Nuprl Lemma : assert_of_eq_int
∀[x,y:ℤ]. uiff(↑(x =z y);x = y ∈ ℤ)
Proof
Definitions occuring in Statement :
assert: ↑b,
eq_int: (i =z j),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
subtype_rel: A ⊆r B,
false: False,
true: True,
bool: 𝔹,
implies: P ⇒ Q,
all: ∀x:A. B[x],
ifthenelse: if b then t else f fi ,
assert: ↑b,
prop: ℙ,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
member: t ∈ T,
uall: ∀[x:A]. B[x],
eq_int: (i =z j),
or: P ∨ Q,
not: ¬A,
guard: {T},
sq_type: SQType(T),
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
bfalse: ff,
btrue: tt
Lemmas referenced :
int_subtype_base,
equal-wf-base,
equal_wf,
false_wf,
true_wf,
bool_wf,
eq_int_wf,
assert_wf,
add-monotonic,
less_than_wf,
subtype_base_sq,
less-trichotomy
Rules used in proof :
because_Cache,
isect_memberEquality,
independent_pairEquality,
productElimination,
applyEquality,
intEquality,
independent_functionElimination,
dependent_functionElimination,
voidElimination,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
unionElimination,
lambdaFormation,
sqequalRule,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
hypothesis,
independent_pairFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
addInverse,
inlFormation,
minusEquality,
independent_isectElimination,
cumulativity,
instantiate,
imageElimination,
int_eqReduceFalseSq,
Error :lambdaFormation_alt,
Error :equalityIsType4,
Error :inhabitedIsType,
baseApply,
closedConclusion,
baseClosed,
hyp_replacement,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
applyLambdaEquality,
setElimination,
rename,
natural_numberEquality,
int_eqReduceTrueSq
Latex:
\mforall{}[x,y:\mBbbZ{}]. uiff(\muparrow{}(x =\msubz{} y);x = y)
Date html generated:
2019_06_20-AM-11_20_10
Last ObjectModification:
2018_10_15-PM-07_38_41
Theory : union
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