Nuprl Lemma : int-decr-map-add-prop
∀[Value:Type]. ∀[m:int-decr-map-type(Value)]. ∀[k1,k2:ℤ]. ∀[v:Value].
(int-decr-map-find(k1;int-decr-map-add(k2;v;m))
= if (k1 =z k2) ∧b (¬bint-decr-map-inDom(k2;m)) then inl v else int-decr-map-find(k1;m) fi
∈ (Value?))
Proof
Definitions occuring in Statement :
int-decr-map-add: int-decr-map-add(k;v;m),
int-decr-map-inDom: int-decr-map-inDom(k;m),
int-decr-map-find: int-decr-map-find(k;m),
int-decr-map-type: int-decr-map-type(Value),
band: p ∧b q,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
unit: Unit,
inl: inl x,
union: left + right,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
int-decr-map-type: int-decr-map-type(Value),
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
pi1: fst(t),
so_apply: x[s1;s2],
gt: i > j,
subtype_rel: A ⊆r B,
or: P ∨ Q,
cons: [a / b],
colength: colength(L),
guard: {T},
decidable: Dec(P),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
int-decr-map-find: int-decr-map-find(k;m),
int-decr-map-add: int-decr-map-add(k;v;m),
find-combine: find-combine(cmp;l),
list_ind: list_ind,
insert-combine: insert-combine(cmp;f;x;l),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
has-value: (a)↓,
ifthenelse: if b then t else f fi ,
btrue: tt,
pi2: snd(t),
uiff: uiff(P;Q),
band: p ∧b q,
bnot: ¬bb,
int-decr-map-inDom: int-decr-map-inDom(k;m),
isl: isl(x),
bfalse: ff,
nequal: a ≠ b ∈ T ,
int-minus-comparison: int-minus-comparison(f),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assert: ↑b,
true: True,
le: A ≤ B,
bool: 𝔹,
unit: Unit
Latex:
\mforall{}[Value:Type]. \mforall{}[m:int-decr-map-type(Value)]. \mforall{}[k1,k2:\mBbbZ{}]. \mforall{}[v:Value].
(int-decr-map-find(k1;int-decr-map-add(k2;v;m))
= if (k1 =\msubz{} k2) \mwedge{}\msubb{} (\mneg{}\msubb{}int-decr-map-inDom(k2;m)) then inl v else int-decr-map-find(k1;m) fi )
Date html generated:
2016_05_17-PM-01_49_51
Last ObjectModification:
2016_01_17-AM-11_38_06
Theory : datatype-signatures
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