- we introduced the pointer_step rc in the perspective of proving

[helm.git] / matita / matita / contribs / lambda_delta / basic_2 / unfold / tpss.ma

diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/tpss.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/tpss.ma
index 16a0d3e4d9fb9a39e6b673c6e3704040a9cc3029..93916208b649a6d301c53ab4ef1e56f9d6417871 100644 (file)
--- a/matita/matita/contribs/lambda_delta/basic_2/unfold/tpss.ma
+++ b/matita/matita/contribs/lambda_delta/basic_2/unfold/tpss.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "Basic_2/substitution/tps.ma".
+include "basic_2/substitution/tps.ma".
 
 (* PARTIAL UNFOLD ON TERMS **************************************************)
 
@@ -25,41 +25,53 @@ interpretation "partial unfold (term)"
 (* Basic eliminators ********************************************************)
 
 lemma tpss_ind: ∀d,e,L,T1. ∀R:predicate term. R T1 →
-                (∀T,T2. L ⊢ T1 [d, e] ▶* T → L ⊢ T [d, e] ▶ T2 → R T → R T2) →
-                ∀T2. L ⊢ T1 [d, e] ▶* T2 → R T2.
-#d #e #L #T1 #R #HT1 #IHT1 #T2 #HT12 @(TC_star_ind … HT1 IHT1 … HT12) //
+                (∀T,T2. L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶ [d, e] T2 → R T → R T2) →
+                ∀T2. L ⊢ T1 ▶* [d, e] T2 → R T2.
+#d #e #L #T1 #R #HT1 #IHT1 #T2 #HT12
+@(TC_star_ind … HT1 IHT1 … HT12) //
+qed-.
+
+lemma tpss_ind_dx: ∀d,e,L,T2. ∀R:predicate term. R T2 →
+                   (∀T1,T. L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → R T → R T1) →
+                   ∀T1. L ⊢ T1 ▶* [d, e] T2 → R T1.
+#d #e #L #T2 #R #HT2 #IHT2 #T1 #HT12
+@(TC_star_ind_dx … HT2 IHT2 … HT12) //
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma tpss_strap: ∀L,T1,T,T2,d,e. 
-                  L ⊢ T1 [d, e] ▶ T → L ⊢ T [d, e] ▶* T2 → L ⊢ T1 [d, e] ▶* T2. 
+lemma tpss_strap1: ∀L,T1,T,T2,d,e.
+                   L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ▶* [d, e] T2. 
+/2 width=3/ qed.
+
+lemma tpss_strap2: ∀L,T1,T,T2,d,e.
+                   L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ▶* [d, e] T2. 
 /2 width=3/ qed.
 
-lemma tpss_lsubs_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ▶* T2 →
-                       ∀L2. L1 [d, e] ≼ L2 → L2 ⊢ T1 [d, e] ▶* T2.
+lemma tpss_lsubs_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶* [d, e] T2 →
+                        ∀L2. L2 ≼ [d, e] L1 → L2 ⊢ T1 ▶* [d, e] T2.
 /3 width=3/ qed.
 
-lemma tpss_refl: ∀d,e,L,T. L ⊢ T [d, e] ▶* T.
+lemma tpss_refl: ∀d,e,L,T. L ⊢ T ▶* [d, e] T.
 /2 width=1/ qed.
 
-lemma tpss_bind: ∀L,V1,V2,d,e. L ⊢ V1 [d, e] ▶* V2 →
-                 ∀I,T1,T2. L. ⓑ{I} V2 ⊢ T1 [d + 1, e] ▶* T2 →
-                 L ⊢ ⓑ{I} V1. T1 [d, e] ▶* ⓑ{I} V2. T2.
+lemma tpss_bind: ∀L,V1,V2,d,e. L ⊢ V1 ▶* [d, e] V2 →
+                 ∀a,I,T1,T2. L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 →
+                 L ⊢ ⓑ{a,I} V1. T1 ▶* [d, e] ⓑ{a,I} V2. T2.
 #L #V1 #V2 #d #e #HV12 elim HV12 -V2
-[ #V2 #HV12 #I #T1 #T2 #HT12 elim HT12 -T2
+[ #V2 #HV12 #a #I #T1 #T2 #HT12 elim HT12 -T2
   [ /3 width=5/
   | #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
   ]
-| #V #V2 #_ #HV12 #IHV #I #T1 #T2 #HT12
-  lapply (tpss_lsubs_conf … HT12 (L. ⓑ{I} V) ?) -HT12 /2 width=1/ #HT12
-  lapply (IHV … HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
+| #V #V2 #_ #HV12 #IHV #a #I #T1 #T2 #HT12
+  lapply (tpss_lsubs_trans … HT12 (L. ⓑ{I} V) ?) -HT12 /2 width=1/ #HT12
+  lapply (IHV a … HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
 ]
 qed.
 
 lemma tpss_flat: ∀L,I,V1,V2,T1,T2,d,e.
-                 L ⊢ V1 [d, e] ▶ * V2 → L ⊢ T1 [d, e] ▶* T2 →
-                 L ⊢ ⓕ{I} V1. T1 [d, e] ▶* ⓕ{I} V2. T2.
+                 L ⊢ V1 ▶* [d, e] V2 → L ⊢ T1 ▶* [d, e] T2 →
+                 L ⊢ ⓕ{I} V1. T1 ▶* [d, e] ⓕ{I} V2. T2.
 #L #I #V1 #V2 #T1 #T2 #d #e #HV12 elim HV12 -V2
 [ #V2 #HV12 #HT12 elim HT12 -T2
   [ /3 width=1/
@@ -70,9 +82,9 @@ lemma tpss_flat: ∀L,I,V1,V2,T1,T2,d,e.
 ]
 qed.
 
