internlm/Lean-Github · Datasets at Hugging Face

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/Theories.lean	formalStarFormal	[173, 1]	[196, 39]	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 h₄ : ~(P&Q) ∈ Γ l₁ : P ∈ fun f => ¬~f ∈ Γ l₂ : Q ∈ fun f => ¬~f ∈ Γ prf₃ : BProof Γ (~P¦~Q) right : ~Q ∈ Γ ⊢ False	no goals
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/Theories.lean	formalStarFormal	[173, 1]	[196, 39]	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
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/Theories.lean	formalStarFormal	[173, 1]	[196, 39]	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 h₄ : ~(P&Q) ∈ Γ l₁ : P ∈ fun f => ¬~f ∈ Γ l₂ : Q ∈ fun f => ¬~f ∈ Γ prf₃ : BProof Γ (~P¦~Q) right : ~Q ∈ Γ ⊢ False	no goals
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/Theories.lean	starInvolution	[220, 1]	[233, 16]	intros Γ	⊢ Function.Involutive primeStarFunction	Γ : Pr ⊢ primeStarFunction (primeStarFunction Γ) = Γ
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/Theories.lean	starInvolution	[220, 1]	[233, 16]	ext f	Γ : Pr ⊢ primeStarFunction (primeStarFunction Γ) = Γ	case a.a.h Γ : Pr f : Form ⊢ f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) ↔ f ∈ ↑↑Γ
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/Theories.lean	starInvolution	[220, 1]	[233, 16]	apply Iff.intro	case a.a.h Γ : Pr f : Form ⊢ f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) ↔ f ∈ ↑↑Γ	case a.a.h.mp Γ : Pr f : Form ⊢ f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) → f ∈ ↑↑Γ case a.a.h.mpr Γ : Pr f : Form ⊢ f ∈ ↑↑Γ → f ∈ ↑↑(primeStarFunction (primeStarFunction Γ))
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/Theories.lean	starInvolution	[220, 1]	[233, 16]	all_goals intros h₁	case a.a.h.mp Γ : Pr f : Form ⊢ f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) → f ∈ ↑↑Γ case a.a.h.mpr Γ : Pr f : Form ⊢ f ∈ ↑↑Γ → f ∈ ↑↑(primeStarFunction (primeStarFunction Γ))	case a.a.h.mp Γ : Pr f : Form h₁ : f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) ⊢ f ∈ ↑↑Γ case a.a.h.mpr Γ : Pr f : Form h₁ : f ∈ ↑↑Γ ⊢ f ∈ ↑↑(primeStarFunction (primeStarFunction Γ))
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/Theories.lean	starInvolution	[220, 1]	[233, 16]	case a.a.h.mpr => intros h₂ have l₁ : ~~f ∈ Γ := Γ.val.property.mpr ⟨BProof.mp (BProof.ax h₁) BTheorem.dni⟩ exact h₂ l₁	Γ : Pr f : Form h₁ : f ∈ ↑↑Γ ⊢ f ∈ ↑↑(primeStarFunction (primeStarFunction Γ))	no goals
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/Theories.lean	starInvolution	[220, 1]	[233, 16]	case a.a.h.mp => apply byContradiction intros h₂ have l₂ : ¬(~~f ∈ Γ) := h₂ ∘ λel => Γ.val.property.mpr ⟨BProof.mp (BProof.ax el) BTheorem.dne⟩ exact h₁ l₂	Γ : Pr f : Form h₁ : f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) ⊢ f ∈ ↑↑Γ	no goals
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/Theories.lean	starInvolution	[220, 1]	[233, 16]	intros h₁	case a.a.h.mpr Γ : Pr f : Form ⊢ f ∈ ↑↑Γ → f ∈ ↑↑(primeStarFunction (primeStarFunction Γ))	case a.a.h.mpr Γ : Pr f : Form h₁ : f ∈ ↑↑Γ ⊢ f ∈ ↑↑(primeStarFunction (primeStarFunction Γ))
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/Theories.lean	starInvolution	[220, 1]	[233, 16]	intros h₂	Γ : Pr f : Form h₁ : f ∈ ↑↑Γ ⊢ f ∈ ↑↑(primeStarFunction (primeStarFunction Γ))	Γ : Pr f : Form h₁ : f ∈ ↑↑Γ h₂ : ~f ∈ ↑↑(primeStarFunction Γ) ⊢ False
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/Theories.lean	starInvolution	[220, 1]	[233, 16]	have l₁ : ~~f ∈ Γ := Γ.val.property.mpr ⟨BProof.mp (BProof.ax h₁) BTheorem.dni⟩	Γ : Pr f : Form h₁ : f ∈ ↑↑Γ h₂ : ~f ∈ ↑↑(primeStarFunction Γ) ⊢ False	Γ : Pr f : Form h₁ : f ∈ ↑↑Γ h₂ : ~f ∈ ↑↑(primeStarFunction Γ) l₁ : ~~f ∈ Γ ⊢ False
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/Theories.lean	starInvolution	[220, 1]	[233, 16]	exact h₂ l₁	Γ : Pr f : Form h₁ : f ∈ ↑↑Γ h₂ : ~f ∈ ↑↑(primeStarFunction Γ) l₁ : ~~f ∈ Γ ⊢ False	no goals
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/Theories.lean	starInvolution	[220, 1]	[233, 16]	apply byContradiction	Γ : Pr f : Form h₁ : f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) ⊢ f ∈ ↑↑Γ	case h Γ : Pr f : Form h₁ : f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) ⊢ ¬f ∈ ↑↑Γ → False
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/Theories.lean	starInvolution	[220, 1]	[233, 16]	intros h₂	case h Γ : Pr f : Form h₁ : f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) ⊢ ¬f ∈ ↑↑Γ → False	case h Γ : Pr f : Form h₁ : f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) h₂ : ¬f ∈ ↑↑Γ ⊢ False
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/Theories.lean	starInvolution	[220, 1]	[233, 16]	have l₂ : ¬(~~f ∈ Γ) := h₂ ∘ λel => Γ.val.property.mpr ⟨BProof.mp (BProof.ax el) BTheorem.dne⟩	case h Γ : Pr f : Form h₁ : f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) h₂ : ¬f ∈ ↑↑Γ ⊢ False	case h Γ : Pr f : Form h₁ : f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) h₂ : ¬f ∈ ↑↑Γ l₂ : ¬~~f ∈ Γ ⊢ False
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/Theories.lean	starInvolution	[220, 1]	[233, 16]	exact h₁ l₂	case h Γ : Pr f : Form h₁ : f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) h₂ : ¬f ∈ ↑↑Γ l₂ : ¬~~f ∈ Γ ⊢ False	no goals
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/Theories.lean	starAntitone	[235, 1]	[237, 19]	intros _ _ h₁ f h₂ h₃	⊢ Antitone primeStarFunction	a✝ b✝ : Pr h₁ : a✝ ≤ b✝ f : Form h₂ : f ∈ (fun a => ↑a) ((fun a => ↑a) (primeStarFunction b✝)) h₃ : ~f ∈ ↑↑a✝ ⊢ False
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/Theories.lean	starAntitone	[235, 1]	[237, 19]	exact h₂ (h₁ h₃)	a✝ b✝ : Pr h₁ : a✝ ≤ b✝ f : Form h₂ : f ∈ (fun a => ↑a) ((fun a => ↑a) (primeStarFunction b✝)) h₃ : ~f ∈ ↑↑a✝ ⊢ False	no goals
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	intros t	⊢ ∀ (t : Th), ↑t = ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }	t : Th ⊢ ↑t = ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	ext x	t : Th ⊢ ↑t = ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }	case h t : Th x : Form ⊢ x ∈ ↑t ↔ x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	apply Iff.intro	case h t : Th x : Form ⊢ x ∈ ↑t ↔ x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }	case h.mp t : Th x : Form ⊢ x ∈ ↑t → x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } case h.mpr t : Th x : Form ⊢ x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } → x ∈ ↑t
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	case h.mp => intros h₁ apply Set.mem_interₛ.mpr intros r h₂ exact h₂.right.left h₁	t : Th x : Form ⊢ x ∈ ↑t → x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }	no goals
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	case h.mpr => intros h₁ apply byContradiction intros h₂ have l₁ : ∀ g : Form, g ∈ generatedDisjunctions x → {g} ⊢ x := by intros g h₃ induction g case or f g ih₁ ih₂ => cases h₃ case inl h₄ => rw [h₄]; exact ⟨BProof.ax rfl⟩ case inr h₄ => have ⟨prf₁⟩ := ih₁ h₄.left have ⟨prf₂⟩ := ih₂ h₄.right have thm₁ := BTheorem.mp (BTheorem.adj prf₁.toTheorem prf₂.toTheorem) BTheorem.orE exact ⟨thm₁.toProof ⟩ all_goals cases h₃; exact ⟨BProof.ax rfl⟩ have l₂ : ↑t ∩ generatedDisjunctions x = ∅ := by apply Set.eq_empty_iff_forall_not_mem.mpr intros y h₃ have ⟨prf₁⟩ := l₁ y h₃.right exact h₂ (t.property.mpr ⟨BProof.monotone (Set.singleton_subset_iff.mpr h₃.left) prf₁⟩) have l₃ : isDisjunctionClosed (generatedDisjunctions x) := by intros f g h₃ exact Or.inr h₃ have l₄ : x ∈ generatedDisjunctions x := by cases x case or f g => exact Or.inl rfl all_goals exact rfl have l₅ := lindenbaumTheorem l₂ l₃ have l₆ := (Set.mem_interₛ.mp h₁) (lindenbaumExtension t (generatedDisjunctions x)) ⟨lindenbaumIsPrime, lindenbaumExtensionExtends, lindenbaumIsFormal⟩ exact Set.eq_empty_iff_forall_not_mem.mp l₅ x ⟨l₆,l₄⟩	t : Th x : Form ⊢ x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } → x ∈ ↑t	no goals
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	intros h₁	t : Th x : Form ⊢ x ∈ ↑t → x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }	t : Th x : Form h₁ : x ∈ ↑t ⊢ x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	apply Set.mem_interₛ.mpr	t : Th x : Form h₁ : x ∈ ↑t ⊢ x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }	t : Th x : Form h₁ : x ∈ ↑t ⊢ ∀ (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	primeAnalysis	[9, 1]	[52, 58]	intros r h₂	t : Th x : Form h₁ : x ∈ ↑t ⊢ ∀ (t_1 : Set Form), t_1 ∈ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } → x ∈ t_1	t : Th x : Form h₁ : x ∈ ↑t 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	primeAnalysis	[9, 1]	[52, 58]	exact h₂.right.left h₁	t : Th x : Form h₁ : x ∈ ↑t 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	primeAnalysis	[9, 1]	[52, 58]	intros h₁	t : Th x : Form ⊢ x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } → x ∈ ↑t	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } ⊢ x ∈ ↑t
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	apply byContradiction	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } ⊢ x ∈ ↑t	case h t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } ⊢ ¬x ∈ ↑t → False
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	intros h₂	case h t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } ⊢ ¬x ∈ ↑t → False	case h t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t ⊢ False
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	have l₁ : ∀ g : Form, g ∈ generatedDisjunctions x → {g} ⊢ x := by intros g h₃ induction g case or f g ih₁ ih₂ => cases h₃ case inl h₄ => rw [h₄]; exact ⟨BProof.ax rfl⟩ case inr h₄ => have ⟨prf₁⟩ := ih₁ h₄.left have ⟨prf₂⟩ := ih₂ h₄.right have thm₁ := BTheorem.mp (BTheorem.adj prf₁.toTheorem prf₂.toTheorem) BTheorem.orE exact ⟨thm₁.toProof ⟩ all_goals cases h₃; exact ⟨BProof.ax rfl⟩	case h t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t ⊢ False	case h t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x ⊢ False
