- the theory of parallel substitution of local environments (ltps) is ready

[helm.git] / matita / matita / contribs / lambda-delta / Basic-2 / substitution / tps.ma

diff --git a/matita/matita/contribs/lambda-delta/Basic-2/substitution/tps.ma b/matita/matita/contribs/lambda-delta/Basic-2/substitution/tps.ma
index fc0851097de3709970bb96a922bcdcbffd8e06d0..7f43bd33218492985a3bb4d6a72a88731bda7ebb 100644 (file)
--- a/matita/matita/contribs/lambda-delta/Basic-2/substitution/tps.ma
+++ b/matita/matita/contribs/lambda-delta/Basic-2/substitution/tps.ma
@@ -17,8 +17,7 @@ include "Basic-2/substitution/drop.ma".
 (* PARALLEL SUBSTITUTION ON TERMS *******************************************)
 
 inductive tps: lenv → term → nat → nat → term → Prop ≝
-| tps_sort : ∀L,k,d,e. tps L (⋆k) d e (⋆k)
-| tps_lref : ∀L,i,d,e. tps L (#i) d e (#i)
+| tps_atom : ∀L,I,d,e. tps L (𝕒{I}) d e (𝕒{I})
 | tps_subst: ∀L,K,V,W,i,d,e. d ≤ i → i < d + e →
              ↓[0, i] L ≡ K. 𝕓{Abbr} V → ↑[0, i + 1] V ≡ W → tps L (#i) d e W
 | tps_bind : ∀L,I,V1,V2,T1,T2,d,e.
@@ -38,7 +37,6 @@ lemma tps_leq_repl: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫ T2 →
                     ∀L2. L1 [d, e] ≈ L2 → L2 ⊢ T1 [d, e] ≫ T2.
 #L1 #T1 #T2 #d #e #H elim H -H L1 T1 T2 d e
 [ //
-| //
 | #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12
   elim (drop_leq_drop1 … HL12 … HLK1 ? ?) -HL12 HLK1 // /2/
 | /4/
@@ -56,7 +54,6 @@ lemma tps_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 [d1, e1] ≫ T2 →
                 L ⊢ T1 [d2, e2] ≫ T2.
 #L #T1 #T #d1 #e1 #H elim H -L T1 T d1 e1
 [ //
-| //
 | #L #K #V #W #i #d1 #e1 #Hid1 #Hide1 #HLK #HVW #d2 #e2 #Hd12 #Hde12
   lapply (transitive_le … Hd12 … Hid1) -Hd12 Hid1 #Hid2
   lapply (lt_to_le_to_lt … Hide1 … Hde12) -Hide1 /2/
@@ -69,7 +66,6 @@ lemma tps_weak_top: ∀L,T1,T2,d,e.
                     L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 [d, |L| - d] ≫ T2.
 #L #T1 #T #d #e #H elim H -L T1 T d e
 [ //
-| //
 | #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW
   lapply (drop_fwd_drop2_length … HLK) #Hi
   lapply (le_to_lt_to_lt … Hdi Hi) #Hd
@@ -90,7 +86,6 @@ lemma tps_split_up: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀i. d ≤ i →
                     ∃∃T. L ⊢ T1 [d, i - d] ≫ T & L ⊢ T [i, d + e - i] ≫ T2.
 #L #T1 #T2 #d #e #H elim H -L T1 T2 d e
 [ /2/
-| /2/
 | #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
   elim (lt_or_ge i j)
   [ -Hide Hjde;
@@ -113,35 +108,54 @@ qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma tps_inv_lref1_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀i. T1 = #i →
-                         T2 = #i ∨
-                         ∃∃K,V. d ≤ i & i < d + e &
-                                ↓[O, i] L ≡ K. 𝕓{Abbr} V &
-                                ↑[O, i + 1] V ≡ T2.
+fact tps_inv_atom1_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀I. T1 = 𝕒{I} →
+                        T2 = 𝕒{I} ∨
+                        ∃∃K,V,i. d ≤ i & i < d + e &
+                                 ↓[O, i] L ≡ K. 𝕓{Abbr} V &
+                                 ↑[O, i + 1] V ≡ T2 &
+                                 I = LRef i.
 #L #T1 #T2 #d #e * -L T1 T2 d e
-[ #L #k #d #e #i #H destruct
-| /2/
-| #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #j #H destruct -i /3/
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
+[ #L #I #d #e #J #H destruct -I /2/
+| #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #I #H destruct -I /3 width=8/
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
 ]
 qed.
 
