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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 *)
10 (* \ / This file is distributed under the terms of the *)
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 include "lambda/notation/relations/seqred_3.ma".
20 include "lambda/xoa/ex_2_2.ma".
22 (* PATH-LABELED SEQUENTIAL REDUCTION (SINGLE STEP) **************************)
24 inductive pl_sred: path → relation term ≝
25 | pl_sred_beta : ∀B,A. pl_sred (◊) (@B.𝛌.A) ([↙B]A)
26 | pl_sred_abst : ∀p,A1,A2. pl_sred p A1 A2 → pl_sred (rc::p) (𝛌.A1) (𝛌.A2)
27 | pl_sred_appl_sn: ∀p,B1,B2,A. pl_sred p B1 B2 → pl_sred (sn::p) (@B1.A) (@B2.A)
28 | pl_sred_appl_dx: ∀p,B,A1,A2. pl_sred p A1 A2 → pl_sred (dx::p) (@B.A1) (@B.A2)
31 interpretation "path-labeled sequential reduction"
32 'SeqRed M p N = (pl_sred p M N).
34 lemma sred_pl_sred: ∀M,N. M ↦ N → ∃p. M ↦[p] N.
37 | #A1 #A2 #_ * /3 width=2/
38 | #B1 #B2 #A #_ * /3 width=2/
39 | #B #A1 #A2 #_ * /3 width=2/
43 lemma pl_sred_inv_sred: ∀p,M,N. M ↦[p] N → M ↦ N.
44 #p #M #N #H elim H -p -M -N // /2 width=1/
47 lemma pl_sred_inv_vref: ∀p,M,N. M ↦[p] N → ∀i. #i = M → ⊥.
48 /3 width=5 by pl_sred_inv_sred, sred_inv_vref/
51 lemma pl_sred_inv_nil: ∀p,M,N. M ↦[p] N → ◊ = p →
52 ∃∃B,A. @B. 𝛌.A = M & [↙B] A = N.
54 [ #B #A #_ destruct /2 width=4/
55 | #p #A1 #A2 #_ #H destruct
56 | #p #B1 #B2 #A #_ #H destruct
57 | #p #B #A1 #A2 #_ #H destruct
61 lemma pl_sred_inv_rc: ∀p,M,N. M ↦[p] N → ∀q. rc::q = p →
62 ∃∃A1,A2. A1 ↦[q] A2 & 𝛌.A1 = M & 𝛌.A2 = N.
64 [ #B #A #q #H destruct
65 | #p #A1 #A2 #HA12 #q #H destruct /2 width=5/
66 | #p #B1 #B2 #A #_ #q #H destruct
67 | #p #B #A1 #A2 #_ #q #H destruct
71 lemma pl_sred_inv_sn: ∀p,M,N. M ↦[p] N → ∀q. sn::q = p →
72 ∃∃B1,B2,A. B1 ↦[q] B2 & @B1.A = M & @B2.A = N.
74 [ #B #A #q #H destruct
75 | #p #A1 #A2 #_ #q #H destruct
76 | #p #B1 #B2 #A #HB12 #q #H destruct /2 width=6/
77 | #p #B #A1 #A2 #_ #q #H destruct
81 lemma pl_sred_inv_dx: ∀p,M,N. M ↦[p] N → ∀q. dx::q = p →
82 ∃∃B,A1,A2. A1 ↦[q] A2 & @B.A1 = M & @B.A2 = N.
84 [ #B #A #q #H destruct
85 | #p #A1 #A2 #_ #q #H destruct
86 | #p #B1 #B2 #A #_ #q #H destruct
87 | #p #B #A1 #A2 #HA12 #q #H destruct /2 width=6/
91 lemma pl_sred_lift: ∀p. liftable (pl_sred p).
92 #p #h #M1 #M2 #H elim H -p -M1 -M2 normalize /2 width=1/
93 #B #A #d <dsubst_lift_le //
96 lemma pl_sred_inv_lift: ∀p. deliftable_sn (pl_sred p).
97 #p #h #N1 #N2 #H elim H -p -N1 -N2
99 elim (lift_inv_appl … H) -H #B #M #H0 #HM #H destruct
100 elim (lift_inv_abst … HM) -HM #A #H0 #H destruct /3 width=3/
101 | #p #C1 #C2 #_ #IHC12 #d #M1 #H
102 elim (lift_inv_abst … H) -H #A1 #HAC1 #H
103 elim (IHC12 … HAC1) -C1 #A2 #HA12 #HAC2 destruct
104 @(ex2_intro … (𝛌.A2)) // /2 width=1/
105 | #p #D1 #D2 #C1 #_ #IHD12 #d #M1 #H
106 elim (lift_inv_appl … H) -H #B1 #A #HBD1 #H1 #H2
107 elim (IHD12 … HBD1) -D1 #B2 #HB12 #HBD2 destruct
108 @(ex2_intro … (@B2.A)) // /2 width=1/
109 | #p #D1 #C1 #C2 #_ #IHC12 #d #M1 #H
110 elim (lift_inv_appl … H) -H #B #A1 #H1 #HAC1 #H2
111 elim (IHC12 … HAC1) -C1 #A2 #HA12 #HAC2 destruct
112 @(ex2_intro … (@B.A2)) // /2 width=1/
116 lemma pl_sred_dsubst: ∀p. dsubstable_dx (pl_sred p).
117 #p #D1 #M1 #M2 #H elim H -p -M1 -M2 normalize /2 width=1/
118 #D2 #A #d >dsubst_dsubst_ge //
121 theorem pl_sred_mono: ∀p. singlevalued … (pl_sred p).
122 #p #M #N1 #H elim H -p -M -N1
123 [ #B #A #N2 #H elim (pl_sred_inv_nil … H …) -H //
124 #D #C #H #HN2 destruct //
125 | #p #A1 #A2 #_ #IHA12 #N2 #H elim (pl_sred_inv_rc … H …) -H [3: // |2: skip ] (**) (* simplify line *)
126 #C1 #C2 #HC12 #H #HN2 destruct /3 width=1/
127 | #p #B1 #B2 #A #_ #IHB12 #N2 #H elim (pl_sred_inv_sn … H …) -H [3: // |2: skip ] (**) (* simplify line *)
128 #D1 #D2 #C #HD12 #H #HN2 destruct /3 width=1/
129 | #p #B #A1 #A2 #_ #IHA12 #N2 #H elim (pl_sred_inv_dx … H …) -H [3: // |2: skip ] (**) (* simplify line *)
130 #D #C1 #C2 #HC12 #H #HN2 destruct /3 width=1/