(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ⬈ [ break term 46 h , break term 46 o ] 𝐒 ⦃ break term 46 T ⦄ )"
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ⬈ * [ break term 46 h , break term 46 o ] 𝐒 ⦃ break term 46 T ⦄ )"
non associative with precedence 45
for @{ 'PRedTyStrong $h $o $G $L $T }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ⬊ ⬊ * [ break term 46 h , break term 46 o ] break term 46 T )"
- non associative with precedence 45
- for @{ 'SNAlt $h $o $G $L $T }.
lapply (cpxs_strap1 … HW1 … HW2) -W
lapply (cpxs_strap1 … HT1 … HT2) -T /3 width=5 by or3_intro0, ex3_2_intro/
qed-.
+
+(* Basic_2A1: removed theorems 1: cpxs_neq_inv_step_sn *)
(* *)
(**************************************************************************)
-include "basic_2/reduction/lpx_drop.ma".
-include "basic_2/computation/cpxs_lift.ma".
+include "basic_2/rt_transition/cpx_lsubr.ma".
+include "basic_2/rt_computation/cpxs.ma".
-(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
+(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS ************)
(* Main properties **********************************************************)
-theorem cpxs_trans: ∀h,o,G,L. Transitive … (cpxs h o G L).
+theorem cpxs_trans: ∀h,G,L. Transitive … (cpxs h G L).
normalize /2 width=3 by trans_TC/ qed-.
-theorem cpxs_bind: ∀h,o,a,I,G,L,V1,V2,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈*[h, o] T2 →
- ⦃G, L⦄ ⊢ V1 ⬈*[h, o] V2 →
- ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ⬈*[h, o] ⓑ{a,I}V2.T2.
-#h #o #a #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2
+theorem cpxs_bind: ∀h,p,I,G,L,V1,V2,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈*[h] T2 →
+ ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
+ ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
+#h #p #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2
/3 width=5 by cpxs_trans, cpxs_bind_dx/
qed.
-theorem cpxs_flat: ∀h,o,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T2 →
- ⦃G, L⦄ ⊢ V1 ⬈*[h, o] V2 →
- ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h, o] ⓕ{I}V2.T2.
-#h #o #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2
+theorem cpxs_flat: ∀h,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 →
+ ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
+ ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h] ⓕ{I}V2.T2.
+#h #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2
/3 width=5 by cpxs_trans, cpxs_flat_dx/
qed.
-theorem cpxs_beta_rc: ∀h,o,a,G,L,V1,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ⬈[h, o] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈*[h, o] T2 → ⦃G, L⦄ ⊢ W1 ⬈*[h, o] W2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ⬈*[h, o] ⓓ{a}ⓝW2.V2.T2.
-#h #o #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cpxs_ind … H) -W2
+theorem cpxs_beta_rc: ∀h,p,G,L,V1,V2,W1,W2,T1,T2.
+ ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ W1 ⬈*[h] W2 →
+ ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈*[h] ⓓ{p}ⓝW2.V2.T2.
+#h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cpxs_ind … H) -W2
/4 width=5 by cpxs_trans, cpxs_beta_dx, cpxs_bind_dx, cpx_pair_sn/
qed.
-theorem cpxs_beta: ∀h,o,a,G,L,V1,V2,W1,W2,T1,T2.
- ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈*[h, o] T2 → ⦃G, L⦄ ⊢ W1 ⬈*[h, o] W2 → ⦃G, L⦄ ⊢ V1 ⬈*[h, o] V2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ⬈*[h, o] ⓓ{a}ⓝW2.V2.T2.
-#h #o #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cpxs_ind … H) -V2
+theorem cpxs_beta: ∀h,p,G,L,V1,V2,W1,W2,T1,T2.
+ ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ W1 ⬈*[h] W2 → ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
+ ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈*[h] ⓓ{p}ⓝW2.V2.T2.
+#h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cpxs_ind … H) -V2
/4 width=5 by cpxs_trans, cpxs_beta_rc, cpxs_bind_dx, cpx_flat/
qed.
-theorem cpxs_theta_rc: ∀h,o,a,G,L,V1,V,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ⬈[h, o] V → ⬆[0, 1] V ≡ V2 →
- ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈*[h, o] T2 → ⦃G, L⦄ ⊢ W1 ⬈*[h, o] W2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ⬈*[h, o] ⓓ{a}W2.ⓐV2.T2.
-#h #o #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H @(cpxs_ind … H) -W2
+theorem cpxs_theta_rc: ∀h,p,G,L,V1,V,V2,W1,W2,T1,T2.
