--- /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/syntax/theq_tdeq.ma".
+include "basic_2/rt_computation/cpxs_lsubr.ma".
+include "basic_2/rt_computation/cpxs_cnx.ma".
+include "basic_2/rt_computation/lfpxs_cpxs.ma".
+
+(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS ************)
+
+(* Forward lemmas with head equivalence for terms ***************************)
+
+lemma cpxs_fwd_sort: ∀h,o,G,L,U,s. ⦃G, L⦄ ⊢ ⋆s ⬈*[h] U →
+ ⋆s ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ ⋆(next h s) ⬈*[h] U.
+#h #o #G #L #U #s #H elim (cpxs_inv_sort1 … H) -H *
+[ #H destruct /2 width=1 by or_introl/
+| #n #H destruct
+ @or_intror >iter_S <(iter_n_Sm … (next h)) // (**)
+]
+qed-.
+
+(* Note: probably this is an inversion lemma *)
+(* Basic_2A1: was: cpxs_fwd_delta *)
+lemma cpxs_fwd_delta_drops: ∀h,o,I,G,L,K,V1,i. ⬇*[i] L ≡ K.ⓑ{I}V1 →
+ ∀V2. ⬆*[⫯i] V1 ≡ V2 →
+ ∀U. ⦃G, L⦄ ⊢ #i ⬈*[h] U →
+ #i ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ V2 ⬈*[h] U.
+#h #o #I #G #L #K #V1 #i #HLK #V2 #HV12 #U #H
+elim (cpxs_inv_lref1_drops … H) -H /2 width=1 by or_introl/
+* #I0 #K0 #V0 #U0 #HLK0 #HVU0 #HU0
+lapply (drops_mono … HLK0 … HLK) -HLK0 #H destruct
+elim (cpxs_lifts … HVU0 (Ⓣ) … L … HV12) -HVU0 -HV12 /2 width=3 by drops_isuni_fwd_drop2/ #X #H
+<(lifts_mono … HU0 … H) -U0 -X /2 width=1 by or_intror/
+qed-.
+
+(* Basic_1: was just: pr3_iso_beta *)
+lemma cpxs_fwd_beta: ∀h,o,p,G,L,V,W,T,U. ⦃G, L⦄ ⊢ ⓐV.ⓛ{p}W.T ⬈*[h] U →
+ ⓐV.ⓛ{p}W.T ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ ⓓ{p}ⓝW.V.T ⬈*[h] U.
+#h #o #p #G #L #V #W #T #U #H elim (cpxs_inv_appl1 … H) -H *
+[ #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or_introl/
+| #b #W0 #T0 #HT0 #HU
+ elim (cpxs_inv_abst1 … HT0) -HT0 #W1 #T1 #HW1 #HT1 #H destruct
+ lapply (lsubr_cpxs_trans … HT1 (L.ⓓⓝW.V) ?) -HT1
+ /5 width=3 by cpxs_trans, cpxs_bind, cpxs_pair_sn, lsubr_beta, or_intror/
+| #b #V1 #V2 #V0 #T1 #_ #_ #HT1 #_
+ elim (cpxs_inv_abst1 … HT1) -HT1 #W2 #T2 #_ #_ #H destruct
+]
+qed-.
+
+lemma cpxs_fwd_theta: ∀h,o,p,G,L,V1,V,T,U. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}V.T ⬈*[h] U →
+ ∀V2. ⬆*[1] V1 ≡ V2 → ⓐV1.ⓓ{p}V.T ⩳[h, o] U ∨
+ ⦃G, L⦄ ⊢ ⓓ{p}V.ⓐV2.T ⬈*[h] U.
+#h #o #p #G #L #V1 #V #T #U #H #V2 #HV12
+elim (cpxs_inv_appl1 … H) -H *
+[ -HV12 #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or_introl/
+| #q #W #T0 #HT0 #HU
+ elim (cpxs_inv_abbr1 … HT0) -HT0 *
+ [ #V3 #T3 #_ #_ #H destruct
+ | #X #HT2 #H #H0 destruct
+ elim (lifts_inv_bind1 … H) -H #W2 #T2 #HW2 #HT02 #H destruct
+ @or_intror @(cpxs_trans … HU) -U (**) (* explicit constructor *)
+ @(cpxs_trans … (+ⓓV.ⓐV2.ⓛ{q}W2.T2)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T
+ @(cpxs_strap2 … (ⓐV1.ⓛ{q}W.T0)) [2: /2 width=1 by cpxs_beta_dx/ ]
+ /4 width=7 by cpx_zeta, lifts_bind, lifts_flat/
+ ]
+| #q #V3 #V4 #V0 #T0 #HV13 #HV34 #HT0 #HU
+ @or_intror @(cpxs_trans … HU) -U (**) (* explicit constructor *)
+ elim (cpxs_inv_abbr1 … HT0) -HT0 *
+ [ #V5 #T5 #HV5 #HT5 #H destruct
+ elim (cpxs_lifts … HV13 (Ⓣ) … (L.ⓓV) … HV12) -V1 /3 width=1 by drops_refl, drops_drop/ #X #H
+ <(lifts_mono … HV34 … H) -V3 -X /3 width=1 by cpxs_flat, cpxs_bind/
+ | #X #HT1 #H #H0 destruct
+ elim (lifts_inv_bind1 … H) -H #V5 #T5 #HV05 #HT05 #H destruct
+ elim (cpxs_lifts … HV13 (Ⓣ) … (L.ⓓV0) … HV12) -HV13 /3 width=1 by drops_refl, drops_drop/ #X #H
+ <(lifts_mono … HV34 … H) -V3 -X #HV24
+ @(cpxs_trans … (+ⓓV.ⓐV2.ⓓ{q}V5.T5)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T
+ @(cpxs_strap2 … (ⓐV1.ⓓ{q}V0.T0)) [ /4 width=7 by cpx_zeta, lifts_bind, lifts_flat/ ] -V -V5 -T5
+ @(cpxs_strap2 … (ⓓ{q}V0.ⓐV2.T0)) /3 width=3 by cpxs_pair_sn, cpxs_bind_dx, cpx_theta/
+ ]
+]
+qed-.
+
+lemma cpxs_fwd_cast: ∀h,o,G,L,W,T,U. ⦃G, L⦄ ⊢ ⓝW.T ⬈*[h] U →
+ ∨∨ ⓝW. T ⩳[h, o] U | ⦃G, L⦄ ⊢ T ⬈*[h] U | ⦃G, L⦄ ⊢ W ⬈*[h] U.
+#h #o #G #L #W #T #U #H
+elim (cpxs_inv_cast1 … H) -H /2 width=1 by or3_intro1, or3_intro2/ *
+#W0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or3_intro0/
+qed-.
+
+lemma cpxs_fwd_cnx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄ →
+ ∀U. ⦃G, L⦄ ⊢ T ⬈*[h] U → T ⩳[h, o] U.
+/3 width=4 by cpxs_inv_cnx1, tdeq_theq/ 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/syntax/theq_simple_vector.ma".
+include "basic_2/relocation/lifts_vector.ma".
+include "basic_2/rt_computation/cpxs_theq.ma".
+
+(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS ************)
+
+(* Vector form of forward lemmas with head equivalence for terms ************)
+
+lemma cpxs_fwd_sort_vector: ∀h,o,G,L,s,Vs,U. ⦃G, L⦄ ⊢ ⒶVs.⋆s ⬈*[h] U →
+ ⒶVs.⋆s ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ ⒶVs.⋆(next h s) ⬈*[h] U.
+#h #o #G #L #s #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_sort/
+#V #Vs #IHVs #U #H
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1 by theq_pair, or_introl/
+| #p #W1 #T1 #HT1 #HU
+ elim (IHVs … HT1) -IHVs -HT1 #HT1
+ [ elim (theq_inv_applv_bind_simple … HT1) //
+ | @or_intror (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -U
+ @(cpxs_strap1 … (ⓐV.ⓛ{p}W1.T1)) /3 width=1 by cpxs_flat_dx, cpx_beta/
+ ]
+| #p #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU
+ elim (IHVs … HT1) -IHVs -HT1 #HT1
+ [ elim (theq_inv_applv_bind_simple … HT1) //
+ | @or_intror (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -U
+ @(cpxs_strap1 … (ⓐV1.ⓓ{p}V3.T1)) /3 width=3 by cpxs_flat, cpx_theta/
+ ]
+]
+qed-.
+
+(* Basic_2A1: was: cpxs_fwd_delta_vector *)
+lemma cpxs_fwd_delta_drops_vector: ∀h,o,I,G,L,K,V1,i. ⬇*[i] L ≡ K.ⓑ{I}V1 →
+ ∀V2. ⬆*[⫯i] V1 ≡ V2 →
+ ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.#i ⬈*[h] U →
+ ⒶVs.#i ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ ⒶVs.V2 ⬈*[h] U.
+#h #o #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs /2 width=5 by cpxs_fwd_delta_drops/
+#V #Vs #IHVs #U #H -K -V1
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or_introl/
+| #q #W0 #T0 #HT0 #HU
+ elim (IHVs … HT0) -IHVs -HT0 #HT0
+ [ elim (theq_inv_applv_bind_simple … HT0) //
+ | @or_intror -i (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -U
+ @(cpxs_strap1 … (ⓐV.ⓛ{q}W0.T0)) /3 width=1 by cpxs_flat_dx, cpx_beta/
+ ]
+| #q #V0 #V1 #V3 #T0 #HV0 #HV01 #HT0 #HU
+ elim (IHVs … HT0) -IHVs -HT0 #HT0
+ [ elim (theq_inv_applv_bind_simple … HT0) //
+ | @or_intror -i (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -U
+ @(cpxs_strap1 … (ⓐV0.ⓓ{q}V3.T0)) /3 width=3 by cpxs_flat, cpx_theta/
+ ]
+]
+qed-.
+
+(* Basic_1: was just: pr3_iso_appls_beta *)
+lemma cpxs_fwd_beta_vector: ∀h,o,p,G,L,Vs,V,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓐV.ⓛ{p}W.T ⬈*[h] U →
+ ⒶVs.ⓐV.ⓛ{p}W. T ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ ⒶVs.ⓓ{p}ⓝW.V.T ⬈*[h] U.
