milestone in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / rt_computation / cpxs_theq.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_theq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_theq.ma
index 4bd1dd0c14cff76f95efd2cc681cf361f6ed3266..58ee196139ccbdc68c1d9fb25f3fdc344ab856a6 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_theq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_theq.ma
@@ -21,22 +21,18 @@ include "basic_2/rt_computation/lpxs_cpxs.ma".
 
 (* Forward lemmas with head equivalence for terms ***************************)
 
-lemma cpxs_fwd_sort: ∀h,o,G,L,U,s. ⦃G, L⦄ ⊢ ⋆s ⬈*[h] U →
-                     ⋆s ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ ⋆(next h s) ⬈*[h] U.
-#h #o #G #L #U #s #H elim (cpxs_inv_sort1 … H) -H *
-[ #H destruct /2 width=1 by or_introl/
-| #n #H destruct
-  @or_intror >iter_S <(iter_n_Sm … (next h)) // (**)
-]
+lemma cpxs_fwd_sort: ∀h,G,L,X2,s1. ⦃G, L⦄ ⊢ ⋆s1 ⬈*[h] X2 → ⋆s1 ⩳ X2.
+#h #G #L #X2 #s1 #H
+elim (cpxs_inv_sort1 … H) -H #s2 #H destruct //
 qed-.
 
 (* Note: probably this is an inversion lemma *)
 (* Basic_2A1: was: cpxs_fwd_delta *)
-lemma cpxs_fwd_delta_drops: ∀h,o,I,G,L,K,V1,i. ⬇*[i] L ≘ K.ⓑ{I}V1 →
+lemma cpxs_fwd_delta_drops: ∀h,I,G,L,K,V1,i. ⬇*[i] L ≘ K.ⓑ{I}V1 →
                             ∀V2. ⬆*[↑i] V1 ≘ V2 →
-                            ∀U. ⦃G, L⦄ ⊢ #i ⬈*[h] U →
-                            #i ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ V2 ⬈*[h] U.
-#h #o #I #G #L #K #V1 #i #HLK #V2 #HV12 #U #H
+                            ∀X2. ⦃G, L⦄ ⊢ #i ⬈*[h] X2 →
+                            ∨∨ #i ⩳ X2 | ⦃G, L⦄ ⊢ V2 ⬈*[h] X2.
+#h #I #G #L #K #V1 #i #HLK #V2 #HV12 #X2 #H
 elim (cpxs_inv_lref1_drops … H) -H /2 width=1 by or_introl/
 * #I0 #K0 #V0 #U0 #HLK0 #HVU0 #HU0
 lapply (drops_mono … HLK0 … HLK) -HLK0 #H destruct
@@ -44,9 +40,9 @@ lapply (drops_mono … HLK0 … HLK) -HLK0 #H destruct
 qed-.
 
 (* Basic_1: was just: pr3_iso_beta *)
-lemma cpxs_fwd_beta: ∀h,o,p,G,L,V,W,T,U. ⦃G, L⦄ ⊢ ⓐV.ⓛ{p}W.T ⬈*[h] U →
-                     â\93\90V.â\93\9b{p}W.T ⩳[h, o] U â\88¨ â¦\83G, Lâ¦\84 â\8a¢ â\93\93{p}â\93\9dW.V.T â¬\88*[h] U.
-#h #o #p #G #L #V #W #T #U #H elim (cpxs_inv_appl1 … H) -H *
+lemma cpxs_fwd_beta: ∀h,p,G,L,V,W,T,X2. ⦃G, L⦄ ⊢ ⓐV.ⓛ{p}W.T ⬈*[h] X2 →
+                     â\88¨â\88¨ â\93\90V.â\93\9b{p}W.T ⩳ X2 | â¦\83G, Lâ¦\84 â\8a¢ â\93\93{p}â\93\9dW.V.T â¬\88*[h] X2.
+#h #p #G #L #V #W #T #X2 #H elim (cpxs_inv_appl1 … H) -H *
 [ #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or_introl/
 | #b #W0 #T0 #HT0 #HU
   elim (cpxs_inv_abst1 … HT0) -HT0 #W1 #T1 #HW1 #HT1 #H destruct
@@ -57,10 +53,10 @@ lemma cpxs_fwd_beta: ∀h,o,p,G,L,V,W,T,U. ⦃G, L⦄ ⊢ ⓐV.ⓛ{p}W.T ⬈*[h]
 ]
 qed-.
 
