(* EXAMPLES *****************************************************************)
-(* A reduction cycle in two steps: the term Omega ***************************)
+(* A reduction cycle in exactly three steps: the term Omega *****************)
-definition Delta: term → term ≝ λW. +ⓛW.ⓐ#0.#0.
+definition Delta (s): term ≝ +ⓛ⋆s.ⓐ#0.#0.
-definition Omega1: term → term ≝ λW. ⓐ(Delta W).(Delta W).
+definition Omega1 (s): term ≝ ⓐ(Delta s).(Delta s).
-definition Omega2: term → term ≝ λW. +ⓓⓝW.(Delta W).ⓐ#0.#0.
+definition Omega2 (s): term ≝ +ⓓⓝ⋆s.(Delta s).ⓐ#0.#0.
+
+definition Omega3 (s): term ≝ +ⓓⓝ⋆s.(Delta s).(Omega1 s).
(* Basic properties *********************************************************)
-lemma Delta_lifts: ∀W1,W2,f. ⬆*[f] W1 ≘ W2 →
- ⬆*[f] (Delta W1) ≘ (Delta W2).
+lemma Delta_lifts (f) (s): ⬆*[f] (Delta s) ≘ (Delta s).
/4 width=1 by lifts_lref, lifts_bind, lifts_flat/ qed.
+(* Basic inversion properties ***********************************************)
+
+lemma cpr_inv_Delta1_body_sn (h) (G) (L) (s):
+ ∀X. ⦃G,L.ⓛ⋆s⦄ ⊢ ⓐ#O.#O ➡[h] X → ⓐ#O.#O = X.
+#h #G #L #s #X #H
+lapply (cpm_inv_appl1 … H) -H * *
+[ #W2 #T2 #HW2 #HT2 #H destruct
+ elim (cpr_inv_zero1 … HW2) -HW2
+ elim (cpr_inv_zero1 … HT2) -HT2
+ [ #H1 #H2 destruct //
+ | * #Y #X1 #X2 #_ #_ #H #_ destruct
+ | #_ * #Y #X1 #X2 #_ #_ #H destruct
+ | #_ * #Y #X1 #X2 #_ #_ #H destruct
+ ]
+| #p #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #H #_ destruct
+| #p #V2 #V #W1 #W2 #T1 #T2 #_ #_ #_ #_ #H #_ destruct
+]
+qed-.
+
+lemma cpr_inv_Delta_sn (h) (G) (L) (s):
+ ∀X. ⦃G, L⦄ ⊢ Delta s ➡[h] X → Delta s = X.
+#h #G #L #s #X #H
+elim (cpm_inv_abst1 … H) -H #X1 #X2 #H1 #H2 #H destruct
+lapply (cpr_inv_sort1 … H1) -H1 #H destruct
+<(cpr_inv_Delta1_body_sn … H2) -X2 //
+qed-.
+
(* Main properties **********************************************************)
-theorem cpr_Omega_12: ∀h,G,L,W. ⦃G, L⦄ ⊢ Omega1 W ➡[h] Omega2 W.
+theorem cpr_Omega_12 (h) (G) (L) (s): ⦃G, L⦄ ⊢ Omega1 s ➡[h] Omega2 s.
/2 width=1 by cpm_beta/ qed.
-theorem cpr_Omega_21: ∀h,G,L,W. ⦃G, L⦄ ⊢ Omega2 W ➡[h] Omega1 W.
-#h #G #L #W1 elim (lifts_total W1 (𝐔❴1❵))
-/5 width=5 by lifts_flat, cpm_zeta, cpm_eps, cpm_appl, cpm_delta, Delta_lifts/
-qed.
+theorem cpr_Omega_23 (h) (G) (L) (s): ⦃G, L⦄ ⊢ Omega2 s ➡[h] Omega3 s.
+/5 width=3 by cpm_eps, cpm_appl, cpm_bind, cpm_delta, Delta_lifts/ qed.
+
+theorem cpr_Omega_31 (h) (G) (L) (s): ⦃G, L⦄ ⊢ Omega3 s ➡[h] Omega1 s.
+/4 width=3 by cpm_zeta, Delta_lifts, lifts_flat/ qed.
