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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 "apps_2/notation/models/fdrop_4.ma".
16 include "apps_2/models/model_veq.ma".
17 include "apps_2/models/model_raise.ma".
19 (* MODEL ********************************************************************)
21 definition lower: ∀M. nat → nat → evaluation M → evaluation M ≝
22 λM,l,m,lv,i. tri … i l (lv i) (lv (i+m)) (lv (i+m)).
24 interpretation "evaluation lower (models)"
25 'FDrop M l m lv = (lower M l m lv).
27 (* Basic properties *********************************************************)
29 lemma lower_lt: ∀M,lv,l,m,i. i < l → (↓[l, m]⦋M⦌ lv) i = lv i.
30 /2 width=1 by tri_lt/ qed-.
32 lemma lower_ge: ∀M,lv,l,m,i. l ≤ i → (↓[l, m]⦋M⦌ lv) i = lv (i+m).
33 #M #lv #l #m #i #Hli elim (le_to_or_lt_eq … Hli) -Hli #Hli destruct
34 [ /2 width=1 by tri_gt/
35 | /2 width=1 by tri_eq/
39 lemma lower_veq: ∀M,v1,v2. v1 ≐⦋M⦌ v2 → ∀l,m. ↓[l, m] v1 ≐ ↓[l, m] v2.
40 #m #v1 #v2 #Hv12 #l #m #i elim (lt_or_ge i l) #Hli
41 [ >lower_lt // >lower_lt //
42 | >lower_ge // >lower_ge //
46 lemma lower_refl: ∀M,v,l. ↓[l, 0] v ≐⦋M⦌ v.
47 #M #v #l #i elim (lt_or_ge … i l) #Hil
53 (* Main properties **********************************************************)
55 theorem lower_lower_le_sym: ∀M,v,l1,l2,m1,m2. l1 ≤ l2 →
56 ↓[l1, m1] ↓[l2+m1, m2] v ≐⦋M⦌ ↓[l2, m2] ↓[l1, m1] v.
57 #M #v #l1 #l2 #m1 #m2 #Hl12 #j elim (lt_or_ge … j l1) #Hjl1
58 [ lapply (lt_to_le_to_lt … Hjl1 Hl12) -Hl12 #Hjl2
59 >lower_lt // >lower_lt /2 width=3 by lt_to_le_to_lt/
60 >lower_lt // >lower_lt //
61 | >lower_ge // elim (lt_or_ge … j l2) #Hjl2 -Hl12
62 [ >lower_lt /2 width=1 by lt_minus_to_plus/
63 >lower_lt // >lower_ge //
64 | >lower_ge /2 width=1 by monotonic_le_plus_l/
65 >lower_ge // >lower_ge /2 width=1 by le_plus/
70 lemma lower_lower_le: ∀M,v,l1,l2,m1,m2. l1 ≤ l2 →
71 ↓[l2, m2] ↓[l1, m1] v ≐⦋M⦌ ↓[l1, m1] ↓[l2+m1, m2] v.
72 /3 width=1 by lower_lower_le_sym, veq_sym/ qed-.
74 (* Properties on raise ******************************************************)
76 lemma lower_raise_lt: ∀M,v,d,l,m,i. i ≤ l → ↓[l+1, m] [i⬐d] v ≐⦋M⦌ [i⬐d] ↓[l, m] v.
77 #M #v #d #l #m #i #Hil #j elim (lt_or_eq_or_gt … j i) #Hij destruct
78 [ lapply (lt_to_le_to_lt … Hij Hil) -Hil #Hjl
79 >lower_lt /2 width=1 by le_S/ >raise_lt // >raise_lt // >lower_lt //
80 | >lower_lt /2 width=1 by le_S_S/ >raise_eq >raise_eq //
81 | lapply (ltn_to_ltO … Hij) #Hj
82 >raise_gt // elim (lt_or_ge … j (l+1)) #Hlj
83 [ -Hil >lower_lt // >lower_lt /2 width=2 by lt_plus_to_minus/ >raise_gt //
84 | >lower_ge // >lower_ge /2 width=1 by le_plus_to_minus_r/
85 >raise_gt /2 width=1 by le_plus/ >plus_minus //
90 lemma raise_lower_lt: ∀M,v,d,l,m,i. i ≤ l → [i⬐d] ↓[l, m] v ≐⦋M⦌ ↓[l+1, m] [i⬐d] v.
91 /3 width=1 by lower_raise_lt, veq_sym/ qed.
93 lemma lower_raise_be: ∀M,v,d,l,m,i. l ≤ i → i ≤ l + m → ↓[l, m+1] [i⬐d] v ≐⦋M⦌ ↓[l, m] v.
94 #M #v #d #l #m #i #Hli #Hilm #j elim (lt_or_ge … j l) #Hlj
95 [ lapply (lt_to_le_to_lt … Hlj Hli) -Hli -Hilm #Hij
96 >lower_lt // >lower_lt // >raise_lt //
97 | lapply (transitive_le … (j+m) Hilm ?) -Hli -Hilm /2 width=1 by monotonic_le_plus_l/ #Hijm
98 >lower_ge // >lower_ge // >raise_gt /2 width=1 by le_S_S/
102 lemma lower_raise_be_sym: ∀M,v,d,l,m,i. l ≤ i → i ≤ l + m → ↓[l, m] v ≐⦋M⦌ ↓[l, m+1] [i⬐d] v.
