Nuprl Lemma : fset-find_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[P:T ⟶ 𝔹]. ∀[s:fset(T)].
(fset-find(P;s) ∈ {x:T| x ∈ s ∧ (↑(P x))} ) supposing
((∀x,y:T. (x ∈ s ⇒ y ∈ s ⇒ (↑(P x)) ⇒ (↑(P y)) ⇒ (x = y ∈ T))) and
(↓∃x:T. (x ∈ s ∧ (↑(P x)))))
Proof
Definitions occuring in Statement :
fset-find: fset-find(P;s),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
so_apply: x[s],
and: P ∧ Q,
fset: fset(T),
quotient: x,y:A//B[x; y],
fset-find: fset-find(P;s),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
fset-member: a ∈ s,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
set-equal: set-equal(T;x;y),
cand: A c∧ B
Lemmas referenced :
all_wf,
fset-member_wf,
assert_wf,
equal_wf,
squash_wf,
exists_wf,
fset_wf,
bool_wf,
deq_wf,
equal-wf-base,
list_wf,
set-equal_wf,
quotient-member-eq,
set-equal-equiv,
true_wf,
iff_weakening_equal,
assert-deq-member,
l_member_wf,
deq-member_wf,
filter_type,
subtype_rel_dep_function,
subtype_rel_self,
set_wf,
list-subtype,
hd_wf,
length-filter-pos,
l_exists_iff,
decidable__le,
length_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
subtype_rel_sets
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
imageElimination,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
lambdaEquality,
functionEquality,
applyEquality,
functionExtensionality,
isect_memberEquality,
because_Cache,
productEquality,
universeEquality,
pointwiseFunctionalityForEquality,
setEquality,
pertypeElimination,
productElimination,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
addLevel,
existsFunctionality,
independent_pairFormation,
natural_numberEquality,
imageMemberEquality,
baseClosed,
promote_hyp,
allFunctionality,
lambdaFormation,
andLevelFunctionality,
setElimination,
rename,
dependent_set_memberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
computeAll,
applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[s:fset(T)].
(fset-find(P;s) \mmember{} \{x:T| x \mmember{} s \mwedge{} (\muparrow{}(P x))\} ) supposing
((\mforall{}x,y:T. (x \mmember{} s {}\mRightarrow{} y \mmember{} s {}\mRightarrow{} (\muparrow{}(P x)) {}\mRightarrow{} (\muparrow{}(P y)) {}\mRightarrow{} (x = y))) and
(\mdownarrow{}\mexists{}x:T. (x \mmember{} s \mwedge{} (\muparrow{}(P x)))))
Date html generated:
2017_02_20-AM-10_49_10
Last ObjectModification:
2017_02_02-PM-05_25_18
Theory : finite!sets
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