]
qed.
-theorem subst_subst_ge: ∀D1,D2,M,d1,d2. d1 ≤ d2 →
- [d2 ⬐ D2] [d1 ⬐ D1] M = [d1 ⬐ [d2 - d1 ⬐ D2] D1] [d2 + 1 ⬐ D2] M.
+theorem dsubst_dsubst_ge: ∀D1,D2,M,d1,d2. d1 ≤ d2 →
+ [d2 ⬐ D2] [d1 ⬐ D1] M = [d1 ⬐ [d2 - d1 ⬐ D2] D1] [d2 + 1 ⬐ D2] M.
#D1 #D2 #M elim M -M
[ #i #d1 #d2 #Hd12 elim (lt_or_eq_or_gt i d1) #Hid1
[ lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
]
qed.
-theorem subst_subst_lt: ∀D1,D2,M,d1,d2. d2 < d1 →
- [d2 ⬐ [d1 - d2 -1 ⬐ D1] D2] [d1 ⬐ D1] M = [d1 - 1 ⬐ D1] [d2 ⬐ D2] M.
+theorem dsubst_dsubst_lt: ∀D1,D2,M,d1,d2. d2 < d1 →
+ [d2 ⬐ [d1 - d2 -1 ⬐ D1] D2] [d1 ⬐ D1] M = [d1 - 1 ⬐ D1] [d2 ⬐ D2] M.
#D1 #D2 #M #d1 #d2 #Hd21
lapply (ltn_to_ltO … Hd21) #Hd1
->
subst_subst_ge in ⊢ (???%); /2 width=1/ <plus_minus_m_m //
+>
dsubst_dsubst_ge in ⊢ (???%); /2 width=1/ <plus_minus_m_m //
qed.
+
+definition dsubstable_dx: predicate (relation term) ≝ λR.
+ ∀D,M1,M2. R M1 M2 → ∀d. R ([d ⬐ D] M1) ([d ⬐ D] M2).
--- /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 "redex_pointer_sequence.ma".
+include "labelled_sequential_reduction.ma".
+
+(* LABELLED SEQUENTIAL COMPUTATION (MULTISTEP) ******************************)
+
+(* Note: this comes from "star term lsred" *)
+inductive lsreds: rpseq → relation term ≝
+| lsreds_nil : ∀M. lsreds (◊) M M
+| lsreds_cons: ∀p,M1,M. M1 ⇀[p] M →
+ ∀s,M2. lsreds s M M2 → lsreds (p::s) M1 M2
+.
+
+interpretation "labelled sequential computation"
+ 'SeqRedStar M s N = (lsreds s M N).
+
+notation "hvbox( M break ⇀* [ term 46 s ] break term 46 N )"
+ non associative with precedence 45
+ for @{ 'SeqRedStar $M $s $N }.
+
+lemma lsred_lsreds: ∀p,M1,M2. M1 ⇀[p] M2 → M1 ⇀*[p::◊] M2.
+/2 width=3/
+qed.
+
+lemma lsreds_inv_nil: ∀s,M1,M2. M1 ⇀*[s] M2 → ◊ = s → M1 = M2.
+#s #M1 #M2 * -s -M1 -M2 //
+#p #M1 #M #_ #s #M2 #_ #H destruct
+qed-.
+
+lemma lsreds_inv_cons: ∀s,M1,M2. M1 ⇀*[s] M2 → ∀q,r. q::r = s →
+ ∃∃M. M1 ⇀[q] M & M ⇀*[r] M2.
+#s #M1 #M2 * -s -M1 -M2
+[ #M #q #r #H destruct
+| #p #M1 #M #HM1 #s #M2 #HM2 #q #r #H destruct /2 width=3/
+]
+qed-.
+
+lemma lsreds_inv_lsred: ∀p,M1,M2. M1 ⇀*[p::◊] M2 → M1 ⇀[p] M2.
+#p #M1 #M2 #H
+elim (lsreds_inv_cons … H ???) -H [4: // |2,3: skip ] #M #HM1 #H (**) (* simplify line *)
+<(lsreds_inv_nil … H ?) -H //
+qed-.
+
+(* Note: "|s|" should be unparetesized *)
+lemma lsreds_fwd_mult: ∀s,M1,M2. M1 ⇀*[s] M2 → #{M2} ≤ #{M1} ^ (2 ^ (|s|)).
+#s #M1 #M2 #H elim H -s -M1 -M2 normalize //
+#p #M1 #M #HM1 #s #M2 #_ #IHM2
+lapply (lsred_fwd_mult … HM1) -p #HM1
+@(transitive_le … IHM2) -M2
+/3 width=1 by le_exp1, lt_O_exp, lt_to_le/ (**) (* auto: slow without trace *)
+qed-.