+#h #o #p #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_beta/
+#V0 #Vs #IHVs #V #W #T #U #H
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1 by theq_pair, or_introl/
+| #q #W1 #T1 #HT1 #HU
+ elim (IHVs … HT1) -IHVs -HT1 #HT1
+ [ elim (theq_inv_applv_bind_simple … HT1) //
+ | @or_intror (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -U
+ @(cpxs_strap1 … (ⓐV0.ⓛ{q}W1.T1)) /3 width=1 by cpxs_flat_dx, cpx_beta/
+ ]
+| #q #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU
+ elim (IHVs … HT1) -IHVs -HT1 #HT1
+ [ elim (theq_inv_applv_bind_simple … HT1) //
+ | @or_intror (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -U
+ @(cpxs_strap1 … (ⓐV1.ⓓ{q}V3.T1)) /3 width=3 by cpxs_flat, cpx_theta/
+ ]
+]
+qed-.
+
+(* Basic_1: was just: pr3_iso_appls_abbr *)
+lemma cpxs_fwd_theta_vector: ∀h,o,G,L,V1b,V2b. ⬆*[1] V1b ≡ V2b →
+ ∀p,V,T,U. ⦃G, L⦄ ⊢ ⒶV1b.ⓓ{p}V.T ⬈*[h] U →
+ ⒶV1b.ⓓ{p}V.T ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ ⓓ{p}V.ⒶV2b.T ⬈*[h] U.
+#h #o #G #L #V1b #V2b * -V1b -V2b /3 width=1 by or_intror/
+#V1b #V2b #V1a #V2a #HV12a #HV12b #p
+generalize in match HV12a; -HV12a
+generalize in match V2a; -V2a
+generalize in match V1a; -V1a
+elim HV12b -V1b -V2b /2 width=1 by cpxs_fwd_theta/
+#V1b #V2b #V1b #V2b #HV12b #_ #IHV12b #V1a #V2a #HV12a #V #T #U #H
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHV12b -HV12a -HV12b #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or_introl/
+| #q #W0 #T0 #HT0 #HU
+ elim (IHV12b … HV12b … HT0) -IHV12b -HT0 #HT0
+ [ -HV12a -HV12b -HU
+ elim (theq_inv_pair1 … HT0) #V1 #T1 #H destruct
+ | @or_intror -V1b (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -U
+ elim (cpxs_inv_abbr1 … HT0) -HT0 *
+ [ -HV12a #V1 #T1 #_ #_ #H destruct
+ | -V1b #X #HT1 #H #H0 destruct
+ elim (lifts_inv_bind1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
+ @(cpxs_trans … (+ⓓV.ⓐV2a.ⓛ{q}W1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2b
+ @(cpxs_strap2 … (ⓐV1a.ⓛ{q}W0.T0))
+ /4 width=7 by cpxs_beta_dx, cpx_zeta, lifts_bind, lifts_flat/
+ ]
+ ]
+| #q #V0a #Va #V0 #T0 #HV10a #HV0a #HT0 #HU
+ elim (IHV12b … HV12b … HT0) -HV12b -IHV12b -HT0 #HT0
+ [ -HV12a -HV10a -HV0a -HU
+ elim (theq_inv_pair1 … HT0) #V1 #T1 #H destruct
+ | @or_intror -V1b -V1b (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -U
+ elim (cpxs_inv_abbr1 … HT0) -HT0 *
+ [ #V1 #T1 #HV1 #HT1 #H destruct
+ elim (cpxs_lifts … HV10a (Ⓣ) … (L.ⓓV) … HV12a) -V1a /3 width=1 by drops_refl, drops_drop/ #X #H
+ <(lifts_mono … HV0a … H) -V0a -X #HV2a
+ @(cpxs_trans … (ⓓ{p}V.ⓐV2a.T1)) /3 width=1 by cpxs_bind, cpxs_pair_sn, cpxs_flat_dx, cpxs_bind_dx/
+ | #X #HT1 #H #H0 destruct
+ elim (lifts_inv_bind1 … H) -H #V1 #T1 #HW01 #HT01 #H destruct
+ elim (cpxs_lifts … HV10a (Ⓣ) … (L.ⓓV0) … HV12a) /3 width=1 by drops_refl, drops_drop/ #X #H
+ <(lifts_mono … HV0a … H) -V0a -X #HV2a
+ @(cpxs_trans … (+ⓓV.ⓐV2a.ⓓ{q}V1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2b
+ @(cpxs_strap2 … (ⓐV1a.ⓓ{q}V0.T0)) [ /4 width=7 by cpx_zeta, lifts_bind, lifts_flat/ ] -V -V1 -T1
+ @(cpxs_strap2 … (ⓓ{q}V0.ⓐV2a.T0)) /3 width=3 by cpxs_pair_sn, cpxs_bind_dx, cpx_theta/
+ ]
+ ]
+]
+qed-.
+
+(* Basic_1: was just: pr3_iso_appls_cast *)
+lemma cpxs_fwd_cast_vector: ∀h,o,G,L,Vs,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓝW.T ⬈*[h] U →
+ ∨∨ ⒶVs. ⓝW. T ⩳[h, o] U
+ | ⦃G, L⦄ ⊢ ⒶVs.T ⬈*[h] U
+ | ⦃G, L⦄ ⊢ ⒶVs.W ⬈*[h] U.
+#h #o #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_cast/
+#V #Vs #IHVs #W #T #U #H
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or3_intro0/
+| #q #W0 #T0 #HT0 #HU elim (IHVs … HT0) -IHVs -HT0 #HT0
+ [ elim (theq_inv_applv_bind_simple … HT0) //
+ | @or3_intro1 -W (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -U
+ @(cpxs_strap1 … (ⓐV.ⓛ{q}W0.T0)) /2 width=1 by cpxs_flat_dx, cpx_beta/
+ | @or3_intro2 -T (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -U
+ @(cpxs_strap1 … (ⓐV.ⓛ{q}W0.T0)) /2 width=1 by cpxs_flat_dx, cpx_beta/
+ ]
+| #q #V0 #V1 #V2 #T0 #HV0 #HV01 #HT0 #HU
+ elim (IHVs … HT0) -IHVs -HT0 #HT0
+ [ elim (theq_inv_applv_bind_simple … HT0) //
+ | @or3_intro1 -W (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -U
+ @(cpxs_strap1 … (ⓐV0.ⓓ{q}V2.T0)) /2 width=3 by cpxs_flat, cpx_theta/
+ | @or3_intro2 -T (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -U
+ @(cpxs_strap1 … (ⓐV0.ⓓ{q}V2.T0)) /2 width=3 by cpxs_flat, cpx_theta/
+ ]
+]
+qed-.
+
+(* Basic_1: was just: nf2_iso_appls_lref *)
+lemma cpxs_fwd_cnx_vector: ∀h,o,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄ →
+ ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.T ⬈*[h] U → ⒶVs.T ⩳[h, o] U.
+#h #o #G #L #T #H1T #H2T #Vs elim Vs -Vs [ @(cpxs_fwd_cnx … H2T) ] (**) (* /2 width=3 by cpxs_fwd_cnx/ does not work *)
+#V #Vs #IHVs #U #H
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair/
+| #p #W0 #T0 #HT0 #HU
+ lapply (IHVs … HT0) -IHVs -HT0 #HT0
+ elim (theq_inv_applv_bind_simple … HT0) //
+| #p #V1 #V2 #V0 #T0 #HV1 #HV12 #HT0 #HU
+ lapply (IHVs … HT0) -IHVs -HT0 #HT0
+ elim (theq_inv_applv_bind_simple … HT0) //
+]
+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/syntax/tsts_tdeq.ma".
-include "basic_2/rt_computation/cpxs_lsubr.ma".
-include "basic_2/rt_computation/cpxs_cnx.ma".
-include "basic_2/rt_computation/lfpxs_cpxs.ma".
-
-(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS ************)
-
-(* Forward lemmas with same top term structure ******************************)
-
-lemma cpxs_fwd_sort: ∀h,o,G,L,U,s. ⦃G, L⦄ ⊢ ⋆s ⬈*[h] U →
- ⋆s ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ ⋆(next h s) ⬈*[h] U.
-#h #o #G #L #U #s #H elim (cpxs_inv_sort1 … H) -H *
-[ #H destruct /2 width=1 by or_introl/
-| #n #H destruct
- @or_intror >iter_S <(iter_n_Sm … (next h)) // (**)
-]
-qed-.
-
-(* Note: probably this is an inversion lemma *)
-(* Basic_2A1: was: cpxs_fwd_delta *)
-lemma cpxs_fwd_delta_drops: ∀h,o,I,G,L,K,V1,i. ⬇*[i] L ≡ K.ⓑ{I}V1 →
- ∀V2. ⬆*[⫯i] V1 ≡ V2 →
- ∀U. ⦃G, L⦄ ⊢ #i ⬈*[h] U →
- #i ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ V2 ⬈*[h] U.
-#h #o #I #G #L #K #V1 #i #HLK #V2 #HV12 #U #H
-elim (cpxs_inv_lref1_drops … H) -H /2 width=1 by or_introl/
-* #I0 #K0 #V0 #U0 #HLK0 #HVU0 #HU0
-lapply (drops_mono … HLK0 … HLK) -HLK0 #H destruct
-elim (cpxs_lifts … HVU0 (Ⓣ) … L … HV12) -HVU0 -HV12 /2 width=3 by drops_isuni_fwd_drop2/ #X #H
-<(lifts_mono … HU0 … H) -U0 -X /2 width=1 by or_intror/
-qed-.
-
-(* Basic_1: was just: pr3_iso_beta *)
-lemma cpxs_fwd_beta: ∀h,o,p,G,L,V,W,T,U. ⦃G, L⦄ ⊢ ⓐV.ⓛ{p}W.T ⬈*[h] U →
- ⓐV.ⓛ{p}W.T ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ ⓓ{p}ⓝW.V.T ⬈*[h] U.
