+++ /dev/null
-(*
- ||M|| This file is part of HELM, an Hypertextual, Electronic
- ||A|| Library of Mathematics, developed at the Computer Science
- ||T|| Department of the University of Bologna, Italy.
- ||I||
- ||T||
- ||A|| This file is distributed under the terms of the
- \ / GNU General Public License Version 2
- \ /
- V_______________________________________________________________ *)
-
-include "lambda-delta/substitution/lift_weight.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: ∀k. ∃∃X. ⋆k ⇒ X & ⋆k ⇒ X.
-/2/ qed.
-
-lemma tpr_conf_lref_lref: ∀i. ∃∃X. #i ⇒ X & #i ⇒ X.
-/2/ qed.
-
-lemma tpr_conf_bind_bind:
- ∀I,V0,V1,T0,T1,V2,T2. (
- ∀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 →
- ∃∃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:
- ∀X2,V0,V1,T0,T1,T. (
- ∀X0:term. #X0 < #V0 + #T0 +1 →
- ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
- ∃∃X. X1 ⇒ X & X2 ⇒ X
- ) →
- 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:
- ∀I,V0,V1,T0,T1,V2,T2. (
- ∀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 →
- ∃∃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:
- ∀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 → 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:
- ∀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
- ) →
- 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:
- ∀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
-[ #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 *)
- #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 *)
- ]
-| #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 *)
- ]
-| #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
- ]
-| #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: ∀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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