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 Mario Pereira committed Aug 31, 2016 1 2 3 4 5 6 7 8 9 `````` (** Leftist heaps (Clark Allan Crane, 1972 && Donald E. Knuth, 1973). Purely applicative implementation, following Okasaki's implementation in his book "Purely Functional Data Structures" (Section 3.1). Author: Mário Pereira (Université Paris Sud) *) `````` 10 11 12 13 14 15 16 17 18 19 20 ``````module Heap use import int.Int type elt predicate le elt elt clone relations.TotalPreOrder with type t = elt, predicate rel = le type heap `````` Jean-Christophe Filliâtre committed Feb 09, 2017 21 `````` val function size heap : int `````` 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 `````` function occ elt heap : int predicate mem (x: elt) (h: heap) = occ x h > 0 function minimum heap : elt predicate is_minimum (x: elt) (h: heap) = mem x h && forall e. mem e h -> le x e axiom min_def: forall h. 0 < size h -> is_minimum (minimum h) h val empty () : heap ensures { size result = 0 } ensures { forall x. occ x result = 0 } val is_empty (h: heap) : bool ensures { result <-> size h = 0 } val merge (h1 h2: heap) : heap ensures { forall x. occ x result = occ x h1 + occ x h2 } ensures { size result = size h1 + size h2 } val insert (x: elt) (h: heap) : heap ensures { occ x result = occ x h + 1 } ensures { forall y. y <> x -> occ y h = occ y result } ensures { size result = size h + 1 } val find_min (h: heap) : elt requires { size h > 0 } ensures { result = minimum h } val delete_min (h: heap) : heap requires { size h > 0 } ensures { let x = minimum h in occ x result = occ x h - 1 } ensures { forall y. y <> minimum h -> occ y result = occ y h } ensures { size result = size h - 1 } end module TreeRank type tree 'a = E | N int (tree 'a) 'a (tree 'a) end module Size use import TreeRank use import int.Int `````` Jean-Christophe Filliâtre committed Feb 09, 2017 74 `````` let rec function size (t: tree 'a) : int = match t with `````` 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 `````` | E -> 0 | N _ l _ r -> 1 + size l + size r end lemma size_nonneg: forall t: tree 'a. 0 <= size t lemma size_empty: forall t: tree 'a. 0 = size t <-> t = E end module Occ use import TreeRank use import int.Int function occ (x: 'a) (t: tree 'a) : int = match t with | E -> 0 | N _ l e r -> (if x = e then 1 else 0) + occ x l + occ x r end lemma occ_nonneg: forall x:'a, t. 0 <= occ x t predicate mem (x: 'a) (t: tree 'a) = 0 < occ x t end module LeftistHeap type elt `````` Jean-Christophe Filliâtre committed Feb 09, 2017 105 `````` val predicate le elt elt `````` 106 107 108 109 110 111 112 113 114 115 116 `````` clone relations.TotalPreOrder with type t = elt, predicate rel = le use import TreeRank use export Size use export Occ use import int.Int use import int.MinMax type t = tree elt (* [e] is no greater than the root of [h], if any *) `````` Jean-Christophe Filliâtre committed Feb 09, 2017 117 `````` let predicate le_root (e: elt) (h: t) = match h with `````` 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 `````` | E -> true | N _ _ x _ -> le e x end lemma le_root_trans: forall x y h. le x y -> le_root y h -> le_root x h (* [h] is a heap *) predicate is_heap (h: t) = match h with | E -> true | N _ l x r -> le_root x l && is_heap l && le_root x r && is_heap r end function minimum t : elt axiom minimum_def: forall l x r s. minimum (N s l x r) = x predicate is_minimum (x: elt) (h: t) = mem x h && forall e. mem e h -> le x e let rec lemma root_is_miminum (h: t) : unit requires { is_heap h && 0 < size h } ensures { is_minimum (minimum h) h } variant { h } = match h with | E -> absurd | N _ l _ r -> `````` Jean-Christophe Filliâtre committed Feb 09, 2017 144 145 `````` match l with E -> root_is_miminum l | _ -> () end; match r with E -> root_is_miminum r | _ -> () end `````` 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 `````` end function rank (h: t) : int = match h with | E -> 0 | N _ l _ r -> 1 + min (rank l) (rank r) end predicate leftist (h: t) = match h with | E -> true | N s l _ r -> s = rank h && leftist l && leftist r && rank l >= rank r end predicate leftist_heap (h: t) = is_heap h && leftist h let empty () : t ensures { size result = 0 } ensures { forall x. occ x result = 0 } ensures { leftist_heap result } = E let is_empty (h: t) : bool ensures { result <-> size h = 0 } `````` Jean-Christophe Filliâtre committed Feb 09, 2017 172 `````` = match h with E -> true | N _ _ _ _ -> false end `````` 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 `````` let rank (h: t) : int requires { leftist_heap h } ensures { result = rank h } = match h with | E -> 0 | N r _ _ _ -> r end let make_n (x: elt) (l r: t) : t requires { leftist_heap r && leftist_heap l } requires { le_root x l && le_root x r } ensures { leftist_heap result } ensures { minimum result = x } ensures { size result = 1 + size l + size r } ensures { occ x result = 1 + occ x l + occ x r } ensures { forall y. x <> y -> occ y result = occ y l + occ y r } = if rank l >= rank r then N (rank r + 1) l x r else N (rank l + 1) r x l let rec merge (h1 h2: t) : t requires { leftist_heap h1 && leftist_heap h2 } ensures { size result = size h1 + size h2 } ensures { forall x. occ x result = occ x h1 + occ x h2 } ensures { leftist_heap result } variant { size h1 + size h2 } = match h1, h2 with | h, E | E, h -> h | N _ l1 x1 r1, N _ l2 x2 r2 -> if le x1 x2 then make_n x1 l1 (merge r1 h2) else make_n x2 l2 (merge h1 r2) end let insert (x: elt) (h: t) : t requires { leftist_heap h } ensures { leftist_heap result } ensures { occ x result = occ x h + 1 } ensures { forall y. x <> y -> occ y result = occ y h } ensures { size result = size h + 1 } = merge (N 1 E x E) h let find_min (h: t) : elt requires { leftist_heap h } requires { 0 < size h } ensures { result = minimum h } = match h with | E -> absurd | N _ _ x _ -> x end let delete_min (h: t) : t requires { 0 < size h } requires { leftist_heap h } ensures { occ (minimum h) result = occ (minimum h) h - 1 } ensures { forall x. x <> minimum h -> occ x result = occ x h } ensures { size result = size h - 1 } ensures { leftist_heap result } = match h with | E -> absurd | N _ l _ r -> merge l r end end``````