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https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | have prf₂ := BProof.monotone (le_trans l₂ l₃) fprf | t : Th
Δ : Ctx
f : Form
h₁ : lindenbaumExtension t Δ⊢f
prf₁ : BProof (lindenbaumExtension t Δ) f
s : Finset Form
l₁ : ↑s ⊆ lindenbaumExtension t Δ
fprf : BProof (↑s) f
i j : ℕ
l₂ : ↑s ⊆ lindenbaumSequence t Δ (i, j)
l₃ : lindenbaumSequence t Δ (i, j) ⊆ lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
⊢ f ∈ lindenbaumExtension t Δ | t : Th
Δ : Ctx
f : Form
h₁ : lindenbaumExtension t Δ⊢f
prf₁ : BProof (lindenbaumExtension t Δ) f
s : Finset Form
l₁ : ↑s ⊆ lindenbaumExtension t Δ
fprf : BProof (↑s) f
i j : ℕ
l₂ : ↑s ⊆ lindenbaumSequence t Δ (i, j)
l₃ : lindenbaumSequence t Δ (i, j) ⊆ lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
prf₂ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) f
⊢ f ∈ lindenbaumExtension t Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | have prf₃ : BProof (lindenbaumSequence t Δ ⟨i+1,Encodable.encode (f,f)⟩) (f ¦ f) := BProof.mp prf₂ BTheorem.orI₁ | t : Th
Δ : Ctx
f : Form
h₁ : lindenbaumExtension t Δ⊢f
prf₁ : BProof (lindenbaumExtension t Δ) f
s : Finset Form
l₁ : ↑s ⊆ lindenbaumExtension t Δ
fprf : BProof (↑s) f
i j : ℕ
l₂ : ↑s ⊆ lindenbaumSequence t Δ (i, j)
l₃ : lindenbaumSequence t Δ (i, j) ⊆ lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
prf₂ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) f
⊢ f ∈ lindenbaumExtension t Δ | t : Th
Δ : Ctx
f : Form
h₁ : lindenbaumExtension t Δ⊢f
prf₁ : BProof (lindenbaumExtension t Δ) f
s : Finset Form
l₁ : ↑s ⊆ lindenbaumExtension t Δ
fprf : BProof (↑s) f
i j : ℕ
l₂ : ↑s ⊆ lindenbaumSequence t Δ (i, j)
l₃ : lindenbaumSequence t Δ (i, j) ⊆ lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
prf₂ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) f
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
⊢ f ∈ lindenbaumExtension t Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | clear s h₁ l₁ l₂ l₃ fprf prf₁ prf₂ | t : Th
Δ : Ctx
f : Form
h₁ : lindenbaumExtension t Δ⊢f
prf₁ : BProof (lindenbaumExtension t Δ) f
s : Finset Form
l₁ : ↑s ⊆ lindenbaumExtension t Δ
fprf : BProof (↑s) f
i j : ℕ
l₂ : ↑s ⊆ lindenbaumSequence t Δ (i, j)
l₃ : lindenbaumSequence t Δ (i, j) ⊆ lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
prf₂ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) f
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
⊢ f ∈ lindenbaumExtension t Δ | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
⊢ f ∈ lindenbaumExtension t Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | have l₄ : f ∈ lindenbaumSequence t Δ ⟨i + 1, Encodable.encode (f,f) + 1⟩ := by
unfold lindenbaumSequence
change
let prev := lindenbaumSequence t Δ (i + 1, Encodable.encode (f,f));
let l := (Denumerable.ofNat (Form × Form) (Encodable.encode (f,f))).fst;
let r := (Denumerable.ofNat (Form × Form) (Encodable.encode (f,f))).snd;
f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev
intros prev l r
have l₅ : l = f := by
change (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst = f
rw [Denumerable.ofNat_encode (f,f)]
have l₆ : r = f := by
change (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd = f
rw [Denumerable.ofNat_encode (f,f)]
split
case inl h₂ =>
split
. rw [l₅]; exact Or.inr rfl
. rw [l₆]; exact Or.inr rfl
case inr h₂ =>
apply False.elim
have l₇ : f¦f ∈ ▲prev := ⟨prf₃⟩
rw [l₅,l₆] at h₂
exact h₂ l₇ | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
⊢ f ∈ lindenbaumExtension t Δ | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
l₄ : f ∈ lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f) + 1)
⊢ f ∈ lindenbaumExtension t Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | exact ⟨⟨i + 1, Encodable.encode (f,f) + 1⟩, l₄⟩ | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
l₄ : f ∈ lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f) + 1)
⊢ f ∈ lindenbaumExtension t Δ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | apply lindenbaumSequenceMonotone | t : Th
Δ : Ctx
f : Form
h₁ : lindenbaumExtension t Δ⊢f
prf₁ : BProof (lindenbaumExtension t Δ) f
s : Finset Form
l₁ : ↑s ⊆ lindenbaumExtension t Δ
fprf : BProof (↑s) f
i j : ℕ
l₂ : ↑s ⊆ lindenbaumSequence t Δ (i, j)
⊢ lindenbaumSequence t Δ (i, j) ⊆ lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f)) | case a
t : Th
Δ : Ctx
f : Form
h₁ : lindenbaumExtension t Δ⊢f
prf₁ : BProof (lindenbaumExtension t Δ) f
s : Finset Form
l₁ : ↑s ⊆ lindenbaumExtension t Δ
fprf : BProof (↑s) f
i j : ℕ
l₂ : ↑s ⊆ lindenbaumSequence t Δ (i, j)
⊢ (i, j) ≤ (i + 1, Encodable.encode (f, f)) |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | apply (Prod.Lex.le_iff (i,j) (i + 1,Encodable.encode (f,f))).mpr $ Or.inl $ Nat.lt_succ_self i | case a
t : Th
Δ : Ctx
f : Form
h₁ : lindenbaumExtension t Δ⊢f
prf₁ : BProof (lindenbaumExtension t Δ) f
s : Finset Form
l₁ : ↑s ⊆ lindenbaumExtension t Δ
fprf : BProof (↑s) f
i j : ℕ
l₂ : ↑s ⊆ lindenbaumSequence t Δ (i, j)
⊢ (i, j) ≤ (i + 1, Encodable.encode (f, f)) | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | unfold lindenbaumSequence | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
⊢ f ∈ lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f) + 1) | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
⊢ f ∈
let prev := lindenbaumSequence t Δ (Nat.succ (Nat.add i 0), Nat.add (Encodable.encode (f, f)) 0);
let l := (Denumerable.ofNat (Form × Form) (Nat.add (Encodable.encode (f, f)) 0)).fst;
let r := (Denumerable.ofNat (Form × Form) (Nat.add (Encodable.encode (f, f)) 0)).snd;
if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | change
let prev := lindenbaumSequence t Δ (i + 1, Encodable.encode (f,f));
let l := (Denumerable.ofNat (Form × Form) (Encodable.encode (f,f))).fst;
let r := (Denumerable.ofNat (Form × Form) (Encodable.encode (f,f))).snd;
f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
⊢ f ∈
let prev := lindenbaumSequence t Δ (Nat.succ (Nat.add i 0), Nat.add (Encodable.encode (f, f)) 0);
let l := (Denumerable.ofNat (Form × Form) (Nat.add (Encodable.encode (f, f)) 0)).fst;
let r := (Denumerable.ofNat (Form × Form) (Nat.add (Encodable.encode (f, f)) 0)).snd;
if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
⊢ let prev := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f));
let l := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst;
let r := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd;
f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | intros prev l r | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
⊢ let prev := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f));
let l := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst;
