V_______________________________________________________________ *)
include "lambda-delta/substitution/lift_weight.ma".
-include "lambda-delta/reduction/tpr_main.ma".
+include "lambda-delta/substitution/tps_tps.ma".
+include "lambda-delta/reduction/tpr_lift.ma".
+include "lambda-delta/reduction/tpr_tps.ma".
(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
(* Confluence lemmas ********************************************************)
-lemma tpr_conf_sort_sort: ∀k1. ∃∃T0. ⋆k1 ⇒ T0 & ⋆k1 ⇒ T0.
+lemma tpr_conf_sort_sort: ∀k. ∃∃X. ⋆k ⇒ X & ⋆k ⇒ X.
/2/ qed.
-lemma tpr_conf_lref_lref: ∀i1. ∃∃T0. #i1 ⇒ T0 & #i1 ⇒ T0.
+lemma tpr_conf_lref_lref: ∀i. ∃∃X. #i ⇒ X & #i ⇒ X.
/2/ qed.
lemma tpr_conf_bind_bind:
- ∀I1,V11,V12,T11,T12,V22,T22. (
- ∀T1. #T1 < #V11 + #T11 + 1 →
- ∀T3,T4. T1 ⇒ T3 → T1 ⇒ T4 →
- ∃∃T0. T3 ⇒ T0 & T4 ⇒ T0
+ ∀I,V0,V1,T0,T1,V2,T2. (
+ ∀X0:term. #X0 < #V0 + #T0 + 1 →
+ ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
+ ∃∃X. X1 ⇒ X & X2 ⇒ X
) →
- V11 ⇒ V12 → T11 ⇒ T12 →
- V11 ⇒ V22 → T11 ⇒ T22 →
- ∃∃T0. 𝕓{I1} V12. T12 ⇒ T0 & 𝕓{I1} V22. T22 ⇒ T0.
-#I1 #V11 #V12 #T11 #T12 #V22 #T22 #IH #HV1 #HT1 #HV2 #HT2
-elim (IH … HV1 … HV2) -HV1 HV2 // #V #HV1 #HV2
-elim (IH … HT1 … HT2) -HT1 HT2 IH /3 width=5/
+ V0 ⇒ V1 → V0 ⇒ V2 → T0 ⇒ T1 → T0 ⇒ T2 →
+ ∃∃X. 𝕓{I} V1. T1 ⇒ X & 𝕓{I} V2. T2 ⇒ X.
+#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/
+qed.
+
+lemma tpr_conf_bind_delta:
+ ∀V0,V1,T0,T1,V2,T2,T. (
+ ∀X0:term. #X0 < #V0 + #T0 + 1 →
+ ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
+ ∃∃X. X1 ⇒ X & X2 ⇒ X
+ ) →
+ V0 ⇒ V1 → V0 ⇒ V2 →
+ T0 ⇒ T1 → T0 ⇒ T2 → ⋆. 𝕓{Abbr} V2 ⊢ T2 [O,1] ≫ T →
+ ∃∃X. 𝕓{Abbr} V1. T1 ⇒ X & 𝕓{Abbr} V2. T ⇒ X.
+#V0 #V1 #T0 #T1 #V2 #T2 #T #IH #HV01 #HV02 #HT01 #HT02 #HT2
+elim (IH … HV01 … HV02) -HV01 HV02 // #V #HV1 #HV2
+elim (IH … HT01 … HT02) -HT01 HT02 IH // -V0 T0 #T0 #HT10 #HT20
+elim (tpr_tps_bind … HV2 HT20 … HT2) -HT20 HT2 /3 width=5/
qed.
lemma tpr_conf_bind_zeta:
- ∀V11,V12,T11,T12,T22,T20. (
- ∀T1. #T1 < #V11 + #T11 +1 →
- ∀T3,T4. T1 ⇒ T3 → T1 ⇒ T4 →
- ∃∃T0. T3 ⇒ T0 & T4 ⇒ T0
+ ∀X2,V0,V1,T0,T1,T. (
+ ∀X0:term. #X0 < #V0 + #T0 +1 →
+ ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
+ ∃∃X. X1 ⇒ X & X2 ⇒ X
) →
- V11 ⇒ V12 → T22 ⇒ T20 → T11 ⇒ T12 → ↑[O, 1] T22 ≡ T11 →
- ∃∃T0. 𝕓{Abbr} V12. T12 ⇒ T0 & T20 ⇒ T0.