+
+(* Main inversion properties ************************************************)
+
+theorem cpr_inv_Omega1_sn (h) (G) (L) (s):
+ ∀X. ⦃G, L⦄ ⊢ Omega1 s ➡[h] X →
+ ∨∨ Omega1 s = X | Omega2 s = X.
+#h #G #L #s #X #H elim (cpm_inv_appl1 … H) -H *
+[ #W2 #T2 #HW2 #HT2 #H destruct
+ <(cpr_inv_Delta_sn … HW2) -W2
+ <(cpr_inv_Delta_sn … HT2) -T2
+ /3 width=1 by refl, or_introl/
+| #p #V2 #W1 #W2 #T1 #T2 #HV #HW #HT whd in ⊢ (??%?→?); #H1 #H2 destruct
+ <(cpr_inv_Delta_sn … HV) -V2
+ >(cpr_inv_sort1 … HW) -W2
+ <(cpr_inv_Delta1_body_sn … HT) -T2 //
+| #p #V2 #V #W1 #W2 #T1 #T2 #_ #_ #_ #_ whd in ⊢ (??%?→?); #H #_ destruct
+]
+qed-.
+
+theorem cpr_Omega_21_false (h) (G) (L) (s): ⦃G, L⦄ ⊢ Omega2 s ➡[h] Omega1 s → ⊥.
+#h #G #L #s #H elim (cpm_inv_bind1 … H) -H *
+[ #W #T #_ #_ whd in ⊢ (??%?→?); #H destruct
+| #X #H #_ #_ #_
+ elim (lifts_inv_flat2 … H) -H #X1 #X2 #H1 #_ #_
+ @(lifts_inv_lref2_uni_lt … H1) -H1 //
+]
+qed-.
(* 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_lifts (f:rtmap):
- ∀W1,W2,s. ⬆*[f] W1 ≘ W2 →
- ⬆*[f] (ApplDelta W1 s) ≘ (ApplDelta W2 s).
+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,W,s. ⦃G, L⦄ ⊢ ApplOmega1 W s ➡[h] ApplOmega2 W s.
+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_23 (h): ∀G,L,W,s. ⦃G, L⦄ ⊢ ApplOmega2 W s ➡[h] ApplOmega3 W s.
-#h #G #L #W1 #s elim (lifts_total W1 (𝐔❴1❵)) #W2 #HW12
-@(cpm_zeta … (ApplOmega3 W2 s)) /4 width=1 by ApplDelta_lifts, lifts_flat/
-@cpm_appl // @cpm_appl @(cpm_delta … (ApplDelta W1 s))
-/2 width=1 by ApplDelta_lifts, cpm_eps/
-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_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): ∀G,L,W,s. ⦃G, L⦄ ⊢ ApplOmega1 W s ⬈*[h] ApplOmega3 W s.
-/4 width=4 by cpxs_strap1, cpm_fwd_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) (o) (G) (L) (s0) (s): ⦃G,L,ApplOmega1 s0 s⦄ >[h,o] ⦃G,L,ApplOmega1 s0 s⦄.
/3 width=5 by fpbs_fpbg_trans, fqup_fpbg, cpxs_fpbs/ qed.
| @(mq … H1M) [4,5: /3 width=2 by seq_sym, me/ |1,2: skip ]
/2 width=1 by/
]
-| #G #L #V #U1 #U2 #T2 #_ #IH #HTU2 #gv #lv #Hlv
+| #G #L #V #U1 #T1 #T2 #HTU1 #_ #IH #gv #lv #Hlv
@(mq … H1M)
[4: /3 width=2 by seq_sym, md/ | skip
|3: @(seq_trans … H1M) [2: @mz // | skip ]
- /3 width=1 by li_abbr/
- |2: skip ] -L -U1
- /3 width=1 by lifts_SO_fwd_vpush, seq_sym/
+ ] /3 width=3 by lifts_SO_fwd_vpush, seq_sym/
| #G #L #V #T1 #T2 #_ #IH #gv #lv #Hlv
@(seq_trans … H1M) [2: @(me … H1M) | skip ]
/2 width=1 by/
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