\ /
V_______________________________________________________________ *)
-include "lambda-delta/substitution/lift_fun.ma".
include "lambda-delta/substitution/lift_weight.ma".
-include "lambda-delta/reduction/tpr_main.ma".
-include "lambda-delta/reduction/tpr_ps.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 *********************************)
#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_ps_bind … HV2 HT20 … HT2) -HT20 HT2 /3 width=5/
+elim (tpr_tps_bind … HV2 HT20 … HT2) -HT20 HT2 /3 width=5/
qed.
lemma tpr_conf_bind_zeta:
| -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_ps_bind … HWW2 HUUU2 … HUU2) -HUU2 HUUU2 #UUU #HUUU2 #HUUU1
+ elim (tpr_tps_bind … HWW2 HUUU2 … HUU2) -HUU2 HUUU2 #UUU #HUUU2 #HUUU1
@ex2_1_intro
- [2: @tpr_theta
+ [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=14/ (**) (* /5 width=14/ is too slow *)
+ |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
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
+ ) →
+ 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:
elim (tpr_inv_bind1 … H1) -H1 *
(* case 3: bind, bind *)
[ #V2 #T2 #HV02 #HT02 #H destruct -X2
- @tpr_conf_bind_bind /2 width=7/ (**) (* /3 width=7/ is too slow *)
+ /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
- @tpr_conf_bind_delta /2 width=9/ (**) (* /3 width=9/ is too slow *)
+ /3 width=9 by tpr_conf_bind_delta/ (**) (* /3 width=9/ is too slow *)
(* case 5: bind, zeta *)
| #T #HT0 #HTX2 #H destruct -I
- @tpr_conf_bind_zeta /2 width=8/ (**) (* /3 width=8/ is too slow *)
+ /3 width=8 by tpr_conf_bind_zeta/ (**) (* /3 width=8/ is too slow *)
]
| #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
- @tpr_conf_flat_flat /2 width=7/ (**) (* /3 width=7/ is too slow *)
+ /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
- @tpr_conf_flat_beta /2 width=8/ (**) (* /3 width=8/ is too slow *)
+ /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
- @tpr_conf_flat_theta /2 width=11/ (**) (* /3 width=11/ is too slow *)
+ /3 width=11 by tpr_conf_flat_theta/ (**) (* /3 width=11/ is too slow *)
(* case 9: flat, tau *)
| #HT02 #H destruct -I
- @tpr_conf_flat_cast /2 width=6/ (**) (* /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;
elim (tpr_inv_appl1 … H1) -H1 *
(* case 10: beta, flat (repeated) *)
[ #V2 #T2 #HV02 #HT02 #H destruct -X2
- @ex2_1_comm @tpr_conf_flat_beta /2 width=8/
+ @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
- @tpr_conf_beta_beta /2 width=8/ (**) (* /3 width=8/ is too slow *)
+ /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
]
-| #V0 #V1 #T0 #T1 #TT1 #HV01 #T01 #HTT1 #H1 #H2 destruct -Y0
+| #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 @tpr_conf_bind_delta /2 width=9/
-
-
-
-lemma tpr_conf_beta_beta:
- ∀V0,V1,W0,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.
-#V0 #V1 #W0 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02
-
-
-
+ @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: ∀T0,T1,T2. T0 ⇒ T1 → T0 ⇒ T2 →
+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/
+#T @(tw_wf_ind … T) -T /3 width=6 by tpr_conf_aux/
qed.
projects / helm.git / blobdiff