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4 (* ||A|| A project by Andrea Asperti *)
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15 include "lambda/terms/multiplicity.ma".
17 include "lambda/notation/relations/seqred_2.ma".
19 (* SEQUENTIAL REDUCTION (SINGLE STEP) ***************************************)
21 (* Note: the application "(A B)" is represented by "@B.A" following:
22 F. Kamareddine and R.P. Nederpelt: "A useful λ-notation".
23 Theoretical Computer Science 155(1), Elsevier (1996), pp. 85-109.
25 inductive sred: relation term ≝
26 | sred_beta : ∀B,A. sred (@B.𝛌.A) ([↙B]A)
27 | sred_abst : ∀A1,A2. sred A1 A2 → sred (𝛌.A1) (𝛌.A2)
28 | sred_appl_sn: ∀B1,B2,A. sred B1 B2 → sred (@B1.A) (@B2.A)
29 | sred_appl_dx: ∀B,A1,A2. sred A1 A2 → sred (@B.A1) (@B.A2)
32 interpretation "sequential reduction"
33 'SeqRed M N = (sred M N).
35 lemma sred_inv_vref: ∀M,N. M ↦ N → ∀i. #i = M → ⊥.
37 [ #B #A #i #H destruct
38 | #A1 #A2 #_ #i #H destruct
39 | #B1 #B2 #A #_ #i #H destruct
40 | #B #A1 #A2 #_ #i #H destruct
44 lemma sred_inv_abst: ∀M,N. M ↦ N → ∀C1. 𝛌.C1 = M →
45 ∃∃C2. C1 ↦ C2 & 𝛌.C2 = N.
47 [ #B #A #C1 #H destruct
48 | #A1 #A2 #HA12 #C1 #H destruct /2 width=3/
49 | #B1 #B2 #A #_ #C1 #H destruct
50 | #B #A1 #A2 #_ #C1 #H destruct
54 lemma sred_inv_appl: ∀M,N. M ↦ N → ∀D,C. @D.C = M →
55 ∨∨ (∃∃C0. 𝛌.C0 = C & [↙D] C0 = N)
56 | (∃∃D0. D ↦ D0 & @D0.C = N)
57 | (∃∃C0. C ↦ C0 & @D.C0 = N).
59 [ #B #A #D #C #H destruct /3 width=3/
60 | #A1 #A2 #_ #D #C #H destruct
61 | #B1 #B2 #A #HB12 #D #C #H destruct /3 width=3/
62 | #B #A1 #A2 #HA12 #D #C #H destruct /3 width=3/
66 lemma sred_fwd_mult: ∀M,N. M ↦ N → ♯{N} < ♯{M} * ♯{M}.
68 [ #B #A @(le_to_lt_to_lt … (♯{A}*♯{B})) //
69 normalize /3 width=1 by lt_minus_to_plus_r, lt_times/ (**) (* auto: too slow without trace *)
72 @(lt_to_le_to_lt … (♯{B}*♯{B}+♯{A})) [ /2 width=1/ ] -D
74 @(lt_to_le_to_lt … (♯{B}+♯{A}*♯{A})) [ /2 width=1/ ] -C
76 @(transitive_le … (♯{B}*♯{B}+♯{A}*♯{A})) [ /2 width=1/ ]
77 >distributive_times_plus normalize /2 width=1/
80 lemma sred_lift: liftable sred.
81 #h #M1 #M2 #H elim H -M1 -M2 normalize /2 width=1/
82 #B #A #d <dsubst_lift_le //
85 lemma sred_inv_lift: deliftable_sn sred.
86 #h #N1 #N2 #H elim H -N1 -N2
88 elim (lift_inv_appl … H) -H #B #M #H0 #HM #H destruct
89 elim (lift_inv_abst … HM) -HM #A #H0 #H destruct /3 width=3/
90 | #C1 #C2 #_ #IHC12 #d #M1 #H
91 elim (lift_inv_abst … H) -H #A1 #HAC1 #H
92 elim (IHC12 … HAC1) -C1 #A2 #HA12 #HAC2 destruct
93 @(ex2_intro … (𝛌.A2)) // /2 width=1/
94 | #D1 #D2 #C1 #_ #IHD12 #d #M1 #H
95 elim (lift_inv_appl … H) -H #B1 #A #HBD1 #H1 #H2
96 elim (IHD12 … HBD1) -D1 #B2 #HB12 #HBD2 destruct
97 @(ex2_intro … (@B2.A)) // /2 width=1/
98 | #D1 #C1 #C2 #_ #IHC12 #d #M1 #H
99 elim (lift_inv_appl … H) -H #B #A1 #H1 #HAC1 #H2
100 elim (IHC12 … HAC1) -C1 #A2 #HA12 #HAC2 destruct
101 @(ex2_intro … (@B.A2)) // /2 width=1/
105 lemma sred_dsubst: dsubstable_dx sred.
106 #D1 #M1 #M2 #H elim H -M1 -M2 normalize /2 width=1/
107 #D2 #A #d >dsubst_dsubst_ge //