some renaming and some typos corrected ...

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / dynamic / lsubsv.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv.ma
index 0d39f32ec5d375d4f56b1928381111ff936c60eb..b2dabf77bef5eec6ee01f7f8388be4f056fcc0ef 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv.ma
@@ -22,8 +22,9 @@ inductive lsubsv (h) (g) (G): relation lenv ≝
 | lsubsv_atom: lsubsv h g G (⋆) (⋆)
 | lsubsv_pair: ∀I,L1,L2,V. lsubsv h g G L1 L2 →
                lsubsv h g G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
-| lsubsv_abbr: ∀L1,L2,W,V,W1,V2,l. ⦃G, L1⦄ ⊢ ⓝW.V ¡[h, g] → ⦃G, L2⦄ ⊢ W ¡[h, g] →
-               ⦃G, L1⦄ ⊢ V •[h, g] ⦃l+1, W1⦄ → ⦃G, L2⦄ ⊢ W •[h, g] ⦃l, V2⦄ →
+| lsubsv_beta: ∀L1,L2,W,V,l. ⦃G, L1⦄ ⊢ W ¡[h, g] → ⦃G, L1⦄ ⊢ V ¡[h, g] →
+               scast h g l G L1 V W → ⦃G, L2⦄ ⊢ W ¡[h, g] →
+               ⦃G, L1⦄ ⊢ V ▪[h, g] l+1 → ⦃G, L2⦄ ⊢ W ▪[h, g] l →
                lsubsv h g G L1 L2 → lsubsv h g G (L1.ⓓⓝW.V) (L2.ⓛW)
 .
 
@@ -33,83 +34,133 @@ interpretation
 
 (* Basic inversion lemmas ***************************************************)
 
-fact lsubsv_inv_atom1_aux: ∀h,g,G,L1,L2. G ⊢ L1 ¡⊑[h, g] L2 → L1 = ⋆ → L2 = ⋆.
+fact lsubsv_inv_atom1_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 → L1 = ⋆ → L2 = ⋆.
 #h #g #G #L1 #L2 * -L1 -L2
 [ //
 | #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #W #V #V1 #V2 #l #_ #_ #_ #_ #_ #H destruct
+| #L1 #L2 #W #V #l #_ #_ #_ #_ #_ #_ #_ #H destruct
 ]
 qed-.
 
-lemma lsubsv_inv_atom1: ∀h,g,G,L2. G ⊢ ⋆ ¡⊑[h, g] L2 → L2 = ⋆.
+lemma lsubsv_inv_atom1: ∀h,g,G,L2. G ⊢ ⋆ ⫃¡[h, g] L2 → L2 = ⋆.
 /2 width=6 by lsubsv_inv_atom1_aux/ qed-.
 
-fact lsubsv_inv_pair1_aux: ∀h,g,G,L1,L2. G ⊢ L1 ¡⊑[h, g] L2 →
+fact lsubsv_inv_pair1_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
                            ∀I,K1,X. L1 = K1.ⓑ{I}X →
-                           (∃∃K2. G ⊢ K1 ¡⊑[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
-                           ∃∃K2,W,V,W1,V2,l. ⦃G, K1⦄ ⊢ X ¡[h, g] & ⦃G, K2⦄ ⊢ W ¡[h, g] &
-                                             ⦃G, K1⦄ ⊢ V •[h, g] ⦃l+1, W1⦄ & ⦃G, K2⦄ ⊢ W •[h, g] ⦃l, V2⦄ &
-                                             G ⊢ K1 ¡⊑[h, g] K2 &
-                                             I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
+                           (∃∃K2. G ⊢ K1 ⫃¡[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
+                           ∃∃K2,W,V,l. ⦃G, K1⦄ ⊢ W ¡[h, g] & ⦃G, K1⦄ ⊢ V ¡[h, g] &
+                                       scast h g l G K1 V W & ⦃G, K2⦄ ⊢ W ¡[h, g] &
+                                       ⦃G, K1⦄ ⊢ V ▪[h, g] l+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] l &
+                                       G ⊢ K1 ⫃¡[h, g] K2 &
+                                       I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
 #h #g #G #L1 #L2 * -L1 -L2
 [ #J #K1 #X #H destruct
 | #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3/
-| #L1 #L2 #W #V #W1 #V2 #l #HV #HW #HW1 #HV2 #HL12 #J #K1 #X #H destruct /3 width=12/
+| #L1 #L2 #W #V #l #H1W #HV #HVW #H2W #H1l #H2l #HL12 #J #K1 #X #H destruct /3 width=13/
 ]
 qed-.
 
