X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambda_delta%2FBasic_2%2Freducibility%2Ftpr_tpr.ma;h=b1e3f3c1b053708642613611d3987edead0444f0;hb=70ac3a792389497103fb80b5a1a144706addb7cb;hp=4217ced9aeb6df8d192c9438292901d7d35ff8ae;hpb=f75be90562ddd964ef7ed43b956eb908f3133e3a;p=helm.git

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 4217ced9a..b1e3f3c1b 100644
--- a/matita/matita/contribs/lambda_delta/Basic_2/reducibility/tpr_tpr.ma
+++ b/matita/matita/contribs/lambda_delta/Basic_2/reducibility/tpr_tpr.ma
@@ -18,35 +18,35 @@ include "Basic_2/reducibility/tpr_tpss.ma".
 
 (* Confluence lemmas ********************************************************)
 
-fact tpr_conf_atom_atom: ∀I. ∃∃X. 𝕒{I} ⇒ X & 𝕒{I} ⇒ X.
-/2/ qed.
+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 →
-      ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
-      ∃∃X. X1 ⇒ X & X2 ⇒ X
+      ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+      ∃∃X. X1 ➡ X & X2 ➡ X
    ) →
-   V0 ⇒ V1 → V0 ⇒ V2 → T0 ⇒ T1 → T0 ⇒ T2 →
-   ∃∃T0. 𝕗{I} V1. T1 ⇒ T0 & 𝕗{I} V2. T2 ⇒ T0.
+   V0 ➡ V1 → V0 ➡ V2 → T0 ➡ T1 → T0 ➡ 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 /3 width=5/
+elim (IH … HV01 … HV02) -HV01 -HV02 // #V #HV1 #HV2
+elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /3 width=5/
 qed.
 
 fact tpr_conf_flat_beta:
    ∀V0,V1,T1,V2,W0,U0,T2. (
       ∀X0:term. #[X0] < #[V0] + (#[W0] + #[U0] + 1) + 1 →
-      ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
-      ∃∃X. X1 ⇒ X & X2 ⇒ X
+      ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+      ∃∃X. X1 ➡ X & X2 ➡ X
    ) →
-   V0 ⇒ V1 → V0 ⇒ V2 →
-   U0 ⇒ T2 → 𝕔{Abst} W0. U0 ⇒ T1 →
-   ∃∃X. 𝕔{Appl} V1. T1 ⇒ X & 𝕔{Abbr} V2. T2 ⇒ X.
+   V0 ➡ V1 → V0 ➡ V2 →
+   U0 ➡ T2 → ⓛW0. U0 ➡ 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 -T1;
-elim (IH … HV01 … HV02) -HV01 HV02 // #V #HV1 #HV2
-elim (IH … HT02 … HU01) -HT02 HU01 IH /3 width=5/
+elim (tpr_inv_abst1 … H) -H #W1 #U1 #HW01 #HU01 #H destruct
+elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
+elim (IH … HT02 … HU01) -HT02 -HU01 -IH /2 width=1/ /3 width=5/
 qed.
 
