(* *)
(**************************************************************************)
-include "Basic-2/substitution/tps_tps.ma".
-include "Basic-2/reduction/tpr_lift.ma".
include "Basic-2/reduction/tpr_tps.ma".
(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
(* Confluence lemmas ********************************************************)
-lemma tpr_conf_sort_sort: ∀k. ∃∃X. ⋆k ⇒ X & ⋆k ⇒ X.
+fact tpr_conf_atom_atom: ∀I. ∃∃X. 𝕒{I} ⇒ X & 𝕒{I} ⇒ X.
/2/ qed.
-lemma tpr_conf_lref_lref: ∀i. ∃∃X. #i ⇒ X & #i ⇒ X.
-/2/ qed.
-
-lemma tpr_conf_flat_flat:
+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
) →
elim (IH … HT01 … HT02) -HT01 HT02 /3 width=5/
qed.
-lemma tpr_conf_flat_beta:
+fact tpr_conf_flat_beta:
∀V0,V1,T1,V2,W0,U0,T2. (
- ∀X0:term. #X0 < #V0 + (#W0 + #U0 + 1) + 1 →
+ ∀X0:term. #[X0] < #[V0] + (#[W0] + #[U0] + 1) + 1 →
∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
∃∃X. X1 ⇒ X & X2 ⇒ X
) →
V0 ⇒ V1 → V0 ⇒ V2 →
- U0 â\87\92 T2 â\86\92 ð\9d\95\93{Abst} W0. U0 ⇒ T1 →
- â\88\83â\88\83X. ð\9d\95\97{Appl} V1. T1 â\87\92 X & ð\9d\95\93{Abbr} V2. T2 ⇒ X.
+ U0 â\87\92 T2 â\86\92 ð\9d\95\94{Abst} W0. U0 ⇒ T1 →
+ â\88\83â\88\83X. ð\9d\95\94{Appl} V1. T1 â\87\92 X & ð\9d\95\94{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:
+(* 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 →
+ ∀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 â\87\92 W2 â\86\92 U0 â\87\92 U2 â\86\92 ð\9d\95\93{Abbr} W0. U0 ⇒ T1 →
- â\88\83â\88\83X. ð\9d\95\97{Appl} V1. T1 â\87\92 X & ð\9d\95\93{Abbr} W2. ð\9d\95\97{Appl} V. U2 ⇒ X.
+ W0 â\87\92 W2 â\86\92 U0 â\87\92 U2 â\86\92 ð\9d\95\94{Abbr} W0. U0 ⇒ T1 →
+ â\88\83â\88\83X. ð\9d\95\94{Appl} V1. T1 â\87\92 X & ð\9d\95\94{Abbr} W2. ð\9d\95\94{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
]
qed.
-lemma tpr_conf_flat_cast:
+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. ð\9d\95\97{Cast} V1. T1 ⇒ X & X2 ⇒ X.
+ â\88\83â\88\83X. ð\9d\95\94{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:
+fact tpr_conf_beta_beta:
∀W0:term. ∀V0,V1,T0,T1,V2,T2. (
- ∀X0:term. #X0 < #V0 + (#W0 + #T0 + 1) + 1 →
+ ∀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 →
- â\88\83â\88\83X. ð\9d\95\93{Abbr} V1. T1 â\87\92X & ð\9d\95\93{Abbr} V2. T2 ⇒ X.
+ â\88\83â\88\83X. ð\9d\95\94{Abbr} V1. T1 â\87\92X & ð\9d\95\94{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:
+(* 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→
+ ∀X0:term. #[X0] < #[V0] + #[T0] + 1 →
∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
∃∃X. X1 ⇒ X & X2 ⇒ X
) →
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
+elim (tps_conf_eq … 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:
+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
) →
elim (IH … HTX2 … HTT2) -HTX2 HTT2 IH /3/
qed.
-lemma tpr_conf_theta_theta:
+(* 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 →
+ ∀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 →
- â\88\83â\88\83X. ð\9d\95\93{Abbr} W1. ð\9d\95\97{Appl} VV1. T1 â\87\92 X & ð\9d\95\93{Abbr} W2. ð\9d\95\97{Appl} VV2. T2 ⇒ X.