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	have l₂ : ↑t ∩ generatedDisjunctions x = ∅ := by apply Set.eq_empty_iff_forall_not_mem.mpr intros y h₃ have ⟨prf₁⟩ := l₁ y h₃.right exact h₂ (t.property.mpr ⟨BProof.monotone (Set.singleton_subset_iff.mpr h₃.left) prf₁⟩)	case h t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x ⊢ False	case h t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ ⊢ False
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	have l₃ : isDisjunctionClosed (generatedDisjunctions x) := by intros f g h₃ exact Or.inr h₃	case h t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ ⊢ False	case h t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions x) ⊢ False
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	have l₄ : x ∈ generatedDisjunctions x := by cases x case or f g => exact Or.inl rfl all_goals exact rfl	case h t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions x) ⊢ False	case h t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions x) l₄ : x ∈ generatedDisjunctions x ⊢ False
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	have l₅ := lindenbaumTheorem l₂ l₃	case h t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions x) l₄ : x ∈ generatedDisjunctions x ⊢ False	case h t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions x) l₄ : x ∈ generatedDisjunctions x l₅ : lindenbaumExtension t (generatedDisjunctions x) ∩ generatedDisjunctions x = ∅ ⊢ False
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	have l₆ := (Set.mem_interₛ.mp h₁) (lindenbaumExtension t (generatedDisjunctions x)) ⟨lindenbaumIsPrime, lindenbaumExtensionExtends, lindenbaumIsFormal⟩	case h t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions x) l₄ : x ∈ generatedDisjunctions x l₅ : lindenbaumExtension t (generatedDisjunctions x) ∩ generatedDisjunctions x = ∅ ⊢ False	case h t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions x) l₄ : x ∈ generatedDisjunctions x l₅ : lindenbaumExtension t (generatedDisjunctions x) ∩ generatedDisjunctions x = ∅ l₆ : x ∈ lindenbaumExtension t (generatedDisjunctions x) ⊢ False
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	exact Set.eq_empty_iff_forall_not_mem.mp l₅ x ⟨l₆,l₄⟩	case h t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions x) l₄ : x ∈ generatedDisjunctions x l₅ : lindenbaumExtension t (generatedDisjunctions x) ∩ generatedDisjunctions x = ∅ l₆ : x ∈ lindenbaumExtension t (generatedDisjunctions x) ⊢ False	no goals
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	intros g h₃	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t ⊢ ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t g : Form h₃ : g ∈ generatedDisjunctions x ⊢ {g}⊢x
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	induction g	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t g : Form h₃ : g ∈ generatedDisjunctions x ⊢ {g}⊢x	case atom t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t a✝ : ℕ h₃ : #a✝ ∈ generatedDisjunctions x ⊢ {#a✝}⊢x case neg t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t a✝ : Form a_ih✝ : a✝ ∈ generatedDisjunctions x → {a✝}⊢x h₃ : ~a✝ ∈ generatedDisjunctions x ⊢ {~a✝}⊢x case and t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t a✝¹ a✝ : Form a_ih✝¹ : a✝¹ ∈ generatedDisjunctions x → {a✝¹}⊢x a_ih✝ : a✝ ∈ generatedDisjunctions x → {a✝}⊢x h₃ : a✝¹&a✝ ∈ generatedDisjunctions x ⊢ {a✝¹&a✝}⊢x case or t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t a✝¹ a✝ : Form a_ih✝¹ : a✝¹ ∈ generatedDisjunctions x → {a✝¹}⊢x a_ih✝ : a✝ ∈ generatedDisjunctions x → {a✝}⊢x h₃ : a✝¹¦a✝ ∈ generatedDisjunctions x ⊢ {a✝¹¦a✝}⊢x case impl t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t a✝¹ a✝ : Form a_ih✝¹ : a✝¹ ∈ generatedDisjunctions x → {a✝¹}⊢x a_ih✝ : a✝ ∈ generatedDisjunctions x → {a✝}⊢x h₃ : a✝¹⊃a✝ ∈ generatedDisjunctions x ⊢ {a✝¹⊃a✝}⊢x
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	case or f g ih₁ ih₂ => cases h₃ case inl h₄ => rw [h₄]; exact ⟨BProof.ax rfl⟩ case inr h₄ => have ⟨prf₁⟩ := ih₁ h₄.left have ⟨prf₂⟩ := ih₂ h₄.right have thm₁ := BTheorem.mp (BTheorem.adj prf₁.toTheorem prf₂.toTheorem) BTheorem.orE exact ⟨thm₁.toProof ⟩	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₃ : f¦g ∈ generatedDisjunctions x ⊢ {f¦g}⊢x	no goals
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	all_goals cases h₃; exact ⟨BProof.ax rfl⟩	case atom t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t a✝ : ℕ h₃ : #a✝ ∈ generatedDisjunctions x ⊢ {#a✝}⊢x case neg t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t a✝ : Form a_ih✝ : a✝ ∈ generatedDisjunctions x → {a✝}⊢x h₃ : ~a✝ ∈ generatedDisjunctions x ⊢ {~a✝}⊢x case and t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t a✝¹ a✝ : Form a_ih✝¹ : a✝¹ ∈ generatedDisjunctions x → {a✝¹}⊢x a_ih✝ : a✝ ∈ generatedDisjunctions x → {a✝}⊢x h₃ : a✝¹&a✝ ∈ generatedDisjunctions x ⊢ {a✝¹&a✝}⊢x case impl t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t a✝¹ a✝ : Form a_ih✝¹ : a✝¹ ∈ generatedDisjunctions x → {a✝¹}⊢x a_ih✝ : a✝ ∈ generatedDisjunctions x → {a✝}⊢x h₃ : a✝¹⊃a✝ ∈ generatedDisjunctions x ⊢ {a✝¹⊃a✝}⊢x	no goals
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	cases h₃	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₃ : f¦g ∈ generatedDisjunctions x ⊢ {f¦g}⊢x	case inl t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h✝ : x = f¦g ⊢ {f¦g}⊢x case inr t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h✝ : generatedDisjunctions x f ∧ generatedDisjunctions x g ⊢ {f¦g}⊢x
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	case inl h₄ => rw [h₄]; exact ⟨BProof.ax rfl⟩	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : x = f¦g ⊢ {f¦g}⊢x	no goals
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	case inr h₄ => have ⟨prf₁⟩ := ih₁ h₄.left have ⟨prf₂⟩ := ih₂ h₄.right have thm₁ := BTheorem.mp (BTheorem.adj prf₁.toTheorem prf₂.toTheorem) BTheorem.orE exact ⟨thm₁.toProof ⟩	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : generatedDisjunctions x f ∧ generatedDisjunctions x g ⊢ {f¦g}⊢x	no goals
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	rw [h₄]	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : x = f¦g ⊢ {f¦g}⊢x	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : x = f¦g ⊢ {f¦g}⊢f¦g
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	exact ⟨BProof.ax rfl⟩	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : x = f¦g ⊢ {f¦g}⊢f¦g	no goals
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	have ⟨prf₁⟩ := ih₁ h₄.left	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : generatedDisjunctions x f ∧ generatedDisjunctions x g ⊢ {f¦g}⊢x	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : generatedDisjunctions x f ∧ generatedDisjunctions x g prf₁ : BProof {f} x ⊢ {f¦g}⊢x
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	have ⟨prf₂⟩ := ih₂ h₄.right	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : generatedDisjunctions x f ∧ generatedDisjunctions x g prf₁ : BProof {f} x ⊢ {f¦g}⊢x	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : generatedDisjunctions x f ∧ generatedDisjunctions x g prf₁ : BProof {f} x prf₂ : BProof {g} x ⊢ {f¦g}⊢x
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	have thm₁ := BTheorem.mp (BTheorem.adj prf₁.toTheorem prf₂.toTheorem) BTheorem.orE	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : generatedDisjunctions x f ∧ generatedDisjunctions x g prf₁ : BProof {f} x prf₂ : BProof {g} x ⊢ {f¦g}⊢x	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : generatedDisjunctions x f ∧ generatedDisjunctions x g prf₁ : BProof {f} x prf₂ : BProof {g} x thm₁ : BTheorem (f¦g⊃x) ⊢ {f¦g}⊢x
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	exact ⟨thm₁.toProof ⟩	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : generatedDisjunctions x f ∧ generatedDisjunctions x g prf₁ : BProof {f} x prf₂ : BProof {g} x thm₁ : BTheorem (f¦g⊃x) ⊢ {f¦g}⊢x	no goals
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	cases h₃	case impl t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t a✝¹ a✝ : Form a_ih✝¹ : a✝¹ ∈ generatedDisjunctions x → {a✝¹}⊢x a_ih✝ : a✝ ∈ generatedDisjunctions x → {a✝}⊢x h₃ : a✝¹⊃a✝ ∈ generatedDisjunctions x ⊢ {a✝¹⊃a✝}⊢x	case impl.refl t : Th a✝¹ a✝ : Form h₁ : a✝¹⊃a✝ ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬a✝¹⊃a✝ ∈ ↑t a_ih✝¹ : a✝¹ ∈ generatedDisjunctions (a✝¹⊃a✝) → {a✝¹}⊢a✝¹⊃a✝ a_ih✝ : a✝ ∈ generatedDisjunctions (a✝¹⊃a✝) → {a✝}⊢a✝¹⊃a✝ ⊢ {a✝¹⊃a✝}⊢a✝¹⊃a✝
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	exact ⟨BProof.ax rfl⟩	case impl.refl t : Th a✝¹ a✝ : Form h₁ : a✝¹⊃a✝ ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬a✝¹⊃a✝ ∈ ↑t a_ih✝¹ : a✝¹ ∈ generatedDisjunctions (a✝¹⊃a✝) → {a✝¹}⊢a✝¹⊃a✝ a_ih✝ : a✝ ∈ generatedDisjunctions (a✝¹⊃a✝) → {a✝}⊢a✝¹⊃a✝ ⊢ {a✝¹⊃a✝}⊢a✝¹⊃a✝	no goals
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	apply Set.eq_empty_iff_forall_not_mem.mpr	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x ⊢ ↑t ∩ generatedDisjunctions x = ∅	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x ⊢ ∀ (x_1 : Form), ¬x_1 ∈ ↑t ∩ generatedDisjunctions x
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	intros y h₃	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x ⊢ ∀ (x_1 : Form), ¬x_1 ∈ ↑t ∩ generatedDisjunctions x	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x y : Form h₃ : y ∈ ↑t ∩ generatedDisjunctions x ⊢ False
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	have ⟨prf₁⟩ := l₁ y h₃.right	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x y : Form h₃ : y ∈ ↑t ∩ generatedDisjunctions x ⊢ False	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x y : Form h₃ : y ∈ ↑t ∩ generatedDisjunctions x prf₁ : BProof {y} x ⊢ False
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	exact h₂ (t.property.mpr ⟨BProof.monotone (Set.singleton_subset_iff.mpr h₃.left) prf₁⟩)	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x y : Form h₃ : y ∈ ↑t ∩ generatedDisjunctions x prf₁ : BProof {y} x ⊢ False	no goals
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	intros f g h₃	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ ⊢ isDisjunctionClosed (generatedDisjunctions x)	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ f g : Form h₃ : f ∈ generatedDisjunctions x ∧ g ∈ generatedDisjunctions x ⊢ f¦g ∈ generatedDisjunctions x
