]

Goal state after:
α : Type u_1
M : Matroid α
K✝ C K : Set α
hC : (M ／ K).IsCircuit C
hKi : M.Indep K
hCE : C ⊆ M.E
hCK : Disjoint C K
C' : Set α
hC'ss : C' ⊆ C ∪ K
hC' : M.IsCircuit C'
⊢ M.Dep (C' \ K ∪ K)
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
M : Matroid α
K✝ C K : Set α
hC : (M ／ K).IsCircuit C
hKi : M.Indep K
hCE : C ⊆ M.E
hCK : Disjoint C K
C' : Set α
hC'ss : C' ⊆ C ∪ K
hC' : M.IsCircuit C'
hdep2 : (M ／ K).Dep (C' \ K)
⊢ C' \ K ⊆ C'

Tactic:
]

Goal state after:
α : Type u_1
M : Matroid α
K✝ C K : Set α
hC : (M ／ K).IsCircuit C
hKi : M.Indep K
hCE : C ⊆ M.E
hCK : Disjoint C K
C' : Set α
hC'ss : C' ⊆ C ∪ K
hC' : M.IsCircuit C'
hdep2 : (M ／ K).Dep (C' \ K)
⊢ C' \ K ⊆ C'
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
M : Matroid α
K C : Set α
hK : (M✶ ＼ C).IsCircuit K
⊢ M.IsCocircuit K

Tactic:
]

Goal state after:
α : Type u_1
M : Matroid α
K C : Set α
hK : (M✶ ＼ C).IsCircuit K
⊢ M.IsCocircuit K
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
M : Matroid α
K D : Set α
hK : M.IsCocircuit K
hD : D ⊂ K
⊢ (M✶ ／ D).IsCircuit (K \ D)

Tactic:
]

Goal state after:
α : Type u_1
M : Matroid α
K D : Set α
hK : M.IsCocircuit K
hD : D ⊂ K
⊢ (M✶ ／ D).IsCircuit (K \ D)
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
M : Matroid α
K X : Set α
hK : M.IsCocircuit K
hXK : X ⊆ K
hX : X.Nonempty
⊢ (M✶ ／ (K \ X)).IsCircuit X

Tactic:
]

Goal state after:
α : Type u_1
M : Matroid α
K X : Set α
hK : M.IsCocircuit K
hXK : X ⊆ K
hX : X.Nonempty
⊢ (M✶ ／ (K \ X)).IsCircuit X
Module: Mathlib.Combinatorics.Nullstellensatz

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
σ✝ : Type u_2
inst✝² : Finite σ✝
inst✝¹ : IsDomain R
σ : Type u_2
inst✝ : Fintype σ
h :
  ∀ (P : MvPolynomial σ R) (S : σ → Finset R),
    (∀ (i : σ), degreeOf i P < #(S i)) → (∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) P = 0) → P = 0
P : MvPolynomial (Option σ) R
S : Option σ → Finset R
Hdeg : ∀ (i : Option σ), degreeOf i P < #(S i)
Heval : ∀ (x : Option σ → R), (∀ (i : Option σ), x i ∈ S i) → (eval x) P = 0
Q : Polynomial (MvPolynomial σ R) := (optionEquivLeft R σ) P
hQ : Q = (optionEquivLeft R σ) P
x : σ → R
hx : ∀ (i : σ), x i ∈ S (Option.some i)
d : ℕ
hd : degreeOf none P < d
⊢ (eval x) (((optionEquivLeft R σ) P).coeff d) = 0

Tactic:
]

Goal state after:
R : Type u_1
inst✝³ : CommRing R
σ✝ : Type u_2
inst✝² : Finite σ✝
inst✝¹ : IsDomain R
σ : Type u_2
inst✝ : Fintype σ
h :
  ∀ (P : MvPolynomial σ R) (S : σ → Finset R),
    (∀ (i : σ), degreeOf i P < #(S i)) → (∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) P = 0) → P = 0
P : MvPolynomial (Option σ) R
S : Option σ → Finset R
Hdeg : ∀ (i : Option σ), degreeOf i P < #(S i)
Heval : ∀ (x : Option σ → R), (∀ (i : Option σ), x i ∈ S i) → (eval x) P = 0
Q : Polynomial (MvPolynomial σ R) := (optionEquivLeft R σ) P
hQ : Q = (optionEquivLeft R σ) P
x : σ → R
hx : ∀ (i : σ), x i ∈ S (Option.some i)
d : ℕ
hd : degreeOf none P < d
⊢ (eval x) (((optionEquivLeft R σ) P).coeff d) = 0
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
M : Matroid α
C D : Set α
⊢ D ∩ (M.E ∩ Cᶜ) = D ∩ (M.E ∩ Cᶜ)

