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https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | exact ⟨S & U, l₁₄, ⟨prf₅⟩⟩ | case a
t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p }
l₂ : ↑t ∩ Δ = ∅
P Q R S : Form
l₆ : S ∈ ↑u
T U : Form
l₁₀ : U ∈ ↑u
prf₅ : BProof {P¦Q} (S&U⊃R¦T)
l₁₄ : S&U ∈ u
⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | apply byContradiction | t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p }
l₄ : lindenbaumExtension t Δ ∩ Δ = ∅
P : Form
h₁ : P ∈ formalApplication (lindenbaumExtension t Δ) ↑u
Q : Form
h₂ : Q ∈ ↑u
h₃ : Q⊃P ∈ lindenbaumExtension t Δ
⊢ formalApplication (▲{Q⊃P}) ↑u ⊆ ↑↑p | case h
t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p }
l₄ : lindenbaumExtension t Δ ∩ Δ = ∅
P : Form
h₁ : P ∈ formalApplication (lindenbaumExtension t Δ) ↑u
Q : Form
h₂ : Q ∈ ↑u
h₃ : Q⊃P ∈ lindenbaumExtension t Δ
⊢ ¬formalApplication (▲{Q⊃P}) ↑u ⊆ ↑↑p → False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | intros h₄ | case h
t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p }
l₄ : lindenbaumExtension t Δ ∩ Δ = ∅
P : Form
h₁ : P ∈ formalApplication (lindenbaumExtension t Δ) ↑u
Q : Form
h₂ : Q ∈ ↑u
h₃ : Q⊃P ∈ lindenbaumExtension t Δ
⊢ ¬formalApplication (▲{Q⊃P}) ↑u ⊆ ↑↑p → False | case h
t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p }
l₄ : lindenbaumExtension t Δ ∩ Δ = ∅
P : Form
h₁ : P ∈ formalApplication (lindenbaumExtension t Δ) ↑u
Q : Form
h₂ : Q ∈ ↑u
h₃ : Q⊃P ∈ lindenbaumExtension t Δ
h₄ : ¬formalApplication (▲{Q⊃P}) ↑u ⊆ ↑↑p
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | exact (Set.eq_empty_iff_forall_not_mem.mp l₄) (Q⊃P) ⟨h₃,h₄⟩ | case h
t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p }
l₄ : lindenbaumExtension t Δ ∩ Δ = ∅
P : Form
h₁ : P ∈ formalApplication (lindenbaumExtension t Δ) ↑u
Q : Form
h₂ : Q ∈ ↑u
h₃ : Q⊃P ∈ lindenbaumExtension t Δ
h₄ : ¬formalApplication (▲{Q⊃P}) ↑u ⊆ ↑↑p
⊢ False | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | intros t u p h₁ | ⊢ ∀ (t u : Th) (p : Pr), formalApplicationFunction t u ≤ ↑p → ∃ r, u ≤ ↑r ∧ formalApplication ↑t ↑↑r ⊆ ↑↑p | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
⊢ ∃ r, u ≤ ↑r ∧ formalApplication ↑t ↑↑r ⊆ ↑↑p |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | let Δ := {f : Form | ¬(formalApplication t (▲{f}) ⊆ p) } | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
⊢ ∃ r, u ≤ ↑r ∧ formalApplication ↑t ↑↑r ⊆ ↑↑p | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
⊢ ∃ r, u ≤ ↑r ∧ formalApplication ↑t ↑↑r ⊆ ↑↑p |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | have l₂ : ↑u ∩ Δ = ∅ := by
apply Set.eq_empty_iff_forall_not_mem.mpr
intros P h₂
have l₃ : ▲{P} ⊆ ↑u := generatedContained (Set.singleton_subset_iff.mpr h₂.left)
have l₄ := formalAppMonotoneLeft ↑t l₃
exact h₂.right $ le_trans l₄ h₁ | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
⊢ ∃ r, u ≤ ↑r ∧ formalApplication ↑t ↑↑r ⊆ ↑↑p | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
⊢ ∃ r, u ≤ ↑r ∧ formalApplication ↑t ↑↑r ⊆ ↑↑p |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | have l₃ : isDisjunctionClosed Δ := by
intros P Q h₁ h₂
have ⟨R,l₄⟩ := nonconstruction h₁.left
have ⟨⟨S,⟨prf₁⟩,l₆⟩,l₈⟩ := nonconstruction l₄
have ⟨T,l₉⟩ := nonconstruction h₁.right
have ⟨⟨U,⟨prf₂⟩,l₁₀⟩,l₁₂⟩ := nonconstruction l₉
clear h₁ l₄ l₉
have l₁₃ : ¬(R¦T ∈ p) := λw => Or.elim (p.property w) l₈ l₁₂
apply l₁₃
apply h₂
clear l₈ l₁₂ l₁₃ h₂
have l₁₄ : S¦U ∈ ▲{P¦Q} := ⟨(BTheorem.orFunctor prf₁.toTheorem prf₂.toTheorem).toProof⟩
have l₁₅ : (S¦U ⊃ R¦T) ∈ t := t.property.mpr ⟨BProof.mp (BProof.adj
(BProof.mp (BProof.ax l₆) (BTheorem.hs BTheorem.taut BTheorem.orI₁))
(BProof.mp (BProof.ax l₁₀) (BTheorem.hs BTheorem.taut BTheorem.orI₂)))
BTheorem.orE⟩
exact ⟨S ¦ U, l₁₄, l₁₅⟩ | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
⊢ ∃ r, u ≤ ↑r ∧ formalApplication ↑t ↑↑r ⊆ ↑↑p | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
l₃ : isDisjunctionClosed Δ
⊢ ∃ r, u ≤ ↑r ∧ formalApplication ↑t ↑↑r ⊆ ↑↑p |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | have l₄ : lindenbaumExtension u Δ ∩ Δ = ∅ := lindenbaumTheorem l₂ l₃ | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
l₃ : isDisjunctionClosed Δ
⊢ ∃ r, u ≤ ↑r ∧ formalApplication ↑t ↑↑r ⊆ ↑↑p | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
l₃ : isDisjunctionClosed Δ
l₄ : lindenbaumExtension u Δ ∩ Δ = ∅
⊢ ∃ r, u ≤ ↑r ∧ formalApplication ↑t ↑↑r ⊆ ↑↑p |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | clear l₂ l₃ | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
l₃ : isDisjunctionClosed Δ
l₄ : lindenbaumExtension u Δ ∩ Δ = ∅
⊢ ∃ r, u ≤ ↑r ∧ formalApplication ↑t ↑↑r ⊆ ↑↑p | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₄ : lindenbaumExtension u Δ ∩ Δ = ∅
⊢ ∃ r, u ≤ ↑r ∧ formalApplication ↑t ↑↑r ⊆ ↑↑p |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | refine ⟨⟨⟨lindenbaumExtension u Δ, lindenbaumIsFormal⟩, lindenbaumIsPrime⟩, lindenbaumExtensionExtends, ?_⟩ | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₄ : lindenbaumExtension u Δ ∩ Δ = ∅
