Nuprl Lemma : fpf-decompose
∀[A:Type]
∀eq:EqDecider(A)
∀[B:A ⟶ Type]
∀f:a:A fp-> B[a]
∃g:a:A fp-> B[a]
∃a:A
∃b:B[a]
((f ⊆ g ⊕ a : b ∧ g ⊕ a : b ⊆ f)
∧ (∀a':A. ¬(a' = a ∈ A) supposing ↑a' ∈ dom(g))
∧ ||fpf-domain(g)|| < ||fpf-domain(f)||)
supposing 0 < ||fpf-domain(f)||
Proof
Definitions occuring in Statement :
fpf-single: x : v
,
fpf-join: f ⊕ g
,
fpf-sub: f ⊆ g
,
fpf-domain: fpf-domain(f)
,
fpf-dom: x ∈ dom(f)
,
fpf: a:A fp-> B[a]
,
length: ||as||
,
deq: EqDecider(T)
,
assert: ↑b
,
less_than: a < b
,
uimplies: b supposing a
,
uall: ∀[x:A]. B[x]
,
so_apply: x[s]
,
all: ∀x:A. B[x]
,
exists: ∃x:A. B[x]
,
not: ¬A
,
and: P ∧ Q
,
function: x:A ⟶ B[x]
,
natural_number: $n
,
universe: Type
,
equal: s = t ∈ T
Definitions unfolded in proof :
rev_implies: P
⇐ Q
,
uiff: uiff(P;Q)
,
iff: P
⇐⇒ Q
,
cons: [a / b]
,
less_than': less_than'(a;b)
,
bfalse: ff
,
ifthenelse: if b then t else f fi
,
assert: ↑b
,
fpf-dom: x ∈ dom(f)
,
eqof: eqof(d)
,
pi1: fst(t)
,
fpf-domain: fpf-domain(f)
,
fpf: a:A fp-> B[a]
,
le: A ≤ B
,
prop: ℙ
,
false: False
,
exists: ∃x:A. B[x]
,
satisfiable_int_formula: satisfiable_int_formula(fmla)
,
implies: P
⇒ Q
,
not: ¬A
,
and: P ∧ Q
,
squash: ↓T
,
less_than: a < b
,
or: P ∨ Q
,
decidable: Dec(P)
,
ge: i ≥ j
,
top: Top
,
so_apply: x[s]
,
so_lambda: λ2x.t[x]
,
subtype_rel: A ⊆r B
,
member: t ∈ T
,
uimplies: b supposing a
,
all: ∀x:A. B[x]
,
uall: ∀[x:A]. B[x]
,
fpf-compatible: f || g
,
rev_uimplies: rev_uimplies(P;Q)
,
cand: A c∧ B
,
fpf-sub: f ⊆ g
,
btrue: tt
,
sq_type: SQType(T)
,
guard: {T}
,
true: True
,
fpf-cap: f(x)?z
,
fpf-join: f ⊕ g
Lemmas referenced :
deq_wf,
fpf_wf,
exists_wf,
not_wf,
fpf-dom_wf,
all_wf,
fpf-single_wf,
fpf-join_wf,
fpf-sub_wf,
fpf-ap_wf,
assert-deq-member,
safe-assert-deq,
assert_of_bor,
iff_weakening_uiff,
member_wf,
or_wf,
deq-member_wf,
bor_wf,
iff_transitivity,
l_member_wf,
deq_member_cons_lemma,
length_of_cons_lemma,
reduce_hd_cons_lemma,
product_subtype_list,
deq_member_nil_lemma,
length_of_nil_lemma,
list-cases,
equal_wf,
less_than_wf,
less_than'_wf,
list_wf,
decidable__assert,
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_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
hd_wf,
eqof_wf,
bnot_wf,
assert_wf,
fpf-split,
top_wf,
subtype-fpf2,
fpf-domain_wf,
length_wf,
member-less_than,
fpf-join-sub,
fpf-sub_transitivity,
fpf-sub-reflexive,
assert_of_bnot,
fpf_ap_single_lemma,
fpf-single-dom,
decidable-equal-deq,
eqff_to_assert,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases,
fpf-join-ap-sq,
true_wf,
squash_wf,
assert_functionality_wrt_uiff,
subtype_rel-equal,
istype-universe,
assert_elim,
fpf-single-dom-sq,
fpf-join-dom,
subtype_rel_self,
iff_weakening_equal,
assert_witness,
istype-assert,
and_wf,
fpf_ap_pair_lemma,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__lt,
proper_sublist_length,
decidable__equal_int,
length_sublist,
member-fpf-domain
Rules used in proof :
universeEquality,
functionEquality,
isectEquality,
instantiate,
functionExtensionality,
productEquality,
orFunctionality,
addLevel,
inlFormation,
hypothesis_subsumption,
promote_hyp,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
independent_pairEquality,
independent_pairFormation,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
productElimination,
imageElimination,
unionElimination,
cumulativity,
dependent_functionElimination,
rename,
because_Cache,
voidEquality,
voidElimination,
isect_memberEquality,
independent_isectElimination,
hypothesis,
lambdaEquality,
sqequalRule,
applyEquality,
hypothesisEquality,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
impliesFunctionality,
baseClosed,
imageMemberEquality,
dependent_set_memberEquality_alt,
equalityIstype,
universeIsType,
lambdaEquality_alt,
inhabitedIsType,
lambdaFormation_alt,
Error :memTop,
productIsType,
applyLambdaEquality,
setElimination,
hyp_replacement,
functionIsType,
dependent_set_memberEquality
Latex:
\mforall{}[A:Type]
\mforall{}eq:EqDecider(A)
\mforall{}[B:A {}\mrightarrow{} Type]
\mforall{}f:a:A fp-> B[a]
\mexists{}g:a:A fp-> B[a]
\mexists{}a:A
\mexists{}b:B[a]
((f \msubseteq{} g \moplus{} a : b \mwedge{} g \moplus{} a : b \msubseteq{} f)
\mwedge{} (\mforall{}a':A. \mneg{}(a' = a) supposing \muparrow{}a' \mmember{} dom(g))
\mwedge{} ||fpf-domain(g)|| < ||fpf-domain(f)||)
supposing 0 < ||fpf-domain(f)||
Date html generated:
2020_05_20-AM-09_03_14
Last ObjectModification:
2020_01_27-PM-04_20_44
Theory : finite!partial!functions
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