 (* basic-1: was:
@@ -56,32 +56,32 @@ qed.
 fact tpr_conf_flat_theta:
    ∀V0,V1,T1,V2,V,W0,W2,U0,U2. (
       ∀X0:term. #[X0] < #[V0] + (#[W0] + #[U0] + 1) + 1 →
-      ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
-      ∃∃X. X1 ⇒ X & X2 ⇒ X
+      ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+      ∃∃X. X1 ➡ X & X2 ➡ X
    ) →
-   V0 ⇒ V1 → V0 ⇒ V2 → ↑[O,1] V2 ≡ V →
-   W0 ⇒ W2 → U0 ⇒ U2 →  𝕔{Abbr} W0. U0 ⇒ T1 →
-   ∃∃X. 𝕔{Appl} V1. T1 ⇒ X & 𝕔{Abbr} W2. 𝕔{Appl} V. U2 ⇒ X.
+   V0 ➡ V1 → V0 ➡ V2 → ⇧[O,1] V2 ≡ V →
+   W0 ➡ W2 → U0 ➡ U2 →  ⓓW0. U0 ➡ 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 // #VV #HVV1 #HVV2
+elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #VV #HVV1 #HVV2
 elim (lift_total VV 0 1) #VVV #HVV
 lapply (tpr_lift … HVV2 … HV2 … HVV) #HVVV
 elim (tpr_inv_abbr1 … H) -H *
 (* case 1: delta *)
-[ -HV2 HVV2 #WW2 #UU2 #UU #HWW2 #HUU02 #HUU2 #H destruct -T1;
-  elim (IH … HW02 … HWW2) -HW02 HWW2 // #W #HW02 #HWW2
-  elim (IH … HU02 … HUU02) -HU02 HUU02 IH // #U #HU2 #HUUU2
-  elim (tpr_tps_bind … HWW2 HUUU2 … HUU2) -HUU2 HUUU2 #UUU #HUUU2 #HUUU1
+[ -HV2 -HVV2 #WW2 #UU2 #UU #HWW2 #HUU02 #HUU2 #H destruct
+  elim (IH … HW02 … HWW2) -HW02 -HWW2 /2 width=1/ #W #HW02 #HWW2
+  elim (IH … HU02 … HUU02) -HU02 -HUU02 -IH /2 width=1/ #U #HU2 #HUUU2
+  elim (tpr_tps_bind … HWW2 HUUU2 … HUU2) -UU2 #UUU #HUUU2 #HUUU1
   @ex2_1_intro
   [2: @tpr_theta [6: @HVV |7: @HWW2 |8: @HUUU2 |1,2,3,4: skip | // ]
   |1:skip
   |3: @tpr_delta [3: @tpr_flat |1: skip ] /2 width=5/
   ] (**) (* /5 width=14/ is too slow *)
 (* case 3: zeta *)
-| -HW02 HVV HVVV #UU1 #HUU10 #HUUT1
+| -HW02 -HVV -HVVV #UU1 #HUU10 #HUUT1
   elim (tpr_inv_lift … HU02 … HUU10) -HU02 #UU #HUU2 #HUU1
   lapply (tw_lift … HUU10) -HUU10 #HUU10
-  elim (IH … HUUT1 … HUU1) -HUUT1 HUU1 IH // -HUU10 #U #HU2 #HUUU2
+  elim (IH … HUUT1 … HUU1) -HUUT1 -HUU1 -IH /2 width=1/ -HUU10 #U #HU2 #HUUU2
   @ex2_1_intro
   [2: @tpr_flat
   |1: skip 
@@ -93,109 +93,109 @@ qed.
 fact tpr_conf_flat_cast:
    ∀X2,V0,V1,T0,T1. (
       ∀X0:term. #[X0] < #[V0] + #[T0] + 1 →
-      ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
-      ∃∃X. X1 ⇒ X & X2 ⇒ X
+      ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+      ∃∃X. X1 ➡ X & X2 ➡ X
    ) →
-   V0 ⇒ V1 → T0 ⇒ T1 → T0 ⇒ X2 →
-   ∃∃X. 𝕔{Cast} V1. T1 ⇒ X & X2 ⇒ X.
+   V0 ➡ V1 → T0 ➡ T1 → T0 ➡ X2 →
+   ∃∃X. ⓣV1. T1 ➡ X & X2 ➡ X.
 #X2 #V0 #V1 #T0 #T1 #IH #_ #HT01 #HT02
-elim (IH … HT01 … HT02) -HT01 HT02 IH /3/
+elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /3 width=3/
 qed.
 
 fact tpr_conf_beta_beta:
    ∀W0:term. ∀V0,V1,T0,T1,V2,T2. (
       ∀X0:term. #[X0] < #[V0] + (#[W0] + #[T0] + 1) + 1 →
-      ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
-      ∃∃X. X1 ⇒ X & X2 ⇒ X
+      ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+      ∃∃X. X1 ➡ X & X2 ➡ X
    ) →
-   V0 ⇒ V1 → V0 ⇒ V2 → T0 ⇒ T1 → T0 ⇒ T2 →
-   ∃∃X. 𝕔{Abbr} V1. T1 ⇒X & 𝕔{Abbr} V2. T2 ⇒ X.
+   V0 ➡ V1 → V0 ➡ V2 → T0 ➡ T1 → T0 ➡ 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 //
-elim (IH … HT01 … HT02) -HT01 HT02 IH /3 width=5/
+elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/
+elim (IH … HT01 … HT02) -HT01 -HT02 -IH /2 width=1/ /3 width=5/
 qed.
 