103 /3 width=1 by lower_raise_be, veq_sym/ qed.
105 lemma lower_raise: ∀M,v,d,l. ↓[l, 1] [l⬐d] v ≐⦋M⦌ v.
106 /3 width=3 by lower_raise_be, veq_trans/ qed.
108 lemma lower_raise_sym: ∀M,v,d,l. v ≐⦋M⦌ ↓[l, 1] [l⬐d] v.
109 /2 width=1 by veq_sym/ qed.
111 lemma raise_lower: ∀M,v,i. [i⬐v i] ↓[i,1] v ≐⦋M⦌ v.
112 #M #V #i #j elim (lt_or_eq_or_gt j i) #Hij destruct
113 [ >raise_lt // >lower_lt //
115 | >raise_gt // >lower_ge /2 width=1 by monotonic_pred/
116 <plus_minus_m_m /2 width=2 by ltn_to_ltO/
120 lemma raise_lower_sym: ∀M,v,i. v ≐⦋M⦌ [i⬐v i] ↓[i,1] v.
121 /3 width=1 by raise_lower, veq_sym/ qed.
123 lemma raise_lower_be: ∀M,v,l,m. ↓[l, m] v ≐⦋M⦌ [l⬐v (l+m)] ↓[l, m+1] v.
124 #M #v #l #m #j elim (lt_or_eq_or_gt j l) #Hlj destruct
125 [ >lower_lt // >raise_lt // >lower_lt //
126 | >lower_ge // >raise_eq //
127 | lapply (ltn_to_ltO … Hlj) #Hj
128 >lower_ge /2 width=1 by lt_to_le/ >raise_gt //
129 >lower_ge /4 width=1 by plus_minus, monotonic_pred, eq_f/
133 lemma raise_lower_be_sym: ∀M,v,l,m. [l⬐v (l+m)] ↓[l, m+1] v ≐⦋M⦌ ↓[l, m] v.
134 /3 width=1 by raise_lower_be, veq_sym/ qed.
136 (* Forward lemmas on raise **************************************************)
138 lemma lower_fwd_raise_be_S: ∀M,v1,v2,d,l,m. ↓[l, m] v1 ≐⦋M⦌ [l⬐d] v2 →
140 #M #v1 #v2 #d #l #m #Hv12 #j elim (lt_or_ge j l) #Hlj
141 [ lapply (Hv12 j) -Hv12
142 >lower_lt // >lower_lt // >raise_lt //
143 | lapply (Hv12 (j+1))
144 >lower_ge /2 width=1 by le_S/ >lower_ge // >raise_gt /2 width=1 by le_S_S/
148 lemma raise_fwd_lower_be_S: ∀M,v1,v2,d,l,m. [l⬐d] v2 ≐⦋M⦌ ↓[l, m] v1 →
150 /3 width=2 by lower_fwd_raise_be_S, veq_sym/ qed-.
152 lemma lower_fwd_raise_be_O: ∀M,v1,v2,d,l,m. ↓[l, m] v1 ≐⦋M⦌ [l⬐d] v2 → v1 (l+m) = d.
153 #M #v1 #v2 #d #l #m #Hv12 lapply (Hv12 l)
154 >lower_ge // >raise_eq //
157 lemma raise_fwd_lower_be_O: ∀M,v1,v2,d,l,m. [l⬐d] v2 ≐⦋M⦌ ↓[l, m] v1 → d = v1 (l+m).
158 /4 width=7 by lower_fwd_raise_be_O, veq_sym, sym_eq/ qed-.
160 (* Inversion lemmas on raise ************************************************)
162 lemma raise_inv_lower_lt: ∀M,v1,v2,d,l,m,i. i ≤ l → [i⬐d] v1 ≐⦋M⦌ ↓[l+1, m] v2 →
163 ∃∃v. v1 ≐ ↓[l, m] v & v2 ≐ [i⬐d] v.
164 #M #v1 #v2 #d #l #m #i #Hil #Hv12
165 lapply (Hv12 i) >raise_eq >lower_lt /2 width=1 by le_S_S/ #H destruct
166 @(ex2_intro … (↓[i, 1] v2)) //
167 @(veq_trans … (↓[i, 1] ↓[l+1, m] v2))
168 /3 width=3 by lower_lower_le_sym, lower_veq, veq_trans/
171 lemma lower_inv_raise_lt: ∀M,v1,v2,d,l,m,i. i ≤ l → ↓[l+1, m] v2 ≐⦋M⦌ [i⬐d] v1 →
172 ∃∃v. v1 ≐ ↓[l, m] v & v2 ≐ [i⬐d] v.
173 /3 width=1 by raise_inv_lower_lt, veq_sym/ qed-.
175 lemma lower_inv_raise_be: ∀M,v1,v2,d,l,m. ↓[l, m] v1 ≐⦋M⦌ [l⬐d] v2 →
176 v1 (l+m) = d ∧ ↓[l, m+1] v1 ≐ v2.
177 /3 width=2 by lower_fwd_raise_be_O, lower_fwd_raise_be_S, conj/ qed-.
179 lemma raise_inv_lower_be: ∀M,v1,v2,d,l,m. [l⬐d] v2 ≐⦋M⦌ ↓[l, m] v1 →
180 d = v1 (l+m) ∧ v2 ≐ ↓[l, m+1] v1.
181 /3 width=2 by raise_fwd_lower_be_O, raise_fwd_lower_be_S, conj/ qed-.