Nuprl Lemma : list-diff-cons
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[as,bs:T List]. ∀[x:T].
([x / as]-bs = if x ∈b bs then as-bs else [x / as-bs] fi ∈ (T List))
Proof
Definitions occuring in Statement :
list-diff: as-bs,
deq-member: x ∈b L,
cons: [a / b],
list: T List,
deq: EqDecider(T),
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
list-diff: as-bs,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
top: Top,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
bfalse: ff,
prop: ℙ,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q
Lemmas referenced :
deq-member_wf,
bool_wf,
eqtt_to_assert,
assert-deq-member,
filter_cons_lemma,
filter_wf5,
l_member_wf,
bnot_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
cons_wf,
list_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
lambdaFormation,
unionElimination,
equalityElimination,
because_Cache,
productElimination,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
axiomEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[as,bs:T List]. \mforall{}[x:T].
([x / as]-bs = if x \mmember{}\msubb{} bs then as-bs else [x / as-bs] fi )
Date html generated:
2017_04_17-AM-09_12_55
Last ObjectModification:
2017_02_27-PM-05_19_42
Theory : decidable!equality
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