 (* Basic_1: was: pr0_cong_delta pr0_delta_delta *)
 fact tpr_conf_delta_delta:
    ∀I1,V0,V1,T0,T1,TT1,V2,T2,TT2. (
       ∀X0:term. #[X0] < #[V0] + #[T0] + 1 →
-      ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
-      ∃∃X. X1 ⇒ X & X2 ⇒ X
+      ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+      ∃∃X. X1 ➡ X & X2 ➡ X
    ) →
-   V0 ⇒ V1 → V0 ⇒ V2 → T0 ⇒ T1 → T0 ⇒ 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 ➡ V1 → V0 ➡ V2 → T0 ➡ T1 → T0 ➡ 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
-elim (tpr_tps_bind … HV1 HT1 … HTT1) -HT1 HTT1 #U1 #TTU1 #HTU1
-elim (tpr_tps_bind … HV2 HT2 … HTT2) -HT2 HTT2 #U2 #TTU2 #HTU2
-elim (tps_conf_eq … HTU1 … HTU2) -HTU1 HTU2 #U #HU1 #HU2
+elim (IH … HV01 … HV02) -HV01 -HV02 // #V #HV1 #HV2
+elim (IH … HT01 … HT02) -HT01 -HT02 -IH // #T #HT1 #HT2
+elim (tpr_tps_bind … HV1 HT1 … HTT1) -T1 #U1 #TTU1 #HTU1
+elim (tpr_tps_bind … HV2 HT2 … HTT2) -T2 #U2 #TTU2 #HTU2
+elim (tps_conf_eq … HTU1 … HTU2) -T #U #HU1 #HU2
 @ex2_1_intro [2,3: @tpr_delta |1: skip ] /width=10/ (**) (* /3 width=10/ is too slow *)
 qed.
 
 fact tpr_conf_delta_zeta:
    ∀X2,V0,V1,T0,T1,TT1,T2. (
       ∀X0:term. #[X0] < #[V0] + #[T0] + 1 →
-      ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
-      ∃∃X. X1 ⇒ X & X2 ⇒ X
+      ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+      ∃∃X. X1 ➡ X & X2 ➡ X
    ) →
-   V0 ⇒ V1 → T0 ⇒ T1 → ⋆. 𝕓{Abbr} V1 ⊢ T1 [O,1] ≫ TT1 →
-   T2 ⇒ X2 → ↑[O, 1] T2 ≡ T0 →
-   ∃∃X. 𝕓{Abbr} V1. TT1 ⇒ X & X2 ⇒ X.
+   V0 ➡ V1 → T0 ➡ T1 → ⋆. ⓓV1 ⊢ T1 [O,1] ▶ TT1 →
+   T2 ➡ X2 → ⇧[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 -T1;
+lapply (tps_inv_lift1_eq … HTT1 … HTT21) -HTT1 #HTT1 destruct
 lapply (tw_lift … HTT20) -HTT20 #HTT20
-elim (IH … HTX2 … HTT2) -HTX2 HTT2 IH /3/
+elim (IH … HTX2 … HTT2) -HTX2 -HTT2 -IH // /3 width=3/
 qed.
 