-#h #o #p #G #L #V #W #T #U #H elim (cpxs_inv_appl1 … H) -H *
-[ #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
-| #b #W0 #T0 #HT0 #HU
- elim (cpxs_inv_abst1 … HT0) -HT0 #W1 #T1 #HW1 #HT1 #H destruct
- lapply (lsubr_cpxs_trans … HT1 (L.ⓓⓝW.V) ?) -HT1
- /5 width=3 by cpxs_trans, cpxs_bind, cpxs_pair_sn, lsubr_beta, or_intror/
-| #b #V1 #V2 #V0 #T1 #_ #_ #HT1 #_
- elim (cpxs_inv_abst1 … HT1) -HT1 #W2 #T2 #_ #_ #H destruct
-]
-qed-.
-
-lemma cpxs_fwd_theta: ∀h,o,p,G,L,V1,V,T,U. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}V.T ⬈*[h] U →
- ∀V2. ⬆*[1] V1 ≡ V2 → ⓐV1.ⓓ{p}V.T ⩳[h, o] U ∨
- ⦃G, L⦄ ⊢ ⓓ{p}V.ⓐV2.T ⬈*[h] U.
-#h #o #p #G #L #V1 #V #T #U #H #V2 #HV12
-elim (cpxs_inv_appl1 … H) -H *
-[ -HV12 #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
-| #q #W #T0 #HT0 #HU
- elim (cpxs_inv_abbr1 … HT0) -HT0 *
- [ #V3 #T3 #_ #_ #H destruct
- | #X #HT2 #H #H0 destruct
- elim (lifts_inv_bind1 … H) -H #W2 #T2 #HW2 #HT02 #H destruct
- @or_intror @(cpxs_trans … HU) -U (**) (* explicit constructor *)
- @(cpxs_trans … (+ⓓV.ⓐV2.ⓛ{q}W2.T2)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T
- @(cpxs_strap2 … (ⓐV1.ⓛ{q}W.T0)) [2: /2 width=1 by cpxs_beta_dx/ ]
- /4 width=7 by cpx_zeta, lifts_bind, lifts_flat/
- ]
-| #q #V3 #V4 #V0 #T0 #HV13 #HV34 #HT0 #HU
- @or_intror @(cpxs_trans … HU) -U (**) (* explicit constructor *)
- elim (cpxs_inv_abbr1 … HT0) -HT0 *
- [ #V5 #T5 #HV5 #HT5 #H destruct
- elim (cpxs_lifts … HV13 (Ⓣ) … (L.ⓓV) … HV12) -V1 /3 width=1 by drops_refl, drops_drop/ #X #H
- <(lifts_mono … HV34 … H) -V3 -X /3 width=1 by cpxs_flat, cpxs_bind/
- | #X #HT1 #H #H0 destruct
- elim (lifts_inv_bind1 … H) -H #V5 #T5 #HV05 #HT05 #H destruct
- elim (cpxs_lifts … HV13 (Ⓣ) … (L.ⓓV0) … HV12) -HV13 /3 width=1 by drops_refl, drops_drop/ #X #H
- <(lifts_mono … HV34 … H) -V3 -X #HV24
- @(cpxs_trans … (+ⓓV.ⓐV2.ⓓ{q}V5.T5)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T
- @(cpxs_strap2 … (ⓐV1.ⓓ{q}V0.T0)) [ /4 width=7 by cpx_zeta, lifts_bind, lifts_flat/ ] -V -V5 -T5
- @(cpxs_strap2 … (ⓓ{q}V0.ⓐV2.T0)) /3 width=3 by cpxs_pair_sn, cpxs_bind_dx, cpx_theta/
- ]
-]
-qed-.
-
-lemma cpxs_fwd_cast: ∀h,o,G,L,W,T,U. ⦃G, L⦄ ⊢ ⓝW.T ⬈*[h] U →
- ∨∨ ⓝW. T ⩳[h, o] U | ⦃G, L⦄ ⊢ T ⬈*[h] U | ⦃G, L⦄ ⊢ W ⬈*[h] U.
-#h #o #G #L #W #T #U #H
-elim (cpxs_inv_cast1 … H) -H /2 width=1 by or3_intro1, or3_intro2/ *
-#W0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or3_intro0/
-qed-.
-
-lemma cpxs_fwd_cnx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄ →
- ∀U. ⦃G, L⦄ ⊢ T ⬈*[h] U → T ⩳[h, o] U.
-/3 width=4 by cpxs_inv_cnx1, tdeq_tsts/ 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/syntax/tsts_simple_vector.ma".
-include "basic_2/relocation/lifts_vector.ma".
-include "basic_2/rt_computation/cpxs_tsts.ma".
-
-(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS ************)
-
-(* Vector form of forward lemmas with same top term structure ***************)
-
-lemma cpxs_fwd_sort_vector: ∀h,o,G,L,s,Vs,U. ⦃G, L⦄ ⊢ ⒶVs.⋆s ⬈*[h] U →
- ⒶVs.⋆s ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ ⒶVs.⋆(next h s) ⬈*[h] U.
-#h #o #G #L #s #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_sort/
-#V #Vs #IHVs #U #H
-elim (cpxs_inv_appl1 … H) -H *
-[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
-| #p #W1 #T1 #HT1 #HU
- elim (IHVs … HT1) -IHVs -HT1 #HT1
- [ elim (tsts_inv_applv_bind_simple … HT1) //
- | @or_intror (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV.ⓛ{p}W1.T1)) /3 width=1 by cpxs_flat_dx, cpx_beta/
- ]
-| #p #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU
- elim (IHVs … HT1) -IHVs -HT1 #HT1
- [ elim (tsts_inv_applv_bind_simple … HT1) //
- | @or_intror (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV1.ⓓ{p}V3.T1)) /3 width=3 by cpxs_flat, cpx_theta/
- ]
-]
-qed-.
-
-(* Basic_2A1: was: cpxs_fwd_delta_vector *)
-lemma cpxs_fwd_delta_drops_vector: ∀h,o,I,G,L,K,V1,i. ⬇*[i] L ≡ K.ⓑ{I}V1 →
- ∀V2. ⬆*[⫯i] V1 ≡ V2 →
- ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.#i ⬈*[h] U →
- ⒶVs.#i ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ ⒶVs.V2 ⬈*[h] U.
-#h #o #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs /2 width=5 by cpxs_fwd_delta_drops/
-#V #Vs #IHVs #U #H -K -V1
-elim (cpxs_inv_appl1 … H) -H *
-[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
-| #q #W0 #T0 #HT0 #HU
- elim (IHVs … HT0) -IHVs -HT0 #HT0
- [ elim (tsts_inv_applv_bind_simple … HT0) //
- | @or_intror -i (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV.ⓛ{q}W0.T0)) /3 width=1 by cpxs_flat_dx, cpx_beta/
- ]
-| #q #V0 #V1 #V3 #T0 #HV0 #HV01 #HT0 #HU
- elim (IHVs … HT0) -IHVs -HT0 #HT0
- [ elim (tsts_inv_applv_bind_simple … HT0) //
- | @or_intror -i (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV0.ⓓ{q}V3.T0)) /3 width=3 by cpxs_flat, cpx_theta/
- ]
-]
-qed-.
-
-(* Basic_1: was just: pr3_iso_appls_beta *)
-lemma cpxs_fwd_beta_vector: ∀h,o,p,G,L,Vs,V,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓐV.ⓛ{p}W.T ⬈*[h] U →
- ⒶVs.ⓐV.ⓛ{p}W. T ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ ⒶVs.ⓓ{p}ⓝW.V.T ⬈*[h] U.
-#h #o #p #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_beta/
-#V0 #Vs #IHVs #V #W #T #U #H
-elim (cpxs_inv_appl1 … H) -H *
-[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
-| #q #W1 #T1 #HT1 #HU
- elim (IHVs … HT1) -IHVs -HT1 #HT1
- [ elim (tsts_inv_applv_bind_simple … HT1) //
- | @or_intror (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV0.ⓛ{q}W1.T1)) /3 width=1 by cpxs_flat_dx, cpx_beta/
- ]
-| #q #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU
- elim (IHVs … HT1) -IHVs -HT1 #HT1
- [ elim (tsts_inv_applv_bind_simple … HT1) //
- | @or_intror (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV1.ⓓ{q}V3.T1)) /3 width=3 by cpxs_flat, cpx_theta/
- ]
-]
-qed-.
-
-(* Basic_1: was just: pr3_iso_appls_abbr *)
-lemma cpxs_fwd_theta_vector: ∀h,o,G,L,V1b,V2b. ⬆*[1] V1b ≡ V2b →
- ∀p,V,T,U. ⦃G, L⦄ ⊢ ⒶV1b.ⓓ{p}V.T ⬈*[h] U →
- ⒶV1b.ⓓ{p}V.T ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ ⓓ{p}V.ⒶV2b.T ⬈*[h] U.
