- main lemmas about abstract reducibility candidates closed

[helm.git] / matita / matita / contribs / lambda_delta / Basic_2 / reducibility / tpr_tpr.ma

diff --git a/matita/matita/contribs/lambda_delta/Basic_2/reducibility/tpr_tpr.ma b/matita/matita/contribs/lambda_delta/Basic_2/reducibility/tpr_tpr.ma
index e541b5777323fc633e2f725ebabd92ad9c46c9e4..b1e3f3c1b053708642613611d3987edead0444f0 100644 (file)
--- a/matita/matita/contribs/lambda_delta/Basic_2/reducibility/tpr_tpr.ma
+++ b/matita/matita/contribs/lambda_delta/Basic_2/reducibility/tpr_tpr.ma
@@ -18,17 +18,17 @@ include "Basic_2/reducibility/tpr_tpss.ma".
 
 (* Confluence lemmas ********************************************************)
 
-fact tpr_conf_atom_atom: ∀I. ∃∃X. 𝕒{I} ⇒ X & 𝕒{I} ⇒ X.
+fact tpr_conf_atom_atom: ∀I. ∃∃X. ⓪{I} ➡ X & ⓪{I} ➡ X.
 /2 width=3/ qed.
 
 fact tpr_conf_flat_flat:
    ∀I,V0,V1,T0,T1,V2,T2. (
       ∀X0:term. #[X0] < #[V0] + #[T0] + 1 →
-      â\88\80X1,X2. X0 â\87\92 X1 â\86\92 X0 â\87\92 X2 →
-      â\88\83â\88\83X. X1 â\87\92 X & X2 â\87\92 X
+      â\88\80X1,X2. X0 â\9e¡ X1 â\86\92 X0 â\9e¡ X2 →
+      â\88\83â\88\83X. X1 â\9e¡ X & X2 â\9e¡ X
    ) →
-   V0 â\87\92 V1 â\86\92 V0 â\87\92 V2 â\86\92 T0 â\87\92 T1 â\86\92 T0 â\87\92 T2 →
-   ∃∃T0. 𝕗{I} V1. T1 ⇒ T0 & 𝕗{I} V2. T2 ⇒ T0.
+   V0 â\9e¡ V1 â\86\92 V0 â\9e¡ V2 â\86\92 T0 â\9e¡ T1 â\86\92 T0 â\9e¡ T2 →
+   ∃∃T0. ⓕ{I} V1. T1 ➡ T0 & ⓕ{I} V2. T2 ➡ T0.
 #I #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02
 elim (IH … HV01 … HV02) -HV01 -HV02 // #V #HV1 #HV2
 elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /3 width=5/
@@ -37,12 +37,12 @@ qed.
 fact tpr_conf_flat_beta:
    ∀V0,V1,T1,V2,W0,U0,T2. (
       ∀X0:term. #[X0] < #[V0] + (#[W0] + #[U0] + 1) + 1 →
-      â\88\80X1,X2. X0 â\87\92 X1 â\86\92 X0 â\87\92 X2 →
-      â\88\83â\88\83X. X1 â\87\92 X & X2 â\87\92 X
+      â\88\80X1,X2. X0 â\9e¡ X1 â\86\92 X0 â\9e¡ X2 →
+      â\88\83â\88\83X. X1 â\9e¡ X & X2 â\9e¡ X
    ) →
-   V0 â\87\92 V1 â\86\92 V0 â\87\92 V2 →
-   U0 â\87\92 T2 â\86\92 ð\9d\95\94{Abst} W0. U0 â\87\92 T1 →
-   ∃∃X. 𝕔{Appl} V1. T1 ⇒ X & 𝕔{Abbr} V2. T2 ⇒ X.
+   V0 â\9e¡ V1 â\86\92 V0 â\9e¡ V2 →
+   U0 â\9e¡ T2 â\86\92 â\93\9bW0. U0 â\9e¡ T1 →
+   ∃∃X. ⓐV1. T1 ➡ X & ⓓV2. T2 ➡ X.
