notational change of lift, drop, and gget

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / reduction / crx.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/crx.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/crx.ma
index 7e7f74763e41e56a7ada0e0c73e0e2bab407431a..fe2ff03f9ba781e26297f972e2b261f1df21c49c 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/crx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/crx.ma
@@ -12,40 +12,43 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "basic_2/notation/relations/predreducible_5.ma".
 include "basic_2/static/sd.ma".
 include "basic_2/reduction/crr.ma".
 
-(* CONTEXT-SENSITIVE EXTENDED REDUCIBLE TERMS *******************************)
+(* REDUCIBLE TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION *****************)
 
+(* activate genv *)
 (* extended reducible terms *)
-inductive crx (h) (g): lenv → predicate term ≝
-| crx_sort   : ∀L,k,l. deg h g k (l+1) → crx h g L (⋆k)
-| crx_delta  : â\88\80I,L,K,V,i. â\87©[0, i] L â\89¡ K.â\93\91{I}V â\86\92 crx h g L (#i)
-| crx_appl_sn: ∀L,V,T. crx h g L V → crx h g L (ⓐV.T)
-| crx_appl_dx: ∀L,V,T. crx h g L T → crx h g L (ⓐV.T)
-| crx_ri2    : ∀I,L,V,T. ri2 I → crx h g L (②{I}V.T)
-| crx_ib2_sn : ∀a,I,L,V,T. ib2 a I → crx h g L V → crx h g L (ⓑ{a,I}V.T)
-| crx_ib2_dx : ∀a,I,L,V,T. ib2 a I → crx h g (L.ⓑ{I}V) T → crx h g L (ⓑ{a,I}V.T)
-| crx_beta   : ∀a,L,V,W,T. crx h g L (ⓐV. ⓛ{a}W.T)
-| crx_theta  : ∀a,L,V,W,T. crx h g L (ⓐV. ⓓ{a}W.T)
+inductive crx (h) (g) (G:genv): relation2 lenv term ≝
+| crx_sort   : ∀L,k,l. deg h g k (l+1) → crx h g G L (⋆k)
+| crx_delta  : â\88\80I,L,K,V,i. â¬\87[i] L â\89¡ K.â\93\91{I}V â\86\92 crx h g G L (#i)
+| crx_appl_sn: ∀L,V,T. crx h g G L V → crx h g G L (ⓐV.T)
+| crx_appl_dx: ∀L,V,T. crx h g G L T → crx h g G L (ⓐV.T)
+| crx_ri2    : ∀I,L,V,T. ri2 I → crx h g G L (②{I}V.T)
+| crx_ib2_sn : ∀a,I,L,V,T. ib2 a I → crx h g G L V → crx h g G L (ⓑ{a,I}V.T)
+| crx_ib2_dx : ∀a,I,L,V,T. ib2 a I → crx h g G (L.ⓑ{I}V) T → crx h g G L (ⓑ{a,I}V.T)
+| crx_beta   : ∀a,L,V,W,T. crx h g G L (ⓐV. ⓛ{a}W.T)
+| crx_theta  : ∀a,L,V,W,T. crx h g G L (ⓐV. ⓓ{a}W.T)
 .
 
 interpretation
-   "context-sensitive extended reducibility (term)"
-   'Reducible h g L T = (crx h g L T).
+   "reducibility for context-sensitive extended reduction (term)"
+   'PRedReducible h g G L T = (crx h g G L T).
 
 (* Basic properties *********************************************************)
 
-lemma crr_crx: ∀h,g,L,T. L ⊢ 𝐑⦃T⦄ → ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄.
-#h #g #L #T #H elim H -L -T // /2 width=1/ /2 width=4/
+lemma crr_crx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄.
+#h #g #G #L #T #H elim H -L -T
+/2 width=4 by crx_delta, crx_appl_sn, crx_appl_dx, crx_ri2, crx_ib2_sn, crx_ib2_dx, crx_beta, crx_theta/
 qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-fact crx_inv_sort_aux: ∀h,g,L,T,k. ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ → T = ⋆k →
+fact crx_inv_sort_aux: ∀h,g,G,L,T,k. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → T = ⋆k →
                        ∃l. deg h g k (l+1).
-#h #g #L #T #k0 * -L -T
-[ #L #k #l #Hkl #H destruct /2 width=2/
+#h #g #G #L #T #k0 * -L -T
+[ #L #k #l #Hkl #H destruct /2 width=2 by ex_intro/
 | #I #L #K #V #i #HLK #H destruct
 | #L #V #T #_ #H destruct
 | #L #V #T #_ #H destruct
@@ -57,14 +60,14 @@ fact crx_inv_sort_aux: ∀h,g,L,T,k. ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ → T = ⋆k
 ]
 qed-.
 
