openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Definability /ExternalWordRewriteFormulas.lean
| import OAI.SetTheory.PartitionPrinciple.Geometry.GroundFiniteWordLaws | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing | |
| open SetRealization | |
| universe u | |
| def wordSingleton (a : ZFSet.{u}) : ZFSet.{u} := {ZFSet.pair (natSetCode 0) a} | |
| def wordDouble (a b : ZFSet.{u}) : ZFSet.{u} := | |
| {ZFSet.pair (natSetCode 0) a, ZFSet.pair (natSetCode 1) b} | |
| theorem finiteWordGraph_singleton {B : Type u} (s : Presentation B) (a : B) : | |
| finiteWordGraph s [a] = wordSingleton (s.encode a) := by | |
| apply ZFSet.ext | |
| intro z | |
| rw [mem_finiteWordGraph, wordSingleton, ZFSet.mem_singleton] | |
| constructor | |
| · rintro ⟨i, eq⟩ | |
| have hi : i.val < 1 := i.isLt | |
| have equal : i = ⟨0, by simp⟩ := Fin.ext (by change i.val = 0; omega) | |
| subst i | |
| exact eq | |
| · intro eq | |
| exact ⟨⟨0, by simp⟩, eq⟩ | |
| theorem finiteWordGraph_double {B : Type u} (s : Presentation B) (a b : B) : | |
| finiteWordGraph s [a, b] = wordDouble (s.encode a) (s.encode b) := by | |
| apply ZFSet.ext | |
| intro z | |
| rw [mem_finiteWordGraph, wordDouble, ZFSet.mem_pair] | |
| constructor | |
| · rintro ⟨i, eq⟩ | |
| have hi : i.val < 2 := i.isLt | |
| have cases : i = ⟨0, by simp⟩ ∨ i = ⟨1, by simp⟩ := by | |
| have values : i.val = 0 ∨ i.val = 1 := by omega | |
| exact values.elim (fun h => Or.inl (Fin.ext h)) (fun h => Or.inr (Fin.ext h)) | |
| rcases cases with rfl | rfl | |
| · exact Or.inl eq | |
| · exact Or.inr eq | |
| · rintro (eq | eq) | |
| · exact ⟨⟨0, by simp⟩, eq⟩ | |
| · exact ⟨⟨1, by simp⟩, eq⟩ | |
| def ExternalWordAtomicCode (G S m one l r : ZFSet.{u}) : Prop := | |
| (one ∈ G ∧ l = wordSingleton (externalActorSymbol one) ∧ r = ∅) ∨ | |
| (∃ g ∈ G, ∃ h ∈ G, ∃ k ∈ G, ZFSet.pair (ZFSet.pair g h) k ∈ m ∧ | |
| l = wordDouble (externalActorSymbol g) (externalActorSymbol h) ∧ | |
| r = wordSingleton (externalActorSymbol k)) ∨ | |
| (∃ s ∈ S, (l = wordDouble (externalLetterSymbol s false) (externalLetterSymbol s true) ∨ | |
| l = wordDouble (externalLetterSymbol s true) (externalLetterSymbol s false)) ∧ r = ∅) | |
| def ExternalWordStepCode (W append G S m one f g : ZFSet.{u}) : Prop := | |
| ∃ pre ∈ W, ∃ post ∈ W, ∃ l ∈ W, ∃ r ∈ W, ∃ pl ∈ W, ∃ pr ∈ W, | |
| ExternalWordAtomicCode G S m one l r ∧ | |
| ZFSet.pair (ZFSet.pair pre l) pl ∈ append ∧ | |
| ZFSet.pair (ZFSet.pair pre r) pr ∈ append ∧ | |
| ((ZFSet.pair (ZFSet.pair pl post) f ∈ append ∧ | |
| ZFSet.pair (ZFSet.pair pr post) g ∈ append) ∨ | |
| (ZFSet.pair (ZFSet.pair pl post) g ∈ append ∧ | |
| ZFSet.pair (ZFSet.pair pr post) f ∈ append)) | |
| theorem ExternalWordStepCode.symm {W append G S m one f g : ZFSet.{u}} | |
| (step : ExternalWordStepCode W append G S m one f g) : | |
| ExternalWordStepCode W append G S m one g f := by | |