-#h #o #G #L #V1b #V2b * -V1b -V2b /3 width=1 by or_intror/
-#V1b #V2b #V1a #V2a #HV12a #HV12b #p
-generalize in match HV12a; -HV12a
-generalize in match V2a; -V2a
-generalize in match V1a; -V1a
-elim HV12b -V1b -V2b /2 width=1 by cpxs_fwd_theta/
-#V1b #V2b #V1b #V2b #HV12b #_ #IHV12b #V1a #V2a #HV12a #V #T #U #H
-elim (cpxs_inv_appl1 … H) -H *
-[ -IHV12b -HV12a -HV12b #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
-| #q #W0 #T0 #HT0 #HU
- elim (IHV12b … HV12b … HT0) -IHV12b -HT0 #HT0
- [ -HV12a -HV12b -HU
- elim (tsts_inv_pair1 … HT0) #V1 #T1 #H destruct
- | @or_intror -V1b (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- elim (cpxs_inv_abbr1 … HT0) -HT0 *
- [ -HV12a #V1 #T1 #_ #_ #H destruct
- | -V1b #X #HT1 #H #H0 destruct
- elim (lifts_inv_bind1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
- @(cpxs_trans … (+ⓓV.ⓐV2a.ⓛ{q}W1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2b
- @(cpxs_strap2 … (ⓐV1a.ⓛ{q}W0.T0))
- /4 width=7 by cpxs_beta_dx, cpx_zeta, lifts_bind, lifts_flat/
- ]
- ]
-| #q #V0a #Va #V0 #T0 #HV10a #HV0a #HT0 #HU
- elim (IHV12b … HV12b … HT0) -HV12b -IHV12b -HT0 #HT0
- [ -HV12a -HV10a -HV0a -HU
- elim (tsts_inv_pair1 … HT0) #V1 #T1 #H destruct
- | @or_intror -V1b -V1b (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- elim (cpxs_inv_abbr1 … HT0) -HT0 *
- [ #V1 #T1 #HV1 #HT1 #H destruct
- elim (cpxs_lifts … HV10a (Ⓣ) … (L.ⓓV) … HV12a) -V1a /3 width=1 by drops_refl, drops_drop/ #X #H
- <(lifts_mono … HV0a … H) -V0a -X #HV2a
- @(cpxs_trans … (ⓓ{p}V.ⓐV2a.T1)) /3 width=1 by cpxs_bind, cpxs_pair_sn, cpxs_flat_dx, cpxs_bind_dx/
- | #X #HT1 #H #H0 destruct
- elim (lifts_inv_bind1 … H) -H #V1 #T1 #HW01 #HT01 #H destruct
- elim (cpxs_lifts … HV10a (Ⓣ) … (L.ⓓV0) … HV12a) /3 width=1 by drops_refl, drops_drop/ #X #H
- <(lifts_mono … HV0a … H) -V0a -X #HV2a
- @(cpxs_trans … (+ⓓV.ⓐV2a.ⓓ{q}V1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2b
- @(cpxs_strap2 … (ⓐV1a.ⓓ{q}V0.T0)) [ /4 width=7 by cpx_zeta, lifts_bind, lifts_flat/ ] -V -V1 -T1
- @(cpxs_strap2 … (ⓓ{q}V0.ⓐV2a.T0)) /3 width=3 by cpxs_pair_sn, cpxs_bind_dx, cpx_theta/
- ]
- ]
-]
-qed-.
-
-(* Basic_1: was just: pr3_iso_appls_cast *)
-lemma cpxs_fwd_cast_vector: ∀h,o,G,L,Vs,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓝW.T ⬈*[h] U →
- ∨∨ ⒶVs. ⓝW. T ⩳[h, o] U
- | ⦃G, L⦄ ⊢ ⒶVs.T ⬈*[h] U
- | ⦃G, L⦄ ⊢ ⒶVs.W ⬈*[h] U.
-#h #o #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_cast/
-#V #Vs #IHVs #W #T #U #H
-elim (cpxs_inv_appl1 … H) -H *
-[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or3_intro0/
-| #q #W0 #T0 #HT0 #HU elim (IHVs … HT0) -IHVs -HT0 #HT0
- [ elim (tsts_inv_applv_bind_simple … HT0) //
- | @or3_intro1 -W (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV.ⓛ{q}W0.T0)) /2 width=1 by cpxs_flat_dx, cpx_beta/
- | @or3_intro2 -T (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV.ⓛ{q}W0.T0)) /2 width=1 by cpxs_flat_dx, cpx_beta/
- ]
-| #q #V0 #V1 #V2 #T0 #HV0 #HV01 #HT0 #HU
- elim (IHVs … HT0) -IHVs -HT0 #HT0
- [ elim (tsts_inv_applv_bind_simple … HT0) //
- | @or3_intro1 -W (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV0.ⓓ{q}V2.T0)) /2 width=3 by cpxs_flat, cpx_theta/
- | @or3_intro2 -T (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV0.ⓓ{q}V2.T0)) /2 width=3 by cpxs_flat, cpx_theta/
- ]
-]
-qed-.
-
-(* Basic_1: was just: nf2_iso_appls_lref *)
-lemma cpxs_fwd_cnx_vector: ∀h,o,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄ →
- ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.T ⬈*[h] U → ⒶVs.T ⩳[h, o] U.
-#h #o #G #L #T #H1T #H2T #Vs elim Vs -Vs [ @(cpxs_fwd_cnx … H2T) ] (**) (* /2 width=3 by cpxs_fwd_cnx/ does not work *)
-#V #Vs #IHVs #U #H
-elim (cpxs_inv_appl1 … H) -H *
-[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair/
-| #p #W0 #T0 #HT0 #HU
- lapply (IHVs … HT0) -IHVs -HT0 #HT0
- elim (tsts_inv_applv_bind_simple … HT0) //
-| #p #V1 #V2 #V0 #T0 #HV1 #HV12 #HT0 #HU
- lapply (IHVs … HT0) -IHVs -HT0 #HT0
- elim (tsts_inv_applv_bind_simple … HT0) //
-]
-qed-.
include "basic_2/computation/gcp_aaa.ma".
include "basic_2/computation/cpxs_aaa.ma".
-include "basic_2/computation/csx_tsts_vector.ma".
+include "basic_2/computation/csx_theq_vector.ma".
(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
(* *)
(**************************************************************************)
-include "basic_2/syntax/tsts_tsts.ma".
include "basic_2/computation/cpxs_cpxs.ma".
include "basic_2/computation/csx_alt.ma".
include "basic_2/computation/csx_lift.ma".
--- /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/syntax/theq_simple.ma".
+include "basic_2/syntax/theq_theq.ma".
+include "basic_2/rt_transition/cpx_simple.ma".
+include "basic_2/rt_computation/cpxs.ma".
+include "basic_2/rt_computation/csx_csx.ma".
+
+(* STRONGLY NORMALIZING TERMS FOR UNCOUNTED PARALLEL RT-TRANSITION **********)
+
+(* Properties with head equivalence for terms *******************************)
+
+(* Basic_1: was just: sn3_appl_appl *)
+(* Basic_2A1: was: csx_appl_simple_tsts *)
+lemma csx_appl_simple_theq: ∀h,o,G,L,V. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ → ∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
+ (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ⩳[h, o] T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓐV.T2⦄) →
+ 𝐒⦃T1⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓐV.T1⦄.
+#h #o #G #L #V #H @(csx_ind … H) -V
+#V #_ #IHV #T1 #H @(csx_ind … H) -T1
+#T1 #H1T1 #IHT1 #H2T1 #H3T1
+@csx_intro #X #HL #H
+elim (cpx_inv_appl1_simple … HL) -HL //
+#V0 #T0 #HLV0 #HLT10 #H0 destruct
+elim (tdneq_inv_pair … H) -H
+[ #H elim H -H //
+| -IHT1 #HV0
+ @(csx_cpx_trans … (ⓐV0.T1)) /2 width=1 by cpx_flat/ -HLT10
+ @IHV -IHV /4 width=3 by csx_cpx_trans, cpx_pair_sn/
+| -IHV -H1T1 #H1T10
+ @(csx_cpx_trans … (ⓐV.T0)) /2 width=1 by cpx_flat/ -HLV0
+ elim (theq_dec h o T1 T0) #H2T10
+ [ @IHT1 -IHT1 /4 width=5 by cpxs_strap2, cpxs_strap1, theq_canc_sn, simple_theq_repl_dx/
+ | -IHT1 -H3T1 -H1T10 /3 width=1 by cpx_cpxs/
+ ]
+]
+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/computation/gcp_cr.ma".
+include "basic_2/computation/cpxs_theq_vector.ma".
+include "basic_2/computation/csx_lpx.ma".
+include "basic_2/computation/csx_vector.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERM VECTORS *************)
+
+(* Advanced properties ******************************************************)
+
+(* Basic_1: was just: sn3_appls_lref *)
+lemma csx_applv_cnx: ∀h,o,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ →
+ ∀Vs. ⦃G, L⦄ ⊢ ⬊*[h, o] Vs → ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.T.
+#h #o #G #L #T #H1T #H2T #Vs elim Vs -Vs [ #_ @(cnx_csx … H2T) ] (**) (* /2 width=1/ does not work *)
+#V #Vs #IHV #H
+elim (csxv_inv_cons … H) -H #HV #HVs
+@csx_appl_simple_theq /2 width=1 by applv_simple/ -IHV -HV -HVs
+#X #H #H0
+lapply (cpxs_fwd_cnx_vector … H) -H // -H1T -H2T #H
+elim (H0) -H0 //
+qed.
+
+lemma csx_applv_sort: ∀h,o,G,L,s,Vs. ⦃G, L⦄ ⊢ ⬊*[h, o] Vs → ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.⋆s.
+#h #o #G #L #s elim (deg_total h o s)
+#d generalize in match s; -s @(nat_ind_plus … d) -d [ /3 width=6 by csx_applv_cnx, cnx_sort, simple_atom/ ]
+#d #IHd #s #Hkd lapply (deg_next_SO … Hkd) -Hkd
+#Hkd #Vs elim Vs -Vs /2 width=1 by/
+#V #Vs #IHVs #HVVs
+elim (csxv_inv_cons … HVVs) #HV #HVs
+@csx_appl_simple_theq /2 width=1 by applv_simple, simple_atom/ -IHVs -HV -HVs
+#X #H #H0
+elim (cpxs_fwd_sort_vector … H) -H #H
+[ elim H0 -H0 //
+| -H0 @(csx_cpxs_trans … (Ⓐ(V@Vs).⋆(next h s))) /2 width=1 by cpxs_flat_dx/
+]
+qed.
+
+(* Basic_1: was just: sn3_appls_beta *)
+lemma csx_applv_beta: ∀h,o,a,G,L,Vs,V,W,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.ⓓ{a}ⓝW.V.T →
+ ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs. ⓐV.ⓛ{a}W.T.
+#h #o #a #G #L #Vs elim Vs -Vs /2 width=1 by csx_appl_beta/
+#V0 #Vs #IHV #V #W #T #H1T
+lapply (csx_fwd_pair_sn … H1T) #HV0
+lapply (csx_fwd_flat_dx … H1T) #H2T
+@csx_appl_simple_theq /2 width=1 by applv_simple, simple_flat/ -IHV -HV0 -H2T
+#X #H #H0
+elim (cpxs_fwd_beta_vector … H) -H #H
+[ -H1T elim H0 -H0 //
+| -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
+]
+qed.
