Nuprl Lemma : insert-int-cons
∀v:ℤ List. ∀u,a:ℤ. (insert-int(a;[u / v]) ~ if u <z a then [u / insert-int(a;v)] else [a; [u / v]] fi )
Proof
Definitions occuring in Statement :
insert-int: insert-int(x;l),
cons: [a / b],
list: T List,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
all: ∀x:A. B[x],
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
insert-int: insert-int(x;l),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
has-value: (a)↓,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
not: ¬A,
false: False,
prop: ℙ,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
subtype_base_sq,
list_wf,
list_subtype_base,
int_subtype_base,
list_ind_cons_lemma,
value-type-has-value,
list-value-type,
insert-int_wf,
subtype_rel_self,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
cons_wf,
eqff_to_assert,
equal_wf,
bool_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
hypothesis,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
callbyvalueReduce,
hypothesisEquality,
because_Cache,
unionElimination,
equalityElimination,
productElimination,
lessCases,
isect_memberFormation,
sqequalAxiom,
independent_pairFormation,
natural_numberEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
promote_hyp
Latex:
\mforall{}v:\mBbbZ{} List. \mforall{}u,a:\mBbbZ{}.
(insert-int(a;[u / v]) \msim{} if u <z a then [u / insert-int(a;v)] else [a; [u / v]] fi )
Date html generated:
2017_09_29-PM-05_48_25
Last ObjectModification:
2017_05_03-PM-03_18_30
Theory : list_0
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