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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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15 include "basic_2/substitution/tps.ma".
17 (* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
19 (* Basic_1: includes: pr0_delta1 *)
20 inductive tpr: relation term ≝
21 | tpr_atom : ∀I. tpr (⓪{I}) (⓪{I})
22 | tpr_flat : ∀I,V1,V2,T1,T2. tpr V1 V2 → tpr T1 T2 →
23 tpr (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
24 | tpr_beta : ∀a,V1,V2,W,T1,T2.
25 tpr V1 V2 → tpr T1 T2 → tpr (ⓐV1. ⓛ{a}W. T1) (ⓓ{a}V2. T2)
26 | tpr_delta: ∀a,I,V1,V2,T1,T,T2.
27 tpr V1 V2 → tpr T1 T → ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 →
28 tpr (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
29 | tpr_theta: ∀a,V,V1,V2,W1,W2,T1,T2.
30 tpr V1 V2 → ⇧[0,1] V2 ≡ V → tpr W1 W2 → tpr T1 T2 →
31 tpr (ⓐV1. ⓓ{a}W1. T1) (ⓓ{a}W2. ⓐV. T2)
32 | tpr_zeta : ∀V,T1,T,T2. tpr T1 T → ⇧[0, 1] T2 ≡ T → tpr (+ⓓV. T1) T2
33 | tpr_tau : ∀V,T1,T2. tpr T1 T2 → tpr (ⓝV. T1) T2
37 "context-free parallel reduction (term)"
38 'PRed T1 T2 = (tpr T1 T2).
40 (* Basic properties *********************************************************)
42 lemma tpr_bind: ∀a,I,V1,V2,T1,T2. V1 ➡ V2 → T1 ➡ T2 → ⓑ{a,I} V1. T1 ➡ ⓑ{a,I} V2. T2.
45 (* Basic_1: was by definition: pr0_refl *)
46 lemma tpr_refl: reflexive … tpr.
48 #I elim I -I /2 width=1/
51 (* Basic inversion lemmas ***************************************************)
53 fact tpr_inv_atom1_aux: ∀U1,U2. U1 ➡ U2 → ∀I. U1 = ⓪{I} → U2 = ⓪{I}.
56 | #I #V1 #V2 #T1 #T2 #_ #_ #k #H destruct
57 | #a #V1 #V2 #W #T1 #T2 #_ #_ #k #H destruct
58 | #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #k #H destruct
59 | #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #k #H destruct
60 | #V #T1 #T #T2 #_ #_ #k #H destruct
61 | #V #T1 #T2 #_ #k #H destruct
65 (* Basic_1: was: pr0_gen_sort pr0_gen_lref *)
66 lemma tpr_inv_atom1: ∀I,U2. ⓪{I} ➡ U2 → U2 = ⓪{I}.
69 fact tpr_inv_bind1_aux: ∀U1,U2. U1 ➡ U2 → ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 →
70 (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T &
71 ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 &
74 ∃∃T. T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr.
76 [ #J #a #I #V #T #H destruct
77 | #I1 #V1 #V2 #T1 #T2 #_ #_ #a #I #V #T #H destruct
78 | #b #V1 #V2 #W #T1 #T2 #_ #_ #a #I #V #T #H destruct
79 | #b #I1 #V1 #V2 #T1 #T #T2 #HV12 #HT1 #HT2 #a #I0 #V0 #T0 #H destruct /3 width=7/
80 | #b #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #a #I0 #V0 #T0 #H destruct
81 | #V #T1 #T #T2 #HT1 #HT2 #a #I0 #V0 #T0 #H destruct /3 width=3/
82 | #V #T1 #T2 #_ #a #I0 #V0 #T0 #H destruct
86 lemma tpr_inv_bind1: ∀V1,T1,U2,a,I. ⓑ{a,I} V1. T1 ➡ U2 →
87 (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T &
88 ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 &
91 ∃∃T. T1 ➡ T & ⇧[0,1] U2 ≡ T & a = true & I = Abbr.
94 (* Basic_1: was pr0_gen_abbr *)
95 lemma tpr_inv_abbr1: ∀a,V1,T1,U2. ⓓ{a}V1. T1 ➡ U2 →
96 (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T &
97 ⋆. ⓓV2 ⊢ T ▶ [0, 1] T2 &
100 ∃∃T. T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true.
102 elim (tpr_inv_bind1 … H) -H * /3 width=7/
105 fact tpr_inv_flat1_aux: ∀U1,U2. U1 ➡ U2 → ∀I,V1,U0. U1 = ⓕ{I} V1. U0 →
106 ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
108 | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
110 U2 = ⓓ{a}V2. T2 & I = Appl
111 | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
114 U2 = ⓓ{a}W2. ⓐV. T2 &
116 | (U0 ➡ U2 ∧ I = Cast).
