Nuprl Lemma : fset-add-as-cons
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[s:fset(T)]. ∀[x:T]. (fset-add(eq;x;s) = [x / s] ∈ fset(T))
Proof
Definitions occuring in Statement :
fset-add: fset-add(eq;x;s),
fset: fset(T),
cons: [a / b],
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
implies: P ⇒ Q,
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
fset-member: a ∈ s,
top: Top,
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
eqof: eqof(d),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
false: False,
not: ¬A,
bor: p ∨bq,
true: True
Lemmas referenced :
fset_wf,
deq_wf,
cons-wf-fset,
fset-extensionality,
fset-add_wf,
fset-member_witness,
or_wf,
equal_wf,
fset-member_wf,
member-fset-add,
uiff_wf,
deq_member_cons_lemma,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
testxxx_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
hypothesisEquality,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
axiomEquality,
because_Cache,
extract_by_obid,
cumulativity,
universeEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
independent_pairFormation,
independent_functionElimination,
rename,
addLevel,
dependent_functionElimination,
independent_pairEquality,
voidElimination,
voidEquality,
applyEquality,
setElimination,
lambdaFormation,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
natural_numberEquality,
inlFormation,
inrFormation
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[s:fset(T)]. \mforall{}[x:T]. (fset-add(eq;x;s) = [x / s])
Date html generated:
2017_04_17-AM-09_19_48
Last ObjectModification:
2017_02_27-PM-05_22_55
Theory : finite!sets
Home
Index