Nuprl Lemma : fset-size-remove
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[s:fset(T)]. ∀[x:T]. ||fset-remove(eq;x;s)|| = (||s|| - 1) ∈ ℤ supposing x ∈ s
Proof
Definitions occuring in Statement :
fset-size: ||s||,
fset-remove: fset-remove(eq;y;s),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
subtract: 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,
istype: istype(T),
true: True,
nat: ℕ,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
fset-member: a ∈ s,
fset-size: ||s||,
fset-remove: fset-remove(eq;y;s),
fset-filter: {x ∈ s | P[x]},
deq: EqDecider(T),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
not: ¬A,
false: False,
eqof: eqof(d),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
sq_type: SQType(T),
cand: A c∧ B,
remove-first: remove-first(P;L),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
bnot: ¬bb,
assert: ↑b,
so_apply: x[s1;s2;s3],
top: Top
Lemmas referenced :
list_wf,
set-equal_wf,
set-equal-reflex,
equal_wf,
squash_wf,
true_wf,
fset-size_wf,
fset-remove_wf,
subtype_rel_self,
istype-nat,
fset_wf,
iff_weakening_equal,
subtract_wf,
istype-int,
fset-member_wf,
deq_wf,
istype-universe,
assert-deq-member,
bnot_wf,
length_wf,
remove-repeats_wf,
length-remove-first,
l_member_wf,
remove-repeats-filter,
l_all_iff,
not_wf,
assert_wf,
member-remove-repeats,
safe-assert-deq,
subtype_base_sq,
int_subtype_base,
remove-repeats-no_repeats,
list_induction,
no_repeats_wf,
filter_wf5,
list_ind_wf,
nil_wf,
bool_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
cons_wf,
filter_nil_lemma,
list_ind_nil_lemma,
filter_cons_lemma,
list_ind_cons_lemma,
eqtt_to_assert,
no_repeats_cons,
cons_member,
filter_trivial,
iff_transitivity,
eqof_wf,
iff_weakening_uiff,
assert_of_bnot,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
Error :universeIsType,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
promote_hyp,
Error :lambdaFormation_alt,
Error :inhabitedIsType,
pointwiseFunctionality,
sqequalRule,
pertypeElimination,
productElimination,
Error :productIsType,
Error :equalityIsType4,
dependent_functionElimination,
applyEquality,
Error :lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
intEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
setElimination,
rename,
independent_isectElimination,
independent_functionElimination,
closedConclusion,
instantiate,
universeEquality,
Error :isect_memberEquality_alt,
axiomEquality,
Error :isectIsTypeImplies,
Error :setIsType,
unionElimination,
voidElimination,
cumulativity,
independent_pairFormation,
Error :equalityIsType1,
lambdaFormation,
lambdaEquality,
functionEquality,
productEquality,
setEquality,
equalityElimination,
dependent_pairFormation,
isect_memberEquality,
voidEquality,
addLevel,
impliesFunctionality,
hyp_replacement,
dependent_set_memberEquality,
applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[s:fset(T)]. \mforall{}[x:T].
||fset-remove(eq;x;s)|| = (||s|| - 1) supposing x \mmember{} s
Date html generated:
2019_06_20-PM-01_59_48
Last ObjectModification:
2018_11_22-AM-10_00_28
Theory : finite!sets
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