X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fstatic%2Flsuba.ma;h=be272d27aad762dcae767606b899a7a3cb35fecc;hb=5a35a42e23b2f343f0241eeb6648bf05f31720db;hp=eeeb35651033f4d6af07743327ecaccc56828813;hpb=65008df95049eb835941ffea1aa682c9253c4c2b;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lsuba.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lsuba.ma
index eeeb35651..be272d27a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/lsuba.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/lsuba.ma
@@ -12,82 +12,133 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/relations/crsubeqa_2.ma".
+include "basic_2/notation/relations/lrsubeqa_3.ma".
+include "basic_2/static/lsubr.ma".
 include "basic_2/static/aaa.ma".
 
 (* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************)
 
-inductive lsuba: relation lenv ≝
-| lsuba_atom: lsuba (⋆) (⋆)
-| lsuba_pair: ∀I,L1,L2,V. lsuba L1 L2 → lsuba (L1. ⓑ{I} V) (L2. ⓑ{I} V)
-| lsuba_abbr: ∀L1,L2,V,W,A. L1 ⊢ V ⁝ A → L2 ⊢ W ⁝ A →
-              lsuba L1 L2 → lsuba (L1. ⓓV) (L2. ⓛW)
+inductive lsuba (G:genv): relation lenv ≝
+| lsuba_atom: lsuba G (⋆) (⋆)
+| lsuba_pair: ∀I,L1,L2,V. lsuba G L1 L2 → lsuba G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+| lsuba_beta: ∀L1,L2,W,V,A. ⦃G, L1⦄ ⊢ ⓝW.V ⁝ A → ⦃G, L2⦄ ⊢ W ⁝ A →
+              lsuba G L1 L2 → lsuba G (L1.ⓓⓝW.V) (L2.ⓛW)
 .
 
 interpretation
-  "local environment refinement (atomic arity assigment)"
-  'CrSubEqA L1 L2 = (lsuba L1 L2).
+  "local environment refinement (atomic arity assignment)"
+  'LRSubEqA G L1 L2 = (lsuba G L1 L2).
 
 (* Basic inversion lemmas ***************************************************)
 
-fact lsuba_inv_atom1_aux: ∀L1,L2. L1 ⁝⊑ L2 → L1 = ⋆ → L2 = ⋆.
-#L1 #L2 * -L1 -L2
+fact lsuba_inv_atom1_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → L1 = ⋆ → L2 = ⋆.
+#G #L1 #L2 * -L1 -L2
 [ //
 | #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V #W #A #_ #_ #_ #H destruct
+| #L1 #L2 #W #V #A #_ #_ #_ #H destruct
 ]
-qed.
+qed-.
+
+lemma lsuba_inv_atom1: ∀G,L2. G ⊢ ⋆ ⫃⁝ L2 → L2 = ⋆.
+/2 width=4 by lsuba_inv_atom1_aux/ qed-.
 
-lemma lsuba_inv_atom1: ∀L2. ⋆ ⁝⊑ L2 → L2 = ⋆.
-/2 width=3/ qed-.
-
-fact lsuba_inv_pair1_aux: ∀L1,L2. L1 ⁝⊑ L2 → ∀I,K1,V. L1 = K1. ⓑ{I} V →
-                          (∃∃K2. K1 ⁝⊑ K2 & L2 = K2. ⓑ{I} V) ∨
-                          ∃∃K2,W,A. K1 ⊢ V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 &
-                                    L2 = K2. ⓛW & I = Abbr.
-#L1 #L2 * -L1 -L2
-[ #I #K1 #V #H destruct
-| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/
-| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K1 #V #H destruct /3 width=7/
+fact lsuba_inv_pair1_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀I,K1,X. L1 = K1.ⓑ{I}X →
+                          (∃∃K2. G ⊢ K1 ⫃⁝ K2 & L2 = K2.ⓑ{I}X) ∨
+                          ∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
+                                      G ⊢ K1 ⫃⁝ K2 & I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
+#G #L1 #L2 * -L1 -L2
+[ #J #K1 #X #H destruct
+| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #W #V #A #HV #HW #HL12 #J #K1 #X #H destruct /3 width=9 by or_intror, ex6_4_intro/
 ]
-qed.
+qed-.
 
-lemma lsuba_inv_pair1: ∀I,K1,L2,V. K1. ⓑ{I} V ⁝⊑ L2 →
-                       (∃∃K2. K1 ⁝⊑ K2 & L2 = K2. ⓑ{I} V) ∨
-                       ∃∃K2,W,A. K1 ⊢ V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 &
-                                 L2 = K2. ⓛW & I = Abbr.
-/2 width=3/ qed-.
+lemma lsuba_inv_pair1: ∀I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⫃⁝ L2 →
+                       (∃∃K2. G ⊢ K1 ⫃⁝ K2 & L2 = K2.ⓑ{I}X) ∨
+                       ∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
+                                   I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
+/2 width=3 by lsuba_inv_pair1_aux/ qed-.
 
