- "functional" component moved to Apps_2

[helm.git] / matita / matita / contribs / lambda_delta / Basic_2 / computation / csn_lcpr.ma

diff --git a/matita/matita/contribs/lambda_delta/Basic_2/computation/csn_lcpr.ma b/matita/matita/contribs/lambda_delta/Basic_2/computation/csn_lcpr.ma
index 8121a0ac1f60910d3a82cf8468fc03677cfc1a3b..98ee1843748ffedabb3571c06618a45f0449d9f9 100644 (file)
--- a/matita/matita/contribs/lambda_delta/Basic_2/computation/csn_lcpr.ma
+++ b/matita/matita/contribs/lambda_delta/Basic_2/computation/csn_lcpr.ma
@@ -12,6 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "Basic_2/grammar/tstc_tstc.ma".
 include "Basic_2/reducibility/lcpr_cpr.ma".
 include "Basic_2/computation/cprs_cprs.ma".
 include "Basic_2/computation/csn_lift.ma".
@@ -35,7 +36,7 @@ elim (cpr_inv_abbr1 … H1) -H1 *
 [ #V0 #V1 #T1 #HLV0 #HLV01 #HLT1 #H destruct
   lapply (cpr_intro … HLV0 HLV01) -HLV01 #HLV1
   lapply (ltpr_cpr_trans (L. ⓓV) … HLT1) /2 width=1/ -V0 #HLT1
-  elim (eq_false_inv_tpair … H2) -H2
+  elim (eq_false_inv_tpair_sn … H2) -H2
   [ #HV1 @IHV // /2 width=1/ -HV1
     @(csn_lcpr_conf (L. ⓓV)) /2 width=1/ -HLV1 /2 width=3/
   | -IHV -HLV1 * #H destruct /3 width=1/
@@ -73,3 +74,74 @@ qed.
 lemma csn_appl_beta: ∀L,W. L ⊢ ⬇* W → ∀V,T. L ⊢ ⬇* (ⓓV. T) → (**)
                      L ⊢ ⬇* ⓐV. ⓛW. T.
 /2 width=3/ qed.
+
+fact csn_appl_theta_aux: ∀L,U. L ⊢ ⬇* U → ∀V1,V2. ⇧[0, 1] V1 ≡ V2 →
+                         ∀V,T. U = ⓓV. ⓐV2. T → L ⊢ ⬇* ⓐV1. ⓓV. T.
+#L #X #H @(csn_ind_cprs … H) -X #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct
+lapply (csn_fwd_pair_sn … HVT) #HV
+lapply (csn_fwd_bind_dx … HVT) -HVT #HVT
+@csn_intro #X #HL #H
+elim (cpr_inv_appl1 … HL) -HL *
+[ -HV #V0 #Y #HLV10 #HL #H0 destruct
+  elim (cpr_inv_abbr1 … HL) -HL *
+  [ #V3 #V4 #T3 #HV3 #HLV34 #HLT3 #H0 destruct
+    lapply (cpr_intro … HV3 HLV34) -HLV34 #HLV34
+    elim (lift_total V0 0 1) #V5 #HV05
+    elim (term_eq_dec (ⓓV.ⓐV2.T) (ⓓV4.ⓐV5.T3))
+    [ -IHVT #H0 destruct
+      elim (eq_false_inv_tpair_sn … H) -H
+      [ -HLV10 -HLV34 -HV3 -HLT3 -HVT
+        >(lift_inj … HV12 … HV05) -V5
+        #H elim (H ?) //
+      | * #_ #H elim (H ?) //
+      ]
+    | -H -HVT #H
+      lapply (cpr_lift (L. ⓓV) … HV12 … HV05 HLV10) -HLV10 -HV12 /2 width=1/ #HV25
+      lapply (ltpr_cpr_trans (L. ⓓV) … HLT3) /2 width=1/ -HLT3 #HLT3
+      @(IHVT … H … HV05) -IHVT // -H -HV05 /3 width=1/
+    ]
+  | -H -IHVT #T0 #HT0 #HLT0
+    @(csn_cpr_trans … (ⓐV1.T0)) /2 width=1/ -V0 -Y
+    @(csn_inv_lift … 0 1 HVT) /2 width=1/
+  ]
+| -HV -HV12 -HVT -IHVT -H #V0 #W0 #T0 #T1 #_ #_ #H destruct
+| -IHVT -H #V0 #V3 #W0 #W1 #T0 #T1 #HLV10 #HLW01 #HLT01 #HV03 #H1 #H2 destruct
+  lapply (cpr_lift (L. ⓓW0) … HV12 … HV03 HLV10) -HLV10 -HV12 -HV03 /2 width=1/ #HLV23
+  lapply (lcpr_cpr_trans (L. ⓓW0) … HLT01) -HLT01 /2 width=1/ #HLT01
+  @csn_abbr /2 width=3/ -HV
+  @(csn_lcpr_conf (L. ⓓW0)) /2 width=1/ -W1
+  @(csn_cprs_trans … HVT) -HVT /2 width=1/
+]
+qed.
+
+(* Basic_1: was: sn3_appl_bind *)
+lemma csn_appl_theta: ∀V1,V2. ⇧[0, 1] V1 ≡ V2 →
+                      ∀L,V,T. L ⊢ ⬇* ⓓV. ⓐV2. T → L ⊢ ⬇* ⓐV1. ⓓV. T.
+/2 width=5/ qed.
+
+(* Basic_1: was only: sn3_appl_appl *)
+lemma csn_appl_simple_tstc: ∀L,V. L ⊢ ⬇* V → ∀T1.
+                            L ⊢ ⬇* T1 →
+                            (∀T2. L ⊢ T1 ➡* T2 → (T1 ≃ T2 → False) → L ⊢ ⬇* ⓐV. T2) →
+                            𝐒[T1] → L ⊢ ⬇* ⓐV. T1.
+#L #V #H @(csn_ind … H) -V #V #_ #IHV #T1 #H @(csn_ind … H) -T1 #T1 #H1T1 #IHT1 #H2T1 #H3T1
+@csn_intro #X #HL #H
+elim (cpr_inv_appl1_simple … HL ?) -HL //
+#V0 #T0 #HLV0 #HLT10 #H0 destruct
+elim (eq_false_inv_tpair_sn … H) -H
+[ -IHT1 #HV0
+  @(csn_cpr_trans … (ⓐV0.T1)) /2 width=1/ -HLT10
+  @IHV -IHV // -H1T1 -H3T1 /2 width=1/ -HV0
+  #T2 #HLT12 #HT12
+  @(csn_cpr_trans … (ⓐV.T2)) /2 width=1/ -HLV0
+  @H2T1 -H2T1 // -HLT12 /2 width=1/
+| -IHV -H1T1 -HLV0 * #H #H1T10 destruct
+  elim (tstc_dec T1 T0) #H2T10
+  [ @IHT1 -IHT1 // /2 width=1/ -H1T10 /2 width=3/ -H3T1
+    #T2 #HLT02 #HT02
+    @H2T1 -H2T1 /2 width=3/ -HLT10 -HLT02 /3 width=3/
+  | -IHT1 -H3T1 -H1T10
+    @H2T1 -H2T1 /2 width=1/
+  ]
+]
+qed.