-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "paths/standard_trace.ma".
-include "paths/labeled_sequential_computation.ma".
-include "paths/labeled_st_reduction.ma".
-
-(* PATH-LABELED STANDARD COMPUTATION (MULTISTEP) ****************************)
-
-(* Note: lstar shuld be replaced by l_sreds *)
-definition pl_sts: trace → relation subterms ≝ lstar … pl_st.
-
-interpretation "path-labeled standard reduction"
- 'StdStar F p G = (pl_sts p F G).
-
-notation "hvbox( F break Ⓡ ↦* [ term 46 p ] break term 46 G )"
- non associative with precedence 45
- for @{ 'StdStar $F $p $G }.
-
-lemma pl_sts_fwd_pl_sreds: ∀s,F1,F2. F1 Ⓡ↦*[s] F2 → ⇓F1 ↦*[s] ⇓F2.
-#s #F1 #F2 #H @(lstar_ind_r … s F2 H) -s -F2 //
-#p #s #F #F2 #_ #HF2 #IHF1
-lapply (pl_st_fwd_pl_sred … HF2) -HF2 /2 width=3/
-qed-.
-
-lemma pl_sts_inv_pl_sreds: ∀s,M1,F2. {⊤}⇑M1 Ⓡ↦*[s] F2 → is_whd s →
- ∃∃M2. M1 ↦*[s] M2 & {⊤}⇑M2 = F2.
-#s #M1 #F2 #H @(lstar_ind_r … s F2 H) -s -F2 /2 width=3/
-#p #s #F #F2 #_ #HF2 #IHF #H
-elim (is_whd_inv_append … H) -H #Hs * #Hp #_
-elim (IHF Hs) -IHF -Hs #M #HM #H destruct
-elim (pl_st_inv_pl_sred … HF2) -HF2 // -Hp #M2 #HM2 #H
-lapply (pl_sreds_step_dx … HM … HM2) -M /2 width=3/
-qed-.
-
-lemma pl_sts_inv_empty: ∀s,M1,F2. {⊥}⇑M1 Ⓡ↦*[s] F2 → ◊ = s ∧ {⊥}⇑M1 = F2.
-#s #M1 #F2 #H @(lstar_ind_r … s F2 H) -s -F2 /2 width=1/ #p #s #F #F2 #_ #HF2 * #_ #H
-elim (pl_st_inv_empty … HF2 … H)
-qed-.
-
-lemma pl_sts_refl: reflexive … (pl_sts (◊)).
-//
-qed.
-
-lemma pl_sts_step_sn: ∀p,F1,F. F1 Ⓡ↦[p] F → ∀s,F2. F Ⓡ↦*[s] F2 → F1 Ⓡ↦*[p::s] F2.
-/2 width=3/
-qed-.
-
-lemma pl_sts_step_dx: ∀s,F1,F. F1 Ⓡ↦*[s] F → ∀p,F2. F Ⓡ↦[p] F2 → F1 Ⓡ↦*[s@p::◊] F2.
-/2 width=3/
-qed-.
-
-lemma pl_sts_step_rc: ∀p,F1,F2. F1 Ⓡ↦[p] F2 → F1 Ⓡ↦*[p::◊] F2.
-/2 width=1/
-qed.
-
-lemma pl_sts_inv_nil: ∀s,F1,F2. F1 Ⓡ↦*[s] F2 → ◊ = s → F1 = F2.
-/2 width=5 by lstar_inv_nil/
-qed-.
-
-lemma pl_sts_inv_cons: ∀s,F1,F2. F1 Ⓡ↦*[s] F2 → ∀q,r. q::r = s →
- ∃∃F. F1 Ⓡ↦[q] F & F Ⓡ↦*[r] F2.
-/2 width=3 by lstar_inv_cons/
-qed-.
-
-lemma pl_sts_inv_step_rc: ∀p,F1,F2. F1 Ⓡ↦*[p::◊] F2 → F1 Ⓡ↦[p] F2.
-/2 width=1 by lstar_inv_step/
-qed-.
-
-lemma pl_sts_inv_pos: ∀s,F1,F2. F1 Ⓡ↦*[s] F2 → 0 < |s| →
- ∃∃p,r,F. p::r = s & F1 Ⓡ↦[p] F & F Ⓡ↦*[r] F2.
-/2 width=1 by lstar_inv_pos/
-qed-.