| obtain ⟨pre, hp, post, hq, l, hl, r, hr, pl, hpl, pr, hpr, atomic, left, right, direction⟩ := step | |
| exact ⟨pre, hp, post, hq, l, hl, r, hr, pl, hpl, pr, hpr, atomic, left, right, direction.symm⟩ | |
| end PartitionPilot.Forcing | |
| namespace PartitionPilot.Forcing.BoundedFormula | |
| open SetFormula | |
| universe u | |
| def actorSingletonWordFormula {n} (w zero g : Fin n) : BoundedFormula n := | |
| exMem w (.conj (nonemptyTupleFormula 2 0 | |
| (Fin.cases zero.succ (Fin.cases zero.succ (Fin.cases g.succ Fin.elim0)))) | |
| (singletonFormula w.succ 0)) | |
| theorem truth_actorSingletonWordFormula {n} (w zero g : Fin n) (env : Fin n → ZFSet.{u}) : | |
| Truth (actorSingletonWordFormula w zero g) env ↔ | |
| env w = {ZFSet.pair (env zero) (ZFSet.pair (env zero) (env g))} := by | |
| simp only [actorSingletonWordFormula, truth_exMem, Truth, | |
| truth_nonemptyTupleFormula, truth_singletonFormula] | |
| change (∃ z ∈ env w, z = ZFSet.pair (env zero) (ZFSet.pair (env zero) (env g)) ∧ | |
| env w = {z}) ↔ _ | |
| constructor | |
| · rintro ⟨z, _, rfl, eq⟩ | |
| exact eq | |
| · intro eq | |
| exact ⟨_, eq.symm ▸ ZFSet.mem_singleton.2 rfl, rfl, eq⟩ | |
| def actorDoubleWordFormula {n} (w zero unit g h : Fin n) : BoundedFormula n := | |
| exMem w (exMem w.succ | |
| (.conj (nonemptyTupleFormula 2 1 | |
| (Fin.cases zero.succ.succ (Fin.cases zero.succ.succ (Fin.cases g.succ.succ Fin.elim0)))) | |
| (.conj (nonemptyTupleFormula 2 0 | |
| (Fin.cases unit.succ.succ (Fin.cases zero.succ.succ (Fin.cases h.succ.succ Fin.elim0)))) | |
| (unorderedPairFormula w.succ.succ 1 0)))) | |
| theorem truth_actorDoubleWordFormula {n} (w zero unit g h : Fin n) (env : Fin n → ZFSet.{u}) : | |
| Truth (actorDoubleWordFormula w zero unit g h) env ↔ | |
| env w = {ZFSet.pair (env zero) (ZFSet.pair (env zero) (env g)), | |
| ZFSet.pair (env unit) (ZFSet.pair (env zero) (env h))} := by | |
| simp only [actorDoubleWordFormula, truth_exMem, Truth, | |
| truth_nonemptyTupleFormula, truth_unorderedPairFormula] | |
| change (∃ z ∈ env w, ∃ v ∈ env w, | |
| z = ZFSet.pair (env zero) (ZFSet.pair (env zero) (env g)) ∧ | |
| v = ZFSet.pair (env unit) (ZFSet.pair (env zero) (env h)) ∧ env w = {z, v}) ↔ _ | |
| constructor | |
| · rintro ⟨z, _, v, _, rfl, rfl, eq⟩ | |
| exact eq | |
| · intro eq | |
| exact ⟨_, eq.symm ▸ ZFSet.mem_pair.2 (Or.inl rfl), | |
| _, eq.symm ▸ ZFSet.mem_pair.2 (Or.inr rfl), rfl, rfl, eq⟩ | |
| def signedDoubleWordFormula {n} (w zero unit s a b : Fin n) : BoundedFormula n := | |
| exMem w (exMem w.succ | |
| (.conj (nonemptyTupleFormula 3 1 | |
| (Fin.cases zero.succ.succ (Fin.cases unit.succ.succ | |
| (Fin.cases s.succ.succ (Fin.cases a.succ.succ Fin.elim0))))) | |
| (.conj (nonemptyTupleFormula 3 0 | |
| (Fin.cases unit.succ.succ (Fin.cases unit.succ.succ | |
| (Fin.cases s.succ.succ (Fin.cases b.succ.succ Fin.elim0))))) | |
| (unorderedPairFormula w.succ.succ 1 0)))) | |