⊢ ∃ r, u ≤ ↑r ∧ formalApplication ↑t ↑↑r ⊆ ↑↑p | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₄ : lindenbaumExtension u Δ ∩ Δ = ∅
⊢ formalApplication ↑t
↑↑{
val :=
{ val := lindenbaumExtension u Δ,
property := (_ : ∀ {f : Form}, f ∈ lindenbaumExtension u Δ ↔ lindenbaumExtension u Δ⊢f) },
property :=
(_ :
∀ {f g : Form},
f¦g ∈ lindenbaumExtension u Δ → f ∈ lindenbaumExtension u Δ ∨ g ∈ lindenbaumExtension u Δ) } ⊆
↑↑p |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | change formalApplication ↑t (lindenbaumExtension u Δ) ⊆ p | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₄ : lindenbaumExtension u Δ ∩ Δ = ∅
⊢ formalApplication ↑t
↑↑{
val :=
{ val := lindenbaumExtension u Δ,
property := (_ : ∀ {f : Form}, f ∈ lindenbaumExtension u Δ ↔ lindenbaumExtension u Δ⊢f) },
property :=
(_ :
∀ {f g : Form},
f¦g ∈ lindenbaumExtension u Δ → f ∈ lindenbaumExtension u Δ ∨ g ∈ lindenbaumExtension u Δ) } ⊆
↑↑p | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₄ : lindenbaumExtension u Δ ∩ Δ = ∅
⊢ formalApplication (↑t) (lindenbaumExtension u Δ) ⊆ ↑↑p |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | intros P h₁ | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₄ : lindenbaumExtension u Δ ∩ Δ = ∅
⊢ formalApplication (↑t) (lindenbaumExtension u Δ) ⊆ ↑↑p | t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₄ : lindenbaumExtension u Δ ∩ Δ = ∅
P : Form
h₁ : P ∈ formalApplication (↑t) (lindenbaumExtension u Δ)
⊢ P ∈ ↑↑p |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | have ⟨Q,h₂,h₃⟩ := h₁ | t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₄ : lindenbaumExtension u Δ ∩ Δ = ∅
P : Form
h₁ : P ∈ formalApplication (↑t) (lindenbaumExtension u Δ)
⊢ P ∈ ↑↑p | t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₄ : lindenbaumExtension u Δ ∩ Δ = ∅
P : Form
h₁ : P ∈ formalApplication (↑t) (lindenbaumExtension u Δ)
Q : Form
h₂ : Q ∈ lindenbaumExtension u Δ
h₃ : Q⊃P ∈ ↑t
⊢ P ∈ ↑↑p |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | have l₄ : formalApplication ↑t (▲{Q}) ⊆ ↑↑p := by
apply byContradiction
intros h₄
exact (Set.eq_empty_iff_forall_not_mem.mp l₄) Q ⟨h₂,h₄⟩ | t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₄ : lindenbaumExtension u Δ ∩ Δ = ∅
P : Form
h₁ : P ∈ formalApplication (↑t) (lindenbaumExtension u Δ)
Q : Form
h₂ : Q ∈ lindenbaumExtension u Δ
h₃ : Q⊃P ∈ ↑t
⊢ P ∈ ↑↑p | t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₄✝ : lindenbaumExtension u Δ ∩ Δ = ∅
P : Form
h₁ : P ∈ formalApplication (↑t) (lindenbaumExtension u Δ)
Q : Form
h₂ : Q ∈ lindenbaumExtension u Δ
h₃ : Q⊃P ∈ ↑t
l₄ : formalApplication (↑t) (▲{Q}) ⊆ ↑↑p
⊢ P ∈ ↑↑p |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | exact l₄ ⟨Q,⟨BProof.ax rfl⟩,h₃⟩ | t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₄✝ : lindenbaumExtension u Δ ∩ Δ = ∅
P : Form
h₁ : P ∈ formalApplication (↑t) (lindenbaumExtension u Δ)
Q : Form
h₂ : Q ∈ lindenbaumExtension u Δ
h₃ : Q⊃P ∈ ↑t
l₄ : formalApplication (↑t) (▲{Q}) ⊆ ↑↑p
⊢ P ∈ ↑↑p | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | apply Set.eq_empty_iff_forall_not_mem.mpr | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
⊢ ↑u ∩ Δ = ∅ | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
⊢ ∀ (x : Form), ¬x ∈ ↑u ∩ Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | intros P h₂ | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
⊢ ∀ (x : Form), ¬x ∈ ↑u ∩ Δ | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
P : Form
h₂ : P ∈ ↑u ∩ Δ
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | have l₃ : ▲{P} ⊆ ↑u := generatedContained (Set.singleton_subset_iff.mpr h₂.left) | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
P : Form
h₂ : P ∈ ↑u ∩ Δ
⊢ False | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
P : Form
h₂ : P ∈ ↑u ∩ Δ
l₃ : ▲{P} ⊆ ↑u
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | have l₄ := formalAppMonotoneLeft ↑t l₃ | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
P : Form
h₂ : P ∈ ↑u ∩ Δ
l₃ : ▲{P} ⊆ ↑u
⊢ False | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
P : Form
h₂ : P ∈ ↑u ∩ Δ
l₃ : ▲{P} ⊆ ↑u
l₄ : formalApplication (↑t) (▲{P}) ≤ formalApplication ↑t ↑u
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | exact h₂.right $ le_trans l₄ h₁ | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
P : Form
h₂ : P ∈ ↑u ∩ Δ
l₃ : ▲{P} ⊆ ↑u
l₄ : formalApplication (↑t) (▲{P}) ≤ formalApplication ↑t ↑u
⊢ False | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | intros P Q h₁ h₂ | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
⊢ isDisjunctionClosed Δ | t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q : Form
h₁ : P ∈ Δ ∧ Q ∈ Δ
h₂ : formalApplication (↑t) (▲{P¦Q}) ⊆ ↑↑p
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | have ⟨R,l₄⟩ := nonconstruction h₁.left | t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q : Form
h₁ : P ∈ Δ ∧ Q ∈ Δ
h₂ : formalApplication (↑t) (▲{P¦Q}) ⊆ ↑↑p