+
+lemma csx_applv_delta: ∀h,o,I,G,L,K,V1,i. ⬇[i] L ≡ K.ⓑ{I}V1 →
+ ∀V2. ⬆[0, i + 1] V1 ≡ V2 →
+ ∀Vs. ⦃G, L⦄ ⊢ ⬊*[h, o] (ⒶVs.V2) → ⦃G, L⦄ ⊢ ⬊*[h, o] (ⒶVs.#i).
+#h #o #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs
+[ /4 width=12 by csx_inv_lift, csx_lref_bind, drop_fwd_drop2/
+| #V #Vs #IHV #H1T
+ lapply (csx_fwd_pair_sn … H1T) #HV
+ lapply (csx_fwd_flat_dx … H1T) #H2T
+ @csx_appl_simple_theq /2 width=1 by applv_simple, simple_atom/ -IHV -HV -H2T
+ #X #H #H0
+ elim (cpxs_fwd_delta_vector … HLK … HV12 … H) -HLK -HV12 -H #H
+ [ -H1T elim H0 -H0 //
+ | -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
+ ]
+]
+qed.
+
+(* Basic_1: was just: sn3_appls_abbr *)
+lemma csx_applv_theta: ∀h,o,a,G,L,V1b,V2b. ⬆[0, 1] V1b ≡ V2b →
+ ∀V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓓ{a}V.ⒶV2b.T →
+ ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶV1b.ⓓ{a}V.T.
+#h #o #a #G #L #V1b #V2b * -V1b -V2b /2 width=1 by/
+#V1b #V2b #V1 #V2 #HV12 #H
+generalize in match HV12; -HV12 generalize in match V2; -V2 generalize in match V1; -V1
+elim H -V1b -V2b /2 width=3 by csx_appl_theta/
+#V1b #V2b #V1 #V2 #HV12 #HV12b #IHV12b #W1 #W2 #HW12 #V #T #H
+lapply (csx_appl_theta … HW12 … H) -H -HW12 #H
+lapply (csx_fwd_pair_sn … H) #HW1
+lapply (csx_fwd_flat_dx … H) #H1
+@csx_appl_simple_theq /2 width=3 by simple_flat/ -IHV12b -HW1 -H1 #X #H1 #H2
+elim (cpxs_fwd_theta_vector … (V2@V2b) … H1) -H1 /2 width=1 by liftv_cons/ -HV12b -HV12
+[ -H #H elim H2 -H2 //
+| -H2 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
+]
+qed.
+
+(* Basic_1: was just: sn3_appls_cast *)
+lemma csx_applv_cast: ∀h,o,G,L,Vs,W,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.W → ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.T →
+ ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.ⓝW.T.
+#h #o #G #L #Vs elim Vs -Vs /2 width=1 by csx_cast/
+#V #Vs #IHV #W #T #H1W #H1T
+lapply (csx_fwd_pair_sn … H1W) #HV
+lapply (csx_fwd_flat_dx … H1W) #H2W
+lapply (csx_fwd_flat_dx … H1T) #H2T
+@csx_appl_simple_theq /2 width=1 by applv_simple, simple_flat/ -IHV -HV -H2W -H2T
+#X #H #H0
+elim (cpxs_fwd_cast_vector … H) -H #H
+[ -H1W -H1T elim H0 -H0 //
+| -H1W -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
+| -H1T -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
+]
+qed.
+
+theorem csx_gcr: ∀h,o. gcr (cpx h o) (eq …) (csx h o) (csx h o).
+#h #o @mk_gcr //
+[ /3 width=1 by csx_applv_cnx/
+|2,3,6: /2 width=1 by csx_applv_beta, csx_applv_sort, csx_applv_cast/
+| /2 width=7 by csx_applv_delta/
+| #G #L #V1b #V2b #HV12b #a #V #T #H #HV
+ @(csx_applv_theta … HV12b) -HV12b
+ @csx_abbr //
+]
+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/syntax/tsts_simple.ma".
-include "basic_2/syntax/tsts_tsts.ma".
-include "basic_2/rt_transition/cpx_simple.ma".
-include "basic_2/rt_computation/cpxs.ma".
-include "basic_2/rt_computation/csx_csx.ma".
-
-(* STRONGLY NORMALIZING TERMS FOR UNCOUNTED PARALLEL RT-TRANSITION **********)
-
-(* Properties with same top term structure **********************************)
-
-(* Basic_1: was just: sn3_appl_appl *)
-lemma csx_appl_simple_tsts: ∀h,o,G,L,V. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ → ∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ⩳[h, o] T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓐV.T2⦄) →
- 𝐒⦃T1⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓐV.T1⦄.
-#h #o #G #L #V #H @(csx_ind … H) -V
-#V #_ #IHV #T1 #H @(csx_ind … H) -T1
-#T1 #H1T1 #IHT1 #H2T1 #H3T1
-@csx_intro #X #HL #H
-elim (cpx_inv_appl1_simple … HL) -HL //
-#V0 #T0 #HLV0 #HLT10 #H0 destruct
-elim (tdneq_inv_pair … H) -H
-[ #H elim H -H //
-| -IHT1 #HV0
- @(csx_cpx_trans … (ⓐV0.T1)) /2 width=1 by cpx_flat/ -HLT10
- @IHV -IHV /4 width=3 by csx_cpx_trans, cpx_pair_sn/
-| -IHV -H1T1 #H1T10
- @(csx_cpx_trans … (ⓐV.T0)) /2 width=1 by cpx_flat/ -HLV0
- elim (tsts_dec h o T1 T0) #H2T10
- [ @IHT1 -IHT1 /4 width=5 by cpxs_strap2, cpxs_strap1, tsts_canc_sn, simple_tsts_repl_dx/
- | -IHT1 -H3T1 -H1T10 /3 width=1 by cpx_cpxs/
- ]
-]
-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/computation/gcp_cr.ma".
-include "basic_2/computation/cpxs_tsts_vector.ma".
-include "basic_2/computation/csx_lpx.ma".
-include "basic_2/computation/csx_vector.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERM VECTORS *************)
-
-(* Advanced properties ******************************************************)
-
-(* Basic_1: was just: sn3_appls_lref *)
-lemma csx_applv_cnx: ∀h,o,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ →
- ∀Vs. ⦃G, L⦄ ⊢ ⬊*[h, o] Vs → ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.T.
-#h #o #G #L #T #H1T #H2T #Vs elim Vs -Vs [ #_ @(cnx_csx … H2T) ] (**) (* /2 width=1/ does not work *)
-#V #Vs #IHV #H
-elim (csxv_inv_cons … H) -H #HV #HVs
-@csx_appl_simple_tsts /2 width=1 by applv_simple/ -IHV -HV -HVs
-#X #H #H0
-lapply (cpxs_fwd_cnx_vector … H) -H // -H1T -H2T #H
-elim (H0) -H0 //
-qed.
-
-lemma csx_applv_sort: ∀h,o,G,L,s,Vs. ⦃G, L⦄ ⊢ ⬊*[h, o] Vs → ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.⋆s.
-#h #o #G #L #s elim (deg_total h o s)
-#d generalize in match s; -s @(nat_ind_plus … d) -d [ /3 width=6 by csx_applv_cnx, cnx_sort, simple_atom/ ]
-#d #IHd #s #Hkd lapply (deg_next_SO … Hkd) -Hkd
-#Hkd #Vs elim Vs -Vs /2 width=1 by/
-#V #Vs #IHVs #HVVs
-elim (csxv_inv_cons … HVVs) #HV #HVs
-@csx_appl_simple_tsts /2 width=1 by applv_simple, simple_atom/ -IHVs -HV -HVs
-#X #H #H0
-elim (cpxs_fwd_sort_vector … H) -H #H
-[ elim H0 -H0 //
-| -H0 @(csx_cpxs_trans … (Ⓐ(V@Vs).⋆(next h s))) /2 width=1 by cpxs_flat_dx/
-]
-qed.
-
-(* Basic_1: was just: sn3_appls_beta *)
-lemma csx_applv_beta: ∀h,o,a,G,L,Vs,V,W,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.ⓓ{a}ⓝW.V.T →
- ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs. ⓐV.ⓛ{a}W.T.
-#h #o #a #G #L #Vs elim Vs -Vs /2 width=1 by csx_appl_beta/
-#V0 #Vs #IHV #V #W #T #H1T
-lapply (csx_fwd_pair_sn … H1T) #HV0
-lapply (csx_fwd_flat_dx … H1T) #H2T
-@csx_appl_simple_tsts /2 width=1 by applv_simple, simple_flat/ -IHV -HV0 -H2T
-#X #H #H0
-elim (cpxs_fwd_beta_vector … H) -H #H
-[ -H1T elim H0 -H0 //
-| -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
-]
-qed.
-
-lemma csx_applv_delta: ∀h,o,I,G,L,K,V1,i. ⬇[i] L ≡ K.ⓑ{I}V1 →
- ∀V2. ⬆[0, i + 1] V1 ≡ V2 →
- ∀Vs. ⦃G, L⦄ ⊢ ⬊*[h, o] (ⒶVs.V2) → ⦃G, L⦄ ⊢ ⬊*[h, o] (ⒶVs.#i).
-#h #o #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs
-[ /4 width=12 by csx_inv_lift, csx_lref_bind, drop_fwd_drop2/
-| #V #Vs #IHV #H1T
- lapply (csx_fwd_pair_sn … H1T) #HV
- lapply (csx_fwd_flat_dx … H1T) #H2T
- @csx_appl_simple_tsts /2 width=1 by applv_simple, simple_atom/ -IHV -HV -H2T
- #X #H #H0
- elim (cpxs_fwd_delta_vector … HLK … HV12 … H) -HLK -HV12 -H #H
- [ -H1T elim H0 -H0 //
- | -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
- ]
-]
-qed.
