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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "redex_pointer.ma".
16 include "multiplicity.ma".
18 (* LABELLED SEQUENTIAL REDUCTION (SINGLE STEP) ******************************)
20 (* Note: the application "(A B)" is represented by "@B.A" following:
21 F. Kamareddine and R.P. Nederpelt: "A useful λ-notation".
22 Theoretical Computer Science 155(1), Elsevier (1996), pp. 85-109.
24 inductive lsred: rpointer → relation term ≝
25 | lsred_beta : ∀A,D. lsred (◊) (@D.𝛌.A) ([⬐D]A)
26 | lsred_abst : ∀p,A,C. lsred p A C → lsred p (𝛌.A) (𝛌.C)
27 | lsred_appl_sn: ∀p,B,D,A. lsred p B D → lsred (true::p) (@B.A) (@D.A)
28 | lsred_appl_dx: ∀p,B,A,C. lsred p A C → lsred (false::p) (@B.A) (@B.C)
31 interpretation "labelled sequential reduction"
32 'SeqRed M p N = (lsred p M N).
34 (* Note: we do not use → since it is reserved by CIC *)
35 notation "hvbox( M break ⇀ [ term 46 p ] break term 46 N )"
36 non associative with precedence 45
37 for @{ 'SeqRed $M $p $N }.
39 lemma lsred_inv_vref: ∀p,M,N. M ⇀[p] N → ∀i. #i = M → ⊥.
41 [ #A #D #i #H destruct
42 | #p #A #C #_ #i #H destruct
43 | #p #B #D #A #_ #i #H destruct
44 | #p #B #A #C #_ #i #H destruct
48 lemma lsred_inv_beta: ∀p,M,N. M ⇀[p] N → ∀D,C. @D.C = M → ◊ = p →
49 ∃∃A. 𝛌.A = C & [⬐D] A = N.
51 [ #A #D #D0 #C0 #H #_ destruct /2 width=3/
52 | #p #A #C #_ #D0 #C0 #H destruct
53 | #p #B #D #A #_ #D0 #C0 #_ #H destruct
54 | #p #B #A #C #_ #D0 #C0 #_ #H destruct
58 lemma lsred_inv_abst: ∀p,M,N. M ⇀[p] N → ∀A. 𝛌.A = M →
59 ∃∃C. A ⇀[p] C & 𝛌.C = N.
61 [ #A #D #A0 #H destruct
62 | #p #A #C #HAC #A0 #H destruct /2 width=3/
63 | #p #B #D #A #_ #A0 #H destruct
64 | #p #B #A #C #_ #A0 #H destruct
68 lemma lsred_inv_appl_sn: ∀p,M,N. M ⇀[p] N → ∀B,A,q. @B.A = M → true::q = p →
69 ∃∃D. B ⇀[q] D & @D.A = N.
71 [ #A #D #B0 #A0 #p0 #_ #H destruct
72 | #p #A #C #_ #B0 #D0 #p0 #H destruct
73 | #p #B #D #A #HBD #B0 #A0 #p0 #H1 #H2 destruct /2 width=3/
74 | #p #B #A #C #_ #B0 #A0 #p0 #_ #H destruct
78 lemma lsred_inv_appl_dx: ∀p,M,N. M ⇀[p] N → ∀B,A,q. @B.A = M → false::q = p →
79 ∃∃C. A ⇀[q] C & @B.C = N.
81 [ #A #D #B0 #A0 #p0 #_ #H destruct
82 | #p #A #C #_ #B0 #D0 #p0 #H destruct
83 | #p #B #D #A #_ #B0 #A0 #p0 #_ #H destruct
84 | #p #B #A #C #HAC #B0 #A0 #p0 #H1 #H2 destruct /2 width=3/
88 lemma lsred_fwd_mult: ∀p,M,N. M ⇀[p] N → #{N} < #{M} * #{M}.
89 #p #M #N #H elim H -p -M -N
90 [ #A #D @(le_to_lt_to_lt … (#{A}*#{D})) //
91 normalize /3 width=1 by lt_minus_to_plus_r, lt_times/ (**) (* auto: too slow without trace *)
93 | #p #B #D #A #_ #IHBD
94 @(lt_to_le_to_lt … (#{B}*#{B}+#{A})) [ /2 width=1/ ] -D -p
95 | #p #B #A #C #_ #IHAC
96 @(lt_to_le_to_lt … (#{B}+#{A}*#{A})) [ /2 width=1/ ] -C -p
98 @(transitive_le … (#{B}*#{B}+#{A}*#{A})) [ /2 width=1/ ]
99 >distributive_times_plus normalize /2 width=1/
102 lemma lsred_lift: ∀p. liftable (lsred p).
103 #p #h #M1 #M2 #H elim H -p -M1 -M2 normalize /2 width=1/
104 #A #D #d <dsubst_lift_le //
107 lemma lsred_inv_lift: ∀p. deliftable (lsred p).
108 #p #h #N1 #N2 #H elim H -p -N1 -N2
110 elim (lift_inv_appl … H) -H #B #M #H0 #HM #H destruct
111 elim (lift_inv_abst … HM) -HM #A #H0 #H destruct /3 width=3/
112 | #p #C1 #C2 #_ #IHC12 #d #M1 #H
113 elim (lift_inv_abst … H) -H #A1 #H0 #H destruct
114 elim (IHC12 ???) -IHC12 [4: // |2,3: skip ] #A2 #HA12 #H destruct (**) (* simplify line *)
115 @(ex2_1_intro … (𝛌.A2)) // /2 width=1/
116 | #p #D1 #D2 #C1 #_ #IHD12 #d #M1 #H
117 elim (lift_inv_appl … H) -H #B1 #A #H1 #H2 #H destruct
118 elim (IHD12 ???) -IHD12 [4: // |2,3: skip ] #B2 #HB12 #H destruct (**) (* simplify line *)
119 @(ex2_1_intro … (@B2.A)) // /2 width=1/
120 | #p #D1 #C1 #C2 #_ #IHC12 #d #M1 #H
121 elim (lift_inv_appl … H) -H #B #A1 #H1 #H2 #H destruct
122 elim (IHC12 ???) -IHC12 [4: // |2,3: skip ] #A2 #HA12 #H destruct (**) (* simplify line *)
123 @(ex2_1_intro … (@B.A2)) // /2 width=1/
127 lemma lsred_dsubst: ∀p. dsubstable_dx (lsred p).
128 #p #D1 #M1 #M2 #H elim H -p -M1 -M2 normalize /2 width=1/
129 #A #D2 #d >dsubst_dsubst_ge //
132 theorem lsred_mono: ∀p. singlevalued … (lsred p).
133 #p #M #N1 #H elim H -p -M -N1
134 [ #A #D #N2 #H elim (lsred_inv_beta … H ????) -H [4,5: // |2,3: skip ] #A0 #H1 #H2 destruct // (**) (* simplify line *)
135 | #p #A #C #_ #IHAC #N2 #H elim (lsred_inv_abst … H ??) -H [3: // |2: skip ] #C0 #HAC #H destruct /3 width=1/ (**) (* simplify line *)
136 | #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 *)
137 | #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 *)