Nuprl Lemma : fset-size-union
∀[T:Type]
∀[eq:EqDecider(T)]. ∀[a,b:fset(T)]. (||a ⋃ b|| = ((||a|| + ||b||) - ||a ⋂ b||) ∈ ℤ) supposing valueall-type(T)
Proof
Definitions occuring in Statement :
fset-size: ||s||,
fset-intersection: a ⋂ b,
fset-union: x ⋃ y,
fset: fset(T),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
subtract: n - m,
add: n + m,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
implies: P ⇒ Q,
prop: ℙ,
guard: {T},
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
cand: A c∧ B
Lemmas referenced :
fset-induction,
all_wf,
fset_wf,
equal_wf,
fset-size_wf,
fset-union_wf,
subtract_wf,
fset-intersection_wf,
nat_wf,
sq_stable__all,
sq_stable__equal,
not_wf,
fset-member_wf,
deq_wf,
valueall-type_wf,
squash_wf,
true_wf,
empty-fset-union,
iff_weakening_equal,
fsize_empty_lemma,
empty_intersect_lemma,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermSubtract_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
empty-fset_wf,
fset-add_wf,
fset-add-as-cons,
fset-add-union,
decidable__fset-member,
and_wf,
fset-extensionality,
sq_stable_from_decidable,
decidable__or,
or_wf,
member-fset-union,
fset-member_witness,
uiff_wf,
member-fset-add,
decidable-equal-deq,
decidable__and2,
iff_weakening_uiff,
member-fset-intersection,
subtract-is-int-iff,
add-is-int-iff,
intformand_wf,
int_formula_prop_and_lemma,
false_wf,
add_functionality_wrt_eq,
fset-size-add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
intEquality,
because_Cache,
applyEquality,
addEquality,
setElimination,
rename,
independent_functionElimination,
lambdaFormation,
axiomEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
voidElimination,
voidEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
hyp_replacement,
applyLambdaEquality,
inrFormation,
dependent_set_memberEquality,
independent_pairFormation,
addLevel,
orFunctionality,
independent_pairEquality,
productEquality,
inlFormation,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
impliesFunctionality
Latex:
\mforall{}[T:Type]
\mforall{}[eq:EqDecider(T)]. \mforall{}[a,b:fset(T)]. (||a \mcup{} b|| = ((||a|| + ||b||) - ||a \mcap{} b||))
supposing valueall-type(T)
Date html generated:
2017_04_17-AM-09_22_54
Last ObjectModification:
2017_02_27-PM-05_25_17
Theory : finite!sets
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