⊢ False | t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q : Form
h₁ : P ∈ Δ ∧ Q ∈ Δ
h₂ : formalApplication (↑t) (▲{P¦Q}) ⊆ ↑↑p
R : Form
l₄ : ¬(R ∈ formalApplication (↑t) (▲{P}) → R ∈ ↑↑p)
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | have ⟨⟨S,⟨prf₁⟩,l₆⟩,l₈⟩ := nonconstruction l₄ | t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q : Form
h₁ : P ∈ Δ ∧ Q ∈ Δ
h₂ : formalApplication (↑t) (▲{P¦Q}) ⊆ ↑↑p
R : Form
l₄ : ¬(R ∈ formalApplication (↑t) (▲{P}) → R ∈ ↑↑p)
⊢ False | t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q : Form
h₁ : P ∈ Δ ∧ Q ∈ Δ
h₂ : formalApplication (↑t) (▲{P¦Q}) ⊆ ↑↑p
R : Form
l₄ : ¬(R ∈ formalApplication (↑t) (▲{P}) → R ∈ ↑↑p)
S : Form
prf₁ : BProof {P} S
l₆ : S⊃R ∈ ↑t
l₈ : ¬R ∈ ↑↑p
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | have ⟨T,l₉⟩ := nonconstruction h₁.right | t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q : Form
h₁ : P ∈ Δ ∧ Q ∈ Δ
h₂ : formalApplication (↑t) (▲{P¦Q}) ⊆ ↑↑p
R : Form
l₄ : ¬(R ∈ formalApplication (↑t) (▲{P}) → R ∈ ↑↑p)
S : Form
prf₁ : BProof {P} S
l₆ : S⊃R ∈ ↑t
l₈ : ¬R ∈ ↑↑p
⊢ False | t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q : Form
h₁ : P ∈ Δ ∧ Q ∈ Δ
h₂ : formalApplication (↑t) (▲{P¦Q}) ⊆ ↑↑p
R : Form
l₄ : ¬(R ∈ formalApplication (↑t) (▲{P}) → R ∈ ↑↑p)
S : Form
prf₁ : BProof {P} S
l₆ : S⊃R ∈ ↑t
l₈ : ¬R ∈ ↑↑p
T : Form
l₉ : ¬(T ∈ formalApplication (↑t) (▲{Q}) → T ∈ ↑↑p)
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | have ⟨⟨U,⟨prf₂⟩,l₁₀⟩,l₁₂⟩ := nonconstruction l₉ | t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q : Form
h₁ : P ∈ Δ ∧ Q ∈ Δ
h₂ : formalApplication (↑t) (▲{P¦Q}) ⊆ ↑↑p
R : Form
l₄ : ¬(R ∈ formalApplication (↑t) (▲{P}) → R ∈ ↑↑p)
S : Form
prf₁ : BProof {P} S
l₆ : S⊃R ∈ ↑t
l₈ : ¬R ∈ ↑↑p
T : Form
l₉ : ¬(T ∈ formalApplication (↑t) (▲{Q}) → T ∈ ↑↑p)
⊢ False | t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q : Form
h₁ : P ∈ Δ ∧ Q ∈ Δ
h₂ : formalApplication (↑t) (▲{P¦Q}) ⊆ ↑↑p
R : Form
l₄ : ¬(R ∈ formalApplication (↑t) (▲{P}) → R ∈ ↑↑p)
S : Form
prf₁ : BProof {P} S
l₆ : S⊃R ∈ ↑t
l₈ : ¬R ∈ ↑↑p
T : Form
l₉ : ¬(T ∈ formalApplication (↑t) (▲{Q}) → T ∈ ↑↑p)
U : Form
prf₂ : BProof {Q} U
l₁₀ : U⊃T ∈ ↑t
l₁₂ : ¬T ∈ ↑↑p
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | clear h₁ l₄ l₉ | t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q : Form
h₁ : P ∈ Δ ∧ Q ∈ Δ
h₂ : formalApplication (↑t) (▲{P¦Q}) ⊆ ↑↑p
R : Form
l₄ : ¬(R ∈ formalApplication (↑t) (▲{P}) → R ∈ ↑↑p)
S : Form
prf₁ : BProof {P} S
l₆ : S⊃R ∈ ↑t
l₈ : ¬R ∈ ↑↑p
T : Form
l₉ : ¬(T ∈ formalApplication (↑t) (▲{Q}) → T ∈ ↑↑p)
U : Form
prf₂ : BProof {Q} U
l₁₀ : U⊃T ∈ ↑t
l₁₂ : ¬T ∈ ↑↑p
⊢ False | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q : Form
h₂ : formalApplication (↑t) (▲{P¦Q}) ⊆ ↑↑p
R S : Form
prf₁ : BProof {P} S
l₆ : S⊃R ∈ ↑t
l₈ : ¬R ∈ ↑↑p
T U : Form
prf₂ : BProof {Q} U
l₁₀ : U⊃T ∈ ↑t
l₁₂ : ¬T ∈ ↑↑p
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | have l₁₃ : ¬(R¦T ∈ p) := λw => Or.elim (p.property w) l₈ l₁₂ | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q : Form
h₂ : formalApplication (↑t) (▲{P¦Q}) ⊆ ↑↑p
R S : Form
prf₁ : BProof {P} S
l₆ : S⊃R ∈ ↑t
l₈ : ¬R ∈ ↑↑p
T U : Form
prf₂ : BProof {Q} U
l₁₀ : U⊃T ∈ ↑t
l₁₂ : ¬T ∈ ↑↑p
⊢ False | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q : Form
h₂ : formalApplication (↑t) (▲{P¦Q}) ⊆ ↑↑p
R S : Form
prf₁ : BProof {P} S
l₆ : S⊃R ∈ ↑t
l₈ : ¬R ∈ ↑↑p
T U : Form
prf₂ : BProof {Q} U
l₁₀ : U⊃T ∈ ↑t
l₁₂ : ¬T ∈ ↑↑p
l₁₃ : ¬R¦T ∈ p
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | apply l₁₃ | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q : Form
h₂ : formalApplication (↑t) (▲{P¦Q}) ⊆ ↑↑p
R S : Form
prf₁ : BProof {P} S
l₆ : S⊃R ∈ ↑t
l₈ : ¬R ∈ ↑↑p
T U : Form
prf₂ : BProof {Q} U
l₁₀ : U⊃T ∈ ↑t
l₁₂ : ¬T ∈ ↑↑p
l₁₃ : ¬R¦T ∈ p
⊢ False | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q : Form
h₂ : formalApplication (↑t) (▲{P¦Q}) ⊆ ↑↑p
R S : Form
prf₁ : BProof {P} S
l₆ : S⊃R ∈ ↑t
l₈ : ¬R ∈ ↑↑p
T U : Form
prf₂ : BProof {Q} U
l₁₀ : U⊃T ∈ ↑t
l₁₂ : ¬T ∈ ↑↑p
l₁₃ : ¬R¦T ∈ p
⊢ R¦T ∈ p |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | apply h₂ | t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q : Form
h₂ : formalApplication (↑t) (▲{P¦Q}) ⊆ ↑↑p
R S : Form
prf₁ : BProof {P} S
l₆ : S⊃R ∈ ↑t
l₈ : ¬R ∈ ↑↑p
T U : Form
prf₂ : BProof {Q} U
l₁₀ : U⊃T ∈ ↑t
l₁₂ : ¬T ∈ ↑↑p
l₁₃ : ¬R¦T ∈ p
⊢ R¦T ∈ p | case a
t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q : Form
h₂ : formalApplication (↑t) (▲{P¦Q}) ⊆ ↑↑p
R S : Form
prf₁ : BProof {P} S
l₆ : S⊃R ∈ ↑t
l₈ : ¬R ∈ ↑↑p
T U : Form
prf₂ : BProof {Q} U
l₁₀ : U⊃T ∈ ↑t
l₁₂ : ¬T ∈ ↑↑p
l₁₃ : ¬R¦T ∈ p
⊢ R¦T ∈ formalApplication (↑t) (▲{P¦Q}) |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | clear l₈ l₁₂ l₁₃ h₂ | case a
t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q : Form
h₂ : formalApplication (↑t) (▲{P¦Q}) ⊆ ↑↑p
R S : Form
prf₁ : BProof {P} S
l₆ : S⊃R ∈ ↑t
l₈ : ¬R ∈ ↑↑p
T U : Form
prf₂ : BProof {Q} U
l₁₀ : U⊃T ∈ ↑t
l₁₂ : ¬T ∈ ↑↑p
l₁₃ : ¬R¦T ∈ p
⊢ R¦T ∈ formalApplication (↑t) (▲{P¦Q}) | case a
