Nuprl Lemma : fset-size-add
∀[T:Type]
∀[eq:EqDecider(T)]. ∀[s:fset(T)]. ∀[x:T]. ||fset-add(eq;x;s)|| = (||s|| + 1) ∈ ℤ supposing ¬x ∈ s
supposing valueall-type(T)
Proof
Definitions occuring in Statement :
fset-size: ||s||,
fset-add: fset-add(eq;x;s),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
fset: fset(T),
all: ∀x:A. B[x],
prop: ℙ,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
squash: ↓T,
subtype_rel: A ⊆r B,
true: True,
nat: ℕ,
fset-add: fset-add(eq;x;s),
fset-size: ||s||,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
not: ¬A,
fset-member: a ∈ s,
guard: {T},
sq_type: SQType(T),
top: Top,
deq: EqDecider(T),
eqof: eqof(d),
uiff: uiff(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
false: False,
or: P ∨ Q,
fset-union: x ⋃ y,
fset-singleton: {x}
Lemmas referenced :
list_wf,
set-equal_wf,
set-equal-reflex,
equal-wf-base,
equal_wf,
squash_wf,
true_wf,
fset-add_wf,
subtype_rel_self,
fset_wf,
fset-size_wf,
not_wf,
fset-member_wf,
deq_wf,
valueall-type_wf,
l_member_wf,
assert-deq-member,
int_subtype_base,
subtype_base_sq,
length_of_cons_lemma,
remove_repeats_cons_lemma,
length_wf,
iff_weakening_equal,
bnot_wf,
remove-repeats-filter,
remove-repeats_wf,
filter_trivial,
safe-assert-deq,
assert_of_bnot,
iff_weakening_uiff,
eqof_wf,
iff_transitivity,
assert_wf,
l_all_iff,
and_wf,
iff_wf,
all_wf,
or_wf,
false_wf,
nil_member,
cons_member,
member-union,
nil_wf,
cons_wf,
l-union_wf,
length-remove-repeats,
or_false_r
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
promote_hyp,
Error :lambdaFormation_alt,
Error :inhabitedIsType,
Error :universeIsType,
equalityTransitivity,
equalitySymmetry,
pointwiseFunctionality,
sqequalRule,
pertypeElimination,
productElimination,
productEquality,
dependent_functionElimination,
applyEquality,
Error :lambdaEquality_alt,
imageElimination,
universeEquality,
intEquality,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
addEquality,
setElimination,
rename,
instantiate,
Error :isect_memberEquality_alt,
axiomEquality,
cumulativity,
independent_functionElimination,
lambdaFormation,
independent_isectElimination,
voidEquality,
voidElimination,
isect_memberEquality,
lambdaEquality,
impliesFunctionality,
independent_pairFormation,
addLevel,
setEquality,
hyp_replacement,
dependent_set_memberEquality,
applyLambdaEquality,
orFunctionality,
allFunctionality,
levelHypothesis
Latex:
\mforall{}[T:Type]
\mforall{}[eq:EqDecider(T)]. \mforall{}[s:fset(T)]. \mforall{}[x:T]. ||fset-add(eq;x;s)|| = (||s|| + 1) supposing \mneg{}x \mmember{} s
supposing valueall-type(T)
Date html generated:
2019_06_20-PM-01_59_53
Last ObjectModification:
2018_09_30-PM-02_47_20
Theory : finite!sets
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