-
-(* Basic_1: was just: sn3_appls_abbr *)
-lemma csx_applv_theta: ∀h,o,a,G,L,V1b,V2b. ⬆[0, 1] V1b ≡ V2b →
- ∀V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓓ{a}V.ⒶV2b.T →
- ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶV1b.ⓓ{a}V.T.
-#h #o #a #G #L #V1b #V2b * -V1b -V2b /2 width=1 by/
-#V1b #V2b #V1 #V2 #HV12 #H
-generalize in match HV12; -HV12 generalize in match V2; -V2 generalize in match V1; -V1
-elim H -V1b -V2b /2 width=3 by csx_appl_theta/
-#V1b #V2b #V1 #V2 #HV12 #HV12b #IHV12b #W1 #W2 #HW12 #V #T #H
-lapply (csx_appl_theta … HW12 … H) -H -HW12 #H
-lapply (csx_fwd_pair_sn … H) #HW1
-lapply (csx_fwd_flat_dx … H) #H1
-@csx_appl_simple_tsts /2 width=3 by simple_flat/ -IHV12b -HW1 -H1 #X #H1 #H2
-elim (cpxs_fwd_theta_vector … (V2@V2b) … H1) -H1 /2 width=1 by liftv_cons/ -HV12b -HV12
-[ -H #H elim H2 -H2 //
-| -H2 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
-]
-qed.
-
-(* Basic_1: was just: sn3_appls_cast *)
-lemma csx_applv_cast: ∀h,o,G,L,Vs,W,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.W → ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.T →
- ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.ⓝW.T.
-#h #o #G #L #Vs elim Vs -Vs /2 width=1 by csx_cast/
-#V #Vs #IHV #W #T #H1W #H1T
-lapply (csx_fwd_pair_sn … H1W) #HV
-lapply (csx_fwd_flat_dx … H1W) #H2W
-lapply (csx_fwd_flat_dx … H1T) #H2T
-@csx_appl_simple_tsts /2 width=1 by applv_simple, simple_flat/ -IHV -HV -H2W -H2T
-#X #H #H0
-elim (cpxs_fwd_cast_vector … H) -H #H
-[ -H1W -H1T elim H0 -H0 //
-| -H1W -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
-| -H1T -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
-]
-qed.
-
-theorem csx_gcr: ∀h,o. gcr (cpx h o) (eq …) (csx h o) (csx h o).
-#h #o @mk_gcr //
-[ /3 width=1 by csx_applv_cnx/
-|2,3,6: /2 width=1 by csx_applv_beta, csx_applv_sort, csx_applv_cast/
-| /2 width=7 by csx_applv_delta/
-| #G #L #V1b #V2b #HV12b #a #V #T #H #HV
- @(csx_applv_theta … HV12b) -HV12b
- @csx_abbr //
-]
-qed.
-cpxs.ma cpxs_tdeq.ma cpxs_tsts.ma cpxs_tsts_vector.ma cpxs_drops.ma cpxs_lsubr.ma cpxs_lfpx.ma cpxs_cnx.ma cpxs_cpxs.ma
+cpxs.ma cpxs_tdeq.ma cpxs_theq.ma cpxs_theq_vector.ma cpxs_drops.ma cpxs_lsubr.ma cpxs_lfpx.ma cpxs_cnx.ma cpxs_cpxs.ma
lfpxs.ma lfpxs_fqup.ma lfpxs_cpxs.ma
-csx.ma csx_simple.ma csx_tsts.ma csx_drops.ma csx_gcp.ma csx_cnx.ma csx_cpxs.ma csx_csx.ma
+csx.ma csx_simple.ma csx_theq.ma csx_drops.ma csx_gcp.ma csx_cnx.ma csx_cpxs.ma csx_csx.ma
csx_vector.ma
--- /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/topiso_4.ma".
+include "basic_2/syntax/item_sd.ma".
+include "basic_2/syntax/term.ma".
+
+(* HEAD EQUIVALENCE FOR TERMS ***********************************************)
+
+(* Basic_2A1: includes: tsts_atom tsts_pair *)
+inductive theq (h) (o): relation term ≝
+| theq_sort: ∀s1,s2,d. deg h o s1 d → deg h o s2 d → theq h o (⋆s1) (⋆s2)
+| theq_lref: ∀i. theq h o (#i) (#i)
+| theq_gref: ∀l. theq h o (§l) (§l)
+| theq_pair: ∀I,V1,V2,T1,T2. theq h o (②{I}V1.T1) (②{I}V2.T2)
+.
+
+interpretation "head equivalence (term)" 'TopIso h o T1 T2 = (theq h o T1 T2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact theq_inv_sort1_aux: ∀h,o,X,Y. X ⩳[h, o] Y → ∀s1. X = ⋆s1 →
+ ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2.
+#h #o #X #Y * -X -Y
+[ #s1 #s2 #d #Hs1 #Hs2 #s #H destruct /2 width=5 by ex3_2_intro/
+| #i #s #H destruct
+| #l #s #H destruct
+| #I #V1 #V2 #T1 #T2 #s #H destruct
+]
+qed-.
+
+(* Basic_1: was just: iso_gen_sort *)
+lemma theq_inv_sort1: ∀h,o,Y,s1. ⋆s1 ⩳[h, o] Y →
+ ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2.
+/2 width=3 by theq_inv_sort1_aux/ qed-.
+
+fact theq_inv_lref1_aux: ∀h,o,X,Y. X ⩳[h, o] Y → ∀i. X = #i → Y = #i.
+#h #o #X #Y * -X -Y //
+[ #s1 #s2 #d #_ #_ #j #H destruct
+| #I #V1 #V2 #T1 #T2 #j #H destruct
+]
+qed-.
+
+(* Basic_1: was: iso_gen_lref *)
+lemma theq_inv_lref1: ∀h,o,Y,i. #i ⩳[h, o] Y → Y = #i.
+/2 width=5 by theq_inv_lref1_aux/ qed-.
+
+fact theq_inv_gref1_aux: ∀h,o,X,Y. X ⩳[h, o] Y → ∀l. X = §l → Y = §l.
+#h #o #X #Y * -X -Y //
+[ #s1 #s2 #d #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #k #H destruct
+]
+qed-.
+
+lemma theq_inv_gref1: ∀h,o,Y,l. §l ⩳[h, o] Y → Y = §l.
+/2 width=5 by theq_inv_gref1_aux/ qed-.
+
+fact theq_inv_pair1_aux: ∀h,o,T1,T2. T1 ⩳[h, o] T2 →
+ ∀J,W1,U1. T1 = ②{J}W1.U1 →
+ ∃∃W2,U2. T2 = ②{J}W2.U2.
+#h #o #T1 #T2 * -T1 -T2
+[ #s1 #s2 #d #_ #_ #J #W1 #U1 #H destruct
+| #i #J #W1 #U1 #H destruct
+| #l #J #W1 #U1 #H destruct
+| #I #V1 #V2 #T1 #T2 #J #W1 #U1 #H destruct /2 width=3 by ex1_2_intro/
+]
+qed-.
+
+(* Basic_1: was: iso_gen_head *)
+(* Basic_2A1: was: tsts_inv_pair1 *)
+lemma theq_inv_pair1: ∀h,o,J,W1,U1,T2. ②{J}W1.U1 ⩳[h, o] T2 →
+ ∃∃W2,U2. T2 = ②{J}W2. U2.
+/2 width=7 by theq_inv_pair1_aux/ qed-.
+
+fact theq_inv_pair2_aux: ∀h,o,T1,T2. T1 ⩳[h, o] T2 →
+ ∀J,W2,U2. T2 = ②{J}W2.U2 →
+ ∃∃W1,U1. T1 = ②{J}W1.U1.
+#h #o #T1 #T2 * -T1 -T2
+[ #s1 #s2 #d #_ #_ #J #W2 #U2 #H destruct
+| #i #J #W2 #U2 #H destruct
+| #l #J #W2 #U2 #H destruct
+| #I #V1 #V2 #T1 #T2 #J #W2 #U2 #H destruct /2 width=3 by ex1_2_intro/
+]
+qed-.
+
+(* Basic_2A1: was: tsts_inv_pair2 *)
+lemma theq_inv_pair2: ∀h,o,J,T1,W2,U2. T1 ⩳[h, o] ②{J}W2.U2 →
+ ∃∃W1,U1. T1 = ②{J}W1.U1.
+/2 width=7 by theq_inv_pair2_aux/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma theq_inv_sort1_deg: ∀h,o,Y,s1. ⋆s1 ⩳[h, o] Y → ∀d. deg h o s1 d →
+ ∃∃s2. deg h o s2 d & Y = ⋆s2.
+#h #o #Y #s1 #H #d #Hs1 elim (theq_inv_sort1 … H) -H
+#s2 #x #Hx <(deg_mono h o … Hx … Hs1) -s1 -d /2 width=3 by ex2_intro/
+qed-.
+
+lemma theq_inv_sort_deg: ∀h,o,s1,s2. ⋆s1 ⩳[h, o] ⋆s2 →
+ ∀d1,d2. deg h o s1 d1 → deg h o s2 d2 →
+ d1 = d2.
+#h #o #s1 #y #H #d1 #d2 #Hs1 #Hy
+elim (theq_inv_sort1_deg … H … Hs1) -s1 #s2 #Hs2 #H destruct
+<(deg_mono h o … Hy … Hs2) -s2 -d1 //
+qed-.
+
+lemma theq_inv_pair: ∀h,o,I1,I2,V1,V2,T1,T2. ②{I1}V1.T1 ⩳[h, o] ②{I2}V2.T2 →
+ I1 = I2.
+#h #o #I1 #I2 #V1 #V2 #T1 #T2 #H elim (theq_inv_pair1 … H) -H
+#V0 #T0 #H destruct //
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: iso_refl *)
+(* Basic_2A1: was: tsts_refl *)
+lemma theq_refl: ∀h,o. reflexive … (theq h o).
+#h #o * //
+* /2 width=1 by theq_lref, theq_gref/
+#s elim (deg_total h o s) /2 width=3 by theq_sort/
+qed.
+
+(* Basic_2A1: was: tsts_sym *)
+lemma theq_sym: ∀h,o. symmetric … (theq h o).
