Nuprl Lemma : quicksort-int-member
∀L:ℤ List. ∀x:ℤ. ((x ∈ L) ⇐⇒ (x ∈ quicksort-int(L)))
Proof
Definitions occuring in Statement :
quicksort-int: quicksort-int(L),
l_member: (x ∈ l),
list: T List,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
guard: {T},
sq_stable: SqStable(P)
Lemmas referenced :
decidable__equal_int,
sq_stable__l_member,
member-permutation,
permutation_wf,
l_member_wf,
le_wf,
sorted-by_wf,
and_wf,
list_wf,
quicksort-int_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
hypothesis,
sqequalRule,
imageMemberEquality,
baseClosed,
setEquality,
isectElimination,
intEquality,
introduction,
because_Cache,
imageElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_functionElimination
Latex:
\mforall{}L:\mBbbZ{} List. \mforall{}x:\mBbbZ{}. ((x \mmember{} L) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} quicksort-int(L)))
Date html generated:
2016_05_15-PM-04_30_01
Last ObjectModification:
2016_01_16-AM-11_14_48
Theory : general
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