-
-lemma pl_sts_inv_rc_abst_dx: ∀b2,s,F1,T2. F1 Ⓡ↦*[s] {b2}𝛌.T2 → ∀r. rc:::r = s →
- ∃∃b1,T1. T1 Ⓡ↦*[r] T2 & {b1}𝛌.T1 = F1.
-#b2 #s #F1 #T2 #H @(lstar_ind_l … s F1 H) -s -F1
-[ #r #H lapply (map_cons_inv_nil … r H) -H #H destruct /2 width=4/
-| #p #s #F1 #F #HF1 #_ #IHF2 #r #H -b2
- elim (map_cons_inv_cons … r H) -H #q #r0 #Hp #Hs #Hr
- elim (pl_st_inv_rc … HF1 … Hp) -HF1 -p #b1 #T1 #T #HT1 #HF1 #HF destruct
- elim (IHF2 …) -IHF2 [3: // |2: skip ] (**) (* simplify line *)
- #b0 #T0 #HT02 #H destruct
- lapply (pl_sts_step_sn … HT1 … HT02) -T /2 width=4/
-]
-qed-.
-
-lemma pl_sts_inv_sn_appl_dx: ∀b2,s,F1,V2,T2. F1 Ⓡ↦*[s] {b2}@V2.T2 → ∀r. sn:::r = s →
- ∃∃b1,V1,T1. V1 Ⓡ↦*[r] V2 & {b1}@V1.T1 = F1.
-#b2 #s #F1 #V2 #T2 #H @(lstar_ind_l … s F1 H) -s -F1
-[ #r #H lapply (map_cons_inv_nil … r H) -H #H destruct /2 width=5/
-| #p #s #F1 #F #HF1 #_ #IHF2 #r #H -b2
- elim (map_cons_inv_cons … r H) -H #q #r0 #Hp #Hs #Hr
- elim (pl_st_inv_sn … HF1 … Hp) -HF1 -p #b1 #V1 #V #T1 #HV1 #HF1 #HF destruct
- elim (IHF2 …) -IHF2 [3: // |2: skip ] (**) (* simplify line *)
- #b0 #V0 #T0 #HV02 #H destruct
- lapply (pl_sts_step_sn … HV1 … HV02) -V /2 width=5/
-]
-qed-.
-
-lemma pl_sts_inv_dx_appl_dx: ∀b,s,F1,V,T2. F1 Ⓡ↦*[s] {b}@V.T2 → ∀r. dx:::r = s →
- ∃∃T1. T1 Ⓡ↦*[r] T2 & {b}@V.T1 = F1.
-#b #s #F1 #V #T2 #H @(lstar_ind_l … s F1 H) -s -F1
-[ #r #H lapply (map_cons_inv_nil … r H) -H #H destruct /2 width=3/
-| #p #s #F1 #F #HF1 #_ #IHF2 #r #H
- elim (map_cons_inv_cons … r H) -H #q #r0 #Hp #Hs #Hr
- elim (pl_st_inv_dx … HF1 … Hp) -HF1 -p #b0 #V0 #T1 #T #HT1 #HF1 #HF destruct
- elim (IHF2 …) -IHF2 [3: // |2: skip ] (**) (* simplify line *)
- #T0 #HT02 #H destruct
- lapply (pl_sts_step_sn … HT1 … HT02) -T /2 width=3/
-]
-qed-.
-
-lemma pl_sts_lift: ∀s. sliftable (pl_sts s).
-/2 width=1/
-qed.
-
-lemma pl_sts_inv_lift: ∀s. sdeliftable_sn (pl_sts s).
-/3 width=3 by lstar_sdeliftable_sn, pl_st_inv_lift/
-qed-.
-
-lemma pl_sts_dsubst: ∀s. sdsubstable_f_dx … (booleanized ⊥) (pl_sts s).
-/2 width=1/
-qed.
-
-theorem pl_sts_mono: ∀s. singlevalued … (pl_sts s).
-/3 width=7 by lstar_singlevalued, pl_st_mono/
-qed-.
-
-theorem pl_sts_trans: ltransitive … pl_sts.
-/2 width=3 by lstar_ltransitive/
-qed-.
-
-lemma pl_sts_inv_trans: inv_ltransitive … pl_sts.
-/2 width=3 by lstar_inv_ltransitive/
-qed-.
-
-lemma pl_sts_fwd_dx_sn_appl_dx: ∀b2,s,r,F1,V2,T2. F1 Ⓡ↦*[(dx:::s)@(sn:::r)] {b2}@V2.T2 →
- ∃∃b1,V1,T1,T0. V1 Ⓡ↦*[r] V2 & T1 Ⓡ↦*[s] T0 & {b1}@V1.T1 = F1.
