File size: 3,563 Bytes
2f887db | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 | 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
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