let r := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd;
f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
⊢ f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | have l₅ : l = f := by
change (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst = f
rw [Denumerable.ofNat_encode (f,f)] | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
⊢ f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
⊢ f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | have l₆ : r = f := by
change (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd = f
rw [Denumerable.ofNat_encode (f,f)] | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
⊢ f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
l₆ : r = f
⊢ f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | split | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
l₆ : r = f
⊢ f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev | case inl
t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
l₆ : r = f
h✝ : l¦r ∈ ▲prev
⊢ f ∈ if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r}
case inr
t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
l₆ : r = f
h✝ : ¬l¦r ∈ ▲prev
⊢ f ∈ prev |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | case inl h₂ =>
split
. rw [l₅]; exact Or.inr rfl
. rw [l₆]; exact Or.inr rfl | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
l₆ : r = f
h₂ : l¦r ∈ ▲prev
⊢ f ∈ if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | case inr h₂ =>
apply False.elim
have l₇ : f¦f ∈ ▲prev := ⟨prf₃⟩
rw [l₅,l₆] at h₂
exact h₂ l₇ | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
l₆ : r = f
h₂ : ¬l¦r ∈ ▲prev
⊢ f ∈ prev | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | change (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst = f | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
⊢ l = f | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
⊢ (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst = f |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | rw [Denumerable.ofNat_encode (f,f)] | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
⊢ (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst = f | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | change (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd = f | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
⊢ r = f | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
⊢ (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd = f |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | rw [Denumerable.ofNat_encode (f,f)] | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
⊢ (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd = f | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | split | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
l₆ : r = f
h₂ : l¦r ∈ ▲prev
⊢ f ∈ if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} | case inl
t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
l₆ : r = f
h₂ : l¦r ∈ ▲prev
h✝ : ▲(prev ∪ {l}) ∩ Δ = ∅
⊢ f ∈ prev ∪ {l}
case inr
t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
l₆ : r = f
h₂ : l¦r ∈ ▲prev
h✝ : ¬▲(prev ∪ {l}) ∩ Δ = ∅
⊢ f ∈ prev ∪ {r} |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | . rw [l₅]; exact Or.inr rfl | case inl
t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
l₆ : r = f
h₂ : l¦r ∈ ▲prev
h✝ : ▲(prev ∪ {l}) ∩ Δ = ∅
⊢ f ∈ prev ∪ {l}
case inr
t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
l₆ : r = f
h₂ : l¦r ∈ ▲prev
h✝ : ¬▲(prev ∪ {l}) ∩ Δ = ∅
⊢ f ∈ prev ∪ {r} | case inr
t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
l₆ : r = f
h₂ : l¦r ∈ ▲prev
h✝ : ¬▲(prev ∪ {l}) ∩ Δ = ∅
⊢ f ∈ prev ∪ {r} |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | . rw [l₆]; exact Or.inr rfl | case inr
t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
l₆ : r = f
h₂ : l¦r ∈ ▲prev
h✝ : ¬▲(prev ∪ {l}) ∩ Δ = ∅
⊢ f ∈ prev ∪ {r} | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | apply False.elim | t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
l₆ : r = f
h₂ : ¬l¦r ∈ ▲prev
⊢ f ∈ prev | case h
t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
l₆ : r = f
h₂ : ¬l¦r ∈ ▲prev
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | have l₇ : f¦f ∈ ▲prev := ⟨prf₃⟩ | case h
t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
l₆ : r = f
h₂ : ¬l¦r ∈ ▲prev
⊢ False | case h
t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
l₆ : r = f
h₂ : ¬l¦r ∈ ▲prev
l₇ : f¦f ∈ ▲prev
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | rw [l₅,l₆] at h₂ | case h
t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
l₆ : r = f
h₂ : ¬l¦r ∈ ▲prev
l₇ : f¦f ∈ ▲prev
⊢ False | case h
t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
l₆ : r = f
h₂ : ¬f¦f ∈ ▲prev
l₇ : f¦f ∈ ▲prev
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsFormal | [107, 1] | [148, 52] | exact h₂ l₇ | case h
t : Th
Δ : Ctx
f : Form
i j : ℕ
prf₃ : BProof (lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))) (f¦f)
prev : Ctx := lindenbaumSequence t Δ (i + 1, Encodable.encode (f, f))
l : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst
r : Form := (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd
l₅ : l = f
l₆ : r = f
h₂ : ¬f¦f ∈ ▲prev
l₇ : f¦f ∈ ▲prev
⊢ False | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | intros f g h₁ | t : Th
Δ : Ctx
⊢ isPrimeTheory (lindenbaumExtension t Δ) | t : Th
Δ : Ctx
f g : Form
h₁ : f¦g ∈ lindenbaumExtension t Δ
⊢ f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | have ⟨⟨i,j⟩,h₂⟩ := h₁ | t : Th
Δ : Ctx
f g : Form
h₁ : f¦g ∈ lindenbaumExtension t Δ
⊢ f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ | t : Th
Δ : Ctx
f g : Form
h₁ : f¦g ∈ lindenbaumExtension t Δ
i j : ℕ
h₂ : f¦g ∈ lindenbaumSequence t Δ (i, j)
⊢ f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | let k := Encodable.encode (f,g) | t : Th
Δ : Ctx
f g : Form
h₁ : f¦g ∈ lindenbaumExtension t Δ
i j : ℕ
h₂ : f¦g ∈ lindenbaumSequence t Δ (i, j)
⊢ f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ | t : Th
Δ : Ctx
f g : Form
h₁ : f¦g ∈ lindenbaumExtension t Δ
i j : ℕ
h₂ : f¦g ∈ lindenbaumSequence t Δ (i, j)
k : ℕ := Encodable.encode (f, g)
⊢ f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | have l₁ : lindenbaumSequence t Δ ⟨i,j⟩ ⊆ lindenbaumSequence t Δ ⟨i + 1,k⟩ := by
apply lindenbaumSequenceMonotone
apply (Prod.Lex.le_iff (i,j) (i + 1,k)).mpr $ Or.inl $ Nat.lt_succ_self i | t : Th
Δ : Ctx
f g : Form
h₁ : f¦g ∈ lindenbaumExtension t Δ
i j : ℕ
h₂ : f¦g ∈ lindenbaumSequence t Δ (i, j)
k : ℕ := Encodable.encode (f, g)
⊢ f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ | t : Th
Δ : Ctx
f g : Form
h₁ : f¦g ∈ lindenbaumExtension t Δ
i j : ℕ
h₂ : f¦g ∈ lindenbaumSequence t Δ (i, j)
k : ℕ := Encodable.encode (f, g)
