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stringclasses 147
values | file_path
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101
| full_name
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stringlengths 6
10
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https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Semantics/Satisfaction.lean | starCompatLeft | [88, 1] | [94, 29] | have ⟨x, l₁⟩ := nonconstruction h₁ | α : Type
inst : Model α
p : Model.primes
f : Form
h₁ : ¬p⊨~f
⊢ p*⊨f | α : Type
inst : Model α
p : Model.primes
f : Form
h₁ : ¬p⊨~f
x : Model.primes
l₁ : ¬(↑p ≤ ↑x → ¬↑(x*)⊨f)
⊢ p*⊨f |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Semantics/Satisfaction.lean | starCompatLeft | [88, 1] | [94, 29] | have ⟨l₂,l₃⟩ := nonconstruction l₁ | α : Type
inst : Model α
p : Model.primes
f : Form
h₁ : ¬p⊨~f
x : Model.primes
l₁ : ¬(↑p ≤ ↑x → ¬↑(x*)⊨f)
⊢ p*⊨f | α : Type
inst : Model α
p : Model.primes
f : Form
h₁ : ¬p⊨~f
x : Model.primes
l₁ : ¬(↑p ≤ ↑x → ¬↑(x*)⊨f)
l₂ : ↑p ≤ ↑x
l₃ : ¬¬↑(x*)⊨f
⊢ p*⊨f |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Semantics/Satisfaction.lean | starCompatLeft | [88, 1] | [94, 29] | have l₄ := byContradiction l₃ | α : Type
inst : Model α
p : Model.primes
f : Form
h₁ : ¬p⊨~f
x : Model.primes
l₁ : ¬(↑p ≤ ↑x → ¬↑(x*)⊨f)
l₂ : ↑p ≤ ↑x
l₃ : ¬¬↑(x*)⊨f
⊢ p*⊨f | α : Type
inst : Model α
p : Model.primes
f : Form
h₁ : ¬p⊨~f
x : Model.primes
l₁ : ¬(↑p ≤ ↑x → ¬↑(x*)⊨f)
l₂ : ↑p ≤ ↑x
l₃ : ¬¬↑(x*)⊨f
l₄ : ↑(x*)⊨f
⊢ p*⊨f |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Semantics/Satisfaction.lean | starCompatLeft | [88, 1] | [94, 29] | have l₅ := inst.starAntitone l₂ | α : Type
inst : Model α
p : Model.primes
f : Form
h₁ : ¬p⊨~f
x : Model.primes
l₁ : ¬(↑p ≤ ↑x → ¬↑(x*)⊨f)
l₂ : ↑p ≤ ↑x
l₃ : ¬¬↑(x*)⊨f
l₄ : ↑(x*)⊨f
⊢ p*⊨f | α : Type
inst : Model α
p : Model.primes
f : Form
h₁ : ¬p⊨~f
x : Model.primes
l₁ : ¬(↑p ≤ ↑x → ¬↑(x*)⊨f)
l₂ : ↑p ≤ ↑x
l₃ : ¬¬↑(x*)⊨f
l₄ : ↑(x*)⊨f
l₅ : x* ≤ p*
⊢ p*⊨f |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Semantics/Satisfaction.lean | starCompatLeft | [88, 1] | [94, 29] | exact upwardsClosure l₅ l₄ | α : Type
inst : Model α
p : Model.primes
f : Form
h₁ : ¬p⊨~f
x : Model.primes
l₁ : ¬(↑p ≤ ↑x → ¬↑(x*)⊨f)
l₂ : ↑p ≤ ↑x
l₃ : ¬¬↑(x*)⊨f
l₄ : ↑(x*)⊨f
l₅ : x* ≤ p*
⊢ p*⊨f | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedFormal | [17, 1] | [35, 36] | unfold formalTheory | ⊢ ∀ (Γ : Ctx), formalTheory (▲Γ) | ⊢ ∀ (Γ : Ctx) {f : Form}, f ∈ ▲Γ ↔ ▲Γ⊢f |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedFormal | [17, 1] | [35, 36] | intros Γ f | ⊢ ∀ (Γ : Ctx) {f : Form}, f ∈ ▲Γ ↔ ▲Γ⊢f | Γ : Ctx
f : Form
⊢ f ∈ ▲Γ ↔ ▲Γ⊢f |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedFormal | [17, 1] | [35, 36] | apply Iff.intro | Γ : Ctx
f : Form
⊢ f ∈ ▲Γ ↔ ▲Γ⊢f | case mp
Γ : Ctx
f : Form
⊢ f ∈ ▲Γ → ▲Γ⊢f
case mpr
Γ : Ctx
f : Form
⊢ ▲Γ⊢f → f ∈ ▲Γ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedFormal | [17, 1] | [35, 36] | case mp =>
intros h₁
exact Nonempty.intro $ BProof.ax h₁ | Γ : Ctx
f : Form
⊢ f ∈ ▲Γ → ▲Γ⊢f | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedFormal | [17, 1] | [35, 36] | case mpr =>
intros h₁
have ⟨w⟩ := h₁
induction w
case ax => assumption
case mp P Q prf thm ih =>
have ⟨prf₂⟩ := ih ⟨prf⟩
exact ⟨BProof.mp prf₂ thm⟩
case adj P Q prf₁ prf₂ ih₁ ih₂ =>
have ⟨prf₃⟩ := ih₁ ⟨prf₁⟩
have ⟨prf₄⟩ := ih₂ ⟨prf₂⟩
exact ⟨BProof.adj prf₃ prf₄⟩ | Γ : Ctx
f : Form
⊢ ▲Γ⊢f → f ∈ ▲Γ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedFormal | [17, 1] | [35, 36] | intros h₁ | Γ : Ctx
f : Form
⊢ f ∈ ▲Γ → ▲Γ⊢f | Γ : Ctx
f : Form
h₁ : f ∈ ▲Γ
⊢ ▲Γ⊢f |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedFormal | [17, 1] | [35, 36] | exact Nonempty.intro $ BProof.ax h₁ | Γ : Ctx
f : Form
h₁ : f ∈ ▲Γ
⊢ ▲Γ⊢f | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedFormal | [17, 1] | [35, 36] | intros h₁ | Γ : Ctx
f : Form
⊢ ▲Γ⊢f → f ∈ ▲Γ | Γ : Ctx
f : Form
h₁ : ▲Γ⊢f
⊢ f ∈ ▲Γ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedFormal | [17, 1] | [35, 36] | have ⟨w⟩ := h₁ | Γ : Ctx
f : Form
h₁ : ▲Γ⊢f
⊢ f ∈ ▲Γ | Γ : Ctx
f : Form
h₁ : ▲Γ⊢f
w : BProof (▲Γ) f
⊢ f ∈ ▲Γ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedFormal | [17, 1] | [35, 36] | induction w | Γ : Ctx
f : Form
h₁ : ▲Γ⊢f
w : BProof (▲Γ) f
⊢ f ∈ ▲Γ | case ax
Γ : Ctx
f p✝ : Form
h✝ : p✝ ∈ ▲Γ
h₁ : ▲Γ⊢p✝
⊢ p✝ ∈ ▲Γ
case mp
Γ : Ctx
f p✝ q✝ : Form
h₁✝ : BProof (▲Γ) p✝
h₂✝ : BTheorem (p✝⊃q✝)
h₁_ih✝ : ▲Γ⊢p✝ → p✝ ∈ ▲Γ
h₁ : ▲Γ⊢q✝
⊢ q✝ ∈ ▲Γ
case adj
Γ : Ctx
f p✝ q✝ : Form
