openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Repetition /StoppedCleanCutEquality.lean
| import OAI.Computability.PerfectCompleteness.Foundations.StoppedProjectedExperiment | |
| import OAI.Computability.PerfectCompleteness.Repetition.StoppedCleanGeometry | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.StoppedCleanCutEquality | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| variable {branch : Nat → Nat} | |
| theorem append_left_injective {N M K : Nat} (pref : Path branch N M) : | |
| Function.Injective (fun suff : Path branch M K => pref.append suff) := by | |
| induction pref with | |
| | refl _ => intro a b hab; exact hab | |
| | step child pref ih => | |
| intro a b hab | |
| apply ih | |
| change Path.step child (pref.append a) = Path.step child (pref.append b) at hab | |
| injection hab | |
| theorem cut_eq_of_path_eq {N : Nat} {upper lower : Nodes branch N} | |
| (left right : OwnInputReference.Cut upper lower) (hpath : left.path = right.path) : | |
| left = right := by | |
| cases left | |
| cases right | |
| cases hpath | |
| rfl | |
| theorem cut_heq_of_fullPath_heq {N : Nat} {upper lower₁ lower₂ : Nodes branch N} | |
| (hlower : lower₁ = lower₂) | |
| (left : OwnInputReference.Cut upper lower₁) (right : OwnInputReference.Cut upper lower₂) | |
| (hpath : HEq (WholeArrayInteriorOwnInputLaw.fullPath upper lower₁ left) | |
| (WholeArrayInteriorOwnInputLaw.fullPath upper lower₂ right)) : HEq left right := by | |
| cases hlower | |
| apply heq_of_eq | |
| apply cut_eq_of_path_eq | |
| exact append_left_injective (Nodes.path upper) (eq_of_heq hpath) | |
| theorem cutOfBelow_heq_selectedCut {N i j : Nat} | |
| (p : Path branch N (j + 1)) (q : Path branch (j + 1) (i + 1)) (hij : i < j) : | |
| HEq (StoppedOwnInputGeometry.cutOfBelow (StoppedProjectedExperiment.below_append hij p q)) | |
| (StoppedCleanGeometry.cut p q hij) := by | |
| refine cut_heq_of_fullPath_heq (StoppedCleanGeometry.lowerNode_descendant p q hij).symm | |
| _ _ ?_ | |
| have hphysical : HEq | |
| (WholeArrayInteriorOwnInputLaw.fullPath (WholeArrayInteriorExterior.upperNode p) | |
| (WholeArrayInteriorExterior.upperNode (p.append q)) | |
| (StoppedOwnInputGeometry.cutOfBelow (StoppedProjectedExperiment.below_append hij p q))) | |
| (p.append q) := by | |
| rw [StoppedOwnInputGeometry.fullPath_cutOfBelow] | |
| exact WholeArrayInteriorExterior.upperNode_path_heq (p.append q) | |
| exact hphysical.trans ((Sigma.mk.inj (StoppedCleanGeometry.fullPath_spec p q hij)).2.symm) | |
| variable {n i j t m : Nat} (rows : Nat → Nat) (hupper : j + 1 ≤ n) (hij : i < j) | |
| (o : StoppedProjectedExperiment.Outer | |
| (branch := branch) (n := n) (j := j) (t := t) (m := m)) | |
| (k : StoppedProjectedExperiment.Inner | |
| (branch := branch) (n := n) (i := i) (j := j) (t := t) rows) | |
| theorem stopped_lower_eq : | |
| StoppedProjectedExperiment.lower rows hupper hij o k = | |
| HierarchicalLeftDecoder.LowerNode (StoppedProjectedExperiment.upper hupper o) (i + 1) | |
| (StoppedCleanGeometry.descendant (StoppedProjectedExperiment.upperPath hupper o) | |
| (StoppedProjectedExperiment.lowerPath rows hij k) hij) := | |
| (StoppedCleanGeometry.lowerNode_descendant (StoppedProjectedExperiment.upperPath hupper o) | |
| (StoppedProjectedExperiment.lowerPath rows hij k) hij).symm | |
| theorem stopped_cut_heq : | |
| HEq (StoppedProjectedExperiment.cut rows hupper hij o k) | |
| (StoppedCleanGeometry.cut (StoppedProjectedExperiment.upperPath hupper o) | |
| (StoppedProjectedExperiment.lowerPath rows hij k) hij) := | |
| cutOfBelow_heq_selectedCut (StoppedProjectedExperiment.upperPath hupper o) | |
| (StoppedProjectedExperiment.lowerPath rows hij k) hij | |
| end PerfectCompleteness.StoppedCleanCutEquality | |
| end | |
| end OAI | |