preservation of stratified vaildity through ordinary reduction and static typing

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / etc / lsubss / lsubss.etc

diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsubss/lsubss.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsubss/lsubss.etc
index 7e9751a9f986a40e92df05957cb426bf3ab8e3ca..8ca46b68ab32def70718618ee8cae48beb7d3e64 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/etc/lsubss/lsubss.etc
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsubss/lsubss.etc
@@ -12,16 +12,24 @@
 (*                                                                        *)
 (**************************************************************************)
 
+notation "hvbox( h ⊢ break term 46 L1 • ⊑ break [ term 46 g ] break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'CrSubEqS $h $g $L1 $L2 }.
+
 include "basic_2/static/ssta.ma".
+include "basic_2/computation/cprs.ma".
+include "basic_2/equivalence/cpcs.ma".
 
 (* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED STATIC TYPE ASSIGNMENT *******)
 
+(* Note: this is not transitive *)
 inductive lsubss (h:sh) (g:sd h): relation lenv ≝
 | lsubss_atom: lsubss h g (⋆) (⋆)
-| lsubss_pair: ∀I,L1,L2,W. lsubss h g L1 L2 →
-               lsubss h g (L1. ⓑ{I} W) (L2. ⓑ{I} W)
-| lsubss_abbr: ∀L1,L2,V,W,l. ⦃h, L1⦄ ⊢ V •[g, l+1] W → ⦃h, L2⦄ ⊢ V •[g, l+1] W →
-               lsubss h g L1 L2 → lsubss h g (L1. ⓓV) (L2. ⓛW)
+| lsubss_pair: ∀I,L1,L2,V. lsubss h g L1 L2 →
+               lsubss h g (L1. ⓑ{I} V) (L2. ⓑ{I} V)
+| lsubss_abbr: ∀L1,L2,V1,V2,W1,W2,l. L1 ⊢ W1 ⬌* W2 →
+               ⦃h, L1⦄ ⊢ V1 •[g] ⦃l+1, W1⦄ → ⦃h, L2⦄ ⊢ W2 •[g] ⦃l, V2⦄ →
+               lsubss h g L1 L2 → lsubss h g (L1. ⓓV1) (L2. ⓛW2)
 .
 
 interpretation
@@ -34,72 +42,74 @@ fact lsubss_inv_atom1_aux: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → L1 = ⋆ →
 #h #g #L1 #L2 * -L1 -L2
 [ //
 | #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V #W #l #_ #_ #_ #H destruct
+| #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #H destruct
 ]
-qed.
+qed-.
 
 lemma lsubss_inv_atom1: ∀h,g,L2. h ⊢ ⋆ •⊑[g] L2 → L2 = ⋆.
-/2 width=5/ qed-.
+/2 width=5 by lsubss_inv_atom1_aux/ qed-.
 
 fact lsubss_inv_pair1_aux: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 →
-                           ∀I,K1,V. L1 = K1. ⓑ{I} V →
-                           (∃∃K2. h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓑ{I} V) ∨
-                           ∃∃K2,W,l. ⦃h, K1⦄ ⊢ V •[g,l+1] W & ⦃h, K2⦄ ⊢ V •[g,l+1] W &
-                                     h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓛW & I = Abbr.
+                           ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
+                           (∃∃K2. h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓑ{I} V1) ∨
+                           ∃∃K2,W1,W2,V2,l. ⦃h, K1⦄ ⊢ V1 •[g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W2 •[g] ⦃l, V2⦄ &
+                                            K1 ⊢ W1 ⬌* W2 & h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr.
 #h #g #L1 #L2 * -L1 -L2
-[ #I #K1 #V #H destruct
-| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/
-| #L1 #L2 #V #W #l #H1VW #H2VW #HL12 #I #K1 #V1 #H destruct /3 width=7/
+[ #J #K1 #U1 #H destruct
+| #I #L1 #L2 #V #HL12 #J #K1 #U1 #H destruct /3 width=3/
+| #L1 #L2 #V1 #V2 #W1 #W2 #l #HW12 #HVW1 #HWV2 #HL12 #J #K1 #U1 #H destruct /3 width=10/
 ]
-qed.
+qed-.
 
