--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 "subterms/booleanized.ma".
+include "paths/labeled_sequential_reduction.ma".
+include "paths/standard_order.ma".
+
+(* PATH-LABELED STANDARD REDUCTION ON SUBTERMS (SINGLE STEP) ****************)
+
+(* Note: this is standard reduction on marked redexes,
+ left residuals are unmarked in the reductum
+*)
+inductive pl_st: path → relation subterms ≝
+| pl_st_beta : ∀V,T. pl_st (◊) ({⊤}@V.{⊤}𝛌.T) ([↙V]T)
+| pl_st_abst : ∀b,p,T1,T2. pl_st p T1 T2 → pl_st (rc::p) ({b}𝛌.T1) ({⊥}𝛌.T2)
+| pl_st_appl_sn: ∀b,p,V1,V2,T. pl_st p V1 V2 → pl_st (sn::p) ({b}@V1.T) ({⊥}@V2.{⊥}⇕T)
+| pl_st_appl_dx: ∀b,p,V,T1,T2. pl_st p T1 T2 → pl_st (dx::p) ({b}@V.T1) ({b}@V.T2)
+.
+
+interpretation "path-labeled standard reduction"
+ 'Std F p G = (pl_st p F G).
+
+notation "hvbox( F break Ⓡ ↦ [ term 46 p ] break term 46 G )"
+ non associative with precedence 45
+ for @{ 'Std $F $p $G }.
+
+lemma pl_st_fwd_pl_sred: ∀p,F1,F2. F1 Ⓡ↦[p] F2 → ⇓F1 ↦[p] ⇓F2.
+#p #F1 #F2 #H elim H -p -F1 -F2 normalize /2 width=1/
+qed-.
+
+lemma pl_st_inv_vref: ∀p,F,G. F Ⓡ↦[p] G → ∀b,i. {b}#i = F → ⊥.
+#p #F #G #HFG #b #i #H
+lapply (pl_st_fwd_pl_sred … HFG) -HFG #HFG
+lapply (eq_f … carrier … H) -H normalize #H
+/2 width=6 by pl_sred_inv_vref/
+qed-.
+
+lemma pl_st_inv_abst: ∀p,F,G. F Ⓡ↦[p] G → ∀c,U1. {c}𝛌.U1 = F →
+ ∃∃q,U2. U1 Ⓡ↦[q] U2 & rc::q = p & {⊥}𝛌.U2 = G.
+#p #F #G * -p -F -G
+[ #V #T #c #U1 #H destruct
+| #b #p #T1 #T2 #HT12 #c #U1 #H destruct /2 width=5/
+| #b #p #V1 #V2 #T #_ #c #U1 #H destruct
+| #b #p #V #T1 #T2 #_ #c #U1 #H destruct
+]
+qed-.
+
+lemma pl_st_inv_appl: ∀p,F,G. F Ⓡ↦[p] G → ∀c,W,U. {c}@W.U = F →
+ ∨∨ (∃∃U0. ⊤ = c & ◊ = p & {⊤}𝛌.U0 = U & [↙W] U0 = G)
+ | (∃∃q,W0. sn::q = p & W Ⓡ↦[q] W0 & {⊥}@W0.{⊥}⇕U = G)
+ | (∃∃q,U0. dx::q = p & U Ⓡ↦[q] U0 & {c}@W.U0 = G).
+#p #F #G * -p -F -G
+[ #V #T #c #W #U #H destruct /3 width=3/
+| #b #p #T1 #T2 #_ #c #W #U #H destruct
+| #b #p #V1 #V2 #T #HV12 #c #W #U #H destruct /3 width=5/
+| #b #p #V #T1 #T2 #HT12 #c #W #U #H destruct /3 width=5/
+]
+qed-.
+
+lemma pl_st_fwd_abst: ∀p,F,G. F Ⓡ↦[p] G → ∀c,U2. {c}𝛌.U2 = G →
+ ◊ = p ∨ ∃q. rc::q = p.
+#p #F #G * -p -F -G
+[ /2 width=1/
+| /3 width=2/
+| #b #p #V1 #V2 #T #_ #c #U2 #H destruct
+| #b #p #V #T1 #T2 #_ #c #U2 #H destruct
+]
+qed-.
+
+lemma pl_st_inv_nil: ∀p,F,G. F Ⓡ↦[p] G → ◊ = p →
+ ∃∃V,T. {⊤}@V.{⊤} 𝛌.T = F & [↙V] T = G.
+#p #F #G * -p -F -G
+[ #V #T #_ destruct /2 width=4/
+| #b #p #T1 #T2 #_ #H destruct
+| #b #p #V1 #V2 #T #_ #H destruct
+| #b #p #V #T1 #T2 #_ #H destruct
+]
+qed-.
