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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "basic_2/notation/relations/predsubtyproper_6.ma".
16 include "static_2/static/feqx.ma".
17 include "basic_2/rt_transition/fpb.ma".
19 (* PROPER PARALLEL RST-TRANSITION FOR CLOSURES ******************************)
21 (* Basic_2A1: uses: fpb *)
22 definition fpbc (G1) (L1) (T1) (G2) (L2) (T2): Prop ≝
23 ∧∧ ❨G1,L1,T1❩ ≽ ❨G2,L2,T2❩
24 & (❨G1,L1,T1❩ ≅ ❨G2,L2,T2❩ → ⊥).
27 "proper parallel rst-transition (closure)"
28 'PRedSubTyProper G1 L1 T1 G2 L2 T2 = (fpbc G1 L1 T1 G2 L2 T2).
30 (* Basic properties *********************************************************)
32 (* Basic_2A1: fpbq_inv_fpb_alt *)
33 lemma fpbc_intro (G1) (L1) (T1) (G2) (L2) (T2):
34 ❨G1,L1,T1❩ ≽ ❨G2,L2,T2❩ → (❨G1,L1,T1❩ ≅ ❨G2,L2,T2❩ → ⊥) →
35 ❨G1,L1,T1❩ ≻ ❨G2,L2,T2❩.
36 /3 width=1 by conj/ qed.
38 lemma rpx_fpbc (G) (T):
39 ∀L1,L2. ❨G,L1❩ ⊢ ⬈[T] L2 → (L1 ≅[T] L2 → ⊥) → ❨G,L1,T❩ ≻ ❨G,L2,T❩.
40 /4 width=4 by fpbc_intro, rpx_fpb, feqg_fwd_reqg_sn/ qed.
42 (* Basic inversion lemmas ***************************************************)
44 (* Basic_2A1: uses: fpb_fpbq_alt *)
45 lemma fpbc_inv_gen (S):
46 ∀G1,G2,L1,L2,T1,T2. ❨G1,L1,T1❩ ≻ ❨G2,L2,T2❩ →
47 ∧∧ ❨G1,L1,T1❩ ≽ ❨G2,L2,T2❩ & (❨G1,L1,T1❩ ≛[S] ❨G2,L2,T2❩ → ⊥).
48 #S #G1 #G2 #L1 #L2 #T1 #T2 *
49 /4 width=2 by feqg_feqx, conj/
52 (* Basic forward lemmas *****************************************************)
54 (* Basic_2A1: uses: fpb_fpbq *)
56 ∀G1,G2,L1,L2,T1,T2. ❨G1,L1,T1❩ ≻ ❨G2,L2,T2❩ →
57 ❨G1,L1,T1❩ ≽ ❨G2,L2,T2❩.
58 #G1 #G2 #L1 #L2 #T1 #T2 * //