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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