Nuprl Lemma : count_diff
∀s:DSet. ∀as,bs:|s| List. ∀c:|s|. ((c #∈ (as - bs)) = ((c #∈ as) -- (c #∈ bs)) ∈ ℤ)
Proof
Definitions occuring in Statement :
diff: as - bs,
count: a #∈ as,
ndiff: a -- b,
list: T List,
all: ∀x:A. B[x],
int: ℤ,
equal: s = t ∈ T,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
dset: DSet,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
prop: ℙ,
top: Top,
nat: ℕ,
and: P ∧ Q,
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
infix_ap: x f y
Lemmas referenced :
list_induction,
set_car_wf,
all_wf,
list_wf,
equal_wf,
count_wf,
diff_wf,
ndiff_wf,
dset_wf,
diff_nil_lemma,
count_nil_lemma,
count_bounds,
le_wf,
squash_wf,
true_wf,
ndiff_id_r,
iff_weakening_equal,
diff_cons_lemma,
count_cons_lemma,
remove1_wf,
b2i_wf,
infix_ap_wf,
bool_wf,
set_eq_wf,
ndiff_ndiff,
count_remove1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
setElimination,
rename,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality,
intEquality,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
productElimination,
natural_numberEquality,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
addEquality
Latex:
\mforall{}s:DSet. \mforall{}as,bs:|s| List. \mforall{}c:|s|. ((c \#\mmember{} (as - bs)) = ((c \#\mmember{} as) -- (c \#\mmember{} bs)))
Date html generated:
2017_10_01-AM-09_56_39
Last ObjectModification:
2017_03_03-PM-00_57_44
Theory : list_2
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