openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /CohenForcing /BoundedForcingCode.lean
| import OAI.Computability.DegreeRigidity.CohenForcing.BoundedForcing | |
| import OAI.Computability.DegreeRigidity.Syntax.AtomicCertificate | |
| namespace OAI | |
| namespace TuringRigidity.BoundedForcing | |
| open Set RecursiveNames TransitiveNameModel BoundedSetTheory AtomicForcing | |
| universe u | |
| def code : BoundedSetTheory.Formula → ℕ → ℕ → ℕ → ℕ → ℕ → (ℕ → ℕ) → ℕ → BoundedSetTheory.Formula | |
| | .equal i j, _d,_c,k,_o,f,v,p => AtomicFormula.triple k f (v i) (v j) p | |
| | .member i j, d,c,k,o,f,v,p => AtomicFormula.membership d c k o f (v i) (v j) p | |
| | .conj φ ψ, d,c,k,o,f,v,p => .conj (code φ d c k o f v p) (code ψ d c k o f v p) | |
| | .neg φ, d,c,k,o,f,v,p => Formula.allMem c | |
| (Formula.imp (Formula.pairMem 0 (p+1) (o+1)) | |
| (.neg (code φ (d+1) (c+1) (k+1) (o+1) (f+1) (fun i => v i+1) 0))) | |
| | .existsMem i φ, d,c,k,o,f,v,p => .existsMem d (.existsMem (c+1) | |
| (.conj (Formula.pairMem 1 0 (v i+2)) (.conj (Formula.pairMem (p+2) 0 (o+2)) | |
| (code φ (d+2) (c+2) (k+2) (o+2) (f+2) (push 1 (fun i => v i+2)) (p+2))))) | |
| variable {c : ZFSet.{u}} [Preorder (Conditions c)] | |
| theorem eval_code (φ : BoundedSetTheory.Formula) {d o f : ZFSet.{u}} (hd : Transitive d) | |
| (ho : ∀ r s : Conditions c, ZFSet.pair (label c r) (label c s) ∈ o ↔ r ≤ s) | |
| (hf : Graph d c o f) (e : ℕ → Name (Conditions c)) | |
| (he : ∀ i, (e i).encode (label c) ∈ d) (p : Conditions c) | |
| (env : ℕ → ZFSet.{u}) (di ci ki oi fi : ℕ) (v : ℕ → ℕ) (pi : ℕ) | |
| (slots : env di = d ∧ env ci = c ∧ env ki = ZFSet.prod d d ∧ env oi = o ∧ env fi = f ∧ env pi = label c p) | |
| (hv : ∀ i, env (v i) = (e i).encode (label c)) : | |
| (code φ di ci ki oi fi v pi).Eval env ↔ Forces e φ p := by | |
| induction φ generalizing e p env di ci ki oi fi v pi with | |
| | equal i j => | |
| simp only [code,AtomicFormula.eval_triple,slots.2.2.1,slots.2.2.2.2.1,slots.2.2.2.2.2,hv,Forces] | |
| exact (triple_pair (he i) (he j)).trans (hf.correct hd ho _ _ (he i) (he j) p) | |
| | member i j => | |
| exact eval_membership hd ho hf _ _ (he i) (he j) p env di ci ki oi fi (v i) (v j) pi | |
| ⟨slots.1,slots.2.1,slots.2.2.1,slots.2.2.2.1,slots.2.2.2.2.1,hv i,hv j,slots.2.2.2.2.2⟩ | |
| | conj φ ψ ihφ ihψ => | |
| exact and_congr (ihφ e he p env di ci ki oi fi v pi slots hv) | |
| (ihψ e he p env di ci ki oi fi v pi slots hv) | |
| | neg φ ih => | |
| simp only [code,Formula.eval_allMem,Formula.eval_imp,Formula.eval_pairMem,Formula.Eval, | |
| cons_zero,cons_succ,slots.2.1,slots.2.2.2.1,slots.2.2.2.2.2] | |
| have sub (q : Conditions c) := ih e he q (cons (label c q) env) | |
| (di+1) (ci+1) (ki+1) (oi+1) (fi+1) (fun i => v i+1) 0 | |
| ⟨slots.1,slots.2.1,slots.2.2.1,slots.2.2.2.1,slots.2.2.2.2.1,rfl⟩ hv | |
| constructor | |
| · intro h q hq hforce | |
| exact h _ (label_mem c q) ((ho q p).mpr hq) ((sub q).mpr hforce) | |
| · intro h q hq hqp hforce | |
| obtain ⟨r,rfl⟩ := label_surjective c hq | |
| exact h r ((ho r p).mp hqp) ((sub r).mp hforce) | |
| | existsMem i φ ih => | |
| simp only [code,Formula.Eval,Formula.eval_pairMem,cons_zero,cons_succ, | |
| slots.1,slots.2.1,slots.2.2.2.1,slots.2.2.2.2.2,hv] | |
| have hej (j) : ∀ n, (push ((e i).child j) e n).encode (label c) ∈ d := by | |
| intro n | |
| cases n with | |
| | zero => | |
| exact names_childClosed d c hd (e i) (he i) _ (child_relation (e i) j) | |
| | succ n => exact he n | |
| have sub (j) (s : Conditions c) := ih (push ((e i).child j) e) (hej j) p | |
| (cons (label c s) (cons (((e i).child j).encode (label c)) env)) | |
| (di+2) (ci+2) (ki+2) (oi+2) (fi+2) (push 1 (fun n => v n+2)) (pi+2) | |
| ⟨slots.1,slots.2.1,slots.2.2.1,slots.2.2.2.1,slots.2.2.2.2.1,slots.2.2.2.2.2⟩ | |
| (by intro n; cases n <;> simp only [push_zero,push_succ,cons_zero,cons_succ,hv]) | |
| constructor | |
| · rintro ⟨x,hx,s,hs,hxs,hps,hφ⟩ | |
| obtain ⟨j,hj,hjs⟩ := (pair_mem_encode _ _ _ _).mp hxs | |
| subst x | |
| subst s | |
| exact ⟨j,(ho p ((e i).tag j)).mp hps,(sub j ((e i).tag j)).mp hφ⟩ | |
| · rintro ⟨j,hj,hφ⟩ | |
| exact ⟨_,hej j 0,_,label_mem c ((e i).tag j), | |
| (pair_mem_encode _ _ _ _).mpr ⟨j,rfl,rfl⟩,(ho p ((e i).tag j)).mpr hj, | |
| (sub j ((e i).tag j)).mpr hφ⟩ | |
| end TuringRigidity.BoundedForcing | |
| end OAI | |