h₁✝ : BProof (▲Γ) p✝
h₂✝ : BProof (▲Γ) q✝
h₁_ih✝ : ▲Γ⊢p✝ → p✝ ∈ ▲Γ
h₂_ih✝ : ▲Γ⊢q✝ → q✝ ∈ ▲Γ
h₁ : ▲Γ⊢p✝&q✝
⊢ p✝&q✝ ∈ ▲Γ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedFormal | [17, 1] | [35, 36] | case ax => assumption | Γ : Ctx
f p✝ : Form
h✝ : p✝ ∈ ▲Γ
h₁ : ▲Γ⊢p✝
⊢ p✝ ∈ ▲Γ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedFormal | [17, 1] | [35, 36] | case mp P Q prf thm ih =>
have ⟨prf₂⟩ := ih ⟨prf⟩
exact ⟨BProof.mp prf₂ thm⟩ | Γ : Ctx
f P Q : Form
prf : BProof (▲Γ) P
thm : BTheorem (P⊃Q)
ih : ▲Γ⊢P → P ∈ ▲Γ
h₁ : ▲Γ⊢Q
⊢ Q ∈ ▲Γ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedFormal | [17, 1] | [35, 36] | case adj P Q prf₁ prf₂ ih₁ ih₂ =>
have ⟨prf₃⟩ := ih₁ ⟨prf₁⟩
have ⟨prf₄⟩ := ih₂ ⟨prf₂⟩
exact ⟨BProof.adj prf₃ prf₄⟩ | Γ : Ctx
f P Q : Form
prf₁ : BProof (▲Γ) P
prf₂ : BProof (▲Γ) Q
ih₁ : ▲Γ⊢P → P ∈ ▲Γ
ih₂ : ▲Γ⊢Q → Q ∈ ▲Γ
h₁ : ▲Γ⊢P&Q
⊢ P&Q ∈ ▲Γ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedFormal | [17, 1] | [35, 36] | assumption | Γ : Ctx
f p✝ : Form
h✝ : p✝ ∈ ▲Γ
h₁ : ▲Γ⊢p✝
⊢ p✝ ∈ ▲Γ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedFormal | [17, 1] | [35, 36] | have ⟨prf₂⟩ := ih ⟨prf⟩ | Γ : Ctx
f P Q : Form
prf : BProof (▲Γ) P
thm : BTheorem (P⊃Q)
ih : ▲Γ⊢P → P ∈ ▲Γ
h₁ : ▲Γ⊢Q
⊢ Q ∈ ▲Γ | Γ : Ctx
f P Q : Form
prf : BProof (▲Γ) P
thm : BTheorem (P⊃Q)
ih : ▲Γ⊢P → P ∈ ▲Γ
h₁ : ▲Γ⊢Q
prf₂ : BProof Γ P
⊢ Q ∈ ▲Γ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedFormal | [17, 1] | [35, 36] | exact ⟨BProof.mp prf₂ thm⟩ | Γ : Ctx
f P Q : Form
prf : BProof (▲Γ) P
thm : BTheorem (P⊃Q)
ih : ▲Γ⊢P → P ∈ ▲Γ
h₁ : ▲Γ⊢Q
prf₂ : BProof Γ P
⊢ Q ∈ ▲Γ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedFormal | [17, 1] | [35, 36] | have ⟨prf₃⟩ := ih₁ ⟨prf₁⟩ | Γ : Ctx
f P Q : Form
prf₁ : BProof (▲Γ) P
prf₂ : BProof (▲Γ) Q
ih₁ : ▲Γ⊢P → P ∈ ▲Γ
ih₂ : ▲Γ⊢Q → Q ∈ ▲Γ
h₁ : ▲Γ⊢P&Q
⊢ P&Q ∈ ▲Γ | Γ : Ctx
f P Q : Form
prf₁ : BProof (▲Γ) P
prf₂ : BProof (▲Γ) Q
ih₁ : ▲Γ⊢P → P ∈ ▲Γ
ih₂ : ▲Γ⊢Q → Q ∈ ▲Γ
h₁ : ▲Γ⊢P&Q
prf₃ : BProof Γ P
⊢ P&Q ∈ ▲Γ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedFormal | [17, 1] | [35, 36] | have ⟨prf₄⟩ := ih₂ ⟨prf₂⟩ | Γ : Ctx
f P Q : Form
prf₁ : BProof (▲Γ) P
prf₂ : BProof (▲Γ) Q
ih₁ : ▲Γ⊢P → P ∈ ▲Γ
ih₂ : ▲Γ⊢Q → Q ∈ ▲Γ
h₁ : ▲Γ⊢P&Q
prf₃ : BProof Γ P
⊢ P&Q ∈ ▲Γ | Γ : Ctx
f P Q : Form
prf₁ : BProof (▲Γ) P
prf₂ : BProof (▲Γ) Q
ih₁ : ▲Γ⊢P → P ∈ ▲Γ
ih₂ : ▲Γ⊢Q → Q ∈ ▲Γ
h₁ : ▲Γ⊢P&Q
prf₃ : BProof Γ P
prf₄ : BProof Γ Q
⊢ P&Q ∈ ▲Γ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedFormal | [17, 1] | [35, 36] | exact ⟨BProof.adj prf₃ prf₄⟩ | Γ : Ctx
f P Q : Form
prf₁ : BProof (▲Γ) P
prf₂ : BProof (▲Γ) Q
ih₁ : ▲Γ⊢P → P ∈ ▲Γ
ih₂ : ▲Γ⊢Q → Q ∈ ▲Γ
h₁ : ▲Γ⊢P&Q
prf₃ : BProof Γ P
prf₄ : BProof Γ Q
⊢ P&Q ∈ ▲Γ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedDisjunction | [49, 1] | [55, 21] | intros h₁ | f g h : Form
⊢ f ∈ ▲{g} ∧ f ∈ ▲{h} → f ∈ ▲{g¦h} | f g h : Form
h₁ : f ∈ ▲{g} ∧ f ∈ ▲{h}
⊢ f ∈ ▲{g¦h} |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedDisjunction | [49, 1] | [55, 21] | have ⟨⟨prf₁⟩,⟨prf₂⟩⟩ := h₁ | f g h : Form
h₁ : f ∈ ▲{g} ∧ f ∈ ▲{h}
⊢ f ∈ ▲{g¦h} | f g h : Form
h₁ : f ∈ ▲{g} ∧ f ∈ ▲{h}
prf₁ : BProof {g} f
prf₂ : BProof {h} f
⊢ f ∈ ▲{g¦h} |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedDisjunction | [49, 1] | [55, 21] | have l₁ := prf₁.toTheorem | f g h : Form
h₁ : f ∈ ▲{g} ∧ f ∈ ▲{h}
prf₁ : BProof {g} f
prf₂ : BProof {h} f
⊢ f ∈ ▲{g¦h} | f g h : Form
h₁ : f ∈ ▲{g} ∧ f ∈ ▲{h}
prf₁ : BProof {g} f
prf₂ : BProof {h} f
l₁ : BTheorem (g⊃f)
⊢ f ∈ ▲{g¦h} |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedDisjunction | [49, 1] | [55, 21] | have l₂ := prf₂.toTheorem | f g h : Form
h₁ : f ∈ ▲{g} ∧ f ∈ ▲{h}
prf₁ : BProof {g} f
prf₂ : BProof {h} f
l₁ : BTheorem (g⊃f)
⊢ f ∈ ▲{g¦h} | f g h : Form
h₁ : f ∈ ▲{g} ∧ f ∈ ▲{h}
prf₁ : BProof {g} f
prf₂ : BProof {h} f
l₁ : BTheorem (g⊃f)
l₂ : BTheorem (h⊃f)
⊢ f ∈ ▲{g¦h} |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedDisjunction | [49, 1] | [55, 21] | have l₃ := (BTheorem.mp (BTheorem.adj l₁ l₂) BTheorem.orE) | f g h : Form
h₁ : f ∈ ▲{g} ∧ f ∈ ▲{h}
prf₁ : BProof {g} f
prf₂ : BProof {h} f
l₁ : BTheorem (g⊃f)
l₂ : BTheorem (h⊃f)
⊢ f ∈ ▲{g¦h} | f g h : Form
h₁ : f ∈ ▲{g} ∧ f ∈ ▲{h}
prf₁ : BProof {g} f
prf₂ : BProof {h} f
l₁ : BTheorem (g⊃f)
l₂ : BTheorem (h⊃f)
l₃ : BTheorem (g¦h⊃f)
⊢ f ∈ ▲{g¦h} |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedDisjunction | [49, 1] | [55, 21] | exact ⟨l₃.toProof⟩ | f g h : Form
h₁ : f ∈ ▲{g} ∧ f ∈ ▲{h}
prf₁ : BProof {g} f
prf₂ : BProof {h} f
l₁ : BTheorem (g⊃f)
l₂ : BTheorem (h⊃f)