+
+lemma lsreds_lift: ∀s. liftable (lsreds s).
+#s #h #M1 #M2 #H elim H -s -M1 -M2 // /3 width=3/
+qed.
+
+lemma lsreds_inv_lift: ∀s. deliftable (lsreds s).
+#s #h #N1 #N2 #H elim H -s -N1 -N2 /2 width=3/
+#p #N1 #N #HN1 #s #N2 #_ #IHN2 #d #M1 #HMN1
+elim (lsred_inv_lift … HN1 … HMN1) -N1 #M #HM1 #HMN
+elim (IHN2 … HMN) -N /3 width=3/
+qed-.
+
+lemma lsreds_dsubst: ∀s. dsubstable_dx (lsreds s).
+#s #D #M1 #M2 #H elim H -s -M1 -M2 // /3 width=3/
+qed.
+
+theorem lsreds_mono: ∀s. singlevalued … (lsreds s).
+#s #M #N1 #H elim H -s -M -N1
+[ /2 width=3 by lsreds_inv_nil/
+| #p #M #M1 #HM1 #s #N1 #_ #IHMN1 #N2 #H
+ elim (lsreds_inv_cons … H ???) -H [4: // |2,3: skip ] #M2 #HM2 #HMN2 (**) (* simplify line *)
+ lapply (lsred_mono … HM1 … HM2) -M #H destruct /2 width=1/
+]
+qed-.
+
+theorem lsreds_trans: ∀s1,M1,M. M1 ⇀*[s1] M →
+ ∀s2,M2. M ⇀*[s2] M2 → M1 ⇀*[s1@s2] M2.
+#s1 #M1 #M #H elim H -s1 -M1 -M normalize // /3 width=3/
+qed-.
include "redex_pointer.ma".
include "multiplicity.ma".
-(* LABELLED SEQUENTIAL REDUCTION (ONE STEP) *********************************)
+(* LABELLED SEQUENTIAL REDUCTION (SINGLE STEP) ******************************)
(* Note: the application "(A B)" is represented by "@B.A" following:
F. Kamareddine and R.P. Nederpelt: "A useful λ-notation".
.
interpretation "labelled sequential reduction"
- 'LablSeqRed M p N = (lsred p M N).
+ 'SeqRed M p N = (lsred p M N).
(* Note: we do not use → since it is reserved by CIC *)
notation "hvbox( M break ⇀ [ term 46 p ] break term 46 N )"
non associative with precedence 45
- for @{ 'LablSeqRed $M $p $N }.
+ for @{ 'SeqRed $M $p $N }.
-theorem lsred_fwd_mult: ∀p,M,N. M ⇀[p] N → #{N} < #{M} * #{M}.
+lemma lsred_inv_vref: ∀p,M,N. M ⇀[p] N → ∀i. #i = M → ⊥.
+#p #M #N * -p -M -N
+[ #A #D #i #H destruct
+| #p #A #C #_ #i #H destruct
+| #p #B #D #A #_ #i #H destruct
+| #p #B #A #C #_ #i #H destruct
+]
+qed-.
+
+lemma lsred_inv_beta: ∀p,M,N. M ⇀[p] N → ∀D,C. @D.C = M → ◊ = p →
+ ∃∃A. 𝛌.A = C & [⬐D] A = N.
+#p #M #N * -p -M -N
+[ #A #D #D0 #C0 #H #_ destruct /2 width=3/
+| #p #A #C #_ #D0 #C0 #H destruct
+| #p #B #D #A #_ #D0 #C0 #_ #H destruct
+| #p #B #A #C #_ #D0 #C0 #_ #H destruct
+]
+qed-.
+
+lemma lsred_inv_abst: ∀p,M,N. M ⇀[p] N → ∀A. 𝛌.A = M →
+ ∃∃C. A ⇀[p] C & 𝛌.C = N.
+#p #M #N * -p -M -N
+[ #A #D #A0 #H destruct
+| #p #A #C #HAC #A0 #H destruct /2 width=3/
+| #p #B #D #A #_ #A0 #H destruct
+| #p #B #A #C #_ #A0 #H destruct
+]
+qed-.
+
+lemma lsred_inv_appl_sn: ∀p,M,N. M ⇀[p] N → ∀B,A,q. @B.A = M → true::q = p →
+ ∃∃D. B ⇀[q] D & @D.A = N.