+ ⦃G, L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≡ V2 →
+ ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ W1 ⬈*[h] W2 →
+ ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈*[h] ⓓ{p}W2.ⓐV2.T2.
+#h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H @(cpxs_ind … H) -W2
/3 width=5 by cpxs_trans, cpxs_theta_dx, cpxs_bind_dx/
qed.
-theorem cpxs_theta: ∀h,o,a,G,L,V1,V,V2,W1,W2,T1,T2.
- ⬆[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ⬈*[h, o] W2 →
- ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈*[h, o] T2 → ⦃G, L⦄ ⊢ V1 ⬈*[h, o] V →
- ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ⬈*[h, o] ⓓ{a}W2.ⓐV2.T2.
-#h #o #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1
+theorem cpxs_theta: ∀h,p,G,L,V1,V,V2,W1,W2,T1,T2.
+ ⬆*[1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ⬈*[h] W2 →
+ ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ V1 ⬈*[h] V →
+ ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈*[h] ⓓ{p}W2.ⓐV2.T2.
+#h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1
/3 width=5 by cpxs_trans, cpxs_theta_rc, cpxs_flat_dx/
qed.
(* Advanced inversion lemmas ************************************************)
-lemma cpxs_inv_appl1: ∀h,o,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓐV1.T1 ⬈*[h, o] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈*[h, o] V2 & ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T2 &
- U2 = ⓐV2. T2
- | ∃∃a,W,T. ⦃G, L⦄ ⊢ T1 ⬈*[h, o] ⓛ{a}W.T & ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V1.T ⬈*[h, o] U2
- | ∃∃a,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ⬈*[h, o] V0 & ⬆[0,1] V0 ≡ V2 &
- ⦃G, L⦄ ⊢ T1 ⬈*[h, o] ⓓ{a}V.T & ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ⬈*[h, o] U2.
-#h #o #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 [ /3 width=5 by or3_intro0, ex3_2_intro/ ]
+lemma cpxs_inv_appl1: ∀h,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓐV1.T1 ⬈*[h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 &
+ U2 = ⓐV2.T2
+ | ∃∃p,W,T. ⦃G, L⦄ ⊢ T1 ⬈*[h] ⓛ{p}W.T & ⦃G, L⦄ ⊢ ⓓ{p}ⓝW.V1.T ⬈*[h] U2
+ | ∃∃p,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ⬈*[h] V0 & ⬆*[1] V0 ≡ V2 &
+ ⦃G, L⦄ ⊢ T1 ⬈*[h] ⓓ{p}V.T & ⦃G, L⦄ ⊢ ⓓ{p}V.ⓐV2.T ⬈*[h] U2.
+#h #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 [ /3 width=5 by or3_intro0, ex3_2_intro/ ]
#U #U2 #_ #HU2 * *
[ #V0 #T0 #HV10 #HT10 #H destruct
elim (cpx_inv_appl1 … HU2) -HU2 *
[ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5 by cpxs_strap1, or3_intro0, ex3_2_intro/
- | #a #V2 #W #W2 #T #T2 #HV02 #HW2 #HT2 #H1 #H2 destruct
+ | #p #V2 #W #W2 #T #T2 #HV02 #HW2 #HT2 #H1 #H2 destruct
lapply (cpxs_strap1 … HV10 … HV02) -V0 #HV12
lapply (lsubr_cpx_trans … HT2 (L.ⓓⓝW.V1) ?) -HT2
/5 width=5 by cpxs_bind, cpxs_flat_dx, cpx_cpxs, lsubr_beta, ex2_3_intro, or3_intro1/
- | #a #V #V2 #W0 #W2 #T #T2 #HV0 #HV2 #HW02 #HT2 #H1 #H2 destruct
+ | #p #V #V2 #W0 #W2 #T #T2 #HV0 #HV2 #HW02 #HT2 #H1 #H2 destruct
/5 width=10 by cpxs_flat_sn, cpxs_bind_dx, cpxs_strap1, ex4_5_intro, or3_intro2/
]
| /4 width=9 by cpxs_strap1, or3_intro1, ex2_3_intro/
| /4 width=11 by cpxs_strap1, or3_intro2, ex4_5_intro/
]
qed-.
-
-(* Properties on sn extended parallel reduction for local environments ******)
-
-lemma lpx_cpx_trans: ∀h,o,G. b_c_transitive … (cpx h o G) (λ_.lpx h o G).