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	exact Or.inr h₃	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ f g : Form h₃ : f ∈ generatedDisjunctions x ∧ g ∈ generatedDisjunctions x ⊢ f¦g ∈ generatedDisjunctions x	no goals
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	cases x	t : Th x : Form h₁ : x ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions x) ⊢ x ∈ generatedDisjunctions x	case atom t : Th a✝ : ℕ h₁ : #a✝ ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬#a✝ ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions #a✝ → {g}⊢#a✝ l₂ : ↑t ∩ generatedDisjunctions #a✝ = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions #a✝) ⊢ #a✝ ∈ generatedDisjunctions #a✝ case neg t : Th a✝ : Form h₁ : ~a✝ ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬~a✝ ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions ~a✝ → {g}⊢~a✝ l₂ : ↑t ∩ generatedDisjunctions ~a✝ = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions ~a✝) ⊢ ~a✝ ∈ generatedDisjunctions ~a✝ case and t : Th a✝¹ a✝ : Form h₁ : a✝¹&a✝ ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬a✝¹&a✝ ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions (a✝¹&a✝) → {g}⊢a✝¹&a✝ l₂ : ↑t ∩ generatedDisjunctions (a✝¹&a✝) = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions (a✝¹&a✝)) ⊢ a✝¹&a✝ ∈ generatedDisjunctions (a✝¹&a✝) case or t : Th a✝¹ a✝ : Form h₁ : a✝¹¦a✝ ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬a✝¹¦a✝ ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions (a✝¹¦a✝) → {g}⊢a✝¹¦a✝ l₂ : ↑t ∩ generatedDisjunctions (a✝¹¦a✝) = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions (a✝¹¦a✝)) ⊢ a✝¹¦a✝ ∈ generatedDisjunctions (a✝¹¦a✝) case impl t : Th a✝¹ a✝ : Form h₁ : a✝¹⊃a✝ ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬a✝¹⊃a✝ ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions (a✝¹⊃a✝) → {g}⊢a✝¹⊃a✝ l₂ : ↑t ∩ generatedDisjunctions (a✝¹⊃a✝) = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions (a✝¹⊃a✝)) ⊢ a✝¹⊃a✝ ∈ generatedDisjunctions (a✝¹⊃a✝)
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	case or f g => exact Or.inl rfl	t : Th f g : Form h₁ : f¦g ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬f¦g ∈ ↑t l₁ : ∀ (g_1 : Form), g_1 ∈ generatedDisjunctions (f¦g) → {g_1}⊢f¦g l₂ : ↑t ∩ generatedDisjunctions (f¦g) = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions (f¦g)) ⊢ f¦g ∈ generatedDisjunctions (f¦g)	no goals
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	all_goals exact rfl	case atom t : Th a✝ : ℕ h₁ : #a✝ ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬#a✝ ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions #a✝ → {g}⊢#a✝ l₂ : ↑t ∩ generatedDisjunctions #a✝ = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions #a✝) ⊢ #a✝ ∈ generatedDisjunctions #a✝ case neg t : Th a✝ : Form h₁ : ~a✝ ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬~a✝ ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions ~a✝ → {g}⊢~a✝ l₂ : ↑t ∩ generatedDisjunctions ~a✝ = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions ~a✝) ⊢ ~a✝ ∈ generatedDisjunctions ~a✝ case and t : Th a✝¹ a✝ : Form h₁ : a✝¹&a✝ ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬a✝¹&a✝ ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions (a✝¹&a✝) → {g}⊢a✝¹&a✝ l₂ : ↑t ∩ generatedDisjunctions (a✝¹&a✝) = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions (a✝¹&a✝)) ⊢ a✝¹&a✝ ∈ generatedDisjunctions (a✝¹&a✝) case impl t : Th a✝¹ a✝ : Form h₁ : a✝¹⊃a✝ ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬a✝¹⊃a✝ ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions (a✝¹⊃a✝) → {g}⊢a✝¹⊃a✝ l₂ : ↑t ∩ generatedDisjunctions (a✝¹⊃a✝) = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions (a✝¹⊃a✝)) ⊢ a✝¹⊃a✝ ∈ generatedDisjunctions (a✝¹⊃a✝)	no goals
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	exact Or.inl rfl	t : Th f g : Form h₁ : f¦g ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬f¦g ∈ ↑t l₁ : ∀ (g_1 : Form), g_1 ∈ generatedDisjunctions (f¦g) → {g_1}⊢f¦g l₂ : ↑t ∩ generatedDisjunctions (f¦g) = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions (f¦g)) ⊢ f¦g ∈ generatedDisjunctions (f¦g)	no goals
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	primeAnalysis	[9, 1]	[52, 58]	exact rfl	case impl t : Th a✝¹ a✝ : Form h₁ : a✝¹⊃a✝ ∈ ⋂₀ { p \| isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬a✝¹⊃a✝ ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions (a✝¹⊃a✝) → {g}⊢a✝¹⊃a✝ l₂ : ↑t ∩ generatedDisjunctions (a✝¹⊃a✝) = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions (a✝¹⊃a✝)) ⊢ a✝¹⊃a✝ ∈ generatedDisjunctions (a✝¹⊃a✝)	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 → ∃ q r, t ≤ ↑q ∧ u ≤ ↑r ∧ formalApplicationFunction (↑q) u ≤ ↑p ∧ formalApplicationFunction t ↑r ≤ ↑p	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p ⊢ ∃ q r, t ≤ ↑q ∧ u ≤ ↑r ∧ formalApplicationFunction (↑q) u ≤ ↑p ∧ formalApplicationFunction t ↑r ≤ ↑p
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	appBoundingFormalApplication	[54, 1]	[141, 38]	have ⟨q,h₂,h₃⟩ := lemma1 t u p h₁	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p ⊢ ∃ q r, t ≤ ↑q ∧ u ≤ ↑r ∧ formalApplicationFunction (↑q) u ≤ ↑p ∧ formalApplicationFunction t ↑r ≤ ↑p	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p q : Pr h₂ : t ≤ ↑q h₃ : formalApplication ↑↑q ↑u ⊆ ↑↑p ⊢ ∃ q r, t ≤ ↑q ∧ u ≤ ↑r ∧ formalApplicationFunction (↑q) u ≤ ↑p ∧ formalApplicationFunction t ↑r ≤ ↑p
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	appBoundingFormalApplication	[54, 1]	[141, 38]	have ⟨r,h₄,h₅⟩ := lemma2 t u p h₁	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p q : Pr h₂ : t ≤ ↑q h₃ : formalApplication ↑↑q ↑u ⊆ ↑↑p ⊢ ∃ q r, t ≤ ↑q ∧ u ≤ ↑r ∧ formalApplicationFunction (↑q) u ≤ ↑p ∧ formalApplicationFunction t ↑r ≤ ↑p	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p q : Pr h₂ : t ≤ ↑q h₃ : formalApplication ↑↑q ↑u ⊆ ↑↑p r : Pr h₄ : u ≤ ↑r h₅ : formalApplication ↑t ↑↑r ⊆ ↑↑p ⊢ ∃ q r, t ≤ ↑q ∧ u ≤ ↑r ∧ formalApplicationFunction (↑q) u ≤ ↑p ∧ formalApplicationFunction t ↑r ≤ ↑p
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	appBoundingFormalApplication	[54, 1]	[141, 38]	exact ⟨q,r,h₂,h₄,h₃,h₅⟩	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p q : Pr h₂ : t ≤ ↑q h₃ : formalApplication ↑↑q ↑u ⊆ ↑↑p r : Pr h₄ : u ≤ ↑r h₅ : formalApplication ↑t ↑↑r ⊆ ↑↑p ⊢ ∃ q r, t ≤ ↑q ∧ u ≤ ↑r ∧ formalApplicationFunction (↑q) u ≤ ↑p ∧ formalApplicationFunction t ↑r ≤ ↑p	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 → ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑p	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑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 (▲{f}) u ⊆ p) }	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑p	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑p
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	appBoundingFormalApplication	[54, 1]	[141, 38]	have l₂ : ↑t ∩ Δ = ∅ := by apply Set.eq_empty_iff_forall_not_mem.mpr intros P h₂ have l₃ : ▲{P} ⊆ ↑t := generatedContained (Set.singleton_subset_iff.mpr h₂.left) have l₄ := formalAppMonotoneRight ↑u l₃ exact h₂.right $ le_trans l₄ h₁	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑p	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑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,l₆,⟨prf₁⟩⟩,l₈⟩ := nonconstruction l₄ have ⟨T,l₉⟩ := nonconstruction h₁.right have ⟨⟨U,l₁₀,⟨prf₂⟩⟩,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 prf₃ : BProof {P} (S & U ⊃ R ¦ T) := BProof.mp (BProof.mp prf₁ (BTheorem.hs BTheorem.taut BTheorem.orI₁)) (BTheorem.hs BTheorem.andE₁ BTheorem.taut) have prf₄ : BProof {Q} (S & U ⊃ R ¦ T) := BProof.mp (BProof.mp prf₂ (BTheorem.hs BTheorem.taut BTheorem.orI₂)) (BTheorem.hs BTheorem.andE₂ BTheorem.taut) have prf₅ : BProof {P ¦ Q} (S & U ⊃ R ¦ T) := BTheorem.toProof $ BTheorem.mp (BTheorem.adj prf₃.toTheorem prf₄.toTheorem) BTheorem.orE clear prf₁ prf₂ prf₃ prf₄ have l₁₄ : S & U ∈ u := u.property.mpr ⟨BProof.adj (BProof.ax l₆) (BProof.ax l₁₀)⟩ exact ⟨S & U, l₁₄, ⟨prf₅⟩⟩	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑p	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ l₃ : isDisjunctionClosed Δ ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑p
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	appBoundingFormalApplication	[54, 1]	[141, 38]	have l₄ : lindenbaumExtension t Δ ∩ Δ = ∅ := lindenbaumTheorem l₂ l₃	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ l₃ : isDisjunctionClosed Δ ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑p	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ l₃ : isDisjunctionClosed Δ l₄ : lindenbaumExtension t Δ ∩ Δ = ∅ ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑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 (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ l₃ : isDisjunctionClosed Δ l₄ : lindenbaumExtension t Δ ∩ Δ = ∅ ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑p	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₄ : lindenbaumExtension t Δ ∩ Δ = ∅ ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑p
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	appBoundingFormalApplication	[54, 1]	[141, 38]	refine ⟨⟨⟨lindenbaumExtension t Δ, lindenbaumIsFormal⟩, lindenbaumIsPrime⟩, lindenbaumExtensionExtends, ?_⟩	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₄ : lindenbaumExtension t Δ ∩ Δ = ∅ ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑p	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₄ : lindenbaumExtension t Δ ∩ Δ = ∅ ⊢ formalApplication ↑↑{ val := { val := lindenbaumExtension t Δ, property := (_ : ∀ {f : Form}, f ∈ lindenbaumExtension t Δ ↔ lindenbaumExtension t Δ⊢f) }, property := (_ : ∀ {f g : Form}, f¦g ∈ lindenbaumExtension t Δ → f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ) } ↑u ⊆ ↑↑p