-lemma cpxs_fwd_theta: ∀h,o,p,G,L,V1,V,T,U. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}V.T ⬈*[h] U →
-                      ∀V2. ⬆*[1] V1 ≘ V2 → ⓐV1.ⓓ{p}V.T ⩳[h, o] U ∨
-                      â¦\83G, Lâ¦\84 â\8a¢ â\93\93{p}V.â\93\90V2.T â¬\88*[h] U.
-#h #o #p #G #L #V1 #V #T #U #H #V2 #HV12
+lemma cpxs_fwd_theta: ∀h,p,G,L,V1,V,T,X2. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}V.T ⬈*[h] X2 →
+                      ∀V2. ⬆*[1] V1 ≘ V2 →
+                      â\88¨â\88¨ â\93\90V1.â\93\93{p}V.T ⩳ X2 | â¦\83G, Lâ¦\84 â\8a¢ â\93\93{p}V.â\93\90V2.T â¬\88*[h] X2.
+#h #p #G #L #V1 #V #T #X2 #H #V2 #HV12
 elim (cpxs_inv_appl1 … H) -H *
 [ -HV12 #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or_introl/
 | #q #W #T0 #HT0 #HU
@@ -68,13 +64,13 @@ elim (cpxs_inv_appl1 … H) -H *
   [ #V3 #T3 #_ #_ #H destruct
   | #X #HT2 #H #H0 destruct
     elim (lifts_inv_bind1 … H) -H #W2 #T2 #HW2 #HT02 #H destruct
-    @or_intror @(cpxs_trans … HU) -U (**) (* explicit constructor *)
+    @or_intror @(cpxs_trans … HU) -X2 (**) (* explicit constructor *)
     @(cpxs_trans … (+ⓓV.ⓐV2.ⓛ{q}W2.T2)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T
     @(cpxs_strap2 … (ⓐV1.ⓛ{q}W.T0)) [2: /2 width=1 by cpxs_beta_dx/ ]
     /4 width=7 by cpx_zeta, lifts_bind, lifts_flat/
   ]
 | #q #V3 #V4 #V0 #T0 #HV13 #HV34 #HT0 #HU
-  @or_intror @(cpxs_trans … HU) -U (**) (* explicit constructor *)
+  @or_intror @(cpxs_trans … HU) -X2 (**) (* explicit constructor *)
   elim (cpxs_inv_abbr1_dx … HT0) -HT0 *
   [ #V5 #T5 #HV5 #HT5 #H destruct
     /6 width=9 by cpxs_lifts_bi, drops_refl, drops_drop, cpxs_flat, cpxs_bind/
@@ -88,13 +84,13 @@ elim (cpxs_inv_appl1 … H) -H *
 ]
 qed-.
 
-lemma cpxs_fwd_cast: ∀h,o,G,L,W,T,U. ⦃G, L⦄ ⊢ ⓝW.T ⬈*[h] U →
-                     ∨∨ ⓝW. T ⩳[h, o] U | ⦃G, L⦄ ⊢ T ⬈*[h] U | ⦃G, L⦄ ⊢ W ⬈*[h] U.
-#h #o #G #L #W #T #U #H
+lemma cpxs_fwd_cast: ∀h,G,L,W,T,X2. ⦃G, L⦄ ⊢ ⓝW.T ⬈*[h] X2 →
+                     ∨∨ ⓝW. T ⩳ X2 | ⦃G, L⦄ ⊢ T ⬈*[h] X2 | ⦃G, L⦄ ⊢ W ⬈*[h] X2.
+#h #G #L #W #T #X2 #H
 elim (cpxs_inv_cast1 … H) -H /2 width=1 by or3_intro1, or3_intro2/ *
 #W0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or3_intro0/
 qed-.
 
-lemma cpxs_fwd_cnx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄ →
-                    ∀U. ⦃G, L⦄ ⊢ T ⬈*[h] U → T ⩳[h, o] U.
-/3 width=4 by cpxs_inv_cnx1, tdeq_theq/ qed-.
+lemma cpxs_fwd_cnx: ∀h,G,L,T1. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T1⦄ →
+                    ∀X2. ⦃G, L⦄ ⊢ T1 ⬈*[h] X2 → T1 ⩳ X2.
+/3 width=5 by cpxs_inv_cnx1, tdeq_theq/ qed-.