-#h #o #G #L2 #T1 #T2 #HT12 elim HT12 -G -L2 -T1 -T2
-[ /2 width=3 by/
-| /3 width=2 by cpx_cpxs, cpx_st/
-| #I #G #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12
- elim (lpx_drop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
- elim (lpx_inv_pair2 … H) -H #K1 #V1 #HK12 #HV10 #H destruct
- /4 width=7 by cpxs_delta, cpxs_strap2/
-|4,9: /4 width=1 by cpxs_beta, cpxs_bind, lpx_pair/
-|5,7,8: /3 width=1 by cpxs_flat, cpxs_ct, cpxs_eps/
-| /4 width=3 by cpxs_zeta, lpx_pair/
-| /4 width=3 by cpxs_theta, cpxs_strap1, lpx_pair/
-]
-qed-.
-
-lemma cpx_bind2: ∀h,o,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h, o] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈[h, o] T2 →
- ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ⬈*[h, o] ⓑ{a,I}V2.T2.
-/4 width=5 by lpx_cpx_trans, cpxs_bind_dx, lpx_pair/ qed.
-
-(* Advanced properties ******************************************************)
-
-lemma lpx_cpxs_trans: ∀h,o,G. b_rs_transitive … (cpx h o G) (λ_.lpx h o G).
-#h #o #G @b_c_trans_LTC1 /2 width=3 by lpx_cpx_trans/ (**) (* full auto fails *)
-qed-.
-
-lemma cpxs_bind2_dx: ∀h,o,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h, o] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h, o] T2 →
- ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ⬈*[h, o] ⓑ{a,I}V2.T2.
-/4 width=5 by lpx_cpxs_trans, cpxs_bind_dx, lpx_pair/ qed.
-
-(* Properties on supclosure *************************************************)
-
-lemma fqu_cpxs_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 → (T2 = U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #V1 #V2 #HV12 #_ elim (lift_total V2 0 1)
- #U2 #HVU2 @(ex3_intro … U2)
- [1,3: /3 width=7 by fqu_drop, cpxs_delta, drop_pair, drop_drop/
- | #H destruct
- lapply (lift_inv_lref2_be … HVU2 ? ?) -HVU2 //
- ]
-| #I #G #L #V1 #T #V2 #HV12 #H @(ex3_intro … (②{I}V2.T))
- [1,3: /2 width=4 by fqu_pair_sn, cpxs_pair_sn/
- | #H0 destruct /2 width=1 by/
- ]
-| #a #I #G #L #V #T1 #T2 #HT12 #H @(ex3_intro … (ⓑ{a,I}V.T2))
- [1,3: /2 width=4 by fqu_bind_dx, cpxs_bind/
- | #H0 destruct /2 width=1 by/
- ]
-| #I #G #L #V #T1 #T2 #HT12 #H @(ex3_intro … (ⓕ{I}V.T2))
- [1,3: /2 width=4 by fqu_flat_dx, cpxs_flat/
- | #H0 destruct /2 width=1 by/
- ]
-| #G #L #K #T1 #U1 #k #HLK #HTU1 #T2 #HT12 #H elim (lift_total T2 0 (k+1))
- #U2 #HTU2 @(ex3_intro … U2)
- [1,3: /2 width=10 by cpxs_lift, fqu_drop/
- | #H0 destruct /3 width=5 by lift_inj/
-]
-qed-.
-
-lemma fquq_cpxs_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 → (T2 = U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fquq_inv_gen … H12) -H12
-[ #H12 elim (fqu_cpxs_trans_neq … H12 … HTU2 H) -T2
- /3 width=4 by fqu_fquq, ex3_intro/
-| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
-]
-qed-.
-
-lemma fqup_cpxs_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 → (T2 = U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
-[ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpxs_trans_neq … H12 … HTU2 H) -T2
- /3 width=4 by fqu_fqup, ex3_intro/
-| #G #G1 #L #L1 #T #T1 #H1 #_ #IH12 #U2 #HTU2 #H elim (IH12 … HTU2 H) -T2
- #U1 #HTU1 #H #H12 elim (fqu_cpxs_trans_neq … H1 … HTU1 H) -T1
- /3 width=8 by fqup_strap2, ex3_intro/
-]
-qed-.
-
-lemma fqus_cpxs_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 → (T2 = U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_gen … H12) -H12
-[ #H12 elim (fqup_cpxs_trans_neq … H12 … HTU2 H) -T2
- /3 width=4 by fqup_fqus, ex3_intro/
-| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
-]
-qed-.
--- /dev/null
+
+include "basic_2/reduction/lpx_drop.ma".
+include "basic_2/computation/cpxs_lift.ma".
+include "basic_2/rt_computation/cpxs_cpxs.ma".
+
+(* Properties on sn extended parallel reduction for local environments ******)
+
+lemma lpx_cpx_trans: ∀h,G. b_c_transitive … (cpx h G) (λ_.lpx h G).