t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q R S : Form
prf₁ : BProof {P} S
l₆ : S⊃R ∈ ↑t
T U : Form
prf₂ : BProof {Q} U
l₁₀ : U⊃T ∈ ↑t
⊢ R¦T ∈ formalApplication (↑t) (▲{P¦Q}) |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | have l₁₄ : S¦U ∈ ▲{P¦Q} := ⟨(BTheorem.orFunctor prf₁.toTheorem prf₂.toTheorem).toProof⟩ | case a
t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q R S : Form
prf₁ : BProof {P} S
l₆ : S⊃R ∈ ↑t
T U : Form
prf₂ : BProof {Q} U
l₁₀ : U⊃T ∈ ↑t
⊢ R¦T ∈ formalApplication (↑t) (▲{P¦Q}) | case a
t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q R S : Form
prf₁ : BProof {P} S
l₆ : S⊃R ∈ ↑t
T U : Form
prf₂ : BProof {Q} U
l₁₀ : U⊃T ∈ ↑t
l₁₄ : S¦U ∈ ▲{P¦Q}
⊢ R¦T ∈ formalApplication (↑t) (▲{P¦Q}) |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | have l₁₅ : (S¦U ⊃ R¦T) ∈ t := t.property.mpr ⟨BProof.mp (BProof.adj
(BProof.mp (BProof.ax l₆) (BTheorem.hs BTheorem.taut BTheorem.orI₁))
(BProof.mp (BProof.ax l₁₀) (BTheorem.hs BTheorem.taut BTheorem.orI₂)))
BTheorem.orE⟩ | case a
t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q R S : Form
prf₁ : BProof {P} S
l₆ : S⊃R ∈ ↑t
T U : Form
prf₂ : BProof {Q} U
l₁₀ : U⊃T ∈ ↑t
l₁₄ : S¦U ∈ ▲{P¦Q}
⊢ R¦T ∈ formalApplication (↑t) (▲{P¦Q}) | case a
t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q R S : Form
prf₁ : BProof {P} S
l₆ : S⊃R ∈ ↑t
T U : Form
prf₂ : BProof {Q} U
l₁₀ : U⊃T ∈ ↑t
l₁₄ : S¦U ∈ ▲{P¦Q}
l₁₅ : S¦U⊃R¦T ∈ t
⊢ R¦T ∈ formalApplication (↑t) (▲{P¦Q}) |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | exact ⟨S ¦ U, l₁₄, l₁₅⟩ | case a
t u : Th
p : Pr
h₁ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₂ : ↑u ∩ Δ = ∅
P Q R S : Form
prf₁ : BProof {P} S
l₆ : S⊃R ∈ ↑t
T U : Form
prf₂ : BProof {Q} U
l₁₀ : U⊃T ∈ ↑t
l₁₄ : S¦U ∈ ▲{P¦Q}
l₁₅ : S¦U⊃R¦T ∈ t
⊢ R¦T ∈ formalApplication (↑t) (▲{P¦Q}) | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | apply byContradiction | t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₄ : lindenbaumExtension u Δ ∩ Δ = ∅
P : Form
h₁ : P ∈ formalApplication (↑t) (lindenbaumExtension u Δ)
Q : Form
h₂ : Q ∈ lindenbaumExtension u Δ
h₃ : Q⊃P ∈ ↑t
⊢ formalApplication (↑t) (▲{Q}) ⊆ ↑↑p | case h
t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₄ : lindenbaumExtension u Δ ∩ Δ = ∅
P : Form
h₁ : P ∈ formalApplication (↑t) (lindenbaumExtension u Δ)
Q : Form
h₂ : Q ∈ lindenbaumExtension u Δ
h₃ : Q⊃P ∈ ↑t
⊢ ¬formalApplication (↑t) (▲{Q}) ⊆ ↑↑p → False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | intros h₄ | case h
t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₄ : lindenbaumExtension u Δ ∩ Δ = ∅
P : Form
h₁ : P ∈ formalApplication (↑t) (lindenbaumExtension u Δ)
Q : Form
h₂ : Q ∈ lindenbaumExtension u Δ
h₃ : Q⊃P ∈ ↑t
⊢ ¬formalApplication (↑t) (▲{Q}) ⊆ ↑↑p → False | case h
t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₄ : lindenbaumExtension u Δ ∩ Δ = ∅
P : Form
h₁ : P ∈ formalApplication (↑t) (lindenbaumExtension u Δ)
Q : Form
h₂ : Q ∈ lindenbaumExtension u Δ
h₃ : Q⊃P ∈ ↑t
h₄ : ¬formalApplication (↑t) (▲{Q}) ⊆ ↑↑p
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | appBoundingFormalApplication | [54, 1] | [141, 38] | exact (Set.eq_empty_iff_forall_not_mem.mp l₄) Q ⟨h₂,h₄⟩ | case h
t u : Th
p : Pr
h₁✝ : formalApplicationFunction t u ≤ ↑p
Δ : Set Form := { f | ¬formalApplication (↑t) (▲{f}) ⊆ ↑↑p }
l₄ : lindenbaumExtension u Δ ∩ Δ = ∅
P : Form
h₁ : P ∈ formalApplication (↑t) (lindenbaumExtension u Δ)
Q : Form
h₂ : Q ∈ lindenbaumExtension u Δ
h₃ : Q⊃P ∈ ↑t
h₄ : ¬formalApplication (↑t) (▲{Q}) ⊆ ↑↑p
⊢ False | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | theoryValuationMonotone | [145, 1] | [147, 14] | intros _ _ h₁ n h₂ | ⊢ Monotone theoryValuation | a✝ b✝ : Th
h₁ : a✝ ≤ b✝
n : ℕ
h₂ : n ∈ theoryValuation a✝
⊢ n ∈ theoryValuation b✝ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | theoryValuationMonotone | [145, 1] | [147, 14] | exact h₁ h₂ | a✝ b✝ : Th
h₁ : a✝ ≤ b✝
n : ℕ
h₂ : n ∈ theoryValuation a✝
⊢ n ∈ theoryValuation b✝ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | theoryValuationBounding | [149, 1] | [155, 54] | intros t x h₁ | ⊢ ∀ (t : Th) (x : ℕ), (∀ (p : Pr), t ≤ ↑p → x ∈ theoryValuation ↑p) → x ∈ theoryValuation t | t : Th
x : ℕ
h₁ : ∀ (p : Pr), t ≤ ↑p → x ∈ theoryValuation ↑p
⊢ x ∈ theoryValuation t |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | theoryValuationBounding | [149, 1] | [155, 54] | change #x ∈ t.val | t : Th
x : ℕ
h₁ : ∀ (p : Pr), t ≤ ↑p → x ∈ theoryValuation ↑p
⊢ x ∈ theoryValuation t | t : Th
x : ℕ
h₁ : ∀ (p : Pr), t ≤ ↑p → x ∈ theoryValuation ↑p
⊢ #x ∈ ↑t |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | theoryValuationBounding | [149, 1] | [155, 54] | rw [primeAnalysis t] | t : Th
x : ℕ
h₁ : ∀ (p : Pr), t ≤ ↑p → x ∈ theoryValuation ↑p
⊢ #x ∈ ↑t | t : Th
x : ℕ
h₁ : ∀ (p : Pr), t ≤ ↑p → x ∈ theoryValuation ↑p
⊢ #x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | theoryValuationBounding | [149, 1] | [155, 54] | apply Set.mem_interₛ.mpr | t : Th