-#b2 #s #r #F1 #V2 #T2 #H
-elim (pl_sts_inv_trans … H) -H #F #HF1 #H
-elim (pl_sts_inv_sn_appl_dx … H …) -H [3: // |2: skip ] (**) (* simplify line *)
-#b #V #T #HV2 #H destruct
-elim (pl_sts_inv_dx_appl_dx … HF1 …) -HF1 [3: // |2: skip ] (**) (* simplify line *)
-#T1 #HT1 #H destruct /2 width=7/
-qed-.
-
-theorem pl_sts_fwd_is_standard: ∀s,F1,F2. F1 Ⓡ↦*[s] F2 → is_standard s.
-#s elim s -s // #p1 * //
-#p2 #s #IHs #F1 #F2 #H
-elim (pl_sts_inv_cons … H …) -H [4: // |2,3: skip ] #F3 #HF13 #H (**) (* simplify line *)
-elim (pl_sts_inv_cons … H …) [2: // |3,4: skip ] #F4 #HF34 #_ (**) (* simplify line *)
-lapply (pl_st_fwd_sle … HF13 … HF34) -F1 -F4 /3 width=3/
-qed-.
-
-lemma pl_sts_fwd_abst_dx: ∀b2,s,F1,T2. F1 Ⓡ↦*[s] {b2}𝛌.T2 →
- ∃∃r1,r2. is_whd r1 & r1@rc:::r2 = s.
-#b2 #s #F1 #T2 #H
-lapply (pl_sts_fwd_is_standard … H)
-@(lstar_ind_l … s F1 H) -s -F1
-[ #_ @(ex2_2_intro … ◊ ◊) // (**) (* auto needs some help here *)
-| #p #s #F1 #F #HF1 #HF2 #IHF1 #Hs
- lapply (is_standard_fwd_cons … Hs) #H
- elim (IHF1 …) // -IHF1 -H #r1 #r2 #Hr1 #H destruct
- elim (in_whd_or_in_inner p) #Hp
- [ -Hs -F1 -F -T2 -b2
- @(ex2_2_intro … (p::r1) r2) // /2 width=1/ (**) (* auto needs some help here *)
- | lapply (is_standard_fwd_append_sn (p::r1) ? Hs) -Hs #H
- lapply (is_standard_fwd_in_inner … H ?) -H // #H
- lapply (is_whd_is_inner_inv … Hr1 ?) -Hr1 // -H #H destruct
- elim (in_inner_inv … Hp) -Hp * #q [3: #IHq ] #Hp
-(* case 1: dx *)
- [ -IHq
- elim (pl_sts_inv_rc_abst_dx … HF2 …) -b2 [3: // |2: skip ] (**) (* simplify line *)
- #b #T #_ #HT -T2
- elim (pl_st_inv_dx … HF1 …) -HF1 [3: // |2: skip ] (**) (* simplify line *)
- #c #V #T1 #T0 #_ #_ #HT0 -q -T1 -F1 destruct
-(* case 2: rc *)
- | destruct -F1 -F -T2 -b2
- @(ex2_2_intro … ◊ (q::r2)) // (**) (* auto needs some help here *)
-(* case 3: sn *)
- | elim (pl_sts_inv_rc_abst_dx … HF2 …) -b2 [3: // |2: skip ] (**) (* simplify line *)
- #b #T #_ #HT -T2
- elim (pl_st_inv_sn … HF1 …) -HF1 [3: // |2: skip ] (**) (* simplify line *)
- #c #V1 #V #T0 #_ #_ #HT0 -c -q -V1 -F1 destruct
- ]
- ]
-]
-qed-.
-
-lemma pl_sts_fwd_appl_dx: ∀b2,s,F1,V2,T2. F1 Ⓡ↦*[s] {b2}@V2.T2 →
- ∃∃r1,r2,r3. is_whd r1 & is_inner r2 &
- r1@(dx:::r2)@sn:::r3 = s.