-lemma lsubsv_inv_pair1: ∀h,g,I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ¡⊑[h, g] L2 →
-                        (∃∃K2. G ⊢ K1 ¡⊑[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
-                        ∃∃K2,W,V,W1,V2,l. ⦃G, K1⦄ ⊢ X ¡[h, g] & ⦃G, K2⦄ ⊢ W ¡[h, g] &
-                                          ⦃G, K1⦄ ⊢ V •[h, g] ⦃l+1, W1⦄ & ⦃G, K2⦄ ⊢ W •[h, g] ⦃l, V2⦄ &
-                                          G ⊢ K1 ¡⊑[h, g] K2 &
-                                          I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
+lemma lsubsv_inv_pair1: ∀h,g,I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⫃¡[h, g] L2 →
+                        (∃∃K2. G ⊢ K1 ⫃¡[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
+                        ∃∃K2,W,V,l. ⦃G, K1⦄ ⊢ W ¡[h, g] & ⦃G, K1⦄ ⊢ V ¡[h, g] &
+                                    scast h g l G K1 V W & ⦃G, K2⦄ ⊢ W ¡[h, g] &
+                                    ⦃G, K1⦄ ⊢ V ▪[h, g] l+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] l &
+                                    G ⊢ K1 ⫃¡[h, g] K2 &
+                                    I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
 /2 width=3 by lsubsv_inv_pair1_aux/ qed-.
 
-fact lsubsv_inv_atom2_aux: ∀h,g,G,L1,L2. G ⊢ L1 ¡⊑[h, g] L2 → L2 = ⋆ → L1 = ⋆.
+fact lsubsv_inv_atom2_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 → L2 = ⋆ → L1 = ⋆.
 #h #g #G #L1 #L2 * -L1 -L2
 [ //
 | #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #W #V #V1 #V2 #l #_ #_ #_ #_ #_ #H destruct
+| #L1 #L2 #W #V #l #_ #_ #_ #_ #_ #_ #_ #H destruct
 ]
 qed-.
 
-lemma lsubsv_inv_atom2: ∀h,g,G,L1. G ⊢ L1 ¡⊑[h, g] ⋆ → L1 = ⋆.
+lemma lsubsv_inv_atom2: ∀h,g,G,L1. G ⊢ L1 ⫃¡[h, g] ⋆ → L1 = ⋆.
 /2 width=6 by lsubsv_inv_atom2_aux/ qed-.
 
-fact lsubsv_inv_pair2_aux: ∀h,g,G,L1,L2. G ⊢ L1 ¡⊑[h, g] L2 →
+fact lsubsv_inv_pair2_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
                            ∀I,K2,W. L2 = K2.ⓑ{I}W →
-                           (∃∃K1. G ⊢ K1 ¡⊑[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
-                           ∃∃K1,V,W1,V2,l. ⦃G, K1⦄ ⊢ ⓝW.V ¡[h, g] & ⦃G, K2⦄ ⊢ W ¡[h, g] &
-                                           ⦃G, K1⦄ ⊢ V •[h, g] ⦃l+1, W1⦄ & ⦃G, K2⦄ ⊢ W •[h, g] ⦃l, V2⦄ &
-                                           G ⊢ K1 ¡⊑[h, g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
+                           (∃∃K1. G ⊢ K1 ⫃¡[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
+                           ∃∃K1,V,l. ⦃G, K1⦄ ⊢ W ¡[h, g] & ⦃G, K1⦄ ⊢ V ¡[h, g] &
+                                     scast h g l G K1 V W & ⦃G, K2⦄ ⊢ W ¡[h, g] &
+                                     ⦃G, K1⦄ ⊢ V ▪[h, g] l+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] l &
+                                     G ⊢ K1 ⫃¡[h, g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
 #h #g #G #L1 #L2 * -L1 -L2
 [ #J #K2 #U #H destruct
 | #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3/
-| #L1 #L2 #W #V #W1 #V2 #l #HV #HW #HW1 #HV2 #HL12 #J #K2 #U #H destruct /3 width=10/
+| #L1 #L2 #W #V #l #H1W #HV #HVW #H2W #H1l #H2l #HL12 #J #K2 #U #H destruct /3 width=10/
 ]
 qed-.
 