+#p #M #N * -p -M -N
+[ #A #D #B0 #A0 #p0 #_ #H destruct
+| #p #A #C #_ #B0 #D0 #p0 #H destruct
+| #p #B #D #A #HBD #B0 #A0 #p0 #H1 #H2 destruct /2 width=3/
+| #p #B #A #C #_ #B0 #A0 #p0 #_ #H destruct
+]
+qed-.
+
+lemma lsred_inv_appl_dx: ∀p,M,N. M ⇀[p] N → ∀B,A,q. @B.A = M → false::q = p →
+ ∃∃C. A ⇀[q] C & @B.C = N.
+#p #M #N * -p -M -N
+[ #A #D #B0 #A0 #p0 #_ #H destruct
+| #p #A #C #_ #B0 #D0 #p0 #H destruct
+| #p #B #D #A #_ #B0 #A0 #p0 #_ #H destruct
+| #p #B #A #C #HAC #B0 #A0 #p0 #H1 #H2 destruct /2 width=3/
+]
+qed-.
+
+lemma lsred_fwd_mult: ∀p,M,N. M ⇀[p] N → #{N} < #{M} * #{M}.
#p #M #N #H elim H -p -M -N
[ #A #D @(le_to_lt_to_lt … (#{A}*#{D})) //
normalize /3 width=1 by lt_minus_to_plus_r, lt_times/ (**) (* auto: too slow without trace *)
@(transitive_le … (#{B}*#{B}+#{A}*#{A})) [ /2 width=1/ ]
>distributive_times_plus normalize /2 width=1/
qed-.
+
+lemma lsred_lift: ∀p. liftable (lsred p).
+#p #h #M1 #M2 #H elim H -p -M1 -M2 normalize /2 width=1/
+#A #D #d <dsubst_lift_le //
+qed.
+
+lemma lsred_inv_lift: ∀p. deliftable (lsred p).
+#p #h #N1 #N2 #H elim H -p -N1 -N2
+[ #C #D #d #M1 #H
+ elim (lift_inv_appl … H) -H #B #M #H0 #HM #H destruct
+ elim (lift_inv_abst … HM) -HM #A #H0 #H destruct /3 width=3/
+| #p #C1 #C2 #_ #IHC12 #d #M1 #H
+ elim (lift_inv_abst … H) -H #A1 #H0 #H destruct
+ elim (IHC12 ???) -IHC12 [4: // |2,3: skip ] #A2 #HA12 #H destruct (**) (* simplify line *)
+ @(ex2_1_intro … (𝛌.A2)) // /2 width=1/
+| #p #D1 #D2 #C1 #_ #IHD12 #d #M1 #H
+ elim (lift_inv_appl … H) -H #B1 #A #H1 #H2 #H destruct
+ elim (IHD12 ???) -IHD12 [4: // |2,3: skip ] #B2 #HB12 #H destruct (**) (* simplify line *)
+ @(ex2_1_intro … (@B2.A)) // /2 width=1/
+| #p #D1 #C1 #C2 #_ #IHC12 #d #M1 #H
+ elim (lift_inv_appl … H) -H #B #A1 #H1 #H2 #H destruct
+ elim (IHC12 ???) -IHC12 [4: // |2,3: skip ] #A2 #HA12 #H destruct (**) (* simplify line *)
+ @(ex2_1_intro … (@B.A2)) // /2 width=1/
+]
+qed-.
+
+lemma lsred_dsubst: ∀p. dsubstable_dx (lsred p).
+#p #D1 #M1 #M2 #H elim H -p -M1 -M2 normalize /2 width=1/
+#A #D2 #d >dsubst_dsubst_ge //
+qed.
+
+theorem lsred_mono: ∀p. singlevalued … (lsred p).
+#p #M #N1 #H elim H -p -M -N1
+[ #A #D #N2 #H elim (lsred_inv_beta … H ????) -H [4,5: // |2,3: skip ] #A0 #H1 #H2 destruct // (**) (* simplify line *)
+| #p #A #C #_ #IHAC #N2 #H elim (lsred_inv_abst … H ??) -H [3: // |2: skip ] #C0 #HAC #H destruct /3 width=1/ (**) (* simplify line *)
+| #p #B #D #A #_ #IHBD #N2 #H elim (lsred_inv_appl_sn … H ?????) -H [5,6: // |2,3,4: skip ] #D0 #HBD #H destruct /3 width=1/ (**) (* simplify line *)
+| #p #B #A #C #_ #IHAC #N2 #H elim (lsred_inv_appl_dx … H ?????) -H [5,6: // |2,3,4: skip ] #C0 #HAC #H destruct /3 width=1/ (**) (* simplify line *)
+]
+qed-.
lapply (sym_eq term … H) -H >(plus_minus_m_m … Hh1d2) in ⊢ (???%→?); -Hh1d2 #H
elim (lift_inv_lift_le … Hd12 … H) -H -Hd12 /2 width=3/
qed-.
