- first piece of the mutual induction for preservation finally closed!

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / reducibility / fpr_cpr.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/fpr_cpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/fpr_cpr.ma
index 0de0b98f58d8c9b622b6286e6596718fe10a886a..007681d0bc410f76fea0d57c3ea7020ebbc939dc 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/reducibility/fpr_cpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/fpr_cpr.ma
@@ -18,10 +18,34 @@ include "basic_2/reducibility/cfpr_cpr.ma".
 
 (* Properties on context-sensitive parallel reduction for terms *************)
 
+lemma ltpr_tpr_fpr: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. T1 ➡ T2 → ⦃L1, T1⦄ ➡ ⦃L2, T2⦄.
+/3 width=4/ qed.
+
 lemma cpr_fpr: ∀L,T1,T2. L ⊢ T1 ➡ T2 → ⦃L, T1⦄ ➡ ⦃L, T2⦄.
 /2 width=4/ qed.
 
-(* Advanced propertis *******************************************************)
+lemma fpr_lift: ∀K1,K2,T1,T2. ⦃K1, T1⦄ ➡ ⦃K2, T2⦄ →
+                ∀d,e,L1. ⇩[d, e] L1 ≡ K1 →
+                ∀U1,U2. ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
+                ∃∃L2. ⦃L1, U1⦄ ➡ ⦃L2, U2⦄ & ⇩[d, e] L2 ≡ K2.
+#K1 #K2 #T1 #T2 #HT12 #d #e #L1 #HLK1 #U1 #U2 #HTU1 #HTU2
+elim (fpr_inv_all … HT12) -HT12 #K #HK1 #HT12 #HK2
+elim (ldrop_ltpr_trans … HLK1 … HK1) -K1 #L #HL1 #HLK
+lapply (cpr_lift … HLK … HTU1 … HTU2 HT12) -T1 -T2 #HU12
+elim (le_or_ge (|K|) d) #Hd
+[ elim (ldrop_ltpss_sn_trans_ge … HLK … HK2 …)
+| elim (ldrop_ltpss_sn_trans_be … HLK … HK2 …)
+] // -Hd #L2 #HL2 #HLK2
+lapply (ltpss_sn_weak_all … HL2) -K /3 width=4/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma fpr_flat_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ∀V1,V2. V1 ➡ V2 →
+                   ∀I. ⦃L1, ⓕ{I}V1.T1⦄ ➡ ⦃L2, ⓕ{I}V2.T2⦄.
+#L1 #L2 #T1 #T2 #HT12
+elim (fpr_inv_all … HT12) -HT12 /4 width=4/
+qed.
 
 lemma fpr_bind_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ → ∀T1,T2. T1 ➡ T2 →
                    ∀a,I. ⦃L1, ⓑ{a,I}V1.T1⦄ ➡ ⦃L2, ⓑ{a,I}V2.T2⦄.
@@ -29,8 +53,24 @@ lemma fpr_bind_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ → ∀T1,T2. T
 elim (fpr_inv_all … H) /3 width=4/
 qed.
 
+lemma fpr_bind2_minus: ∀I,L1,L2,V1,T1,U2. ⦃L1, -ⓑ{I}V1.T1⦄ ➡ ⦃L2, U2⦄ →
+                       ∃∃V2,T2. ⦃L1.ⓑ{I}V1, T1⦄ ➡ ⦃L2.ⓑ{I}V2, T2⦄ &
+                                U2 = -ⓑ{I}V2.T2.
+#I1 #L1 #L2 #V1 #T1 #U2 #H
+elim (fpr_inv_all … H) -H #L #HL1 #H #HL2
+elim (cpr_fwd_bind1_minus … H false) -H /4 width=4/
+qed-.
+
 (* Advanced forward lemmas **************************************************)
 
+lemma fpr_fwd_bind2_minus: ∀I,L1,L,V1,T1,T. ⦃L1, -ⓑ{I}V1.T1⦄ ➡ ⦃L, T⦄ → ∀b.
+                           ∃∃V2,T2. ⦃L1, ⓑ{b,I}V1.T1⦄ ➡ ⦃L, ⓑ{b,I}V2.T2⦄ &
+                                    T = -ⓑ{I}V2.T2.
+#I #L1 #L #V1 #T1 #T #H1 #b
+elim (fpr_inv_all … H1) -H1 #L0 #HL10 #HT1 #HL0
+elim (cpr_fwd_bind1_minus … HT1 b) -HT1 /3 width=4/
+qed-.
+
 lemma fpr_fwd_shift_bind_minus: ∀I1,I2,L1,L2,V1,V2,T1,T2.
                                 ⦃L1, -ⓑ{I1}V1.T1⦄ ➡ ⦃L2, -ⓑ{I2}V2.T2⦄ →
                                 ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ ∧ I1 = I2.
@@ -45,11 +85,19 @@ elim (fpr_inv_all … H) -H #L #HL1 #H #HL2
 ]
 qed-.
 
+lemma fpr_fwd_abst2: ∀a,L1,L2,V1,T1,U2. ⦃L1, ⓛ{a}V1.T1⦄ ➡ ⦃L2, U2⦄ → ∀b,I,W.
+                     ∃∃V2,T2. ⦃L1, ⓑ{b,I}W.T1⦄ ➡ ⦃L2, ⓑ{b,I}W.T2⦄ &
+                              U2 = ⓛ{a}V2.T2.
+#a #L1 #L2 #V1 #T1 #U2 #H
+elim (fpr_inv_all … H) #L #HL1 #H #HL2 #b #I #W
+elim (cpr_fwd_abst1 … H b I W) -H /3 width=4/
+qed-.
+
 (* Advanced inversion lemmas ************************************************)
 
 lemma fpr_inv_pair1: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡ ⦃L2, T2⦄ →
                      ∃∃K2,V2. ⦃K1, V1⦄  ➡ ⦃K2, V2⦄ &
-                              ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄  &
+                              ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ &
                               L2 = K2.ⓑ{I}V2.
 #I1 #K1 #X #V1 #T1 #T2 #H
 elim (fpr_fwd_pair1 … H) -H #I2 #K2 #V2 #HT12 #H destruct
@@ -58,9 +106,20 @@ qed-.
 
 lemma fpr_inv_pair3: ∀I,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡ ⦃K2.ⓑ{I}V2, T2⦄ →
                      ∃∃K1,V1. ⦃K1, V1⦄  ➡ ⦃K2, V2⦄ &
-                              ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄  &
+                              ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ &
                               L1 = K1.ⓑ{I}V1.
 #I2 #X #K2 #V2 #T1 #T2 #H
 elim (fpr_fwd_pair3 … H) -H #I1 #K1 #V1 #HT12 #H destruct
 elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/
 qed-.
+
+(* More advanced forward lemmas *********************************************)
+
+lemma fpr_fwd_pair1_full: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡ ⦃L2, T2⦄ →
+                          ∀b. ∃∃K2,V2. ⦃K1, V1⦄  ➡ ⦃K2, V2⦄ &
+                                       ⦃K1, ⓑ{b,I}V1.T1⦄ ➡ ⦃K2, ⓑ{b,I}V2.T2⦄ &
+                                       L2 = K2.ⓑ{I}V2.
+#I #K1 #L2 #V1 #T1 #T2 #H #b
+elim (fpr_inv_pair1 … H) -H #K2 #V2 #HV12 #HT12 #H destruct
+elim (fpr_fwd_bind2_minus … HT12 b) -HT12 #W1 #U1 #HTU1 #H destruct /2 width=5/
+qed-.