1 (**************************************************************************)
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 *)
13 (**************************************************************************)
15 include "ground_2/insert_eq/insert_eq_0.ma".
16 include "basic_2/rt_transition/cpm.ma".
18 (* CONTEXT-SENSITIVE PARALLEL R-TRANSITION FOR TERMS ************************)
20 (* Basic properties *********************************************************)
22 (* Note: cpr_flat: does not hold in basic_1 *)
23 (* Basic_1: includes: pr2_thin_dx *)
24 lemma cpr_flat: ∀h,I,G,L,V1,V2,T1,T2.
25 ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[h] T2 →
26 ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡[h] ⓕ{I}V2.T2.
27 #h * /2 width=1 by cpm_cast, cpm_appl/
30 (* Basic_1: was: pr2_head_1 *)
31 lemma cpr_pair_sn: ∀h,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
32 ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡[h] ②{I}V2.T.
33 #h * /2 width=1 by cpm_bind, cpr_flat/
36 (* Basic inversion properties ***********************************************)
38 lemma cpr_inv_atom1: ∀h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[h] T2 →
40 | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 & ⬆*[1] V2 ≘ T2 &
41 L = K.ⓓV1 & J = LRef 0
42 | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ➡[h] T & ⬆*[1] T ≘ T2 &
43 L = K.ⓘ{I} & J = LRef (↑i).
44 #h #J #G #L #T2 #H elim (cpm_inv_atom1 … H) -H *
45 [2,4:|*: /3 width=8 by or3_intro0, or3_intro1, or3_intro2, ex4_4_intro, ex4_3_intro/ ]
46 [ #n #_ #_ #H destruct
47 | #n #K #V1 #V2 #_ #_ #_ #_ #H destruct
51 (* Basic_1: includes: pr0_gen_sort pr2_gen_sort *)
52 lemma cpr_inv_sort1: ∀h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡[h] T2 → T2 = ⋆s.
53 #h #G #L #T2 #s #H elim (cpm_inv_sort1 … H) -H //
56 lemma cpr_inv_zero1: ∀h,G,L,T2. ⦃G, L⦄ ⊢ #0 ➡[h] T2 →
58 | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 & ⬆*[1] V2 ≘ T2 &
60 #h #G #L #T2 #H elim (cpm_inv_zero1 … H) -H *
61 /3 width=6 by ex3_3_intro, or_introl, or_intror/
62 #n #K #V1 #V2 #_ #_ #_ #H destruct
65 lemma cpr_inv_lref1: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #↑i ➡[h] T2 →
67 | ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ➡[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
68 #h #G #L #T2 #i #H elim (cpm_inv_lref1 … H) -H *
69 /3 width=6 by ex3_3_intro, or_introl, or_intror/
72 lemma cpr_inv_gref1: ∀h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ➡[h] T2 → T2 = §l.
73 #h #G #L #T2 #l #H elim (cpm_inv_gref1 … H) -H //
76 (* Basic_1: includes: pr0_gen_cast pr2_gen_cast *)
77 lemma cpr_inv_cast1: ∀h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝ V1.U1 ➡[h] U2 →
78 ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[h] T2 &
80 | ⦃G, L⦄ ⊢ U1 ➡[h] U2.
81 #h #G #L #V1 #U1 #U2 #H elim (cpm_inv_cast1 … H) -H
82 /2 width=1 by or_introl, or_intror/ * #n #_ #H destruct
85 lemma cpr_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡[h] U2 →
86 ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[h] T2 &
88 | (⦃G, L⦄ ⊢ U1 ➡[h] U2 ∧ I = Cast)
89 | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ W1 ➡[h] W2 &
90 ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h] T2 & U1 = ⓛ{p}W1.T1 &
91 U2 = ⓓ{p}ⓝW2.V2.T2 & I = Appl
92 | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V & ⬆*[1] V ≘ V2 &
93 ⦃G, L⦄ ⊢ W1 ➡[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h] T2 &
95 U2 = ⓓ{p}W2.ⓐV2.T2 & I = Appl.
96 #h * #G #L #V1 #U1 #U2 #H
97 [ elim (cpm_inv_appl1 … H) -H *
98 /3 width=13 by or4_intro0, or4_intro2, or4_intro3, ex7_7_intro, ex6_6_intro, ex3_2_intro/
99 | elim (cpr_inv_cast1 … H) -H [ * ]
100 /3 width=5 by or4_intro0, or4_intro1, ex3_2_intro, conj/
104 (* Basic eliminators ********************************************************)
106 lemma cpr_ind (h): ∀Q:relation4 genv lenv term term.
107 (∀I,G,L. Q G L (⓪{I}) (⓪{I})) →
108 (∀G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 → Q G K V1 V2 →
109 ⬆*[1] V2 ≘ W2 → Q G (K.ⓓV1) (#0) W2
110 ) → (∀I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ➡[h] T → Q G K (#i) T →
111 ⬆*[1] T ≘ U → Q G (K.ⓘ{I}) (#↑i) (U)
112 ) → (∀p,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[h] T2 →
113 Q G L V1 V2 → Q G (L.ⓑ{I}V1) T1 T2 → Q G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
114 ) → (∀I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[h] T2 →
115 Q G L V1 V2 → Q G L T1 T2 → Q G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
116 ) → (∀G,L,V,T1,T,T2. ⬆*[1] T ≘ T1 → ⦃G, L⦄ ⊢ T ➡[h] T2 →
117 Q G L T T2 → Q G L (+ⓓV.T1) T2
118 ) → (∀G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → Q G L T1 T2 →
120 ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h] T2 →
121 Q G L V1 V2 → Q G L W1 W2 → Q G (L.ⓛW1) T1 T2 →
122 Q G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
123 ) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h] T2 →
124 Q G L V1 V → Q G L W1 W2 → Q G (L.ⓓW1) T1 T2 →
125 ⬆*[1] V ≘ V2 → Q G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
127 ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → Q G L T1 T2.
128 #h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #G #L #T1 #T2
129 @(insert_eq_0 … 0) #n #H
130 @(cpm_ind … H) -G -L -T1 -T2 -n [2,4,11:|*: /3 width=4 by/ ]
131 [ #G #L #s #H destruct
132 | #n #G #K #V1 #V2 #W2 #_ #_ #_ #H destruct
133 | #n #G #L #U1 #U2 #T #_ #_ #H destruct
137 (* Basic_1: removed theorems 12:
138 pr0_subst0_back pr0_subst0_fwd pr0_subst0
140 pr2_head_2 pr2_cflat clear_pr2_trans
141 pr2_gen_csort pr2_gen_cflat pr2_gen_cbind
142 pr2_gen_ctail pr2_ctail