Nuprl Lemma : less_sqle
∀[a,b,x1,y1,x2,y2:Base].
if (a) < (b) then x1 else y1 ≤ if (a) < (b) then x2 else y2
supposing ((a ∈ ℤ) ∧ (b ∈ ℤ)) ⇒ ((a < b ⇒ (x1 ≤ x2)) ∧ ((¬a < b) ⇒ (y1 ≤ y2)))
Proof
Definitions occuring in Statement :
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
less: if (a) < (b) then c else d,
int: ℤ,
base: Base,
sqle: s ≤ t
Definitions unfolded in proof :
has-value: (a)↓,
member: t ∈ T,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
prop: ℙ,
uimplies: b supposing a,
cand: A c∧ B,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
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,
decidable: Dec(P)
Lemmas referenced :
has-value_wf_base,
is-exception_wf,
equal-wf-base,
less_than_wf,
sqle_wf_base,
not_wf,
base_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
iff_weakening_uiff,
assert_of_bnot,
decidable__lt
Rules used in proof :
divergentSqle,
cut,
callbyvalueLess,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
hypothesis,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
productElimination,
thin,
introduction,
lessExceptionCases,
axiomSqleEquality,
exceptionSqequal,
sqleReflexivity,
extract_by_obid,
isectElimination,
functionEquality,
productEquality,
because_Cache,
isect_memberFormation,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
independent_pairFormation,
lambdaFormation,
unionElimination,
equalityElimination,
independent_isectElimination,
lessCases,
sqequalAxiom,
voidElimination,
voidEquality,
natural_numberEquality,
imageMemberEquality,
imageElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
impliesFunctionality,
exceptionLess
Latex:
\mforall{}[a,b,x1,y1,x2,y2:Base].
if (a) < (b) then x1 else y1 \mleq{} if (a) < (b) then x2 else y2
supposing ((a \mmember{} \mBbbZ{}) \mwedge{} (b \mmember{} \mBbbZ{})) {}\mRightarrow{} ((a < b {}\mRightarrow{} (x1 \mleq{} x2)) \mwedge{} ((\mneg{}a < b) {}\mRightarrow{} (y1 \mleq{} y2)))
Date html generated:
2017_04_14-AM-07_21_57
Last ObjectModification:
2017_02_27-PM-02_55_20
Theory : computation
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