theorem cpxs_trans: ∀h,o,G,L. Transitive … (cpxs h o G L).
normalize /2 width=3 by trans_TC/ qed-.
-theorem cpxs_bind: â\88\80h,o,a,I,G,L,V1,V2,T1,T2. â¦\83G, L.â\93\91{I}V1â¦\84 â\8a¢ T1 â\9e¡*[h, o] T2 →
- â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡*[h, o] V2 →
- â¦\83G, Lâ¦\84 â\8a¢ â\93\91{a,I}V1.T1 â\9e¡*[h, o] ⓑ{a,I}V2.T2.
+theorem cpxs_bind: â\88\80h,o,a,I,G,L,V1,V2,T1,T2. â¦\83G, L.â\93\91{I}V1â¦\84 â\8a¢ T1 â¬\88*[h, o] T2 →
+ â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88*[h, o] V2 →
+ â¦\83G, Lâ¦\84 â\8a¢ â\93\91{a,I}V1.T1 â¬\88*[h, o] ⓑ{a,I}V2.T2.
#h #o #a #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: â\88\80h,o,I,G,L,V1,V2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[h, o] T2 →
- â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡*[h, o] V2 →
- â¦\83G, Lâ¦\84 â\8a¢ â\93\95{I}V1.T1 â\9e¡*[h, o] ⓕ{I}V2.T2.
+theorem cpxs_flat: â\88\80h,o,I,G,L,V1,V2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h, o] T2 →
+ â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88*[h, o] V2 →
+ â¦\83G, Lâ¦\84 â\8a¢ â\93\95{I}V1.T1 â¬\88*[h, o] ⓕ{I}V2.T2.
#h #o #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.
- â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h, o] V2 â\86\92 â¦\83G, L.â\93\9bW1â¦\84 â\8a¢ T1 â\9e¡*[h, o] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ W1 â\9e¡*[h, o] W2 →
- â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.â\93\9b{a}W1.T1 â\9e¡*[h, o] ⓓ{a}ⓝW2.V2.T2.
+ â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h, o] V2 â\86\92 â¦\83G, L.â\93\9bW1â¦\84 â\8a¢ T1 â¬\88*[h, o] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ W1 â¬\88*[h, o] W2 →
+ â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.â\93\9b{a}W1.T1 â¬\88*[h, o] ⓓ{a}ⓝW2.V2.T2.
#h #o #a #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.
- â¦\83G, L.â\93\9bW1â¦\84 â\8a¢ T1 â\9e¡*[h, o] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ W1 â\9e¡*[h, o] W2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡*[h, o] V2 →
- â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.â\93\9b{a}W1.T1 â\9e¡*[h, o] ⓓ{a}ⓝW2.V2.T2.
+ â¦\83G, L.â\93\9bW1â¦\84 â\8a¢ T1 â¬\88*[h, o] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ W1 â¬\88*[h, o] W2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88*[h, o] V2 →
+ â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.â\93\9b{a}W1.T1 â¬\88*[h, o] ⓓ{a}ⓝW2.V2.T2.
#h #o #a #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.
- â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h, o] V → ⬆[0, 1] V ≡ V2 →
- â¦\83G, L.â\93\93W1â¦\84 â\8a¢ T1 â\9e¡*[h, o] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ W1 â\9e¡*[h, o] W2 →
- â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.â\93\93{a}W1.T1 â\9e¡*[h, o] ⓓ{a}W2.ⓐV2.T2.
+ â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h, o] V → ⬆[0, 1] V ≡ V2 →
+ â¦\83G, L.â\93\93W1â¦\84 â\8a¢ T1 â¬\88*[h, o] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ W1 â¬\88*[h, o] W2 →
+ â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.â\93\93{a}W1.T1 â¬\88*[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
/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.
- â¬\86[0, 1] V â\89¡ V2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ W1 â\9e¡*[h, o] W2 →
- â¦\83G, L.â\93\93W1â¦\84 â\8a¢ T1 â\9e¡*[h, o] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡*[h, o] V →
- â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.â\93\93{a}W1.T1 â\9e¡*[h, o] ⓓ{a}W2.ⓐV2.T2.
+ â¬\86[0, 1] V â\89¡ V2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ W1 â¬\88*[h, o] W2 →
+ â¦\83G, L.â\93\93W1â¦\84 â\8a¢ T1 â¬\88*[h, o] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88*[h, o] V →
+ â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.â\93\93{a}W1.T1 â¬\88*[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
/3 width=5 by cpxs_trans, cpxs_theta_rc, cpxs_flat_dx/
qed.