 (* Basic_1: was: pr0_upsilon_upsilon *)
 fact tpr_conf_theta_theta:
    ∀VV1,V0,V1,W0,W1,T0,T1,V2,VV2,W2,T2. (
       ∀X0:term. #[X0] < #[V0] + (#[W0] + #[T0] + 1) + 1 →
-      ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
-      ∃∃X. X1 ⇒ X & X2 ⇒ X
+      ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+      ∃∃X. X1 ➡ X & X2 ➡ X
    ) →
-   V0 ⇒ V1 → V0 ⇒ V2 → W0 ⇒ W1 → W0 ⇒ W2 → T0 ⇒ T1 → T0 ⇒ T2 →
-   ↑[O, 1] V1 ≡ VV1 → ↑[O, 1] V2 ≡ VV2 →
-   ∃∃X. 𝕔{Abbr} W1. 𝕔{Appl} VV1. T1 ⇒ X & 𝕔{Abbr} W2. 𝕔{Appl} VV2. T2 ⇒ X.
+   V0 ➡ V1 → V0 ➡ V2 → W0 ➡ W1 → W0 ➡ W2 → T0 ➡ T1 → T0 ➡ T2 →
+   ⇧[O, 1] V1 ≡ VV1 → ⇧[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 // #V #HV1 #HV2
-elim (IH … HW01 … HW02) -HW01 HW02 // #W #HW1 #HW2
-elim (IH … HT01 … HT02) -HT01 HT02 IH // #T #HT1 #HT2
+elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
+elim (IH … HW01 … HW02) -HW01 -HW02 /2 width=1/ #W #HW1 #HW2
+elim (IH … HT01 … HT02) -HT01 -HT02 -IH /2 width=1/ #T #HT1 #HT2
 elim (lift_total V 0 1) #VV #HVV
-lapply (tpr_lift … HV1 … HVV1 … HVV) -HV1 HVV1 #HVV1
-lapply (tpr_lift … HV2 … HVV2 … HVV) -HV2 HVV2 HVV #HVV2
+lapply (tpr_lift … HV1 … HVV1 … HVV) -V1 #HVV1
+lapply (tpr_lift … HV2 … HVV2 … HVV) -V2 -HVV #HVV2
 @ex2_1_intro [2,3: @tpr_bind |1:skip ] /2 width=5/ (**) (* /4 width=5/ is too slow *)
 qed.
 
 fact tpr_conf_zeta_zeta:
    ∀V0:term. ∀X2,TT0,T0,T1,T2. (
       ∀X0:term. #[X0] < #[V0] + #[TT0] + 1 →
-      ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
-      ∃∃X. X1 ⇒ X & X2 ⇒ X
+      ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+      ∃∃X. X1 ➡ X & X2 ➡ X
    ) →
-   T0 ⇒ T1 → T2 ⇒ X2 →
-   ↑[O, 1] T0 ≡ TT0 → ↑[O, 1] T2 ≡ TT0 →
-   ∃∃X. T1 ⇒ X & X2 ⇒ X.
+   T0 ➡ T1 → T2 ➡ X2 →
+   ⇧[O, 1] T0 ≡ TT0 → ⇧[O, 1] T2 ≡ TT0 →
+   ∃∃X. T1 ➡ X & X2 ➡ X.
 #V0 #X2 #TT0 #T0 #T1 #T2 #IH #HT01 #HTX2 #HTT0 #HTT20
-lapply (lift_inj … HTT0 … HTT20) -HTT0 #H destruct -T0;
+lapply (lift_inj … HTT0 … HTT20) -HTT0 #H destruct
 lapply (tw_lift … HTT20) -HTT20 #HTT20
-elim (IH … HT01 … HTX2) -HT01 HTX2 IH /2/
+elim (IH … HT01 … HTX2) -HT01 -HTX2 -IH // /2 width=3/
 qed.
 
 fact tpr_conf_tau_tau:
    ∀V0,T0:term. ∀X2,T1. (
       ∀X0:term. #[X0] < #[V0] + #[T0] + 1 →
-      ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
-      ∃∃X. X1 ⇒ X & X2 ⇒ X
+      ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+      ∃∃X. X1 ➡ X & X2 ➡ X
    ) →
-   T0 ⇒ T1 → T0 ⇒ X2 →
-   ∃∃X. T1 ⇒ X & X2 ⇒ X.
+   T0 ➡ T1 → T0 ➡ X2 →
+   ∃∃X. T1 ➡ X & X2 ➡ X.
 #V0 #T0 #X2 #T1 #IH #HT01 #HT02
-elim (IH … HT01 … HT02) -HT01 HT02 IH /2/
+elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /2 width=3/
 qed.
 