-lemma crx_inv_sort: ∀h,g,L,k. ⦃h, L⦄ ⊢ 𝐑[g]⦃⋆k⦄ → ∃l. deg h g k (l+1).
-/2 width=4 by crx_inv_sort_aux/ qed-.
+lemma crx_inv_sort: ∀h,g,G,L,k. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃⋆k⦄ → ∃l. deg h g k (l+1).
+/2 width=5 by crx_inv_sort_aux/ qed-.
 
-fact crx_inv_lref_aux: ∀h,g,L,T,i. ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ → T = #i →
-                       â\88\83â\88\83I,K,V. â\87©[0, i] L ≡ K.ⓑ{I}V.
-#h #g #L #T #j * -L -T
+fact crx_inv_lref_aux: ∀h,g,G,L,T,i. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → T = #i →
+                       â\88\83â\88\83I,K,V. â¬\87[i] L ≡ K.ⓑ{I}V.
+#h #g #G #L #T #j * -L -T
 [ #L #k #l #_ #H destruct
-| #I #L #K #V #i #HLK #H destruct /2 width=4/
+| #I #L #K #V #i #HLK #H destruct /2 width=4 by ex1_3_intro/
 | #L #V #T #_ #H destruct
 | #L #V #T #_ #H destruct
 | #I #L #V #T #_ #H destruct
@@ -75,11 +78,11 @@ fact crx_inv_lref_aux: ∀h,g,L,T,i. ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ → T = #i 
 ]
 qed-.
 
-lemma crx_inv_lref: ∀h,g,L,i. ⦃h, L⦄ ⊢ 𝐑[g]⦃#i⦄ → ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V.
-/2 width=5 by crx_inv_lref_aux/ qed-.
+lemma crx_inv_lref: ∀h,g,G,L,i. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃#i⦄ → ∃∃I,K,V. ⬇[i] L ≡ K.ⓑ{I}V.
+/2 width=6 by crx_inv_lref_aux/ qed-.
 
-fact crx_inv_gref_aux: ∀h,g,L,T,p. ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ → T = §p → ⊥.
-#h #g #L #T #q * -L -T
+fact crx_inv_gref_aux: ∀h,g,G,L,T,p. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → T = §p → ⊥.
+#h #g #G #L #T #q * -L -T
 [ #L #k #l #_ #H destruct
 | #I #L #K #V #i #HLK #H destruct
 | #L #V #T #_ #H destruct
@@ -92,22 +95,22 @@ fact crx_inv_gref_aux: ∀h,g,L,T,p. ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ → T = §p 
 ]
 qed-.
 
-lemma crx_inv_gref: ∀h,g,L,p. ⦃h, L⦄ ⊢ 𝐑[g]⦃§p⦄ → ⊥.
-/2 width=7 by crx_inv_gref_aux/ qed-.
+lemma crx_inv_gref: ∀h,g,G,L,p. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃§p⦄ → ⊥.
+/2 width=8 by crx_inv_gref_aux/ qed-.
 
-lemma trx_inv_atom: ∀h,g,I. ⦃h, ⋆⦄ ⊢ 𝐑[g]⦃⓪{I}⦄ →
+lemma trx_inv_atom: ∀h,g,I,G. ⦃G, ⋆⦄ ⊢ ➡[h, g] 𝐑⦃⓪{I}⦄ →
                     ∃∃k,l. deg h g k (l+1) & I = Sort k.
-#h #g * #i #H
-[ elim (crx_inv_sort … H) -H /2 width=4/
+#h #g * #i #G #H
+[ elim (crx_inv_sort … H) -H /2 width=4 by ex2_2_intro/
 | elim (crx_inv_lref … H) -H #I #L #V #H
-  elim (ldrop_inv_atom1 … H) -H #H destruct
+  elim (drop_inv_atom1 … H) -H #H destruct
 | elim (crx_inv_gref … H)
 ]
 qed-.
 