| theorem truth_signedDoubleWordFormula {n} (w zero unit s a b : Fin n) | |
| (env : Fin n → ZFSet.{u}) : | |
| Truth (signedDoubleWordFormula w zero unit s a b) env ↔ | |
| env w = {ZFSet.pair (env zero) (ZFSet.pair (env unit) (ZFSet.pair (env s) (env a))), | |
| ZFSet.pair (env unit) (ZFSet.pair (env unit) (ZFSet.pair (env s) (env b)))} := by | |
| simp only [signedDoubleWordFormula, truth_exMem, Truth, | |
| truth_nonemptyTupleFormula, truth_unorderedPairFormula] | |
| change (∃ z ∈ env w, ∃ v ∈ env w, | |
| z = ZFSet.pair (env zero) (ZFSet.pair (env unit) (ZFSet.pair (env s) (env a))) ∧ | |
| v = ZFSet.pair (env unit) (ZFSet.pair (env unit) (ZFSet.pair (env s) (env b))) ∧ | |
| env w = {z, v}) ↔ _ | |
| constructor | |
| · rintro ⟨z, _, v, _, rfl, rfl, eq⟩ | |
| exact eq | |
| · intro eq | |
| exact ⟨_, eq.symm ▸ ZFSet.mem_pair.2 (Or.inl rfl), | |
| _, eq.symm ▸ ZFSet.mem_pair.2 (Or.inr rfl), rfl, rfl, eq⟩ | |
| def externalWordAtomicFormula {n} (zero unit G S m one l r : Fin n) : BoundedFormula n := | |
| disj (.conj (.mem one G) (.conj (actorSingletonWordFormula l zero one) (emptyFormula r))) | |
| (disj (exMem G (exMem G.succ (exMem G.succ.succ | |
| (.conj (tripleMembershipFormula 2 1 0 m.succ.succ.succ) | |
| (.conj (actorDoubleWordFormula l.succ.succ.succ zero.succ.succ.succ unit.succ.succ.succ 2 1) | |
| (actorSingletonWordFormula r.succ.succ.succ zero.succ.succ.succ 0)))))) | |
| (exMem S (.conj | |
| (disj (signedDoubleWordFormula l.succ zero.succ unit.succ 0 zero.succ unit.succ) | |
| (signedDoubleWordFormula l.succ zero.succ unit.succ 0 unit.succ zero.succ)) (emptyFormula r.succ)))) | |
| theorem truth_externalWordAtomicFormula {n} (zero unit G S m one l r : Fin n) | |
| (env : Fin n → ZFSet.{u}) (zeroValue : env zero = natSetCode 0) | |
| (unitValue : env unit = natSetCode 1) : | |
| Truth (externalWordAtomicFormula zero unit G S m one l r) env ↔ | |
| ExternalWordAtomicCode (env G) (env S) (env m) (env one) (env l) (env r) := by | |
| simp only [externalWordAtomicFormula, truth_disj, truth_exMem, Truth, | |
| truth_actorSingletonWordFormula, truth_actorDoubleWordFormula, truth_signedDoubleWordFormula, | |
| truth_emptyFormula, truth_tripleMembershipFormula] | |
| change ((env one ∈ env G ∧ env l = {ZFSet.pair (env zero) (ZFSet.pair (env zero) (env one))} ∧ env r = ∅) ∨ | |
| (∃ g ∈ env G, ∃ h ∈ env G, ∃ k ∈ env G, ZFSet.pair (ZFSet.pair g h) k ∈ env m ∧ | |
| env l = {ZFSet.pair (env zero) (ZFSet.pair (env zero) g), ZFSet.pair (env unit) (ZFSet.pair (env zero) h)} ∧ | |
| env r = {ZFSet.pair (env zero) (ZFSet.pair (env zero) k)}) ∨ | |
| (∃ s ∈ env S, | |
| (env l = {ZFSet.pair (env zero) (ZFSet.pair (env unit) (ZFSet.pair s (env zero))), | |
| ZFSet.pair (env unit) (ZFSet.pair (env unit) (ZFSet.pair s (env unit)))} ∨ | |
| env l = {ZFSet.pair (env zero) (ZFSet.pair (env unit) (ZFSet.pair s (env unit))), | |
| ZFSet.pair (env unit) (ZFSet.pair (env unit) (ZFSet.pair s (env zero)))}) ∧ env r = ∅)) ↔ _ | |