 (* Confluence ***************************************************************)
@@ -203,75 +203,75 @@ qed.
 fact tpr_conf_aux:
    ∀Y0:term. (
       ∀X0:term. #[X0] < #[Y0] →
-      ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
-      ∃∃X. X1 ⇒ X & X2 ⇒ X
+      ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+      ∃∃X. X1 ➡ X & X2 ➡ X
          ) →
-   ∀X0,X1,X2. X0 ⇒ X1 → X0 ⇒ X2 → X0 = Y0 →
-   ∃∃X. X1 ⇒ X & X2 ⇒ X.
-#Y0 #IH #X0 #X1 #X2 * -X0 X1
-[ #I1 #H1 #H2 destruct -Y0;
+   ∀X0,X1,X2. X0 ➡ X1 → X0 ➡ X2 → X0 = Y0 →
+   ∃∃X. X1 ➡ X & X2 ➡ X.
+#Y0 #IH #X0 #X1 #X2 * -X0 -X1
+[ #I1 #H1 #H2 destruct
   lapply (tpr_inv_atom1 … H1) -H1
 (* case 1: atom, atom *)
-  #H1 destruct -X2 //
-| #I #V0 #V1 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct -Y0;
+  #H1 destruct //
+| #I #V0 #V1 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct
   elim (tpr_inv_flat1 … H1) -H1 *
 (* case 2: flat, flat *)
-  [ #V2 #T2 #HV02 #HT02 #H destruct -X2
+  [ #V2 #T2 #HV02 #HT02 #H destruct
     /3 width=7 by tpr_conf_flat_flat/ (**) (* /3 width=7/ is too slow *)
 (* case 3: flat, beta *)
-  | #V2 #W #U0 #T2 #HV02 #HT02 #H1 #H2 #H3 destruct -T0 X2 I
+  | #V2 #W #U0 #T2 #HV02 #HT02 #H1 #H2 #H3 destruct
     /3 width=8 by tpr_conf_flat_beta/ (**) (* /3 width=8/ is too slow *)
 (* case 4: flat, theta *)
-  | #V2 #V #W0 #W2 #U0 #U2 #HV02 #HW02 #HT02 #HV2 #H1 #H2 #H3 destruct -T0 X2 I
+  | #V2 #V #W0 #W2 #U0 #U2 #HV02 #HW02 #HT02 #HV2 #H1 #H2 #H3 destruct
     /3 width=11 by tpr_conf_flat_theta/ (**) (* /3 width=11/ is too slow *)
 (* case 5: flat, tau *)
-  | #HT02 #H destruct -I
+  | #HT02 #H destruct
     /3 width=6 by tpr_conf_flat_cast/ (**) (* /3 width=6/ is too slow *)
   ]
-| #V0 #V1 #W0 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct -Y0;
+| #V0 #V1 #W0 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct
   elim (tpr_inv_appl1 … H1) -H1 *
 (* case 6: beta, flat (repeated) *)
-  [ #V2 #T2 #HV02 #HT02 #H destruct -X2
+  [ #V2 #T2 #HV02 #HT02 #H destruct
     @ex2_1_comm /3 width=8 by tpr_conf_flat_beta/
 (* case 7: beta, beta *)
-  | #V2 #WW0 #TT0 #T2 #HV02 #HT02 #H1 #H2 destruct -W0 T0 X2
+  | #V2 #WW0 #TT0 #T2 #HV02 #HT02 #H1 #H2 destruct
     /3 width=8 by tpr_conf_beta_beta/ (**) (* /3 width=8/ is too slow *)
 (* case 8, beta, theta (excluded) *)
   | #V2 #VV2 #WW0 #W2 #TT0 #T2 #_ #_ #_ #_ #H destruct
   ]