l₃ : BTheorem (g¦h⊃f)
⊢ f ∈ ▲{g¦h} | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedContained | [57, 1] | [71, 48] | intros h₁ f h₂ | Γ : Ctx
Δ : Th
⊢ Γ ⊆ ↑Δ → ▲Γ ⊆ ↑Δ | Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f : Form
h₂ : f ∈ ▲Γ
⊢ f ∈ ↑Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedContained | [57, 1] | [71, 48] | have ⟨prf⟩ := h₂ | Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f : Form
h₂ : f ∈ ▲Γ
⊢ f ∈ ↑Δ | Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f : Form
h₂ : f ∈ ▲Γ
prf : BProof Γ f
⊢ f ∈ ↑Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedContained | [57, 1] | [71, 48] | induction prf | Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f : Form
h₂ : f ∈ ▲Γ
prf : BProof Γ f
⊢ f ∈ ↑Δ | case ax
Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f p✝ : Form
h✝ : p✝ ∈ Γ
h₂ : p✝ ∈ ▲Γ
⊢ p✝ ∈ ↑Δ
case mp
Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f p✝ q✝ : Form
h₁✝ : BProof Γ p✝
h₂✝ : BTheorem (p✝⊃q✝)
h₁_ih✝ : p✝ ∈ ▲Γ → p✝ ∈ ↑Δ
h₂ : q✝ ∈ ▲Γ
⊢ q✝ ∈ ↑Δ
case adj
Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f p✝ q✝ : Form
h₁✝ : BProof Γ p✝
h₂✝ : BProof Γ q✝
h₁_ih✝ : p✝ ∈ ▲Γ → p✝ ∈ ↑Δ
h₂_ih✝ : q✝ ∈ ▲Γ → q✝ ∈ ↑Δ
h₂ : p✝&q✝ ∈ ▲Γ
⊢ p✝&q✝ ∈ ↑Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedContained | [57, 1] | [71, 48] | case ax p ih => exact h₁ ih | Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f p : Form
ih : p ∈ Γ
h₂ : p ∈ ▲Γ
⊢ p ∈ ↑Δ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedContained | [57, 1] | [71, 48] | case mp p q prf₁ prf₂ ih =>
have l₁ := ih ⟨prf₁⟩
have ⟨prf₃⟩ := Δ.property.mp l₁
exact Δ.property.mpr ⟨BProof.mp prf₃ prf₂⟩ | Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f p q : Form
prf₁ : BProof Γ p
prf₂ : BTheorem (p⊃q)
ih : p ∈ ▲Γ → p ∈ ↑Δ
h₂ : q ∈ ▲Γ
⊢ q ∈ ↑Δ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedContained | [57, 1] | [71, 48] | case adj p q prf₁ prf₂ ih₁ ih₂ =>
have l₁ := ih₁ ⟨prf₁⟩
have l₂ := ih₂ ⟨prf₂⟩
have ⟨prf₃⟩ := Δ.property.mp l₁
have ⟨prf₄⟩ := Δ.property.mp l₂
exact Δ.property.mpr ⟨BProof.adj prf₃ prf₄⟩ | Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f p q : Form
prf₁ : BProof Γ p
prf₂ : BProof Γ q
ih₁ : p ∈ ▲Γ → p ∈ ↑Δ
ih₂ : q ∈ ▲Γ → q ∈ ↑Δ
h₂ : p&q ∈ ▲Γ
⊢ p&q ∈ ↑Δ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedContained | [57, 1] | [71, 48] | exact h₁ ih | Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f p : Form
ih : p ∈ Γ
h₂ : p ∈ ▲Γ
⊢ p ∈ ↑Δ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedContained | [57, 1] | [71, 48] | have l₁ := ih ⟨prf₁⟩ | Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f p q : Form
prf₁ : BProof Γ p
prf₂ : BTheorem (p⊃q)
ih : p ∈ ▲Γ → p ∈ ↑Δ
h₂ : q ∈ ▲Γ
⊢ q ∈ ↑Δ | Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f p q : Form
prf₁ : BProof Γ p
prf₂ : BTheorem (p⊃q)
ih : p ∈ ▲Γ → p ∈ ↑Δ
h₂ : q ∈ ▲Γ
l₁ : p ∈ ↑Δ
⊢ q ∈ ↑Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedContained | [57, 1] | [71, 48] | have ⟨prf₃⟩ := Δ.property.mp l₁ | Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f p q : Form
prf₁ : BProof Γ p
prf₂ : BTheorem (p⊃q)
ih : p ∈ ▲Γ → p ∈ ↑Δ
h₂ : q ∈ ▲Γ
l₁ : p ∈ ↑Δ
⊢ q ∈ ↑Δ | Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f p q : Form
prf₁ : BProof Γ p
prf₂ : BTheorem (p⊃q)
ih : p ∈ ▲Γ → p ∈ ↑Δ
h₂ : q ∈ ▲Γ
l₁ : p ∈ ↑Δ
prf₃ : BProof (↑Δ) p
⊢ q ∈ ↑Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedContained | [57, 1] | [71, 48] | exact Δ.property.mpr ⟨BProof.mp prf₃ prf₂⟩ | Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f p q : Form
prf₁ : BProof Γ p
prf₂ : BTheorem (p⊃q)
ih : p ∈ ▲Γ → p ∈ ↑Δ
h₂ : q ∈ ▲Γ
l₁ : p ∈ ↑Δ
prf₃ : BProof (↑Δ) p
⊢ q ∈ ↑Δ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedContained | [57, 1] | [71, 48] | have l₁ := ih₁ ⟨prf₁⟩ | Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f p q : Form
prf₁ : BProof Γ p
prf₂ : BProof Γ q
ih₁ : p ∈ ▲Γ → p ∈ ↑Δ
ih₂ : q ∈ ▲Γ → q ∈ ↑Δ
h₂ : p&q ∈ ▲Γ
⊢ p&q ∈ ↑Δ | Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f p q : Form
prf₁ : BProof Γ p
prf₂ : BProof Γ q
ih₁ : p ∈ ▲Γ → p ∈ ↑Δ
ih₂ : q ∈ ▲Γ → q ∈ ↑Δ
h₂ : p&q ∈ ▲Γ
l₁ : p ∈ ↑Δ
⊢ p&q ∈ ↑Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedContained | [57, 1] | [71, 48] | have l₂ := ih₂ ⟨prf₂⟩ | Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f p q : Form
prf₁ : BProof Γ p
prf₂ : BProof Γ q
ih₁ : p ∈ ▲Γ → p ∈ ↑Δ
ih₂ : q ∈ ▲Γ → q ∈ ↑Δ
h₂ : p&q ∈ ▲Γ
l₁ : p ∈ ↑Δ
⊢ p&q ∈ ↑Δ | Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f p q : Form
prf₁ : BProof Γ p
prf₂ : BProof Γ q
