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],
cons: [a / b],
nil: [],
list: T List,
deq: EqDecider(T),
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
Definitions unfolded in proof :
fpf-vals: fpf-vals(eq;P;f),
let: let,
fpf: a:A fp-> B[a],
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
prop: ℙ,
and: P ∧ Q,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
fpf-dom: x ∈ dom(f),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
not: ¬A,
false: False,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
nil: [],
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b)
Lemmas referenced :
fpf_ap_pair_lemma,
all_wf,
iff_wf,
assert_wf,
equal_wf,
fpf-dom_wf,
subtype-fpf2,
top_wf,
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,
nil_wf,
iff_transitivity,
iff_weakening_uiff,
eqtt_to_assert,
eqff_to_assert,
assert_of_bnot,
list_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,
and_wf,
remove-repeats_wf,
bool_cases,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_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,
ge_wf,
less_than_wf,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
equal-wf-base-T,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
reduce_hd_cons_lemma,
hd_wf,
squash_wf,
length_wf,
length_cons_ge_one,
subtype_rel_list,
null_nil_lemma,
btrue_wf,
reduce_tl_cons_lemma,
tl_wf,
null_wf3,
null_cons_lemma,
bfalse_wf,
btrue_neq_bfalse,
map_cons_lemma,
map_nil_lemma,
zip_cons_cons_lemma,
zip_nil_lemma,
member-remove-repeats
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isectElimination,
cumulativity,
hypothesisEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
because_Cache,
independent_isectElimination,
lambdaFormation,
functionEquality,
universeEquality,
isect_memberFormation,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
independent_pairFormation,
hyp_replacement,
applyLambdaEquality,
baseClosed,
unionElimination,
equalityElimination,
impliesFunctionality,
promote_hyp,
setEquality,
setElimination,
rename,
dependent_pairFormation,
instantiate,
dependent_set_memberEquality,
inrFormation,
inlFormation,
intWeakElimination,
natural_numberEquality,
int_eqEquality,
intEquality,
computeAll,
hypothesis_subsumption,
addEquality,
imageElimination,
imageMemberEquality,
productEquality,
dependent_pairEquality
Latex:
\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:
2018_05_21-PM-09_26_08
Last ObjectModification:
2018_02_09-AM-10_21_37
Theory : finite!partial!functions
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