+#h #G #L2 #T1 #T2 #HT12 elim HT12 -G -L2 -T1 -T2
+[ /2 width=3 by/
+| /3 width=2 by cpx_cpxs, cpx_st/
+| #I #G #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12
+ elim (lpx_drop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
+ elim (lpx_inv_pair2 … H) -H #K1 #V1 #HK12 #HV10 #H destruct
+ /4 width=7 by cpxs_delta, cpxs_strap2/
+|4,9: /4 width=1 by cpxs_beta, cpxs_bind, lpx_pair/
+|5,7,8: /3 width=1 by cpxs_flat, cpxs_ct, cpxs_eps/
+| /4 width=3 by cpxs_zeta, lpx_pair/
+| /4 width=3 by cpxs_theta, cpxs_strap1, lpx_pair/
+]
+qed-.
+
+lemma cpx_bind2: ∀h,o,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h, o] V2 →
+ ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈[h, o] T2 →
+ ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ⬈*[h, o] ⓑ{a,I}V2.T2.
+/4 width=5 by lpx_cpx_trans, cpxs_bind_dx, lpx_pair/ qed.
+
+(* Advanced properties ******************************************************)
+
+lemma lpx_cpxs_trans: ∀h,o,G. b_rs_transitive … (cpx h o G) (λ_.lpx h o G).
+#h #o #G @b_c_trans_LTC1 /2 width=3 by lpx_cpx_trans/ (**) (* full auto fails *)
+qed-.
+
+lemma cpxs_bind2_dx: ∀h,o,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h, o] V2 →
+ ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h, o] T2 →
+ ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ⬈*[h, o] ⓑ{a,I}V2.T2.
+/4 width=5 by lpx_cpxs_trans, cpxs_bind_dx, lpx_pair/ qed.
+
+(* Properties on supclosure *************************************************)
+
+lemma fqu_cpxs_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 → (T2 = U2 → ⊥) →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G #L #V1 #V2 #HV12 #_ elim (lift_total V2 0 1)
+ #U2 #HVU2 @(ex3_intro … U2)
+ [1,3: /3 width=7 by fqu_drop, cpxs_delta, drop_pair, drop_drop/
+ | #H destruct
+ lapply (lift_inv_lref2_be … HVU2 ? ?) -HVU2 //
+ ]
+| #I #G #L #V1 #T #V2 #HV12 #H @(ex3_intro … (②{I}V2.T))
+ [1,3: /2 width=4 by fqu_pair_sn, cpxs_pair_sn/
+ | #H0 destruct /2 width=1 by/
+ ]
+| #a #I #G #L #V #T1 #T2 #HT12 #H @(ex3_intro … (ⓑ{a,I}V.T2))
+ [1,3: /2 width=4 by fqu_bind_dx, cpxs_bind/
+ | #H0 destruct /2 width=1 by/
+ ]
+| #I #G #L #V #T1 #T2 #HT12 #H @(ex3_intro … (ⓕ{I}V.T2))
+ [1,3: /2 width=4 by fqu_flat_dx, cpxs_flat/
+ | #H0 destruct /2 width=1 by/
+ ]
+| #G #L #K #T1 #U1 #k #HLK #HTU1 #T2 #HT12 #H elim (lift_total T2 0 (k+1))
+ #U2 #HTU2 @(ex3_intro … U2)
+ [1,3: /2 width=10 by cpxs_lift, fqu_drop/
+ | #H0 destruct /3 width=5 by lift_inj/
+]
+qed-.
+
+lemma fquq_cpxs_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 → (T2 = U2 → ⊥) →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fquq_inv_gen … H12) -H12
+[ #H12 elim (fqu_cpxs_trans_neq … H12 … HTU2 H) -T2
+ /3 width=4 by fqu_fquq, ex3_intro/
+| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
+]
+qed-.
+
+lemma fqup_cpxs_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 → (T2 = U2 → ⊥) →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
+[ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpxs_trans_neq … H12 … HTU2 H) -T2
+ /3 width=4 by fqu_fqup, ex3_intro/
+| #G #G1 #L #L1 #T #T1 #H1 #_ #IH12 #U2 #HTU2 #H elim (IH12 … HTU2 H) -T2
+ #U1 #HTU1 #H #H12 elim (fqu_cpxs_trans_neq … H1 … HTU1 H) -T1
+ /3 width=8 by fqup_strap2, ex3_intro/
+]
+qed-.
+
+lemma fqus_cpxs_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 → (T2 = U2 → ⊥) →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_gen … H12) -H12
+[ #H12 elim (fqup_cpxs_trans_neq … H12 … HTU2 H) -T2
+ /3 width=4 by fqup_fqus, ex3_intro/
+| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
+]
+qed-.
