Nuprl Lemma : member-fpf-vals
∀[A:Type]
∀eq:EqDecider(A)
∀[B:A ─→ Type]
∀P:A ─→ 𝔹. ∀f:x:A fp-> B[x]. ∀x:A. ∀v:B[x].
((<x, v> ∈ fpf-vals(eq;P;f)) ⇐⇒ {((↑x ∈ dom(f)) ∧ (↑(P x))) ∧ (v = f(x) ∈ B[x])})
Proof
Definitions occuring in Statement :
fpf-vals: fpf-vals(eq;P;f),
fpf-ap: f(x),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
l_member: (x ∈ l),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ─→ B[x],
pair: <a, b>,
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Lemmas :
l_member_wf,
member-remove-repeats,
list_induction,
filter_nil_lemma,
deq_member_nil_lemma,
zip_nil_lemma,
filter_cons_lemma,
deq_member_cons_lemma,
all_wf,
iff_wf,
assert_wf,
deq-member_wf,
list_wf,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
equal-wf-T-base,
colength_wf_list,
list-cases,
nil_wf,
product_subtype_list,
spread_cons_lemma,
sq_stable__le,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
nat_wf,
decidable__le,
false_wf,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-commutes,
le_wf,
subtract_wf,
not-ge-2,
less-iff-le,
minus-minus,
add-swap,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
bool_wf,
eqtt_to_assert,
map_cons_lemma,
zip_cons_cons_lemma,
cons_wf,
cons_member,
subtype_rel_dep_function,
subtype_rel_sets,
equal_wf,
subtype_rel_self,
set_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
assert-deq-member,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
assert_witness,
list-subtype,
subtype_rel_list_set,
bnot_wf,
not_wf,
uiff_transitivity,
assert_of_bnot,
subtype_rel_list,
subtype_rel_product,
bor_wf,
iff_transitivity,
eqof_wf,
or_wf,
iff_weakening_uiff,
assert_of_bor,
safe-assert-deq,
assert_elim,
and_wf,
subtype_rel_wf,
not_assert_elim,
true_wf
\mforall{}[A:Type]
\mforall{}eq:EqDecider(A)
\mforall{}[B:A {}\mrightarrow{} Type]
\mforall{}P:A {}\mrightarrow{} \mBbbB{}. \mforall{}f:x:A fp-> B[x]. \mforall{}x:A. \mforall{}v:B[x].
((<x, v> \mmember{} fpf-vals(eq;P;f)) \mLeftarrow{}{}\mRightarrow{} \{((\muparrow{}x \mmember{} dom(f)) \mwedge{} (\muparrow{}(P x))) \mwedge{} (v = f(x))\})
Date html generated:
2015_07_17-AM-11_09_27
Last ObjectModification:
2015_01_28-AM-07_54_18
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