Nuprl Lemma : less_sqequal
∀[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,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
and: P ∧ Q,
prop: ℙ
Lemmas referenced :
base_wf,
equal-wf-base,
not_wf,
less_than_wf,
is-exception_wf,
has-value_wf_base,
less_sqle
Rules used in proof :
sqequalSqle,
cut,
lemma_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
independent_isectElimination,
lambdaFormation,
independent_functionElimination,
hypothesis,
independent_pairFormation,
productElimination,
promote_hyp,
divergentSqle,
sqleReflexivity,
productEquality,
because_Cache,
functionEquality,
intEquality,
sqequalIntensionalEquality,
isect_memberFormation,
introduction,
sqequalAxiom,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[a,b,x1,y1,x2,y2:Base].
if (a) < (b) then x1 else y1 \msim{} if (a) < (b) then x2 else y2
supposing ((a \mmember{} \mBbbZ{}) \mwedge{} (b \mmember{} \mBbbZ{})) {}\mRightarrow{} ((a < b {}\mRightarrow{} (x1 \msim{} x2)) \mwedge{} ((\mneg{}a < b) {}\mRightarrow{} (y1 \msim{} y2)))
Date html generated:
2016_05_13-PM-03_45_42
Last ObjectModification:
2016_01_14-PM-07_06_36
Theory : computation
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