(**************************************************************************)
include "basic_2/syntax/tdeq_tdeq.ma".
-include "basic_2/rt_computation/cpxs.ma".
-include "basic_2/rt_transition/cpx_lfdeq.ma".
include "basic_2/rt_transition/lfpx_fqup.ma".
+include "basic_2/rt_transition/lfpx_lfdeq.ma".
+include "basic_2/rt_computation/cpxs.ma".
(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS ************)
+(* Properties with degree-based equivalence for terms ***********************)
+
lemma tdeq_cpxs_trans: ∀h,o,U1,T1. U1 ≡[h, o] T1 → ∀G,L,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 →
∃∃U2. ⦃G, L⦄ ⊢ U1 ⬈*[h] U2 & U2 ≡[h, o] T2.
#h #o #U1 #T1 #HUT1 #G #L #T2 #HT12 @(cpxs_ind … HT12) -T2 /2 width=3 by ex2_intro/
]
]
qed-.
-
-(* Basic_2A1: removed theorems 1: cpxs_neq_inv_step_sn *)
(* Basic eliminators ********************************************************)
lemma csx_ind: ∀h,o,G,L. ∀R:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃T1⦄ →
+ (∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
(∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≡[h, o] T2 → ⊥) → R T2) →
R T1
) →
- ∀T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃T⦄ → R T.
+ ∀T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → R T.
#h #o #G #L #R #H0 #T1 #H elim H -T1
/5 width=1 by SN_intro/
qed-.
(* Basic_1: was just: sn3_pr2_intro *)
lemma csx_intro: ∀h,o,G,L,T1.
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≡[h, o] T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃T2⦄) →
- ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃T1⦄.
+ (∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≡[h, o] T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄) →
+ ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄.
/4 width=1 by SN_intro/ qed.
-lemma csx_sort: ∀h,o,G,L,s. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃⋆s⦄.
+lemma csx_sort: ∀h,o,G,L,s. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃⋆s⦄.
#h #o #G #L #s elim (deg_total h o s)
#d generalize in match s; -s elim d -d
[ #s1 #Hs1 @csx_intro #X #H #HX elim HX -HX
(* Basic forward lemmas *****************************************************)
-fact csx_fwd_pair_sn_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃U⦄ →
- ∀I,V,T. U = ②{I}V.T → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃V⦄.
+fact csx_fwd_pair_sn_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃U⦄ →
+ ∀I,V,T. U = ②{I}V.T → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄.
#h #o #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
@csx_intro #V2 #HLV2 #HV2
@(IH (②{I}V2.T)) -IH /2 width=3 by cpx_pair_sn/ -HLV2
qed-.
(* Basic_1: was just: sn3_gen_head *)
-lemma csx_fwd_pair_sn: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃②{I}V.T⦄ → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃V⦄.
+lemma csx_fwd_pair_sn: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃②{I}V.T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄.
/2 width=5 by csx_fwd_pair_sn_aux/ qed-.
-fact csx_fwd_bind_dx_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃U⦄ →
- ∀p,I,V,T. U = ⓑ{p,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈[h, o] 𝐒⦃T⦄.
+fact csx_fwd_bind_dx_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃U⦄ →
+ ∀p,I,V,T. U = ⓑ{p,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
#h #o #G #L #U #H elim H -H #U0 #_ #IH #p #I #V #T #H destruct
@csx_intro #T2 #HLT2 #HT2
@(IH (ⓑ{p,I}V.T2)) -IH /2 width=3 by cpx_bind/ -HLT2
qed-.
(* Basic_1: was just: sn3_gen_bind *)
-lemma csx_fwd_bind_dx: ∀h,o,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃ⓑ{p,I}V.T⦄ → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈[h, o] 𝐒⦃T⦄.
+lemma csx_fwd_bind_dx: ∀h,o,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓑ{p,I}V.T⦄ → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
/2 width=4 by csx_fwd_bind_dx_aux/ qed-.
-fact csx_fwd_flat_dx_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃U⦄ →
- ∀I,V,T. U = ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃T⦄.
+fact csx_fwd_flat_dx_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃U⦄ →
+ ∀I,V,T. U = ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
#h #o #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
@csx_intro #T2 #HLT2 #HT2
@(IH (ⓕ{I}V.T2)) -IH /2 width=3 by cpx_flat/ -HLT2
qed-.
(* Basic_1: was just: sn3_gen_flat *)
-lemma csx_fwd_flat_dx: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃ⓕ{I}V.T⦄ → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃T⦄.
