Nuprl Lemma : ifthenelse_sqle
∀[a,x1,y1,x2,y2:Base].
if a then x1 else y1 fi ≤ if a then x2 else y2 fi
supposing ((∃z:Base. (a ~ inl z)) ⇒ (x1 ≤ x2)) ∧ ((∃z:Base. (a ~ inr z )) ⇒ (y1 ≤ y2))
Proof
Definitions occuring in Statement :
ifthenelse: if b then t else f fi ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
inr: inr x ,
inl: inl x,
base: Base,
sqle: s ≤ t,
sqequal: s ~ t
Definitions unfolded in proof :
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
has-value: (a)↓,
all: ∀x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
btrue: tt,
exists: ∃x:A. B[x],
bfalse: ff,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
sqle_wf_base,
base_wf,
exists_wf,
and_wf,
assert_of_bnot,
eqff_to_assert,
is-exception_wf,
has-value_wf_base,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases,
top_wf,
isl_wf,
injection-eta
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
divergentSqle,
callbyvalueDecide,
sqequalHypSubstitution,
hypothesis,
lemma_by_obid,
dependent_functionElimination,
thin,
equalityTransitivity,
equalitySymmetry,
isectElimination,
because_Cache,
unionElimination,
instantiate,
cumulativity,
independent_isectElimination,
independent_functionElimination,
productElimination,
dependent_pairFormation,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
sqequalIntensionalEquality,
decideExceptionCases,
axiomSqleEquality,
exceptionSqequal,
sqleReflexivity,
functionEquality,
lambdaEquality,
isect_memberEquality
Latex:
\mforall{}[a,x1,y1,x2,y2:Base].
if a then x1 else y1 fi \mleq{} if a then x2 else y2 fi
supposing ((\mexists{}z:Base. (a \msim{} inl z)) {}\mRightarrow{} (x1 \mleq{} x2)) \mwedge{} ((\mexists{}z:Base. (a \msim{} inr z )) {}\mRightarrow{} (y1 \mleq{} y2))
Date html generated:
2016_05_13-PM-03_45_17
Last ObjectModification:
2016_01_14-PM-07_08_11
Theory : computation
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