 #V0 #V1 #T1 #V2 #W0 #U0 #T2 #IH #HV01 #HV02 #HT02 #H
 elim (tpr_inv_abst1 … H) -H #W1 #U1 #HW01 #HU01 #H destruct
 elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
@@ -56,12 +56,12 @@ qed.
 fact tpr_conf_flat_theta:
    ∀V0,V1,T1,V2,V,W0,W2,U0,U2. (
       ∀X0:term. #[X0] < #[V0] + (#[W0] + #[U0] + 1) + 1 →
-      â\88\80X1,X2. X0 â\87\92 X1 â\86\92 X0 â\87\92 X2 →
-      â\88\83â\88\83X. X1 â\87\92 X & X2 â\87\92 X
+      â\88\80X1,X2. X0 â\9e¡ X1 â\86\92 X0 â\9e¡ X2 →
+      â\88\83â\88\83X. X1 â\9e¡ X & X2 â\9e¡ X
    ) →
-   V0 â\87\92 V1 â\86\92 V0 â\87\92 V2 â\86\92 â\87\91[O,1] V2 ≡ V →
-   W0 â\87\92 W2 â\86\92 U0 â\87\92 U2 â\86\92  ð\9d\95\94{Abbr} W0. U0 â\87\92 T1 →
-   ∃∃X. 𝕔{Appl} V1. T1 ⇒ X & 𝕔{Abbr} W2. 𝕔{Appl} V. U2 ⇒ X.
+   V0 â\9e¡ V1 â\86\92 V0 â\9e¡ V2 â\86\92 â\87§[O,1] V2 ≡ V →
+   W0 â\9e¡ W2 â\86\92 U0 â\9e¡ U2 â\86\92  â\93\93W0. U0 â\9e¡ T1 →
+   ∃∃X. ⓐV1. T1 ➡ X & ⓓW2. ⓐV. U2 ➡ X.
 #V0 #V1 #T1 #V2 #V #W0 #W2 #U0 #U2 #IH #HV01 #HV02 #HV2 #HW02 #HU02 #H
 elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #VV #HVV1 #HVV2
 elim (lift_total VV 0 1) #VVV #HVV
@@ -93,11 +93,11 @@ qed.
 fact tpr_conf_flat_cast:
    ∀X2,V0,V1,T0,T1. (
       ∀X0:term. #[X0] < #[V0] + #[T0] + 1 →
-      â\88\80X1,X2. X0 â\87\92 X1 â\86\92 X0 â\87\92 X2 →
-      â\88\83â\88\83X. X1 â\87\92 X & X2 â\87\92 X
+      â\88\80X1,X2. X0 â\9e¡ X1 â\86\92 X0 â\9e¡ X2 →
+      â\88\83â\88\83X. X1 â\9e¡ X & X2 â\9e¡ X
    ) →
-   V0 â\87\92 V1 â\86\92 T0 â\87\92 T1 â\86\92 T0 â\87\92 X2 →
-   ∃∃X. 𝕔{Cast} V1. T1 ⇒ X & X2 ⇒ X.
+   V0 â\9e¡ V1 â\86\92 T0 â\9e¡ T1 â\86\92 T0 â\9e¡ X2 →
+   ∃∃X. ⓣV1. T1 ➡ X & X2 ➡ X.
 #X2 #V0 #V1 #T0 #T1 #IH #_ #HT01 #HT02
 elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /3 width=3/
 qed.
@@ -105,11 +105,11 @@ qed.
 fact tpr_conf_beta_beta:
    ∀W0:term. ∀V0,V1,T0,T1,V2,T2. (
       ∀X0:term. #[X0] < #[V0] + (#[W0] + #[T0] + 1) + 1 →
-      â\88\80X1,X2. X0 â\87\92 X1 â\86\92 X0 â\87\92 X2 →
-      â\88\83â\88\83X. X1 â\87\92 X & X2 â\87\92 X
+      â\88\80X1,X2. X0 â\9e¡ X1 â\86\92 X0 â\9e¡ X2 →
+      â\88\83â\88\83X. X1 â\9e¡ X & X2 â\9e¡ X
    ) →
-   V0 â\87\92 V1 â\86\92 V0 â\87\92 V2 â\86\92 T0 â\87\92 T1 â\86\92 T0 â\87\92 T2 →
-   ∃∃X. 𝕔{Abbr} V1. T1 ⇒X & 𝕔{Abbr} V2. T2 ⇒ X.
