Basic_2: - we addedsome files

[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 4217ced9aeb6df8d192c9438292901d7d35ff8ae..e541b5777323fc633e2f725ebabd92ad9c46c9e4 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
@@ -19,7 +19,7 @@ include "Basic_2/reducibility/tpr_tpss.ma".
 (* Confluence lemmas ********************************************************)
 
 fact tpr_conf_atom_atom: ∀I. ∃∃X. 𝕒{I} ⇒ X & 𝕒{I} ⇒ X.
 (* Confluence lemmas ********************************************************)
 
 fact tpr_conf_atom_atom: ∀I. ∃∃X. 𝕒{I} ⇒ X & 𝕒{I} ⇒ X.

-/2/ qed.
+/2 width=3/ qed.

 
 fact tpr_conf_flat_flat:
    ∀I,V0,V1,T0,T1,V2,T2. (
 
 fact tpr_conf_flat_flat:
    ∀I,V0,V1,T0,T1,V2,T2. (


@@ -30,8 +30,8 @@ fact tpr_conf_flat_flat:
    V0 ⇒ V1 → V0 ⇒ V2 → T0 ⇒ T1 → T0 ⇒ T2 →
    ∃∃T0. 𝕗{I} V1. T1 ⇒ T0 & 𝕗{I} V2. T2 ⇒ T0.
 #I #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02
    V0 ⇒ V1 → V0 ⇒ V2 → T0 ⇒ T1 → T0 ⇒ T2 →
    ∃∃T0. 𝕗{I} V1. T1 ⇒ T0 & 𝕗{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:
 qed.
 
 fact tpr_conf_flat_beta:


@@ -44,9 +44,9 @@ fact tpr_conf_flat_beta:
    U0 ⇒ T2 → 𝕔{Abst} W0. U0 ⇒ T1 →
    ∃∃X. 𝕔{Appl} V1. T1 ⇒ X & 𝕔{Abbr} V2. T2 ⇒ X.
 #V0 #V1 #T1 #V2 #W0 #U0 #T2 #IH #HV01 #HV02 #HT02 #H
    U0 ⇒ T2 → 𝕔{Abst} W0. U0 ⇒ T1 →
    ∃∃X. 𝕔{Appl} V1. T1 ⇒ X & 𝕔{Abbr} 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:
 qed.
 
 (* basic-1: was:


@@ -59,29 +59,29 @@ fact tpr_conf_flat_theta:
       ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
       ∃∃X. X1 ⇒ X & X2 ⇒ X
    ) →
       ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
       ∃∃X. X1 ⇒ X & X2 ⇒ X
    ) →

-   V0 â\87\92 V1 â\86\92 V0 â\87\92 V2 â\86\92 â\86\91[O,1] V2 â\89¡ V â\86\92
+   V0 â\87\92 V1 â\86\92 V0 â\87\92 V2 â\86\92 â\87\91[O,1] V2 â\89¡ V â\86\92

    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 #T1 #V2 #V #W0 #W2 #U0 #U2 #IH #HV01 #HV02 #HV2 #HW02 #HU02 #H
    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 #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 *)
 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 *)
   @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 (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 
   @ex2_1_intro
   [2: @tpr_flat
   |1: skip 


@@ -99,7 +99,7 @@ fact tpr_conf_flat_cast:
    V0 ⇒ V1 → T0 ⇒ T1 → T0 ⇒ X2 →
    ∃∃X. 𝕔{Cast} V1. T1 ⇒ X & X2 ⇒ X.
 #X2 #V0 #V1 #T0 #T1 #IH #_ #HT01 #HT02
    V0 ⇒ V1 → T0 ⇒ T1 → T0 ⇒ X2 →
    ∃∃X. 𝕔{Cast} 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:
 qed.
 
 fact tpr_conf_beta_beta:


@@ -111,8 +111,8 @@ fact tpr_conf_beta_beta:
    V0 ⇒ V1 → V0 ⇒ V2 → T0 ⇒ T1 → T0 ⇒ T2 →
    ∃∃X. 𝕔{Abbr} V1. T1 ⇒X & 𝕔{Abbr} V2. T2 ⇒ X.
 #W0 #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02
    V0 ⇒ V1 → V0 ⇒ V2 → T0 ⇒ T1 → T0 ⇒ T2 →
    ∃∃X. 𝕔{Abbr} V1. T1 ⇒X & 𝕔{Abbr} 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 *)
 qed.
 
 (* Basic_1: was: pr0_cong_delta pr0_delta_delta *)


@@ -127,11 +127,11 @@ fact tpr_conf_delta_delta:
    ⋆. 𝕓{I1} V2 ⊢ T2 [O, 1] ≫ TT2 →
    ∃∃X. 𝕓{I1} V1. TT1 ⇒ X & 𝕓{I1} V2. TT2 ⇒ X.
 #I1 #V0 #V1 #T0 #T1 #TT1 #V2 #T2 #TT2 #IH #HV01 #HV02 #HT01 #HT02 #HTT1 #HTT2
    ⋆. 𝕓{I1} V2 ⊢ T2 [O, 1] ≫ TT2 →
    ∃∃X. 𝕓{I1} V1. TT1 ⇒ X & 𝕓{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.
 
