Nuprl Lemma : list-diff-cons-single
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[as:T List]. ∀[b,x:T]. [x / as]-[b] = [x / as-[b]] ∈ (T List) supposing ¬(x = b ∈ T)
Proof
Definitions occuring in Statement :
list-diff: as-bs,
cons: [a / b],
nil: [],
list: T List,
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
top: Top,
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
list-diff-cons,
cons_wf,
nil_wf,
list-diff_wf,
iff_weakening_equal,
deq_member_cons_lemma,
deq_member_nil_lemma,
bor_wf,
bfalse_wf,
bool_wf,
eqtt_to_assert,
assert-deq-member,
eqff_to_assert,
deq-member_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
l_member_wf,
not_wf,
list_wf,
deq_wf,
member_singleton
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
because_Cache,
cumulativity,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
setElimination,
rename,
lambdaFormation,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
axiomEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[as:T List]. \mforall{}[b,x:T].
[x / as]-[b] = [x / as-[b]] supposing \mneg{}(x = b)
Date html generated:
2017_04_17-AM-09_12_58
Last ObjectModification:
2017_02_27-PM-05_19_52
Theory : decidable!equality
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