-#V11 #V12 #T11 #T12 #T22 #T20 #IH #HV112 #HT202 #HT112 #HT
-elim (tpr_inv_lift … HT112 … HT) -HT112 #T #HT12 #HT22
-lapply (tw_lift … HT) -HT #HT
-elim (IH … HT202 … HT22) -HT202 HT22 IH /3/
+ V0 ⇒ V1 → T0 ⇒ T1 → T ⇒ X2 → ↑[O, 1] T ≡ T0 →
+ ∃∃X. 𝕓{Abbr} V1. T1 ⇒ X & X2 ⇒ X.
+#X2 #V0 #V1 #T0 #T1 #T #IH #HV01 #HT01 #HTX2 #HT0
+elim (tpr_inv_lift … HT01 … HT0) -HT01 #U #HUT1 #HTU
+lapply (tw_lift … HT0) -HT0 #HT0
+elim (IH … HTX2 … HTU) -HTX2 HTU IH /3/
qed.
lemma tpr_conf_flat_flat:
- ∀I1,V11,V12,T11,T12,V22,T22. (
- ∀T1. #T1 < #V11 + #T11 + 1 →
- ∀T3,T4. T1 ⇒ T3 → T1 ⇒ T4 →
- ∃∃T0. T3 ⇒ T0 & T4 ⇒ T0
+ ∀I,V0,V1,T0,T1,V2,T2. (
+ ∀X0:term. #X0 < #V0 + #T0 + 1 →
+ ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
+ ∃∃X. X1 ⇒ X & X2 ⇒ X
) →
- V11 ⇒ V12 → T11 ⇒ T12 →
- V11 ⇒ V22 → T11 ⇒ T22 →
- ∃∃T0. 𝕗{I1} V12. T12 ⇒ T0 & 𝕗{I1} V22. T22 ⇒ T0.
-#I1 #V11 #V12 #T11 #T12 #V22 #T22 #IH #HV1 #HT1 #HV2 #HT2
-elim (IH … HV1 … HV2) -HV1 HV2 // #V #HV1 #HV2
-elim (IH … HT1 … HT2) -HT1 HT2 /3 width=5/
+ 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/
qed.
lemma tpr_conf_flat_beta:
- ∀V11,V12,T12,V22,W2,T21,T22. (
- ∀T1. #T1 < #V11 + (#W2 + #T21 + 1) + 1 →
- ∀T3,T4. T1 ⇒ T3 → T1 ⇒ T4 →
- ∃∃T0. T3 ⇒ T0 & T4 ⇒ T0
+ ∀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
) →
- V11 ⇒ V12 → V11 ⇒ V22 →
- T21 ⇒ T22 → 𝕓{Abst} W2. T21 ⇒ T12 →
- ∃∃T0. 𝕗{Appl} V12. T12 ⇒ T0 & 𝕓{Abbr} V22. T22 ⇒ T0.
-#V11 #V12 #T12 #V22 #W2 #T21 #T22 #IH #HV1 #HV2 #HT1 #HT2
-elim (tpr_inv_abst1 … HT2) -HT2 #W1 #T1 #HW21 #HT21 #H destruct -T12;
-elim (IH … HV1 … HV2) -HV1 HV2 // #V #HV12 #HV22
-elim (IH … HT21 … HT1) -HT21 HT1 IH /3 width=5/
+ V0 ⇒ V1 → V0 ⇒ V2 →
+ U0 ⇒ T2 → 𝕓{Abst} W0. U0 ⇒ T1 →
+ ∃∃X. 𝕗{Appl} V1. T1 ⇒ X & 𝕓{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/
qed.
