-lemma cpm_ind (h): ∀R:relation5 nat genv lenv term term.
- (∀I,G,L. R 0 G L (⓪{I}) (⓪{I})) →
- (∀G,L,s. R 1 G L (⋆s) (⋆(next h s))) →
- (â\88\80n,G,K,V1,V2,W2. â¦\83G, Kâ¦\84 â\8a¢ V1 â\9e¡[n, h] V2 â\86\92 R n G K V1 V2 →
- â¬\86*[1] V2 â\89\98 W2 â\86\92 R n G (K.ⓓV1) (#0) W2
- ) â\86\92 (â\88\80n,G,K,V1,V2,W2. â¦\83G, Kâ¦\84 â\8a¢ V1 â\9e¡[n, h] V2 â\86\92 R n G K V1 V2 →
- â¬\86*[1] V2 â\89\98 W2 â\86\92 R (↑n) G (K.ⓛV1) (#0) W2
- ) â\86\92 (â\88\80n,I,G,K,T,U,i. â¦\83G, Kâ¦\84 â\8a¢ #i â\9e¡[n, h] T â\86\92 R n G K (#i) T →
- â¬\86*[1] T â\89\98 U â\86\92 R n G (K.â\93\98{I}) (#↑i) (U)
- ) â\86\92 (â\88\80n,p,I,G,L,V1,V2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h] V2 â\86\92 â¦\83G, L.â\93\91{I}V1â¦\84 â\8a¢ T1 â\9e¡[n, h] T2 →
- R 0 G L V1 V2 → R n G (L.ⓑ{I}V1) T1 T2 → R n G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
- ) â\86\92 (â\88\80n,G,L,V1,V2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h] V2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[n, h] T2 →
- R 0 G L V1 V2 → R n G L T1 T2 → R n G L (ⓐV1.T1) (ⓐV2.T2)
- ) â\86\92 (â\88\80n,G,L,V1,V2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[n, h] V2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[n, h] T2 →
- R n G L V1 V2 → R n G L T1 T2 → R n G L (ⓝV1.T1) (ⓝV2.T2)
- ) â\86\92 (â\88\80n,G,L,V,T1,T,T2. â¦\83G, L.â\93\93Vâ¦\84 â\8a¢ T1 â\9e¡[n, h] T â\86\92 R n G (L.â\93\93V) T1 T →
- ⬆*[1] T2 ≘ T → R n G L (+ⓓV.T1) T2
- ) â\86\92 (â\88\80n,G,L,V,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[n, h] T2 →
- R n G L T1 T2 → R n G L (ⓝV.T1) T2
- ) â\86\92 (â\88\80n,G,L,V1,V2,T. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[n, h] V2 →
- R n G L V1 V2 → R (↑n) G L (ⓝV1.T) V2
- ) â\86\92 (â\88\80n,p,G,L,V1,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h] V2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ W1 â\9e¡[h] W2 â\86\92 â¦\83G, L.â\93\9bW1â¦\84 â\8a¢ T1 â\9e¡[n, h] T2 →
- R 0 G L V1 V2 → R 0 G L W1 W2 → R n G (L.ⓛW1) T1 T2 →
- R n G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
- ) â\86\92 (â\88\80n,p,G,L,V1,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h] V â\86\92 â¦\83G, Lâ¦\84 â\8a¢ W1 â\9e¡[h] W2 â\86\92 â¦\83G, L.â\93\93W1â¦\84 â\8a¢ T1 â\9e¡[n, h] T2 →
- R 0 G L V1 V → R 0 G L W1 W2 → R n G (L.ⓓW1) T1 T2 →
- â¬\86*[1] V â\89\98 V2 â\86\92 R n G L (â\93\90V1.â\93\93{p}W1.T1) (â\93\93{p}W2.ⓐV2.T2)
+lemma cpm_ind (h): ∀Q:relation5 nat genv lenv term term.