| rw [zeroValue, unitValue] | |
| rfl | |
| def externalWordStepFormula {n} (W append zero unit G S m one f g : Fin n) : BoundedFormula n := | |
| exMem W (exMem W.succ (exMem W.succ.succ (exMem W.succ.succ.succ | |
| (exMem W.succ.succ.succ.succ (exMem W.succ.succ.succ.succ.succ | |
| (.conj (externalWordAtomicFormula zero.succ.succ.succ.succ.succ.succ | |
| unit.succ.succ.succ.succ.succ.succ G.succ.succ.succ.succ.succ.succ | |
| S.succ.succ.succ.succ.succ.succ m.succ.succ.succ.succ.succ.succ | |
| one.succ.succ.succ.succ.succ.succ 3 2) | |
| (.conj (tripleMembershipFormula 5 3 1 append.succ.succ.succ.succ.succ.succ) | |
| (.conj (tripleMembershipFormula 5 2 0 append.succ.succ.succ.succ.succ.succ) | |
| (disj (.conj (tripleMembershipFormula 1 4 f.succ.succ.succ.succ.succ.succ append.succ.succ.succ.succ.succ.succ) | |
| (tripleMembershipFormula 0 4 g.succ.succ.succ.succ.succ.succ append.succ.succ.succ.succ.succ.succ)) | |
| (.conj (tripleMembershipFormula 1 4 g.succ.succ.succ.succ.succ.succ append.succ.succ.succ.succ.succ.succ) | |
| (tripleMembershipFormula 0 4 f.succ.succ.succ.succ.succ.succ append.succ.succ.succ.succ.succ.succ))))))))))) | |
| theorem truth_externalWordStepFormula {n} (W append zero unit G S m one f g : Fin n) | |
| (env : Fin n → ZFSet.{u}) (zeroValue : env zero = natSetCode 0) | |
| (unitValue : env unit = natSetCode 1) : | |
| Truth (externalWordStepFormula W append zero unit G S m one f g) env ↔ | |
| ExternalWordStepCode (env W) (env append) (env G) (env S) (env m) (env one) (env f) (env g) := by | |
| simp only [externalWordStepFormula, truth_exMem, Truth, truth_tripleMembershipFormula, truth_disj, | |
| ExternalWordStepCode] | |
| constructor | |
| · rintro ⟨pre, hp, post, hq, l, hl, r, hr, pl, hpl, pr, hpr, atomic, left, right, direction⟩ | |
| exact ⟨pre, hp, post, hq, l, hl, r, hr, pl, hpl, pr, hpr, | |
| (truth_externalWordAtomicFormula zero.succ.succ.succ.succ.succ.succ | |
| unit.succ.succ.succ.succ.succ.succ G.succ.succ.succ.succ.succ.succ | |
| S.succ.succ.succ.succ.succ.succ m.succ.succ.succ.succ.succ.succ | |
| one.succ.succ.succ.succ.succ.succ 3 2 | |
| (extend pr (extend pl (extend r (extend l (extend post (extend pre env)))))) zeroValue unitValue).1 atomic, left, right, direction⟩ | |
| · rintro ⟨pre, hp, post, hq, l, hl, r, hr, pl, hpl, pr, hpr, atomic, left, right, direction⟩ | |
| exact ⟨pre, hp, post, hq, l, hl, r, hr, pl, hpl, pr, hpr, | |
| (truth_externalWordAtomicFormula zero.succ.succ.succ.succ.succ.succ | |
| unit.succ.succ.succ.succ.succ.succ G.succ.succ.succ.succ.succ.succ | |
| S.succ.succ.succ.succ.succ.succ m.succ.succ.succ.succ.succ.succ | |
| one.succ.succ.succ.succ.succ.succ 3 2 | |
| (extend pr (extend pl (extend r (extend l (extend post (extend pre env)))))) zeroValue unitValue).2 atomic, left, right, direction⟩ | |
| end PartitionPilot.Forcing.BoundedFormula | |
| end | |
| end OAI | |