ih₁ : p ∈ ▲Γ → p ∈ ↑Δ
ih₂ : q ∈ ▲Γ → q ∈ ↑Δ
h₂ : p&q ∈ ▲Γ
l₁ : p ∈ ↑Δ
l₂ : q ∈ ↑Δ
⊢ p&q ∈ ↑Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedContained | [57, 1] | [71, 48] | have ⟨prf₃⟩ := Δ.property.mp l₁ | Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f p q : Form
prf₁ : BProof Γ p
prf₂ : BProof Γ q
ih₁ : p ∈ ▲Γ → p ∈ ↑Δ
ih₂ : q ∈ ▲Γ → q ∈ ↑Δ
h₂ : p&q ∈ ▲Γ
l₁ : p ∈ ↑Δ
l₂ : q ∈ ↑Δ
⊢ p&q ∈ ↑Δ | Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f p q : Form
prf₁ : BProof Γ p
prf₂ : BProof Γ q
ih₁ : p ∈ ▲Γ → p ∈ ↑Δ
ih₂ : q ∈ ▲Γ → q ∈ ↑Δ
h₂ : p&q ∈ ▲Γ
l₁ : p ∈ ↑Δ
l₂ : q ∈ ↑Δ
prf₃ : BProof (↑Δ) p
⊢ p&q ∈ ↑Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedContained | [57, 1] | [71, 48] | have ⟨prf₄⟩ := Δ.property.mp l₂ | Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f p q : Form
prf₁ : BProof Γ p
prf₂ : BProof Γ q
ih₁ : p ∈ ▲Γ → p ∈ ↑Δ
ih₂ : q ∈ ▲Γ → q ∈ ↑Δ
h₂ : p&q ∈ ▲Γ
l₁ : p ∈ ↑Δ
l₂ : q ∈ ↑Δ
prf₃ : BProof (↑Δ) p
⊢ p&q ∈ ↑Δ | Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f p q : Form
prf₁ : BProof Γ p
prf₂ : BProof Γ q
ih₁ : p ∈ ▲Γ → p ∈ ↑Δ
ih₂ : q ∈ ▲Γ → q ∈ ↑Δ
h₂ : p&q ∈ ▲Γ
l₁ : p ∈ ↑Δ
l₂ : q ∈ ↑Δ
prf₃ : BProof (↑Δ) p
prf₄ : BProof (↑Δ) q
⊢ p&q ∈ ↑Δ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | generatedContained | [57, 1] | [71, 48] | exact Δ.property.mpr ⟨BProof.adj prf₃ prf₄⟩ | Γ : Ctx
Δ : Th
h₁ : Γ ⊆ ↑Δ
f p q : Form
prf₁ : BProof Γ p
prf₂ : BProof Γ q
ih₁ : p ∈ ▲Γ → p ∈ ↑Δ
ih₂ : q ∈ ▲Γ → q ∈ ↑Δ
h₂ : p&q ∈ ▲Γ
l₁ : p ∈ ↑Δ
l₂ : q ∈ ↑Δ
prf₃ : BProof (↑Δ) p
prf₄ : BProof (↑Δ) q
⊢ p&q ∈ ↑Δ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalFixed | [73, 1] | [82, 20] | intros h₁ | Γ : Ctx
⊢ formalTheory Γ → ▲Γ = Γ | Γ : Ctx
h₁ : formalTheory Γ
⊢ ▲Γ = Γ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalFixed | [73, 1] | [82, 20] | funext | Γ : Ctx
h₁ : formalTheory Γ
⊢ ▲Γ = Γ | case h
Γ : Ctx
h₁ : formalTheory Γ
x✝ : Form
⊢ (▲Γ) x✝ = Γ x✝ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalFixed | [73, 1] | [82, 20] | ext | Γ : Ctx
h₁ : formalTheory Γ
x : Form
⊢ (▲Γ) x = Γ x | case a
Γ : Ctx
h₁ : formalTheory Γ
x : Form
⊢ (▲Γ) x ↔ Γ x |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalFixed | [73, 1] | [82, 20] | apply Iff.intro | case a
Γ : Ctx
h₁ : formalTheory Γ
x : Form
⊢ (▲Γ) x ↔ Γ x | case a.mp
Γ : Ctx
h₁ : formalTheory Γ
x : Form
⊢ (▲Γ) x → Γ x
case a.mpr
Γ : Ctx
h₁ : formalTheory Γ
x : Form
⊢ Γ x → (▲Γ) x |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | BisFormal | [84, 1] | [100, 33] | intros f | ⊢ formalTheory BTheory | f : Form
⊢ f ∈ BTheory ↔ BTheory⊢f |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | BisFormal | [84, 1] | [100, 33] | apply Iff.intro | f : Form
⊢ f ∈ BTheory ↔ BTheory⊢f | case mp
f : Form
⊢ f ∈ BTheory → BTheory⊢f
case mpr
f : Form
⊢ BTheory⊢f → f ∈ BTheory |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | BisFormal | [84, 1] | [100, 33] | have l₁ := ih ⟨prf₁⟩ | f P Q : Form
prf₁ : BProof BTheory P
thm₁ : BTheorem (P⊃Q)
ih : BTheory⊢P → P ∈ BTheory
h₁ : BTheory⊢Q
⊢ Q ∈ BTheory | f P Q : Form
prf₁ : BProof BTheory P
thm₁ : BTheorem (P⊃Q)
ih : BTheory⊢P → P ∈ BTheory
h₁ : BTheory⊢Q
l₁ : P ∈ BTheory
⊢ Q ∈ BTheory |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | BisFormal | [84, 1] | [100, 33] | have ⟨thm₂⟩ := l₁ | f P Q : Form
prf₁ : BProof BTheory P
thm₁ : BTheorem (P⊃Q)
ih : BTheory⊢P → P ∈ BTheory
h₁ : BTheory⊢Q
l₁ : P ∈ BTheory
⊢ Q ∈ BTheory | f P Q : Form
prf₁ : BProof BTheory P
thm₁ : BTheorem (P⊃Q)
ih : BTheory⊢P → P ∈ BTheory
h₁ : BTheory⊢Q
l₁ : P ∈ BTheory
thm₂ : BTheorem P
⊢ Q ∈ BTheory |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | BisFormal | [84, 1] | [100, 33] | exact ⟨BTheorem.mp thm₂ thm₁⟩ | f P Q : Form
prf₁ : BProof BTheory P
thm₁ : BTheorem (P⊃Q)
ih : BTheory⊢P → P ∈ BTheory
h₁ : BTheory⊢Q
l₁ : P ∈ BTheory
thm₂ : BTheorem P
⊢ Q ∈ BTheory | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | BisFormal | [84, 1] | [100, 33] | have ⟨l₁⟩ := ih₁ ⟨prf₁⟩ | f P Q : Form
prf₁ : BProof BTheory P
prf₂ : BProof BTheory Q
ih₁ : BTheory⊢P → P ∈ BTheory
ih₂ : BTheory⊢Q → Q ∈ BTheory
h₁ : BTheory⊢P&Q
⊢ P&Q ∈ BTheory | f P Q : Form
prf₁ : BProof BTheory P
prf₂ : BProof BTheory Q
ih₁ : BTheory⊢P → P ∈ BTheory
ih₂ : BTheory⊢Q → Q ∈ BTheory
h₁ : BTheory⊢P&Q
l₁ : BTheorem P
⊢ P&Q ∈ BTheory |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | BisFormal | [84, 1] | [100, 33] | have ⟨l₂⟩ := ih₂ ⟨prf₂⟩ | f P Q : Form
prf₁ : BProof BTheory P
prf₂ : BProof BTheory Q
ih₁ : BTheory⊢P → P ∈ BTheory
