(* *)
(**************************************************************************)
-include "basic_2/reducibility/tpr_tpss.ma".
+include "basic_2/reducibility/ltpr_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
) →
fact tpr_conf_flat_beta:
∀a,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
) →
*)
fact tpr_conf_flat_theta:
∀a,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
) →
[ -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
+ elim (tpr_tps_conf_bind … HWW2 HUUU2 … HUU2) -UU2 #UUU #HUUU2 #HUUU1
@ex2_intro
[2: @tpr_theta [6: @HVV |7: @HWW2 |8: @HUUU2 |1,2,3,4: skip | // ]
|1:skip
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
) →
fact tpr_conf_beta_beta:
∀a. ∀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
) →
(* Basic_1: was: pr0_cong_delta pr0_delta_delta *)
fact tpr_conf_delta_delta:
∀a,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
) →
#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
-elim (tpr_tps_bind … HV2 HT2 … HTT2) -T2 #U2 #TTU2 #HTU2
+elim (tpr_tps_conf_bind … HV1 HT1 … HTT1) -T1 #U1 #TTU1 #HTU1
+elim (tpr_tps_conf_bind … HV2 HT2 … HTT2) -T2 #U2 #TTU2 #HTU2
elim (tps_conf_eq … HTU1 … HTU2) -T #U #HU1 #HU2
@ex2_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 →
+ ∀X0:term. ♯{X0} < ♯{V0} + ♯{T0} + 1 →
∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
∃∃X. X1 ➡ X & X2 ➡ X
) →
∃∃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_tps_conf_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:
∀a,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
) →
fact tpr_conf_zeta_zeta:
∀V0:term. ∀X2,TT0,T0,T1,TT2. (
- ∀X0:term. #{X0} < #{V0} + #{TT0} + 1 →
+ ∀X0:term. ♯{X0} < ♯{V0} + ♯{TT0} + 1 →
∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
∃∃X. X1 ➡ X & X2 ➡ X
) →
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 ***************************************************************)
-fact tpr_conf_aux:
- ∀Y0:term. (
- ∀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
-[ #I1 #H1 #H2 destruct
- lapply (tpr_inv_atom1 … H1) -H1
+(* Basic_1: was: pr0_confluence *)
+theorem tpr_conf: ∀T0:term. ∀T1,T2. T0 ➡ T1 → T0 ➡ T2 →
+ ∃∃T. T1 ➡ T & T2 ➡ T.
+#T0 @(f_ind … tw … T0) -T0 #n #IH *
+[ #I #_ #X1 #X2 #H1 #H2 -n
+ >(tpr_inv_atom1 … H1) -X1
+ >(tpr_inv_atom1 … H2) -X2
(* case 1: atom, atom *)
- #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
- /3 width=7 by tpr_conf_flat_flat/ (**) (* /3 width=7/ is too slow *)
-(* case 3: flat, beta *)
- | #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 *)
- | #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 *)
- ]
-| #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 *)
- | #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) *)
- | #b #V2 #VV2 #WW0 #W2 #TT0 #T2 #_ #_ #_ #_ #H 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, zeta *)
- | #T2 #HT20 #HTX2 #H1 #H2 destruct
- /3 width=10 by tpr_conf_delta_zeta/ (**) (* /3 width=10/ is too slow *)
- ]
-| #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) *)
- | #b #V2 #WW0 #TT0 #T2 #_ #_ #H destruct
+ //
+| * [ #a ] #I #V0 #T0 #Hn #X1 #X2 #H1 #H2
+ [ elim (tpr_inv_bind1 … H1) -H1 *
+ [ #V1 #T1 #U1 #HV01 #HT01 #HTU1 #H1
+ | #T1 #HT01 #HXT1 #H11 #H12
+ ]
+ elim (tpr_inv_bind1 … H2) -H2 *
+ [1,3: #V2 #T2 #U2 #HV02 #HT02 #HTU2 #H2
+ |2,4: #T2 #HT02 #HXT2 #H21 #H22
+ ] destruct
+(* case 2: delta, delta *)
+ [ /3 width=11 by tpr_conf_delta_delta/ (**) (* /3 width=11/ is too slow *)
+(* case 3: zeta, delta (repeated) *)
+ | @ex2_commute /3 width=10 by tpr_conf_delta_zeta/
+(* case 4: delta, zeta *)
+ | /3 width=10 by tpr_conf_delta_zeta/ (**) (* /3 width=10/ is too slow *)
+(* case 5: zeta, zeta *)
+ | /3 width=9 by tpr_conf_zeta_zeta/ (**) (* /3 width=9/ is too slow *)
+ ]
+ | elim (tpr_inv_flat1 … H1) -H1 *
+ [ #V1 #T1 #HV01 #HT01 #H1
+ | #b1 #V1 #W1 #U1 #T1 #HV01 #HUT1 #H11 #H12 #H13
+ | #b1 #V1 #Y1 #W1 #Z1 #U1 #T1 #HV01 #HWZ1 #HUT1 #HVY1 #H11 #H12 #H13
+ | #HX1 #H1
+ ]
+ elim (tpr_inv_flat1 … H2) -H2 *
+ [1,5,9,13: #V2 #T2 #HV02 #HT02 #H2
+ |2,6,10,14: #b2 #V2 #W2 #U2 #T2 #HV02 #HUT2 #H21 #H22 #H23
+ |3,7,11,15: #b2 #V2 #Y2 #W2 #Z2 #U2 #T2 #HV02 #HWZ2 #HUT2 #HVY2 #H21 #H22 #H23
+ |4,8,12,16: #HX2 #H2
+ ] destruct
+(* case 6: flat, flat *)
+ [ /3 width=7 by tpr_conf_flat_flat/ (**) (* /3 width=7/ is too slow *)
+(* case 7: beta, flat (repeated) *)
+ | @ex2_commute /3 width=8 by tpr_conf_flat_beta/
+(* case 8: theta, flat (repeated) *)
+ | @ex2_commute /3 width=11 by tpr_conf_flat_theta/
+(* case 9: tau, flat (repeated) *)
+ | @ex2_commute /3 width=6 by tpr_conf_flat_cast/
+(* case 10: flat, beta *)
+ | /3 width=8 by tpr_conf_flat_beta/ (**) (* /3 width=8/ is too slow *)
+(* case 11: beta, beta *)
+ | /3 width=8 by tpr_conf_beta_beta/ (**) (* /3 width=8/ is too slow *)
+(* case 12: flat, theta *)
+ | /3 width=11 by tpr_conf_flat_theta/ (**) (* /3 width=11/ is too slow *)
(* case 13: theta, theta *)
- | #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 #TT2 #T2 #HV02 #HTT02 #HTT2 #H destruct
- @ex2_1_comm /3 width=10 by tpr_conf_delta_zeta/
-(* case 15: zeta, zeta *)
- | #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) *)
- [ * #V2 #T2 #HV02 #HT02 #H destruct
- @ex2_1_comm /3 width=6 by tpr_conf_flat_cast/
-(* case 17: tau, tau *)
- | #HT02
- /3 width=5 by tpr_conf_tau_tau/
+ | /3 width=14 by tpr_conf_theta_theta/ (**) (* /3 width=14/ is too slow *)
+(* case 14: flat, tau *)
+ | /3 width=6 by tpr_conf_flat_cast/ (**) (* /3 width=6/ is too slow *)
+(* case 15: tau, tau *)
+ | /3 width=5 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_ind … T) -T /3 width=6 by tpr_conf_aux/
-qed.
+qed-.
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