+   V0 â\9e¡ V1 â\86\92 V0 â\9e¡ V2 â\86\92 T0 â\9e¡ T1 â\86\92 T0 â\9e¡ T2 →
+   ∃∃X. ⓓV1. T1 ➡X & ⓓV2. T2 ➡ X.
 #W0 #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02
 elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/
 elim (IH … HT01 … HT02) -HT01 -HT02 -IH /2 width=1/ /3 width=5/
@@ -119,13 +119,13 @@ qed.
 fact tpr_conf_delta_delta:
    ∀I1,V0,V1,T0,T1,TT1,V2,T2,TT2. (
       ∀X0:term. #[X0] < #[V0] + #[T0] + 1 →
-      â\88\80X1,X2. X0 â\87\92 X1 â\86\92 X0 â\87\92 X2 →
-      â\88\83â\88\83X. X1 â\87\92 X & X2 â\87\92 X
+      â\88\80X1,X2. X0 â\9e¡ X1 â\86\92 X0 â\9e¡ X2 →
+      â\88\83â\88\83X. X1 â\9e¡ X & X2 â\9e¡ X
    ) →
-   V0 â\87\92 V1 â\86\92 V0 â\87\92 V2 â\86\92 T0 â\87\92 T1 â\86\92 T0 â\87\92 T2 →
-   ⋆. 𝕓{I1} V1 ⊢ T1 [O, 1] ≫ TT1 →
-   ⋆. 𝕓{I1} V2 ⊢ T2 [O, 1] ≫ TT2 →
-   ∃∃X. 𝕓{I1} V1. TT1 ⇒ X & 𝕓{I1} V2. TT2 ⇒ X.
+   V0 â\9e¡ V1 â\86\92 V0 â\9e¡ V2 â\86\92 T0 â\9e¡ T1 â\86\92 T0 â\9e¡ T2 →
+   ⋆. ⓑ{I1} V1 ⊢ T1 [O, 1] ▶ TT1 →
+   ⋆. ⓑ{I1} V2 ⊢ T2 [O, 1] ▶ TT2 →
+   ∃∃X. ⓑ{I1} V1. TT1 ➡ X & ⓑ{I1} V2. TT2 ➡ X.
 #I1 #V0 #V1 #T0 #T1 #TT1 #V2 #T2 #TT2 #IH #HV01 #HV02 #HT01 #HT02 #HTT1 #HTT2
 elim (IH … HV01 … HV02) -HV01 -HV02 // #V #HV1 #HV2
 elim (IH … HT01 … HT02) -HT01 -HT02 -IH // #T #HT1 #HT2
@@ -138,12 +138,12 @@ qed.
 fact tpr_conf_delta_zeta:
    ∀X2,V0,V1,T0,T1,TT1,T2. (
       ∀X0:term. #[X0] < #[V0] + #[T0] + 1 →
-      â\88\80X1,X2. X0 â\87\92 X1 â\86\92 X0 â\87\92 X2 →
-      â\88\83â\88\83X. X1 â\87\92 X & X2 â\87\92 X
+      â\88\80X1,X2. X0 â\9e¡ X1 â\86\92 X0 â\9e¡ X2 →
+      â\88\83â\88\83X. X1 â\9e¡ X & X2 â\9e¡ X
    ) →
-   V0 â\87\92 V1 â\86\92 T0 â\87\92 T1 â\86\92 â\8b\86. ð\9d\95\93{Abbr} V1 â\8a¢ T1 [O,1] â\89« TT1 →
-   T2 â\87\92 X2 â\86\92 â\87\91[O, 1] T2 ≡ T0 →
-   ∃∃X. 𝕓{Abbr} V1. TT1 ⇒ X & X2 ⇒ X.
