Nuprl Lemma : l_member-permutation
∀[T:Type]. ∀L:T List. ∀x:T. ((x ∈ L) ⇒ (∃L':T List. permutation(T;L;[x / L'])))
Proof
Definitions occuring in Statement :
permutation: permutation(T;L1;L2),
l_member: (x ∈ l),
cons: [a / b],
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
not: ¬A,
false: False,
iff: P ⇐⇒ Q,
and: P ∧ Q,
or: P ∨ Q,
exists: ∃x:A. B[x],
l_member: (x ∈ l),
cand: A c∧ B,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3]
Lemmas referenced :
list_induction,
all_wf,
l_member_wf,
exists_wf,
list_wf,
permutation_wf,
cons_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
cons_member,
permutation_weakening,
and_wf,
equal_wf,
append_functionality_wrt_permutation,
list_ind_cons_lemma,
list_ind_nil_lemma,
permutation_transitivity,
permutation-rotate
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
hypothesis,
independent_functionElimination,
rename,
because_Cache,
dependent_functionElimination,
universeEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
voidElimination,
productElimination,
unionElimination,
dependent_pairFormation,
dependent_set_memberEquality,
independent_pairFormation,
applyEquality,
setElimination,
setEquality,
isect_memberEquality,
voidEquality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}x:T. ((x \mmember{} L) {}\mRightarrow{} (\mexists{}L':T List. permutation(T;L;[x / L'])))
Date html generated:
2016_05_14-PM-02_21_44
Last ObjectModification:
2015_12_26-PM-04_28_00
Theory : list_1
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