-fact crx_inv_ib2_aux: ∀h,g,a,I,L,W,U,T. ib2 a I → ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ →
-                      T = ⓑ{a,I}W.U → ⦃h, L⦄ ⊢ 𝐑[g]⦃W⦄ ∨ ⦃h, L.ⓑ{I}W⦄ ⊢ 𝐑[g]⦃U⦄.
-#h #g #b #J #L #W0 #U #T #HI * -L -T
+fact crx_inv_ib2_aux: ∀h,g,a,I,G,L,W,U,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ →
+                      T = ⓑ{a,I}W.U → ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃W⦄ ∨ ⦃G, L.ⓑ{I}W⦄ ⊢ ➡[h, g] 𝐑⦃U⦄.
+#h #g #b #J #G #L #W0 #U #T #HI * -L -T
 [ #L #k #l #_ #H destruct
 | #I #L #K #V #i #_ #H destruct
 | #L #V #T #_ #H destruct
@@ -115,24 +118,24 @@ fact crx_inv_ib2_aux: ∀h,g,a,I,L,W,U,T. ib2 a I → ⦃h, L⦄ ⊢ 𝐑[g]⦃T
 | #I #L #V #T #H1 #H2 destruct
   elim H1 -H1 #H destruct
   elim HI -HI #H destruct
-| #a #I #L #V #T #_ #HV #H destruct /2 width=1/
-| #a #I #L #V #T #_ #HT #H destruct /2 width=1/
+| #a #I #L #V #T #_ #HV #H destruct /2 width=1 by or_introl/
+| #a #I #L #V #T #_ #HT #H destruct /2 width=1 by or_intror/
 | #a #L #V #W #T #H destruct
 | #a #L #V #W #T #H destruct
 ]
 qed-.
 
-lemma crx_inv_ib2: ∀h,g,a,I,L,W,T. ib2 a I → ⦃h, L⦄ ⊢ 𝐑[g]⦃ⓑ{a,I}W.T⦄ →
-                   ⦃h, L⦄ ⊢ 𝐑[g]⦃W⦄ ∨ ⦃h, L.ⓑ{I}W⦄ ⊢ 𝐑[g]⦃T⦄.
+lemma crx_inv_ib2: ∀h,g,a,I,G,L,W,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃ⓑ{a,I}W.T⦄ →
+                   ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃W⦄ ∨ ⦃G, L.ⓑ{I}W⦄ ⊢ ➡[h, g] 𝐑⦃T⦄.
 /2 width=5 by crx_inv_ib2_aux/ qed-.
 
-fact crx_inv_appl_aux: ∀h,g,L,W,U,T. ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ → T = ⓐW.U →
-                       ∨∨ ⦃h, L⦄ ⊢ 𝐑[g]⦃W⦄ | ⦃h, L⦄ ⊢ 𝐑[g]⦃U⦄ | (𝐒⦃U⦄ → ⊥).
-#h #g #L #W0 #U #T * -L -T
+fact crx_inv_appl_aux: ∀h,g,G,L,W,U,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → T = ⓐW.U →
+                       ∨∨ ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃W⦄ | ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃U⦄ | (𝐒⦃U⦄ → ⊥).
+#h #g #G #L #W0 #U #T * -L -T
 [ #L #k #l #_ #H destruct
 | #I #L #K #V #i #_ #H destruct
-| #L #V #T #HV #H destruct /2 width=1/
-| #L #V #T #HT #H destruct /2 width=1/
+| #L #V #T #HV #H destruct /2 width=1 by or3_intro0/
+| #L #V #T #HT #H destruct /2 width=1 by or3_intro1/
 | #I #L #V #T #H1 #H2 destruct
   elim H1 -H1 #H destruct
 | #a #I #L #V #T #_ #_ #H destruct
@@ -144,6 +147,6 @@ fact crx_inv_appl_aux: ∀h,g,L,W,U,T. ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ → T = ⓐ
 ]
 qed-.
 
-lemma crx_inv_appl: ∀h,g,L,V,T. ⦃h, L⦄ ⊢ 𝐑[g]⦃ⓐV.T⦄ →
-                    ∨∨ ⦃h, L⦄ ⊢ 𝐑[g]⦃V⦄ | ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ | (𝐒⦃T⦄ → ⊥).
+lemma crx_inv_appl: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃ⓐV.T⦄ →
+                    ∨∨ ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃V⦄ | ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ | (𝐒⦃T⦄ → ⊥).
 /2 width=3 by crx_inv_appl_aux/ qed-.