- some renaming according to the written version of basic_2

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

diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cix.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cix.ma
index d7da41971f5237fceaa8491f7c753d0d5244a50a..9532052b875e10cca7dce554d016ed07fdb454e5 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cix.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cix.ma
@@ -12,49 +12,49 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/relations/notreducible_5.ma".
+include "basic_2/notation/relations/prednotreducible_5.ma".
 include "basic_2/reduction/cir.ma".
 include "basic_2/reduction/crx.ma".
 
-(* CONTEXT-SENSITIVE EXTENDED IRREDUCIBLE TERMS *****************************)
+(* IRREDUCIBLE TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION ***************)
 
 definition cix: ∀h. sd h → relation3 genv lenv term ≝
-                λh,g,G,L,T. ⦃G, L⦄ ⊢ 𝐑[h, g]⦃T⦄ → ⊥.
+                λh,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → ⊥.
 
-interpretation "context-sensitive extended irreducibility (term)"
-   'NotReducible h g G L T = (cix h g G L T).
+interpretation "irreducibility for context-sensitive extended reduction (term)"
+   'PRedNotReducible h g G L T = (cix h g G L T).
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma cix_inv_sort: ∀h,g,G,L,k,l. deg h g k (l+1) → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃⋆k⦄ → ⊥.
+lemma cix_inv_sort: ∀h,g,G,L,k,d. deg h g k (d+1) → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃⋆k⦄ → ⊥.
 /3 width=2 by crx_sort/ qed-.
 
-lemma cix_inv_delta: â\88\80h,g,I,G,L,K,V,i. â\87©[i] L â\89¡ K.â\93\91{I}V â\86\92 â¦\83G, Lâ¦\84 â\8a¢ ð\9d\90\88[h, g]⦃#i⦄ → ⊥.
+lemma cix_inv_delta: â\88\80h,g,I,G,L,K,V,i. â¬\87[i] L â\89¡ K.â\93\91{I}V â\86\92 â¦\83G, Lâ¦\84 â\8a¢ â\9e¡[h, g] ð\9d\90\88⦃#i⦄ → ⊥.
 /3 width=4 by crx_delta/ qed-.
 
-lemma cix_inv_ri2: ∀h,g,I,G,L,V,T. ri2 I → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃②{I}V.T⦄ → ⊥.
+lemma cix_inv_ri2: ∀h,g,I,G,L,V,T. ri2 I → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃②{I}V.T⦄ → ⊥.
 /3 width=1 by crx_ri2/ qed-.
 
-lemma cix_inv_ib2: ∀h,g,a,I,G,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓑ{a,I}V.T⦄ →
-                   ⦃G, L⦄ ⊢ 𝐈[h, g]⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ 𝐈[h, g]⦃T⦄.
+lemma cix_inv_ib2: ∀h,g,a,I,G,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃ⓑ{a,I}V.T⦄ →
+                   ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ➡[h, g] 𝐈⦃T⦄.
 /4 width=1 by crx_ib2_sn, crx_ib2_dx, conj/ qed-.
 
-lemma cix_inv_bind: ∀h,g,a,I,G,L,V,T. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓑ{a,I}V.T⦄ →
-                    ∧∧ ⦃G, L⦄ ⊢ 𝐈[h, g]⦃V⦄ & ⦃G, L.ⓑ{I}V⦄ ⊢ 𝐈[h, g]⦃T⦄ & ib2 a I.
+lemma cix_inv_bind: ∀h,g,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃ⓑ{a,I}V.T⦄ →
+                    ∧∧ ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃V⦄ & ⦃G, L.ⓑ{I}V⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ & ib2 a I.
 #h #g #a * [ elim a -a ]
 #G #L #V #T #H [ elim H -H /3 width=1 by crx_ri2, or_introl/ ]
 elim (cix_inv_ib2 … H) -H /3 width=1 by and3_intro, or_introl/
 qed-.
 