x : ℕ
h₁ : ∀ (p : Pr), t ≤ ↑p → x ∈ theoryValuation ↑p
⊢ #x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } | t : Th
x : ℕ
h₁ : ∀ (p : Pr), t ≤ ↑p → x ∈ theoryValuation ↑p
⊢ ∀ (t_1 : Set Form), t_1 ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } → #x ∈ t_1 |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | theoryValuationBounding | [149, 1] | [155, 54] | intros r h₂ | t : Th
x : ℕ
h₁ : ∀ (p : Pr), t ≤ ↑p → x ∈ theoryValuation ↑p
⊢ ∀ (t_1 : Set Form), t_1 ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } → #x ∈ t_1 | t : Th
x : ℕ
h₁ : ∀ (p : Pr), t ≤ ↑p → x ∈ theoryValuation ↑p
r : Set Form
h₂ : r ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
⊢ #x ∈ r |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | theoryValuationBounding | [149, 1] | [155, 54] | exact h₁ ⟨⟨r,h₂.right.right⟩,h₂.left⟩ h₂.right.left | t : Th
x : ℕ
h₁ : ∀ (p : Pr), t ≤ ↑p → x ∈ theoryValuation ↑p
r : Set Form
h₂ : r ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
⊢ #x ∈ r | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | intros t f | ⊢ ∀ {t : Th} {f : Form}, t⊨f ↔ f ∈ ↑t | t : Th
f : Form
⊢ t⊨f ↔ f ∈ ↑t |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | cases f | t : Th
f : Form
⊢ t⊨f ↔ f ∈ ↑t | case atom
t : Th
a✝ : ℕ
⊢ t⊨#a✝ ↔ #a✝ ∈ ↑t
case neg
t : Th
a✝ : Form
⊢ t⊨~a✝ ↔ ~a✝ ∈ ↑t
case and
t : Th
a✝¹ a✝ : Form
⊢ t⊨a✝¹&a✝ ↔ a✝¹&a✝ ∈ ↑t
case or
t : Th
a✝¹ a✝ : Form
⊢ t⊨a✝¹¦a✝ ↔ a✝¹¦a✝ ∈ ↑t
case impl
t : Th
a✝¹ a✝ : Form
⊢ t⊨a✝¹⊃a✝ ↔ a✝¹⊃a✝ ∈ ↑t |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | all_goals apply Iff.intro <;> intros h₁ | case atom
t : Th
a✝ : ℕ
⊢ t⊨#a✝ ↔ #a✝ ∈ ↑t
case neg
t : Th
a✝ : Form
⊢ t⊨~a✝ ↔ ~a✝ ∈ ↑t
case and
t : Th
a✝¹ a✝ : Form
⊢ t⊨a✝¹&a✝ ↔ a✝¹&a✝ ∈ ↑t
case or
t : Th
a✝¹ a✝ : Form
⊢ t⊨a✝¹¦a✝ ↔ a✝¹¦a✝ ∈ ↑t
case impl
t : Th
a✝¹ a✝ : Form
⊢ t⊨a✝¹⊃a✝ ↔ a✝¹⊃a✝ ∈ ↑t | case atom.mp
t : Th
a✝ : ℕ
h₁ : t⊨#a✝
⊢ #a✝ ∈ ↑t
case atom.mpr
t : Th
a✝ : ℕ
h₁ : #a✝ ∈ ↑t
⊢ t⊨#a✝
case neg.mp
t : Th
a✝ : Form
h₁ : t⊨~a✝
⊢ ~a✝ ∈ ↑t
case neg.mpr
t : Th
a✝ : Form
h₁ : ~a✝ ∈ ↑t
⊢ t⊨~a✝
case and.mp
t : Th
a✝¹ a✝ : Form
h₁ : t⊨a✝¹&a✝
⊢ a✝¹&a✝ ∈ ↑t
case and.mpr
t : Th
a✝¹ a✝ : Form
h₁ : a✝¹&a✝ ∈ ↑t
⊢ t⊨a✝¹&a✝
case or.mp
t : Th
a✝¹ a✝ : Form
h₁ : t⊨a✝¹¦a✝
⊢ a✝¹¦a✝ ∈ ↑t
case or.mpr
t : Th
a✝¹ a✝ : Form
h₁ : a✝¹¦a✝ ∈ ↑t
⊢ t⊨a✝¹¦a✝
case impl.mp
t : Th
a✝¹ a✝ : Form
h₁ : t⊨a✝¹⊃a✝
⊢ a✝¹⊃a✝ ∈ ↑t
case impl.mpr
t : Th
a✝¹ a✝ : Form
h₁ : a✝¹⊃a✝ ∈ ↑t
⊢ t⊨a✝¹⊃a✝ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | case atom.mp => exact h₁ | t : Th
a✝ : ℕ
h₁ : t⊨#a✝
⊢ #a✝ ∈ ↑t | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | case atom.mpr => exact h₁ | t : Th
a✝ : ℕ
h₁ : #a✝ ∈ ↑t
⊢ t⊨#a✝ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | case and.mp f g =>
have l₁ : f ∈ t := canonicalSatisfaction.mp h₁.left
have l₂ : g ∈ t := canonicalSatisfaction.mp h₁.right
exact t.property.mpr ⟨BProof.adj (BProof.ax l₁) (BProof.ax l₂)⟩ | t : Th
f g : Form
h₁ : t⊨f&g
⊢ f&g ∈ ↑t | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | case and.mpr f g =>
have l₁ : t ⊨ f := canonicalSatisfaction.mpr $ t.property.mpr ⟨BProof.mp (BProof.ax h₁) BTheorem.andE₁⟩
have l₂ : t ⊨ g := canonicalSatisfaction.mpr $ t.property.mpr ⟨BProof.mp (BProof.ax h₁) BTheorem.andE₂⟩
exact ⟨l₁,l₂⟩ | t : Th
f g : Form
h₁ : f&g ∈ ↑t
⊢ t⊨f&g | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | case or.mp f g =>
rw [primeAnalysis]
intros p h₂
let pr : Pr := ⟨⟨p, h₂.right.right⟩,h₂.left⟩
have l₁ := @h₁ pr h₂.right.left
cases l₁
case inl h₃ =>
have l₂ := canonicalSatisfaction.mp h₃
exact h₂.right.right.mpr ⟨BProof.mp (BProof.ax l₂) BTheorem.orI₁⟩
case inr h₃ =>
have l₂ := canonicalSatisfaction.mp h₃
exact h₂.right.right.mpr ⟨BProof.mp (BProof.ax l₂) BTheorem.orI₂⟩ | t : Th
f g : Form
h₁ : t⊨f¦g
⊢ f¦g ∈ ↑t | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | case or.mpr f g =>
intros p h₂
rw [primeAnalysis] at h₁
have l₁ : f ¦ g ∈ p.val.val := (Set.mem_interₛ.mp h₁ p) ⟨p.property, h₂, p.val.property⟩
have l₂ : f ∈ p.val.val ∨ g ∈ p.val.val := p.property l₁
cases l₂
case inl h₂ => exact Or.inl $ canonicalSatisfaction.mpr h₂
case inr h₂ => exact Or.inr $ canonicalSatisfaction.mpr h₂ | t : Th
f g : Form
h₁ : f¦g ∈ ↑t
⊢ t⊨f¦g | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | case neg.mp f =>
rw [primeAnalysis]
intros p h₂
let pr : Pr := ⟨⟨p, h₂.right.right⟩,h₂.left⟩
have l₁ := @h₁ pr h₂.right.left
have l₂ := l₁ ∘ canonicalSatisfaction.mpr
apply byContradiction
exact l₂ | t : Th
f : Form
h₁ : t⊨~f
⊢ ~f ∈ ↑t | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | case neg.mpr f =>
intros p h₂ h₃
rw [primeAnalysis] at h₁
have l₁ : ~f ∈ p.val.val := (Set.mem_interₛ.mp h₁ p) ⟨p.property, h₂, p.val.property⟩
exact canonicalSatisfaction.mp h₃ l₁ | t : Th
f : Form
h₁ : ~f ∈ ↑t
⊢ t⊨~f | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | case impl.mp f g =>
apply byContradiction
intros h₂
let Δ : Th := ⟨▲{f}, generatedFormal {f}⟩
have l₁ : ¬(g ∈ (formalApplicationFunction t Δ).val) := by
