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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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11 (* v GNU General Public License Version 2 *)
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15 include "lambda/paths/path.ma".
16 include "lambda/terms/sequential_reduction.ma".
18 (* PATH-LABELED SEQUENTIAL REDUCTION (SINGLE STEP) **************************)
20 inductive pl_sred: path → relation term ≝
21 | pl_sred_beta : ∀B,A. pl_sred (◊) (@B.𝛌.A) ([↙B]A)
22 | pl_sred_abst : ∀p,A1,A2. pl_sred p A1 A2 → pl_sred (rc::p) (𝛌.A1) (𝛌.A2)
23 | pl_sred_appl_sn: ∀p,B1,B2,A. pl_sred p B1 B2 → pl_sred (sn::p) (@B1.A) (@B2.A)
24 | pl_sred_appl_dx: ∀p,B,A1,A2. pl_sred p A1 A2 → pl_sred (dx::p) (@B.A1) (@B.A2)
27 interpretation "path-labeled sequential reduction"
28 'SeqRed M p N = (pl_sred p M N).
30 lemma sred_pl_sred: ∀M,N. M ↦ N → ∃p. M ↦[p] N.
33 | #A1 #A2 #_ * /3 width=2/
34 | #B1 #B2 #A #_ * /3 width=2/
35 | #B #A1 #A2 #_ * /3 width=2/
39 lemma pl_sred_inv_sred: ∀p,M,N. M ↦[p] N → M ↦ N.
40 #p #M #N #H elim H -p -M -N // /2 width=1/
43 lemma pl_sred_inv_vref: ∀p,M,N. M ↦[p] N → ∀i. #i = M → ⊥.
44 /3 width=5 by pl_sred_inv_sred, sred_inv_vref/
47 lemma pl_sred_inv_nil: ∀p,M,N. M ↦[p] N → ◊ = p →
48 ∃∃B,A. @B. 𝛌.A = M & [↙B] A = N.
50 [ #B #A #_ destruct /2 width=4/
51 | #p #A1 #A2 #_ #H destruct
52 | #p #B1 #B2 #A #_ #H destruct
53 | #p #B #A1 #A2 #_ #H destruct
57 lemma pl_sred_inv_rc: ∀p,M,N. M ↦[p] N → ∀q. rc::q = p →
58 ∃∃A1,A2. A1 ↦[q] A2 & 𝛌.A1 = M & 𝛌.A2 = N.
60 [ #B #A #q #H destruct
61 | #p #A1 #A2 #HA12 #q #H destruct /2 width=5/
62 | #p #B1 #B2 #A #_ #q #H destruct
63 | #p #B #A1 #A2 #_ #q #H destruct
67 lemma pl_sred_inv_sn: ∀p,M,N. M ↦[p] N → ∀q. sn::q = p →
68 ∃∃B1,B2,A. B1 ↦[q] B2 & @B1.A = M & @B2.A = N.
70 [ #B #A #q #H destruct
71 | #p #A1 #A2 #_ #q #H destruct
72 | #p #B1 #B2 #A #HB12 #q #H destruct /2 width=6/
73 | #p #B #A1 #A2 #_ #q #H destruct
77 lemma pl_sred_inv_dx: ∀p,M,N. M ↦[p] N → ∀q. dx::q = p →
78 ∃∃B,A1,A2. A1 ↦[q] A2 & @B.A1 = M & @B.A2 = N.
80 [ #B #A #q #H destruct
81 | #p #A1 #A2 #_ #q #H destruct
82 | #p #B1 #B2 #A #_ #q #H destruct
83 | #p #B #A1 #A2 #HA12 #q #H destruct /2 width=6/
87 lemma pl_sred_lift: ∀p. liftable (pl_sred p).
88 #p #h #M1 #M2 #H elim H -p -M1 -M2 normalize /2 width=1/
89 #B #A #d <dsubst_lift_le //
92 lemma pl_sred_inv_lift: ∀p. deliftable_sn (pl_sred p).
93 #p #h #N1 #N2 #H elim H -p -N1 -N2
95 elim (lift_inv_appl … H) -H #B #M #H0 #HM #H destruct
96 elim (lift_inv_abst … HM) -HM #A #H0 #H destruct /3 width=3/
97 | #p #C1 #C2 #_ #IHC12 #d #M1 #H
98 elim (lift_inv_abst … H) -H #A1 #HAC1 #H
99 elim (IHC12 … HAC1) -C1 #A2 #HA12 #HAC2 destruct
100 @(ex2_intro … (𝛌.A2)) // /2 width=1/
101 | #p #D1 #D2 #C1 #_ #IHD12 #d #M1 #H
102 elim (lift_inv_appl … H) -H #B1 #A #HBD1 #H1 #H2
103 elim (IHD12 … HBD1) -D1 #B2 #HB12 #HBD2 destruct
104 @(ex2_intro … (@B2.A)) // /2 width=1/
105 | #p #D1 #C1 #C2 #_ #IHC12 #d #M1 #H
106 elim (lift_inv_appl … H) -H #B #A1 #H1 #HAC1 #H2
107 elim (IHC12 … HAC1) -C1 #A2 #HA12 #HAC2 destruct
108 @(ex2_intro … (@B.A2)) // /2 width=1/
112 lemma pl_sred_dsubst: ∀p. dsubstable_dx (pl_sred p).
113 #p #D1 #M1 #M2 #H elim H -p -M1 -M2 normalize /2 width=1/
114 #D2 #A #d >dsubst_dsubst_ge //
117 theorem pl_sred_mono: ∀p. singlevalued … (pl_sred p).
118 #p #M #N1 #H elim H -p -M -N1
119 [ #B #A #N2 #H elim (pl_sred_inv_nil … H …) -H //
120 #D #C #H #HN2 destruct //
121 | #p #A1 #A2 #_ #IHA12 #N2 #H elim (pl_sred_inv_rc … H …) -H [3: // |2: skip ] (**) (* simplify line *)
122 #C1 #C2 #HC12 #H #HN2 destruct /3 width=1/
123 | #p #B1 #B2 #A #_ #IHB12 #N2 #H elim (pl_sred_inv_sn … H …) -H [3: // |2: skip ] (**) (* simplify line *)
124 #D1 #D2 #C #HD12 #H #HN2 destruct /3 width=1/
125 | #p #B #A1 #A2 #_ #IHA12 #N2 #H elim (pl_sred_inv_dx … H …) -H [3: // |2: skip ] (**) (* simplify line *)
126 #D #C1 #C2 #HC12 #H #HN2 destruct /3 width=1/