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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "subterms/booleanize.ma".
16 include "paths/standard_order.ma".
18 (* PATH-LABELED STANDARD REDUCTION ON SUBTERMS (SINGLE STEP) ****************)
20 (* Note: this is standard reduction on marked redexes,
21 left residuals are unmarked in the reductum
23 inductive pl_st: path → relation subterms ≝
24 | pl_st_beta : ∀V,T. pl_st (◊) ({⊤}@V.{⊤}𝛌.T) ([↙V]T)
25 | pl_st_abst : ∀b,p,T1,T2. pl_st p T1 T2 → pl_st (rc::p) ({b}𝛌.T1) ({⊥}𝛌.T2)
26 | pl_st_appl_sn: ∀b,p,V1,V2,T. pl_st p V1 V2 → pl_st (sn::p) ({b}@V1.T) ({⊥}@V2.{⊥}⇕T)
27 | pl_st_appl_dx: ∀b,p,V,T1,T2. pl_st p T1 T2 → pl_st (dx::p) ({b}@V.T1) ({b}@V.T2)
30 interpretation "path-labeled standard reduction"
31 'Std F p G = (pl_st p F G).
33 notation "hvbox( F break Ⓡ ↦ [ term 46 p ] break term 46 G )"
34 non associative with precedence 45
35 for @{ 'Std $F $p $G }.
37 lemma pl_st_inv_pl_sred: ∀p,F,G. F Ⓡ↦[p] G → ⇓F ↦[p] ⇓G.
38 #p #F #G #H elim H -p -F -G // /2 width=1/
41 lemma pl_st_inv_vref: ∀p,F,G. F Ⓡ↦[p] G → ∀b,i. {b}#i = F → ⊥.
42 /3 width=5 by pl_st_inv_st, st_inv_vref/
45 lemma pl_st_inv_abst: ∀p,F,G. F Ⓡ↦[p] G → ∀b0,U1. {b0}𝛌.U1 = F →
46 ∃∃q,U2. U1 Ⓡ↦[q] U2 & rc::q = p & {⊥}𝛌.U2 = G.
48 [ #V #T #b0 #U1 #H destruct
49 | #b #p #T1 #T2 #HT12 #b0 #U1 #H destruct /2 width=5/
50 | #b #p #V1 #V2 #T #_ #b0 #U1 #H destruct
51 | #b #p #V #T1 #T2 #_ #b0 #U1 #H destruct
55 lemma pl_st_inv_appl: ∀p,F,G. F Ⓡ↦[p] G → ∀b0,W,U. {b0}@W.U = F →
56 ∨∨ (∃∃U0. ⊤ = b0 & ◊ = p & {⊤}𝛌.U0 = U & [↙W] U0 = G)
57 | (∃∃q,W0. sn::q = p & W Ⓡ↦[q] W0 & {⊥}@W0.{⊥}⇕U = G)
58 | (∃∃q,U0. dx::q = p & U Ⓡ↦[q] U0 & {b0}@W.U0 = G).
60 [ #V #T #b0 #W #U #H destruct /3 width=3/
61 | #b #p #T1 #T2 #_ #b0 #W #U #H destruct
62 | #b #p #V1 #V2 #T #HV12 #b0 #W #U #H destruct /3 width=5/
63 | #b #p #V #T1 #T2 #HT12 #b0 #W #U #H destruct /3 width=5/
67 lemma pl_st_fwd_abst: ∀p,F,G. F Ⓡ↦[p] G → ∀b0,U2. {b0}𝛌.U2 = G →
68 ◊ = p ∨ ∃q. rc::q = p.
72 | #b #p #V1 #V2 #T #_ #b0 #U2 #H destruct
73 | #b #p #V #T1 #T2 #_ #b0 #U2 #H destruct
77 lemma pl_st_inv_nil: ∀p,F,G. F Ⓡ↦[p] G → ◊ = p →
78 ∃∃V,T. {⊤}@V.{⊤} 𝛌.T = F & [↙V] T = G.
80 [ #V #T #_ destruct /2 width=4/
81 | #b #p #T1 #T2 #_ #H destruct
82 | #b #p #V1 #V2 #T #_ #H destruct
83 | #b #p #V #T1 #T2 #_ #H destruct
87 lemma pl_st_inv_rc: ∀p,F,G. F Ⓡ↦[p] G → ∀q. rc::q = p →
88 ∃∃b,T1,T2. T1 Ⓡ↦[q] T2 & {b}𝛌.T1 = F & {⊥}𝛌.T2 = G.
90 [ #V #T #q #H destruct
91 | #b #p #T1 #T2 #HT12 #q #H destruct /2 width=6/
92 | #b #p #V1 #V2 #T #_ #q #H destruct
93 | #b #p #V #T1 #T2 #_ #q #H destruct
97 lemma pl_st_inv_sn: ∀p,F,G. F Ⓡ↦[p] G → ∀q. sn::q = p →
98 ∃∃b,V1,V2,T. V1 Ⓡ↦[q] V2 & {b}@V1.T = F & {⊥}@V2.{⊥}⇕T = G.
100 [ #V #T #q #H destruct
101 | #b #p #T1 #T2 #_ #q #H destruct
102 | #b #p #V1 #V2 #T #HV12 #q #H destruct /2 width=7/
103 | #b #p #V #T1 #T2 #_ #q #H destruct
107 lemma pl_st_inv_dx: ∀p,F,G. F Ⓡ↦[p] G → ∀q. dx::q = p →
108 ∃∃b,V,T1,T2. T1 Ⓡ↦[q] T2 & {b}@V.T1 = F & {b}@V.T2 = G.
