- probe: critical bug fixed (all objects were deleted due to wrong test)

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / static / lsuba.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lsuba.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lsuba.ma
index eeeb35651033f4d6af07743327ecaccc56828813..f3e13c10544761b187a3ff1a0ade2c28c2779648 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/static/lsuba.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/lsuba.ma
@@ -12,21 +12,22 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/relations/crsubeqa_2.ma".
+include "basic_2/notation/relations/lrsubeqa_2.ma".
+include "basic_2/substitution/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)
+| lsuba_pair: ∀I,L1,L2,V. lsuba L1 L2 → lsuba (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+| lsuba_abbr: ∀L1,L2,W,V,A. L1 ⊢ ⓝW.V ⁝ A → L2 ⊢ W ⁝ A →
+              lsuba L1 L2 → lsuba (L1.ⓓⓝW.V) (L2.ⓛW)
 .
 
 interpretation
   "local environment refinement (atomic arity assigment)"
-  'CrSubEqA L1 L2 = (lsuba L1 L2).
+  'LRSubEqA L1 L2 = (lsuba L1 L2).
 
 (* Basic inversion lemmas ***************************************************)
 
@@ -34,57 +35,63 @@ fact lsuba_inv_atom1_aux: ∀L1,L2. L1 ⁝⊑ L2 → L1 = ⋆ → L2 = ⋆.
 #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: ∀L2. ⋆ ⁝⊑ L2 → L2 = ⋆.
-/2 width=3/ qed-.
+/2 width=3 by lsuba_inv_atom1_aux/ 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.
+fact lsuba_inv_pair1_aux: ∀L1,L2. L1 ⁝⊑ L2 → ∀I,K1,X. L1 = K1.ⓑ{I}X →
+                          (∃∃K2. K1 ⁝⊑ K2 & L2 = K2.ⓑ{I}X) ∨
+                          ∃∃K2,W,V,A. K1 ⊢ ⓝW.V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 &
+                                      I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
 #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/
+[ #J #K1 #X #H destruct
+| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3/
+| #L1 #L2 #W #V #A #HV #HW #HL12 #J #K1 #X #H destruct /3 width=9/
 ]
-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,K1,L2,X. K1.ⓑ{I}X ⁝⊑ L2 →
+                       (∃∃K2. K1 ⁝⊑ K2 & L2 = K2.ⓑ{I}X) ∨
+                       ∃∃K2,W,V,A. K1 ⊢ ⓝW.V ⁝ A & K2 ⊢ W ⁝ A & 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
 [ //
 | #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: ∀L1. L1 ⁝⊑ ⋆ → L1 = ⋆.
-/2 width=3/ qed-.
+/2 width=3 by lsuba_inv_atom2_aux/ 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.
+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 ⊢ ⓝW.V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 &
+                                    I = Abst & L1 = K1.ⓓⓝW.V.
 #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/
+[ #J #K2 #U #H destruct
+| #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3/
+| #L1 #L2 #W #V #A #HV #HW #HL12 #J #K2 #U #H destruct /3 width=7/
 ]
-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 ⊢ ⓝW.V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 &
+                                 I = Abst & L1 = K1.ⓓⓝW.V.
+/2 width=3 by lsuba_inv_pair2_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
 
-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_fwd_lsubr: ∀L1,L2. L1 ⁝⊑ L2 → L1 ⊑ L2.
+#L1 #L2 #H elim H -L1 -L2 // /2 width=1/
+qed-.
 
 (* Basic properties *********************************************************)