Nuprl Lemma : remove-combine-cons
∀[T:Type]. ∀[cmp:T ⟶ ℤ]. ∀[x:T]. ∀[l:T List].
(remove-combine(cmp;[x / l]) ~ if (cmp x =z 0) then l
if 0 <z cmp x then [x / l]
else [x / remove-combine(cmp;l)]
fi )
Proof
Definitions occuring in Statement :
remove-combine: remove-combine(cmp;l),
cons: [a / b],
list: T List,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
remove-combine: remove-combine(cmp;l),
all: ∀x:A. B[x],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
has-value: (a)↓,
uimplies: b supposing a
Lemmas referenced :
list_ind_cons_lemma,
value-type-has-value,
int-value-type,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
callbyvalueReduce,
isectElimination,
intEquality,
independent_isectElimination,
applyEquality,
hypothesisEquality,
sqequalAxiom,
because_Cache,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[cmp:T {}\mrightarrow{} \mBbbZ{}]. \mforall{}[x:T]. \mforall{}[l:T List].
(remove-combine(cmp;[x / l]) \msim{} if (cmp x =\msubz{} 0) then l
if 0 <z cmp x then [x / l]
else [x / remove-combine(cmp;l)]
fi )
Date html generated:
2016_05_14-PM-02_42_23
Last ObjectModification:
2015_12_26-PM-02_42_34
Theory : list_1
Home
Index