+#h #o #T1 #T2 * -T1 -T2 /2 width=3 by theq_sort/
+qed-.
+
+(* Basic_2A1: was: tsts_dec *)
+lemma theq_dec: ∀h,o,T1,T2. Decidable (T1 ⩳[h, o] T2).
+#h #o * [ * #s1 | #I1 #V1 #T1 ] * [1,3,5,7: * #s2 |*: #I2 #V2 #T2 ]
+[ elim (deg_total h o s1) #d1 #H1
+ elim (deg_total h o s2) #d2 #H2
+ elim (eq_nat_dec d1 d2) #Hd12 destruct /3 width=3 by theq_sort, or_introl/
+ @or_intror #H
+ lapply (theq_inv_sort_deg … H … H1 H2) -H -H1 -H2 /2 width=1 by/
+|2,3,13:
+ @or_intror #H
+ elim (theq_inv_sort1 … H) -H #x1 #x2 #_ #_ #H destruct
+|4,6,14:
+ @or_intror #H
+ lapply (theq_inv_lref1 … H) -H #H destruct
+|5:
+ elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
+ @or_intror #H
+ lapply (theq_inv_lref1 … H) -H #H destruct /2 width=1 by/
+|7,8,15:
+ @or_intror #H
+ lapply (theq_inv_gref1 … H) -H #H destruct
+|9:
+ elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
+ @or_intror #H
+ lapply (theq_inv_gref1 … H) -H #H destruct /2 width=1 by/
+|10,11,12:
+ @or_intror #H
+ elim (theq_inv_pair1 … H) -H #X1 #X2 #H destruct
+|16:
+ elim (eq_item2_dec I1 I2) #HI12 destruct
+ [ /3 width=1 by theq_pair, or_introl/ ]
+ @or_intror #H
+ lapply (theq_inv_pair … H) -H /2 width=1 by/
+]
+qed-.
+
+(* Basic_2A1: removed theorems 2:
+ tsts_inv_atom1 tsts_inv_atom2
+*)
--- /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/syntax/term_simple.ma".
+include "basic_2/syntax/theq.ma".
+
+(* HEAD EQUIVALENCE FOR TERMS ***********************************************)
+
+(* Properies with simple (neutral) terms ************************************)
+
+(* Basic_2A1: was: simple_tsts_repl_dx *)
+lemma simple_theq_repl_dx: ∀h,o,T1,T2. T1 ⩳[h, o] T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
+#h #o #T1 #T2 * -T1 -T2 //
+#I #V1 #V2 #T1 #T2 #H
+elim (simple_inv_pair … H) -H #J #H destruct //
+qed-.
+
+(* Basic_2A1: was: simple_tsts_repl_sn *)
+lemma simple_theq_repl_sn: ∀h,o,T1,T2. T1 ⩳[h, o] T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
+/3 width=5 by simple_theq_repl_dx, theq_sym/ 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/syntax/term_vector.ma".
+include "basic_2/syntax/theq_simple.ma".
+
+(* HEAD EQUIVALENCE FOR TERMS ***********************************************)
+
+(* Advanced inversion lemmas with simple (neutral) terms ********************)
+
+(* Basic_1: was only: iso_flats_lref_bind_false iso_flats_flat_bind_false *)
+(* Basic_2A1: was: tsts_inv_bind_applv_simple *)
+lemma theq_inv_applv_bind_simple: ∀h,o,p,I,Vs,V2,T1,T2. ⒶVs.T1 ⩳[h, o] ⓑ{p,I}V2.T2 →
+ 𝐒⦃T1⦄ → ⊥.
+#h #o #p #I #Vs #V2 #T1 #T2 #H elim (theq_inv_pair2 … H) -H
+#V0 #T0 elim Vs -Vs normalize
+[ #H destruct #H /2 width=5 by simple_inv_bind/
+| #V #Vs #_ #H destruct
+]
+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/syntax/tdeq.ma".
+include "basic_2/syntax/theq.ma".
+
+(* HEAD EQUIVALENCE FOR TERMS ***********************************************)
+
+(* Properties with degree-based equivalence for terms ***********************)
+
+lemma tdeq_theq: ∀h,o,T1,T2. T1 ≡[h, o] T2 → T1 ⩳[h, o] T2.
+#h #o #T1 #T2 * -T1 -T2 /2 width=3 by theq_sort, theq_pair/
+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/syntax/theq.ma".
+
+(* HEAD EQUIVALENCE FOR TERMS ***********************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: iso_trans *)
+(* Basic_2A1: was: tsts_trans *)
+theorem theq_trans: ∀h,o. Transitive … (theq h o).
+#h #o #T1 #T * -T1 -T
+[ #s1 #s #d #Hs1 #Hs #X #H
+ elim (theq_inv_sort1_deg … H … Hs) -s /2 width=3 by theq_sort/
+| #i1 #i #H <(theq_inv_lref1 … H) -H //
+| #l1 #l #H <(theq_inv_gref1 … H) -H //
+| #I #V1 #V #T1 #T #X #H
+ elim (theq_inv_pair1 … H) -H #V2 #T2 #H destruct //
+]
+qed-.
+
+(* Basic_2A1: was: tsts_canc_sn *)
+theorem theq_canc_sn: ∀h,o. left_cancellable … (theq h o).
+/3 width=3 by theq_trans, theq_sym/ qed-.
+
+(* Basic_2A1: was: tsts_canc_dx *)
+theorem theq_canc_dx: ∀h,o. right_cancellable … (theq h o).
+/3 width=3 by theq_trans, theq_sym/ 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/notation/relations/topiso_4.ma".
-include "basic_2/syntax/item_sd.ma".
-include "basic_2/syntax/term.ma".
-
-(* SAME TOP TERM STRUCTURE **************************************************)
-
-(* Basic_2A1: includes: tsts_atom *)
-inductive tsts (h) (o): relation term ≝
-| tsts_sort: ∀s1,s2,d. deg h o s1 d → deg h o s2 d → tsts h o (⋆s1) (⋆s2)
-| tsts_lref: ∀i. tsts h o (#i) (#i)
-| tsts_gref: ∀l. tsts h o (§l) (§l)
-| tsts_pair: ∀I,V1,V2,T1,T2. tsts h o (②{I}V1.T1) (②{I}V2.T2)
-.
-
-interpretation "same top structure (term)" 'TopIso h o T1 T2 = (tsts h o T1 T2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact tsts_inv_sort1_aux: ∀h,o,X,Y. X ⩳[h, o] Y → ∀s1. X = ⋆s1 →
- ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2.
-#h #o #X #Y * -X -Y
-[ #s1 #s2 #d #Hs1 #Hs2 #s #H destruct /2 width=5 by ex3_2_intro/
-| #i #s #H destruct
-| #l #s #H destruct
-| #I #V1 #V2 #T1 #T2 #s #H destruct
-]
-qed-.
-
-(* Basic_1: was just: iso_gen_sort *)
-lemma tsts_inv_sort1: ∀h,o,Y,s1. ⋆s1 ⩳[h, o] Y →
- ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2.
-/2 width=3 by tsts_inv_sort1_aux/ qed-.
-
-fact tsts_inv_lref1_aux: ∀h,o,X,Y. X ⩳[h, o] Y → ∀i. X = #i → Y = #i.
-#h #o #X #Y * -X -Y //
-[ #s1 #s2 #d #_ #_ #j #H destruct
-| #I #V1 #V2 #T1 #T2 #j #H destruct
-]
-qed-.
-
-(* Basic_1: was: iso_gen_lref *)
-lemma tsts_inv_lref1: ∀h,o,Y,i. #i ⩳[h, o] Y → Y = #i.
-/2 width=5 by tsts_inv_lref1_aux/ qed-.
-
-fact tsts_inv_gref1_aux: ∀h,o,X,Y. X ⩳[h, o] Y → ∀l. X = §l → Y = §l.
-#h #o #X #Y * -X -Y //
-[ #s1 #s2 #d #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #k #H destruct
-]
-qed-.
-
-lemma tsts_inv_gref1: ∀h,o,Y,l. §l ⩳[h, o] Y → Y = §l.
-/2 width=5 by tsts_inv_gref1_aux/ qed-.
-
-fact tsts_inv_pair1_aux: ∀h,o,T1,T2. T1 ⩳[h, o] T2 →
- ∀J,W1,U1. T1 = ②{J}W1.U1 →
- ∃∃W2,U2. T2 = ②{J}W2.U2.
-#h #o #T1 #T2 * -T1 -T2
-[ #s1 #s2 #d #_ #_ #J #W1 #U1 #H destruct
-| #i #J #W1 #U1 #H destruct
-| #l #J #W1 #U1 #H destruct
-| #I #V1 #V2 #T1 #T2 #J #W1 #U1 #H destruct /2 width=3 by ex1_2_intro/
-]
-qed-.
-
-(* Basic_1: was: iso_gen_head *)
-lemma tsts_inv_pair1: ∀h,o,J,W1,U1,T2. ②{J}W1.U1 ⩳[h, o] T2 →
- ∃∃W2,U2. T2 = ②{J}W2. U2.
-/2 width=7 by tsts_inv_pair1_aux/ qed-.
-
-fact tsts_inv_pair2_aux: ∀h,o,T1,T2. T1 ⩳[h, o] T2 →
- ∀J,W2,U2. T2 = ②{J}W2.U2 →
- ∃∃W1,U1. T1 = ②{J}W1.U1.
-#h #o #T1 #T2 * -T1 -T2
-[ #s1 #s2 #d #_ #_ #J #W2 #U2 #H destruct
-| #i #J #W2 #U2 #H destruct
-| #l #J #W2 #U2 #H destruct
-| #I #V1 #V2 #T1 #T2 #J #W2 #U2 #H destruct /2 width=3 by ex1_2_intro/
-]
-qed-.
-
-lemma tsts_inv_pair2: ∀h,o,J,T1,W2,U2. T1 ⩳[h, o] ②{J}W2.U2 →
- ∃∃W1,U1. T1 = ②{J}W1.U1.
