(* *)
(**************************************************************************)
+include "ground_2/insert_eq/insert_eq_0.ma".
include "basic_2/rt_transition/cpm.ma".
(* CONTEXT-SENSITIVE PARALLEL R-TRANSITION FOR TERMS ************************)
(* Basic eliminators ********************************************************)
-lemma cpr_ind (h): ∀R:relation4 genv lenv term term.
- (∀I,G,L. R G L (⓪{I}) (⓪{I})) →
- (∀G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 → R G K V1 V2 →
- ⬆*[1] V2 ≘ W2 → R G (K.ⓓV1) (#0) W2
- ) → (∀I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ➡[h] T → R G K (#i) T →
- ⬆*[1] T ≘ U → R G (K.ⓘ{I}) (#↑i) (U)
+lemma cpr_ind (h): ∀Q:relation4 genv lenv term term.
+ (∀I,G,L. Q G L (⓪{I}) (⓪{I})) →
+ (∀G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 → Q G K V1 V2 →
+ ⬆*[1] V2 ≘ W2 → Q G (K.ⓓV1) (#0) W2
+ ) → (∀I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ➡[h] T → Q G K (#i) T →
+ ⬆*[1] T ≘ U → Q G (K.ⓘ{I}) (#↑i) (U)
) → (∀p,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[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)
+ Q G L V1 V2 → Q G (L.ⓑ{I}V1) T1 T2 → Q G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
) → (∀I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[h] T2 →
- R G L V1 V2 → R G L T1 T2 → R G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
- ) → (∀G,L,V,T1,T,T2. ⦃G, L.ⓓV⦄ ⊢ T1 ➡[h] T → R G (L.ⓓV) T1 T →
- ⬆*[1] T2 ≘ T → R G L (+ⓓV.T1) T2
- ) → (∀G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → R G L T1 T2 →
- R G L (ⓝV.T1) T2
+ Q G L V1 V2 → Q G L T1 T2 → Q G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+ ) → (∀G,L,V,T1,T,T2. ⦃G, L.ⓓV⦄ ⊢ T1 ➡[h] T → Q G (L.ⓓV) T1 T →
+ ⬆*[1] T2 ≘ T → Q G L (+ⓓV.T1) T2
+ ) → (∀G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → Q G L T1 T2 →
+ Q G L (ⓝV.T1) T2
) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[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)
+ 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)
) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h] T2 →
- R G L V1 V → R G L W1 W2 → R G (L.ⓓW1) T1 T2 →
- ⬆*[1] V ≘ V2 → R G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
+ Q G L V1 V → Q G L W1 W2 → Q G (L.ⓓW1) T1 T2 →
+ ⬆*[1] V ≘ V2 → Q G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
) →
- ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → R G L T1 T2.
-#h #R #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #G #L #T1 #T2
-* #c #HC #H generalize in match HC; -HC
-elim H -c -G -L -T1 -T2
-[ /2 width=3 by ex2_intro/
-| #G #L #s #H
- lapply (isrt_inv_01 … H) -H #H destruct
-| /3 width=4 by ex2_intro/
-| #c #G #L #V1 #V2 #W2 #_ #_ #_ #H
- elim (isrt_inv_plus_SO_dx … H) -H // #n #_ #H destruct
-| /3 width=4 by ex2_intro/
-| #cV #cT #p #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #IHV #IHT #H
- elim (isrt_O_inv_max … H) -H #HcV #HcT
- /4 width=3 by isr_inv_shift, ex2_intro/
-| #cV #cT #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #IHV #IHT #H
- elim (isrt_O_inv_max … H) -H #HcV #HcT
- /4 width=3 by isr_inv_shift, ex2_intro/
-| #cU #cT #G #L #U1 #U2 #T1 #T2 #HUT #HU12 #HT12 #IHU #IHT #H
- elim (isrt_O_inv_max … H) -H #HcV #HcT
- /3 width=3 by ex2_intro/
-| /4 width=4 by isrt_inv_plus_O_dx, ex2_intro/
-| /4 width=4 by isrt_inv_plus_O_dx, ex2_intro/
-| #c #G #L #U1 #U2 #T #_ #_ #H
- elim (isrt_inv_plus_SO_dx … H) -H // #n #_ #H destruct
-| #cV #cW #cT #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #IHV #IHW #IHT #H
- lapply (isrt_inv_plus_O_dx … H ?) -H // #H
- elim (isrt_O_inv_max … H) -H #H #HcT
- elim (isrt_O_inv_max … H) -H #HcV #HcW
- /4 width=3 by isr_inv_shift, ex2_intro/
-| #cV #cW #cT #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #IHV #IHW #IHT #H
- lapply (isrt_inv_plus_O_dx … H ?) -H // #H
- elim (isrt_O_inv_max … H) -H #H #HcT
- elim (isrt_O_inv_max … H) -H #HcV #HcW
- /4 width=4 by isr_inv_shift, ex2_intro/
+ ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → Q G L T1 T2.
+#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #G #L #T1 #T2
+@(insert_eq_0 … 0) #n #H
+@(cpm_ind … H) -G -L -T1 -T2 -n /3 width=4 by/
+[ #G #L #s #H destruct
+| #n #G #K #V1 #V2 #W2 #_ #_ #_ #H destruct
+| #n #G #L #U1 #U2 #T #_ #_ #H destruct
]
qed-.