l₁ : lindenbaumSequence t Δ (i, j) ⊆ lindenbaumSequence t Δ (i + 1, k)
⊢ f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | have l₂ : f ¦ g ∈ lindenbaumSequence t Δ ⟨i + 1, k⟩ := l₁ h₂ | t : Th
Δ : Ctx
f g : Form
h₁ : f¦g ∈ lindenbaumExtension t Δ
i j : ℕ
h₂ : f¦g ∈ lindenbaumSequence t Δ (i, j)
k : ℕ := Encodable.encode (f, g)
l₁ : lindenbaumSequence t Δ (i, j) ⊆ lindenbaumSequence t Δ (i + 1, k)
⊢ f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ | t : Th
Δ : Ctx
f g : Form
h₁ : f¦g ∈ lindenbaumExtension t Δ
i j : ℕ
h₂ : f¦g ∈ lindenbaumSequence t Δ (i, j)
k : ℕ := Encodable.encode (f, g)
l₁ : lindenbaumSequence t Δ (i, j) ⊆ lindenbaumSequence t Δ (i + 1, k)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
⊢ f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | clear l₁ h₁ h₂ | t : Th
Δ : Ctx
f g : Form
h₁ : f¦g ∈ lindenbaumExtension t Δ
i j : ℕ
h₂ : f¦g ∈ lindenbaumSequence t Δ (i, j)
k : ℕ := Encodable.encode (f, g)
l₁ : lindenbaumSequence t Δ (i, j) ⊆ lindenbaumSequence t Δ (i + 1, k)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
⊢ f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
⊢ f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | have l₃ : f ∈ lindenbaumSequence t Δ ⟨i + 1, k + 1⟩ ∨ g ∈ lindenbaumSequence t Δ ⟨i + 1, k + 1⟩ := by
unfold lindenbaumSequence
change
let prev := lindenbaumSequence t Δ (i + 1, k);
let l := (Denumerable.ofNat (Form × Form) k).fst;
let r := (Denumerable.ofNat (Form × Form) k).snd;
(f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev) ∨
(g ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev)
intros prev l r
have l₄ : Denumerable.ofNat (Form × Form) k = (f,g) := Denumerable.ofNat_encode (f,g)
have l₅ : l = f := by
change (Denumerable.ofNat (Form × Form) k).fst = f
rw [l₄]
have l₆ : r = g := by
change (Denumerable.ofNat (Form × Form) k).snd = g
rw [l₄]
repeat rw [l₅,l₆]
clear l r l₅ l₆
cases Classical.em (▲(prev ∪ {f}) ∩ Δ = ∅)
case' inl h₁ => apply Or.inl
case' inr h₁ => apply Or.inr
all_goals
split
case inl => exact Or.inr rfl
case inr h₂ => exact False.elim $ h₂ ⟨BProof.ax l₂⟩ | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
⊢ f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
l₃ : f ∈ lindenbaumSequence t Δ (i + 1, k + 1) ∨ g ∈ lindenbaumSequence t Δ (i + 1, k + 1)
⊢ f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | apply Or.elim l₃ | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
l₃ : f ∈ lindenbaumSequence t Δ (i + 1, k + 1) ∨ g ∈ lindenbaumSequence t Δ (i + 1, k + 1)
⊢ f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ | case left
t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
l₃ : f ∈ lindenbaumSequence t Δ (i + 1, k + 1) ∨ g ∈ lindenbaumSequence t Δ (i + 1, k + 1)
⊢ f ∈ lindenbaumSequence t Δ (i + 1, k + 1) → f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ
case right
t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
l₃ : f ∈ lindenbaumSequence t Δ (i + 1, k + 1) ∨ g ∈ lindenbaumSequence t Δ (i + 1, k + 1)
⊢ g ∈ lindenbaumSequence t Δ (i + 1, k + 1) → f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | case left => intros h₁; exact Or.inl ⟨⟨i+1,k+1⟩,h₁⟩ | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
l₃ : f ∈ lindenbaumSequence t Δ (i + 1, k + 1) ∨ g ∈ lindenbaumSequence t Δ (i + 1, k + 1)
⊢ f ∈ lindenbaumSequence t Δ (i + 1, k + 1) → f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | case right => intros h₁; exact Or.inr ⟨⟨i+1,k+1⟩,h₁⟩ | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
l₃ : f ∈ lindenbaumSequence t Δ (i + 1, k + 1) ∨ g ∈ lindenbaumSequence t Δ (i + 1, k + 1)
⊢ g ∈ lindenbaumSequence t Δ (i + 1, k + 1) → f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | apply lindenbaumSequenceMonotone | t : Th
Δ : Ctx
f g : Form
h₁ : f¦g ∈ lindenbaumExtension t Δ
i j : ℕ
h₂ : f¦g ∈ lindenbaumSequence t Δ (i, j)
k : ℕ := Encodable.encode (f, g)
⊢ lindenbaumSequence t Δ (i, j) ⊆ lindenbaumSequence t Δ (i + 1, k) | case a
t : Th
Δ : Ctx
f g : Form
h₁ : f¦g ∈ lindenbaumExtension t Δ
i j : ℕ
h₂ : f¦g ∈ lindenbaumSequence t Δ (i, j)
k : ℕ := Encodable.encode (f, g)
⊢ (i, j) ≤ (i + 1, k) |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | apply (Prod.Lex.le_iff (i,j) (i + 1,k)).mpr $ Or.inl $ Nat.lt_succ_self i | case a
t : Th
Δ : Ctx
f g : Form
h₁ : f¦g ∈ lindenbaumExtension t Δ
i j : ℕ
h₂ : f¦g ∈ lindenbaumSequence t Δ (i, j)
k : ℕ := Encodable.encode (f, g)
⊢ (i, j) ≤ (i + 1, k) | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | unfold lindenbaumSequence | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
⊢ f ∈ lindenbaumSequence t Δ (i + 1, k + 1) ∨ g ∈ lindenbaumSequence t Δ (i + 1, k + 1) | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
⊢ (f ∈
let prev := lindenbaumSequence t Δ (Nat.succ (Nat.add i 0), Nat.add k 0);
let l := (Denumerable.ofNat (Form × Form) (Nat.add k 0)).fst;
let r := (Denumerable.ofNat (Form × Form) (Nat.add k 0)).snd;
if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev) ∨
g ∈
let prev := lindenbaumSequence t Δ (Nat.succ (Nat.add i 0), Nat.add k 0);
let l := (Denumerable.ofNat (Form × Form) (Nat.add k 0)).fst;
let r := (Denumerable.ofNat (Form × Form) (Nat.add k 0)).snd;
if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | change
let prev := lindenbaumSequence t Δ (i + 1, k);
let l := (Denumerable.ofNat (Form × Form) k).fst;
let r := (Denumerable.ofNat (Form × Form) k).snd;
(f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev) ∨
(g ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev) | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
⊢ (f ∈
let prev := lindenbaumSequence t Δ (Nat.succ (Nat.add i 0), Nat.add k 0);
let l := (Denumerable.ofNat (Form × Form) (Nat.add k 0)).fst;
let r := (Denumerable.ofNat (Form × Form) (Nat.add k 0)).snd;
if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev) ∨
g ∈
let prev := lindenbaumSequence t Δ (Nat.succ (Nat.add i 0), Nat.add k 0);
let l := (Denumerable.ofNat (Form × Form) (Nat.add k 0)).fst;
let r := (Denumerable.ofNat (Form × Form) (Nat.add k 0)).snd;
if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
⊢ let prev := lindenbaumSequence t Δ (i + 1, k);
let l := (Denumerable.ofNat (Form × Form) k).fst;
let r := (Denumerable.ofNat (Form × Form) k).snd;
(f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev) ∨
g ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | intros prev l r | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
⊢ let prev := lindenbaumSequence t Δ (i + 1, k);
let l := (Denumerable.ofNat (Form × Form) k).fst;
let r := (Denumerable.ofNat (Form × Form) k).snd;
(f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev) ∨
g ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l : Form := (Denumerable.ofNat (Form × Form) k).fst
