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| author | Chloe Brown <chloe.brown.00@outlook.com> | 2023-06-01 17:07:41 +0100 |
|---|---|---|
| committer | Chloe Brown <chloe.brown.00@outlook.com> | 2023-06-01 17:07:41 +0100 |
| commit | d5f8497dbb6de72d9664f48d6acbc9772de77be3 (patch) | |
| tree | e1a4aeb598d263af3d888f8c0c68c07f15d13f14 /src/Total/Reduction.idr | |
| parent | 3eb03e2347b35432f7eae7d6847ec9ecf1dff19e (diff) | |
Give a logical relation template.
Diffstat (limited to 'src/Total/Reduction.idr')
| -rw-r--r-- | src/Total/Reduction.idr | 32 |
1 files changed, 32 insertions, 0 deletions
diff --git a/src/Total/Reduction.idr b/src/Total/Reduction.idr index efa6f74..4870f30 100644 --- a/src/Total/Reduction.idr +++ b/src/Total/Reduction.idr @@ -1,5 +1,6 @@ module Total.Reduction +import Syntax.PreorderReasoning import Total.Term public export @@ -63,3 +64,34 @@ export RecCong3' : v1 >= v2 -> Rec t u v1 >= Rec t u v2 RecCong3' [<] = [<] RecCong3' (steps :< step) = RecCong3' steps :< RecCong3 step + +-- Properties ------------------------------------------------------------------ + +lemma : + (t : Term (ctx :< ty) ty') -> + (thin : ctx `Thins` ctx') -> + (u : Term ctx ty) -> + subst (wkn t (keep thin)) (Base Id :< wkn u thin) = wkn (subst t (Base Id :< u)) thin +lemma t thin u = Calc $ + |~ subst (wkn t (keep thin)) (Base Id :< wkn u thin) + ~~ subst t (restrict (Base Id :< wkn u thin) (keep thin)) + ...(substWkn t (keep thin) (Base Id :< wkn u thin)) + ~~ subst t (Base thin :< wkn u thin) + ...(cong (subst t) $ restrictKeep (Base Id) (wkn u thin) thin) + ~~ subst t (Base (thin . Id) :< wkn u thin) + ...(sym $ cong (subst t . (:< wkn u thin) . Base) $ identityRight thin) + ~~ wkn (subst t (Base Id :< u)) thin + ...(sym $ wknSubst t (Base Id :< u) thin) + +export +wknStep : t > u -> wkn t thin > wkn u thin +wknStep (AbsCong step) = AbsCong (wknStep step) +wknStep (AppCong1 step) = AppCong1 (wknStep step) +wknStep (AppCong2 step) = AppCong2 (wknStep step) +wknStep (AppBeta {t, u}) {thin} = rewrite sym $ lemma t thin u in AppBeta +wknStep (SucCong step) = SucCong (wknStep step) +wknStep (RecCong1 step) = RecCong1 (wknStep step) +wknStep (RecCong2 step) = RecCong2 (wknStep step) +wknStep (RecCong3 step) = RecCong3 (wknStep step) +wknStep RecZero = RecZero +wknStep RecSuc = RecSuc |