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	appBoundingFormalApplication	[54, 1]	[141, 38]	change formalApplication (lindenbaumExtension t Δ) ↑u ⊆ p	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₄ : lindenbaumExtension t Δ ∩ Δ = ∅ ⊢ formalApplication ↑↑{ val := { val := lindenbaumExtension t Δ, property := (_ : ∀ {f : Form}, f ∈ lindenbaumExtension t Δ ↔ lindenbaumExtension t Δ⊢f) }, property := (_ : ∀ {f g : Form}, f¦g ∈ lindenbaumExtension t Δ → f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ) } ↑u ⊆ ↑↑p	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₄ : lindenbaumExtension t Δ ∩ Δ = ∅ ⊢ formalApplication (lindenbaumExtension t Δ) ↑u ⊆ ↑↑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 (▲{f}) ↑u ⊆ ↑↑p } l₄ : lindenbaumExtension t Δ ∩ Δ = ∅ ⊢ formalApplication (lindenbaumExtension t Δ) ↑u ⊆ ↑↑p	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 ⊢ 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 (▲{f}) ↑u ⊆ ↑↑p } l₄ : lindenbaumExtension t Δ ∩ Δ = ∅ P : Form h₁ : P ∈ formalApplication (lindenbaumExtension t Δ) ↑u ⊢ P ∈ ↑↑p	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 ⊢ P ∈ ↑↑p
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	appBoundingFormalApplication	[54, 1]	[141, 38]	have l₄ : formalApplication (▲{Q⊃P}) ↑u ⊆ ↑↑p := by apply byContradiction intros h₄ exact (Set.eq_empty_iff_forall_not_mem.mp l₄) (Q⊃P) ⟨h₃,h₄⟩	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 ⊢ P ∈ ↑↑p	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 l₄ : formalApplication (▲{Q⊃P}) ↑u ⊆ ↑↑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,h₂,⟨BProof.ax rfl⟩⟩	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 l₄ : formalApplication (▲{Q⊃P}) ↑u ⊆ ↑↑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 (▲{f}) ↑u ⊆ ↑↑p } ⊢ ↑t ∩ Δ = ∅	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } ⊢ ∀ (x : Form), ¬x ∈ ↑t ∩ 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 (▲{f}) ↑u ⊆ ↑↑p } ⊢ ∀ (x : Form), ¬x ∈ ↑t ∩ Δ	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } P : Form h₂ : P ∈ ↑t ∩ Δ ⊢ False
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	appBoundingFormalApplication	[54, 1]	[141, 38]	have l₃ : ▲{P} ⊆ ↑t := generatedContained (Set.singleton_subset_iff.mpr h₂.left)	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } P : Form h₂ : P ∈ ↑t ∩ Δ ⊢ False	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } P : Form h₂ : P ∈ ↑t ∩ Δ l₃ : ▲{P} ⊆ ↑t ⊢ False
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	appBoundingFormalApplication	[54, 1]	[141, 38]	have l₄ := formalAppMonotoneRight ↑u l₃	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } P : Form h₂ : P ∈ ↑t ∩ Δ l₃ : ▲{P} ⊆ ↑t ⊢ False	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } P : Form h₂ : P ∈ ↑t ∩ Δ l₃ : ▲{P} ⊆ ↑t l₄ : flip formalApplication (↑u) (▲{P}) ≤ flip formalApplication ↑u ↑t ⊢ 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 (▲{f}) ↑u ⊆ ↑↑p } P : Form h₂ : P ∈ ↑t ∩ Δ l₃ : ▲{P} ⊆ ↑t l₄ : flip formalApplication (↑u) (▲{P}) ≤ flip formalApplication ↑u ↑t ⊢ 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 (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ ⊢ isDisjunctionClosed Δ	t u : Th p : Pr h₁✝ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₁ : P ∈ Δ ∧ Q ∈ Δ h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑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 (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₁ : P ∈ Δ ∧ Q ∈ Δ h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p ⊢ False	t u : Th p : Pr h₁✝ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₁ : P ∈ Δ ∧ Q ∈ Δ h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R : Form l₄ : ¬(R ∈ formalApplication (▲{P}) ↑u → R ∈ ↑↑p) ⊢ False
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	appBoundingFormalApplication	[54, 1]	[141, 38]	have ⟨⟨S,l₆,⟨prf₁⟩⟩,l₈⟩ := nonconstruction l₄	t u : Th p : Pr h₁✝ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₁ : P ∈ Δ ∧ Q ∈ Δ h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R : Form l₄ : ¬(R ∈ formalApplication (▲{P}) ↑u → R ∈ ↑↑p) ⊢ False	t u : Th p : Pr h₁✝ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₁ : P ∈ Δ ∧ Q ∈ Δ h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R : Form l₄ : ¬(R ∈ formalApplication (▲{P}) ↑u → R ∈ ↑↑p) S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) 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 (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₁ : P ∈ Δ ∧ Q ∈ Δ h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R : Form l₄ : ¬(R ∈ formalApplication (▲{P}) ↑u → R ∈ ↑↑p) S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p ⊢ False	t u : Th p : Pr h₁✝ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₁ : P ∈ Δ ∧ Q ∈ Δ h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R : Form l₄ : ¬(R ∈ formalApplication (▲{P}) ↑u → R ∈ ↑↑p) S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T : Form l₉ : ¬(T ∈ formalApplication (▲{Q}) ↑u → T ∈ ↑↑p) ⊢ False
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	appBoundingFormalApplication	[54, 1]	[141, 38]	have ⟨⟨U,l₁₀,⟨prf₂⟩⟩,l₁₂⟩ := nonconstruction l₉	t u : Th p : Pr h₁✝ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₁ : P ∈ Δ ∧ Q ∈ Δ h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R : Form l₄ : ¬(R ∈ formalApplication (▲{P}) ↑u → R ∈ ↑↑p) S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T : Form l₉ : ¬(T ∈ formalApplication (▲{Q}) ↑u → T ∈ ↑↑p) ⊢ False	t u : Th p : Pr h₁✝ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₁ : P ∈ Δ ∧ Q ∈ Δ h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R : Form l₄ : ¬(R ∈ formalApplication (▲{P}) ↑u → R ∈ ↑↑p) S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T : Form l₉ : ¬(T ∈ formalApplication (▲{Q}) ↑u → T ∈ ↑↑p) U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃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 (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₁ : P ∈ Δ ∧ Q ∈ Δ h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R : Form l₄ : ¬(R ∈ formalApplication (▲{P}) ↑u → R ∈ ↑↑p) S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T : Form l₉ : ¬(T ∈ formalApplication (▲{Q}) ↑u → T ∈ ↑↑p) U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) l₁₂ : ¬T ∈ ↑↑p ⊢ False	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃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 (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) l₁₂ : ¬T ∈ ↑↑p ⊢ False	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃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 (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) l₁₂ : ¬T ∈ ↑↑p l₁₃ : ¬R¦T ∈ p ⊢ False	t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f \| ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃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 (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃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 (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) l₁₂ : ¬T ∈ ↑↑p l₁₃ : ¬R¦T ∈ p ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u
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 (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) l₁₂ : ¬T ∈ ↑↑p l₁₃ : ¬R¦T ∈ p ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u	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 prf₁ : BProof {P} (S⊃R) T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	appBoundingFormalApplication	[54, 1]	[141, 38]	have prf₃ : BProof {P} (S & U ⊃ R ¦ T) := BProof.mp (BProof.mp prf₁ (BTheorem.hs BTheorem.taut BTheorem.orI₁)) (BTheorem.hs BTheorem.andE₁ BTheorem.taut)	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 prf₁ : BProof {P} (S⊃R) T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u	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 prf₁ : BProof {P} (S⊃R) T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) prf₃ : BProof {P} (S&U⊃R¦T) ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	appBoundingFormalApplication	[54, 1]	[141, 38]	have prf₄ : BProof {Q} (S & U ⊃ R ¦ T) := BProof.mp (BProof.mp prf₂ (BTheorem.hs BTheorem.taut BTheorem.orI₂)) (BTheorem.hs BTheorem.andE₂ BTheorem.taut)	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 prf₁ : BProof {P} (S⊃R) T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) prf₃ : BProof {P} (S&U⊃R¦T) ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u	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 prf₁ : BProof {P} (S⊃R) T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) prf₃ : BProof {P} (S&U⊃R¦T) prf₄ : BProof {Q} (S&U⊃R¦T) ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	appBoundingFormalApplication	[54, 1]	[141, 38]	have prf₅ : BProof {P ¦ Q} (S & U ⊃ R ¦ T) := BTheorem.toProof $ BTheorem.mp (BTheorem.adj prf₃.toTheorem prf₄.toTheorem) BTheorem.orE	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 prf₁ : BProof {P} (S⊃R) T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) prf₃ : BProof {P} (S&U⊃R¦T) prf₄ : BProof {Q} (S&U⊃R¦T) ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u	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 prf₁ : BProof {P} (S⊃R) T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) prf₃ : BProof {P} (S&U⊃R¦T) prf₄ : BProof {Q} (S&U⊃R¦T) prf₅ : BProof {P¦Q} (S&U⊃R¦T) ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	appBoundingFormalApplication	[54, 1]	[141, 38]	clear prf₁ prf₂ prf₃ prf₄	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 prf₁ : BProof {P} (S⊃R) T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) prf₃ : BProof {P} (S&U⊃R¦T) prf₄ : BProof {Q} (S&U⊃R¦T) prf₅ : BProof {P¦Q} (S&U⊃R¦T) ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u	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) ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git	0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f	Fine/SystemB/CanonicalModel.lean	appBoundingFormalApplication	[54, 1]	[141, 38]	have l₁₄ : S & U ∈ u := u.property.mpr ⟨BProof.adj (BProof.ax l₆) (BProof.ax l₁₀)⟩	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) ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u	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

