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
Home
Index