-| #I1 #V0 #V1 #T0 #T1 #TT1 #HV01 #HT01 #HTT1 #H1 #H2 destruct -Y0;
+| #I1 #V0 #V1 #T0 #T1 #TT1 #HV01 #HT01 #HTT1 #H1 #H2 destruct
   elim (tpr_inv_bind1 … H1) -H1 *
 (* case 9: delta, delta *)
-  [ #V2 #T2 #TT2 #HV02 #HT02 #HTT2 #H destruct -X2
+  [ #V2 #T2 #TT2 #HV02 #HT02 #HTT2 #H destruct
     /3 width=11 by tpr_conf_delta_delta/ (**) (* /3 width=11/ is too slow *)
 (* case 10: delta, zata *)
-  | #T2 #HT20 #HTX2 #H destruct -I1;
+  | #T2 #HT20 #HTX2 #H destruct
     /3 width=10 by tpr_conf_delta_zeta/ (**) (* /3 width=10/ is too slow *)
   ]
-| #VV1 #V0 #V1 #W0 #W1 #T0 #T1 #HV01 #HVV1 #HW01 #HT01 #H1 #H2 destruct -Y0;
+| #VV1 #V0 #V1 #W0 #W1 #T0 #T1 #HV01 #HVV1 #HW01 #HT01 #H1 #H2 destruct
   elim (tpr_inv_appl1 … H1) -H1 *
 (* case 11: theta, flat (repeated) *)
-  [ #V2 #T2 #HV02 #HT02 #H destruct -X2
+  [ #V2 #T2 #HV02 #HT02 #H destruct
     @ex2_1_comm /3 width=11 by tpr_conf_flat_theta/
 (* case 12: theta, beta (repeated) *)
   | #V2 #WW0 #TT0 #T2 #_ #_ #H destruct
 (* case 13: theta, theta *)
-  | #V2 #VV2 #WW0 #W2 #TT0 #T2 #V02 #HW02 #HT02 #HVV2 #H1 #H2 destruct -W0 T0 X2
+  | #V2 #VV2 #WW0 #W2 #TT0 #T2 #V02 #HW02 #HT02 #HVV2 #H1 #H2 destruct
     /3 width=14 by tpr_conf_theta_theta/ (**) (* /3 width=14/ is too slow *)
   ]
-| #V0 #TT0 #T0 #T1 #HTT0 #HT01 #H1 #H2 destruct -Y0;
+| #V0 #TT0 #T0 #T1 #HTT0 #HT01 #H1 #H2 destruct
   elim (tpr_inv_abbr1 … H1) -H1 *
 (* case 14: zeta, delta (repeated) *)
-  [ #V2 #T2 #TT2 #HV02 #HT02 #HTT2 #H destruct -X2
+  [ #V2 #T2 #TT2 #HV02 #HT02 #HTT2 #H destruct
     @ex2_1_comm /3 width=10 by tpr_conf_delta_zeta/
 (* case 15: zeta, zeta *)
   | #T2 #HTT20 #HTX2
     /3 width=9 by tpr_conf_zeta_zeta/ (**) (* /3 width=9/ is too slow *)
   ] 
-| #V0 #T0 #T1 #HT01 #H1 #H2 destruct -Y0;
+| #V0 #T0 #T1 #HT01 #H1 #H2 destruct
   elim (tpr_inv_cast1 … H1) -H1
 (* case 16: tau, flat (repeated) *)
-  [ * #V2 #T2 #HV02 #HT02 #H destruct -X2
+  [ * #V2 #T2 #HV02 #HT02 #H destruct
     @ex2_1_comm /3 width=6 by tpr_conf_flat_cast/
 (* case 17: tau, tau *)
   | #HT02
@@ -281,7 +281,7 @@ fact tpr_conf_aux:
 qed.
 
 (* Basic_1: was: pr0_confluence *)
-theorem tpr_conf: ∀T0:term. ∀T1,T2. T0 ⇒ T1 → T0 ⇒ T2 →
-                  ∃∃T. T1 ⇒ T & T2 ⇒ T.
+theorem tpr_conf: ∀T0:term. ∀T1,T2. T0 ➡ T1 → T0 ➡ T2 →
+                  ∃∃T. T1 ➡ T & T2 ➡ T.
 #T @(tw_wf_ind … T) -T /3 width=6 by tpr_conf_aux/
 qed.