ih₂ : BTheory⊢Q → Q ∈ BTheory
h₁ : BTheory⊢P&Q
l₁ : BTheorem P
⊢ P&Q ∈ BTheory | f P Q : Form
prf₁ : BProof BTheory P
prf₂ : BProof BTheory Q
ih₁ : BTheory⊢P → P ∈ BTheory
ih₂ : BTheory⊢Q → Q ∈ BTheory
h₁ : BTheory⊢P&Q
l₁ : BTheorem P
l₂ : BTheorem Q
⊢ P&Q ∈ BTheory |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | BisFormal | [84, 1] | [100, 33] | exact ⟨BTheorem.adj l₁ l₂⟩ | f P Q : Form
prf₁ : BProof BTheory P
prf₂ : BProof BTheory Q
ih₁ : BTheory⊢P → P ∈ BTheory
ih₂ : BTheory⊢Q → Q ∈ BTheory
h₁ : BTheory⊢P&Q
l₁ : BTheorem P
l₂ : BTheorem Q
⊢ P&Q ∈ BTheory | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalAppMonotoneLeft | [106, 1] | [109, 34] | intros Γ a b h₁ A h₂ | ⊢ ∀ (Γ : Ctx), Monotone (formalApplication Γ) | Γ a b : Ctx
h₁ : a ≤ b
A : Form
h₂ : A ∈ formalApplication Γ a
⊢ A ∈ formalApplication Γ b |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalAppMonotoneLeft | [106, 1] | [109, 34] | have ⟨g,h₃⟩ := h₂ | Γ a b : Ctx
h₁ : a ≤ b
A : Form
h₂ : A ∈ formalApplication Γ a
⊢ A ∈ formalApplication Γ b | Γ a b : Ctx
h₁ : a ≤ b
A : Form
h₂ : A ∈ formalApplication Γ a
g : Form
h₃ : g ∈ a ∧ g⊃A ∈ Γ
⊢ A ∈ formalApplication Γ b |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalAppMonotoneLeft | [106, 1] | [109, 34] | exact ⟨g, h₁ h₃.left, h₃.right⟩ | Γ a b : Ctx
h₁ : a ≤ b
A : Form
h₂ : A ∈ formalApplication Γ a
g : Form
h₃ : g ∈ a ∧ g⊃A ∈ Γ
⊢ A ∈ formalApplication Γ b | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalAppMonotoneRight | [111, 1] | [114, 34] | intros Γ a b h₁ A h₂ | ⊢ ∀ (Γ : Ctx), Monotone (flip formalApplication Γ) | Γ a b : Ctx
h₁ : a ≤ b
A : Form
h₂ : A ∈ flip formalApplication Γ a
⊢ A ∈ flip formalApplication Γ b |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalAppMonotoneRight | [111, 1] | [114, 34] | have ⟨g,h₃⟩ := h₂ | Γ a b : Ctx
h₁ : a ≤ b
A : Form
h₂ : A ∈ flip formalApplication Γ a
⊢ A ∈ flip formalApplication Γ b | Γ a b : Ctx
h₁ : a ≤ b
A : Form
h₂ : A ∈ flip formalApplication Γ a
g : Form
h₃ : g ∈ Γ ∧ g⊃A ∈ a
⊢ A ∈ flip formalApplication Γ b |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalAppMonotoneRight | [111, 1] | [114, 34] | exact ⟨g, h₃.left, h₁ h₃.right⟩ | Γ a b : Ctx
h₁ : a ≤ b
A : Form
h₂ : A ∈ flip formalApplication Γ a
g : Form
h₃ : g ∈ Γ ∧ g⊃A ∈ a
⊢ A ∈ flip formalApplication Γ b | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalAppFunctionMonotoneRight | [153, 1] | [155, 36] | intros Γ _ _ h₁ | ⊢ ∀ (Γ : Th), Monotone (flip formalApplicationFunction Γ) | Γ a✝ b✝ : Th
h₁ : a✝ ≤ b✝
⊢ flip formalApplicationFunction Γ a✝ ≤ flip formalApplicationFunction Γ b✝ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalAppFunctionMonotoneRight | [153, 1] | [155, 36] | exact formalAppMonotoneRight Γ h₁ | Γ a✝ b✝ : Th
h₁ : a✝ ≤ b✝
⊢ flip formalApplicationFunction Γ a✝ ≤ flip formalApplicationFunction Γ b✝ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalAppFunctionMonotoneLeft | [157, 1] | [159, 35] | intros Γ _ _ h₁ | ⊢ ∀ (Γ : Th), Monotone (formalApplicationFunction Γ) | Γ a✝ b✝ : Th
h₁ : a✝ ≤ b✝
⊢ formalApplicationFunction Γ a✝ ≤ formalApplicationFunction Γ b✝ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalAppFunctionMonotoneLeft | [157, 1] | [159, 35] | exact formalAppMonotoneLeft Γ h₁ | Γ a✝ b✝ : Th
h₁ : a✝ ≤ b✝
⊢ formalApplicationFunction Γ a✝ ≤ formalApplicationFunction Γ b✝ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalAppIdentLeft | [161, 1] | [171, 35] | intros Γ | ⊢ ∀ (Γ : Th), formalApplicationFunction BTh Γ = Γ | Γ : Th
⊢ formalApplicationFunction BTh Γ = Γ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalAppIdentLeft | [161, 1] | [171, 35] | ext f | Γ : Th
⊢ formalApplicationFunction BTh Γ = Γ | case a.h
Γ : Th
f : Form
⊢ f ∈ ↑(formalApplicationFunction BTh Γ) ↔ f ∈ ↑Γ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalAppIdentLeft | [161, 1] | [171, 35] | apply Iff.intro | case a.h
Γ : Th
f : Form
⊢ f ∈ ↑(formalApplicationFunction BTh Γ) ↔ f ∈ ↑Γ | case a.h.mp
Γ : Th
f : Form
⊢ f ∈ ↑(formalApplicationFunction BTh Γ) → f ∈ ↑Γ
case a.h.mpr
Γ : Th
f : Form
⊢ f ∈ ↑Γ → f ∈ ↑(formalApplicationFunction BTh Γ) |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalAppIdentLeft | [161, 1] | [171, 35] | case a.h.mp =>
intros h₁
have ⟨g,h₂,⟨h₃⟩⟩ := h₁
exact Γ.property.mpr ⟨BProof.mp (BProof.ax h₂) h₃⟩ | Γ : Th
f : Form
⊢ f ∈ ↑(formalApplicationFunction BTh Γ) → f ∈ ↑Γ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalAppIdentLeft | [161, 1] | [171, 35] | case a.h.mpr =>
intros h₁