110 [ #V #T #q #H destruct
111 | #b #p #T1 #T2 #_ #q #H destruct
112 | #b #p #V1 #V2 #T #_ #q #H destruct
113 | #b #p #V #T1 #T2 #HT12 #q #H destruct /2 width=7/
120 lemma pl_st_lift: ∀p. sliftable (pl_st p).
121 #p #h #F1 #F2 #H elim H -p -F1 -F2 normalize /2 width=1/
122 [ #V #T #d <sdsubst_slift_le //
123 | #b #p #V1 #V2 #T #_ #IHV12 #d
126 lemma pl_st_inv_lift: ∀p. deliftable_sn (pl_st p).
127 #p #h #G1 #G2 #H elim H -p -G1 -G2
129 elim (lift_inv_appl … H) -H #V #F #H0 #HF #H destruct
130 elim (lift_inv_abst … HF) -HF #T #H0 #H destruct /3 width=3/
131 | #p #U1 #U2 #_ #IHU12 #d #F1 #H
132 elim (lift_inv_abst … H) -H #T1 #HTU1 #H
133 elim (IHU12 … HTU1) -U1 #T2 #HT12 #HTU2 destruct
134 @(ex2_intro … (𝛌.T2)) // /2 width=1/
135 | #p #W1 #W2 #U1 #_ #IHW12 #d #F1 #H
136 elim (lift_inv_appl … H) -H #V1 #T #HVW1 #H1 #H2
137 elim (IHW12 … HVW1) -W1 #V2 #HV12 #HVW2 destruct
138 @(ex2_intro … (@V2.T)) // /2 width=1/
139 | #p #W1 #U1 #U2 #_ #IHU12 #d #F1 #H
140 elim (lift_inv_appl … H) -H #V #T1 #H1 #HTU1 #H2
141 elim (IHU12 … HTU1) -U1 #T2 #HT12 #HTU2 destruct
142 @(ex2_intro … (@V.T2)) // /2 width=1/
146 lemma pl_st_dsubst: ∀p. sdsubstable_dx (pl_st p).
147 #p #W1 #F1 #F2 #H elim H -p -F1 -F2 normalize /2 width=1/
148 #W2 #T #d >dsubst_dsubst_ge //
152 lemma pl_st_inv_empty: ∀p,G1,G2. G1 Ⓡ↦[p] G2 → ∀F1. {⊥}⇕F1 = G1 → ⊥.
153 #p #F1 #F2 #H elim H -p -F1 -F2
155 elim (mk_boolean_inv_appl … H) -H #b0 #W #U #H destruct
156 | #b #p #T1 #T2 #_ #IHT12 #F1 #H
157 elim (mk_boolean_inv_abst … H) -H /2 width=2/
158 | #b #p #V1 #V2 #T #_ #IHV12 #F1 #H
159 elim (mk_boolean_inv_appl … H) -H /2 width=2/
160 | #b #p #V #T1 #T2 #_ #IHT12 #F1 #H
161 elim (mk_boolean_inv_appl … H) -H /2 width=2/
165 theorem pl_st_mono: ∀p. singlevalued … (pl_st p).
166 #p #F #G1 #H elim H -p -F -G1
167 [ #V #T #G2 #H elim (pl_st_inv_nil … H ?) -H //
168 #W #U #H #HG2 destruct //
169 | #b #p #T1 #T2 #_ #IHT12 #G2 #H elim (pl_st_inv_rc … H ??) -H [3: // |2: skip ] (**) (* simplify line *)
170 #b0 #U1 #U2 #HU12 #H #HG2 destruct /3 width=1/
171 | #b #p #V1 #V2 #T #_ #IHV12 #G2 #H elim (pl_st_inv_sn … H ??) -H [3: // |2: skip ] (**) (* simplify line *)
172 #b0 #W1 #W2 #U #HW12 #H #HG2 destruct /3 width=1/
173 | #b #p #V #T1 #T2 #_ #IHT12 #G2 #H elim (pl_st_inv_dx … H ??) -H [3: // |2: skip ] (**) (* simplify line *)
174 #b0 #W #U1 #U2 #HU12 #H #HG2 destruct /3 width=1/
178 theorem pl_st_inv_is_standard: ∀p1,F1,F. F1 Ⓡ↦[p1] F →
179 ∀p2,F2. F Ⓡ↦[p2] F2 → p1 ≤ p2.
180 #p1 #F1 #F #H elim H -p1 -F1 -F //
181 [ #b #p #T1 #T #_ #IHT1 #p2 #F2 #H elim (pl_st_inv_abst … H ???) -H [3: // |2,4: skip ] (**) (* simplify line *)
182 #q #T2 #HT2 #H1 #H2 destruct /3 width=2/
183 | #b #p #V1 #V #T #_ #IHV1 #p2 #F2 #H elim (pl_st_inv_appl … H ????) -H [7: // |2,3,4: skip ] * (**) (* simplify line *)
185 | #q #V2 #H1 #HV2 #H2 destruct /3 width=2/
186 | #q #U #_ #H elim (pl_st_inv_empty … H ??) [ // | skip ] (**) (* simplify line *)
188 | #b #p #V #T1 #T #HT1 #IHT1 #p2 #F2 #H elim (pl_st_inv_appl … H ????) -H [7: // |2,3,4: skip ] * (**) (* simplify line *)
189 [ #U #_ #H1 #H2 #_ -b -V -F2 -IHT1
190 elim (pl_st_fwd_abst … HT1 … H2) // -H1 * #q #H
191 elim (pl_st_inv_rc … HT1 … H) -HT1 -H #b #U1 #U2 #_ #_ #H -b -q -T1 -U1 destruct
192 | #q #V2 #H1 #_ #_ -b -F2 -T1 -T -V -V2 destruct //
193 | #q #T2 #H1 #HT2 #H2 -b -F2 -T1 -V /3 width=2/