lemma tpr_conf_flat_theta:
- ∀V11,V12,T12,V2,V22,W21,W22,T21,T22. (
- ∀T1. #T1 < #V11 + (#W21 + #T21 + 1) + 1 →
- ∀T3,T4. T1 ⇒ T3 → T1 ⇒ T4 →
- ∃∃T0. T3 ⇒ T0 & T4 ⇒T0
+ ∀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
+ ) →
+ V0 ⇒ V1 → V0 ⇒ V2 → ↑[O,1] V2 ≡ V →
+ W0 ⇒ W2 → U0 ⇒ U2 → 𝕓{Abbr} W0. U0 ⇒ T1 →
+ ∃∃X. 𝕗{Appl} V1. T1 ⇒ X & 𝕓{Abbr} W2. 𝕗{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 (lift_total VV 0 1) #VVV #HVV
+lapply (tpr_lift … HVV2 … HV2 … HVV) #HVVV
+elim (tpr_inv_abbr1 … H) -H *
+(* case 1: bind *)
+[ -HV2 HVV2 #WW #UU #HWW0 #HUU0 #H destruct -T1;
+ elim (IH … HW02 … HWW0) -HW02 HWW0 // #W #HW2 #HWW
+ elim (IH … HU02 … HUU0) -HU02 HUU0 IH // #U #HU2 #HUU
+ @ex2_1_intro [2: @tpr_theta |1:skip |3: @tpr_bind ] /2 width=7/ (**) (* /4 width=7/ is too slow *)
+(* case 2: 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
+ @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
+ 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
+ @ex2_1_intro
+ [2: @tpr_flat
+ |1: skip
+ |3: @tpr_zeta [2: @lift_flat |1: skip |3: @tpr_flat ]
+ ] /2 width=5/ (**) (* /5 width=5/ is too slow *)
+]
+qed.
+
+lemma 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
+ ) →
+ V0 ⇒ V1 → T0 ⇒ T1 → T0 ⇒ X2 →
+ ∃∃X. 𝕗{Cast} V1. T1 ⇒ X & X2 ⇒ X.
+#X2 #V0 #V1 #T0 #T1 #IH #_ #HT01 #HT02
+elim (IH … HT01 … HT02) -HT01 HT02 IH /3/
+qed.
+
+lemma 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
+ ) →
+ V0 ⇒ V1 → V0 ⇒ V2 → T0 ⇒ T1 → T0 ⇒ T2 →
+ ∃∃X. 𝕓{Abbr} V1. T1 ⇒X & 𝕓{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/
+qed.
+
+lemma tpr_conf_delta_delta:
+ ∀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
) →
- V11 ⇒ V12 → V11 ⇒ V22 → ↑[O,1] V22 ≡ V2 →
- W21 ⇒ W22 → T21 ⇒ T22 → 𝕓{Abbr} W21. T21 ⇒ T12 →
- ∃∃T0. 𝕗{Appl} V12. T12 ⇒ T0 & 𝕓{Abbr} W22. 𝕗{Appl} V2. T22 ⇒T0.
+ V0 ⇒ V1 → V0 ⇒ V2 → T0 ⇒ T1 → T0 ⇒ T2 →
+ ⋆. 𝕓{Abbr} V1 ⊢ T1 [O, 1] ≫ TT1 →
+ ⋆. 𝕓{Abbr} V2 ⊢ T2 [O, 1] ≫ TT2 →
+ ∃∃X. 𝕓{Abbr} V1. TT1 ⇒ X & 𝕓{Abbr} V2. TT2 ⇒ X.
+#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 … HTU1 … HTU2) -HTU1 HTU2 #U #HU1 #HU2
+@ex2_1_intro [2,3: @tpr_delta |1: skip ] /width=10/ (**) (* /3 width=10/ is too slow *)
+qed.
+lemma 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
+ ) →
+ 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.
+#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 (tw_lift … HTT20) -HTT20 #HTT20
+elim (IH … HTX2 … HTT2) -HTX2 HTT2 IH /3/
+qed.
+
+lemma 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
+ ) →
+ V0 ⇒ V1 → V0 ⇒ V2 → W0 ⇒ W1 → W0 ⇒ W2 → T0 ⇒ T1 → T0 ⇒ T2 →
+ ↑[O, 1] V1 ≡ VV1 → ↑[O, 1] V2 ≡ VV2 →
+ ∃∃X. 𝕓{Abbr} W1. 𝕗{Appl} VV1. T1 ⇒ X & 𝕓{Abbr} W2. 𝕗{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 (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
+@ex2_1_intro [2,3: @tpr_bind |1:skip ] /2 width=5/ (**) (* /4 width=5/ is too slow *)
+qed.
+
+lemma 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
+ ) →
+ 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 (tw_lift … HTT20) -HTT20 #HTT20
+elim (IH … HT01 … HTX2) -HT01 HTX2 IH /2/
+qed.
+
+lemma 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
+ ) →
+ T0 ⇒ T1 → T0 ⇒ X2 →
+ ∃∃X. T1 ⇒ X & X2 ⇒ X.
+#V0 #T0 #X2 #T1 #IH #HT01 #HT02
+elim (IH … HT01 … HT02) -HT01 HT02 IH /2/
+qed.