r : Form := (Denumerable.ofNat (Form × Form) k).snd
⊢ (f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev) ∨
g ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | have l₄ : Denumerable.ofNat (Form × Form) k = (f,g) := Denumerable.ofNat_encode (f,g) | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l : Form := (Denumerable.ofNat (Form × Form) k).fst
r : Form := (Denumerable.ofNat (Form × Form) k).snd
⊢ (f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev) ∨
g ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l : Form := (Denumerable.ofNat (Form × Form) k).fst
r : Form := (Denumerable.ofNat (Form × Form) k).snd
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
⊢ (f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev) ∨
g ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | have l₅ : l = f := by
change (Denumerable.ofNat (Form × Form) k).fst = f
rw [l₄] | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l : Form := (Denumerable.ofNat (Form × Form) k).fst
r : Form := (Denumerable.ofNat (Form × Form) k).snd
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
⊢ (f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev) ∨
g ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l : Form := (Denumerable.ofNat (Form × Form) k).fst
r : Form := (Denumerable.ofNat (Form × Form) k).snd
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
l₅ : l = f
⊢ (f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev) ∨
g ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | have l₆ : r = g := by
change (Denumerable.ofNat (Form × Form) k).snd = g
rw [l₄] | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l : Form := (Denumerable.ofNat (Form × Form) k).fst
r : Form := (Denumerable.ofNat (Form × Form) k).snd
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
l₅ : l = f
⊢ (f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev) ∨
g ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l : Form := (Denumerable.ofNat (Form × Form) k).fst
r : Form := (Denumerable.ofNat (Form × Form) k).snd
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
l₅ : l = f
l₆ : r = g
⊢ (f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev) ∨
g ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | repeat rw [l₅,l₆] | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l : Form := (Denumerable.ofNat (Form × Form) k).fst
r : Form := (Denumerable.ofNat (Form × Form) k).snd
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
l₅ : l = f
l₆ : r = g
⊢ (f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev) ∨
g ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l : Form := (Denumerable.ofNat (Form × Form) k).fst
r : Form := (Denumerable.ofNat (Form × Form) k).snd
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
l₅ : l = f
l₆ : r = g
⊢ (f ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev) ∨
g ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | clear l r l₅ l₆ | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l : Form := (Denumerable.ofNat (Form × Form) k).fst
r : Form := (Denumerable.ofNat (Form × Form) k).snd
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
l₅ : l = f
l₆ : r = g
⊢ (f ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev) ∨
g ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
⊢ (f ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev) ∨
g ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | cases Classical.em (▲(prev ∪ {f}) ∩ Δ = ∅) | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
⊢ (f ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev) ∨
g ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev | case inl
t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h✝ : ▲(prev ∪ {f}) ∩ Δ = ∅
⊢ (f ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev) ∨
g ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev
case inr
t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h✝ : ¬▲(prev ∪ {f}) ∩ Δ = ∅
⊢ (f ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev) ∨
g ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | case' inl h₁ => apply Or.inl | case inl
t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h✝ : ▲(prev ∪ {f}) ∩ Δ = ∅
⊢ (f ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev) ∨
g ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev
case inr
t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h✝ : ¬▲(prev ∪ {f}) ∩ Δ = ∅
⊢ (f ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev) ∨
g ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev | case inl
t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h₁ : ▲(prev ∪ {f}) ∩ Δ = ∅
⊢ f ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev
case inr
t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h✝ : ¬▲(prev ∪ {f}) ∩ Δ = ∅
⊢ (f ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev) ∨
g ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | case' inr h₁ => apply Or.inr | case inl
t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h₁ : ▲(prev ∪ {f}) ∩ Δ = ∅
⊢ f ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev
case inr
t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h✝ : ¬▲(prev ∪ {f}) ∩ Δ = ∅
⊢ (f ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev) ∨
g ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev | case inr
t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h₁ : ¬▲(prev ∪ {f}) ∩ Δ = ∅
⊢ g ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev
case inl
t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h₁ : ▲(prev ∪ {f}) ∩ Δ = ∅
⊢ f ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | all_goals
split
case inl => exact Or.inr rfl
case inr h₂ => exact False.elim $ h₂ ⟨BProof.ax l₂⟩ | case inr
t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h₁ : ¬▲(prev ∪ {f}) ∩ Δ = ∅
⊢ g ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev
case inl
t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h₁ : ▲(prev ∪ {f}) ∩ Δ = ∅
⊢ f ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | change (Denumerable.ofNat (Form × Form) k).fst = f | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l : Form := (Denumerable.ofNat (Form × Form) k).fst
r : Form := (Denumerable.ofNat (Form × Form) k).snd
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
⊢ l = f | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l : Form := (Denumerable.ofNat (Form × Form) k).fst
r : Form := (Denumerable.ofNat (Form × Form) k).snd
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
⊢ (Denumerable.ofNat (Form × Form) k).fst = f |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | rw [l₄] | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l : Form := (Denumerable.ofNat (Form × Form) k).fst
r : Form := (Denumerable.ofNat (Form × Form) k).snd
