(* 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
+lemma cpxs_fwd_sort_vector: ∀h,G,L,s,Vs,X2. ⦃G, L⦄ ⊢ ⒶVs.⋆s ⬈*[h] X2 → ⒶVs.⋆s ⩳ X2.
+#h #G #L #s #Vs elim Vs -Vs /2 width=4 by cpxs_fwd_sort/
+#V #Vs #IHVs #X2 #H
elim (cpxs_inv_appl1 … H) -H *
-[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1 by theq_pair, or_introl/
+[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1 by theq_pair/
| #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/
- ]
+ lapply (IHVs … HT1) -IHVs -HT1 #HT1
+ elim (theq_inv_applv_bind_simple … HT1) //
| #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/
- ]
+ lapply (IHVs … HT1) -IHVs -HT1 #HT1
+ elim (theq_inv_applv_bind_simple … HT1) //
]
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 →
+lemma cpxs_fwd_delta_drops_vector: ∀h,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
+ ∀Vs,X2. ⦃G, L⦄ ⊢ ⒶVs.#i ⬈*[h] X2 →
+ ∨∨ ⒶVs.#i ⩳ X2 | ⦃G, L⦄ ⊢ ⒶVs.V2 ⬈*[h] X2.
+#h #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs
+elim Vs -Vs /2 width=5 by cpxs_fwd_delta_drops/
+#V #Vs #IHVs #X2 #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_trans … HU) -X2
@(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_trans … HU) -X2
@(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 →
- â\92¶Vs.â\93\90V.â\93\9b{p}W. T ⩳[h, o] U â\88¨ â¦\83G, Lâ¦\84 â\8a¢ â\92¶Vs.â\93\93{p}â\93\9dW.V.T â¬\88*[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
+lemma cpxs_fwd_beta_vector: ∀h,p,G,L,Vs,V,W,T,X2. ⦃G, L⦄ ⊢ ⒶVs.ⓐV.ⓛ{p}W.T ⬈*[h] X2 →
+ â\88¨â\88¨ â\92¶Vs.â\93\90V.â\93\9b{p}W. T ⩳ X2 | â¦\83G, Lâ¦\84 â\8a¢ â\92¶Vs.â\93\93{p}â\93\9dW.V.T â¬\88*[h] X2.
+#h #p #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_beta/
+#V0 #Vs #IHVs #V #W #T #X2 #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_trans … HU) -X2
@(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_trans … HU) -X2
@(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 →
- â\92¶V1b.â\93\93{p}V.T ⩳[h, o] U â\88¨ â¦\83G, Lâ¦\84 â\8a¢ â\93\93{p}V.â\92¶V2b.T â¬\88*[h] U.
-#h #o #G #L #V1b #V2b * -V1b -V2b /3 width=1 by or_intror/
+lemma cpxs_fwd_theta_vector: ∀h,G,L,V1b,V2b. ⬆*[1] V1b ≘ V2b →
+ ∀p,V,T,X2. ⦃G, L⦄ ⊢ ⒶV1b.ⓓ{p}V.T ⬈*[h] X2 →
+ â\88¨â\88¨ â\92¶V1b.â\93\93{p}V.T ⩳ X2 | â¦\83G, Lâ¦\84 â\8a¢ â\93\93{p}V.â\92¶V2b.T â¬\88*[h] X2.
+#h #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
+#V1b #V2b #V1b #V2b #HV12b #_ #IHV12b #V1a #V2a #HV12a #V #T #X2 #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
[ -HV12a -HV12b -HU
elim (theq_inv_pair1 … HT0) #V1 #T1 #H destruct
| @or_intror -V1b (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
+ @(cpxs_trans … HU) -X2
elim (cpxs_inv_abbr1_dx … HT0) -HT0 *
[ -HV12a #V1 #T1 #_ #_ #H destruct
| -V1b #X #HT1 #H #H0 destruct
[ -HV12a -HV10a -HV0a -HU
elim (theq_inv_pair1 … HT0) #V1 #T1 #H destruct
| @or_intror -V1b -V1b (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
+ @(cpxs_trans … HU) -X2
elim (cpxs_inv_abbr1_dx … HT0) -HT0 *
[ #V1 #T1 #HV1 #HT1 #H destruct
lapply (cpxs_lifts_bi … HV10a (Ⓣ) … (L.ⓓV) … HV12a … HV0a) -V1a -V0a /3 width=1 by drops_refl, drops_drop/ #HV2a
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
+lemma cpxs_fwd_cast_vector: ∀h,G,L,Vs,W,T,X2. ⦃G, L⦄ ⊢ ⒶVs.ⓝW.T ⬈*[h] X2 →
+ ∨∨ ⒶVs. ⓝW. T ⩳ X2
+ | ⦃G, L⦄ ⊢ ⒶVs.T ⬈*[h] X2
+ | ⦃G, L⦄ ⊢ ⒶVs.W ⬈*[h] X2.
+#h #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_cast/
+#V #Vs #IHVs #W #T #X2 #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_trans … HU) -X2
@(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_trans … HU) -X2
@(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_trans … HU) -X2
@(cpxs_strap1 … (ⓐV0.ⓓ{q}V2.T0)) /2 width=3 by cpxs_flat, cpx_theta/
| @or3_intro2 -T (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
+ @(cpxs_trans … HU) -X2
@(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
+lemma cpxs_fwd_cnx_vector: ∀h,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ →
+ ∀Vs,X2. ⦃G, L⦄ ⊢ ⒶVs.T ⬈*[h] X2 → ⒶVs.T ⩳ X2.
+#h #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 #X2 #H
elim (cpxs_inv_appl1 … H) -H *
[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair/
| #p #W0 #T0 #HT0 #HU
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