+lemma csx_fwd_flat_dx: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓕ{I}V.T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
/2 width=5 by csx_fwd_flat_dx_aux/ qed-.
-lemma csx_fwd_bind: ∀h,o,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃ⓑ{p,I}V.T⦄ →
- ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈[h, o] 𝐒⦃T⦄.
+lemma csx_fwd_bind: ∀h,o,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓑ{p,I}V.T⦄ →
+ ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
/3 width=3 by csx_fwd_pair_sn, csx_fwd_bind_dx, conj/ qed-.
-lemma csx_fwd_flat: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃ⓕ{I}V.T⦄ →
- ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃V⦄ ∧ ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃T⦄.
+lemma csx_fwd_flat: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓕ{I}V.T⦄ →
+ ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ ∧ ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
/3 width=3 by csx_fwd_pair_sn, csx_fwd_flat_dx, conj/ qed-.
(* Basic_1: removed theorems 14:
sn3_appl_cast sn3_appl_beta sn3_appl_lref sn3_appl_abbr
sn3_appl_appls sn3_bind sn3_appl_bind sn3_appls_bind
*)
+(* Basic_2A1: removed theorems 6:
+ csxa_ind csxa_intro csxa_cpxs_trans csxa_intro_cpx
+ csx_csxa csxa_csx
+*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/snalt_5.ma".
-include "basic_2/computation/cpxs.ma".
-include "basic_2/computation/csx.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
-
-(* alternative definition of csx *)
-definition csxa: ∀h. sd h → relation3 genv lenv term ≝
- λh,o,G,L. SN … (cpxs h o G L) (eq …).
-
-interpretation
- "context-sensitive extended strong normalization (term) alternative"
- 'SNAlt h o G L T = (csxa h o G L T).
-
-(* Basic eliminators ********************************************************)
-
-lemma csxa_ind: ∀h,o,G,L. ∀R:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T1 →
- (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → (T1 = T2 → ⊥) → R T2) → R T1
- ) →
- ∀T. ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T → R T.
-#h #o #G #L #R #H0 #T1 #H elim H -T1 /5 width=1 by SN_intro/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma csx_intro_cpxs: ∀h,o,G,L,T1.
- (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, o] T2) →
- ⦃G, L⦄ ⊢ ⬊*[h, o] T1.
-/4 width=1 by cpx_cpxs, csx_intro/ qed.
-
-(* Basic_1: was just: sn3_intro *)
-lemma csxa_intro: ∀h,o,G,L,T1.
- (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T2) →
- ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T1.
-/4 width=1 by SN_intro/ qed.
-
-fact csxa_intro_aux: ∀h,o,G,L,T1. (
- ∀T,T2. ⦃G, L⦄ ⊢ T ➡*[h, o] T2 → T1 = T → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T2
- ) → ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T1.
-/4 width=3 by csxa_intro/ qed-.
-
-(* Basic_1: was just: sn3_pr3_trans (old version) *)
-lemma csxa_cpxs_trans: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T1 →
- ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T2.
-#h #o #G #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
-@csxa_intro #T #HLT2 #HT2
-elim (eq_term_dec T1 T2) #HT12
-[ -IHT1 -HLT12 destruct /3 width=1 by/
-| -HT1 -HT2 /3 width=4 by/
-qed.
-
-(* Basic_1: was just: sn3_pr2_intro (old version) *)
-lemma csxa_intro_cpx: ∀h,o,G,L,T1. (
- ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T2
- ) → ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T1.
-#h #o #G #L #T1 #H
-@csxa_intro_aux #T #T2 #H @(cpxs_ind_dx … H) -T
-[ -H #H destruct #H
- elim H //
-| #T0 #T #HLT1 #HLT2 #IHT #HT10 #HT12 destruct
- elim (eq_term_dec T0 T) #HT0
- [ -HLT1 -HLT2 -H /3 width=1 by/
- | -IHT -HT12 /4 width=3 by csxa_cpxs_trans/
- ]
-]
-qed.
-
-(* Main properties **********************************************************)
-
-theorem csx_csxa: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T.
-#h #o #G #L #T #H @(csx_ind … H) -T /4 width=1 by csxa_intro_cpx/
-qed.
-
-theorem csxa_csx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T → ⦃G, L⦄ ⊢ ⬊*[h, o] T.
-#h #o #G #L #T #H @(csxa_ind … H) -T /4 width=1 by cpx_cpxs, csx_intro/
-qed.
-
-(* Basic_1: was just: sn3_pr3_trans *)
-lemma csx_cpxs_trans: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, o] T1 →
- ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → ⦃G, L⦄ ⊢ ⬊*[h, o] T2.