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/Theories.lean

formalStarFormal

[173, 1]

[196, 39]

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 h₄ : ~(P&Q) ∈ Γ l₁ : P ∈ fun f => ¬~f ∈ Γ l₂ : Q ∈ fun f => ¬~f ∈ Γ prf₃ : BProof Γ (~P¦~Q) right : ~Q ∈ Γ ⊢ False

no goals

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/Theories.lean

formalStarFormal

[173, 1]

[196, 39]

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

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/Theories.lean

formalStarFormal

[173, 1]

[196, 39]

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 h₄ : ~(P&Q) ∈ Γ l₁ : P ∈ fun f => ¬~f ∈ Γ l₂ : Q ∈ fun f => ¬~f ∈ Γ prf₃ : BProof Γ (~P¦~Q) right : ~Q ∈ Γ ⊢ False

no goals

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/Theories.lean

starInvolution

[220, 1]

[233, 16]

intros Γ

⊢ Function.Involutive primeStarFunction

Γ : Pr ⊢ primeStarFunction (primeStarFunction Γ) = Γ

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/Theories.lean

starInvolution

[220, 1]

[233, 16]

ext f

Γ : Pr ⊢ primeStarFunction (primeStarFunction Γ) = Γ

case a.a.h Γ : Pr f : Form ⊢ f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) ↔ f ∈ ↑↑Γ

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/Theories.lean

starInvolution

[220, 1]

[233, 16]

apply Iff.intro

case a.a.h Γ : Pr f : Form ⊢ f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) ↔ f ∈ ↑↑Γ

case a.a.h.mp Γ : Pr f : Form ⊢ f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) → f ∈ ↑↑Γ case a.a.h.mpr Γ : Pr f : Form ⊢ f ∈ ↑↑Γ → f ∈ ↑↑(primeStarFunction (primeStarFunction Γ))

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/Theories.lean

starInvolution

[220, 1]

[233, 16]

all_goals intros h₁

case a.a.h.mp Γ : Pr f : Form ⊢ f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) → f ∈ ↑↑Γ case a.a.h.mpr Γ : Pr f : Form ⊢ f ∈ ↑↑Γ → f ∈ ↑↑(primeStarFunction (primeStarFunction Γ))

case a.a.h.mp Γ : Pr f : Form h₁ : f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) ⊢ f ∈ ↑↑Γ case a.a.h.mpr Γ : Pr f : Form h₁ : f ∈ ↑↑Γ ⊢ f ∈ ↑↑(primeStarFunction (primeStarFunction Γ))

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/Theories.lean

starInvolution

[220, 1]

[233, 16]

case a.a.h.mpr => intros h₂ have l₁ : ~~f ∈ Γ := Γ.val.property.mpr ⟨BProof.mp (BProof.ax h₁) BTheorem.dni⟩ exact h₂ l₁

Γ : Pr f : Form h₁ : f ∈ ↑↑Γ ⊢ f ∈ ↑↑(primeStarFunction (primeStarFunction Γ))

no goals

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/Theories.lean

starInvolution

[220, 1]

[233, 16]

case a.a.h.mp => apply byContradiction intros h₂ have l₂ : ¬(~~f ∈ Γ) := h₂ ∘ λel => Γ.val.property.mpr ⟨BProof.mp (BProof.ax el) BTheorem.dne⟩ exact h₁ l₂

Γ : Pr f : Form h₁ : f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) ⊢ f ∈ ↑↑Γ

no goals

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/Theories.lean

starInvolution

[220, 1]

[233, 16]

intros h₁

case a.a.h.mpr Γ : Pr f : Form ⊢ f ∈ ↑↑Γ → f ∈ ↑↑(primeStarFunction (primeStarFunction Γ))

case a.a.h.mpr Γ : Pr f : Form h₁ : f ∈ ↑↑Γ ⊢ f ∈ ↑↑(primeStarFunction (primeStarFunction Γ))

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/Theories.lean

starInvolution

[220, 1]

[233, 16]

intros h₂

Γ : Pr f : Form h₁ : f ∈ ↑↑Γ ⊢ f ∈ ↑↑(primeStarFunction (primeStarFunction Γ))

Γ : Pr f : Form h₁ : f ∈ ↑↑Γ h₂ : ~f ∈ ↑↑(primeStarFunction Γ) ⊢ False

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/Theories.lean

starInvolution

[220, 1]

[233, 16]

have l₁ : ~~f ∈ Γ := Γ.val.property.mpr ⟨BProof.mp (BProof.ax h₁) BTheorem.dni⟩

Γ : Pr f : Form h₁ : f ∈ ↑↑Γ h₂ : ~f ∈ ↑↑(primeStarFunction Γ) ⊢ False

Γ : Pr f : Form h₁ : f ∈ ↑↑Γ h₂ : ~f ∈ ↑↑(primeStarFunction Γ) l₁ : ~~f ∈ Γ ⊢ False

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/Theories.lean

starInvolution

[220, 1]

[233, 16]

exact h₂ l₁

Γ : Pr f : Form h₁ : f ∈ ↑↑Γ h₂ : ~f ∈ ↑↑(primeStarFunction Γ) l₁ : ~~f ∈ Γ ⊢ False

no goals

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/Theories.lean

starInvolution

[220, 1]

[233, 16]

apply byContradiction

Γ : Pr f : Form h₁ : f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) ⊢ f ∈ ↑↑Γ

case h Γ : Pr f : Form h₁ : f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) ⊢ ¬f ∈ ↑↑Γ → False

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/Theories.lean

starInvolution

[220, 1]

[233, 16]

intros h₂

case h Γ : Pr f : Form h₁ : f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) ⊢ ¬f ∈ ↑↑Γ → False

case h Γ : Pr f : Form h₁ : f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) h₂ : ¬f ∈ ↑↑Γ ⊢ False

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/Theories.lean

starInvolution

[220, 1]

[233, 16]

have l₂ : ¬(~~f ∈ Γ) := h₂ ∘ λel => Γ.val.property.mpr ⟨BProof.mp (BProof.ax el) BTheorem.dne⟩

case h Γ : Pr f : Form h₁ : f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) h₂ : ¬f ∈ ↑↑Γ ⊢ False

case h Γ : Pr f : Form h₁ : f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) h₂ : ¬f ∈ ↑↑Γ l₂ : ¬~~f ∈ Γ ⊢ False

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/Theories.lean

starInvolution

[220, 1]

[233, 16]

exact h₁ l₂

case h Γ : Pr f : Form h₁ : f ∈ ↑↑(primeStarFunction (primeStarFunction Γ)) h₂ : ¬f ∈ ↑↑Γ l₂ : ¬~~f ∈ Γ ⊢ False

no goals

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/Theories.lean

starAntitone

[235, 1]

[237, 19]

intros _ _ h₁ f h₂ h₃

⊢ Antitone primeStarFunction

a✝ b✝ : Pr h₁ : a✝ ≤ b✝ f : Form h₂ : f ∈ (fun a => ↑a) ((fun a => ↑a) (primeStarFunction b✝)) h₃ : ~f ∈ ↑↑a✝ ⊢ False

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/Theories.lean

starAntitone

[235, 1]

[237, 19]

exact h₂ (h₁ h₃)

a✝ b✝ : Pr h₁ : a✝ ≤ b✝ f : Form h₂ : f ∈ (fun a => ↑a) ((fun a => ↑a) (primeStarFunction b✝)) h₃ : ~f ∈ ↑↑a✝ ⊢ False

no goals

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

intros t

⊢ ∀ (t : Th), ↑t = ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }

t : Th ⊢ ↑t = ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

ext x

t : Th ⊢ ↑t = ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }

case h t : Th x : Form ⊢ x ∈ ↑t ↔ x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

apply Iff.intro

case h t : Th x : Form ⊢ x ∈ ↑t ↔ x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }

case h.mp t : Th x : Form ⊢ x ∈ ↑t → x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } case h.mpr t : Th x : Form ⊢ x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } → x ∈ ↑t

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

case h.mp => intros h₁ apply Set.mem_interₛ.mpr intros r h₂ exact h₂.right.left h₁

t : Th x : Form ⊢ x ∈ ↑t → x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }

no goals

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

case h.mpr => intros h₁ apply byContradiction intros h₂ have l₁ : ∀ g : Form, g ∈ generatedDisjunctions x → {g} ⊢ x := by intros g h₃ induction g case or f g ih₁ ih₂ => cases h₃ case inl h₄ => rw [h₄]; exact ⟨BProof.ax rfl⟩ case inr h₄ => have ⟨prf₁⟩ := ih₁ h₄.left have ⟨prf₂⟩ := ih₂ h₄.right have thm₁ := BTheorem.mp (BTheorem.adj prf₁.toTheorem prf₂.toTheorem) BTheorem.orE exact ⟨thm₁.toProof ⟩ all_goals cases h₃; exact ⟨BProof.ax rfl⟩ have l₂ : ↑t ∩ generatedDisjunctions x = ∅ := by apply Set.eq_empty_iff_forall_not_mem.mpr intros y h₃ have ⟨prf₁⟩ := l₁ y h₃.right exact h₂ (t.property.mpr ⟨BProof.monotone (Set.singleton_subset_iff.mpr h₃.left) prf₁⟩) have l₃ : isDisjunctionClosed (generatedDisjunctions x) := by intros f g h₃ exact Or.inr h₃ have l₄ : x ∈ generatedDisjunctions x := by cases x case or f g => exact Or.inl rfl all_goals exact rfl have l₅ := lindenbaumTheorem l₂ l₃ have l₆ := (Set.mem_interₛ.mp h₁) (lindenbaumExtension t (generatedDisjunctions x)) ⟨lindenbaumIsPrime, lindenbaumExtensionExtends, lindenbaumIsFormal⟩ exact Set.eq_empty_iff_forall_not_mem.mp l₅ x ⟨l₆,l₄⟩

t : Th x : Form ⊢ x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } → x ∈ ↑t

no goals

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

intros h₁

t : Th x : Form ⊢ x ∈ ↑t → x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }

t : Th x : Form h₁ : x ∈ ↑t ⊢ x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

apply Set.mem_interₛ.mpr

t : Th x : Form h₁ : x ∈ ↑t ⊢ x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p }

t : Th x : Form h₁ : x ∈ ↑t ⊢ ∀ (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

primeAnalysis

[9, 1]

[52, 58]

intros r h₂

t : Th x : Form h₁ : x ∈ ↑t ⊢ ∀ (t_1 : Set Form), t_1 ∈ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } → x ∈ t_1

t : Th x : Form h₁ : x ∈ ↑t 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

primeAnalysis

[9, 1]

[52, 58]

exact h₂.right.left h₁

t : Th x : Form h₁ : x ∈ ↑t 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

primeAnalysis

[9, 1]

[52, 58]

intros h₁

t : Th x : Form ⊢ x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } → x ∈ ↑t

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } ⊢ x ∈ ↑t

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

apply byContradiction

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } ⊢ x ∈ ↑t

case h t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } ⊢ ¬x ∈ ↑t → False

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

intros h₂

case h t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } ⊢ ¬x ∈ ↑t → False

case h t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t ⊢ False

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

have l₁ : ∀ g : Form, g ∈ generatedDisjunctions x → {g} ⊢ x := by intros g h₃ induction g case or f g ih₁ ih₂ => cases h₃ case inl h₄ => rw [h₄]; exact ⟨BProof.ax rfl⟩ case inr h₄ => have ⟨prf₁⟩ := ih₁ h₄.left have ⟨prf₂⟩ := ih₂ h₄.right have thm₁ := BTheorem.mp (BTheorem.adj prf₁.toTheorem prf₂.toTheorem) BTheorem.orE exact ⟨thm₁.toProof ⟩ all_goals cases h₃; exact ⟨BProof.ax rfl⟩

case h t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t ⊢ False

case h t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x ⊢ False

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

have l₂ : ↑t ∩ generatedDisjunctions x = ∅ := by apply Set.eq_empty_iff_forall_not_mem.mpr intros y h₃ have ⟨prf₁⟩ := l₁ y h₃.right exact h₂ (t.property.mpr ⟨BProof.monotone (Set.singleton_subset_iff.mpr h₃.left) prf₁⟩)

case h t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x ⊢ False

case h t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ ⊢ False

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

have l₃ : isDisjunctionClosed (generatedDisjunctions x) := by intros f g h₃ exact Or.inr h₃

case h t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ ⊢ False

case h t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions x) ⊢ False

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

have l₄ : x ∈ generatedDisjunctions x := by cases x case or f g => exact Or.inl rfl all_goals exact rfl

case h t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions x) ⊢ False

case h t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions x) l₄ : x ∈ generatedDisjunctions x ⊢ False

