Nuprl Lemma : assert-is_int
∀[T:Type]. ∀[x:T]. uiff(↑is_int(x);x ∈ ℤ) supposing value-type(T) ∧ (T ⊆r Base)
Proof
Definitions occuring in Statement :
is_int: is_int(x),
value-type: value-type(T),
assert: ↑b,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
int: ℤ,
base: Base,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
is_int: is_int(x),
has-value: (a)↓,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
top: Top,
bfalse: ff,
false: False,
cand: A c∧ B,
true: True,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
sq_type: SQType(T),
all: ∀x:A. B[x],
guard: {T}
Lemmas referenced :
value-type-has-value,
has-value_wf_base,
is-exception_wf,
istype-top,
istype-void,
istype-assert,
is_int_wf,
int-value-type,
assert_witness,
istype-int,
value-type_wf,
subtype_rel_wf,
base_wf,
istype-universe,
subtype_base_sq,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
sqequalRule,
callbyvalueReduce,
extract_by_obid,
isectElimination,
hypothesisEquality,
independent_isectElimination,
hypothesis,
isintCases,
divergentSqle,
baseClosed,
because_Cache,
isintReduceTrue,
equalityTransitivity,
equalitySymmetry,
axiomSqEquality,
Error :inhabitedIsType,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
voidElimination,
axiomEquality,
intEquality,
natural_numberEquality,
independent_functionElimination,
Error :equalityIstype,
applyEquality,
sqequalBase,
independent_pairEquality,
Error :universeIsType,
Error :productIsType,
instantiate,
universeEquality,
cumulativity,
dependent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[x:T]. uiff(\muparrow{}is\_int(x);x \mmember{} \mBbbZ{}) supposing value-type(T) \mwedge{} (T \msubseteq{}r Base)
Date html generated:
2019_06_20-AM-11_33_09
Last ObjectModification:
2019_02_07-PM-00_00_13
Theory : int_1
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