-#h #o #G #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2 /2 width=3 by csx_cpx_trans/
-qed-.
-
-(* Main eliminators *********************************************************)
-
-lemma csx_ind_alt: ∀h,o,G,L. ∀R:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ ⬊*[h, o] T1 →
- (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → (T1 = T2 → ⊥) → R T2) → R T1
- ) →
- ∀T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → R T.
-#h #o #G #L #R #H0 #T1 #H @(csxa_ind … (csx_csxa … H)) -T1 /4 width=1 by csxa_csx/
-qed-.
(* Properties with normal terms for uncounted parallel rt-transition ********)
(* Basic_1: was just: sn3_nf2 *)
-lemma cnx_csx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃T⦄.
+lemma cnx_csx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
/2 width=1 by NF_to_SN/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/cpxs_tdeq.ma".
+include "basic_2/rt_computation/cpxs_cpxs.ma".
+include "basic_2/rt_computation/csx_csx.ma".
+
+(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS ************)
+
+(* Properties with uncounted context-sensitive rt-computation for terms *****)
+
+(* Basic_1: was just: sn3_intro *)
+lemma csx_intro_cpxs: ∀h,o,G,L,T1.
+ (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≡[h, o] T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄) →
+ ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄.
+/4 width=1 by cpx_cpxs, csx_intro/ qed-.
+
+(* Basic_1: was just: sn3_pr3_trans *)
+lemma csx_cpxs_trans: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
+ ∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄.
+#h #o #G #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2
+/2 width=3 by csx_cpx_trans/
+qed-.
+
+(* Eliminators with uncounted context-sensitive rt-computation for terms ****)
+
+lemma csx_ind_cpxs_tdeq: ∀h,o,G,L. ∀R:predicate term.
+ (∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
+ (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≡[h, o] T2 → ⊥) → R T2) → R T1
+ ) →
+ ∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
+ ∀T0. ⦃G, L⦄ ⊢ T1 ⬈*[h] T0 → ∀T2. T0 ≡[h, o] T2 → R T2.
+#h #o #G #L #R #IH #T1 #H @(csx_ind … H) -T1
+#T1 #HT1 #IH1 #T0 #HT10 #T2 #HT02
+@IH -IH /3 width=3 by csx_cpxs_trans, csx_tdeq_trans/ -HT1 #V2 #HTV2 #HnTV2
+lapply (tdeq_tdneq_trans … HT02 … HnTV2) -HnTV2 #H
+elim (tdeq_cpxs_trans … HT02 … HTV2) -T2 #V0 #HTV0 #HV02
+lapply (tndeq_tdeq_canc_dx … H … HV02) -H #HnTV0
+elim (tdeq_dec h o T1 T0) #H
+[ lapply (tdeq_tdneq_trans … H … HnTV0) -H -HnTV0 #Hn10
+ lapply (cpxs_trans … HT10 … HTV0) -T0 #H10
+ elim (cpxs_tdneq_inv_step_sn … H10 … Hn10) -H10 -Hn10
+ /3 width=8 by tdeq_trans/
+| elim (cpxs_tdneq_inv_step_sn … HT10 … H) -HT10 -H #T #V #HT1 #HnT1 #HTV #HVT0
+ elim (tdeq_cpxs_trans … HVT0 … HTV0) -T0
+ /3 width=8 by cpxs_trans, tdeq_trans/
+]
+qed-.
+
+lemma csx_ind_alt: ∀h,o,G,L. ∀R:predicate term.
+ (∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
+ (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≡[h, o] T2 → ⊥) → R T2) → R T1
+ ) →
+ ∀T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → R T.
+#h #o #G #L #R #IH #T #HT
+@(csx_ind_cpxs_tdeq … IH … HT) -IH -HT // (**) (* full auto fails *)
+qed-.
(* Advanced properties ******************************************************)
-lemma csx_tdeq_trans: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃T1⦄ →
- ∀T2. T1 ≡[h, o] T2 → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃T2⦄.
+lemma csx_tdeq_trans: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
+ ∀T2. T1 ≡[h, o] T2 → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄.
#h #o #G #L #T1 #H @(csx_ind … H) -T1 #T #_ #IH #T2 #HT2
@csx_intro #T1 #HT21 #HnT21 elim (tdeq_cpx_trans … HT2 … HT21) -HT21
/4 width=5 by tdeq_repl/
qed-.
-lemma csx_cpx_trans: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃T1⦄ →
- ∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃T2⦄.
+lemma csx_cpx_trans: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
+ ∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄.