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
⊢ (Denumerable.ofNat (Form × Form) k).fst = f | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | change (Denumerable.ofNat (Form × Form) k).snd = g | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l : Form := (Denumerable.ofNat (Form × Form) k).fst
r : Form := (Denumerable.ofNat (Form × Form) k).snd
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
l₅ : l = f
⊢ r = g | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l : Form := (Denumerable.ofNat (Form × Form) k).fst
r : Form := (Denumerable.ofNat (Form × Form) k).snd
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
l₅ : l = f
⊢ (Denumerable.ofNat (Form × Form) k).snd = g |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | rw [l₄] | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l : Form := (Denumerable.ofNat (Form × Form) k).fst
r : Form := (Denumerable.ofNat (Form × Form) k).snd
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
l₅ : l = f
⊢ (Denumerable.ofNat (Form × Form) k).snd = g | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | rw [l₅,l₆] | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l : Form := (Denumerable.ofNat (Form × Form) k).fst
r : Form := (Denumerable.ofNat (Form × Form) k).snd
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
l₅ : l = f
l₆ : r = g
⊢ (f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev) ∨
g ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l : Form := (Denumerable.ofNat (Form × Form) k).fst
r : Form := (Denumerable.ofNat (Form × Form) k).snd
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
l₅ : l = f
l₆ : r = g
⊢ (f ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev) ∨
g ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | apply Or.inl | case inl
t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h₁ : ▲(prev ∪ {f}) ∩ Δ = ∅
⊢ (f ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev) ∨
g ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev | case inl.h
t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h₁ : ▲(prev ∪ {f}) ∩ Δ = ∅
⊢ f ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | apply Or.inr | case inr
t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h₁ : ¬▲(prev ∪ {f}) ∩ Δ = ∅
⊢ (f ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev) ∨
g ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev | case inr.h
t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h₁ : ¬▲(prev ∪ {f}) ∩ Δ = ∅
⊢ g ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | split | case inl
t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h₁ : ▲(prev ∪ {f}) ∩ Δ = ∅
⊢ f ∈ if f¦g ∈ ▲prev then if ▲(prev ∪ {f}) ∩ Δ = ∅ then prev ∪ {f} else prev ∪ {g} else prev | case inl.inl
t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h₁ : ▲(prev ∪ {f}) ∩ Δ = ∅
h✝ : f¦g ∈ ▲prev
⊢ f ∈ prev ∪ {f}
case inl.inr
t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h₁ : ▲(prev ∪ {f}) ∩ Δ = ∅
h✝ : ¬f¦g ∈ ▲prev
⊢ f ∈ prev |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | case inl => exact Or.inr rfl | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h₁ : ▲(prev ∪ {f}) ∩ Δ = ∅
h✝ : f¦g ∈ ▲prev
⊢ f ∈ prev ∪ {f} | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | case inr h₂ => exact False.elim $ h₂ ⟨BProof.ax l₂⟩ | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h₁ : ▲(prev ∪ {f}) ∩ Δ = ∅
h₂ : ¬f¦g ∈ ▲prev
⊢ f ∈ prev | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | exact Or.inr rfl | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h₁ : ▲(prev ∪ {f}) ∩ Δ = ∅
h✝ : f¦g ∈ ▲prev
⊢ f ∈ prev ∪ {f} | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | exact False.elim $ h₂ ⟨BProof.ax l₂⟩ | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
prev : Ctx := lindenbaumSequence t Δ (i + 1, k)
l₄ : Denumerable.ofNat (Form × Form) k = (f, g)
h₁ : ▲(prev ∪ {f}) ∩ Δ = ∅
h₂ : ¬f¦g ∈ ▲prev
⊢ f ∈ prev | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | intros h₁ | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
l₃ : f ∈ lindenbaumSequence t Δ (i + 1, k + 1) ∨ g ∈ lindenbaumSequence t Δ (i + 1, k + 1)
⊢ f ∈ lindenbaumSequence t Δ (i + 1, k + 1) → f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
l₃ : f ∈ lindenbaumSequence t Δ (i + 1, k + 1) ∨ g ∈ lindenbaumSequence t Δ (i + 1, k + 1)
h₁ : f ∈ lindenbaumSequence t Δ (i + 1, k + 1)
⊢ f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | exact Or.inl ⟨⟨i+1,k+1⟩,h₁⟩ | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
l₃ : f ∈ lindenbaumSequence t Δ (i + 1, k + 1) ∨ g ∈ lindenbaumSequence t Δ (i + 1, k + 1)
h₁ : f ∈ lindenbaumSequence t Δ (i + 1, k + 1)
⊢ f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | intros h₁ | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
l₃ : f ∈ lindenbaumSequence t Δ (i + 1, k + 1) ∨ g ∈ lindenbaumSequence t Δ (i + 1, k + 1)
⊢ g ∈ lindenbaumSequence t Δ (i + 1, k + 1) → f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
l₃ : f ∈ lindenbaumSequence t Δ (i + 1, k + 1) ∨ g ∈ lindenbaumSequence t Δ (i + 1, k + 1)
h₁ : g ∈ lindenbaumSequence t Δ (i + 1, k + 1)
⊢ f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumIsPrime | [150, 1] | [186, 55] | exact Or.inr ⟨⟨i+1,k+1⟩,h₁⟩ | t : Th
Δ : Ctx
f g : Form
i j : ℕ
k : ℕ := Encodable.encode (f, g)
l₂ : f¦g ∈ lindenbaumSequence t Δ (i + 1, k)
l₃ : f ∈ lindenbaumSequence t Δ (i + 1, k + 1) ∨ g ∈ lindenbaumSequence t Δ (i + 1, k + 1)
h₁ : g ∈ lindenbaumSequence t Δ (i + 1, k + 1)
⊢ f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | have l₁ := formalFixed t.property | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
⊢ ▲lindenbaumSequence t Δ (0, 0) ∩ Δ = ∅ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
l₁ : ▲↑t = ↑t
⊢ ▲lindenbaumSequence t Δ (0, 0) ∩ Δ = ∅ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | rw [←l₁] at h₁ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
l₁ : ▲↑t = ↑t
⊢ ▲lindenbaumSequence t Δ (0, 0) ∩ Δ = ∅ | t : Th
Δ : Ctx
h₁ : ▲↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
l₁ : ▲↑t = ↑t
⊢ ▲lindenbaumSequence t Δ (0, 0) ∩ Δ = ∅ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | exact h₁ | t : Th
Δ : Ctx
h₁ : ▲↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
l₁ : ▲↑t = ↑t
⊢ ▲lindenbaumSequence t Δ (0, 0) ∩ Δ = ∅ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | change
▲{ f : Form | ∃j : Nat, f ∈ lindenbaumSequence t Δ ⟨i, j⟩ } ∩ Δ = ∅ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i : ℕ
⊢ ▲lindenbaumSequence t Δ (i + 1, 0) ∩ Δ = ∅ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i : ℕ
⊢ ▲{ f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) } ∩ Δ = ∅ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | apply Set.not_nonempty_iff_eq_empty.mp | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i : ℕ