exact ⟨f, h₁, ⟨BTheorem.taut⟩⟩ | Γ : Th
f : Form
⊢ f ∈ ↑Γ → f ∈ ↑(formalApplicationFunction BTh Γ) | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalAppIdentLeft | [161, 1] | [171, 35] | intros h₁ | Γ : Th
f : Form
⊢ f ∈ ↑(formalApplicationFunction BTh Γ) → f ∈ ↑Γ | Γ : Th
f : Form
h₁ : f ∈ ↑(formalApplicationFunction BTh Γ)
⊢ f ∈ ↑Γ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalAppIdentLeft | [161, 1] | [171, 35] | have ⟨g,h₂,⟨h₃⟩⟩ := h₁ | Γ : Th
f : Form
h₁ : f ∈ ↑(formalApplicationFunction BTh Γ)
⊢ f ∈ ↑Γ | Γ : Th
f : Form
h₁ : f ∈ ↑(formalApplicationFunction BTh Γ)
g : Form
h₂ : g ∈ ↑Γ
h₃ : BTheorem (g⊃f)
⊢ f ∈ ↑Γ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalAppIdentLeft | [161, 1] | [171, 35] | exact Γ.property.mpr ⟨BProof.mp (BProof.ax h₂) h₃⟩ | Γ : Th
f : Form
h₁ : f ∈ ↑(formalApplicationFunction BTh Γ)
g : Form
h₂ : g ∈ ↑Γ
h₃ : BTheorem (g⊃f)
⊢ f ∈ ↑Γ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalAppIdentLeft | [161, 1] | [171, 35] | intros h₁ | Γ : Th
f : Form
⊢ f ∈ ↑Γ → f ∈ ↑(formalApplicationFunction BTh Γ) | Γ : Th
f : Form
h₁ : f ∈ ↑Γ
⊢ f ∈ ↑(formalApplicationFunction BTh Γ) |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalAppIdentLeft | [161, 1] | [171, 35] | exact ⟨f, h₁, ⟨BTheorem.taut⟩⟩ | Γ : Th
f : Form
h₁ : f ∈ ↑Γ
⊢ f ∈ ↑(formalApplicationFunction BTh Γ) | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | intros F | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
⊢ formalTheory (FormalDual Γ) | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F : Form
⊢ F ∈ FormalDual Γ ↔ FormalDual Γ⊢F |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | apply Iff.intro <;> intros h₃ <;> unfold FormalDual | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F : Form
⊢ F ∈ FormalDual Γ ↔ FormalDual Γ⊢F | case mp
Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F : Form
h₃ : F ∈ FormalDual Γ
⊢ (fun f => ¬~f ∈ Γ)⊢F
case mpr
Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F : Form
h₃ : FormalDual Γ⊢F
⊢ F ∈ fun f => ¬~f ∈ Γ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | case mp => exact ⟨BProof.ax h₃⟩ | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F : Form
h₃ : F ∈ FormalDual Γ
⊢ (fun f => ¬~f ∈ Γ)⊢F | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | case mpr =>
have ⟨prf₁⟩ := h₃
induction prf₁
case ax => assumption
case mp P Q prf₂ thm₁ ih₁ =>
intros h₄
have l₁ := ih₁ ⟨prf₂⟩
unfold FormalDual at l₁
have thm₂ : BTheorem (~Q ⊃ ~P) := BTheorem.cp $ BTheorem.transitivity thm₁ (BTheorem.cp BTheorem.taut)
have prf₂ := BProof.mp (BProof.ax h₄) thm₂
exact l₁ (h₁.mpr ⟨prf₂⟩)
case adj P Q prf₁ prf₂ ih₁ ih₂ =>
intros h₄
have l₁ := ih₁ ⟨prf₁⟩
have l₂ := ih₂ ⟨prf₂⟩
have prf₃ := BProof.mp (BProof.ax h₄) BTheorem.demorgansLaw3
have l₃ := h₂ (h₁.mpr ⟨prf₃⟩)
cases l₃
case inl left => exact l₁ left
case inr right => exact l₂ right | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F : Form
h₃ : FormalDual Γ⊢F
⊢ F ∈ fun f => ¬~f ∈ Γ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | exact ⟨BProof.ax h₃⟩ | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F : Form
h₃ : F ∈ FormalDual Γ
⊢ (fun f => ¬~f ∈ Γ)⊢F | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | have ⟨prf₁⟩ := h₃ | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F : Form
h₃ : FormalDual Γ⊢F
⊢ F ∈ fun f => ¬~f ∈ Γ | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F : Form
h₃ : FormalDual Γ⊢F
prf₁ : BProof (FormalDual Γ) F
⊢ F ∈ fun f => ¬~f ∈ Γ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | induction prf₁ | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F : Form
h₃ : FormalDual Γ⊢F
prf₁ : BProof (FormalDual Γ) F
⊢ F ∈ fun f => ¬~f ∈ Γ | case ax
Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F p✝ : Form
h✝ : p✝ ∈ FormalDual Γ
h₃ : FormalDual Γ⊢p✝
⊢ p✝ ∈ fun f => ¬~f ∈ Γ
case mp
Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F p✝ q✝ : Form
h₁✝ : BProof (FormalDual Γ) p✝
h₂✝ : BTheorem (p✝⊃q✝)
h₁_ih✝ : FormalDual Γ⊢p✝ → p✝ ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢q✝
⊢ q✝ ∈ fun f => ¬~f ∈ Γ
case adj
Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F p✝ q✝ : Form
h₁✝ : BProof (FormalDual Γ) p✝
h₂✝ : BProof (FormalDual Γ) q✝
h₁_ih✝ : FormalDual Γ⊢p✝ → p✝ ∈ fun f => ¬~f ∈ Γ
h₂_ih✝ : FormalDual Γ⊢q✝ → q✝ ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢p✝&q✝
⊢ p✝&q✝ ∈ fun f => ¬~f ∈ Γ |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | case ax => assumption | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F p✝ : Form