 @ex2_1_intro [2,3: @tpr_delta |1: skip ] /width=10/ (**) (* /3 width=10/ is too slow *)
 qed.
 


@@ -142,13 +142,13 @@ fact tpr_conf_delta_zeta:
       ∃∃X. X1 ⇒ X & X2 ⇒ X
    ) →
    V0 ⇒ V1 → T0 ⇒ T1 → ⋆. 𝕓{Abbr} V1 ⊢ T1 [O,1] ≫ TT1 →
       ∃∃X. X1 ⇒ X & X2 ⇒ X
    ) →
    V0 ⇒ V1 → T0 ⇒ T1 → ⋆. 𝕓{Abbr} V1 ⊢ T1 [O,1] ≫ TT1 →

-   T2 â\87\92 X2 â\86\92 â\86\91[O, 1] T2 â\89¡ T0 â\86\92
+   T2 â\87\92 X2 â\86\92 â\87\91[O, 1] T2 â\89¡ T0 â\86\92

    ∃∃X. 𝕓{Abbr} 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
    ∃∃X. 𝕓{Abbr} 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
 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 *)
 qed.
 
 (* Basic_1: was: pr0_upsilon_upsilon *)


@@ -159,15 +159,15 @@ fact tpr_conf_theta_theta:
       ∃∃X. X1 ⇒ X & X2 ⇒ X
    ) →
    V0 ⇒ V1 → V0 ⇒ V2 → W0 ⇒ W1 → W0 ⇒ W2 → T0 ⇒ T1 → T0 ⇒ T2 →
       ∃∃X. X1 ⇒ X & X2 ⇒ X
    ) →
    V0 ⇒ V1 → V0 ⇒ V2 → W0 ⇒ W1 → W0 ⇒ W2 → T0 ⇒ T1 → T0 ⇒ T2 →

-   â\86\91[O, 1] V1 â\89¡ VV1 â\86\92 â\86\91[O, 1] V2 â\89¡ VV2 â\86\92
+   â\87\91[O, 1] V1 â\89¡ VV1 â\86\92 â\87\91[O, 1] V2 â\89¡ VV2 â\86\92

    ∃∃X. 𝕔{Abbr} W1. 𝕔{Appl} VV1. T1 ⇒ X & 𝕔{Abbr} W2. 𝕔{Appl} VV2. T2 ⇒ X.
 #VV1 #V0 #V1 #W0 #W1 #T0 #T1 #V2 #VV2 #W2 #T2 #IH #HV01 #HV02 #HW01 #HW02 #HT01 #HT02 #HVV1 #HVV2
    ∃∃X. 𝕔{Abbr} W1. 𝕔{Appl} VV1. T1 ⇒ X & 𝕔{Abbr} W2. 𝕔{Appl} 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
 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.
 
 @ex2_1_intro [2,3: @tpr_bind |1:skip ] /2 width=5/ (**) (* /4 width=5/ is too slow *)
 qed.
 


@@ -178,12 +178,12 @@ fact tpr_conf_zeta_zeta:
       ∃∃X. X1 ⇒ X & X2 ⇒ X
    ) →
    T0 ⇒ T1 → T2 ⇒ X2 →
       ∃∃X. X1 ⇒ X & X2 ⇒ X
    ) →
    T0 ⇒ T1 → T2 ⇒ X2 →

-   â\86\91[O, 1] T0 â\89¡ TT0 â\86\92 â\86\91[O, 1] T2 â\89¡ TT0 â\86\92
+   â\87\91[O, 1] T0 â\89¡ TT0 â\86\92 â\87\91[O, 1] T2 â\89¡ TT0 â\86\92

    ∃∃X. T1 ⇒ X & X2 ⇒ X.
 #V0 #X2 #TT0 #T0 #T1 #T2 #IH #HT01 #HTX2 #HTT0 #HTT20
    ∃∃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
 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:
 qed.
 
 fact tpr_conf_tau_tau:


@@ -195,7 +195,7 @@ fact tpr_conf_tau_tau:
    T0 ⇒ T1 → T0 ⇒ X2 →
    ∃∃X. T1 ⇒ X & X2 ⇒ X.
 #V0 #T0 #X2 #T1 #IH #HT01 #HT02
    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 ***************************************************************)
 qed.
 
 (* Confluence ***************************************************************)


@@ -208,70 +208,70 @@ fact tpr_conf_aux:
          ) →
    ∀X0,X1,X2. X0 ⇒ X1 → X0 ⇒ X2 → X0 = Y0 →
    ∃∃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;
+#Y0 #IH #X0 #X1 #X2 * -X0 -X1
+[ #I1 #H1 #H2 destruct

   lapply (tpr_inv_atom1 … H1) -H1
 (* case 1: atom, atom *)
   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 *)
   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 *)
     /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 *)
     /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 *)
     /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 *)
   ]
     /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) *)
   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 *)
     @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
   ]
     /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 *)
   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 *)
     /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 *)
   ]
     /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) *)
   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 *)
     @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 *)
   ]
     /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) *)
   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 *)
   ] 
     @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) *)
   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
     @ex2_1_comm /3 width=6 by tpr_conf_flat_cast/
 (* case 17: tau, tau *)
   | #HT02