-#b2 #s #F1 #V2 #T2 #H
-lapply (pl_sts_fwd_is_standard … H)
-@(lstar_ind_l … s F1 H) -s -F1
-[ #_ @(ex3_3_intro … ◊ ◊ ◊) // (**) (* auto needs some help here *)
-| #p #s #F1 #F #HF1 #HF2 #IHF1 #Hs
- lapply (is_standard_fwd_cons … Hs) #H
- elim (IHF1 …) // -IHF1 -H #r1 #r2 #r3 #Hr1 #Hr2 #H destruct
- elim (in_whd_or_in_inner p) #Hp
- [ -Hs -F1 -F -V2 -T2 -b2
- @(ex3_3_intro … (p::r1) r2 r3) // /2 width=1/ (**) (* auto needs some help here *)
- | lapply (is_standard_fwd_append_sn (p::r1) ? Hs) -Hs #H
- lapply (is_standard_fwd_in_inner … H ?) -H // #H
- lapply (is_whd_is_inner_inv … Hr1 ?) -Hr1 // -H #H destruct
- elim (in_inner_inv … Hp) -Hp * #q [3: #IHq ] #Hp
-(* case 1: dx *)
- [ destruct -F1 -F -V2 -T2 -b2
- @(ex3_3_intro … ◊ (q::r2) r3) // /2 width=1/ (**) (* auto needs some help here *)
-(* case 2: rc *)
- | -Hr2
- elim (pl_sts_fwd_dx_sn_appl_dx … HF2) -b2 #b #V #T #T0 #_ #_ #HT -V2 -T2 -T0
- elim (pl_st_inv_rc … HF1 … Hp) -HF1 #c #V0 #T0 #_ #_ #HT0 -c -V0 -q -F1 destruct
-(* case 3: sn *)
- | -Hr2
- elim (pl_sts_fwd_dx_sn_appl_dx … HF2) -b2 #b #V #T #T0 #_ #HT0 #HT -V2 -T2
- elim (pl_st_inv_sn … HF1 … Hp) -HF1 #c #V1 #V0 #T1 #_ #_ #H -c -V1 -F1 destruct -V
- elim (pl_sts_inv_empty … HT0) -HT0 #H #_ -T0 -T1 destruct
- @(ex3_3_intro … ◊ ◊ (q::r3)) // (**) (* auto needs some help here *)
- ]
- ]
-]
-qed-.
-
-lemma pl_sred_is_standard_pl_st: ∀p,M,M2. M ↦[p] M2 → ∀F. ⇓F = M →
- ∀s,M1.{⊤}⇑ M1 Ⓡ↦*[s] F →
- is_standard (s@(p::◊)) →
- ∃∃F2. F Ⓡ↦[p] F2 & ⇓F2 = M2.
-#p #M #M2 #H elim H -p -M -M2
-[ #B #A #F #HF #s #M1 #HM1 #Hs
- lapply (is_standard_fwd_is_whd … Hs) -Hs // #Hs
- elim (pl_sts_inv_pl_sreds … HM1 Hs) -HM1 -Hs #M #_ #H -s -M1 destruct
- >carrier_boolean in HF; #H destruct normalize /2 width=3/
-| #p #A1 #A2 #_ #IHA12 #F #HF #s #M1 #HM1 #Hs
- elim (carrier_inv_abst … HF) -HF #b #T #HT #HF destruct
- elim (pl_sts_fwd_abst_dx … HM1) #r1 #r2 #Hr1 #H destruct
- elim (pl_sts_inv_trans … HM1) -HM1 #F0 #HM1 #HT
- elim (pl_sts_inv_pl_sreds … HM1 …) // #M0 #_ #H -M1 -Hr1 destruct
- >associative_append in Hs; #Hs
- lapply (is_standard_fwd_append_dx … Hs) -r1
- <(map_cons_append … r2 (p::◊)) #H
- lapply (is_standard_inv_compatible_rc … H) -H #Hp
- elim (pl_sts_inv_rc_abst_dx … HT …) -HT [3: // |2: skip ] #b0 #T0 #HT02 #H (**) (* simplify line *)
- elim (boolean_inv_abst … (sym_eq … H)) -H #A0 #_ #H #_ -b0 -M0 destruct
- elim (IHA12 … HT02 …) // -r2 -A0 -IHA12 #F2 #HF2 #H
- @(ex2_intro … ({⊥}𝛌.F2)) normalize // /2 width=1/ (**) (* auto needs some help here *)
-| #p #B1 #B2 #A #_ #IHB12 #F #HF #s #M1 #HM1 #Hs
- elim (carrier_inv_appl … HF) -HF #b #V #T #HV #HT #HF destruct
- elim (pl_sts_fwd_appl_dx … HM1) #r1 #r2 #r3 #Hr1 #_ #H destruct
- elim (pl_sts_inv_trans … HM1) -HM1 #F0 #HM1 #HT