intros h₃
have ⟨q,⟨prf₁⟩,l₂⟩ := h₃
have ⟨prf₂⟩ := t.property.mp l₂
have prf₃ := BProof.mp prf₂ (BTheorem.hs prf₁.toTheorem BTheorem.taut)
exact h₂ $ t.property.mpr ⟨prf₃⟩
have l₂ : Δ ⊨ f := canonicalSatisfaction.mpr ⟨BProof.ax rfl⟩
exact l₁ $ canonicalSatisfaction.mp (h₁ l₂) | t : Th
f g : Form
h₁ : t⊨f⊃g
⊢ f⊃g ∈ ↑t | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | case impl.mpr f g =>
intros r h₂
have l₁ := canonicalSatisfaction.mp h₂
have l₂ : g ∈ (formalApplicationFunction t r).val := ⟨f, l₁, h₁⟩
exact canonicalSatisfaction.mpr l₂ | t : Th
f g : Form
h₁ : f⊃g ∈ ↑t
⊢ t⊨f⊃g | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | apply Iff.intro <;> intros h₁ | case impl
t : Th
a✝¹ a✝ : Form
⊢ t⊨a✝¹⊃a✝ ↔ a✝¹⊃a✝ ∈ ↑t | case impl.mp
t : Th
a✝¹ a✝ : Form
h₁ : t⊨a✝¹⊃a✝
⊢ a✝¹⊃a✝ ∈ ↑t
case impl.mpr
t : Th
a✝¹ a✝ : Form
h₁ : a✝¹⊃a✝ ∈ ↑t
⊢ t⊨a✝¹⊃a✝ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | exact h₁ | t : Th
a✝ : ℕ
h₁ : t⊨#a✝
⊢ #a✝ ∈ ↑t | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | exact h₁ | t : Th
a✝ : ℕ
h₁ : #a✝ ∈ ↑t
⊢ t⊨#a✝ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | have l₁ : f ∈ t := canonicalSatisfaction.mp h₁.left | t : Th
f g : Form
h₁ : t⊨f&g
⊢ f&g ∈ ↑t | t : Th
f g : Form
h₁ : t⊨f&g
l₁ : f ∈ t
⊢ f&g ∈ ↑t |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | have l₂ : g ∈ t := canonicalSatisfaction.mp h₁.right | t : Th
f g : Form
h₁ : t⊨f&g
l₁ : f ∈ t
⊢ f&g ∈ ↑t | t : Th
f g : Form
h₁ : t⊨f&g
l₁ : f ∈ t
l₂ : g ∈ t
⊢ f&g ∈ ↑t |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | exact t.property.mpr ⟨BProof.adj (BProof.ax l₁) (BProof.ax l₂)⟩ | t : Th
f g : Form
h₁ : t⊨f&g
l₁ : f ∈ t
l₂ : g ∈ t
⊢ f&g ∈ ↑t | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | have l₁ : t ⊨ f := canonicalSatisfaction.mpr $ t.property.mpr ⟨BProof.mp (BProof.ax h₁) BTheorem.andE₁⟩ | t : Th
f g : Form
h₁ : f&g ∈ ↑t
⊢ t⊨f&g | t : Th
f g : Form
h₁ : f&g ∈ ↑t
l₁ : t⊨f
⊢ t⊨f&g |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | have l₂ : t ⊨ g := canonicalSatisfaction.mpr $ t.property.mpr ⟨BProof.mp (BProof.ax h₁) BTheorem.andE₂⟩ | t : Th
f g : Form
h₁ : f&g ∈ ↑t
l₁ : t⊨f
⊢ t⊨f&g | t : Th
f g : Form
h₁ : f&g ∈ ↑t
l₁ : t⊨f
l₂ : t⊨g
⊢ t⊨f&g |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | exact ⟨l₁,l₂⟩ | t : Th
f g : Form
h₁ : f&g ∈ ↑t
l₁ : t⊨f
l₂ : t⊨g
⊢ t⊨f&g | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | rw [primeAnalysis] | t : Th
f g : Form
h₁ : t⊨f¦g
⊢ f¦g ∈ ↑t | t : Th
f g : Form
h₁ : t⊨f¦g
⊢ f¦g ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | intros p h₂ | t : Th
f g : Form
h₁ : t⊨f¦g
⊢ f¦g ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } | t : Th
f g : Form
h₁ : t⊨f¦g
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
⊢ f¦g ∈ p |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | let pr : Pr := ⟨⟨p, h₂.right.right⟩,h₂.left⟩ | t : Th
f g : Form
h₁ : t⊨f¦g
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
⊢ f¦g ∈ p | t : Th
f g : Form
h₁ : t⊨f¦g
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
pr : Pr :=
{ val := { val := p, property := (_ : ∀ {f : Form}, f ∈ p ↔ p⊢f) },
property := (_ : ∀ {f g : Form}, f¦g ∈ p → f ∈ p ∨ g ∈ p) }
⊢ f¦g ∈ p |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | have l₁ := @h₁ pr h₂.right.left | t : Th
f g : Form
h₁ : t⊨f¦g
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
pr : Pr :=
{ val := { val := p, property := (_ : ∀ {f : Form}, f ∈ p ↔ p⊢f) },
property := (_ : ∀ {f g : Form}, f¦g ∈ p → f ∈ p ∨ g ∈ p) }
⊢ f¦g ∈ p | t : Th
f g : Form
h₁ : t⊨f¦g
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
pr : Pr :=
{ val := { val := p, property := (_ : ∀ {f : Form}, f ∈ p ↔ p⊢f) },
property := (_ : ∀ {f g : Form}, f¦g ∈ p → f ∈ p ∨ g ∈ p) }
l₁ : ↑pr⊨f ∨ ↑pr⊨g
⊢ f¦g ∈ p |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | cases l₁ | t : Th
f g : Form
h₁ : t⊨f¦g
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
pr : Pr :=
{ val := { val := p, property := (_ : ∀ {f : Form}, f ∈ p ↔ p⊢f) },
property := (_ : ∀ {f g : Form}, f¦g ∈ p → f ∈ p ∨ g ∈ p) }
l₁ : ↑pr⊨f ∨ ↑pr⊨g
⊢ f¦g ∈ p | case inl
t : Th
f g : Form
h₁ : t⊨f¦g
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
pr : Pr :=
{ val := { val := p, property := (_ : ∀ {f : Form}, f ∈ p ↔ p⊢f) },
property := (_ : ∀ {f g : Form}, f¦g ∈ p → f ∈ p ∨ g ∈ p) }
h✝ : ↑pr⊨f
⊢ f¦g ∈ p
case inr
t : Th
f g : Form
h₁ : t⊨f¦g
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
pr : Pr :=
{ val := { val := p, property := (_ : ∀ {f : Form}, f ∈ p ↔ p⊢f) },
property := (_ : ∀ {f g : Form}, f¦g ∈ p → f ∈ p ∨ g ∈ p) }
h✝ : ↑pr⊨g
⊢ f¦g ∈ p |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | case inl h₃ =>
have l₂ := canonicalSatisfaction.mp h₃
exact h₂.right.right.mpr ⟨BProof.mp (BProof.ax l₂) BTheorem.orI₁⟩ | t : Th
f g : Form
h₁ : t⊨f¦g
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
pr : Pr :=
{ val := { val := p, property := (_ : ∀ {f : Form}, f ∈ p ↔ p⊢f) },
property := (_ : ∀ {f g : Form}, f¦g ∈ p → f ∈ p ∨ g ∈ p) }
