Nuprl Lemma : count_remove1
∀s:DSet. ∀as:|s| List. ∀b,c:|s|. ((c #∈ (as \ b)) = ((c #∈ as) -- b2i(b (=b) c)) ∈ ℤ)
Proof
Definitions occuring in Statement :
count: a #∈ as,
remove1: as \ a,
ndiff: a -- b,
list: T List,
b2i: b2i(b),
infix_ap: x f y,
all: ∀x:A. B[x],
int: ℤ,
equal: s = t ∈ T,
dset: DSet,
set_eq: =b,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
dset: DSet,
or: P ∨ Q,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
subtype_rel: A ⊆r B,
infix_ap: x f y,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
b2i: b2i(b)
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
set_car_wf,
list-cases,
remove1_nil_lemma,
count_nil_lemma,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
le_wf,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
remove1_cons_lemma,
count_cons_lemma,
nat_wf,
list_wf,
dset_wf,
ndiff_ann_l,
b2i_wf,
set_eq_wf,
b2i_bounds,
equal-wf-T-base,
bool_wf,
assert_wf,
equal_wf,
bnot_wf,
not_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_dset_eq,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
count_wf,
count_bounds,
istype-universe,
ndiff_inv,
iff_weakening_equal,
ndiff_wf,
add_com,
squash_wf,
true_wf,
add_functionality_wrt_eq,
subtype_rel_self,
equal-wf-base,
infix_ap_wf,
le_weakening2,
non_neg_length,
remove1_wf,
decidable__lt,
ndiff_id_r,
zero-add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
because_Cache,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityIsType1,
dependent_set_memberEquality_alt,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
imageElimination,
equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
equalityElimination,
universeEquality,
imageMemberEquality,
hyp_replacement,
addEquality
Latex:
\mforall{}s:DSet. \mforall{}as:|s| List. \mforall{}b,c:|s|. ((c \#\mmember{} (as \mbackslash{} b)) = ((c \#\mmember{} as) -- b2i(b (=\msubb{}) c)))
Date html generated:
2019_10_16-PM-01_04_16
Last ObjectModification:
2018_10_08-AM-11_19_46
Theory : list_2
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