-lemma tpss_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 [d1, e1] ▶* T2 →
+lemma tpss_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 ▶* [d1, e1] T2 →
                  ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
-                 L ⊢ T1 [d2, e2] ▶* T2.
+                 L ⊢ T1 ▶* [d2, e2] T2.
 #L #T1 #T2 #d1 #e1 #H #d1 #d2 #Hd21 #Hde12 @(tpss_ind … H) -T2
 [ //
 | #T #T2 #_ #HT12 #IHT
@@ -81,7 +93,7 @@ lemma tpss_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 [d1, e1] ▶* T2 →
 qed.
 
 lemma tpss_weak_top: ∀L,T1,T2,d,e.
-                     L ⊢ T1 [d, e] ▶* T2 → L ⊢ T1 [d, |L| - d] ▶* T2.
+                     L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶* [d, |L| - d] T2.
 #L #T1 #T2 #d #e #H @(tpss_ind … H) -T2
 [ //
 | #T #T2 #_ #HT12 #IHT
@@ -90,16 +102,21 @@ lemma tpss_weak_top: ∀L,T1,T2,d,e.
 qed.
 
 lemma tpss_weak_all: ∀L,T1,T2,d,e.
-                     L ⊢ T1 [d, e] ▶* T2 → L ⊢ T1 [0, |L|] ▶* T2.
+                     L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶* [0, |L|] T2.
 #L #T1 #T2 #d #e #HT12
 lapply (tpss_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12
 lapply (tpss_weak_top … HT12) //
 qed.
 
+lemma tpss_append: ∀K,T1,T2,d,e. K ⊢ T1 ▶* [d, e] T2 →
+                   ∀L. L @@ K ⊢ T1 ▶* [d, e] T2.
+#K #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // /3 width=3/
+qed.
+
 (* Basic inversion lemmas ***************************************************)
 
 (* Note: this can be derived from tpss_inv_atom1 *)
-lemma tpss_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k [d, e] ▶* T2 → T2 = ⋆k.
+lemma tpss_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k ▶* [d, e] T2 → T2 = ⋆k.
 #L #T2 #k #d #e #H @(tpss_ind … H) -T2
 [ //
 | #T #T2 #_ #HT2 #IHT destruct
@@ -108,7 +125,7 @@ lemma tpss_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k [d, e] ▶* T2 → T2 = ⋆k.
 qed-.
 
 (* Note: this can be derived from tpss_inv_atom1 *)
-lemma tpss_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p [d, e] ▶* T2 → T2 = §p.
+lemma tpss_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p ▶* [d, e] T2 → T2 = §p.
 #L #T2 #p #d #e #H @(tpss_ind … H) -T2
 [ //
 | #T #T2 #_ #HT2 #IHT destruct
@@ -116,20 +133,20 @@ lemma tpss_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p [d, e] ▶* T2 → T2 = §p.
 ]
 qed-.
 
-lemma tpss_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓑ{I} V1. T1 [d, e] ▶* U2 →
-                      ∃∃V2,T2. L ⊢ V1 [d, e] ▶* V2 & 
-                               L. ⓑ{I} V2 ⊢ T1 [d + 1, e] ▶* T2 &
-                               U2 =  ⓑ{I} V2. T2.
-#d #e #L #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2
+lemma tpss_inv_bind1: ∀d,e,L,a,I,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ▶* [d, e] U2 →
+                      ∃∃V2,T2. L ⊢ V1 ▶* [d, e] V2 & 
+                               L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 &
+                               U2 = ⓑ{a,I} V2. T2.
+#d #e #L #a #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2
 [ /2 width=5/
 | #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
   elim (tps_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H
-  lapply (tpss_lsubs_conf … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
+  lapply (tpss_lsubs_trans … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
 ]
 qed-.
 
-lemma tpss_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 [d, e] ▶* U2 →
-                      ∃∃V2,T2. L ⊢ V1 [d, e] ▶* V2 & L ⊢ T1 [d, e] ▶* T2 &
+lemma tpss_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ▶* [d, e] U2 →
+                      ∃∃V2,T2. L ⊢ V1 ▶* [d, e] V2 & L ⊢ T1 ▶* [d, e] T2 &
                                U2 =  ⓕ{I} V2. T2.
 #d #e #L #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2
 [ /2 width=5/
@@ -138,9 +155,28 @@ lemma tpss_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 [d, e] ▶* U2 
 ]
 qed-.
 
-lemma tpss_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 [d, 0] ▶* T2 → T1 = T2.
+lemma tpss_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 ▶* [d, 0] T2 → T1 = T2.
 #L #T1 #T2 #d #H @(tpss_ind … H) -T2
 [ //
 | #T #T2 #_ #HT2 #IHT <(tps_inv_refl_O2 … HT2) -HT2 //
 ]
 qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma tpss_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → #{T1} ≤ #{T2}.
+#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1
+lapply (tps_fwd_tw … HT2) -HT2 #HT2
+@(transitive_le … IHT1) //
+qed-.
+
+lemma tpss_fwd_shift1: ∀L,L1,T1,T,d,e. L ⊢ L1 @@ T1 ▶*[d, e] T →
+                       ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
+#L #L1 #T1 #T #d #e #H @(tpss_ind … H) -T
+[ /2 width=4/
+| #T #X #_ #H0 * #L0 #T0 #HL10 #H destruct
+  elim (tps_fwd_shift1 … H0) -H0 #L2 #T2 #HL02 #H destruct /2 width=4/
+]
+qed-.
+  
\ No newline at end of file