- elim (pl_sts_inv_pl_sreds … HM1 …) // #M0 #_ #H -M1 -Hr1 destruct
- >associative_append in Hs; #Hs
- lapply (is_standard_fwd_append_dx … Hs) -r1
- >associative_append #Hs
- lapply (is_standard_fwd_append_dx … Hs) -Hs
- <(map_cons_append … r3 (p::◊)) #H
- lapply (is_standard_inv_compatible_sn … H) -H #Hp
- elim (pl_sts_fwd_dx_sn_appl_dx … HT) -HT #b0 #V0 #T0 #T1 #HV0 #_ #H -T1 -r2
- elim (boolean_inv_appl … (sym_eq … H)) -H #B0 #A0 #_ #H #_ #_ -b0 -M0 -T0 destruct
- elim (IHB12 … HV0 …) // -r3 -B0 -IHB12 #G2 #HG2 #H
- @(ex2_intro … ({⊥}@G2.{⊥}⇕T)) normalize // /2 width=1/ (**) (* auto needs some help here *)
-| #p #B #A1 #A2 #_ #IHA12 #F #HF #s #M1 #HM1 #Hs
- elim (carrier_inv_appl … HF) -HF #b #V #T #HV #HT #HF destruct
- elim (pl_sts_fwd_appl_dx … HM1) #r1 #r2 #r3 #Hr1 #Hr2 #H destruct
- elim (pl_sts_inv_trans … HM1) -HM1 #F0 #HM1 #HT
- elim (pl_sts_inv_pl_sreds … HM1 …) // #M0 #_ #H -M1 -Hr1 destruct
- >associative_append in Hs; #Hs
- lapply (is_standard_fwd_append_dx … Hs) -r1
- >associative_append #Hs
- elim (list_inv … r3)
- [ #H destruct
- elim (in_whd_or_in_inner p) #Hp
- [ lapply (is_standard_fwd_is_whd … Hs) -Hs /2 width=1/ -Hp #Hs
- lapply (is_whd_inv_dx … Hs) -Hs #H
- lapply (is_whd_is_inner_inv … Hr2) -Hr2 // -H #H destruct
- lapply (pl_sts_inv_nil … HT ?) -HT // #H
- elim (boolean_inv_appl … H) -H #B0 #A0 #_ #_ #H #_ -M0 -B0 destruct
- elim (IHA12 … A0 …) -IHA12 [3,5,6: // |2,4: skip ] (* simplify line *)
- #F2 #HF2 #H
- @(ex2_intro … ({b}@V.F2)) normalize // /2 width=1/ (**) (* auto needs some help here *)
- | <(map_cons_append … r2 (p::◊)) in Hs; #H
- lapply (is_standard_inv_compatible_dx … H ?) -H /3 width=1/ -Hp #Hp
- >append_nil in HT; #HT
- elim (pl_sts_inv_dx_appl_dx … HT …) -HT [3: // |2: skip ] (* simplify line *)
- #T0 #HT0 #H
- elim (boolean_inv_appl … (sym_eq … H)) -H #B0 #A0 #_ #_ #H #_ -M0 -B0 destruct
- elim (IHA12 … HT0 …) // -r2 -A0 -IHA12 #F2 #HF2 #H
- @(ex2_intro … ({b}@V.F2)) normalize // /2 width=1/ (**) (* auto needs some help here *)
- ]
- | -IHA12 -Hr2 -M0 * #q #r #H destruct
- lapply (is_standard_fwd_append_dx … Hs) -r2 #Hs
- lapply (is_standard_fwd_sle … Hs) -r #H
- elim (sle_inv_sn … H …) -H [3: // |2: skip ] (**) (* simplify line *)
- #q0 #_ #H destruct
- ]
-]
-qed-.
-
-theorem pl_sreds_is_standard_pl_sts: ∀s,M1,M2. M1 ↦*[s] M2 → is_standard s →
- ∃∃F2. {⊤}⇑ M1 Ⓡ↦*[s] F2 & ⇓F2 = M2.
-#s #M1 #M2 #H @(lstar_ind_r … s M2 H) -s -M2 /2 width=3/
-#p #s #M #M2 #_ #HM2 #IHM1 #Hsp
-lapply (is_standard_fwd_append_sn … Hsp) #Hs
-elim (IHM1 Hs) -IHM1 -Hs #F #HM1 #H
-elim (pl_sred_is_standard_pl_st … HM2 … HM1 …) -HM2 // -M -Hsp #F2 #HF2 #HFM2
-lapply (pl_sts_step_dx … HM1 … HF2) -F
-#H @(ex2_intro … F2) // (**) (* auto needs some help here *)
-qed-.