h✝ : p✝ ∈ FormalDual Γ
h₃ : FormalDual Γ⊢p✝
⊢ p✝ ∈ fun f => ¬~f ∈ Γ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | case mp P Q prf₂ thm₁ ih₁ =>
intros h₄
have l₁ := ih₁ ⟨prf₂⟩
unfold FormalDual at l₁
have thm₂ : BTheorem (~Q ⊃ ~P) := BTheorem.cp $ BTheorem.transitivity thm₁ (BTheorem.cp BTheorem.taut)
have prf₂ := BProof.mp (BProof.ax h₄) thm₂
exact l₁ (h₁.mpr ⟨prf₂⟩) | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₂ : BProof (FormalDual Γ) P
thm₁ : BTheorem (P⊃Q)
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢Q
⊢ Q ∈ fun f => ¬~f ∈ Γ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | case adj P Q prf₁ prf₂ ih₁ ih₂ =>
intros h₄
have l₁ := ih₁ ⟨prf₁⟩
have l₂ := ih₂ ⟨prf₂⟩
have prf₃ := BProof.mp (BProof.ax h₄) BTheorem.demorgansLaw3
have l₃ := h₂ (h₁.mpr ⟨prf₃⟩)
cases l₃
case inl left => exact l₁ left
case inr right => exact l₂ right | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₁ : BProof (FormalDual Γ) P
prf₂ : BProof (FormalDual Γ) Q
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
ih₂ : FormalDual Γ⊢Q → Q ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢P&Q
⊢ P&Q ∈ fun f => ¬~f ∈ Γ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | assumption | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F p✝ : Form
h✝ : p✝ ∈ FormalDual Γ
h₃ : FormalDual Γ⊢p✝
⊢ p✝ ∈ fun f => ¬~f ∈ Γ | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | intros h₄ | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₂ : BProof (FormalDual Γ) P
thm₁ : BTheorem (P⊃Q)
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢Q
⊢ Q ∈ fun f => ¬~f ∈ Γ | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₂ : BProof (FormalDual Γ) P
thm₁ : BTheorem (P⊃Q)
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢Q
h₄ : ~Q ∈ Γ
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | have l₁ := ih₁ ⟨prf₂⟩ | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₂ : BProof (FormalDual Γ) P
thm₁ : BTheorem (P⊃Q)
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢Q
h₄ : ~Q ∈ Γ
⊢ False | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₂ : BProof (FormalDual Γ) P
thm₁ : BTheorem (P⊃Q)
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢Q
h₄ : ~Q ∈ Γ
l₁ : P ∈ fun f => ¬~f ∈ Γ
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | have thm₂ : BTheorem (~Q ⊃ ~P) := BTheorem.cp $ BTheorem.transitivity thm₁ (BTheorem.cp BTheorem.taut) | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₂ : BProof (FormalDual Γ) P
thm₁ : BTheorem (P⊃Q)
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢Q
h₄ : ~Q ∈ Γ
l₁ : P ∈ fun f => ¬~f ∈ Γ
⊢ False | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₂ : BProof (FormalDual Γ) P
thm₁ : BTheorem (P⊃Q)
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢Q
h₄ : ~Q ∈ Γ
l₁ : P ∈ fun f => ¬~f ∈ Γ
thm₂ : BTheorem (~Q⊃~P)
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | have prf₂ := BProof.mp (BProof.ax h₄) thm₂ | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₂ : BProof (FormalDual Γ) P
thm₁ : BTheorem (P⊃Q)
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢Q
h₄ : ~Q ∈ Γ
l₁ : P ∈ fun f => ¬~f ∈ Γ
thm₂ : BTheorem (~Q⊃~P)
⊢ False | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₂✝ : BProof (FormalDual Γ) P
thm₁ : BTheorem (P⊃Q)
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢Q
h₄ : ~Q ∈ Γ
l₁ : P ∈ fun f => ¬~f ∈ Γ
thm₂ : BTheorem (~Q⊃~P)
prf₂ : BProof Γ ~P
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | exact l₁ (h₁.mpr ⟨prf₂⟩) | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₂✝ : BProof (FormalDual Γ) P
thm₁ : BTheorem (P⊃Q)
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢Q
h₄ : ~Q ∈ Γ
l₁ : P ∈ fun f => ¬~f ∈ Γ
thm₂ : BTheorem (~Q⊃~P)
prf₂ : BProof Γ ~P
⊢ False | no goals |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | intros h₄ | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₁ : BProof (FormalDual Γ) P
prf₂ : BProof (FormalDual Γ) Q
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
ih₂ : FormalDual Γ⊢Q → Q ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢P&Q
⊢ P&Q ∈ fun f => ¬~f ∈ Γ | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₁ : BProof (FormalDual Γ) P
prf₂ : BProof (FormalDual Γ) Q
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
ih₂ : FormalDual Γ⊢Q → Q ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢P&Q
h₄ : ~(P&Q) ∈ Γ
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | have l₁ := ih₁ ⟨prf₁⟩ | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₁ : BProof (FormalDual Γ) P