(* Confluence ***************************************************************)
lemma tpr_conf_aux:
- ∀T. (
- ∀T1. #T1 < #T → ∀T3,T4. T1 ⇒ T3 → T1 ⇒ T4 →
- ∃∃T0. T3 ⇒ T0 & T4 ⇒ T0
+ ∀Y0:term. (
+ ∀X0:term. #X0 < #Y0 → ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
+ ∃∃X. X1 ⇒ X & X2 ⇒ X
) →
- ∀U1,T1,U2,T2. U1 ⇒ T1 → U2 ⇒ T2 →
- U1 = T → U2 = T →
- ∃∃T0. T1 ⇒ T0 & T2 ⇒ T0.
-#T #IH #U1 #T1 #U2 #T2
-* -U1 T1
-[ #k1 * -U2 T2
+ ∀X0,X1,X2. X0 ⇒ X1 → X0 ⇒ X2 → X0 = Y0 →
+ ∃∃X. X1 ⇒ X & X2 ⇒ X.
+#Y0 #IH #X0 #X1 #X2 * -X0 X1
+[ #k1 #H1 #H2 destruct -Y0;
+ lapply (tpr_inv_sort1 … H1) -H1
(* case 1: sort, sort *)
- [ #k2 #H1 #H2 destruct -T k2 //
-(* case 2: sort, lref (excluded) *)
- | #i2 #H1 #H2 destruct
-(* case 3: sort, bind (excluded) *)
- | #I2 #V21 #V22 #T21 #T22 #_ #_ #H1 #H2 destruct
-(* case 4: sort, flat (excluded) *)
- | #I2 #V21 #V22 #T21 #T22 #_ #_ #H1 #H2 destruct
-(* case 5: sort, beta (excluded) *)
- | #V21 #V22 #W2 #T21 #T22 #_ #_ #H1 #H2 destruct
-(* case 6: sort, delta (excluded) *)
- | #V21 #V22 #T21 #T22 #T20 #_ #_ #_ #H1 #H2 destruct
-(* case 7: sort, theta (excluded) *)
- | #V2 #V21 #V22 #W21 #W22 #T21 #T22 #_ #_ #_ #_ #H1 #H2 destruct
-(* case 8: sort, zeta (excluded) *)
- | #V2 #T21 #T22 #T20 #_ #_ #H1 #H2 destruct
-(* case 9: sort, tau (excluded) *)
- | #V2 #T21 #T22 #_ #H1 #H2 destruct
+ #H1 destruct -X2 //
+| #i1 #H1 #H2 destruct -Y0;
+ lapply (tpr_inv_lref1 … H1) -H1
+(* case 2: lref, lref *)
+ #H1 destruct -X2 //
+| #I #V0 #V1 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct -Y0;
+ elim (tpr_inv_bind1 … H1) -H1 *
+(* case 3: bind, bind *)
+ [ #V2 #T2 #HV02 #HT02 #H destruct -X2
+ /3 width=7 by tpr_conf_bind_bind/ (**) (* /3 width=7/ is too slow *)
+(* case 4: bind, delta *)
+ | #V2 #T2 #T #HV02 #HT02 #HT2 #H1 #H2 destruct -X2 I
+ /3 width=9 by tpr_conf_bind_delta/ (**) (* /3 width=9/ is too slow *)
+(* case 5: bind, zeta *)
+ | #T #HT0 #HTX2 #H destruct -I
+ /3 width=8 by tpr_conf_bind_zeta/ (**) (* /3 width=8/ is too slow *)
]
-| #i1 * -U2 T2
-(* case 10: lref, sort (excluded) broken *)
- [ #k2 #H1 #H2 destruct
-(* case 11: lref, sort (excluded) *)
- | #i2 #H1 #H2 destruct -T i2 //
-(* case 12: lref, bind (excluded) *)
- | #I2 #V21 #V22 #T21 #T22 #_ #_ #H1 #H2 destruct
-(* case 13: lref, flat (excluded) *)
- | #I2 #V21 #V22 #T21 #T22 #_ #_ #H1 #H2 destruct
-(* case 14: lref, beta (excluded) *)
- | #V21 #V22 #W2 #T21 #T22 #_ #_ #H1 #H2 destruct
-(* case 15: lref, delta (excluded) *)
- | #V21 #V22 #T21 #T22 #T20 #_ #_ #_ #H1 #H2 destruct
-(* case 16: lref, theta (excluded) *)
