--- /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 "terms/size.ma".
+include "terms/sequential_reduction.ma".
+
+(* PARALLEL REDUCTION (SINGLE STEP) *****************************************)
+
+(* Note: the application "(A B)" is represented by "@B.A"
+ as for sequential reduction
+*)
+inductive pred: relation term ≝
+| pred_vref: ∀i. pred (#i) (#i)
+| pred_abst: ∀A1,A2. pred A1 A2 → pred (𝛌.A1) (𝛌.A2)
+| pred_appl: ∀B1,B2,A1,A2. pred B1 B2 → pred A1 A2 → pred (@B1.A1) (@B2.A2)
+| pred_beta: ∀B1,B2,A1,A2. pred B1 B2 → pred A1 A2 → pred (@B1.𝛌.A1) ([↙B2]A2)
+.
+
+interpretation "parallel reduction"
+ 'ParRed M N = (pred M N).
+
+lemma pred_refl: reflexive … pred.
+#M elim M -M // /2 width=1/
+qed.
+
+lemma pred_inv_vref: ∀M,N. M ⤇ N → ∀i. #i = M → #i = N.
+#M #N * -M -N //
+[ #A1 #A2 #_ #i #H destruct
+| #B1 #B2 #A1 #A2 #_ #_ #i #H destruct
+| #B1 #B2 #A1 #A2 #_ #_ #i #H destruct
+]
+qed-.
+
+lemma pred_inv_abst: ∀M,N. M ⤇ N → ∀A. 𝛌.A = M →
+ ∃∃C. A ⤇ C & 𝛌.C = N.
+#M #N * -M -N
+[ #i #A0 #H destruct
+| #A1 #A2 #HA12 #A0 #H destruct /2 width=3/
+| #B1 #B2 #A1 #A2 #_ #_ #A0 #H destruct
+| #B1 #B2 #A1 #A2 #_ #_ #A0 #H destruct
+]
+qed-.
+
+lemma pred_inv_appl: ∀M,N. M ⤇ N → ∀B,A. @B.A = M →
+ (∃∃D,C. B ⤇ D & A ⤇ C & @D.C = N) ∨
+ ∃∃A0,D,C0. B ⤇ D & A0 ⤇ C0 & 𝛌.A0 = A & [↙D]C0 = N.
+#M #N * -M -N
+[ #i #B0 #A0 #H destruct
+| #A1 #A2 #_ #B0 #A0 #H destruct
+| #B1 #B2 #A1 #A2 #HB12 #HA12 #B0 #A0 #H destruct /3 width=5/
+| #B1 #B2 #A1 #A2 #HB12 #HA12 #B0 #A0 #H destruct /3 width=7/
+]
+qed-.
+
+lemma pred_lift: liftable pred.
+#h #M1 #M2 #H elim H -M1 -M2 normalize // /2 width=1/
+#B1 #B2 #A1 #A2 #_ #_ #IHB12 #IHC12 #d <dsubst_lift_le // /2 width=1/
+qed.
+
+lemma pred_inv_lift: deliftable_sn pred.
+#h #N1 #N2 #H elim H -N1 -N2 /2 width=3/
+[ #C1 #C2 #_ #IHC12 #d #M1 #H
+ elim (lift_inv_abst … H) -H #A1 #HAC1 #H
+ elim (IHC12 … HAC1) -C1 #A2 #HA12 #HAC2 destruct
+ @(ex2_intro … (𝛌.A2)) // /2 width=1/
+| #D1 #D2 #C1 #C2 #_ #_ #IHD12 #IHC12 #d #M1 #H
+ elim (lift_inv_appl … H) -H #B1 #A1 #HBD1 #HAC1 #H
+ elim (IHD12 … HBD1) -D1 #B2 #HB12 #HBD2
+ elim (IHC12 … HAC1) -C1 #A2 #HA12 #HAC2 destruct
+ @(ex2_intro … (@B2.A2)) // /2 width=1/
+| #D1 #D2 #C1 #C2 #_ #_ #IHD12 #IHC12 #d #M1 #H
+ elim (lift_inv_appl … H) -H #B1 #M #HBD1 #HM #H1
+ elim (lift_inv_abst … HM) -HM #A1 #HAC1 #H
+ elim (IHD12 … HBD1) -D1 #B2 #HB12 #HBD2
+ elim (IHC12 … HAC1) -C1 #A2 #HA12 #HAC2 destruct
+ @(ex2_intro … ([↙B2]A2)) /2 width=1/
+]
+qed-.