h₃ : ↑pr⊨f
⊢ f¦g ∈ p | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | case inr h₃ =>
have l₂ := canonicalSatisfaction.mp h₃
exact h₂.right.right.mpr ⟨BProof.mp (BProof.ax l₂) BTheorem.orI₂⟩ | t : Th
f g : Form
h₁ : t⊨f¦g
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
pr : Pr :=
{ val := { val := p, property := (_ : ∀ {f : Form}, f ∈ p ↔ p⊢f) },
property := (_ : ∀ {f g : Form}, f¦g ∈ p → f ∈ p ∨ g ∈ p) }
h₃ : ↑pr⊨g
⊢ f¦g ∈ p | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | have l₂ := canonicalSatisfaction.mp h₃ | t : Th
f g : Form
h₁ : t⊨f¦g
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
pr : Pr :=
{ val := { val := p, property := (_ : ∀ {f : Form}, f ∈ p ↔ p⊢f) },
property := (_ : ∀ {f g : Form}, f¦g ∈ p → f ∈ p ∨ g ∈ p) }
h₃ : ↑pr⊨f
⊢ f¦g ∈ p | t : Th
f g : Form
h₁ : t⊨f¦g
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
pr : Pr :=
{ val := { val := p, property := (_ : ∀ {f : Form}, f ∈ p ↔ p⊢f) },
property := (_ : ∀ {f g : Form}, f¦g ∈ p → f ∈ p ∨ g ∈ p) }
h₃ : ↑pr⊨f
l₂ : f ∈ ↑↑pr
⊢ f¦g ∈ p |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | exact h₂.right.right.mpr ⟨BProof.mp (BProof.ax l₂) BTheorem.orI₁⟩ | t : Th
f g : Form
h₁ : t⊨f¦g
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
pr : Pr :=
{ val := { val := p, property := (_ : ∀ {f : Form}, f ∈ p ↔ p⊢f) },
property := (_ : ∀ {f g : Form}, f¦g ∈ p → f ∈ p ∨ g ∈ p) }
h₃ : ↑pr⊨f
l₂ : f ∈ ↑↑pr
⊢ f¦g ∈ p | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | have l₂ := canonicalSatisfaction.mp h₃ | t : Th
f g : Form
h₁ : t⊨f¦g
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
pr : Pr :=
{ val := { val := p, property := (_ : ∀ {f : Form}, f ∈ p ↔ p⊢f) },
property := (_ : ∀ {f g : Form}, f¦g ∈ p → f ∈ p ∨ g ∈ p) }
h₃ : ↑pr⊨g
⊢ f¦g ∈ p | t : Th
f g : Form
h₁ : t⊨f¦g
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
pr : Pr :=
{ val := { val := p, property := (_ : ∀ {f : Form}, f ∈ p ↔ p⊢f) },
property := (_ : ∀ {f g : Form}, f¦g ∈ p → f ∈ p ∨ g ∈ p) }
h₃ : ↑pr⊨g
l₂ : g ∈ ↑↑pr
⊢ f¦g ∈ p |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | exact h₂.right.right.mpr ⟨BProof.mp (BProof.ax l₂) BTheorem.orI₂⟩ | t : Th
f g : Form
h₁ : t⊨f¦g
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
pr : Pr :=
{ val := { val := p, property := (_ : ∀ {f : Form}, f ∈ p ↔ p⊢f) },
property := (_ : ∀ {f g : Form}, f¦g ∈ p → f ∈ p ∨ g ∈ p) }
h₃ : ↑pr⊨g
l₂ : g ∈ ↑↑pr
⊢ f¦g ∈ p | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | intros p h₂ | t : Th
f g : Form
h₁ : f¦g ∈ ↑t
⊢ t⊨f¦g | t : Th
f g : Form
h₁ : f¦g ∈ ↑t
p : Model.primes
h₂ : t ≤ ↑p
⊢ ↑p⊨f ∨ ↑p⊨g |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | rw [primeAnalysis] at h₁ | t : Th
f g : Form
h₁ : f¦g ∈ ↑t
p : Model.primes
h₂ : t ≤ ↑p
⊢ ↑p⊨f ∨ ↑p⊨g | t : Th
f g : Form
h₁ : f¦g ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
p : Model.primes
h₂ : t ≤ ↑p
⊢ ↑p⊨f ∨ ↑p⊨g |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | have l₁ : f ¦ g ∈ p.val.val := (Set.mem_interₛ.mp h₁ p) ⟨p.property, h₂, p.val.property⟩ | t : Th
f g : Form
h₁ : f¦g ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
p : Model.primes
h₂ : t ≤ ↑p
⊢ ↑p⊨f ∨ ↑p⊨g | t : Th
f g : Form
h₁ : f¦g ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
p : Model.primes
h₂ : t ≤ ↑p
l₁ : f¦g ∈ ↑↑p
⊢ ↑p⊨f ∨ ↑p⊨g |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | have l₂ : f ∈ p.val.val ∨ g ∈ p.val.val := p.property l₁ | t : Th
f g : Form
h₁ : f¦g ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
p : Model.primes
h₂ : t ≤ ↑p
l₁ : f¦g ∈ ↑↑p
⊢ ↑p⊨f ∨ ↑p⊨g | t : Th
f g : Form
h₁ : f¦g ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
p : Model.primes
h₂ : t ≤ ↑p
l₁ : f¦g ∈ ↑↑p
l₂ : f ∈ ↑↑p ∨ g ∈ ↑↑p
⊢ ↑p⊨f ∨ ↑p⊨g |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | cases l₂ | t : Th
f g : Form
h₁ : f¦g ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
p : Model.primes
h₂ : t ≤ ↑p
l₁ : f¦g ∈ ↑↑p
l₂ : f ∈ ↑↑p ∨ g ∈ ↑↑p
⊢ ↑p⊨f ∨ ↑p⊨g | case inl
t : Th
f g : Form
h₁ : f¦g ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
p : Model.primes
h₂ : t ≤ ↑p
l₁ : f¦g ∈ ↑↑p
h✝ : f ∈ ↑↑p
⊢ ↑p⊨f ∨ ↑p⊨g
case inr
t : Th
f g : Form
h₁ : f¦g ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
p : Model.primes
h₂ : t ≤ ↑p
l₁ : f¦g ∈ ↑↑p
h✝ : g ∈ ↑↑p
⊢ ↑p⊨f ∨ ↑p⊨g |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | case inl h₂ => exact Or.inl $ canonicalSatisfaction.mpr h₂ | t : Th
f g : Form
h₁ : f¦g ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
p : Model.primes
h₂✝ : t ≤ ↑p
l₁ : f¦g ∈ ↑↑p
h₂ : f ∈ ↑↑p
⊢ ↑p⊨f ∨ ↑p⊨g | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | case inr h₂ => exact Or.inr $ canonicalSatisfaction.mpr h₂ | t : Th
f g : Form
h₁ : f¦g ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
p : Model.primes
h₂✝ : t ≤ ↑p
l₁ : f¦g ∈ ↑↑p
h₂ : g ∈ ↑↑p
⊢ ↑p⊨f ∨ ↑p⊨g | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | exact Or.inl $ canonicalSatisfaction.mpr h₂ | t : Th
f g : Form
