(* Vector form of forward lemmas involving same top term structure **********)
(* Basic_1: was just: nf2_iso_appls_lref *)
-lemma cpxs_fwd_cnx_vector: ∀h,g,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ →
- ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.T ➡*[h, g] U → ⒶVs.T ≂ U.
-#h #g #G #L #T #H1T #H2T #Vs elim Vs -Vs [ @(cpxs_fwd_cnx … H2T) ] (**) (* /2 width=3 by cpxs_fwd_cnx/ does not work *)
+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, o] U → ⒶVs.T ≂ 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/
]
qed-.
-lemma cpxs_fwd_sort_vector: ∀h,g,G,L,k,Vs,U. ⦃G, L⦄ ⊢ ⒶVs.⋆k ➡*[h, g] U →
- ⒶVs.⋆k ≂ U ∨ ⦃G, L⦄ ⊢ ⒶVs.⋆(next h k) ➡*[h, g] U.
-#h #g #G #L #k #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_sort/
+lemma cpxs_fwd_sort_vector: ∀h,o,G,L,s,Vs,U. ⦃G, L⦄ ⊢ ⒶVs.⋆s ➡*[h, o] U →
+ ⒶVs.⋆s ≂ U ∨ ⦃G, L⦄ ⊢ ⒶVs.⋆(next h s) ➡*[h, o] 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/
(* Basic_1: was just: pr3_iso_appls_beta *)
-lemma cpxs_fwd_beta_vector: ∀h,g,a,G,L,Vs,V,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓐV.ⓛ{a}W.T ➡*[h, g] U →
- ⒶVs. ⓐV. ⓛ{a}W. T ≂ U ∨ ⦃G, L⦄ ⊢ ⒶVs.ⓓ{a}ⓝW.V.T ➡*[h, g] U.
-#h #g #a #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_beta/
+lemma cpxs_fwd_beta_vector: ∀h,o,a,G,L,Vs,V,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓐV.ⓛ{a}W.T ➡*[h, o] U →
+ ⒶVs. ⓐV. ⓛ{a}W. T ≂ U ∨ ⦃G, L⦄ ⊢ ⒶVs.ⓓ{a}ⓝW.V.T ➡*[h, o] U.
+#h #o #a #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/
]
qed-.
-lemma cpxs_fwd_delta_vector: ∀h,g,I,G,L,K,V1,i. ⬇[i] L ≡ K.ⓑ{I}V1 →
+lemma cpxs_fwd_delta_vector: ∀h,o,I,G,L,K,V1,i. ⬇[i] L ≡ K.ⓑ{I}V1 →
∀V2. ⬆[0, i + 1] V1 ≡ V2 →
- ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.#i ➡*[h, g] U →
- ⒶVs.#i ≂ U ∨ ⦃G, L⦄ ⊢ ⒶVs.V2 ➡*[h, g] U.
-#h #g #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs /2 width=5 by cpxs_fwd_delta/
+ ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.#i ➡*[h, o] U →
+ ⒶVs.#i ≂ U ∨ ⦃G, L⦄ ⊢ ⒶVs.V2 ➡*[h, o] U.
+#h #o #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs /2 width=5 by cpxs_fwd_delta/
#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/
qed-.
(* Basic_1: was just: pr3_iso_appls_abbr *)
-lemma cpxs_fwd_theta_vector: ∀h,g,G,L,V1s,V2s. ⬆[0, 1] V1s ≡ V2s →
- ∀a,V,T,U. ⦃G, L⦄ ⊢ ⒶV1s.ⓓ{a}V.T ➡*[h, g] U →
- ⒶV1s. ⓓ{a}V. T ≂ U ∨ ⦃G, L⦄ ⊢ ⓓ{a}V.ⒶV2s.T ➡*[h, g] U.
-#h #g #G #L #V1s #V2s * -V1s -V2s /3 width=1 by or_intror/
-#V1s #V2s #V1a #V2a #HV12a #HV12s #a
+lemma cpxs_fwd_theta_vector: ∀h,o,G,L,V1c,V2c. ⬆[0, 1] V1c ≡ V2c →
+ ∀a,V,T,U. ⦃G, L⦄ ⊢ ⒶV1c.ⓓ{a}V.T ➡*[h, o] U →
+ ⒶV1c. ⓓ{a}V. T ≂ U ∨ ⦃G, L⦄ ⊢ ⓓ{a}V.ⒶV2c.T ➡*[h, o] U.
