more lemmas and some generated logical constants for them

[helm.git] / matita / matita / lib / lambda-delta / substitution / subst_defs.ma

diff --git a/matita/matita/lib/lambda-delta/substitution/subst_defs.ma b/matita/matita/lib/lambda-delta/substitution/subst_defs.ma
index 27de6a5ce4fbd9b1fa1e91f15bf2892ff93f141a..c563157f49aa68768d667954fb85bf3538ec8fed 100644 (file)
--- a/matita/matita/lib/lambda-delta/substitution/subst_defs.ma
+++ b/matita/matita/lib/lambda-delta/substitution/subst_defs.ma
@@ -16,10 +16,10 @@ include "lambda-delta/substitution/drop_defs.ma".
 inductive subst: lenv → term → nat → nat → term → Prop ≝
 | subst_sort   : ∀L,k,d,e. subst L (⋆k) d e (⋆k)
 | subst_lref_lt: ∀L,i,d,e. i < d → subst L (#i) d e (#i)
-| subst_lref_be: ∀L,K,V,U,i,d,e.
+| subst_lref_be: ∀L,K,V,U1,U2,i,d,e.
                  d ≤ i → i < d + e →
-                 ↑[0, i] K. 𝕓{Abbr} V ≡ L → subst K V d (d + e - i - 1) U →
-                 subst L (#i) d e U
+                 ↑[0, i] K. 𝕓{Abbr} V ≡ L → subst K V 0 (d + e - i - 1) U1 →
+                 ↑[0, d] U1 ≡ U2 → subst L (#i) d e U2
 | subst_lref_ge: ∀L,i,d,e. d + e ≤ i → subst L (#i) d e (#(i - e))
 | subst_bind   : ∀L,I,V1,V2,T1,T2,d,e.
                  subst L V1 d e V2 → subst (L. 𝕓{I} V1) T1 (d + 1) e T2 →
@@ -31,8 +31,61 @@ inductive subst: lenv → term → nat → nat → term → Prop ≝
 
 interpretation "telescopic substritution" 'RSubst L T1 d e T2 = (subst L T1 d e T2).
 
+(* The basic properties *****************************************************)
+
+lemma subst_lift_inv: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀L. L ⊢ ↓[d,e] T2 ≡ T1.
+#d #e #T1 #T2 #H elim H -H d e T1 T2 /2/
+#i #d #e #Hdi #L >(minus_plus_m_m i e) in ⊢ (? ? ? ? ? %) /3/
+qed.
+(*
+| subst_lref_O : ∀L,V1,V2,e. subst L V1 0 e V2 →
+                 subst (L. 𝕓{Abbr} V1) #0 0 (e + 1) V2
+| subst_lref_S : ∀L,I,V,i,T1,T2,d,e.
+                 d ≤ i → i < d + e → subst L #i d e T1 → ↑[d,1] T2 ≡ T1 →
+                 subst (L. 𝕓{I} V) #(i + 1) (d + 1) e T2
+*)
 (* The basic inversion lemmas ***********************************************)
 
+lemma subst_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ ↓[d, e] U1 ≡ U2 →
+                           ∀I,V1,T1. U1 = 𝕓{I} V1. T1 →
+                           ∃∃V2,T2. subst L V1 d e V2 & 
+                                    subst (L. 𝕓{I} V1) T1 (d + 1) e T2 &
+                                    U2 =  𝕓{I} V2. T2.
+#d #e #L #U1 #U2 #H elim H -H d e L U1 U2
+[ #L #k #d #e #I #V1 #T1 #H destruct
+| #L #i #d #e #_ #I #V1 #T1 #H destruct
+| #L #K #V #U1 #U2 #i #d #e #_ #_ #_ #_ #_ #_ #I #V1 #T1 #H destruct
+| #L #i #d #e #_ #I #V1 #T1 #H destruct
+| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #_ #_ #I #V #T #H destruct /2 width=5/
+| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #I #V #T #H destruct
+]
+qed.
+
+lemma subst_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ ↓[d, e] 𝕓{I} V1. T1 ≡ U2 →
+                       ∃∃V2,T2. subst L V1 d e V2 & 
+                                subst (L. 𝕓{I} V1) T1 (d + 1) e T2 &
+                                U2 =  𝕓{I} V2. T2.
+/2/ qed.
+
+lemma subst_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ ↓[d, e] U1 ≡ U2 →
+                           ∀I,V1,T1. U1 = 𝕗{I} V1. T1 →
+                           ∃∃V2,T2. subst L V1 d e V2 & subst L T1 d e T2 &
+                                    U2 =  𝕗{I} V2. T2.
+#d #e #L #U1 #U2 #H elim H -H d e L U1 U2
+[ #L #k #d #e #I #V1 #T1 #H destruct
+| #L #i #d #e #_ #I #V1 #T1 #H destruct
+| #L #K #V #U1 #U2 #i #d #e #_ #_ #_ #_ #_ #_ #I #V1 #T1 #H destruct
+| #L #i #d #e #_ #I #V1 #T1 #H destruct
+| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #I #V #T #H destruct
+| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #_ #_ #I #V #T #H destruct /2 width=5/
+]
+qed.
+
+lemma subst_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ ↓[d, e] 𝕗{I} V1. T1 ≡ U2 →
+                       ∃∃V2,T2. subst L V1 d e V2 & subst L T1 d e T2 &
+                                U2 =  𝕗{I} V2. T2.
+/2/ qed.
+(*
 lemma subst_inv_lref1_be_aux: ∀d,e,L,T,U. L ⊢ ↓[d, e] T ≡ U →
                               ∀i. d ≤ i → i < d + e → T = #i →
                               ∃∃K,V. ↑[0, i] K. 𝕓{Abbr} V ≡ L &
@@ -56,17 +109,4 @@ lemma subst_inv_lref1_be: ∀d,e,i,L,U. L ⊢ ↓[d, e] #i ≡ U →
                           ∃∃K,V. ↑[0, i] K. 𝕓{Abbr} V ≡ L &
                                  K ⊢ ↓[d, d + e - i - 1] V ≡ U.
 /2/ qed.
-
-(* The basic properties *****************************************************)
-
-lemma subst_lift_inv: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀L. L ⊢ ↓[d,e] T2 ≡ T1.
-#d #e #T1 #T2 #H elim H -H d e T1 T2 /2/
-#i #d #e #Hdi #L >(minus_plus_m_m i e) in ⊢ (? ? ? ? ? %) /3/
-qed.
-(*
-| subst_lref_O : ∀L,V1,V2,e. subst L V1 0 e V2 →
-                 subst (L. 𝕓{Abbr} V1) #0 0 (e + 1) V2
-| subst_lref_S : ∀L,I,V,i,T1,T2,d,e.
-                 d ≤ i → i < d + e → subst L #i d e T1 → ↑[d,1] T2 ≡ T1 →
-                 subst (L. 𝕓{I} V) #(i + 1) (d + 1) e T2
 *)