(* *)
(**************************************************************************)
-include "Basic_2/reducibility/tpr_tpss.ma".
+include "basic_2/reducibility/tpr_tpss.ma".
(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
fact tpr_conf_flat_flat:
∀I,V0,V1,T0,T1,V2,T2. (
- ∀X0:term. #[X0] < #[V0] + #[T0] + 1 →
+ ∀X0:term. #{X0} < #{V0} + #{T0} + 1 →
∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
∃∃X. X1 ➡ X & X2 ➡ X
) →
qed.
fact tpr_conf_flat_beta:
- ∀V0,V1,T1,V2,W0,U0,T2. (
- ∀X0:term. #[X0] < #[V0] + (#[W0] + #[U0] + 1) + 1 →
+ ∀a,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
) →
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
+ U0 ➡ T2 → ⓛ{a}W0. U0 ➡ T1 →
+ ∃∃X. ⓐV1. T1 ➡ X & ⓓ{a}V2. T2 ➡ X.
+#a #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
elim (IH … HT02 … HU01) -HT02 -HU01 -IH /2 width=1/ /3 width=5/
qed.
-(* basic-1: was:
+(* Basic-1: was:
pr0_cong_upsilon_refl pr0_cong_upsilon_zeta
pr0_cong_upsilon_cong pr0_cong_upsilon_delta
*)
fact tpr_conf_flat_theta:
- ∀V0,V1,T1,V2,V,W0,W2,U0,U2. (
- ∀X0:term. #[X0] < #[V0] + (#[W0] + #[U0] + 1) + 1 →
+ ∀a,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 → ⓓ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
+ W0 ➡ W2 → U0 ➡ U2 → ⓓ{a}W0. U0 ➡ T1 →
+ ∃∃X. ⓐV1. T1 ➡ X & ⓓ{a}W2. ⓐV. U2 ➡ X.
+#a #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
lapply (tpr_lift … HVV2 … HV2 … HVV) #HVVV
|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 /2 width=1/ -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 *)
+| -HV2 -HW02 -HVV2 #U1 #HU01 #HTU1
+ elim (IH … HU01 … HU02) -HU01 -HU02 -IH // -U0 #U #HU1 #HU2
+ elim (tpr_inv_lift1 … HU1 … HTU1) -U1 #UU #HUU #HT1UU #H destruct
+ @(ex2_1_intro … (ⓐVV.UU)) /2 width=1/ /3 width=5/ (**) (* /4 width=9/ is too slow *)
]
qed.
fact tpr_conf_flat_cast:
∀X2,V0,V1,T0,T1. (
- ∀X0:term. #[X0] < #[V0] + #[T0] + 1 →
+ ∀X0:term. #{X0} < #{V0} + #{T0} + 1 →
∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
∃∃X. X1 ➡ X & X2 ➡ X
) →
V0 ➡ V1 → T0 ➡ T1 → T0 ➡ X2 →
- â\88\83â\88\83X. â\93£V1. T1 ➡ X & X2 ➡ X.
+ â\88\83â\88\83X. â\93\9dV1. T1 ➡ X & X2 ➡ X.
#X2 #V0 #V1 #T0 #T1 #IH #_ #HT01 #HT02
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 →
+ ∀a. ∀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. ⓓV1. T1 ➡X & ⓓV2. T2 ➡ X.
-#W0 #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02
+ ∃∃X. ⓓ{a}V1. T1 ➡X & ⓓ{a}V2. T2 ➡ X.
+#a #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/
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 →
+ ∀a,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
) →
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
+ ⋆. ⓑ{I1} V1 ⊢ T1 ▶ [O, 1] TT1 →
+ ⋆. ⓑ{I1} V2 ⊢ T2 ▶ [O, 1] TT2 →
+ ∃∃X. ⓑ{a,I1} V1. TT1 ➡ X & ⓑ{a,I1} V2. TT2 ➡ X.
+#a #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) -T1 #U1 #TTU1 #HTU1
fact tpr_conf_delta_zeta:
∀X2,V0,V1,T0,T1,TT1,T2. (
- ∀X0:term. #[X0] < #[V0] + #[T0] + 1 →
+ ∀X0:term. #{X0} < #{V0} + #{T0} + 1 →
∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
∃∃X. X1 ➡ 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
-lapply (tw_lift … HTT20) -HTT20 #HTT20
-elim (IH … HTX2 … HTT2) -HTX2 -HTT2 -IH // /3 width=3/
+ V0 ➡ V1 → T0 ➡ T1 → ⋆. ⓓV1 ⊢ T1 ▶ [O,1] TT1 →
+ T0 ➡ T2 → ⇧[O, 1] X2 ≡ T2 →
+ ∃∃X. +ⓓV1. TT1 ➡ X & X2 ➡ X.