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

have l₅ := lindenbaumTheorem l₂ l₃

case h t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions x) l₄ : x ∈ generatedDisjunctions x ⊢ False

case h t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions x) l₄ : x ∈ generatedDisjunctions x l₅ : lindenbaumExtension t (generatedDisjunctions x) ∩ generatedDisjunctions x = ∅ ⊢ False

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

have l₆ := (Set.mem_interₛ.mp h₁) (lindenbaumExtension t (generatedDisjunctions x)) ⟨lindenbaumIsPrime, lindenbaumExtensionExtends, lindenbaumIsFormal⟩

case h t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions x) l₄ : x ∈ generatedDisjunctions x l₅ : lindenbaumExtension t (generatedDisjunctions x) ∩ generatedDisjunctions x = ∅ ⊢ False

case h t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions x) l₄ : x ∈ generatedDisjunctions x l₅ : lindenbaumExtension t (generatedDisjunctions x) ∩ generatedDisjunctions x = ∅ l₆ : x ∈ lindenbaumExtension t (generatedDisjunctions x) ⊢ False

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

exact Set.eq_empty_iff_forall_not_mem.mp l₅ x ⟨l₆,l₄⟩

case h t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions x) l₄ : x ∈ generatedDisjunctions x l₅ : lindenbaumExtension t (generatedDisjunctions x) ∩ generatedDisjunctions x = ∅ l₆ : x ∈ lindenbaumExtension t (generatedDisjunctions x) ⊢ False

no goals

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

intros g h₃

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t ⊢ ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t g : Form h₃ : g ∈ generatedDisjunctions x ⊢ {g}⊢x

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

induction g

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t g : Form h₃ : g ∈ generatedDisjunctions x ⊢ {g}⊢x

case atom t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t a✝ : ℕ h₃ : #a✝ ∈ generatedDisjunctions x ⊢ {#a✝}⊢x case neg t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t a✝ : Form a_ih✝ : a✝ ∈ generatedDisjunctions x → {a✝}⊢x h₃ : ~a✝ ∈ generatedDisjunctions x ⊢ {~a✝}⊢x case and t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t a✝¹ a✝ : Form a_ih✝¹ : a✝¹ ∈ generatedDisjunctions x → {a✝¹}⊢x a_ih✝ : a✝ ∈ generatedDisjunctions x → {a✝}⊢x h₃ : a✝¹&a✝ ∈ generatedDisjunctions x ⊢ {a✝¹&a✝}⊢x case or t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t a✝¹ a✝ : Form a_ih✝¹ : a✝¹ ∈ generatedDisjunctions x → {a✝¹}⊢x a_ih✝ : a✝ ∈ generatedDisjunctions x → {a✝}⊢x h₃ : a✝¹¦a✝ ∈ generatedDisjunctions x ⊢ {a✝¹¦a✝}⊢x case impl t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t a✝¹ a✝ : Form a_ih✝¹ : a✝¹ ∈ generatedDisjunctions x → {a✝¹}⊢x a_ih✝ : a✝ ∈ generatedDisjunctions x → {a✝}⊢x h₃ : a✝¹⊃a✝ ∈ generatedDisjunctions x ⊢ {a✝¹⊃a✝}⊢x

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

case or f g ih₁ ih₂ => cases h₃ case inl h₄ => rw [h₄]; exact ⟨BProof.ax rfl⟩ case inr h₄ => have ⟨prf₁⟩ := ih₁ h₄.left have ⟨prf₂⟩ := ih₂ h₄.right have thm₁ := BTheorem.mp (BTheorem.adj prf₁.toTheorem prf₂.toTheorem) BTheorem.orE exact ⟨thm₁.toProof ⟩

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₃ : f¦g ∈ generatedDisjunctions x ⊢ {f¦g}⊢x

no goals

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

all_goals cases h₃; exact ⟨BProof.ax rfl⟩

case atom t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t a✝ : ℕ h₃ : #a✝ ∈ generatedDisjunctions x ⊢ {#a✝}⊢x case neg t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t a✝ : Form a_ih✝ : a✝ ∈ generatedDisjunctions x → {a✝}⊢x h₃ : ~a✝ ∈ generatedDisjunctions x ⊢ {~a✝}⊢x case and t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t a✝¹ a✝ : Form a_ih✝¹ : a✝¹ ∈ generatedDisjunctions x → {a✝¹}⊢x a_ih✝ : a✝ ∈ generatedDisjunctions x → {a✝}⊢x h₃ : a✝¹&a✝ ∈ generatedDisjunctions x ⊢ {a✝¹&a✝}⊢x case impl t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t a✝¹ a✝ : Form a_ih✝¹ : a✝¹ ∈ generatedDisjunctions x → {a✝¹}⊢x a_ih✝ : a✝ ∈ generatedDisjunctions x → {a✝}⊢x h₃ : a✝¹⊃a✝ ∈ generatedDisjunctions x ⊢ {a✝¹⊃a✝}⊢x

no goals

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

cases h₃

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₃ : f¦g ∈ generatedDisjunctions x ⊢ {f¦g}⊢x

case inl t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h✝ : x = f¦g ⊢ {f¦g}⊢x case inr t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h✝ : generatedDisjunctions x f ∧ generatedDisjunctions x g ⊢ {f¦g}⊢x

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

case inl h₄ => rw [h₄]; exact ⟨BProof.ax rfl⟩

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : x = f¦g ⊢ {f¦g}⊢x

no goals

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

case inr h₄ => have ⟨prf₁⟩ := ih₁ h₄.left have ⟨prf₂⟩ := ih₂ h₄.right have thm₁ := BTheorem.mp (BTheorem.adj prf₁.toTheorem prf₂.toTheorem) BTheorem.orE exact ⟨thm₁.toProof ⟩

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : generatedDisjunctions x f ∧ generatedDisjunctions x g ⊢ {f¦g}⊢x

no goals

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

rw [h₄]

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : x = f¦g ⊢ {f¦g}⊢x

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : x = f¦g ⊢ {f¦g}⊢f¦g

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

exact ⟨BProof.ax rfl⟩

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : x = f¦g ⊢ {f¦g}⊢f¦g

no goals

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

have ⟨prf₁⟩ := ih₁ h₄.left

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : generatedDisjunctions x f ∧ generatedDisjunctions x g ⊢ {f¦g}⊢x

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : generatedDisjunctions x f ∧ generatedDisjunctions x g prf₁ : BProof {f} x ⊢ {f¦g}⊢x

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

have ⟨prf₂⟩ := ih₂ h₄.right

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : generatedDisjunctions x f ∧ generatedDisjunctions x g prf₁ : BProof {f} x ⊢ {f¦g}⊢x

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : generatedDisjunctions x f ∧ generatedDisjunctions x g prf₁ : BProof {f} x prf₂ : BProof {g} x ⊢ {f¦g}⊢x

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

have thm₁ := BTheorem.mp (BTheorem.adj prf₁.toTheorem prf₂.toTheorem) BTheorem.orE

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : generatedDisjunctions x f ∧ generatedDisjunctions x g prf₁ : BProof {f} x prf₂ : BProof {g} x ⊢ {f¦g}⊢x

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : generatedDisjunctions x f ∧ generatedDisjunctions x g prf₁ : BProof {f} x prf₂ : BProof {g} x thm₁ : BTheorem (f¦g⊃x) ⊢ {f¦g}⊢x

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

exact ⟨thm₁.toProof ⟩

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t f g : Form ih₁ : f ∈ generatedDisjunctions x → {f}⊢x ih₂ : g ∈ generatedDisjunctions x → {g}⊢x h₄ : generatedDisjunctions x f ∧ generatedDisjunctions x g prf₁ : BProof {f} x prf₂ : BProof {g} x thm₁ : BTheorem (f¦g⊃x) ⊢ {f¦g}⊢x

no goals

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

cases h₃

case impl t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t a✝¹ a✝ : Form a_ih✝¹ : a✝¹ ∈ generatedDisjunctions x → {a✝¹}⊢x a_ih✝ : a✝ ∈ generatedDisjunctions x → {a✝}⊢x h₃ : a✝¹⊃a✝ ∈ generatedDisjunctions x ⊢ {a✝¹⊃a✝}⊢x

case impl.refl t : Th a✝¹ a✝ : Form h₁ : a✝¹⊃a✝ ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬a✝¹⊃a✝ ∈ ↑t a_ih✝¹ : a✝¹ ∈ generatedDisjunctions (a✝¹⊃a✝) → {a✝¹}⊢a✝¹⊃a✝ a_ih✝ : a✝ ∈ generatedDisjunctions (a✝¹⊃a✝) → {a✝}⊢a✝¹⊃a✝ ⊢ {a✝¹⊃a✝}⊢a✝¹⊃a✝

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

exact ⟨BProof.ax rfl⟩

case impl.refl t : Th a✝¹ a✝ : Form h₁ : a✝¹⊃a✝ ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬a✝¹⊃a✝ ∈ ↑t a_ih✝¹ : a✝¹ ∈ generatedDisjunctions (a✝¹⊃a✝) → {a✝¹}⊢a✝¹⊃a✝ a_ih✝ : a✝ ∈ generatedDisjunctions (a✝¹⊃a✝) → {a✝}⊢a✝¹⊃a✝ ⊢ {a✝¹⊃a✝}⊢a✝¹⊃a✝

no goals

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

apply Set.eq_empty_iff_forall_not_mem.mpr

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x ⊢ ↑t ∩ generatedDisjunctions x = ∅

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x ⊢ ∀ (x_1 : Form), ¬x_1 ∈ ↑t ∩ generatedDisjunctions x

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

intros y h₃

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x ⊢ ∀ (x_1 : Form), ¬x_1 ∈ ↑t ∩ generatedDisjunctions x

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x y : Form h₃ : y ∈ ↑t ∩ generatedDisjunctions x ⊢ False

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

have ⟨prf₁⟩ := l₁ y h₃.right

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x y : Form h₃ : y ∈ ↑t ∩ generatedDisjunctions x ⊢ False

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x y : Form h₃ : y ∈ ↑t ∩ generatedDisjunctions x prf₁ : BProof {y} x ⊢ False

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

exact h₂ (t.property.mpr ⟨BProof.monotone (Set.singleton_subset_iff.mpr h₃.left) prf₁⟩)

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x y : Form h₃ : y ∈ ↑t ∩ generatedDisjunctions x prf₁ : BProof {y} x ⊢ False

no goals

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

intros f g h₃

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ ⊢ isDisjunctionClosed (generatedDisjunctions x)

