Nuprl Lemma : test-int-cmp-normalize
∀[x,y,a,b:Top].
(if (a) < (b)
then <if (a) < (b) then 1 else 2, if (b) < (a) then 1 else 2, if b=a then 1 else 2>
else <if (a) < (b) then 1 else 2, if (b) < (a) then 1 else 2, if b=a then 1 else 2> ~ if (a) < (b)
then <1, 2, 2>
else <2
, if (b) < (a)
then 1
else 2
, if b=a
then 1
else 2>)
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
top: Top,
less: if (a) < (b) then c else d,
int_eq: if a=b then c else d,
pair: <a, b>,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
has-value: (a)↓,
member: t ∈ T,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
prop: ℙ,
le: A ≤ B,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
less-iff-le,
add_functionality_wrt_le,
add-associates,
add-swap,
add-commutes,
le-add-cancel,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
eq_int_wf,
assert_of_eq_int,
le_antisymmetry_iff,
equal-wf-base,
has-value_wf_base,
is-exception_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
cut,
sqequalSqle,
divergentSqle,
callbyvalueLess,
sqequalHypSubstitution,
sqequalTransitivity,
computationStep,
hypothesis,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
productElimination,
thin,
introduction,
extract_by_obid,
isectElimination,
lambdaFormation,
unionElimination,
equalityElimination,
because_Cache,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
lessCases,
isect_memberFormation,
sqequalAxiom,
isect_memberEquality,
independent_pairFormation,
voidElimination,
voidEquality,
natural_numberEquality,
imageMemberEquality,
imageElimination,
independent_functionElimination,
dependent_functionElimination,
addEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
impliesFunctionality,
int_eqReduceTrueSq,
intEquality,
int_eqReduceFalseSq,
sqleReflexivity,
lessExceptionCases,
axiomSqleEquality,
exceptionSqequal,
exceptionLess
Latex:
\mforall{}[x,y,a,b:Top].
(if (a) < (b)
then <if (a) < (b) then 1 else 2, if (b) < (a) then 1 else 2, if b=a then 1 else 2>
else <if (a) < (b) then 1 else 2, if (b) < (a) then 1 else 2, if b=a then 1 else 2>
\msim{} if (a) < (b) then ə, 2, 2> else ɚ, if (b) < (a) then 1 else 2, if b=a then 1 else 2>)
Date html generated:
2017_04_14-AM-07_16_34
Last ObjectModification:
2017_02_27-PM-02_51_50
Theory : arithmetic
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