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Nullstellensatz

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
σ✝ : Type u_2
inst✝² : Finite σ✝
inst✝¹ : IsDomain R
σ : Type u_2
inst✝ : Fintype σ
h :
  ∀ (P : MvPolynomial σ R) (S : σ → Finset R),
    (∀ (i : σ), degreeOf i P < #(S i)) → (∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) P = 0) → P = 0
P : MvPolynomial (Option σ) R
S : Option σ → Finset R
Hdeg : ∀ (i : Option σ), degreeOf i P < #(S i)
Heval : ∀ (x : Option σ → R), (∀ (i : Option σ), x i ∈ S i) → (eval x) P = 0
Q : Polynomial (MvPolynomial σ R) := (optionEquivLeft R σ) P
hQ : Q = (optionEquivLeft R σ) P
x : σ → R
hx : ∀ (i : σ), x i ∈ S (Option.some i)
d : ℕ
hd : degreeOf none P < d
m : σ →₀ ℕ
n : Option σ →₀ ℕ := (embDomain Embedding.some m).update none d
hn : n none = d ∧ n.some = m
⊢ coeff m (((optionEquivLeft R σ) P).coeff d) = 0

Tactic:
]

Goal state after:
R : Type u_1
inst✝³ : CommRing R
σ✝ : Type u_2
inst✝² : Finite σ✝
inst✝¹ : IsDomain R
σ : Type u_2
inst✝ : Fintype σ
h :
  ∀ (P : MvPolynomial σ R) (S : σ → Finset R),
    (∀ (i : σ), degreeOf i P < #(S i)) → (∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) P = 0) → P = 0
P : MvPolynomial (Option σ) R
S : Option σ → Finset R
Hdeg : ∀ (i : Option σ), degreeOf i P < #(S i)
Heval : ∀ (x : Option σ → R), (∀ (i : Option σ), x i ∈ S i) → (eval x) P = 0
Q : Polynomial (MvPolynomial σ R) := (optionEquivLeft R σ) P
hQ : Q = (optionEquivLeft R σ) P
x : σ → R
hx : ∀ (i : σ), x i ∈ S (Option.some i)
d : ℕ
hd : degreeOf none P < d
m : σ →₀ ℕ
n : Option σ →₀ ℕ := (embDomain Embedding.some m).update none d
hn : n none = d ∧ n.some = m
⊢ coeff m (((optionEquivLeft R σ) P).coeff d) = 0
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
R C : Set α
M : Matroid α
h : Disjoint C R
I : Set α
hI : I ⊆ R
J : Set α
hJ : (M ↾ (R ∪ C)).IsBasis' J C
hJ' : M.IsBasis' J C
⊢ (M.Indep (I ∪ J) ∧ Disjoint C I) ∧ I ⊆ R ↔ (M.Indep (I ∪ J) ∧ I ∪ J ⊆ R ∪ C) ∧ Disjoint C I

Tactic:
]

Goal state after:
α : Type u_1
R C : Set α
M : Matroid α
h : Disjoint C R
I : Set α
hI : I ⊆ R
J : Set α
hJ : (M ↾ (R ∪ C)).IsBasis' J C
hJ' : M.IsBasis' J C
⊢ (M.Indep (I ∪ J) ∧ Disjoint C I) ∧ I ⊆ R ↔ (M.Indep (I ∪ J) ∧ I ∪ J ⊆ R ∪ C) ∧ Disjoint C I
Module: Mathlib.Combinatorics.Matroid.Sum

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
h : ∀ (i : ι), (M i).Finitary
I : Set ((i : ι) × α i)
hI : ∀ J ⊆ I, J.Finite → ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' J)
i : ι
J : Set (α i)
hJI : J ⊆ Sigma.mk i ⁻¹' I
hJ : J.Finite
⊢ J = J

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
R C : Set α
M : Matroid α
hCR : C ⊆ R
⊢ (M ↾ R) ／ C = (M ↾ (R \ C ∪ C)) ／ C

Tactic:
]

Goal state after:
α : Type u_1
R C : Set α
M : Matroid α
hCR : C ⊆ R
⊢ (M ↾ R) ／ C = (M ↾ (R \ C ∪ C)) ／ C
Module: Mathlib.Combinatorics.Matroid.Sum

Goal state before:
pp: α : Type u_1
ι : Type u_2
M : ι → Matroid α
I : Set (ι × α)
⊢ (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' (fun a ↦ ⟨a.1, a.2⟩) '' I)) ↔ ∀ (i : ι), (M i).Indep (Prod.mk i ⁻¹' I)

Tactic:
convert! Iff.rfl

Goal state after:
case e'_2
α : Type u_1
ι : Type u_2
M : ι → Matroid α
I : Set (ι × α)
a✝ : ι
⊢ Prod.mk a✝ ⁻¹' I = Sigma.mk a✝ ⁻¹' (fun a ↦ ⟨a.1, a.2⟩) '' I