-lemma cpr_ind (h): ∀R:relation4 genv lenv term term.
- (∀I,G,L. R G L (⓪{I}) (⓪{I})) →
- (â\88\80G,K,V1,V2,W2. â¦\83G, Kâ¦\84 â\8a¢ V1 â\9e¡[h] V2 â\86\92 R G K V1 V2 →
- â¬\86*[1] V2 â\89\98 W2 â\86\92 R G (K.ⓓV1) (#0) W2
- ) â\86\92 (â\88\80I,G,K,T,U,i. â¦\83G, Kâ¦\84 â\8a¢ #i â\9e¡[h] T â\86\92 R G K (#i) T →
- â¬\86*[1] T â\89\98 U â\86\92 R G (K.â\93\98{I}) (#↑i) (U)
- ) â\86\92 (â\88\80p,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¡[h] T2 →
- R G L V1 V2 → R G (L.ⓑ{I}V1) T1 T2 → R G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
- ) â\86\92 (â\88\80I,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¡[h] T2 →
- R G L V1 V2 → R G L T1 T2 → R G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
- ) â\86\92 (â\88\80G,L,V,T1,T,T2. â¦\83G, L.â\93\93Vâ¦\84 â\8a¢ T1 â\9e¡[h] T â\86\92 R G (L.â\93\93V) T1 T →
- ⬆*[1] T2 ≘ T → R G L (+ⓓV.T1) T2
- ) â\86\92 (â\88\80G,L,V,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[h] T2 â\86\92 R G L T1 T2 →
- R G L (ⓝV.T1) T2
- ) â\86\92 (â\88\80p,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¡[h] T2 →
- R G L V1 V2 → R G L W1 W2 → R G (L.ⓛW1) T1 T2 →
- R G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
- ) â\86\92 (â\88\80p,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¡[h] T2 →
- R G L V1 V → R G L W1 W2 → R G (L.ⓓW1) T1 T2 →
- â¬\86*[1] V â\89\98 V2 â\86\92 R G L (â\93\90V1.â\93\93{p}W1.T1) (â\93\93{p}W2.ⓐV2.T2)
+lemma cpr_ind (h): ∀Q:relation4 genv lenv term term.
+ (∀I,G,L. Q G L (⓪[I]) (⓪[I])) →
+ (â\88\80G,K,V1,V2,W2. â\9dªG,Kâ\9d« â\8a¢ V1 â\9e¡[h,0] V2 â\86\92 Q G K V1 V2 →
+ â\87§[1] V2 â\89\98 W2 â\86\92 Q G (K.ⓓV1) (#0) W2
+ ) â\86\92 (â\88\80I,G,K,T,U,i. â\9dªG,Kâ\9d« â\8a¢ #i â\9e¡[h,0] T â\86\92 Q G K (#i) T →
+ â\87§[1] T â\89\98 U â\86\92 Q G (K.â\93\98[I]) (#↑i) (U)
+ ) â\86\92 (â\88\80p,I,G,L,V1,V2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h,0] V2 â\86\92 â\9dªG,L.â\93\91[I]V1â\9d« â\8a¢ T1 â\9e¡[h,0] T2 →
+ Q G L V1 V2 → Q G (L.ⓑ[I]V1) T1 T2 → Q G L (ⓑ[p,I]V1.T1) (ⓑ[p,I]V2.T2)
+ ) â\86\92 (â\88\80I,G,L,V1,V2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h,0] V2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[h,0] T2 →
+ Q G L V1 V2 → Q G L T1 T2 → Q G L (ⓕ[I]V1.T1) (ⓕ[I]V2.T2)
+ ) â\86\92 (â\88\80G,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¡[h,0] T2 →
+ Q G L T T2 → Q G L (+ⓓV.T1) T2
+ ) â\86\92 (â\88\80G,L,V,T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[h,0] T2 â\86\92 Q G L T1 T2 →
+ Q G L (ⓝV.T1) T2
+ ) â\86\92 (â\88\80p,G,L,V1,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h,0] V2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ W1 â\9e¡[h,0] W2 â\86\92 â\9dªG,L.â\93\9bW1â\9d« â\8a¢ T1 â\9e¡[h,0] T2 →
+ Q G L V1 V2 → Q G L W1 W2 → Q G (L.ⓛW1) T1 T2 →
+ Q G L (ⓐV1.ⓛ[p]W1.T1) (ⓓ[p]ⓝW2.V2.T2)
+ ) â\86\92 (â\88\80p,G,L,V1,V,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h,0] V â\86\92 â\9dªG,Lâ\9d« â\8a¢ W1 â\9e¡[h,0] W2 â\86\92 â\9dªG,L.â\93\93W1â\9d« â\8a¢ T1 â\9e¡[h,0] T2 →
+ Q G L V1 V → Q G L W1 W2 → Q G (L.ⓓW1) T1 T2 →
+ â\87§[1] V â\89\98 V2 â\86\92 Q G L (â\93\90V1.â\93\93[p]W1.T1) (â\93\93[p]W2.ⓐV2.T2)