Theorem: Profinite.NobelingProof.Products.prop_of_isGood_of_contained
File: Mathlib/Topology/Category/Profinite/Nobeling.lean

Proof steps:
state_before: I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
h : isGood C l
hsC : contained C o
i : I
hi : i ∈ ↑l
⊢ ord I i < o
state_after: I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
h : isGood C l
hsC : contained C o
i : I
hi : i ∈ ↑l
h' : ¬ord I i < o
⊢ False
tactic: by_contra h'
state_before: I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
h : isGood C l
hsC : contained C o
i : I
hi : i ∈ ↑l
h' : ¬ord I i < o
⊢ False
state_after: I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
h : isGood C l
hsC : contained C o
i : I
hi : i ∈ ↑l
h' : ¬ord I i < o
⊢ eval C l ∈ Submodule.span ℤ (eval C '' {m | m < l})
tactic: apply h
state_before: I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
h : isGood C l
hsC : contained C o
i : I
hi : i ∈ ↑l
h' : ¬ord I i < o
⊢ eval C l ∈ Submodule.span ℤ (eval C '' {m | m < l})
state_after: I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
h : isGood C l
hsC : contained C o
i : I
hi : i ∈ ↑l
h' : ¬ord I i < o
⊢ eval C l = 0
tactic: suffices eval C l = 0 by simp [this, Submodule.zero_mem]
state_before: I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
h : isGood C l
hsC : contained C o
i : I
hi : i ∈ ↑l
h' : ¬ord I i < o
⊢ eval C l = 0
state_after: case h
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
h : isGood C l
hsC : contained C o
i : I
hi : i ∈ ↑l
h' : ¬ord I i < o
x : ↑C
⊢ (eval C l) x = 0 x
tactic: ext x
state_before: case h
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
h : isGood C l
hsC : contained C o
i : I
hi : i ∈ ↑l
h' : ¬ord I i < o
x : ↑C
⊢ (eval C l) x = 0 x
state_after: case h
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
h : isGood C l
hsC : contained C o
i : I
hi : i ∈ ↑l
h' : ¬ord I i < o
x : ↑C
⊢ (∀ i ∈ ↑l, ↑x i = true) → False
tactic: simp only [eval_eq, LocallyConstant.coe_zero, Pi.zero_apply, ite_eq_right_iff, one_ne_zero]
state_before: case h
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
h : isGood C l
hsC : contained C o
i : I
hi : i ∈ ↑l
h' : ¬ord I i < o
x : ↑C
⊢ (∀ i ∈ ↑l, ↑x i = true) → False
state_after: case h
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
h : isGood C l
hsC : contained C o
i : I
hi : i ∈ ↑l
x : ↑C
h' : (∀ i ∈ ↑l, ↑x i = true) ∧ ¬False
⊢ ord I i < o
tactic: contrapose! h'
state_before: case h
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o : Ordinal.{u}
h : isGood C l
hsC : contained C o
i : I
hi : i ∈ ↑l
x : ↑C
h' : (∀ i ∈ ↑l, ↑x i = true) ∧ ¬False
⊢ ord I i < o
state_after: no goals
tactic: exact hsC x.val x.prop i (h'.1 i hi)