- | #V2 #V21 #V22 #W21 #W22 #T21 #T22 #_ #_ #_ #_ #H1 #H2 destruct
-(* case 17: lref, zeta (excluded) *)
- | #V2 #T21 #T22 #T20 #_ #_ #H1 #H2 destruct
-(* case 18: lref, tau (excluded) *)
- | #V2 #T21 #T22 #_ #H1 #H2 destruct
+| #I #V0 #V1 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct -Y0;
+ elim (tpr_inv_flat1 … H1) -H1 *
+(* case 6: flat, flat *)
+ [ #V2 #T2 #HV02 #HT02 #H destruct -X2
+ /3 width=7 by tpr_conf_flat_flat/ (**) (* /3 width=7/ is too slow *)
+(* case 7: flat, beta *)
+ | #V2 #W #U0 #T2 #HV02 #HT02 #H1 #H2 #H3 destruct -T0 X2 I
+ /3 width=8 by tpr_conf_flat_beta/ (**) (* /3 width=8/ is too slow *)
+(* case 8: flat, theta *)
+ | #V2 #V #W0 #W2 #U0 #U2 #HV02 #HW02 #HT02 #HV2 #H1 #H2 #H3 destruct -T0 X2 I
+ /3 width=11 by tpr_conf_flat_theta/ (**) (* /3 width=11/ is too slow *)
+(* case 9: flat, tau *)
+ | #HT02 #H destruct -I
+ /3 width=6 by tpr_conf_flat_cast/ (**) (* /3 width=6/ is too slow *)
]
-| #I1 #V11 #V12 #T11 #T12 #HV112 #HT112 * -U2 T2
-(* case 19: bind, sort (excluded) *)
- [ #k2 #H1 #H2 destruct
-(* case 20: bind, lref (excluded) *)
- | #i2 #H1 #H2 destruct
-(* case 21: bind, bind *)
- | #I2 #V21 #V22 #T21 #T22 #HV212 #HT212 #H1 #H2
- destruct -T I2 V21 T21 /3 width=7/
-(* case 22: bind, flat (excluded) *)
- | #I2 #V21 #V22 #T21 #T22 #_ #_ #H1 #H2 destruct
-(* case 23: bind, beta (excluded) *)
- | #V21 #V22 #W2 #T21 #T22 #_ #_ #H1 #H2 destruct
-(* case 24: bind, delta (excluded) *)
- | #V21 #V22 #T21 #T22 #T20 #_ #_ #_ #H1 #H2 destruct
-(* case 25: bind, theta (excluded) *)
- | #V2 #V21 #V22 #W21 #W22 #T21 #T22 #_ #_ #_ #_ #H1 #H2 destruct
-(* case 26: bind, zeta *)
- | #V2 #T21 #T22 #T20 #HT212 #HT220 #H1 #H2
- destruct -I1 V2 T21 T /3 width=8/
-(* case 27: bind, tau (excluded) *)
- | #V2 #T21 #T22 #_ #H1 #H2 destruct
+| #V0 #V1 #W0 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct -Y0;
+ elim (tpr_inv_appl1 … H1) -H1 *
+(* case 10: beta, flat (repeated) *)
+ [ #V2 #T2 #HV02 #HT02 #H destruct -X2
+ @ex2_1_comm /3 width=8 by tpr_conf_flat_beta/
+(* case 11: beta, beta *)
+ | #V2 #WW0 #TT0 #T2 #HV02 #HT02 #H1 #H2 destruct -W0 T0 X2
+ /3 width=8 by tpr_conf_beta_beta/ (**) (* /3 width=8/ is too slow *)
+(* case 12, beta, theta (excluded) *)
+ | #V2 #VV2 #WW0 #W2 #TT0 #T2 #_ #_ #_ #_ #H destruct
]
-| #I1 #V11 #V12 #T11 #T12 #HV112 #HT112 * -U2 T2
-(* case 28: flat, sort (excluded) *)
- [ #k2 #H1 #H2 destruct
-(* case 29: flat, lref (excluded) *)
- | #i2 #H1 #H2 destruct
-(* case 30: flat, bind (excluded) *)
- | #I2 #V21 #V22 #T21 #T22 #_ #_ #H1 #H2 destruct
-(* case 31: flat, flat *)