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ f g : Form h₃ : f ∈ generatedDisjunctions x ∧ g ∈ generatedDisjunctions x ⊢ f¦g ∈ generatedDisjunctions x

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

exact Or.inr h₃

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ f g : Form h₃ : f ∈ generatedDisjunctions x ∧ g ∈ generatedDisjunctions x ⊢ f¦g ∈ generatedDisjunctions x

no goals

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

cases x

t : Th x : Form h₁ : x ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬x ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions x → {g}⊢x l₂ : ↑t ∩ generatedDisjunctions x = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions x) ⊢ x ∈ generatedDisjunctions x

case atom t : Th a✝ : ℕ h₁ : #a✝ ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬#a✝ ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions #a✝ → {g}⊢#a✝ l₂ : ↑t ∩ generatedDisjunctions #a✝ = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions #a✝) ⊢ #a✝ ∈ generatedDisjunctions #a✝ case neg t : Th a✝ : Form h₁ : ~a✝ ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬~a✝ ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions ~a✝ → {g}⊢~a✝ l₂ : ↑t ∩ generatedDisjunctions ~a✝ = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions ~a✝) ⊢ ~a✝ ∈ generatedDisjunctions ~a✝ case and t : Th a✝¹ a✝ : Form h₁ : a✝¹&a✝ ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬a✝¹&a✝ ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions (a✝¹&a✝) → {g}⊢a✝¹&a✝ l₂ : ↑t ∩ generatedDisjunctions (a✝¹&a✝) = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions (a✝¹&a✝)) ⊢ a✝¹&a✝ ∈ generatedDisjunctions (a✝¹&a✝) case or t : Th a✝¹ a✝ : Form h₁ : a✝¹¦a✝ ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬a✝¹¦a✝ ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions (a✝¹¦a✝) → {g}⊢a✝¹¦a✝ l₂ : ↑t ∩ generatedDisjunctions (a✝¹¦a✝) = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions (a✝¹¦a✝)) ⊢ a✝¹¦a✝ ∈ generatedDisjunctions (a✝¹¦a✝) case impl t : Th a✝¹ a✝ : Form h₁ : a✝¹⊃a✝ ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬a✝¹⊃a✝ ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions (a✝¹⊃a✝) → {g}⊢a✝¹⊃a✝ l₂ : ↑t ∩ generatedDisjunctions (a✝¹⊃a✝) = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions (a✝¹⊃a✝)) ⊢ a✝¹⊃a✝ ∈ generatedDisjunctions (a✝¹⊃a✝)

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

case or f g => exact Or.inl rfl

t : Th f g : Form h₁ : f¦g ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬f¦g ∈ ↑t l₁ : ∀ (g_1 : Form), g_1 ∈ generatedDisjunctions (f¦g) → {g_1}⊢f¦g l₂ : ↑t ∩ generatedDisjunctions (f¦g) = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions (f¦g)) ⊢ f¦g ∈ generatedDisjunctions (f¦g)

no goals

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

all_goals exact rfl

case atom t : Th a✝ : ℕ h₁ : #a✝ ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬#a✝ ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions #a✝ → {g}⊢#a✝ l₂ : ↑t ∩ generatedDisjunctions #a✝ = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions #a✝) ⊢ #a✝ ∈ generatedDisjunctions #a✝ case neg t : Th a✝ : Form h₁ : ~a✝ ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬~a✝ ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions ~a✝ → {g}⊢~a✝ l₂ : ↑t ∩ generatedDisjunctions ~a✝ = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions ~a✝) ⊢ ~a✝ ∈ generatedDisjunctions ~a✝ case and t : Th a✝¹ a✝ : Form h₁ : a✝¹&a✝ ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬a✝¹&a✝ ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions (a✝¹&a✝) → {g}⊢a✝¹&a✝ l₂ : ↑t ∩ generatedDisjunctions (a✝¹&a✝) = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions (a✝¹&a✝)) ⊢ a✝¹&a✝ ∈ generatedDisjunctions (a✝¹&a✝) case impl t : Th a✝¹ a✝ : Form h₁ : a✝¹⊃a✝ ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬a✝¹⊃a✝ ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions (a✝¹⊃a✝) → {g}⊢a✝¹⊃a✝ l₂ : ↑t ∩ generatedDisjunctions (a✝¹⊃a✝) = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions (a✝¹⊃a✝)) ⊢ a✝¹⊃a✝ ∈ generatedDisjunctions (a✝¹⊃a✝)

no goals

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

exact Or.inl rfl

t : Th f g : Form h₁ : f¦g ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬f¦g ∈ ↑t l₁ : ∀ (g_1 : Form), g_1 ∈ generatedDisjunctions (f¦g) → {g_1}⊢f¦g l₂ : ↑t ∩ generatedDisjunctions (f¦g) = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions (f¦g)) ⊢ f¦g ∈ generatedDisjunctions (f¦g)

no goals

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

primeAnalysis

[9, 1]

[52, 58]

exact rfl

case impl t : Th a✝¹ a✝ : Form h₁ : a✝¹⊃a✝ ∈ ⋂₀ { p | isPrimeTheory p ∧ ↑t ≤ p ∧ formalTheory p } h₂ : ¬a✝¹⊃a✝ ∈ ↑t l₁ : ∀ (g : Form), g ∈ generatedDisjunctions (a✝¹⊃a✝) → {g}⊢a✝¹⊃a✝ l₂ : ↑t ∩ generatedDisjunctions (a✝¹⊃a✝) = ∅ l₃ : isDisjunctionClosed (generatedDisjunctions (a✝¹⊃a✝)) ⊢ a✝¹⊃a✝ ∈ generatedDisjunctions (a✝¹⊃a✝)

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 → ∃ q r, t ≤ ↑q ∧ u ≤ ↑r ∧ formalApplicationFunction (↑q) u ≤ ↑p ∧ formalApplicationFunction t ↑r ≤ ↑p

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p ⊢ ∃ q r, t ≤ ↑q ∧ u ≤ ↑r ∧ formalApplicationFunction (↑q) u ≤ ↑p ∧ formalApplicationFunction t ↑r ≤ ↑p

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

appBoundingFormalApplication

[54, 1]

[141, 38]

have ⟨q,h₂,h₃⟩ := lemma1 t u p h₁

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p ⊢ ∃ q r, t ≤ ↑q ∧ u ≤ ↑r ∧ formalApplicationFunction (↑q) u ≤ ↑p ∧ formalApplicationFunction t ↑r ≤ ↑p

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p q : Pr h₂ : t ≤ ↑q h₃ : formalApplication ↑↑q ↑u ⊆ ↑↑p ⊢ ∃ q r, t ≤ ↑q ∧ u ≤ ↑r ∧ formalApplicationFunction (↑q) u ≤ ↑p ∧ formalApplicationFunction t ↑r ≤ ↑p

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

appBoundingFormalApplication

[54, 1]

[141, 38]

have ⟨r,h₄,h₅⟩ := lemma2 t u p h₁

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p q : Pr h₂ : t ≤ ↑q h₃ : formalApplication ↑↑q ↑u ⊆ ↑↑p ⊢ ∃ q r, t ≤ ↑q ∧ u ≤ ↑r ∧ formalApplicationFunction (↑q) u ≤ ↑p ∧ formalApplicationFunction t ↑r ≤ ↑p

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p q : Pr h₂ : t ≤ ↑q h₃ : formalApplication ↑↑q ↑u ⊆ ↑↑p r : Pr h₄ : u ≤ ↑r h₅ : formalApplication ↑t ↑↑r ⊆ ↑↑p ⊢ ∃ q r, t ≤ ↑q ∧ u ≤ ↑r ∧ formalApplicationFunction (↑q) u ≤ ↑p ∧ formalApplicationFunction t ↑r ≤ ↑p

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

appBoundingFormalApplication

[54, 1]

[141, 38]

exact ⟨q,r,h₂,h₄,h₃,h₅⟩

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p q : Pr h₂ : t ≤ ↑q h₃ : formalApplication ↑↑q ↑u ⊆ ↑↑p r : Pr h₄ : u ≤ ↑r h₅ : formalApplication ↑t ↑↑r ⊆ ↑↑p ⊢ ∃ q r, t ≤ ↑q ∧ u ≤ ↑r ∧ formalApplicationFunction (↑q) u ≤ ↑p ∧ formalApplicationFunction t ↑r ≤ ↑p

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 → ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑p

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑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 (▲{f}) u ⊆ p) }

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑p

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑p

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

appBoundingFormalApplication

[54, 1]

[141, 38]

have l₂ : ↑t ∩ Δ = ∅ := by apply Set.eq_empty_iff_forall_not_mem.mpr intros P h₂ have l₃ : ▲{P} ⊆ ↑t := generatedContained (Set.singleton_subset_iff.mpr h₂.left) have l₄ := formalAppMonotoneRight ↑u l₃ exact h₂.right $ le_trans l₄ h₁

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑p

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑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,l₆,⟨prf₁⟩⟩,l₈⟩ := nonconstruction l₄ have ⟨T,l₉⟩ := nonconstruction h₁.right have ⟨⟨U,l₁₀,⟨prf₂⟩⟩,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 prf₃ : BProof {P} (S & U ⊃ R ¦ T) := BProof.mp (BProof.mp prf₁ (BTheorem.hs BTheorem.taut BTheorem.orI₁)) (BTheorem.hs BTheorem.andE₁ BTheorem.taut) have prf₄ : BProof {Q} (S & U ⊃ R ¦ T) := BProof.mp (BProof.mp prf₂ (BTheorem.hs BTheorem.taut BTheorem.orI₂)) (BTheorem.hs BTheorem.andE₂ BTheorem.taut) have prf₅ : BProof {P ¦ Q} (S & U ⊃ R ¦ T) := BTheorem.toProof $ BTheorem.mp (BTheorem.adj prf₃.toTheorem prf₄.toTheorem) BTheorem.orE clear prf₁ prf₂ prf₃ prf₄ have l₁₄ : S & U ∈ u := u.property.mpr ⟨BProof.adj (BProof.ax l₆) (BProof.ax l₁₀)⟩ exact ⟨S & U, l₁₄, ⟨prf₅⟩⟩

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑p

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ l₃ : isDisjunctionClosed Δ ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑p

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

appBoundingFormalApplication

[54, 1]

[141, 38]

have l₄ : lindenbaumExtension t Δ ∩ Δ = ∅ := lindenbaumTheorem l₂ l₃

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ l₃ : isDisjunctionClosed Δ ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑p

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ l₃ : isDisjunctionClosed Δ l₄ : lindenbaumExtension t Δ ∩ Δ = ∅ ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑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 (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ l₃ : isDisjunctionClosed Δ l₄ : lindenbaumExtension t Δ ∩ Δ = ∅ ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑p

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₄ : lindenbaumExtension t Δ ∩ Δ = ∅ ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑p

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

appBoundingFormalApplication

[54, 1]

[141, 38]

refine ⟨⟨⟨lindenbaumExtension t Δ, lindenbaumIsFormal⟩, lindenbaumIsPrime⟩, lindenbaumExtensionExtends, ?_⟩