⊢ ▲{ f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) } ∩ Δ = ∅ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i : ℕ
⊢ ¬Set.Nonempty (▲{ f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) } ∩ Δ) |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | intros h₃ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i : ℕ
⊢ ¬Set.Nonempty (▲{ f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) } ∩ Δ) | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i : ℕ
h₃ : Set.Nonempty (▲{ f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) } ∩ Δ)
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | have ⟨w,⟨prf₁⟩,l₁⟩ := h₃ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i : ℕ
h₃ : Set.Nonempty (▲{ f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) } ∩ Δ)
⊢ False | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i : ℕ
h₃ : Set.Nonempty (▲{ f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) } ∩ Δ)
w : Form
prf₁ : BProof { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) } w
l₁ : w ∈ Δ
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | clear h₃ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i : ℕ
h₃ : Set.Nonempty (▲{ f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) } ∩ Δ)
w : Form
prf₁ : BProof { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) } w
l₁ : w ∈ Δ
⊢ False | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i : ℕ
w : Form
prf₁ : BProof { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) } w
l₁ : w ∈ Δ
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | have ⟨s,l₂,prf₂⟩ := BProof.compactness prf₁ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i : ℕ
w : Form
prf₁ : BProof { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) } w
l₁ : w ∈ Δ
⊢ False | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i : ℕ
w : Form
prf₁ : BProof { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) } w
l₁ : w ∈ Δ
s : Finset Form
l₂ : ↑s ⊆ { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) }
prf₂ : BProof (↑s) w
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | have ⟨j, l₃⟩ := finiteExhaustion lindenbaumStageMonotone l₂ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i : ℕ
w : Form
prf₁ : BProof { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) } w
l₁ : w ∈ Δ
s : Finset Form
l₂ : ↑s ⊆ { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) }
prf₂ : BProof (↑s) w
⊢ False | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i : ℕ
w : Form
prf₁ : BProof { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) } w
l₁ : w ∈ Δ
s : Finset Form
l₂ : ↑s ⊆ { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) }
prf₂ : BProof (↑s) w
j : ℕ
l₃ : ↑s ⊆ lindenbaumSequence t Δ (i, j)
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | have l₄ := lindenbaumAvoids h₁ h₂ ⟨i,j⟩ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i : ℕ
w : Form
prf₁ : BProof { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) } w
l₁ : w ∈ Δ
s : Finset Form
l₂ : ↑s ⊆ { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) }
prf₂ : BProof (↑s) w
j : ℕ
l₃ : ↑s ⊆ lindenbaumSequence t Δ (i, j)
⊢ False | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i : ℕ
w : Form
prf₁ : BProof { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) } w
l₁ : w ∈ Δ
s : Finset Form
l₂ : ↑s ⊆ { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) }
prf₂ : BProof (↑s) w
j : ℕ
l₃ : ↑s ⊆ lindenbaumSequence t Δ (i, j)
l₄ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | have prf₃ := BProof.monotone l₃ prf₂ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i : ℕ
w : Form
prf₁ : BProof { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) } w
l₁ : w ∈ Δ
s : Finset Form
l₂ : ↑s ⊆ { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) }
prf₂ : BProof (↑s) w
j : ℕ
l₃ : ↑s ⊆ lindenbaumSequence t Δ (i, j)
l₄ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
⊢ False | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i : ℕ
w : Form
prf₁ : BProof { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) } w
l₁ : w ∈ Δ
s : Finset Form
l₂ : ↑s ⊆ { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) }
prf₂ : BProof (↑s) w
j : ℕ
l₃ : ↑s ⊆ lindenbaumSequence t Δ (i, j)
l₄ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
prf₃ : BProof (lindenbaumSequence t Δ (i, j)) w
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | have l₅ := Set.not_nonempty_iff_eq_empty.mpr l₄ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i : ℕ
w : Form
prf₁ : BProof { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) } w
l₁ : w ∈ Δ
s : Finset Form
l₂ : ↑s ⊆ { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) }
prf₂ : BProof (↑s) w
j : ℕ
l₃ : ↑s ⊆ lindenbaumSequence t Δ (i, j)
l₄ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
prf₃ : BProof (lindenbaumSequence t Δ (i, j)) w
⊢ False | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i : ℕ
w : Form
prf₁ : BProof { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) } w
l₁ : w ∈ Δ
s : Finset Form
l₂ : ↑s ⊆ { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) }
prf₂ : BProof (↑s) w
j : ℕ
l₃ : ↑s ⊆ lindenbaumSequence t Δ (i, j)
l₄ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
prf₃ : BProof (lindenbaumSequence t Δ (i, j)) w
l₅ : ¬Set.Nonempty (▲lindenbaumSequence t Δ (i, j) ∩ Δ)
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | exact l₅ ⟨w, ⟨⟨prf₃⟩,l₁⟩⟩ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i : ℕ
w : Form
prf₁ : BProof { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) } w
l₁ : w ∈ Δ
s : Finset Form
l₂ : ↑s ⊆ { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) }
prf₂ : BProof (↑s) w
j : ℕ
l₃ : ↑s ⊆ lindenbaumSequence t Δ (i, j)
l₄ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
prf₃ : BProof (lindenbaumSequence t Δ (i, j)) w
l₅ : ¬Set.Nonempty (▲lindenbaumSequence t Δ (i, j) ∩ Δ)
⊢ False | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | apply Set.not_nonempty_iff_eq_empty.mp | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
⊢ ▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ = ∅ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
⊢ ¬Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ) |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | intros h₃ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
⊢ ¬Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ) | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | have ⟨w₁,l₁,l₂⟩ := h₃ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
⊢ False | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₁ : w₁ ∈ ▲lindenbaumSequence t Δ (i, j + 1)
l₂ : w₁ ∈ Δ