+   V0 â\9e¡ V1 â\86\92 T0 â\9e¡ T1 â\86\92 â\8b\86. â\93\93V1 â\8a¢ T1 [O,1] â\96¶ TT1 →
+   T2 â\9e¡ X2 â\86\92 â\87§[O, 1] T2 ≡ T0 →
+   ∃∃X. ⓓV1. TT1 ➡ X & X2 ➡ X.
 #X2 #V0 #V1 #T0 #T1 #TT1 #T2 #IH #_ #HT01 #HTT1 #HTX2 #HTT20
 elim (tpr_inv_lift … HT01 … HTT20) -HT01 #TT2 #HTT21 #HTT2
 lapply (tps_inv_lift1_eq … HTT1 … HTT21) -HTT1 #HTT1 destruct
@@ -155,12 +155,12 @@ qed.
 fact tpr_conf_theta_theta:
    ∀VV1,V0,V1,W0,W1,T0,T1,V2,VV2,W2,T2. (
       ∀X0:term. #[X0] < #[V0] + (#[W0] + #[T0] + 1) + 1 →
-      â\88\80X1,X2. X0 â\87\92 X1 â\86\92 X0 â\87\92 X2 →
-      â\88\83â\88\83X. X1 â\87\92 X & X2 â\87\92 X
+      â\88\80X1,X2. X0 â\9e¡ X1 â\86\92 X0 â\9e¡ X2 →
+      â\88\83â\88\83X. X1 â\9e¡ X & X2 â\9e¡ X
    ) →
-   V0 â\87\92 V1 â\86\92 V0 â\87\92 V2 â\86\92 W0 â\87\92 W1 â\86\92 W0 â\87\92 W2 â\86\92 T0 â\87\92 T1 â\86\92 T0 â\87\92 T2 →
-   â\87\91[O, 1] V1 â\89¡ VV1 â\86\92 â\87\91[O, 1] V2 ≡ VV2 →
-   ∃∃X. 𝕔{Abbr} W1. 𝕔{Appl} VV1. T1 ⇒ X & 𝕔{Abbr} W2. 𝕔{Appl} VV2. T2 ⇒ X.
+   V0 â\9e¡ V1 â\86\92 V0 â\9e¡ V2 â\86\92 W0 â\9e¡ W1 â\86\92 W0 â\9e¡ W2 â\86\92 T0 â\9e¡ T1 â\86\92 T0 â\9e¡ T2 →
+   â\87§[O, 1] V1 â\89¡ VV1 â\86\92 â\87§[O, 1] V2 ≡ VV2 →
+   ∃∃X. ⓓW1. ⓐVV1. T1 ➡ X & ⓓW2. ⓐVV2. T2 ➡ X.
 #VV1 #V0 #V1 #W0 #W1 #T0 #T1 #V2 #VV2 #W2 #T2 #IH #HV01 #HV02 #HW01 #HW02 #HT01 #HT02 #HVV1 #HVV2
 elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
 elim (IH … HW01 … HW02) -HW01 -HW02 /2 width=1/ #W #HW1 #HW2
@@ -174,12 +174,12 @@ qed.
 fact tpr_conf_zeta_zeta:
    ∀V0:term. ∀X2,TT0,T0,T1,T2. (
       ∀X0:term. #[X0] < #[V0] + #[TT0] + 1 →
-      â\88\80X1,X2. X0 â\87\92 X1 â\86\92 X0 â\87\92 X2 →
-      â\88\83â\88\83X. X1 â\87\92 X & X2 â\87\92 X
+      â\88\80X1,X2. X0 â\9e¡ X1 â\86\92 X0 â\9e¡ X2 →
+      â\88\83â\88\83X. X1 â\9e¡ X & X2 â\9e¡ X
    ) →
-   T0 â\87\92 T1 â\86\92 T2 â\87\92 X2 →
-   â\87\91[O, 1] T0 â\89¡ TT0 â\86\92 â\87\91[O, 1] T2 ≡ TT0 →
-   â\88\83â\88\83X. T1 â\87\92 X & X2 â\87\92 X.