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₄ : lindenbaumExtension t Δ ∩ Δ = ∅ ⊢ ∃ q, t ≤ ↑q ∧ formalApplication ↑↑q ↑u ⊆ ↑↑p

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₄ : lindenbaumExtension t Δ ∩ Δ = ∅ ⊢ formalApplication ↑↑{ val := { val := lindenbaumExtension t Δ, property := (_ : ∀ {f : Form}, f ∈ lindenbaumExtension t Δ ↔ lindenbaumExtension t Δ⊢f) }, property := (_ : ∀ {f g : Form}, f¦g ∈ lindenbaumExtension t Δ → f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ) } ↑u ⊆ ↑↑p

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

appBoundingFormalApplication

[54, 1]

[141, 38]

change formalApplication (lindenbaumExtension t Δ) ↑u ⊆ p

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₄ : lindenbaumExtension t Δ ∩ Δ = ∅ ⊢ formalApplication ↑↑{ val := { val := lindenbaumExtension t Δ, property := (_ : ∀ {f : Form}, f ∈ lindenbaumExtension t Δ ↔ lindenbaumExtension t Δ⊢f) }, property := (_ : ∀ {f g : Form}, f¦g ∈ lindenbaumExtension t Δ → f ∈ lindenbaumExtension t Δ ∨ g ∈ lindenbaumExtension t Δ) } ↑u ⊆ ↑↑p

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₄ : lindenbaumExtension t Δ ∩ Δ = ∅ ⊢ formalApplication (lindenbaumExtension t Δ) ↑u ⊆ ↑↑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 (▲{f}) ↑u ⊆ ↑↑p } l₄ : lindenbaumExtension t Δ ∩ Δ = ∅ ⊢ formalApplication (lindenbaumExtension t Δ) ↑u ⊆ ↑↑p

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 ⊢ 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 (▲{f}) ↑u ⊆ ↑↑p } l₄ : lindenbaumExtension t Δ ∩ Δ = ∅ P : Form h₁ : P ∈ formalApplication (lindenbaumExtension t Δ) ↑u ⊢ P ∈ ↑↑p

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 ⊢ P ∈ ↑↑p

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

appBoundingFormalApplication

[54, 1]

[141, 38]

have l₄ : formalApplication (▲{Q⊃P}) ↑u ⊆ ↑↑p := by apply byContradiction intros h₄ exact (Set.eq_empty_iff_forall_not_mem.mp l₄) (Q⊃P) ⟨h₃,h₄⟩

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 ⊢ P ∈ ↑↑p

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 l₄ : formalApplication (▲{Q⊃P}) ↑u ⊆ ↑↑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,h₂,⟨BProof.ax rfl⟩⟩

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 l₄ : formalApplication (▲{Q⊃P}) ↑u ⊆ ↑↑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 (▲{f}) ↑u ⊆ ↑↑p } ⊢ ↑t ∩ Δ = ∅

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } ⊢ ∀ (x : Form), ¬x ∈ ↑t ∩ 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 (▲{f}) ↑u ⊆ ↑↑p } ⊢ ∀ (x : Form), ¬x ∈ ↑t ∩ Δ

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } P : Form h₂ : P ∈ ↑t ∩ Δ ⊢ False

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

appBoundingFormalApplication

[54, 1]

[141, 38]

have l₃ : ▲{P} ⊆ ↑t := generatedContained (Set.singleton_subset_iff.mpr h₂.left)

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } P : Form h₂ : P ∈ ↑t ∩ Δ ⊢ False

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } P : Form h₂ : P ∈ ↑t ∩ Δ l₃ : ▲{P} ⊆ ↑t ⊢ False

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

appBoundingFormalApplication

[54, 1]

[141, 38]

have l₄ := formalAppMonotoneRight ↑u l₃

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } P : Form h₂ : P ∈ ↑t ∩ Δ l₃ : ▲{P} ⊆ ↑t ⊢ False

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } P : Form h₂ : P ∈ ↑t ∩ Δ l₃ : ▲{P} ⊆ ↑t l₄ : flip formalApplication (↑u) (▲{P}) ≤ flip formalApplication ↑u ↑t ⊢ 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 (▲{f}) ↑u ⊆ ↑↑p } P : Form h₂ : P ∈ ↑t ∩ Δ l₃ : ▲{P} ⊆ ↑t l₄ : flip formalApplication (↑u) (▲{P}) ≤ flip formalApplication ↑u ↑t ⊢ 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 (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ ⊢ isDisjunctionClosed Δ

t u : Th p : Pr h₁✝ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₁ : P ∈ Δ ∧ Q ∈ Δ h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑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 (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₁ : P ∈ Δ ∧ Q ∈ Δ h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p ⊢ False

t u : Th p : Pr h₁✝ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₁ : P ∈ Δ ∧ Q ∈ Δ h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R : Form l₄ : ¬(R ∈ formalApplication (▲{P}) ↑u → R ∈ ↑↑p) ⊢ False

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

appBoundingFormalApplication

[54, 1]

[141, 38]

have ⟨⟨S,l₆,⟨prf₁⟩⟩,l₈⟩ := nonconstruction l₄

t u : Th p : Pr h₁✝ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₁ : P ∈ Δ ∧ Q ∈ Δ h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R : Form l₄ : ¬(R ∈ formalApplication (▲{P}) ↑u → R ∈ ↑↑p) ⊢ False

t u : Th p : Pr h₁✝ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₁ : P ∈ Δ ∧ Q ∈ Δ h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R : Form l₄ : ¬(R ∈ formalApplication (▲{P}) ↑u → R ∈ ↑↑p) S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) 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 (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₁ : P ∈ Δ ∧ Q ∈ Δ h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R : Form l₄ : ¬(R ∈ formalApplication (▲{P}) ↑u → R ∈ ↑↑p) S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p ⊢ False

t u : Th p : Pr h₁✝ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₁ : P ∈ Δ ∧ Q ∈ Δ h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R : Form l₄ : ¬(R ∈ formalApplication (▲{P}) ↑u → R ∈ ↑↑p) S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T : Form l₉ : ¬(T ∈ formalApplication (▲{Q}) ↑u → T ∈ ↑↑p) ⊢ False

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

appBoundingFormalApplication

[54, 1]

[141, 38]

have ⟨⟨U,l₁₀,⟨prf₂⟩⟩,l₁₂⟩ := nonconstruction l₉

t u : Th p : Pr h₁✝ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₁ : P ∈ Δ ∧ Q ∈ Δ h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R : Form l₄ : ¬(R ∈ formalApplication (▲{P}) ↑u → R ∈ ↑↑p) S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T : Form l₉ : ¬(T ∈ formalApplication (▲{Q}) ↑u → T ∈ ↑↑p) ⊢ False

t u : Th p : Pr h₁✝ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₁ : P ∈ Δ ∧ Q ∈ Δ h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R : Form l₄ : ¬(R ∈ formalApplication (▲{P}) ↑u → R ∈ ↑↑p) S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T : Form l₉ : ¬(T ∈ formalApplication (▲{Q}) ↑u → T ∈ ↑↑p) U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃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 (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₁ : P ∈ Δ ∧ Q ∈ Δ h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R : Form l₄ : ¬(R ∈ formalApplication (▲{P}) ↑u → R ∈ ↑↑p) S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T : Form l₉ : ¬(T ∈ formalApplication (▲{Q}) ↑u → T ∈ ↑↑p) U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) l₁₂ : ¬T ∈ ↑↑p ⊢ False

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃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 (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) l₁₂ : ¬T ∈ ↑↑p ⊢ False

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃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 (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) l₁₂ : ¬T ∈ ↑↑p l₁₃ : ¬R¦T ∈ p ⊢ False

t u : Th p : Pr h₁ : formalApplicationFunction t u ≤ ↑p Δ : Set Form := { f | ¬formalApplication (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃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 (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃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 (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) l₁₂ : ¬T ∈ ↑↑p l₁₃ : ¬R¦T ∈ p ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u

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 (▲{f}) ↑u ⊆ ↑↑p } l₂ : ↑t ∩ Δ = ∅ P Q : Form h₂ : formalApplication (▲{P¦Q}) ↑u ⊆ ↑↑p R S : Form l₆ : S ∈ ↑u prf₁ : BProof {P} (S⊃R) l₈ : ¬R ∈ ↑↑p T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) l₁₂ : ¬T ∈ ↑↑p l₁₃ : ¬R¦T ∈ p ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u

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 prf₁ : BProof {P} (S⊃R) T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

appBoundingFormalApplication

[54, 1]

[141, 38]

have prf₃ : BProof {P} (S & U ⊃ R ¦ T) := BProof.mp (BProof.mp prf₁ (BTheorem.hs BTheorem.taut BTheorem.orI₁)) (BTheorem.hs BTheorem.andE₁ BTheorem.taut)

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 prf₁ : BProof {P} (S⊃R) T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u

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 prf₁ : BProof {P} (S⊃R) T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) prf₃ : BProof {P} (S&U⊃R¦T) ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

appBoundingFormalApplication

[54, 1]

[141, 38]

have prf₄ : BProof {Q} (S & U ⊃ R ¦ T) := BProof.mp (BProof.mp prf₂ (BTheorem.hs BTheorem.taut BTheorem.orI₂)) (BTheorem.hs BTheorem.andE₂ BTheorem.taut)

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 prf₁ : BProof {P} (S⊃R) T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) prf₃ : BProof {P} (S&U⊃R¦T) ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u

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 prf₁ : BProof {P} (S⊃R) T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) prf₃ : BProof {P} (S&U⊃R¦T) prf₄ : BProof {Q} (S&U⊃R¦T) ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

appBoundingFormalApplication

[54, 1]

[141, 38]

have prf₅ : BProof {P ¦ Q} (S & U ⊃ R ¦ T) := BTheorem.toProof $ BTheorem.mp (BTheorem.adj prf₃.toTheorem prf₄.toTheorem) BTheorem.orE

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 prf₁ : BProof {P} (S⊃R) T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) prf₃ : BProof {P} (S&U⊃R¦T) prf₄ : BProof {Q} (S&U⊃R¦T) ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u

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 prf₁ : BProof {P} (S⊃R) T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) prf₃ : BProof {P} (S&U⊃R¦T) prf₄ : BProof {Q} (S&U⊃R¦T) prf₅ : BProof {P¦Q} (S&U⊃R¦T) ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

appBoundingFormalApplication

[54, 1]

[141, 38]

clear prf₁ prf₂ prf₃ prf₄

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 prf₁ : BProof {P} (S⊃R) T U : Form l₁₀ : U ∈ ↑u prf₂ : BProof {Q} (U⊃T) prf₃ : BProof {P} (S&U⊃R¦T) prf₄ : BProof {Q} (S&U⊃R¦T) prf₅ : BProof {P¦Q} (S&U⊃R¦T) ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u

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) ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u

https://github.com/gleachkr/Completeness-For-Fine-Semantics.git

0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f

Fine/SystemB/CanonicalModel.lean

appBoundingFormalApplication

[54, 1]

[141, 38]

have l₁₄ : S & U ∈ u := u.property.mpr ⟨BProof.adj (BProof.ax l₆) (BProof.ax l₁₀)⟩

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) ⊢ R¦T ∈ formalApplication (▲{P¦Q}) ↑u

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