+
+definition liftable: predicate (relation term) ≝ λR.
+ ∀h,M1,M2. R M1 M2 → ∀d. R (↑[d, h] M1) (↑[d, h] M2).
+
+definition deliftable: predicate (relation term) ≝ λR.
+ ∀h,N1,N2. R N1 N2 → ∀d,M1. ↑[d, h] M1 = N1 →
+ ∃∃M2. R M1 M2 & ↑[d, h] M2 = N2.
--- /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 "multiplicity.ma".
+
+(* PARALLEL REDUCTION (SINGLE STEP) *****************************************)
+
+(* Note: the application "(A B)" is represented by "@B.A"
+ as for labelled sequential reduction
+*)
+inductive pred: relation term ≝
+| pred_vref: ∀i. pred (#i) (#i)
+| pred_abst: ∀A,C. pred A C → pred (𝛌.A) (𝛌.C)
+| pred_appl: ∀B,D,A,C. pred B D → pred A C → pred (@B.A) (@D.C)
+| pred_beta: ∀B,D,A,C. pred B D → pred A C → pred (@B.𝛌.A) ([⬐D]C)
+.
+
+interpretation "parallel reduction"
+ 'ParRed M N = (pred M N).
+
+notation "hvbox( M break ⥤ break term 46 N )"
+ non associative with precedence 45
+ for @{ 'ParRed $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 //
+[ #A #C #_ #i #H destruct
+| #B #D #A #C #_ #_ #i #H destruct
+| #B #D #A #C #_ #_ #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
+| #A #C #HAC #A0 #H destruct /2 width=3/
+| #B #D #A #C #_ #_ #A0 #H destruct
+| #B #D #A #C #_ #_ #A0 #H destruct
+]
+qed-.
+
+lemma pred_lift: liftable pred.
+#h #M1 #M2 #H elim H -M1 -M2 normalize // /2 width=1/
+#D #D #A #C #_ #_ #IHBD #IHAC #d <dsubst_lift_le // /2 width=1/
+qed.
+
+lemma pred_inv_lift: deliftable 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_1_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_1_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_1_intro … ([⬐B2]A2)) /2 width=1/
+]
+qed-.
--- /dev/null
+NAMING CONVENTIONS FOR METAVARIABLES
+
+A, B, C, D: term
+H : transient premise
+IH : inductive premise
+M, N : term
+
+a, b : boolean
+d, e : variable reference depth
+h : relocation height
+i, j : de Bruijn index
+k : relocation height
+p, q : redex pointer
+r, s : redex pointer sequence
include "basics/star.ma".
include "basics/lists/list.ma".
-include "arithmetics/nat.ma".
+include "arithmetics/exp.ma".
include "xoa_notation.ma".
include "xoa.ma".
interpretation "'less than' on redex pointers"
'lt p q = (TC rpointer rpprec p q).
-(* Note: this is p ≤ q, that *really* is p < q ∨ p = q *)
+(* Note: this is p ≤ q *)
interpretation "'less or equal to' on redex pointers"
- 'leq x y = (RC rpointer (TC rpointer rpprec) x y).
+ 'leq x y = (star rpointer x y).
--- /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 "redex_pointer.ma".
+
+(* REDEX POINTER SEQUENCE ***************************************************)
+
+(* Policy: pointer sequence metavariables: r, s *)
+
+definition rpseq: Type[0] \def list rpointer.
[ nil ⇒ 0
| cons a tl ⇒ S (length A tl)].
-notation "|M|" non associative with precedence 65 for @{'norm $M}.
-interpretation "norm" 'norm l = (length ? l).
+interpretation "list length" 'card l = (length ? l).
lemma length_tail: ∀A,l. length ? (tail A l) = pred (length ? l).
#A #l elim l //
definition antisymmetric: ∀A.∀R:relation A.Prop
≝ λA.λR.∀x,y:A. R x y → ¬(R y x).
+definition singlevalued: ∀A,B. predicate (relation2 A B) ≝ λA,B,R.
+ ∀a,b1. R a b1 → ∀b2. R a b2 → b1 = b2.
+
(* Reflexive closure ************)
definition RC: ∀A:Type[0]. relation A → relation A ≝
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