prf₂ : BProof (FormalDual Γ) Q
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
ih₂ : FormalDual Γ⊢Q → Q ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢P&Q
h₄ : ~(P&Q) ∈ Γ
⊢ False | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₁ : BProof (FormalDual Γ) P
prf₂ : BProof (FormalDual Γ) Q
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
ih₂ : FormalDual Γ⊢Q → Q ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢P&Q
h₄ : ~(P&Q) ∈ Γ
l₁ : P ∈ fun f => ¬~f ∈ Γ
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | have l₂ := ih₂ ⟨prf₂⟩ | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₁ : BProof (FormalDual Γ) P
prf₂ : BProof (FormalDual Γ) Q
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
ih₂ : FormalDual Γ⊢Q → Q ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢P&Q
h₄ : ~(P&Q) ∈ Γ
l₁ : P ∈ fun f => ¬~f ∈ Γ
⊢ False | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₁ : BProof (FormalDual Γ) P
prf₂ : BProof (FormalDual Γ) Q
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
ih₂ : FormalDual Γ⊢Q → Q ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢P&Q
h₄ : ~(P&Q) ∈ Γ
l₁ : P ∈ fun f => ¬~f ∈ Γ
l₂ : Q ∈ fun f => ¬~f ∈ Γ
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | have prf₃ := BProof.mp (BProof.ax h₄) BTheorem.demorgansLaw3 | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₁ : BProof (FormalDual Γ) P
prf₂ : BProof (FormalDual Γ) Q
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
ih₂ : FormalDual Γ⊢Q → Q ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢P&Q
h₄ : ~(P&Q) ∈ Γ
l₁ : P ∈ fun f => ¬~f ∈ Γ
l₂ : Q ∈ fun f => ¬~f ∈ Γ
⊢ False | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₁ : BProof (FormalDual Γ) P
prf₂ : BProof (FormalDual Γ) Q
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
ih₂ : FormalDual Γ⊢Q → Q ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢P&Q
h₄ : ~(P&Q) ∈ Γ
l₁ : P ∈ fun f => ¬~f ∈ Γ
l₂ : Q ∈ fun f => ¬~f ∈ Γ
prf₃ : BProof Γ (~P¦~Q)
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | have l₃ := h₂ (h₁.mpr ⟨prf₃⟩) | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₁ : BProof (FormalDual Γ) P
prf₂ : BProof (FormalDual Γ) Q
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
ih₂ : FormalDual Γ⊢Q → Q ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢P&Q
h₄ : ~(P&Q) ∈ Γ
l₁ : P ∈ fun f => ¬~f ∈ Γ
l₂ : Q ∈ fun f => ¬~f ∈ Γ
prf₃ : BProof Γ (~P¦~Q)
⊢ False | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₁ : BProof (FormalDual Γ) P
prf₂ : BProof (FormalDual Γ) Q
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
ih₂ : FormalDual Γ⊢Q → Q ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢P&Q
h₄ : ~(P&Q) ∈ Γ
l₁ : P ∈ fun f => ¬~f ∈ Γ
l₂ : Q ∈ fun f => ¬~f ∈ Γ
prf₃ : BProof Γ (~P¦~Q)
l₃ : ~P ∈ Γ ∨ ~Q ∈ Γ
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | cases l₃ | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₁ : BProof (FormalDual Γ) P
prf₂ : BProof (FormalDual Γ) Q
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
ih₂ : FormalDual Γ⊢Q → Q ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢P&Q
h₄ : ~(P&Q) ∈ Γ
l₁ : P ∈ fun f => ¬~f ∈ Γ
l₂ : Q ∈ fun f => ¬~f ∈ Γ
prf₃ : BProof Γ (~P¦~Q)
l₃ : ~P ∈ Γ ∨ ~Q ∈ Γ
⊢ False | case inl
Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₁ : BProof (FormalDual Γ) P
prf₂ : BProof (FormalDual Γ) Q
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
ih₂ : FormalDual Γ⊢Q → Q ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢P&Q
h₄ : ~(P&Q) ∈ Γ
l₁ : P ∈ fun f => ¬~f ∈ Γ
l₂ : Q ∈ fun f => ¬~f ∈ Γ
prf₃ : BProof Γ (~P¦~Q)
h✝ : ~P ∈ Γ
⊢ False
case inr
Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₁ : BProof (FormalDual Γ) P
prf₂ : BProof (FormalDual Γ) Q
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
ih₂ : FormalDual Γ⊢Q → Q ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢P&Q
h₄ : ~(P&Q) ∈ Γ
l₁ : P ∈ fun f => ¬~f ∈ Γ
l₂ : Q ∈ fun f => ¬~f ∈ Γ
prf₃ : BProof Γ (~P¦~Q)
h✝ : ~Q ∈ Γ
⊢ False |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Theories.lean | formalStarFormal | [173, 1] | [196, 39] | case inl left => exact l₁ left | Γ : Ctx
h₁ : formalTheory Γ
h₂ : isPrimeTheory Γ
F P Q : Form
prf₁ : BProof (FormalDual Γ) P
prf₂ : BProof (FormalDual Γ) Q
ih₁ : FormalDual Γ⊢P → P ∈ fun f => ¬~f ∈ Γ
ih₂ : FormalDual Γ⊢Q → Q ∈ fun f => ¬~f ∈ Γ
h₃ : FormalDual Γ⊢P&Q
h₄ : ~(P&Q) ∈ Γ
l₁ : P ∈ fun f => ¬~f ∈ Γ
l₂ : Q ∈ fun f => ¬~f ∈ Γ
prf₃ : BProof Γ (~P¦~Q)
left : ~P ∈ Γ
⊢ False | no goals |
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