milestone in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / apps_2 / examples / ex_fpbg_refl.ma

diff --git a/matita/matita/contribs/lambdadelta/apps_2/examples/ex_fpbg_refl.ma b/matita/matita/contribs/lambdadelta/apps_2/examples/ex_fpbg_refl.ma
index e37f8831f0689d63dfb49c72c875ad3da8ab7fd0..6ee55d8d0634d8c347c9f3b55e42a88a09fa5c34 100644 (file)
--- a/matita/matita/contribs/lambdadelta/apps_2/examples/ex_fpbg_refl.ma
+++ b/matita/matita/contribs/lambdadelta/apps_2/examples/ex_fpbg_refl.ma
@@ -12,43 +12,46 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/computation/fpbg_fpbs.ma".
+include "basic_2/rt_computation/fpbs_cpxs.ma".
+include "basic_2/rt_computation/fpbg_fqup.ma".
+include "basic_2/rt_computation/fpbg_fpbs.ma".
 
 (* EXAMPLES *****************************************************************)
 
-(* Reflexivity of proper qrst-computation: the term ApplOmega ***************)
+(* Reflexivity of proper rst-computation: the term ApplOmega ****************)
 
-definition ApplDelta: term → nat → term ≝ λW,s. +ⓛW.ⓐ⋆s.ⓐ#0.#0.
+definition ApplDelta (s0) (s): term ≝ +ⓛ⋆s0.ⓐ⋆s.ⓐ#0.#0.
 
-definition ApplOmega1: term → nat → term ≝ λW,s. ⓐ(ApplDelta W s).(ApplDelta W s).
+definition ApplOmega1 (s0) (s): term ≝ ⓐ(ApplDelta s0 s).(ApplDelta s0 s).
 
-definition ApplOmega2: term → nat → term ≝ λW,s. +ⓓⓝW.(ApplDelta W s).ⓐ⋆s.ⓐ#0.#0.
+definition ApplOmega2 (s0) (s): term ≝ +ⓓⓝ⋆s0.(ApplDelta s0 s).ⓐ⋆s.ⓐ#0.#0.
 
-definition ApplOmega3: term → nat → term ≝ λW,s. ⓐ⋆s.(ApplOmega1 W s).
+definition ApplOmega3 (s0) (s): term ≝ +ⓓⓝ⋆s0.(ApplDelta s0 s).ⓐ⋆s.(ApplOmega1 s0 s).
+
+definition ApplOmega4 (s0) (s): term ≝ ⓐ⋆s.(ApplOmega1 s0 s).
 
 (* Basic properties *********************************************************)
 
-lemma ApplDelta_lift: ∀W1,W2,s,l,k. ⬆[l, k] W1 ≡ W2 →
-                      ⬆[l, k] (ApplDelta W1 s) ≡ (ApplDelta W2 s).
-/5 width=1 by lift_flat, lift_bind, lift_lref_lt/ qed.
+lemma ApplDelta_lifts (f:rtmap) (s0) (s):
+                      ⬆*[f] (ApplDelta s0 s) ≘ (ApplDelta s0 s).
+/5 width=1 by lifts_sort, lifts_lref, lifts_bind, lifts_flat/ qed.
+
+lemma cpr_ApplOmega_12 (h) (G) (L) (s0) (s): ⦃G, L⦄ ⊢ ApplOmega1 s0 s ➡[h] ApplOmega2 s0 s.
+/2 width=1 by cpm_beta/ qed.
 
-lemma cpr_ApplOmega_12: ∀G,L,W,s. ⦃G, L⦄ ⊢ ApplOmega1 W s ➡ ApplOmega2 W s.
-/2 width=1 by cpr_beta/ qed.
+lemma cpr_ApplOmega_23 (h) (G) (L) (s0) (s): ⦃G, L⦄ ⊢ ApplOmega2 s0 s ➡[h] ApplOmega3 s0 s.
+/6 width=3 by cpm_eps, cpm_appl, cpm_bind, cpm_delta, ApplDelta_lifts/ qed.
 
-lemma cpr_ApplOmega_23: ∀G,L,W,s. ⦃G, L⦄ ⊢ ApplOmega2 W s ➡ ApplOmega3 W s.
-#G #L #W1 #s elim (lift_total W1 0 1) #W2 #HW12
-@(cpr_zeta … (ApplOmega3 W2 s)) /4 width=1 by ApplDelta_lift, lift_flat/
-@cpr_flat // @cpr_flat @(cpr_delta … (ApplDelta W1 s) ? 0)
-[3,5,8,10: /2 width=2 by ApplDelta_lift/ |4,9: /2 width=1 by cpr_eps/ |*: skip ]
-qed.
+lemma cpr_ApplOmega_34 (h) (G) (L) (s0) (s): ⦃G, L⦄ ⊢ ApplOmega3 s0 s ➡[h] ApplOmega4 s0 s.
+/4 width=3 by cpm_zeta, ApplDelta_lifts, lifts_sort, lifts_flat/ qed.
 
-lemma cpxs_ApplOmega_13: ∀h,o,G,L,W,s. ⦃G, L⦄ ⊢ ApplOmega1 W s ➡*[h,o] ApplOmega3 W s.
-/4 width=3 by cpxs_strap1, cpr_cpx/ qed.
+lemma cpxs_ApplOmega_14 (h) (G) (L) (s0) (s): ⦃G, L⦄ ⊢ ApplOmega1 s0 s ⬈*[h] ApplOmega4 s0 s.
+/5 width=4 by cpxs_strap1, cpm_fwd_cpx/ qed.
 
-lemma fqup_ApplOmega_13: ∀G,L,W,s. ⦃G, L, ApplOmega3 W s⦄ ⊐+ ⦃G, L, ApplOmega1 W s⦄.
+lemma fqup_ApplOmega_41 (G) (L) (s0) (s): ⦃G,L,ApplOmega4 s0 s⦄ ⊐+ ⦃G,L,ApplOmega1 s0 s⦄.
 /2 width=1 by/ qed.
 
 (* Main properties **********************************************************)
 
-theorem fpbg_refl: ∀h,o,G,L,W,s. ⦃G, L, ApplOmega1 W s⦄ >≡[h,o] ⦃G, L, ApplOmega1 W s⦄.
+theorem fpbg_refl (h) (G) (L) (s0) (s): ⦃G,L,ApplOmega1 s0 s⦄ >[h] ⦃G,L,ApplOmega1 s0 s⦄.
 /3 width=5 by fpbs_fpbg_trans, fqup_fpbg, cpxs_fpbs/ qed.