Nuprl Lemma : count_append
∀s:DSet. ∀as,bs:|s| List. ∀c:|s|. ((c #∈ (as @ bs)) = ((c #∈ as) + (c #∈ bs)) ∈ ℤ)
Proof
Definitions occuring in Statement :
count: a #∈ as,
append: as @ bs,
list: T List,
all: ∀x:A. B[x],
add: n + m,
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,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
uiff: uiff(P;Q),
and: P ∧ Q,
subtype_rel: A ⊆r B,
guard: {T},
infix_ap: x f y
Lemmas referenced :
list_induction,
set_car_wf,
equal_wf,
count_wf,
append_wf,
list_wf,
list_ind_nil_lemma,
count_nil_lemma,
list_ind_cons_lemma,
count_cons_lemma,
dset_wf,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
add-is-int-iff,
b2i_wf,
infix_ap_wf,
bool_wf,
set_eq_wf,
false_wf
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,
addEquality,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
productElimination,
applyEquality,
hyp_replacement,
applyLambdaEquality
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_01_09-AM-08_37_30
Last ObjectModification:
2016_07_12-PM-01_12_32
Theory : list_2
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