+#X2 #V0 #V1 #T0 #T1 #TT1 #T2 #IH #_ #HT01 #HTT1 #HT02 #HXT2
+elim (IH … HT01 … HT02) -IH -HT01 -HT02 // -V0 -T0 #T #HT1 #HT2
+elim (tpr_tps_bind ? ? V1 … HT1 HTT1) -T1 // #TT #HTT1 #HTT
+elim (tpr_inv_lift1 … HT2 … HXT2) -T2 #X #HXT #HX2
+lapply (tps_inv_lift1_eq … HTT … HXT) -HTT #H destruct /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 →
+ ∀a,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. ⓓ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
+ ∃∃X. ⓓ{a}W1. ⓐVV1. T1 ➡ X & ⓓ{a}W2. ⓐVV2. T2 ➡ X.
+#a #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
elim (IH … HT01 … HT02) -HT01 -HT02 -IH /2 width=1/ #T #HT1 #HT2
qed.
fact tpr_conf_zeta_zeta:
- ∀V0:term. ∀X2,TT0,T0,T1,T2. (
- ∀X0:term. #[X0] < #[V0] + #[TT0] + 1 →
+ ∀V0:term. ∀X2,TT0,T0,T1,TT2. (
+ ∀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 →
+ TT0 ➡ T0 → ⇧[O, 1] T1 ≡ T0 →
+ TT0 ➡ TT2 → ⇧[O, 1] X2 ≡ TT2 →
∃∃X. T1 ➡ X & X2 ➡ 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
-elim (IH … HT01 … HTX2) -HT01 -HTX2 -IH // /2 width=3/
+#V0 #X2 #TT0 #T0 #T1 #TT2 #IH #HTT0 #HT10 #HTT02 #HXTT2
+elim (IH … HTT0 … HTT02) -IH -HTT0 -HTT02 // -V0 -TT0 #T #HT0 #HTT2
+elim (tpr_inv_lift1 … HT0 … HT10) -T0 #T0 #HT0 #HT10
+elim (tpr_inv_lift1 … HTT2 … HXTT2) -TT2 #TT2 #HTT2 #HXTT2
+lapply (lift_inj … HTT2 … HT0) -HTT2 #H destruct /2 width=3/
qed.
fact tpr_conf_tau_tau:
∀V0,T0:term. ∀X2,T1. (
- ∀X0:term. #[X0] < #[V0] + #[T0] + 1 →
+ ∀X0:term. #{X0} < #{V0} + #{T0} + 1 →
∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
∃∃X. X1 ➡ X & X2 ➡ X
) →
fact tpr_conf_aux:
∀Y0:term. (
- ∀X0:term. #[X0] < #[Y0] →
+ ∀X0:term. #{X0} < #{Y0} →
∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
∃∃X. X1 ➡ X & X2 ➡ X
) →
[ #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
+ | #b #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
+ | #b #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
/3 width=6 by tpr_conf_flat_cast/ (**) (* /3 width=6/ is too slow *)
]
-| #V0 #V1 #W0 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct
+| #a #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
@ex2_1_comm /3 width=8 by tpr_conf_flat_beta/
(* case 7: beta, beta *)
- | #V2 #WW0 #TT0 #T2 #HV02 #HT02 #H1 #H2 destruct
+ | #b #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
+ | #b #V2 #VV2 #WW0 #W2 #TT0 #T2 #_ #_ #_ #_ #H destruct
]
-| #I1 #V0 #V1 #T0 #T1 #TT1 #HV01 #HT01 #HTT1 #H1 #H2 destruct
+| #a #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
/3 width=11 by tpr_conf_delta_delta/ (**) (* /3 width=11/ is too slow *)
-(* case 10: delta, zata *)
- | #T2 #HT20 #HTX2 #H destruct
+(* case 10: delta, zeta *)
+ | #T2 #HT20 #HTX2 #H1 #H2 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
+| #a #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
@ex2_1_comm /3 width=11 by tpr_conf_flat_theta/
(* case 12: theta, beta (repeated) *)
- | #V2 #WW0 #TT0 #T2 #_ #_ #H destruct
+ | #b #V2 #WW0 #TT0 #T2 #_ #_ #H destruct
(* case 13: theta, theta *)
- | #V2 #VV2 #WW0 #W2 #TT0 #T2 #V02 #HW02 #HT02 #HVV2 #H1 #H2 destruct
+ | #b #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
elim (tpr_inv_abbr1 … H1) -H1 *
(* case 14: zeta, delta (repeated) *)
- [ #V2 #T2 #TT2 #HV02 #HT02 #HTT2 #H destruct
+ [ #V2 #TT2 #T2 #HV02 #HTT02 #HTT2 #H destruct
@ex2_1_comm /3 width=10 by tpr_conf_delta_zeta/
(* case 15: zeta, zeta *)
- | #T2 #HTT20 #HTX2
+ | #TT2 #HTT02 #HXTT2
/3 width=9 by tpr_conf_zeta_zeta/ (**) (* /3 width=9/ is too slow *)
- ]
+ ]
| #V0 #T0 #T1 #HT01 #H1 #H2 destruct
elim (tpr_inv_cast1 … H1) -H1
(* case 16: tau, flat (repeated) *)
(* Basic_1: was: pr0_confluence *)
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/
+#T @(tw_ind … T) -T /3 width=6 by tpr_conf_aux/
qed.
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