-/2 width=7 by tsts_inv_pair2_aux/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma tsts_inv_sort1_deg: ∀h,o,Y,s1. ⋆s1 ⩳[h, o] Y → ∀d. deg h o s1 d →
- ∃∃s2. deg h o s2 d & Y = ⋆s2.
-#h #o #Y #s1 #H #d #Hs1 elim (tsts_inv_sort1 … H) -H
-#s2 #x #Hx <(deg_mono h o … Hx … Hs1) -s1 -d /2 width=3 by ex2_intro/
-qed-.
-
-lemma tsts_inv_sort_deg: ∀h,o,s1,s2. ⋆s1 ⩳[h, o] ⋆s2 →
- ∀d1,d2. deg h o s1 d1 → deg h o s2 d2 →
- d1 = d2.
-#h #o #s1 #y #H #d1 #d2 #Hs1 #Hy
-elim (tsts_inv_sort1_deg … H … Hs1) -s1 #s2 #Hs2 #H destruct
-<(deg_mono h o … Hy … Hs2) -s2 -d1 //
-qed-.
-
-lemma tsts_inv_pair: ∀h,o,I1,I2,V1,V2,T1,T2. ②{I1}V1.T1 ⩳[h, o] ②{I2}V2.T2 →
- I1 = I2.
-#h #o #I1 #I2 #V1 #V2 #T1 #T2 #H elim (tsts_inv_pair1 … H) -H
-#V0 #T0 #H destruct //
-qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: iso_refl *)
-lemma tsts_refl: ∀h,o. reflexive … (tsts h o).
-#h #o * //
-* /2 width=1 by tsts_lref, tsts_gref/
-#s elim (deg_total h o s) /2 width=3 by tsts_sort/
-qed.
-
-lemma tsts_sym: ∀h,o. symmetric … (tsts h o).
-#h #o #T1 #T2 * -T1 -T2 /2 width=3 by tsts_sort/
-qed-.
-
-lemma tsts_dec: ∀h,o,T1,T2. Decidable (T1 ⩳[h, o] T2).
-#h #o * [ * #s1 | #I1 #V1 #T1 ] * [1,3,5,7: * #s2 |*: #I2 #V2 #T2 ]
-[ elim (deg_total h o s1) #d1 #H1
- elim (deg_total h o s2) #d2 #H2
- elim (eq_nat_dec d1 d2) #Hd12 destruct /3 width=3 by tsts_sort, or_introl/
- @or_intror #H
- lapply (tsts_inv_sort_deg … H … H1 H2) -H -H1 -H2 /2 width=1 by/
-|2,3,13:
- @or_intror #H
- elim (tsts_inv_sort1 … H) -H #x1 #x2 #_ #_ #H destruct
-|4,6,14:
- @or_intror #H
- lapply (tsts_inv_lref1 … H) -H #H destruct
-|5:
- elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
- @or_intror #H
- lapply (tsts_inv_lref1 … H) -H #H destruct /2 width=1 by/
-|7,8,15:
- @or_intror #H
- lapply (tsts_inv_gref1 … H) -H #H destruct
-|9:
- elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
- @or_intror #H
- lapply (tsts_inv_gref1 … H) -H #H destruct /2 width=1 by/
-|10,11,12:
- @or_intror #H
- elim (tsts_inv_pair1 … H) -H #X1 #X2 #H destruct
-|16:
- elim (eq_item2_dec I1 I2) #HI12 destruct
- [ /3 width=1 by tsts_pair, or_introl/ ]
- @or_intror #H
- lapply (tsts_inv_pair … H) -H /2 width=1 by/
-]
-qed-.
-
-(* Basic_2A1: removed theorems 4:
- tsts_inv_atom1 tsts_inv_atom2
-*)
+++ /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/syntax/term_simple.ma".
-include "basic_2/syntax/tsts.ma".
-
-(* SAME TOP TERM STRUCTURE **************************************************)
-
-(* Properies with simple (neutral) terms ************************************)
-
-lemma simple_tsts_repl_dx: ∀h,o,T1,T2. T1 ⩳[h, o] T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
-#h #o #T1 #T2 * -T1 -T2 //
-#I #V1 #V2 #T1 #T2 #H
-elim (simple_inv_pair … H) -H #J #H destruct //
-qed-.
-
-lemma simple_tsts_repl_sn: ∀h,o,T1,T2. T1 ⩳[h, o] T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
-/3 width=5 by simple_tsts_repl_dx, tsts_sym/ 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/syntax/term_vector.ma".
-include "basic_2/syntax/tsts_simple.ma".
-
-(* SAME TOP TERM STRUCTURE **************************************************)
-
-(* Advanced inversion lemmas with simple (neutral) terms ********************)
-
-(* Basic_1: was only: iso_flats_lref_bind_false iso_flats_flat_bind_false *)
-(* Basic_2A1: was: tsts_inv_bind_applv_simple *)
-lemma tsts_inv_applv_bind_simple: ∀h,o,p,I,Vs,V2,T1,T2. ⒶVs.T1 ⩳[h, o] ⓑ{p,I}V2.T2 →
- 𝐒⦃T1⦄ → ⊥.
-#h #o #p #I #Vs #V2 #T1 #T2 #H elim (tsts_inv_pair2 … H) -H
-#V0 #T0 elim Vs -Vs normalize
-[ #H destruct #H /2 width=5 by simple_inv_bind/
-| #V #Vs #_ #H destruct
-]
-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/syntax/tdeq.ma".
-include "basic_2/syntax/tsts.ma".
-
-(* SAME TOP TERM STRUCTURE **************************************************)
-
-(* Properties with degree-based equivalence for terms ***********************)
-
-lemma tdeq_tsts: ∀h,o,T1,T2. T1 ≡[h, o] T2 → T1 ⩳[h, o] T2.
-#h #o #T1 #T2 * -T1 -T2 /2 width=3 by tsts_sort, tsts_pair/
-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/syntax/tsts.ma".
-
-(* SAME TOP TERM STRUCTURE **************************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: iso_trans *)
-theorem tsts_trans: ∀h,o. Transitive … (tsts h o).
-#h #o #T1 #T * -T1 -T
-[ #s1 #s #d #Hs1 #Hs #X #H
- elim (tsts_inv_sort1_deg … H … Hs) -s /2 width=3 by tsts_sort/
-| #i1 #i #H <(tsts_inv_lref1 … H) -H //
-| #l1 #l #H <(tsts_inv_gref1 … H) -H //
-| #I #V1 #V #T1 #T #X #H
- elim (tsts_inv_pair1 … H) -H #V2 #T2 #H destruct //
-]
-qed-.
-
-theorem tsts_canc_sn: ∀h,o. left_cancellable … (tsts h o).
-/3 width=3 by tsts_trans, tsts_sym/ qed-.
-
-theorem tsts_canc_dx: ∀h,o. right_cancellable … (tsts h o).
-/3 width=3 by tsts_trans, tsts_sym/ qed-.
[ { "strongly normalizing rt-computation" * } {
[ "lcosx ( ? ⊢ ~⬊*[?,?,?] ? )" "lcosx_cpx" * ]
[ "lsx ( ? ⊢ ⬊*[?,?,?,?] ? )" "lsx_alt ( ? ⊢ ⬊⬊*[?,?,?,?] ? )" "lsx_drop" + "lsx_lpx" + "lsx_lpxs" + "llsx_csx" * ]
- [ "csx_vector ( ⦃?,?⦄ ⊢ ⬊*[?,?] ? )" "csx_tsts_vector" + "csx_aaa" * ]
+ [ "csx_vector ( ⦃?,?⦄ ⊢ ⬊*[?,?] ? )" "csx_theq_vector" + "csx_aaa" * ]
[ "csx ( ⦃?,?⦄ ⊢ ⬊*[?,?] ? )" "csx_alt ( ⦃?,?⦄ ⊢ ⬊⬊*[?,?] ? )" "csx_lift" + "csx_lleq" + "csx_lpx" + "csx_lpxs" + "csx_fpbs" * ]
}
]
[ "cpxs_lreq" + "cpxs_lleq" + "cpxs_aaa" * ]
*)
[ "csx_vector ( ⦃?,?⦄ ⊢ ⬈*[?,?] 𝐒⦃?⦄ )" * ]
- [ "csx ( ⦃?,?⦄ ⊢ ⬈*[?,?] 𝐒⦃?⦄ )" "csx_simple" + "csx_tsts" + "csx_drops" + "csx_gcp" + "csx_cnx" + "csx_cpxs" + "csx_csx" * ]
+ [ "csx ( ⦃?,?⦄ ⊢ ⬈*[?,?] 𝐒⦃?⦄ )" "csx_simple" + "csx_theq" + "csx_drops" + "csx_gcp" + "csx_cnx" + "csx_cpxs" + "csx_csx" * ]
[ "lfpxs ( ⦃?,?⦄ ⊢ ⬈*[?,?] ? )" "lfpxs_length" + "lfpxs_fqup" + "lfpxs_cpxs" * ]
- [ "cpxs ( ⦃?,?⦄ ⊢ ? ⬈*[?] ? )" "cpxs_tdeq" + "cpxs_tsts" + "cpxs_tsts_vector" + "cpxs_drops" + "cpxs_lsubr" + "cpxs_lfpx" + "cpxs_cnx" + "cpxs_cpxs" * ]
+ [ "cpxs ( ⦃?,?⦄ ⊢ ? ⬈*[?] ? )" "cpxs_tdeq" + "cpxs_theq" + "cpxs_theq_vector" + "cpxs_drops" + "cpxs_lsubr" + "cpxs_lfpx" + "cpxs_cnx" + "cpxs_cpxs" * ]
}
]
}
[ "append ( ? @@ ? )" "append_length" * ]
}
]
- [ { "same top term structure" * } {
- [ "tsts ( ? ⩳[?,?] ? )" "tsts_simple" + "tsts_tdeq" + "tsts_tsts" + "tsts_simple_vector" * ]
+ [ { "head equivalence for terms" * } {
+ [ "theq ( ? ⩳[?,?] ? )" "theq_simple" + "theq_tdeq" + "theq_theq" + "theq_simple_vector" * ]
}
]
[ { "degree-based equivalence for terms" * } {
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