+
+lemma pl_st_inv_rc: ∀p,F,G. F Ⓡ↦[p] G → ∀q. rc::q = p →
+ ∃∃b,T1,T2. T1 Ⓡ↦[q] T2 & {b}𝛌.T1 = F & {⊥}𝛌.T2 = G.
+#p #F #G * -p -F -G
+[ #V #T #q #H destruct
+| #b #p #T1 #T2 #HT12 #q #H destruct /2 width=6/
+| #b #p #V1 #V2 #T #_ #q #H destruct
+| #b #p #V #T1 #T2 #_ #q #H destruct
+]
+qed-.
+
+lemma pl_st_inv_sn: ∀p,F,G. F Ⓡ↦[p] G → ∀q. sn::q = p →
+ ∃∃b,V1,V2,T. V1 Ⓡ↦[q] V2 & {b}@V1.T = F & {⊥}@V2.{⊥}⇕T = G.
+#p #F #G * -p -F -G
+[ #V #T #q #H destruct
+| #b #p #T1 #T2 #_ #q #H destruct
+| #b #p #V1 #V2 #T #HV12 #q #H destruct /2 width=7/
+| #b #p #V #T1 #T2 #_ #q #H destruct
+]
+qed-.
+
+lemma pl_st_inv_dx: ∀p,F,G. F Ⓡ↦[p] G → ∀q. dx::q = p →
+ ∃∃b,V,T1,T2. T1 Ⓡ↦[q] T2 & {b}@V.T1 = F & {b}@V.T2 = G.
+#p #F #G * -p -F -G
+[ #V #T #q #H destruct
+| #b #p #T1 #T2 #_ #q #H destruct
+| #b #p #V1 #V2 #T #_ #q #H destruct
+| #b #p #V #T1 #T2 #HT12 #q #H destruct /2 width=7/
+]
+qed-.
+
+lemma pl_st_inv_pl_sred: ∀p. in_whd p → ∀M1,F2. {⊤}⇑M1 Ⓡ↦[p] F2 →
+ ∃∃M2. M1 ↦[p] M2 & {⊤}⇑M2 = F2.
+#p @(in_whd_ind … p) -p
+[ #M1 #F2 #H
+ elim (pl_st_inv_nil … H …) -H // #V #T #HM1 #H
+ elim (boolean_inv_appl … (sym_eq … HM1)) -HM1 #B #N #_ #HB #HN #HM1
+ elim (boolean_inv_abst … HN) -HN #A #_ #HA #HN destruct /2 width=3/
+| #p #_ #IHp #M1 #F2 #H
+ elim (pl_st_inv_dx … H …) -H [3: // |2:skip ] #b #V #T1 #T2 #HT12 #HM1 #H (**) (* simplify line *)
+ elim (boolean_inv_appl … (sym_eq … HM1)) -HM1 #B #A #Hb #HB #HA #HM1 destruct
+ elim (IHp … HT12) -IHp -HT12 #C #HAC #H destruct
+ @(ex2_intro … (@B.C)) // /2 width=1/ (**) (* auto needs some help here *)
+]
+qed-.
+
+lemma pl_st_lift: ∀p. sliftable (pl_st p).
+#p #h #F1 #F2 #H elim H -p -F1 -F2 /2 width=1/
+[ #V #T #d normalize <sdsubst_slift_le //
+| #b #p #V1 #V2 #T #_ #IHV12 #d
+ whd in ⊢ (??%%); <booleanized_lift /2 width=1/ (**) (* auto needs some help here *)
+]
+qed.
+
+lemma pl_st_inv_lift: ∀p. sdeliftable_sn (pl_st p).
+#p #h #G1 #G2 #H elim H -p -G1 -G2
+[ #W #U #d #F1 #H
+ elim (slift_inv_appl … H) -H #V #F #H0 #HF #H destruct
+ elim (slift_inv_abst … HF) -HF #T #H0 #H destruct /3 width=3/
+| #b #p #U1 #U2 #_ #IHU12 #d #F1 #H
+ elim (slift_inv_abst … H) -H #T1 #HTU1 #H
+ elim (IHU12 … HTU1) -U1 #T2 #HT12 #HTU2 destruct
+ @(ex2_intro … ({⊥}𝛌.T2)) // /2 width=1/
+| #b #p #W1 #W2 #U1 #_ #IHW12 #d #F1 #H
+ elim (slift_inv_appl … H) -H #V1 #T #HVW1 #H1 #H2
+ elim (IHW12 … HVW1) -W1 #V2 #HV12 #HVW2 destruct
+ @(ex2_intro … ({⊥}@V2.{⊥}⇕T)) [ /2 width=1/ ]
+ whd in ⊢ (??%%); // (**) (* auto needs some help here *)
+| #b #p #W1 #U1 #U2 #_ #IHU12 #d #F1 #H
+ elim (slift_inv_appl … H) -H #V #T1 #H1 #HTU1 #H2
+ elim (IHU12 … HTU1) -U1 #T2 #HT12 #HTU2 destruct
+ @(ex2_intro … ({b}@V.T2)) // /2 width=1/
+]
+qed-.
