Add uninterpreted Map implementation to Prelude.lean (#4734)

Tackling #4725, this PR adds an uninterpreted implementation for the
`Map` sort.

This PR's goal is to iron out how we want these uninterpreted
implementations to be before proceeding with the rest of the hooked
functions in `domains.md`.

---------

Co-authored-by: Tamás Tóth <[email protected]>

pyk/src/pyk/k2lean4/Prelude.lean

@@ -1,10 +1,105 @@
-abbrev SortBool : Type := Int
-abbrev SortBytes: Type := ByteArray
-abbrev SortId : Type := String
-abbrev SortInt : Type := Int
-abbrev SortString : Type := String
+/-
+K Prelude in Lean 4
+
+Functions with the `hook` attribute need to have a manual implementation in the backends.
+This file contains the Lean 4 definitions of the hooked functions in `domains.md`.
+
+Currently we translate hooked functions as uninterpreted functions together with axioms asserting their behavior.
+The current definition can be put into three levels:
+
+1. Signature Level:
+The signature of the hooks, this includes aliases for Sorts and function symbols for hooked functions.
+
+2. Rule Level:
+The behavior of the uninterpreted symbols can be asserted through axioms or theorems.
+Inconsistencies can arise from them, so it falls under the user to make sure axioms are consistent and/or theorems provable.
+
+3. Simplification Level:
+With the theory defined through function rules, simplifications can be stated as theorems.
+These theorems should be provable directly from the function rules and the semantics of the Sorts.
+ -/
+
+-- Basic K types
+abbrev SortBool         : Type := Int
+abbrev SortBytes        : Type := ByteArray
+abbrev SortId           : Type := String
+abbrev SortInt          : Type := Int
+abbrev SortString       : Type := String
 abbrev SortStringBuffer : Type := String
 
+namespace MapHookDef
+/-
+The `Map` sort represents a generalized associative array.
+Each key can be paired with an arbitrary value, and can be used to reference its associated value.
+Multiple bindings for the same key are not allowed.
+Note that both keys and values will always be KItems.
+ -/
+
+-- Signature to be instantiated by map implementations
+structure MapHookSig (K V : Type) where
+  map        : Type -- Carrier, such as List (KItem × KItem)
+  unit       : map
+  cons       : K → V → map → map
+  lookup     : map → K → Option V
+  lookup?    : map → K → V -- lookup with default
+  update     : K → V → map → map
+  delete     : map → K → map
+  concat     : map → map → Option map
+  difference : map → map → map
+  updateMap  : map → map → map
+  removeAll  : map → List K → map
+  keys       : map → List K
+  in_keys    : map → K → Bool
+  values     : map → List V
+  size       : map → Nat
+  includes   : map → map → Bool -- map inclusion
+  choice     : map → K -- arbitrary key from a map
+  nodup      : forall al : map, List.Nodup (keys al)
+
+-- We use axioms to have uninterpreted functions
+variable (K V : Type)
+axiom mapCAx       : Type -- Map Carrier
+axiom unitAx       : mapCAx
+axiom consAx       : K → V → mapCAx → mapCAx
+axiom lookupAx     : mapCAx → K → Option V
+axiom lookupAx?    : mapCAx → K → V -- lookup with default
+axiom updateAx     : K → V → mapCAx → mapCAx
+axiom deleteAx     : mapCAx → K → mapCAx
+axiom concatAx     : mapCAx → mapCAx → Option mapCAx
+axiom differenceAx : mapCAx → mapCAx → mapCAx
+axiom updateMapAx  : mapCAx → mapCAx → mapCAx
+axiom removeAllAx  : mapCAx → List K → mapCAx
+axiom keysAx       : mapCAx → List K
+axiom in_keysAx    : mapCAx → K → Bool
+axiom valuesAx     : mapCAx → List V
+axiom sizeAx       : mapCAx → Nat
+axiom includesAx   : mapCAx → mapCAx → Bool -- map inclusion
+axiom choiceAx     : mapCAx → K -- arbitrary key from a map
+axiom nodupAx      : forall m, List.Nodup (keysAx K m)
+
+-- Uninterpreted Map implementation
+noncomputable def MapHook (K V : Type) : MapHookSig K V :=
+  { map        := mapCAx,
+    unit       := unitAx,
+    cons       := (consAx K V),
+    lookup     := (lookupAx K V),
+    lookup?    := (lookupAx? K V),
+    update     := (updateAx K V),
+    delete     := (deleteAx K),
+    concat     := concatAx,
+    difference := differenceAx,
+    updateMap  := updateMapAx,
+    removeAll  := (removeAllAx K),
+    keys       := (keysAx K),
+    in_keys    := (in_keysAx K),
+    values     := (valuesAx V),
+    size       := sizeAx,
+    includes   := includesAx,
+    choice     := (choiceAx K),
+    nodup      := (nodupAx K) }
+
+end MapHookDef
+
 class Inj (From To : Type) : Type where
   inj (x : From) : To
 

pyk/src/pyk/k2lean4/k2lean4.py

@@ -156,7 +156,7 @@ def _collection(self, sort: str) -> Structure:
                 val = Term(f'List {item}')
             case CollectionKind.MAP:
                 key, value = sorts
-                val = Term(f'List ({key} × {value})')
+                val = Term(f'(MapHook {key} {value}).map')
         field = ExplBinder(('coll',), val)
         return Structure(sort, Signature((), Term('Type')), ctor=StructCtor((field,)))