+ â\88\83â\88\83X. ð\9d\95\94{Abbr} W1. ð\9d\95\94{Appl} VV1. T1 â\87\92 X & ð\9d\95\94{Abbr} W2. ð\9d\95\94{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
@ex2_1_intro [2,3: @tpr_bind |1:skip ] /2 width=5/ (**) (* /4 width=5/ is too slow *)
qed.
-lemma tpr_conf_zeta_zeta:
+fact tpr_conf_zeta_zeta:
∀V0:term. ∀X2,TT0,T0,T1,T2. (
- ∀X0:term. #X0 < #V0 + #TT0 + 1 →
+ ∀X0:term. #[X0] < #[V0] + #[TT0] + 1 →
∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
∃∃X. X1 ⇒ X & X2 ⇒ X
) →
elim (IH … HT01 … HTX2) -HT01 HTX2 IH /2/
qed.
-lemma tpr_conf_tau_tau:
+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
) →
(* Confluence ***************************************************************)
-lemma tpr_conf_aux:
+fact tpr_conf_aux:
∀Y0:term. (
- ∀X0:term. #X0 < #Y0 → ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
+ ∀X0:term. #[X0] < #[Y0] →
+ ∀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
-[ #k1 #H1 #H2 destruct -Y0;
- lapply (tpr_inv_sort1 … H1) -H1
-(* case 1: sort, sort *)
- #H1 destruct -X2 //
-| #i1 #H1 #H2 destruct -Y0;
- lapply (tpr_inv_lref1 … H1) -H1
-(* case 2: lref, lref *)
+[ #I1 #H1 #H2 destruct -Y0;
+ lapply (tpr_inv_atom1 … H1) -H1
+(* case 1: atom, atom *)
#H1 destruct -X2 //
| #I #V0 #V1 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct -Y0;
elim (tpr_inv_flat1 … H1) -H1 *
-(* case 3: flat, flat *)
+(* case 2: flat, flat *)
[ #V2 #T2 #HV02 #HT02 #H destruct -X2
/3 width=7 by tpr_conf_flat_flat/ (**) (* /3 width=7/ is too slow *)
-(* case 4: flat, beta *)
+(* case 3: 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 5: flat, theta *)
+(* case 4: 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 6: flat, tau *)
+(* case 5: flat, tau *)
| #HT02 #H destruct -I
/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;
elim (tpr_inv_appl1 … H1) -H1 *
-(* case 7: beta, flat (repeated) *)
+(* case 6: beta, flat (repeated) *)
[ #V2 #T2 #HV02 #HT02 #H destruct -X2
@ex2_1_comm /3 width=8 by tpr_conf_flat_beta/
-(* case 8: beta, beta *)
+(* case 7: 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 9, beta, theta (excluded) *)
+(* 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;
elim (tpr_inv_bind1 … H1) -H1 *
-(* case 10: delta, delta *)
+(* case 9: 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 11: delta, zata *)
+(* case 10: delta, zata *)
| #T2 #HT20 #HTX2 #H destruct -I1;
/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 12: theta, flat (repeated) *)
+(* case 11: theta, flat (repeated) *)
[ #V2 #T2 #HV02 #HT02 #H destruct -X2
@ex2_1_comm /3 width=11 by tpr_conf_flat_theta/
-(* case 13: theta, beta (repeated) *)
+(* case 12: theta, beta (repeated) *)
| #V2 #WW0 #TT0 #T2 #_ #_ #H destruct
-(* case 14: theta, theta *)
+(* case 13: 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 15: zeta, delta (repeated) *)
+(* case 14: zeta, delta (repeated) *)
[ #V2 #T2 #TT2 #HV02 #HT02 #HTT2 #H destruct -X2
@ex2_1_comm /3 width=10 by tpr_conf_delta_zeta/
-(* case 16: zeta, 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;
elim (tpr_inv_cast1 … H1) -H1
-(* case 17: tau, flat (repeated) *)
+(* case 16: tau, flat (repeated) *)
[ * #V2 #T2 #HV02 #HT02 #H destruct -X2
@ex2_1_comm /3 width=6 by tpr_conf_flat_cast/
-(* case 18: tau, tau *)
+(* case 17: tau, tau *)
| #HT02
/2 by tpr_conf_tau_tau/
]
]
qed.
+(* 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/
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