h₁ : f¦g ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
p : Model.primes
h₂✝ : t ≤ ↑p
l₁ : f¦g ∈ ↑↑p
h₂ : f ∈ ↑↑p
⊢ ↑p⊨f ∨ ↑p⊨g | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | exact Or.inr $ canonicalSatisfaction.mpr h₂ | t : Th
f g : Form
h₁ : f¦g ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
p : Model.primes
h₂✝ : t ≤ ↑p
l₁ : f¦g ∈ ↑↑p
h₂ : g ∈ ↑↑p
⊢ ↑p⊨f ∨ ↑p⊨g | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | rw [primeAnalysis] | t : Th
f : Form
h₁ : t⊨~f
⊢ ~f ∈ ↑t | t : Th
f : Form
h₁ : t⊨~f
⊢ ~f ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | intros p h₂ | t : Th
f : Form
h₁ : t⊨~f
⊢ ~f ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } | t : Th
f : Form
h₁ : t⊨~f
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
⊢ ~f ∈ p |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | let pr : Pr := ⟨⟨p, h₂.right.right⟩,h₂.left⟩ | t : Th
f : Form
h₁ : t⊨~f
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
⊢ ~f ∈ p | t : Th
f : Form
h₁ : t⊨~f
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
pr : Pr :=
{ val := { val := p, property := (_ : ∀ {f : Form}, f ∈ p ↔ p⊢f) },
property := (_ : ∀ {f g : Form}, f¦g ∈ p → f ∈ p ∨ g ∈ p) }
⊢ ~f ∈ p |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | have l₁ := @h₁ pr h₂.right.left | t : Th
f : Form
h₁ : t⊨~f
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
pr : Pr :=
{ val := { val := p, property := (_ : ∀ {f : Form}, f ∈ p ↔ p⊢f) },
property := (_ : ∀ {f g : Form}, f¦g ∈ p → f ∈ p ∨ g ∈ p) }
⊢ ~f ∈ p | t : Th
f : Form
h₁ : t⊨~f
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
pr : Pr :=
{ val := { val := p, property := (_ : ∀ {f : Form}, f ∈ p ↔ p⊢f) },
property := (_ : ∀ {f g : Form}, f¦g ∈ p → f ∈ p ∨ g ∈ p) }
l₁ : ¬↑(pr*)⊨f
⊢ ~f ∈ p |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | have l₂ := l₁ ∘ canonicalSatisfaction.mpr | t : Th
f : Form
h₁ : t⊨~f
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
pr : Pr :=
{ val := { val := p, property := (_ : ∀ {f : Form}, f ∈ p ↔ p⊢f) },
property := (_ : ∀ {f g : Form}, f¦g ∈ p → f ∈ p ∨ g ∈ p) }
l₁ : ¬↑(pr*)⊨f
⊢ ~f ∈ p | t : Th
f : Form
h₁ : t⊨~f
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
pr : Pr :=
{ val := { val := p, property := (_ : ∀ {f : Form}, f ∈ p ↔ p⊢f) },
property := (_ : ∀ {f g : Form}, f¦g ∈ p → f ∈ p ∨ g ∈ p) }
l₁ : ¬↑(pr*)⊨f
l₂ : f ∈ ↑↑(pr*) → False
⊢ ~f ∈ p |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | apply byContradiction | t : Th
f : Form
h₁ : t⊨~f
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
pr : Pr :=
{ val := { val := p, property := (_ : ∀ {f : Form}, f ∈ p ↔ p⊢f) },
property := (_ : ∀ {f g : Form}, f¦g ∈ p → f ∈ p ∨ g ∈ p) }
l₁ : ¬↑(pr*)⊨f
l₂ : f ∈ ↑↑(pr*) → False
⊢ ~f ∈ p | case h
t : Th
f : Form
h₁ : t⊨~f
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
pr : Pr :=
{ val := { val := p, property := (_ : ∀ {f : Form}, f ∈ p ↔ p⊢f) },
property := (_ : ∀ {f g : Form}, f¦g ∈ p → f ∈ p ∨ g ∈ p) }
l₁ : ¬↑(pr*)⊨f
l₂ : f ∈ ↑↑(pr*) → False
⊢ ¬~f ∈ p → False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | exact l₂ | case h
t : Th
f : Form
h₁ : t⊨~f
p : Set Form
h₂ : p ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
pr : Pr :=
{ val := { val := p, property := (_ : ∀ {f : Form}, f ∈ p ↔ p⊢f) },
property := (_ : ∀ {f g : Form}, f¦g ∈ p → f ∈ p ∨ g ∈ p) }
l₁ : ¬↑(pr*)⊨f
l₂ : f ∈ ↑↑(pr*) → False
⊢ ¬~f ∈ p → False | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | intros p h₂ h₃ | t : Th
f : Form
h₁ : ~f ∈ ↑t
⊢ t⊨~f | t : Th
f : Form
h₁ : ~f ∈ ↑t
p : Model.primes
h₂ : t ≤ ↑p
h₃ : ↑(p*)⊨f
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | rw [primeAnalysis] at h₁ | t : Th
f : Form
h₁ : ~f ∈ ↑t
p : Model.primes
h₂ : t ≤ ↑p
h₃ : ↑(p*)⊨f
⊢ False | t : Th
f : Form
h₁ : ~f ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
p : Model.primes
h₂ : t ≤ ↑p
h₃ : ↑(p*)⊨f
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | have l₁ : ~f ∈ p.val.val := (Set.mem_interₛ.mp h₁ p) ⟨p.property, h₂, p.val.property⟩ | t : Th
f : Form
h₁ : ~f ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
p : Model.primes
h₂ : t ≤ ↑p
h₃ : ↑(p*)⊨f
⊢ False | t : Th
f : Form
h₁ : ~f ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
p : Model.primes
h₂ : t ≤ ↑p
h₃ : ↑(p*)⊨f
l₁ : ~f ∈ ↑↑p
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | exact canonicalSatisfaction.mp h₃ l₁ | t : Th
f : Form
h₁ : ~f ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
p : Model.primes
h₂ : t ≤ ↑p
h₃ : ↑(p*)⊨f
l₁ : ~f ∈ ↑↑p
⊢ False | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | apply byContradiction | t : Th
f g : Form
h₁ : t⊨f⊃g
⊢ f⊃g ∈ ↑t | case h
t : Th
f g : Form
h₁ : t⊨f⊃g
⊢ ¬f⊃g ∈ ↑t → False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/CanonicalModel.lean | canonicalSatisfaction | [172, 1] | [235, 39] | intros h₂ | case h
t : Th
f g : Form
h₁ : t⊨f⊃g
⊢ ¬f⊃g ∈ ↑t → False | case h
t : Th
f g : Form
h₁ : t⊨f⊃g
h₂ : ¬f⊃g ∈ ↑t
⊢ False |
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