- | #I2 #V21 #V22 #T21 #T22 #HV212 #HT212 #H1 #H2
- destruct -T I2 V21 T21 /3 width=7/
-(* case 32: flat, beta *)
- | #V21 #V22 #W2 #T21 #T22 #HV212 #HT212 #H1 #H2
- destruct -I1 V21 T11 T /3 width=8/ (**) (* slow *)
-(* case 33: flat, delta (excluded) *)
- | #V21 #V22 #T21 #T22 #T20 #_ #_ #_ #H1 #H2 destruct
-(* case 34: flat, theta *)
- | #V2 #V21 #V22 #W21 #W22 #T21 #T22 #H212 #HV222 #HW212 #HT212 #H1 #H2
- destruct -I1 V21 T11 T //
-
-lemma tpr_conf_flat_theta:
- ∀V11,V12,T12,V2,V22,W21,W22,T21,T22. (
- ∀T1. #T1 < #V11 + (#W21 + #T21 + 1) + 1 →
- ∀T3,T4. T1 ⇒ T3 → T1 ⇒ T4 →
- ∃∃T0. T3 ⇒ T0 & T4 ⇒T0
- ) →
- V11 ⇒ V12 → V11 ⇒ V22 → ↑[O,1] V22 ≡ V2 →
- W21 ⇒ W22 → T21 ⇒ T22 → 𝕓{Abbr} W21. T21 ⇒ T12 →
- ∃∃T0. 𝕗{Appl} V12. T12 ⇒ T0 & 𝕓{Abbr} W22. 𝕗{Appl} V2. T22 ⇒T0.
+| #V0 #V1 #T0 #T1 #TT1 #HV01 #HT01 #HTT1 #H1 #H2 destruct -Y0;
+ elim (tpr_inv_abbr1 … H1) -H1 *
+(* case 13: delta, bind (repeated) *)
+ [ #V2 #T2 #HV02 #T02 #H destruct -X2
+ @ex2_1_comm /3 width=9 by tpr_conf_bind_delta/
+(* case 14: delta, delta *)
+ | #V2 #T2 #TT2 #HV02 #HT02 #HTT2 #H destruct -X2
+ /3 width=11 by tpr_conf_delta_delta/ (**) (* /3 width=11/ is too slow *)
+(* case 15: delta, zata *)
+ | #T2 #HT20 #HTX2
+ /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;
+ elim (tpr_inv_appl1 … H1) -H1 *
+(* case 16: theta, flat (repeated) *)
+ [ #V2 #T2 #HV02 #HT02 #H destruct -X2
+ @ex2_1_comm /3 width=11 by tpr_conf_flat_theta/
+(* case 17: theta, beta (repeated) *)
+ | #V2 #WW0 #TT0 #T2 #_ #_ #H destruct
+(* case 18: theta, theta *)
+ | #V2 #VV2 #WW0 #W2 #TT0 #T2 #V02 #HW02 #HT02 #HVV2 #H1 #H2 destruct -W0 T0 X2
+ /3 width=14 by tpr_conf_theta_theta/ (**) (* /3 width=14/ is too slow *)
+ ]
+| #V0 #TT0 #T0 #T1 #HTT0 #HT01 #H1 #H2 destruct -Y0;
+ elim (tpr_inv_abbr1 … H1) -H1 *
+(* case 19: zeta, bind (repeated) *)
+ [ #V2 #T2 #HV02 #T02 #H destruct -X2
+ @ex2_1_comm /3 width=8 by tpr_conf_bind_zeta/
+(* case 20: zeta, delta (repeated) *)
+ | #V2 #T2 #TT2 #HV02 #HT02 #HTT2 #H destruct -X2
+ @ex2_1_comm /3 width=10 by tpr_conf_delta_zeta/
+(* case 21: 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;
+ elim (tpr_inv_cast1 … H1) -H1
+(* case 22: tau, flat (repeated) *)
+ [ * #V2 #T2 #HV02 #HT02 #H destruct -X2
+ @ex2_1_comm /3 width=6 by tpr_conf_flat_cast/
+(* case 23: tau, tau *)
+ | #HT02
+ /2 by tpr_conf_tau_tau/
+ ]
+]
+qed.
-theorem tpr_conf: ∀T,T1,T2. T ⇒ T1 → T ⇒ T2 →
- ∃∃T0. T1 ⇒ T0 & T2 ⇒ T0.
-#T @(tw_wf_ind … T) -T /3 width=8/
+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.
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