-lemma cix_inv_appl: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓐV.T⦄ →
-                    ∧∧ ⦃G, L⦄ ⊢ 𝐈[h, g]⦃V⦄ & ⦃G, L⦄ ⊢ 𝐈[h, g]⦃T⦄ & 𝐒⦃T⦄.
-#h #g #G #L #V #T #HVT @and3_intro /3 width=1/
+lemma cix_inv_appl: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃ⓐV.T⦄ →
+                    ∧∧ ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ & 𝐒⦃T⦄.
+#h #g #G #L #V #T #HVT @and3_intro /3 width=1 by crx_appl_sn, crx_appl_dx/
 generalize in match HVT; -HVT elim T -T //
 * // #a * #U #T #_ #_ #H elim H -H /2 width=1 by crx_beta, crx_theta/
 qed-.
 
-lemma cix_inv_flat: ∀h,g,I,G,L,V,T. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓕ{I}V.T⦄ →
-                    ∧∧ ⦃G, L⦄ ⊢ 𝐈[h, g]⦃V⦄ & ⦃G, L⦄ ⊢ 𝐈[h, g]⦃T⦄ & 𝐒⦃T⦄ & I = Appl.
+lemma cix_inv_flat: ∀h,g,I,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃ⓕ{I}V.T⦄ →
+                    ∧∧ ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ & 𝐒⦃T⦄ & I = Appl.
 #h #g * #G #L #V #T #H
 [ elim (cix_inv_appl … H) -H /2 width=1 by and4_intro/
 | elim (cix_inv_ri2 … H) -H //
@@ -63,31 +63,31 @@ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma cix_inv_cir: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃T⦄ → ⦃G, L⦄ ⊢ 𝐈⦃T⦄.
+lemma cix_inv_cir: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄.
 /3 width=1 by crr_crx/ qed-.
 
 (* Basic properties *********************************************************)
 
-lemma cix_sort: ∀h,g,G,L,k. deg h g k 0 → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃⋆k⦄.
-#h #g #G #L #k #Hk #H elim (crx_inv_sort … H) -L #l #Hkl
-lapply (deg_mono … Hk Hkl) -h -k <plus_n_Sm #H destruct
+lemma cix_sort: ∀h,g,G,L,k. deg h g k 0 → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃⋆k⦄.
+#h #g #G #L #k #Hk #H elim (crx_inv_sort … H) -L #d #Hkd
+lapply (deg_mono … Hk Hkd) -h -k <plus_n_Sm #H destruct
 qed.
 
-lemma tix_lref: ∀h,g,G,i. ⦃G, ⋆⦄ ⊢ 𝐈[h, g]⦃#i⦄.
-#h #g #G #i #H elim (trx_inv_atom … H) -H #k #l #_ #H destruct
+lemma tix_lref: ∀h,g,G,i. ⦃G, ⋆⦄ ⊢ ➡[h, g] 𝐈⦃#i⦄.
+#h #g #G #i #H elim (trx_inv_atom … H) -H #k #d #_ #H destruct
 qed.
 
-lemma cix_gref: ∀h,g,G,L,p. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃§p⦄.
+lemma cix_gref: ∀h,g,G,L,p. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃§p⦄.
 #h #g #G #L #p #H elim (crx_inv_gref … H)
 qed.
 
-lemma cix_ib2: ∀h,g,a,I,G,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃V⦄ → ⦃G, L.ⓑ{I}V⦄ ⊢ 𝐈[h, g]⦃T⦄ →
-                               ⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓑ{a,I}V.T⦄.
+lemma cix_ib2: ∀h,g,a,I,G,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃V⦄ → ⦃G, L.ⓑ{I}V⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ →
+                               ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃ⓑ{a,I}V.T⦄.
 #h #g #a #I #G #L #V #T #HI #HV #HT #H
 elim (crx_inv_ib2 … HI H) -HI -H /2 width=1 by/
 qed.
 
-lemma cix_appl: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃V⦄ → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃T⦄ →  𝐒⦃T⦄ → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓐV.T⦄.
+lemma cix_appl: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃V⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ →  𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃ⓐV.T⦄.
 #h #g #G #L #V #T #HV #HT #H1 #H2
 elim (crx_inv_appl … H2) -H2 /2 width=1 by/
 qed.