+
+lemma pred_dsubst: dsubstable pred.
+#N1 #N2 #HN12 #M1 #M2 #H elim H -M1 -M2
+[ #i #d elim (lt_or_eq_or_gt i d) #Hid
+ [ >(dsubst_vref_lt … Hid) >(dsubst_vref_lt … Hid) //
+ | destruct >dsubst_vref_eq >dsubst_vref_eq /2 width=1/
+ | >(dsubst_vref_gt … Hid) >(dsubst_vref_gt … Hid) //
+ ]
+| normalize /2 width=1/
+| normalize /2 width=1/
+| normalize #B1 #B2 #A1 #A2 #_ #_ #IHB12 #IHC12 #d
+ >dsubst_dsubst_ge // /2 width=1/
+]
+qed.
+
+lemma pred_conf1_vref: ∀i. confluent1 … pred (#i).
+#i #M1 #H1 #M2 #H2
+<(pred_inv_vref … H1) -H1 [3: // |2: skip ] (**) (* simplify line *)
+<(pred_inv_vref … H2) -H2 [3: // |2: skip ] (**) (* simplify line *)
+/2 width=3/
+qed-.
+
+lemma pred_conf1_abst: ∀A. confluent1 … pred A → confluent1 … pred (𝛌.A).
+#A #IH #M1 #H1 #M2 #H2
+elim (pred_inv_abst … H1 …) -H1 [3: // |2: skip ] #A1 #HA1 #H destruct (**) (* simplify line *)
+elim (pred_inv_abst … H2 …) -H2 [3: // |2: skip ] #A2 #HA2 #H destruct (**) (* simplify line *)
+elim (IH … HA1 … HA2) -A /3 width=3/
+qed-.
+
+lemma pred_conf1_appl_beta: ∀B,B1,B2,C,C2,M1.
+ (∀M0. |M0| < |B|+|𝛌.C|+1 → confluent1 ? pred M0) → (**) (* ? needed in place of … *)
+ B ⤇ B1 → B ⤇ B2 → 𝛌.C ⤇ M1 → C ⤇ C2 →
+ ∃∃M. @B1.M1 ⤇ M & [↙B2]C2 ⤇ M.
+#B #B1 #B2 #C #C2 #M1 #IH #HB1 #HB2 #H1 #HC2
+elim (pred_inv_abst … H1 …) -H1 [3: // |2: skip ] #C1 #HC1 #H destruct (**) (* simplify line *)
+elim (IH B … HB1 … HB2) -HB1 -HB2 //
+elim (IH C … HC1 … HC2) normalize // -B -C /3 width=5/
+qed-.
+
+theorem pred_conf: confluent … pred.
+#M @(f_ind … size … M) -M #n #IH * normalize
+[ /2 width=3 by pred_conf1_vref/
+| /3 width=4 by pred_conf1_abst/
+| #B #A #H #M1 #H1 #M2 #H2 destruct
+ elim (pred_inv_appl … H1 …) -H1 [5: // |2,3: skip ] * (**) (* simplify line *)
+ elim (pred_inv_appl … H2 …) -H2 [5,10: // |2,3,7,8: skip ] * (**) (* simplify line *)
+ [ #B2 #A2 #HB2 #HA2 #H2 #B1 #A1 #HB1 #HA1 #H1 destruct
+ elim (IH A … HA1 … HA2) -HA1 -HA2 //
+ elim (IH B … HB1 … HB2) // -A -B /3 width=5/
+ | #C #B2 #C2 #HB2 #HC2 #H2 #HM2 #B1 #N #HB1 #H #HM1 destruct
+ @(pred_conf1_appl_beta … IH) // (**) (* /2 width=7 by pred_conf1_appl_beta/ does not work *)
+ | #B2 #N #B2 #H #HM2 #C #B1 #C1 #HB1 #HC1 #H1 #HM1 destruct
+ @ex2_commute @(pred_conf1_appl_beta … IH) //
+ | #C #B2 #C2 #HB2 #HC2 #H2 #HM2 #C0 #B1 #C1 #HB1 #HC1 #H1 #HM1 destruct
+ elim (IH B … HB1 … HB2) -HB1 -HB2 //
+ elim (IH C … HC1 … HC2) normalize // -B -C /3 width=5/
+ ]
+]
+qed-.
+
+lemma sred_pred: ∀M,N. M ↦ N → M ⤇ N.
+#M #N #H elim H -M -N /2 width=1/
+qed.
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