+lemma tps_inv_atom1: ∀L,T2,I,d,e. L ⊢ 𝕒{I} [d, e] ≫ T2 →
+                     T2 = 𝕒{I} ∨
+                     ∃∃K,V,i. d ≤ i & i < d + e &
+                              ↓[O, i] L ≡ K. 𝕓{Abbr} V &
+                              ↑[O, i + 1] V ≡ T2 &
+                              I = LRef i.
+/2/ qed.
+
+
+(* Basic-1: was: subst1_gen_sort *)
+lemma tps_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k [d, e] ≫ T2 → T2 = ⋆k.
+#L #T2 #k #d #e #H
+elim (tps_inv_atom1 … H) -H //
+* #K #V #i #_ #_ #_ #_ #H destruct
+qed.
+
+(* Basic-1: was: subst1_gen_lref *)
 lemma tps_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i [d, e] ≫ T2 →
                      T2 = #i ∨
                      ∃∃K,V. d ≤ i & i < d + e &
                             ↓[O, i] L ≡ K. 𝕓{Abbr} V &
                             ↑[O, i + 1] V ≡ T2.
-/2/ qed.
+#L #T2 #i #d #e #H
+elim (tps_inv_atom1 … H) -H /2/
+* #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct -i /3/
+qed.
 
-lemma tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
-                         ∀I,V1,T1. U1 = 𝕓{I} V1. T1 →
-                         ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & 
-                                  L. 𝕓{I} V1 ⊢ T1 [d + 1, e] ≫ T2 &
-                                  U2 =  𝕓{I} V2. T2.
+fact tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
+                        ∀I,V1,T1. U1 = 𝕓{I} V1. T1 →
+                        ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & 
+                                 L. 𝕓{I} V1 ⊢ T1 [d + 1, e] ≫ T2 &
+                                 U2 =  𝕓{I} V2. T2.
 #d #e #L #U1 #U2 * -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 #W #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
@@ -154,13 +168,12 @@ lemma tps_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕓{I} V1. T1 [d, e] ≫ U2 →
                               U2 =  𝕓{I} V2. T2.
 /2/ qed.
 
-lemma tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
-                         ∀I,V1,T1. U1 = 𝕗{I} V1. T1 →
-                         ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &
-                                  U2 =  𝕗{I} V2. T2.
+fact tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
+                        ∀I,V1,T1. U1 = 𝕗{I} V1. T1 →
+                        ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &
+                                 U2 =  𝕗{I} V2. T2.
 #d #e #L #U1 #U2 * -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 #W #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/
@@ -171,3 +184,27 @@ lemma tps_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕗{I} V1. T1 [d, e] ≫ U2 →
                      ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &
                               U2 =  𝕗{I} V2. T2.
 /2/ qed.
+
+fact tps_inv_refl_O2_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → e = 0 → T1 = T2.
+#L #T1 #T2 #d #e #H elim H -H L T1 T2 d e
+[ //
+| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct -e;
+  lapply (le_to_lt_to_lt … Hdi … Hide) -Hdi Hide <plus_n_O #Hdd
+  elim (lt_refl_false … Hdd)
+| /3/
+| /3/
+]
+qed.
+
+lemma tps_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 [d, 0] ≫ T2 → T1 = T2.
+/2 width=6/ qed.
+
+(* Basic-1: removed theorems 23:
+            subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
+            subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
+           subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
+            subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
+           subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
+            subst0_confluence_lift subst0_tlt
+            subst1_head subst1_gen_head  
+*)