-lemma lsubsv_inv_pair2: ∀h,g,I,G,L1,K2,W. G ⊢ L1 ¡⊑[h, g] K2.ⓑ{I}W →
-                        (∃∃K1. G ⊢ K1 ¡⊑[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
-                        ∃∃K1,V,W1,V2,l. ⦃G, K1⦄ ⊢ ⓝW.V ¡[h, g] & ⦃G, K2⦄ ⊢ W ¡[h, g] &
-                                        ⦃G, K1⦄ ⊢ V •[h, g] ⦃l+1, W1⦄ & ⦃G, K2⦄ ⊢ W •[h, g] ⦃l, V2⦄ &
-                                        G ⊢ K1 ¡⊑[h, g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
+lemma lsubsv_inv_pair2: ∀h,g,I,G,L1,K2,W. G ⊢ L1 ⫃¡[h, g] K2.ⓑ{I}W →
+                        (∃∃K1. G ⊢ K1 ⫃¡[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
+                        ∃∃K1,V,l. ⦃G, K1⦄ ⊢ W ¡[h, g] & ⦃G, K1⦄ ⊢ V ¡[h, g] &
+                                  scast h g l G K1 V W & ⦃G, K2⦄ ⊢ W ¡[h, g] &
+                                  ⦃G, K1⦄ ⊢ V ▪[h, g] l+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] l &
+                                  G ⊢ K1 ⫃¡[h, g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
 /2 width=3 by lsubsv_inv_pair2_aux/ qed-.
 
-(* Basic_forward lemmas *****************************************************)
+(* Basic forward lemmas *****************************************************)
 
-lemma lsubsv_fwd_lsubr: ∀h,g,G,L1,L2. G ⊢ L1 ¡⊑[h, g] L2 → L1 ⊑ L2.
+lemma lsubsv_fwd_lsubr: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 → L1 ⫃ L2.
 #h #g #G #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma lsubsv_refl: ∀h,g,G,L. G ⊢ L ¡⊑[h, g] L.
+lemma lsubsv_refl: ∀h,g,G,L. G ⊢ L ⫃¡[h, g] L.
 #h #g #G #L elim L -L // /2 width=1/
 qed.
 
-lemma lsubsv_cprs_trans: ∀h,g,G,L1,L2. G ⊢ L1 ¡⊑[h, g] L2 →
+lemma lsubsv_cprs_trans: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
                          ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ➡* T2 → ⦃G, L1⦄ ⊢ T1 ➡* T2.
 /3 width=6 by lsubsv_fwd_lsubr, lsubr_cprs_trans/
 qed-.
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsubsv_drop_O1_conf: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
+                           ∀K1,s,e. ⇩[s, 0, e] L1 ≡ K1 →
+                           ∃∃K2. G ⊢ K1 ⫃¡[h, g] K2 & ⇩[s, 0, e] L2 ≡ K2.
+#h #g #G #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #_ #IHL12 #K1 #s #e #H
+  elim (drop_inv_O1_pair1 … H) -H * #He #HLK1
+  [ destruct
+    elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+    <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubsv_pair, drop_pair, ex2_intro/
+  | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+| #L1 #L2 #W #V #l #H1W #HV #HVW #H2W #H1l #H2l #_ #IHL12 #K1 #s #e #H
+  elim (drop_inv_O1_pair1 … H) -H * #He #HLK1
+  [ destruct
+    elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+    <(drop_inv_O2 … H) in HL12; -H /3 width=4 by lsubsv_beta, drop_pair, ex2_intro/
+  | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+]
+qed-.
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsubsv_drop_O1_trans: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
+                            ∀K2,s, e. ⇩[s, 0, e] L2 ≡ K2 →
+                            ∃∃K1. G ⊢ K1 ⫃¡[h, g] K2 & ⇩[s, 0, e] L1 ≡ K1.
+#h #g #G #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #_ #IHL12 #K2 #s #e #H
+  elim (drop_inv_O1_pair1 … H) -H * #He #HLK2
+  [ destruct
+    elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+    <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubsv_pair, drop_pair, ex2_intro/
+  | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+| #L1 #L2 #W #V #l #H1W #HV #HVW #H2W #H1l #H2l #_ #IHL12 #K2 #s #e #H
+  elim (drop_inv_O1_pair1 … H) -H * #He #HLK2
+  [ destruct
+    elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+    <(drop_inv_O2 … H) in HL12; -H /3 width=4 by lsubsv_beta, drop_pair, ex2_intro/
+  | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+]
+qed-.