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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 include "basic_2/computation/fprs_cprs.ma".
16 include "basic_2/equivalence/cpcs_cpcs.ma".
17 include "basic_2/equivalence/fpcs_fpcs.ma".
19 (* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
21 (* Advanced properties ******************************************************)
23 lemma fpcs_shift: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1.ⓑ{I}V1, T1⦄ ⬌* ⦃L2.ⓑ{I}V2, T2⦄ →
24 ⦃L1, -ⓑ{I}V1.T1⦄ ⬌* ⦃L2, -ⓑ{I}V2.T2⦄.
25 #I #L1 #L2 #V1 #V2 #T1 #T2 #H12
26 elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2
27 elim (fprs_inv_pair1 … H1) -H1 #K1 #U1 #_ #HTU1 #H destruct
28 elim (fprs_inv_pair1 … H2) -H2 #K2 #U2 #_ #HTU2 #H destruct /2 width=4/
31 lemma fpcs_bind_minus: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1, -ⓑ{I}V1.T1⦄ ⬌* ⦃L2, -ⓑ{I}V2.T2⦄ →
32 ∀b. ⦃L1, ⓑ{b,I}V1.T1⦄ ⬌* ⦃L2, ⓑ{b,I}V2.T2⦄.
33 #I #L1 #L2 #V1 #V2 #T1 #T2 #H12 #b
34 elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2
35 elim (fprs_fwd_bind2_minus … H1 b) -H1 #W1 #U1 #HTU1 #H destruct
36 elim (fprs_fwd_bind2_minus … H2 b) -H2 #W2 #U2 #HTU2 #H destruct /2 width=4/
39 lemma fpcs_shift_full: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1.ⓑ{I}V1, T1⦄ ⬌* ⦃L2.ⓑ{I}V2, T2⦄ →
40 ∀b. ⦃L1, ⓑ{b,I}V1.T1⦄ ⬌* ⦃L2, ⓑ{b,I}V2.T2⦄.
43 (* Properties on context-sensitive parallel equivalence for terms ***********)
45 lemma cpcs_fpcs: ∀L,T1,T2. L ⊢ T1 ⬌* T2 → ⦃L, T1⦄ ⬌* ⦃L, T2⦄.
47 elim (cpcs_inv_cprs … H) -H /3 width=4 by fprs_div, cprs_fprs/ (**) (* too slow without trace *)
50 (* Inversion lemmas on context-sensitive parallel equivalence for terms *****)
52 lemma fpcs_inv_cpcs: ∀L,T1,T2. ⦃L, T1⦄ ⬌* ⦃L, T2⦄ → L ⊢ T1 ⬌* T2.
54 elim (fpcs_inv_fprs … H) -H /3 width=4 by cprs_div, fprs_fwd_cprs/