+ (∀I,G,L. Q 0 G L (⓪[I]) (⓪[I])) →
+ (∀G,L,s. Q 1 G L (⋆s) (⋆(⫯[h]s))) →
+ (â\88\80n,G,K,V1,V2,W2. â\9dªG,Kâ\9d« â\8a¢ V1 â\9e¡[n,h] V2 â\86\92 Q n G K V1 V2 →
+ â\87§[1] V2 â\89\98 W2 â\86\92 Q n G (K.ⓓV1) (#0) W2
+ ) â\86\92 (â\88\80n,G,K,V1,V2,W2. â\9dªG,Kâ\9d« â\8a¢ V1 â\9e¡[n,h] V2 â\86\92 Q n G K V1 V2 →
+ â\87§[1] V2 â\89\98 W2 â\86\92 Q (↑n) G (K.ⓛV1) (#0) W2
+ ) â\86\92 (â\88\80n,I,G,K,T,U,i. â\9dªG,Kâ\9d« â\8a¢ #i â\9e¡[n,h] T â\86\92 Q n G K (#i) T →
+ â\87§[1] T â\89\98 U â\86\92 Q n G (K.â\93\98[I]) (#↑i) (U)
+ ) â\86\92 (â\88\80n,p,I,G,L,V1,V2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h] V2 â\86\92 â\9dªG,L.â\93\91[I]V1â\9d« â\8a¢ T1 â\9e¡[n,h] T2 →
+ Q 0 G L V1 V2 → Q n G (L.ⓑ[I]V1) T1 T2 → Q n G L (ⓑ[p,I]V1.T1) (ⓑ[p,I]V2.T2)
+ ) â\86\92 (â\88\80n,G,L,V1,V2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h] V2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[n,h] T2 →
+ Q 0 G L V1 V2 → Q n G L T1 T2 → Q n G L (ⓐV1.T1) (ⓐV2.T2)
+ ) â\86\92 (â\88\80n,G,L,V1,V2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[n,h] V2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[n,h] T2 →
+ Q n G L V1 V2 → Q n G L T1 T2 → Q n G L (ⓝV1.T1) (ⓝV2.T2)
+ ) â\86\92 (â\88\80n,G,L,V,T1,T,T2. â\87§[1] T â\89\98 T1 â\86\92 â\9dªG,Lâ\9d« â\8a¢ T â\9e¡[n,h] T2 →
+ Q n G L T T2 → Q n G L (+ⓓV.T1) T2
+ ) â\86\92 (â\88\80n,G,L,V,T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[n,h] T2 →
+ Q n G L T1 T2 → Q n G L (ⓝV.T1) T2
+ ) â\86\92 (â\88\80n,G,L,V1,V2,T. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[n,h] V2 →
+ Q n G L V1 V2 → Q (↑n) G L (ⓝV1.T) V2
+ ) â\86\92 (â\88\80n,p,G,L,V1,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h] V2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ W1 â\9e¡[h] W2 â\86\92 â\9dªG,L.â\93\9bW1â\9d« â\8a¢ T1 â\9e¡[n,h] T2 →
+ Q 0 G L V1 V2 → Q 0 G L W1 W2 → Q n G (L.ⓛW1) T1 T2 →
+ Q n G L (ⓐV1.ⓛ[p]W1.T1) (ⓓ[p]ⓝW2.V2.T2)
+ ) â\86\92 (â\88\80n,p,G,L,V1,V,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h] V â\86\92 â\9dªG,Lâ\9d« â\8a¢ W1 â\9e¡[h] W2 â\86\92 â\9dªG,L.â\93\93W1â\9d« â\8a¢ T1 â\9e¡[n,h] T2 →
+ Q 0 G L V1 V → Q 0 G L W1 W2 → Q n G (L.ⓓW1) T1 T2 →
+ â\87§[1] V â\89\98 V2 â\86\92 Q n G L (â\93\90V1.â\93\93[p]W1.T1) (â\93\93[p]W2.ⓐV2.T2)