#h #o #G #L #T1 #H @(csx_ind … H) -T1 #T1 #HT1 #IHT1 #T2 #HLT12
elim (tdeq_dec h o T1 T2) /3 width=4 by csx_tdeq_trans/
qed-.
(* Basic_1: was just: sn3_cast *)
-lemma csx_cast: ∀h,o,G,L,W. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃W⦄ →
- ∀T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃ⓝW.T⦄.
+lemma csx_cast: ∀h,o,G,L,W. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃W⦄ →
+ ∀T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓝW.T⦄.
#h #o #G #L #W #HW @(csx_ind … HW) -W
#W #HW #IHW #T #HT @(csx_ind … HT) -T
#T #HT #IHT @csx_intro
#X #H1 #H2 elim (cpx_inv_cast1 … H1) -H1
[ * #W0 #T0 #HLW0 #HLT0 #H destruct
- elim (tdeq_false_inv_pair … H2) -H2
+ elim (tdneq_inv_pair … H2) -H2
[ -W -T #H elim H -H //
| -HW -IHT /3 width=3 by csx_cpx_trans/
| -HW -HT -IHW /4 width=3 by csx_cpx_trans, cpx_pair_sn/
-cpxs.ma
-csx.ma csx_cnx.ma csx_csx.ma
+cpxs.ma cpxs_tdeq.ma cpxs_cpxs.ma
+csx.ma csx_cnx.ma csx_cpxs.ma csx_csx.ma
(* Negated inversion lemmas *************************************************)
-lemma tdeq_false_inv_pair: ∀h,o,I1,I2,V1,V2,T1,T2.
- (②{I1}V1.T1 ≡[h, o] ②{I2}V2.T2 → ⊥) →
- ∨∨ I1 = I2 → ⊥
- | V1 ≡[h, o] V2 → ⊥
- | (T1 ≡[h, o] T2 → ⊥).
+lemma tdneq_inv_pair: ∀h,o,I1,I2,V1,V2,T1,T2.
+ (②{I1}V1.T1 ≡[h, o] ②{I2}V2.T2 → ⊥) →
+ ∨∨ I1 = I2 → ⊥
+ | V1 ≡[h, o] V2 → ⊥
+ | (T1 ≡[h, o] T2 → ⊥).
#h #o #I1 #I2 #V1 #V2 #T1 #T2 #H12
elim (eq_item2_dec I1 I2) /3 width=1 by or3_intro0/ #H destruct
elim (tdeq_dec h o V1 V2) /3 width=1 by or3_intro1/
theorem tdeq_repl: ∀h,o,T1,T2. T1 ≡[h, o] T2 →
∀U1. T1 ≡[h, o] U1 → ∀U2. T2 ≡[h, o] U2 → U1 ≡[h, o] U2.
/3 width=3 by tdeq_canc_sn, tdeq_trans/ qed-.
+
+(* Negated main properies ***************************************************)
+
+theorem tdeq_tdneq_trans: ∀h,o,T1,T. T1 ≡[h, o] T → ∀T2. (T ≡[h, o] T2 → ⊥) →
+ T1 ≡[h, o] T2 → ⊥.
+/3 width=3 by tdeq_canc_sn/ qed-.
+
+theorem tndeq_tdeq_canc_dx: ∀h,o,T1,T. (T1 ≡[h, o] T → ⊥) → ∀T2. T2 ≡[h, o] T →
+ T1 ≡[h, o] T2 → ⊥.
+/3 width=3 by tdeq_trans/ qed-.
[ "lpxs ( ⦃?,?⦄ ⊢ ➡*[?,?] ? )" "lpxs_drop" + "lpxs_lleq" + "lpxs_aaa" + "lpxs_cpxs" + "lpxs_lpxs" * ]
[ "cpxs_tsts" + "cpxs_tsts_vector" + "cpxs_lreq" + "cpxs_lift" + "cpxs_lleq" + "cpxs_aaa" + "cpxs_cpxs" * ]
*)
- [ "csx ( ⦃?,?⦄ ⊢ ⬈[?,?] 𝐒⦃?⦄ )" "csx_cnx" + "csx_csx" * ]
- [ "cpxs ( ⦃?,?⦄ ⊢ ? ⬈*[?] ? )" * ]
+ [ "csx ( ⦃?,?⦄ ⊢ ⬈*[?,?] 𝐒⦃?⦄ )" "csx_cnx" + "csx_cpxs" + "csx_csx" * ]
+ [ "cpxs ( ⦃?,?⦄ ⊢ ? ⬈*[?] ? )" "cpxs_tdeq" + "cpxs_cpxs" * ]
}
]
}
projects / helm.git / commitdiff