-lemma lsubss_inv_pair1: ∀h,g,I,K1,L2,V. h ⊢ K1. ⓑ{I} V •⊑[g] L2 →
-                        (∃∃K2. h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓑ{I} V) ∨
-                        ∃∃K2,W,l. ⦃h, K1⦄ ⊢ V •[g,l+1] W & ⦃h, K2⦄ ⊢ V •[g,l+1] W &
-                                  h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓛW & I = Abbr.
-/2 width=3/ qed-.
+lemma lsubss_inv_pair1: ∀h,g,I,K1,L2,V1. h ⊢ K1. ⓑ{I} V1 •⊑[g] L2 →
+                        (∃∃K2. h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓑ{I} V1) ∨
+                        ∃∃K2,W1,W2,V2,l. ⦃h, K1⦄ ⊢ V1 •[g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W2 •[g] ⦃l, V2⦄ &
+                                         K1 ⊢ W1 ⬌* W2 & h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr.
+/2 width=3 by lsubss_inv_pair1_aux/ qed-.
 
 fact lsubss_inv_atom2_aux: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → L2 = ⋆ → L1 = ⋆.
 #h #g #L1 #L2 * -L1 -L2
 [ //
 | #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V #W #l #_ #_ #_ #H destruct
+| #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #H destruct
 ]
-qed.
+qed-.
 
 lemma lsubss_inv_atom2: ∀h,g,L1. h ⊢ L1 •⊑[g] ⋆ → L1 = ⋆.
-/2 width=5/ qed-.
+/2 width=5 by lsubss_inv_atom2_aux/ qed-.
 
 fact lsubss_inv_pair2_aux: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 →
-                           ∀I,K2,W. L2 = K2. ⓑ{I} W →
-                           (∃∃K1. h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓑ{I} W) ∨
-                           ∃∃K1,V,l. ⦃h, K1⦄ ⊢ V •[g,l+1] W & ⦃h, K2⦄ ⊢ V •[g,l+1] W &
-                                     h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓓV & I = Abst.
+                           ∀I,K2,W2. L2 = K2. ⓑ{I} W2 →
+                           (∃∃K1. h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓑ{I} W2) ∨
+                           ∃∃K1,W1,V1,V2,l. ⦃h, K1⦄ ⊢ V1 •[g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W2 •[g] ⦃l, V2⦄ &
+                                            K1 ⊢ W1 ⬌* W2 & h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst.
 #h #g #L1 #L2 * -L1 -L2
-[ #I #K2 #W #H destruct
-| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/
-| #L1 #L2 #V #W #l #H1VW #H2VW #HL12 #I #K2 #W2 #H destruct /3 width=7/
+[ #J #K2 #U2 #H destruct
+| #I #L1 #L2 #V #HL12 #J #K2 #U2 #H destruct /3 width=3/
+| #L1 #L2 #V1 #V2 #W1 #W2 #l #HW12 #HVW1 #HWV2 #HL12 #J #K2 #U2 #H destruct /3 width=10/
 ]
-qed.
+qed-.
 
-lemma lsubss_inv_pair2: ∀h,g,I,L1,K2,W. h ⊢ L1 •⊑[g] K2. ⓑ{I} W →
-                        (∃∃K1. h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓑ{I} W) ∨
-                        ∃∃K1,V,l. ⦃h, K1⦄ ⊢ V •[g,l+1] W & ⦃h, K2⦄ ⊢ V •[g,l+1] W &
-                                  h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓓV & I = Abst.
-/2 width=3/ qed-.
+lemma lsubss_inv_pair2: ∀h,g,I,L1,K2,W2. h ⊢ L1 •⊑[g] K2. ⓑ{I} W2 →
+                        (∃∃K1. h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓑ{I} W2) ∨
+                        ∃∃K1,W1,V1,V2,l. ⦃h, K1⦄ ⊢ V1 •[g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W2 •[g] ⦃l, V2⦄ &
+                                         K1 ⊢ W1 ⬌* W2 & h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst.
+/2 width=3 by lsubss_inv_pair2_aux/ qed-.
 
 (* Basic_forward lemmas *****************************************************)
 
-lemma lsubss_fwd_lsubs1: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → L1 ≼[0, |L1|] L2.
-#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
-qed-.
-
-lemma lsubss_fwd_lsubs2: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → L1 ≼[0, |L2|] L2.
+axiom lsubss_fwd_lsubx: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → L1 ⓝ⊑ L2.
+(*
 #h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
 qed-.
-
+*)
 (* Basic properties *********************************************************)
 
 lemma lsubss_refl: ∀h,g,L. h ⊢ L •⊑[g] L.
 #h #g #L elim L -L // /2 width=1/
 qed.
+
+lemma lsubss_cprs_trans: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 →
+                         ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2.
+/3 width=5 by lsubss_fwd_lsubx, lsubx_cprs_trans/
+qed-.