Nuprl Lemma : fpf-vals-singleton
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ─→ Type]. ∀[P:A ─→ 𝔹]. ∀[f:x:A fp-> B[x]]. ∀[a:A].
(fpf-vals(eq;P;f) = [<a, f(a)>] ∈ ((x:A × B[x]) List)) supposing ((∀b:A. (↑(P b) ⇐⇒ b = a ∈ A)) and (↑a ∈ dom(f)))
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),
cons: [a / b],
nil: [],
list: T List,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: 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 :
fpf_ap_pair_lemma,
all_wf,
iff_wf,
assert_wf,
equal_wf,
fpf-dom_wf,
subtype-fpf2,
top_wf,
subtype_top,
fpf_wf,
bool_wf,
deq_wf,
remove-repeats_property,
assert-deq-member,
deq-member_wf,
equal-wf-T-base,
l_member_wf,
bnot_wf,
not_wf,
cons_wf,
iff_transitivity,
iff_weakening_uiff,
eqtt_to_assert,
eqff_to_assert,
assert_of_bnot,
nil_wf,
nil_member,
false_wf,
filter_nil_lemma,
no_repeats_wf,
cons_member,
filter_cons_lemma,
no_repeats_cons,
uiff_transitivity,
or_wf,
list_induction,
filter_wf5,
subtype_rel_dep_function,
subtype_rel_self,
set_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
list_wf,
and_wf,
remove-repeats_wf,
bool_cases,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
colength_wf_list,
list-cases,
equal-wf-base-T,
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,
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,
set_subtype_base,
int_subtype_base,
reduce_hd_cons_lemma,
hd_wf,
squash_wf,
length_wf,
listp_properties,
cons_wf_listp,
null_nil_lemma,
btrue_wf,
reduce_tl_cons_lemma,
tl_wf,
null_wf3,
subtype_rel_list,
null_cons_lemma,
bfalse_wf,
btrue_neq_bfalse,
map_cons_lemma,
map_nil_lemma,
zip_cons_cons_lemma,
zip_nil_lemma,
member-remove-repeats
\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{}[a:A].
(fpf-vals(eq;P;f) = [<a, f(a)>]) supposing ((\mforall{}b:A. (\muparrow{}(P b) \mLeftarrow{}{}\mRightarrow{} b = a)) and (\muparrow{}a \mmember{} dom(f)))
Date html generated:
2015_07_17-AM-11_09_52
Last ObjectModification:
2015_01_28-AM-07_48_31
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