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | unfold lindenbaumSequence at l₁ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₁ : w₁ ∈ ▲lindenbaumSequence t Δ (i, j + 1)
l₂ : w₁ ∈ Δ
⊢ False | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
l₁ :
w₁ ∈
▲match (i, j + 1) with
| (0, 0) => ↑t
| (Nat.succ i, 0) => { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) }
| (i, Nat.succ j) =>
let prev := lindenbaumSequence t Δ (i, j);
let l := (Denumerable.ofNat (Form × Form) j).fst;
let r := (Denumerable.ofNat (Form × Form) j).snd;
if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | split at l₁ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
l₁ :
w₁ ∈
▲match (i, j + 1) with
| (0, 0) => ↑t
| (Nat.succ i, 0) => { f | ∃ j, f ∈ lindenbaumSequence t Δ (i, j) }
| (i, Nat.succ j) =>
let prev := lindenbaumSequence t Δ (i, j);
let l := (Denumerable.ofNat (Form × Form) j).fst;
let r := (Denumerable.ofNat (Form × Form) j).snd;
if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev
⊢ False | case h_1
t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
x✝ : Lex (ℕ × ℕ)
heq✝ : (i, j + 1) = (0, 0)
l₁ : w₁ ∈ ▲↑t
⊢ False
case h_2
t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
x✝ : Lex (ℕ × ℕ)
i✝ : ℕ
heq✝ : (i, j + 1) = (Nat.succ i✝, 0)
l₁ : w₁ ∈ ▲{ f | ∃ j, f ∈ lindenbaumSequence t Δ (i✝, j) }
⊢ False
case h_3
t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
x✝ : Lex (ℕ × ℕ)
i✝ j✝ : ℕ
heq✝ : (i, j + 1) = (i✝, Nat.succ j✝)
l₁ :
w₁ ∈
▲let prev := lindenbaumSequence t Δ (i✝, j✝);
let l := (Denumerable.ofNat (Form × Form) j✝).fst;
let r := (Denumerable.ofNat (Form × Form) j✝).snd;
if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | case h_1 x heq => injection heq with heq; contradiction | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
x : Lex (ℕ × ℕ)
heq : (i, j + 1) = (0, 0)
l₁ : w₁ ∈ ▲↑t
⊢ False | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | case h_2 x heq => injection heq with heq; contradiction | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
x✝ : Lex (ℕ × ℕ)
x : ℕ
heq : (i, j + 1) = (Nat.succ x, 0)
l₁ : w₁ ∈ ▲{ f | ∃ j, f ∈ lindenbaumSequence t Δ (x, j) }
⊢ False | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | case h_3 x n m heq =>
have l₃ := lindenbaumAvoids h₁ h₂ ⟨i,j⟩
injection heq with heq₁ heq₂
injection heq₂ with heq₂
rw [←heq₁,←heq₂] at l₁
clear n m x heq₁ heq₂ h₃
dsimp at l₁ j
split at l₁
case inr h₄ => exact (Set.not_nonempty_iff_eq_empty.mpr l₃) ⟨w₁, l₁, l₂⟩
case inl h₄ =>
split at l₁
case inl h₅ => exact (Set.not_nonempty_iff_eq_empty.mpr h₅) ⟨w₁, l₁, l₂⟩
case inr h₅ =>
have ⟨prf₁⟩ := l₁
have ⟨w₂,⟨⟨prf₂⟩,l₄⟩⟩ := Set.nonempty_iff_ne_empty.mpr h₅
have l₅ : w₁¦w₂ ∈ Δ := h₂ ⟨l₂, l₄⟩
clear l₁ l₂ l₄ h₅
have ⟨lst₁,l₆,prf₃⟩ := BProof.sentenceCompactness (Set.union_singleton ▸ prf₁)
have ⟨lst₂,l₇,prf₄⟩ := BProof.sentenceCompactness (Set.union_singleton ▸ prf₂)
have thm₁ := BTheorem.transitivity prf₃.toTheorem (BTheorem.orI₁ : BTheorem (w₁ ⊃ w₁ ¦ w₂))
have thm₂ := BTheorem.transitivity prf₄.toTheorem (BTheorem.orI₂ : BTheorem (w₂ ⊃ w₁ ¦ w₂))
have thm₃ := BTheorem.mp (BTheorem.adj thm₂ thm₁) BTheorem.orE
have ⟨prf₅⟩ := h₄
clear h₄ thm₁ thm₂ prf₁ prf₂ prf₃ prf₄
cases lst₁
all_goals
cases lst₂
case' nil.nil =>
have : w₁¦w₂ ∈ ▲lindenbaumSequence t Δ (i, j) ∩ Δ := ⟨⟨BProof.mp prf₅ thm₃⟩, l₅⟩
exact (Set.not_nonempty_iff_eq_empty.mpr l₃) ⟨w₁¦w₂, this⟩
case' nil.cons head tail =>
have := BProof.proveList l₇
have := BProof.mp (BProof.adj prf₅ this) BTheorem.distRight
have := BProof.mp this (BTheorem.orFunctor BTheorem.taut BTheorem.andE₁)
have : w₁¦w₂ ∈ ▲lindenbaumSequence t Δ (i, j) ∩ Δ := ⟨⟨BProof.mp this thm₃⟩, l₅⟩
exact (Set.not_nonempty_iff_eq_empty.mpr l₃) ⟨w₁¦w₂, this⟩
case' cons.nil head tail =>
have := BProof.proveList l₆
have := BProof.mp (BProof.adj prf₅ this) BTheorem.distRight
have := BProof.mp this (BTheorem.orFunctor BTheorem.andE₁ BTheorem.taut)
have : w₁¦w₂ ∈ ▲lindenbaumSequence t Δ (i, j) ∩ Δ := ⟨⟨BProof.mp this thm₃⟩, l₅⟩
exact (Set.not_nonempty_iff_eq_empty.mpr l₃) ⟨w₁¦w₂, this⟩
case' cons.cons head tail head' tail'=>
have prf₆ := BProof.proveList l₆
have prf₇ := BProof.proveList l₇
have := BProof.mp (BProof.adj prf₅ prf₇) BTheorem.distRight
have prf₈ := BProof.mp this (BTheorem.orFunctor BTheorem.taut BTheorem.andE₁)
have := BProof.mp (BProof.adj prf₈ prf₆) BTheorem.distRight
have prf₉ := BProof.mp this (BTheorem.orFunctor BTheorem.andE₁ BTheorem.taut)
have : w₁¦w₂ ∈ ▲lindenbaumSequence t Δ (i, j) ∩ Δ := ⟨⟨BProof.mp prf₉ thm₃⟩, l₅⟩
exact (Set.not_nonempty_iff_eq_empty.mpr l₃) ⟨w₁¦w₂, this⟩ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
x : Lex (ℕ × ℕ)
n m : ℕ
heq : (i, j + 1) = (n, Nat.succ m)
l₁ :
w₁ ∈
▲let prev := lindenbaumSequence t Δ (n, m);
let l := (Denumerable.ofNat (Form × Form) m).fst;
let r := (Denumerable.ofNat (Form × Form) m).snd;
if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev
⊢ False | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | injection heq with heq | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
x : Lex (ℕ × ℕ)
heq : (i, j + 1) = (0, 0)
l₁ : w₁ ∈ ▲↑t
⊢ False | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
x : Lex (ℕ × ℕ)
l₁ : w₁ ∈ ▲↑t
heq : i = 0
snd_eq✝ : j + 1 = 0
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | contradiction | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
x : Lex (ℕ × ℕ)
l₁ : w₁ ∈ ▲↑t
heq : i = 0
snd_eq✝ : j + 1 = 0
⊢ False | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | injection heq with heq | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
x✝ : Lex (ℕ × ℕ)
x : ℕ
heq : (i, j + 1) = (Nat.succ x, 0)
l₁ : w₁ ∈ ▲{ f | ∃ j, f ∈ lindenbaumSequence t Δ (x, j) }
⊢ False | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
x✝ : Lex (ℕ × ℕ)
x : ℕ
l₁ : w₁ ∈ ▲{ f | ∃ j, f ∈ lindenbaumSequence t Δ (x, j) }
heq : i = Nat.succ x
snd_eq✝ : j + 1 = 0
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | contradiction | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
x✝ : Lex (ℕ × ℕ)
x : ℕ
l₁ : w₁ ∈ ▲{ f | ∃ j, f ∈ lindenbaumSequence t Δ (x, j) }
heq : i = Nat.succ x
snd_eq✝ : j + 1 = 0
⊢ False | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | have l₃ := lindenbaumAvoids h₁ h₂ ⟨i,j⟩ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
x : Lex (ℕ × ℕ)
n m : ℕ
heq : (i, j + 1) = (n, Nat.succ m)
l₁ :
w₁ ∈
▲let prev := lindenbaumSequence t Δ (n, m);
let l := (Denumerable.ofNat (Form × Form) m).fst;
let r := (Denumerable.ofNat (Form × Form) m).snd;
if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev
⊢ False | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
x : Lex (ℕ × ℕ)
n m : ℕ
heq : (i, j + 1) = (n, Nat.succ m)
l₁ :
w₁ ∈
▲let prev := lindenbaumSequence t Δ (n, m);
let l := (Denumerable.ofNat (Form × Form) m).fst;
let r := (Denumerable.ofNat (Form × Form) m).snd;