+
+lemma pl_st_dsubst: ∀p. sdsubstable_f_dx … (booleanized ⊥) (pl_st p).
+#p #W1 #F1 #F2 #H elim H -p -F1 -F2 /2 width=1/
+[ #W2 #T #d normalize >sdsubst_sdsubst_ge //
+| #b #p #V1 #V2 #T #_ #IHV12 #d
+ whd in ⊢ (??%%); <(booleanized_booleanized ⊥) in ⊢ (???(???%)); <booleanized_dsubst /2 width=1/ (**) (* auto needs some help here *)
+]
+qed.
+
+lemma pl_st_inv_empty: ∀p,F1,F2. F1 Ⓡ↦[p] F2 → ∀M1. {⊥}⇑M1 = F1 → ⊥.
+#p #F1 #F2 #H elim H -p -F1 -F2
+[ #V #T #M1 #H
+ elim (boolean_inv_appl … H) -H #B #A #H destruct
+| #b #p #T1 #T2 #_ #IHT12 #M1 #H
+ elim (boolean_inv_abst … H) -H /2 width=2/
+| #b #p #V1 #V2 #T #_ #IHV12 #M1 #H
+ elim (boolean_inv_appl … H) -H /2 width=2/
+| #b #p #V #T1 #T2 #_ #IHT12 #M1 #H
+ elim (boolean_inv_appl … H) -H /2 width=2/
+]
+qed-.
+
+theorem pl_st_mono: ∀p. singlevalued … (pl_st p).
+#p #F #G1 #H elim H -p -F -G1
+[ #V #T #G2 #H elim (pl_st_inv_nil … H …) -H //
+ #W #U #H #HG2 destruct //
+| #b #p #T1 #T2 #_ #IHT12 #G2 #H elim (pl_st_inv_rc … H …) -H [3: // |2: skip ] (**) (* simplify line *)
+ #c #U1 #U2 #HU12 #H #HG2 destruct /3 width=1/
+| #b #p #V1 #V2 #T #_ #IHV12 #G2 #H elim (pl_st_inv_sn … H …) -H [3: // |2: skip ] (**) (* simplify line *)
+ #c #W1 #W2 #U #HW12 #H #HG2 destruct /3 width=1/
+| #b #p #V #T1 #T2 #_ #IHT12 #G2 #H elim (pl_st_inv_dx … H …) -H [3: // |2: skip ] (**) (* simplify line *)
+ #c #W #U1 #U2 #HU12 #H #HG2 destruct /3 width=1/
+]
+qed-.
+
+theorem pl_st_fwd_sle: ∀p1,F1,F. F1 Ⓡ↦[p1] F →
+ ∀p2,F2. F Ⓡ↦[p2] F2 → p1 ≤ p2.
+#p1 #F1 #F #H elim H -p1 -F1 -F //
+[ #b #p #T1 #T #_ #IHT1 #p2 #F2 #H elim (pl_st_inv_abst … H …) -H [3: // |2,4: skip ] (**) (* simplify line *)
+ #q #T2 #HT2 #H1 #H2 destruct /3 width=2/
+| #b #p #V1 #V #T #_ #IHV1 #p2 #F2 #H elim (pl_st_inv_appl … H …) -H [7: // |2,3,4: skip ] * (**) (* simplify line *)
+ [ #U #H destruct
+ | #q #V2 #H1 #HV2 #H2 destruct /3 width=2/
+ | #q #U #_ #H elim (pl_st_inv_empty … H …) [ // | skip ] (**) (* simplify line *)
+ ]
+| #b #p #V #T1 #T #HT1 #IHT1 #p2 #F2 #H elim (pl_st_inv_appl … H …) -H [7: // |2,3,4: skip ] * (**) (* simplify line *)
+ [ #U #_ #H1 #H2 #_ -b -V -F2 -IHT1
+ elim (pl_st_fwd_abst … HT1 … H2) // -H1 * #q #H
+ elim (pl_st_inv_rc … HT1 … H) -HT1 -H #b #U1 #U2 #_ #_ #H -b -q -T1 -U1 destruct
+ | #q #V2 #H1 #_ #_ -b -F2 -T1 -T -V -V2 destruct //
+ | #q #T2 #H1 #HT2 #H2 -b -F2 -T1 -V /3 width=2/
+ ]
+]
+qed-.
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