confluence of parallel substitution (tps) started ...

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

diff --git a/matita/matita/lib/lambda-delta/substitution/tps_tps.ma b/matita/matita/lib/lambda-delta/substitution/tps_tps.ma
index 55e8be812464b15b7ec29c89d58d92a384ade7ba..8d801d50eae17970bd4fa4bb0eedfca9e24716ab 100644 (file)
--- a/matita/matita/lib/lambda-delta/substitution/tps_tps.ma
+++ b/matita/matita/lib/lambda-delta/substitution/tps_tps.ma
@@ -11,12 +11,15 @@
 
 include "lambda-delta/substitution/tps_split.ma".
 
+lemma arith_i2: ∀a,c1,c2. c1 + c2 ≤ a → c1 + c2 + (a - c1 - c2) = a.
+/2/ qed.
+
 (* PARTIAL SUBSTITUTION ON TERMS ********************************************)
 
 (* Main properties **********************************************************)
-
-lemma tps_trans: ∀L,T1,T,d,e. L ⊢ T1 [d, e] ≫ T → ∀T2. L ⊢ T [d, e] ≫ T2 →
-                 L ⊢ T1 [d, e] ≫ T2.
+(*
+theorem tps_trans: ∀L,T1,T,d,e. L ⊢ T1 [d, e] ≫ T → ∀T2. L ⊢ T [d, e] ≫ T2 →
+                   L ⊢ T1 [d, e] ≫ T2.
 #L #T1 #T #d #e #H elim H -L T1 T d e
 [ //
 | //
@@ -31,9 +34,60 @@ lemma tps_trans: ∀L,T1,T,d,e. L ⊢ T1 [d, e] ≫ T → ∀T2. L ⊢ T [d, e]
   elim (tps_inv_flat1 … HX) -HX #V #T #HV2 #HT2 #HX destruct -X /3/
 ]
 qed.
+*)
 
-axiom tps_conf: ∀L,T0,d,e,T1. L ⊢ T0 [d, e] ≫ T1 → ∀T2. L ⊢ T0 [d, e] ≫ T2 →
+axiom tps_conf_subst_subst_lt: ∀L,K1,V1,W1,T1,i1,d,e,T2,K2,V2,W2,i2.
+   ↓[O, i1] L ≡ K1. 𝕓{Abbr} V1 → ↓[O, i2] L≡ K2. 𝕓{Abbr} V2 →
+   K1 ⊢ V1 [O, d + e - i1 - 1] ≫ W1 → K2 ⊢ V2 [O, d + e - i2 - 1] ≫ W2 →
+   ↑[O, i1 + 1] W1 ≡ T1 → ↑[O, i2 + 1] W2 ≡ T2 → 
+   d ≤ i1 → i1 < d + e → d ≤ i2 → i2 < d + e → i1 < i2 →
+   ∃∃T. L ⊢ T1 [d, e] ≫ T & L ⊢ T2 [d, e] ≫ T.  
+(*
+#L #K1 #V1 #W1 #T1 #i1 #d #e #T2 #K2 #V2 #W2 #i2
+#HLK1 #HLK2 #HVW1 #HVW2 #HWT1 #HWT2 #Hdi1 #Hi1de #Hdi2 #Hi2de #Hi12
+*)
+
+theorem tps_conf: ∀L,T0,T1,d,e. L ⊢ T0 [d, e] ≫ T1 → ∀T2. L ⊢ T0 [d, e] ≫ T2 →
                   ∃∃T. L ⊢ T1 [d, e] ≫ T & L ⊢ T2 [d, e] ≫ T.
+#L #T0 #T1 #d #e #H elim H -H L T0 T1 d e
+[ /2/
+| /2/
+| #L #K1 #V1 #W1 #T1 #i1 #d #e #Hdi1 #Hi1de #HLK1 #HVW1 #HWT1 #IHVW1 #T2 #H
+  elim (tps_inv_lref1 … H) -H
+  [ -IHVW1 #HX destruct -T2
+    @ex2_1_intro [2: // | skip ] /2 width=6/ (**) (* /3 width=9/ is slow *)
+  | * #K2 #V2 #W2 #i2 #Hdi2 #Hi2de #HLK2 #HVW2 #HWT2
+    elim (lt_or_eq_or_gt i1 i2) #Hi12
+    [ @tps_conf_subst_subst_lt /width=19/
+    | -HVW1; destruct -i2;
+      lapply (transitive_le ? ? (i1 + 1) Hdi2 ?) -Hdi2 // #Hdi2
+      lapply (drop_mono … HLK1 … HLK2) -HLK1 Hdi1 Hi1de #H destruct -V1 K1;
+      elim (IHVW1 … HVW2) -IHVW1 HVW2 #W #HW1 #HW2
+      lapply (drop_fwd_drop2 … HLK2) -HLK2 #HLK2
+      elim (lift_total W 0 (i1 + 1)) #T #HWT
+      lapply (tps_lift_ge … HW1 … HLK2 HWT1 HWT ?) -HW1 HWT1 //
+      lapply (tps_lift_ge … HW2 … HLK2 HWT2 HWT ?) -HW2 HWT2 HLK2 HWT // normalize #HT2 #HT1
+      lapply (tps_weak … HT1 d e ? ?) -HT1 [ >arith_i2 // | // ]
+      lapply (tps_weak … HT2 d e ? ?) -HT2 [ >arith_i2 // | // ]
+      /2/
+    | @ex2_1_comm @tps_conf_subst_subst_lt /width=19/
+    ]
+  ]
+| #L #I #V0 #V1 #T0 #T1 #d #e #_ #_ #IHV01 #IHT01 #X #HX
+  elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X; 
+  elim (IHV01 … HV02) -IHV01 HV02 #V #HV1 #HV2
+  elim (IHT01 … HT02) -IHT01 HT02 #T #HT1 #HT2 
+  @ex2_1_intro
+  [2: @tps_bind [4: @(tps_leq_repl … HT1) /2/ |2: skip ]
+  |1: skip
+  |3: @tps_bind [2: @(tps_leq_repl … HT2) /2/ ]
+  ] // (**) (* /5/ is too slow *)
+| #L #I #V0 #V1 #T0 #T1 #d #e #_ #_ #IHV01 #IHT01 #X #HX
+  elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X; 
+  elim (IHV01 … HV02) -IHV01 HV02;
+  elim (IHT01 … HT02) -IHT01 HT02 /3 width=5/
+]
+qed.
 
 (*
       Theorem subst0_subst0: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->