if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev
l₃ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | injection heq with heq₁ heq₂ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
x : Lex (ℕ × ℕ)
n m : ℕ
heq : (i, j + 1) = (n, Nat.succ m)
l₁ :
w₁ ∈
▲let prev := lindenbaumSequence t Δ (n, m);
let l := (Denumerable.ofNat (Form × Form) m).fst;
let r := (Denumerable.ofNat (Form × Form) m).snd;
if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev
l₃ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
⊢ False | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
x : Lex (ℕ × ℕ)
n m : ℕ
l₁ :
w₁ ∈
▲let prev := lindenbaumSequence t Δ (n, m);
let l := (Denumerable.ofNat (Form × Form) m).fst;
let r := (Denumerable.ofNat (Form × Form) m).snd;
if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev
l₃ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
heq₁ : i = n
heq₂ : j + 1 = Nat.succ m
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | injection heq₂ with heq₂ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
x : Lex (ℕ × ℕ)
n m : ℕ
l₁ :
w₁ ∈
▲let prev := lindenbaumSequence t Δ (n, m);
let l := (Denumerable.ofNat (Form × Form) m).fst;
let r := (Denumerable.ofNat (Form × Form) m).snd;
if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev
l₃ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
heq₁ : i = n
heq₂ : j + 1 = Nat.succ m
⊢ False | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
x : Lex (ℕ × ℕ)
n m : ℕ
l₁ :
w₁ ∈
▲let prev := lindenbaumSequence t Δ (n, m);
let l := (Denumerable.ofNat (Form × Form) m).fst;
let r := (Denumerable.ofNat (Form × Form) m).snd;
if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev
l₃ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
heq₁ : i = n
heq₂ : Nat.add j 0 = m
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | rw [←heq₁,←heq₂] at l₁ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
x : Lex (ℕ × ℕ)
n m : ℕ
l₁ :
w₁ ∈
▲let prev := lindenbaumSequence t Δ (n, m);
let l := (Denumerable.ofNat (Form × Form) m).fst;
let r := (Denumerable.ofNat (Form × Form) m).snd;
if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev
l₃ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
heq₁ : i = n
heq₂ : Nat.add j 0 = m
⊢ False | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
x : Lex (ℕ × ℕ)
n m : ℕ
l₁ :
w₁ ∈
▲let prev := lindenbaumSequence t Δ (i, Nat.add j 0);
let l := (Denumerable.ofNat (Form × Form) (Nat.add j 0)).fst;
let r := (Denumerable.ofNat (Form × Form) (Nat.add j 0)).snd;
if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev
l₃ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
heq₁ : i = n
heq₂ : Nat.add j 0 = m
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | clear n m x heq₁ heq₂ h₃ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
h₃ : Set.Nonempty (▲lindenbaumSequence t Δ (i, j + 1) ∩ Δ)
w₁ : Form
l₂ : w₁ ∈ Δ
x : Lex (ℕ × ℕ)
n m : ℕ
l₁ :
w₁ ∈
▲let prev := lindenbaumSequence t Δ (i, Nat.add j 0);
let l := (Denumerable.ofNat (Form × Form) (Nat.add j 0)).fst;
let r := (Denumerable.ofNat (Form × Form) (Nat.add j 0)).snd;
if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev
l₃ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
heq₁ : i = n
heq₂ : Nat.add j 0 = m
⊢ False | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
w₁ : Form
l₂ : w₁ ∈ Δ
l₁ :
w₁ ∈
▲let prev := lindenbaumSequence t Δ (i, Nat.add j 0);
let l := (Denumerable.ofNat (Form × Form) (Nat.add j 0)).fst;
let r := (Denumerable.ofNat (Form × Form) (Nat.add j 0)).snd;
if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev
l₃ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | dsimp at l₁ j | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
w₁ : Form
l₂ : w₁ ∈ Δ
l₁ :
w₁ ∈
▲let prev := lindenbaumSequence t Δ (i, Nat.add j 0);
let l := (Denumerable.ofNat (Form × Form) (Nat.add j 0)).fst;
let r := (Denumerable.ofNat (Form × Form) (Nat.add j 0)).snd;
if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev
l₃ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
⊢ False | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
w₁ : Form
l₂ : w₁ ∈ Δ
l₁ :
w₁ ∈
▲if
(Denumerable.ofNat (Form × Form) (j + 0)).fst¦(Denumerable.ofNat (Form × Form) (j + 0)).snd ∈
▲lindenbaumSequence t Δ (i, j + 0) then
if ▲(lindenbaumSequence t Δ (i, j + 0) ∪ {(Denumerable.ofNat (Form × Form) (j + 0)).fst}) ∩ Δ = ∅ then
lindenbaumSequence t Δ (i, j + 0) ∪ {(Denumerable.ofNat (Form × Form) (j + 0)).fst}
else lindenbaumSequence t Δ (i, j + 0) ∪ {(Denumerable.ofNat (Form × Form) (j + 0)).snd}
else lindenbaumSequence t Δ (i, j + 0)
l₃ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | split at l₁ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
w₁ : Form
l₂ : w₁ ∈ Δ
l₁ :
w₁ ∈
▲if
(Denumerable.ofNat (Form × Form) (j + 0)).fst¦(Denumerable.ofNat (Form × Form) (j + 0)).snd ∈
▲lindenbaumSequence t Δ (i, j + 0) then
if ▲(lindenbaumSequence t Δ (i, j + 0) ∪ {(Denumerable.ofNat (Form × Form) (j + 0)).fst}) ∩ Δ = ∅ then
lindenbaumSequence t Δ (i, j + 0) ∪ {(Denumerable.ofNat (Form × Form) (j + 0)).fst}
else lindenbaumSequence t Δ (i, j + 0) ∪ {(Denumerable.ofNat (Form × Form) (j + 0)).snd}
else lindenbaumSequence t Δ (i, j + 0)
l₃ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
⊢ False | case inl
t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
w₁ : Form
l₂ : w₁ ∈ Δ
l₃ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
h✝ :
(Denumerable.ofNat (Form × Form) (j + 0)).fst¦(Denumerable.ofNat (Form × Form) (j + 0)).snd ∈
▲lindenbaumSequence t Δ (i, j + 0)
l₁ :
w₁ ∈
▲if ▲(lindenbaumSequence t Δ (i, j + 0) ∪ {(Denumerable.ofNat (Form × Form) (j + 0)).fst}) ∩ Δ = ∅ then
lindenbaumSequence t Δ (i, j + 0) ∪ {(Denumerable.ofNat (Form × Form) (j + 0)).fst}
else lindenbaumSequence t Δ (i, j + 0) ∪ {(Denumerable.ofNat (Form × Form) (j + 0)).snd}
⊢ False
case inr
t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
w₁ : Form
l₂ : w₁ ∈ Δ
l₃ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
h✝ :
¬(Denumerable.ofNat (Form × Form) (j + 0)).fst¦(Denumerable.ofNat (Form × Form) (j + 0)).snd ∈
▲lindenbaumSequence t Δ (i, j + 0)
l₁ : w₁ ∈ ▲lindenbaumSequence t Δ (i, j + 0)
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Lindenbaum.lean | lindenbaumAvoids | [188, 1] | [265, 62] | case inr h₄ => exact (Set.not_nonempty_iff_eq_empty.mpr l₃) ⟨w₁, l₁, l₂⟩ | t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
w₁ : Form
l₂ : w₁ ∈ Δ
l₃ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
h₄ :
¬(Denumerable.ofNat (Form × Form) (j + 0)).fst¦(Denumerable.ofNat (Form × Form) (j + 0)).snd ∈
▲lindenbaumSequence t Δ (i, j + 0)
l₁ : w₁ ∈ ▲lindenbaumSequence t Δ (i, j + 0)
⊢ False | no goals |
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