+   T0 â\9e¡ T1 â\86\92 T2 â\9e¡ X2 →
+   â\87§[O, 1] T0 â\89¡ TT0 â\86\92 â\87§[O, 1] T2 ≡ TT0 →
+   â\88\83â\88\83X. T1 â\9e¡ X & X2 â\9e¡ X.
 #V0 #X2 #TT0 #T0 #T1 #T2 #IH #HT01 #HTX2 #HTT0 #HTT20
 lapply (lift_inj … HTT0 … HTT20) -HTT0 #H destruct
 lapply (tw_lift … HTT20) -HTT20 #HTT20
@@ -189,11 +189,11 @@ qed.
 fact tpr_conf_tau_tau:
    ∀V0,T0:term. ∀X2,T1. (
       ∀X0:term. #[X0] < #[V0] + #[T0] + 1 →
-      â\88\80X1,X2. X0 â\87\92 X1 â\86\92 X0 â\87\92 X2 →
-      â\88\83â\88\83X. X1 â\87\92 X & X2 â\87\92 X
+      â\88\80X1,X2. X0 â\9e¡ X1 â\86\92 X0 â\9e¡ X2 →
+      â\88\83â\88\83X. X1 â\9e¡ X & X2 â\9e¡ X
    ) →
-   T0 â\87\92 T1 â\86\92 T0 â\87\92 X2 →
-   â\88\83â\88\83X. T1 â\87\92 X & X2 â\87\92 X.
+   T0 â\9e¡ T1 â\86\92 T0 â\9e¡ X2 →
+   â\88\83â\88\83X. T1 â\9e¡ X & X2 â\9e¡ X.
 #V0 #T0 #X2 #T1 #IH #HT01 #HT02
 elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /2 width=3/
 qed.
@@ -203,11 +203,11 @@ qed.
 fact tpr_conf_aux:
    ∀Y0:term. (
       ∀X0:term. #[X0] < #[Y0] →
-      â\88\80X1,X2. X0 â\87\92 X1 â\86\92 X0 â\87\92 X2 →
-      â\88\83â\88\83X. X1 â\87\92 X & X2 â\87\92 X
+      â\88\80X1,X2. X0 â\9e¡ X1 â\86\92 X0 â\9e¡ X2 →
+      â\88\83â\88\83X. X1 â\9e¡ X & X2 â\9e¡ X
          ) →
-   â\88\80X0,X1,X2. X0 â\87\92 X1 â\86\92 X0 â\87\92 X2 → X0 = Y0 →
-   â\88\83â\88\83X. X1 â\87\92 X & X2 â\87\92 X.
+   â\88\80X0,X1,X2. X0 â\9e¡ X1 â\86\92 X0 â\9e¡ X2 → X0 = Y0 →
+   â\88\83â\88\83X. X1 â\9e¡ X & X2 â\9e¡ X.
 #Y0 #IH #X0 #X1 #X2 * -X0 -X1
 [ #I1 #H1 #H2 destruct
   lapply (tpr_inv_atom1 … H1) -H1
@@ -281,7 +281,7 @@ fact tpr_conf_aux:
 qed.
 
 (* Basic_1: was: pr0_confluence *)
-theorem tpr_conf: â\88\80T0:term. â\88\80T1,T2. T0 â\87\92 T1 â\86\92 T0 â\87\92 T2 →
-                  â\88\83â\88\83T. T1 â\87\92 T & T2 â\87\92 T.
+theorem tpr_conf: â\88\80T0:term. â\88\80T1,T2. T0 â\9e¡ T1 â\86\92 T0 â\9e¡ T2 →
+                  â\88\83â\88\83T. T1 â\9e¡ T & T2 â\9e¡ T.
 #T @(tw_wf_ind … T) -T /3 width=6 by tpr_conf_aux/
 qed.