-fact lsuba_inv_atom2_aux: ∀L1,L2. L1 ⁝⊑ L2 → L2 = ⋆ → L1 = ⋆.
-#L1 #L2 * -L1 -L2
+fact lsuba_inv_atom2_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → L2 = ⋆ → L1 = ⋆.
+#G #L1 #L2 * -L1 -L2
 [ //
 | #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V #W #A #_ #_ #_ #H destruct
+| #L1 #L2 #W #V #A #_ #_ #_ #H destruct
 ]
-qed.
+qed-.
+
+lemma lsubc_inv_atom2: ∀G,L1. G ⊢ L1 ⫃⁝ ⋆ → L1 = ⋆.
+/2 width=4 by lsuba_inv_atom2_aux/ qed-.
 
-lemma lsubc_inv_atom2: ∀L1. L1 ⁝⊑ ⋆ → L1 = ⋆.
-/2 width=3/ qed-.
-
-fact lsuba_inv_pair2_aux: ∀L1,L2. L1 ⁝⊑ L2 → ∀I,K2,W. L2 = K2. ⓑ{I} W →
-                          (∃∃K1. K1 ⁝⊑ K2 & L1 = K1. ⓑ{I} W) ∨
-                          ∃∃K1,V,A. K1 ⊢ V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 &
-                                    L1 = K1. ⓓV & I = Abst.
-#L1 #L2 * -L1 -L2
-[ #I #K2 #W #H destruct
-| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/
-| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K2 #W #H destruct /3 width=7/
+fact lsuba_inv_pair2_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀I,K2,W. L2 = K2.ⓑ{I}W →
+                          (∃∃K1. G ⊢ K1 ⫃⁝ K2 & L1 = K1.ⓑ{I}W) ∨
+                          ∃∃K1,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
+                                    G ⊢ K1 ⫃⁝ K2 & I = Abst & L1 = K1.ⓓⓝW.V.
+#G #L1 #L2 * -L1 -L2
+[ #J #K2 #U #H destruct
+| #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #W #V #A #HV #HW #HL12 #J #K2 #U #H destruct /3 width=7 by or_intror, ex5_3_intro/
 ]
-qed.
+qed-.
 
-lemma lsuba_inv_pair2: ∀I,L1,K2,W. L1 ⁝⊑ K2. ⓑ{I} W →
-                       (∃∃K1. K1 ⁝⊑ K2 & L1 = K1. ⓑ{I} W) ∨
-                       ∃∃K1,V,A. K1 ⊢ V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 &
-                                 L1 = K1. ⓓV & I = Abst.
-/2 width=3/ qed-.
+lemma lsuba_inv_pair2: ∀I,G,L1,K2,W. G ⊢ L1 ⫃⁝ K2.ⓑ{I}W →
+                       (∃∃K1. G ⊢ K1 ⫃⁝ K2 & L1 = K1.ⓑ{I}W) ∨
+                       ∃∃K1,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
+                                 I = Abst & L1 = K1.ⓓⓝW.V.
+/2 width=3 by lsuba_inv_pair2_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lsuba_fwd_lsubr: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → L1 ⫃ L2.
+#G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_pair, lsubr_beta/
+qed-.
 
 (* Basic properties *********************************************************)
 
-lemma lsuba_refl: ∀L. L ⁝⊑ L.
-#L elim L -L // /2 width=1/
+lemma lsuba_refl: ∀G,L. G ⊢ L ⫃⁝ L.
+#G #L elim L -L /2 width=1 by lsuba_atom, lsuba_pair/
 qed.
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsuba_drop_O1_conf: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀K1,s,e. ⬇[s, 0, e] L1 ≡ K1 →
+                          ∃∃K2. G ⊢ K1 ⫃⁝ K2 & ⬇[s, 0, e] L2 ≡ K2.
+#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 lsuba_pair, drop_pair, ex2_intro/
+  | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+| #L1 #L2 #W #V #A #HV #HW #_ #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 lsuba_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 lsuba_drop_O1_trans: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀K2,s,e. ⬇[s, 0, e] L2 ≡ K2 →
+                           ∃∃K1. G ⊢ K1 ⫃⁝ K2 & ⬇[s, 0, e] L1 ≡ K1.
+#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 lsuba_pair, drop_pair, ex2_intro/
+  | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+| #L1 #L2 #W #V #A #HV #HW #_ #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 lsuba_beta, drop_pair, ex2_intro/
+  | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+]
+qed-.