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],
l_member: (x ∈ l),
deq: EqDecider(T),
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
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
fpf: a:A fp-> B[a],
fpf-ap: f(x),
fpf-dom: x ∈ dom(f),
fpf-vals: fpf-vals(eq;P;f),
pi1: fst(t),
pi2: snd(t),
let: let,
member: t ∈ T,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
sq_type: SQType(T),
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
istype: istype(T),
uiff: uiff(P;Q),
btrue: tt,
unit: Unit,
bool: 𝔹,
subtype_rel: A ⊆r B,
decidable: Dec(P),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
squash: ↓T,
less_than: a < b,
it: ⋅,
nil: [],
colength: colength(L),
less_than': less_than'(a;b),
le: A ≤ B,
cons: [a / b],
or: P ∨ Q,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
ge: i ≥ j ,
false: False,
nat: ℕ,
cand: A c∧ B,
deq: EqDecider(T),
eqof: eqof(d),
true: True,
label: ...$L... t,
rev_uimplies: rev_uimplies(P;Q),
bor: p ∨bq
Lemmas referenced :
subtype_base_sq,
bool_wf,
bool_subtype_base,
iff_imp_equal_bool,
deq-member_wf,
remove-repeats_wf,
member-remove-repeats,
l_member_wf,
assert-deq-member,
istype-assert,
fpf_wf,
deq_wf,
istype-universe,
list_induction,
filter_nil_lemma,
istype-void,
deq_member_nil_lemma,
zip_nil_lemma,
filter_cons_lemma,
deq_member_cons_lemma,
equal_wf,
assert_wf,
iff_wf,
all_wf,
nat_wf,
assert-bnot,
bool_cases_sqequal,
eqff_to_assert,
subtype_rel_sets,
subtype_rel_dep_function,
cons_member,
cons_wf,
zip_cons_cons_lemma,
map_cons_lemma,
eqtt_to_assert,
decidable__le,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermAdd_wf,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__equal_int,
spread_cons_lemma,
int_subtype_base,
set_subtype_base,
int_formula_prop_eq_lemma,
intformeq_wf,
subtract-1-ge-0,
le_wf,
istype-false,
colength_wf_list,
colength-cons-not-zero,
product_subtype_list,
nil_wf,
list-cases,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
istype-int,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
false_wf,
guard_wf,
assert_witness,
btrue_neq_bfalse,
member-implies-null-eq-bfalse,
btrue_wf,
null_nil_lemma,
list-subtype,
subtype_rel_list,
not_wf,
bnot_wf,
equal-wf-T-base,
list_wf,
set_wf,
uiff_transitivity,
assert_of_bnot,
subtype_rel_product,
bor_wf,
iff_transitivity,
eqof_wf,
iff_weakening_uiff,
assert_of_bor,
safe-assert-deq,
assert_elim,
iff_weakening_equal,
subtype_rel_self,
true_wf,
squash_wf,
assert_functionality_wrt_uiff,
or_wf,
pi1_wf_top,
subtype_rel-equal,
not_assert_elim
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
cut,
instantiate,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesis,
independent_isectElimination,
hypothesisEquality,
independent_pairFormation,
dependent_functionElimination,
independent_functionElimination,
universeIsType,
because_Cache,
promote_hyp,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
equalityIstype,
applyEquality,
lambdaEquality_alt,
functionIsType,
universeEquality,
setIsType,
dependent_set_memberEquality_alt,
rename,
setElimination,
functionExtensionality_alt,
applyLambdaEquality,
hyp_replacement,
isect_memberEquality_alt,
voidElimination,
dependent_pairEquality_alt,
productEquality,
setEquality,
functionEquality,
inrFormation_alt,
inlFormation_alt,
equalityElimination,
intEquality,
baseClosed,
closedConclusion,
baseApply,
equalityIsType4,
imageElimination,
equalityIsType1,
hypothesis_subsumption,
unionElimination,
functionIsTypeImplies,
axiomEquality,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
natural_numberEquality,
intWeakElimination,
independent_pairEquality,
unionEquality,
unionIsType,
imageMemberEquality,
productIsType
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{}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:
2019_10_16-AM-11_26_03
Last ObjectModification:
2019_06_25-PM-03_26_35
Theory : finite!partial!functions
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