(* Advanced inversion lemmas ************************************************)
-lemma cpxs_inv_appl1: â\88\80h,o,G,L,V1,T1,U2. â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.T1 â\9e¡*[h, o] U2 →
- â\88¨â\88¨ â\88\83â\88\83V2,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡*[h, o] V2 & â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[h, o] T2 &
+lemma cpxs_inv_appl1: â\88\80h,o,G,L,V1,T1,U2. â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.T1 â¬\88*[h, o] U2 →
+ â\88¨â\88¨ â\88\83â\88\83V2,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88*[h, o] V2 & â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h, o] T2 &
U2 = ⓐV2. T2
- | â\88\83â\88\83a,W,T. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[h, o] â\93\9b{a}W.T & â¦\83G, Lâ¦\84 â\8a¢ â\93\93{a}â\93\9dW.V1.T â\9e¡*[h, o] U2
- | â\88\83â\88\83a,V0,V2,V,T. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡*[h, o] V0 & ⬆[0,1] V0 ≡ V2 &
- â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[h, o] â\93\93{a}V.T & â¦\83G, Lâ¦\84 â\8a¢ â\93\93{a}V.â\93\90V2.T â\9e¡*[h, o] U2.
+ | â\88\83â\88\83a,W,T. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h, o] â\93\9b{a}W.T & â¦\83G, Lâ¦\84 â\8a¢ â\93\93{a}â\93\9dW.V1.T â¬\88*[h, o] U2
+ | â\88\83â\88\83a,V0,V2,V,T. â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88*[h, o] V0 & ⬆[0,1] V0 ≡ V2 &
+ â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h, o] â\93\93{a}V.T & â¦\83G, Lâ¦\84 â\8a¢ â\93\93{a}V.â\93\90V2.T â¬\88*[h, o] U2.
#h #o #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
]
qed-.
-lemma cpx_bind2: â\88\80h,o,G,L,V1,V2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h, o] V2 →
- â\88\80I,T1,T2. â¦\83G, L.â\93\91{I}V2â¦\84 â\8a¢ T1 â\9e¡[h, o] T2 →
- â\88\80a. â¦\83G, Lâ¦\84 â\8a¢ â\93\91{a,I}V1.T1 â\9e¡*[h, o] ⓑ{a,I}V2.T2.
+lemma cpx_bind2: â\88\80h,o,G,L,V1,V2. â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h, o] V2 →
+ â\88\80I,T1,T2. â¦\83G, L.â\93\91{I}V2â¦\84 â\8a¢ T1 â¬\88[h, o] T2 →
+ â\88\80a. â¦\83G, Lâ¦\84 â\8a¢ â\93\91{a,I}V1.T1 â¬\88*[h, o] ⓑ{a,I}V2.T2.
/4 width=5 by lpx_cpx_trans, cpxs_bind_dx, lpx_pair/ qed.
(* Advanced properties ******************************************************)
#h #o #G @b_c_trans_LTC1 /2 width=3 by lpx_cpx_trans/ (**) (* full auto fails *)
qed-.
-lemma cpxs_bind2_dx: â\88\80h,o,G,L,V1,V2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h, o] V2 →
- â\88\80I,T1,T2. â¦\83G, L.â\93\91{I}V2â¦\84 â\8a¢ T1 â\9e¡*[h, o] T2 →
- â\88\80a. â¦\83G, Lâ¦\84 â\8a¢ â\93\91{a,I}V1.T1 â\9e¡*[h, o] ⓑ{a,I}V2.T2.
+lemma cpxs_bind2_dx: â\88\80h,o,G,L,V1,V2. â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h, o] V2 →
+ â\88\80I,T1,T2. â¦\83G, L.â\93\91{I}V2â¦\84 â\8a¢ T1 â¬\88*[h, o] T2 →
+ â\88\80a. â¦\83G, Lâ¦\84 â\8a¢ â\93\91{a,I}V1.T1 â¬\88*[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⦄ →
- â\88\80U2. â¦\83G2, L2â¦\84 â\8a¢ T2 â\9e¡*[h, o] U2 → (T2 = U2 → ⊥) →
- â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+ â\88\80U2. â¦\83G2, L2â¦\84 â\8a¢ T2 â¬\88*[h, o] U2 → (T2 = U2 → ⊥) →
+ â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â¬\88*[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)
qed-.
lemma fquq_cpxs_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- â\88\80U2. â¦\83G2, L2â¦\84 â\8a¢ T2 â\9e¡*[h, o] U2 → (T2 = U2 → ⊥) →
- â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+ â\88\80U2. â¦\83G2, L2â¦\84 â\8a¢ T2 â¬\88*[h, o] U2 → (T2 = U2 → ⊥) →
+ â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â¬\88*[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/
qed-.
lemma fqup_cpxs_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- â\88\80U2. â¦\83G2, L2â¦\84 â\8a¢ T2 â\9e¡*[h, o] U2 → (T2 = U2 → ⊥) →
- â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
+ â\88\80U2. â¦\83G2, L2â¦\84 â\8a¢ T2 â¬\88*[h, o] U2 → (T2 = U2 → ⊥) →
+ â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â¬\88*[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/
qed-.
lemma fqus_cpxs_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- â\88\80U2. â¦\83G2, L2â¦\84 â\8a¢ T2 â\9e¡*[h, o] U2 → (T2 = U2 → ⊥) →
- â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+ â\88\80U2. â¦\83G2, L2â¦\84 â\8a¢ T2 â¬\88*[h, o] U2 → (T2 = U2 → ⊥) →
+ â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â¬\88*[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/