118 [ #I #J #V #T #H destruct
119 | #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #V #T #H destruct /3 width=5/
120 | #a #V1 #V2 #W #T1 #T2 #HV12 #HT12 #J #V #T #H destruct /3 width=9/
121 | #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #J #V0 #T0 #H destruct
122 | #a #V #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HV2 #HW12 #HT12 #J #V0 #T0 #H destruct /3 width=13/
123 | #V #T1 #T #T2 #_ #_ #J #V0 #T0 #H destruct
124 | #V #T1 #T2 #HT12 #J #V0 #T0 #H destruct /3 width=1/
128 lemma tpr_inv_flat1: ∀V1,U0,U2,I. ⓕ{I} V1. U0 ➡ U2 →
129 ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
131 | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
133 U2 = ⓓ{a}V2. T2 & I = Appl
134 | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
137 U2 = ⓓ{a}W2. ⓐV. T2 &
139 | (U0 ➡ U2 ∧ I = Cast).
142 (* Basic_1: was pr0_gen_appl *)
143 lemma tpr_inv_appl1: ∀V1,U0,U2. ⓐV1. U0 ➡ U2 →
144 ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
146 | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
149 | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
154 elim (tpr_inv_flat1 … H) -H *
155 /3 width=5/ /3 width=9/ /3 width=13/
159 (* Note: the main property of simple terms *)
160 lemma tpr_inv_appl1_simple: ∀V1,T1,U. ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
161 ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 &
164 elim (tpr_inv_appl1 … H) -H *
166 | #a #V2 #W #W1 #W2 #_ #_ #H #_ destruct
167 elim (simple_inv_bind … HT1)
168 | #a #V2 #V #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
169 elim (simple_inv_bind … HT1)
173 (* Basic_1: was: pr0_gen_cast *)
174 lemma tpr_inv_cast1: ∀V1,T1,U2. ⓝV1. T1 ➡ U2 →
175 (∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓝV2. T2)
178 elim (tpr_inv_flat1 … H) -H * /3 width=5/ #a #V2 #W #W1 #W2
179 [ #_ #_ #_ #_ #H destruct
180 | #T2 #U1 #_ #_ #_ #_ #_ #_ #H destruct
184 fact tpr_inv_lref2_aux: ∀T1,T2. T1 ➡ T2 → ∀i. T2 = #i →
186 | ∃∃V,T. T ➡ #(i+1) & T1 = +ⓓV. T
187 | ∃∃V,T. T ➡ #i & T1 = ⓝV. T.
189 [ #I #i #H destruct /2 width=1/
190 | #I #V1 #V2 #T1 #T2 #_ #_ #i #H destruct
191 | #a #V1 #V2 #W #T1 #T2 #_ #_ #i #H destruct
192 | #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #i #H destruct
193 | #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #i #H destruct
194 | #V #T1 #T #T2 #HT1 #HT2 #i #H destruct
195 lapply (lift_inv_lref1_ge … HT2 ?) -HT2 // #H destruct /3 width=4/
196 | #V #T1 #T2 #HT12 #i #H destruct /3 width=4/
200 lemma tpr_inv_lref2: ∀T1,i. T1 ➡ #i →
202 | ∃∃V,T. T ➡ #(i+1) & T1 = +ⓓV. T
203 | ∃∃V,T. T ➡ #i & T1 = ⓝV. T.
206 (* Basic forward lemmas *****************************************************)
208 lemma tpr_fwd_shift1: ∀L1,T1,T. L1 @@ T1 ➡ T →
209 ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
210 #L1 @(lenv_ind_dx … L1) -L1 normalize
212 @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
213 | #I #L1 #V1 #IH #T1 #X
214 >shift_append_assoc normalize #H
215 elim (tpr_inv_bind1 … H) -H *
216 [ #V0 #T0 #X0 #_ #HT10 #H0 #H destruct
217 elim (IH … HT10) -IH -T1 #L #T #HL1 #H destruct
218 elim (tps_fwd_shift1 … H0) -T #L2 #T2 #HL2 #H destruct
219 >append_length >HL1 >HL2 -L1 -L
220 @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
221 | #T #_ #_ #H destruct
226 (* Basic_1: removed theorems 3:
227 pr0_subst0_back pr0_subst0_fwd pr0_subst0
228 Basic_1: removed local theorems: 1: pr0_delta_tau