+#h #o #G #L #V1c #V2c * -V1c -V2c /3 width=1 by or_intror/
+#V1c #V2c #V1a #V2a #HV12a #HV12c #a
generalize in match HV12a; -HV12a
generalize in match V2a; -V2a
generalize in match V1a; -V1a
-elim HV12s -V1s -V2s /2 width=1 by cpxs_fwd_theta/
-#V1s #V2s #V1b #V2b #HV12b #_ #IHV12s #V1a #V2a #HV12a #V #T #U #H
+elim HV12c -V1c -V2c /2 width=1 by cpxs_fwd_theta/
+#V1c #V2c #V1b #V2b #HV12b #_ #IHV12c #V1a #V2a #HV12a #V #T #U #H
elim (cpxs_inv_appl1 … H) -H *
-[ -IHV12s -HV12a -HV12b #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
+[ -IHV12c -HV12a -HV12b #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
| #b #W0 #T0 #HT0 #HU
- elim (IHV12s … HV12b … HT0) -IHV12s -HT0 #HT0
+ elim (IHV12c … HV12b … HT0) -IHV12c -HT0 #HT0
[ -HV12a -HV12b -HU
elim (tsts_inv_pair1 … HT0) #V1 #T1 #H destruct
- | @or_intror -V1s (**) (* explicit constructor *)
+ | @or_intror -V1c (**) (* explicit constructor *)
@(cpxs_trans … HU) -U
elim (cpxs_inv_abbr1 … HT0) -HT0 *
[ -HV12a -HV12b #V1 #T1 #_ #_ #H destruct
| -V1b #X #HT1 #H #H0 destruct
elim (lift_inv_bind1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
- @(cpxs_trans … (+ⓓV.ⓐV2a.ⓛ{b}W1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2s
+ @(cpxs_trans … (+ⓓV.ⓐV2a.ⓛ{b}W1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2c
@(cpxs_strap2 … (ⓐV1a.ⓛ{b}W0.T0))
/4 width=7 by cpxs_beta_dx, cpx_zeta, lift_bind, lift_flat/
]
]
| #b #V0a #Va #V0 #T0 #HV10a #HV0a #HT0 #HU
- elim (IHV12s … HV12b … HT0) -HV12b -IHV12s -HT0 #HT0
+ elim (IHV12c … HV12b … HT0) -HV12b -IHV12c -HT0 #HT0
[ -HV12a -HV10a -HV0a -HU
elim (tsts_inv_pair1 … HT0) #V1 #T1 #H destruct
- | @or_intror -V1s -V1b (**) (* explicit constructor *)
+ | @or_intror -V1c -V1b (**) (* explicit constructor *)
@(cpxs_trans … HU) -U
elim (cpxs_inv_abbr1 … HT0) -HT0 *
[ #V1 #T1 #HV1 #HT1 #H destruct
| #X #HT1 #H #H0 destruct
elim (lift_inv_bind1 … H) -H #V1 #T1 #HW01 #HT01 #H destruct
lapply (cpxs_lift … HV10a (L.ⓓV0) (Ⓕ) … HV12a … HV0a) -V0a [ /2 width=1 by drop_drop/ ] #HV2a
- @(cpxs_trans … (+ⓓV.ⓐV2a.ⓓ{b}V1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2s
+ @(cpxs_trans … (+ⓓV.ⓐV2a.ⓓ{b}V1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2c
@(cpxs_strap2 … (ⓐV1a.ⓓ{b}V0.T0)) [ /4 width=7 by cpx_zeta, lift_bind, lift_flat/ ] -V -V1 -T1
@(cpxs_strap2 … (ⓓ{b}V0.ⓐV2a.T0)) /3 width=3 by cpxs_pair_sn, cpxs_bind_dx, cpr_cpx, cpr_theta/
]
qed-.
(* Basic_1: was just: pr3_iso_appls_cast *)
-lemma cpxs_fwd_cast_vector: ∀h,g,G,L,Vs,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓝW.T ➡*[h, g] U →
+lemma cpxs_fwd_cast_vector: ∀h,o,G,L,Vs,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓝW.T ➡*[h, o] U →
∨∨ ⒶVs. ⓝW. T ≂ U
- | ⦃G, L⦄ ⊢ ⒶVs.T ➡*[h, g] U
- | ⦃G, L⦄ ⊢ ⒶVs.W ➡*[h, g] U.
-#h #g #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_cast/
+ | ⦃G, L⦄ ⊢ ⒶVs.T ➡*[h, o] U
+ | ⦃G, L⦄ ⊢ ⒶVs.W ➡*[h, o] 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/
projects / helm.git / blobdiff