### Nuprl Lemma : fpf-split

`∀[A:Type]`
`  ∀eq:EqDecider(A)`
`    ∀[B:A ⟶ Type]`
`      ∀f:a:A fp-> B[a]`
`        ∀[P:A ⟶ ℙ]`
`          ((∀a:A. Dec(P[a]))`
`          `` (∃fp,fnp:a:A fp-> B[a]`
`               ((f ⊆ fp ⊕ fnp ∧ fp ⊕ fnp ⊆ f)`
`               ∧ ((∀a:A. P[a] supposing ↑a ∈ dom(fp)) ∧ (∀a:A. ¬P[a] supposing ↑a ∈ dom(fnp)))`
`               ∧ fpf-domain(fp) ⊆ fpf-domain(f)`
`               ∧ fpf-domain(fnp) ⊆ fpf-domain(f))))`

Proof

Definitions occuring in Statement :  fpf-join: `f ⊕ g` fpf-sub: `f ⊆ g` fpf-domain: `fpf-domain(f)` fpf-dom: `x ∈ dom(f)` fpf: `a:A fp-> B[a]` sublist: `L1 ⊆ L2` deq: `EqDecider(T)` assert: `↑b` decidable: `Dec(P)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` implies: `P `` Q` and: `P ∧ Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` fpf: `a:A fp-> B[a]` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` uimplies: `b supposing a` iff: `P `⇐⇒` Q` and: `P ∧ Q` exists: `∃x:A. B[x]` cand: `A c∧ B` top: `Top` not: `¬A` false: `False` guard: `{T}` fpf-join: `f ⊕ g` fpf-sub: `f ⊆ g` pi1: `fst(t)` fpf-cap: `f(x)?z` fpf-dom: `x ∈ dom(f)` rev_implies: `P `` Q` or: `P ∨ Q` decidable: `Dec(P)` fpf-domain: `fpf-domain(f)`
Lemmas referenced :  all_wf decidable_wf fpf_wf deq_wf l_member_wf filter_wf5 dcdr-to-bool_wf subtype_rel_dep_function subtype_rel_sets member_filter_2 subtype_rel_self set_wf bnot_wf assert_witness fpf-dom_wf top_wf assert_wf fpf-sub_wf fpf-join_wf isect_wf subtype-fpf2 not_wf sublist_wf fpf-domain_wf exists_wf fpf_ap_pair_lemma assert-deq-member append_wf deq-member_wf trivial-ifthenelse trivial-equal member_append member_filter or_wf dcdr-to-bool-equivalence iff_transitivity iff_weakening_uiff assert_of_bnot filter_is_sublist
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalHypSubstitution productElimination thin rename cut introduction extract_by_obid isectElimination hypothesisEquality sqequalRule lambdaEquality applyEquality hypothesis functionEquality cumulativity universeEquality dependent_pairEquality setEquality setElimination functionExtensionality because_Cache independent_isectElimination dependent_functionElimination independent_functionElimination dependent_pairFormation equalityTransitivity equalitySymmetry independent_pairFormation isect_memberEquality voidElimination voidEquality instantiate productEquality independent_pairEquality axiomEquality addLevel orFunctionality impliesFunctionality andLevelFunctionality impliesLevelFunctionality unionElimination inlFormation inrFormation dependent_set_memberEquality promote_hyp

Latex:
\mforall{}[A:Type]
\mforall{}eq:EqDecider(A)
\mforall{}[B:A  {}\mrightarrow{}  Type]
\mforall{}f:a:A  fp->  B[a]
\mforall{}[P:A  {}\mrightarrow{}  \mBbbP{}]
((\mforall{}a:A.  Dec(P[a]))
{}\mRightarrow{}  (\mexists{}fp,fnp:a:A  fp->  B[a]
((f  \msubseteq{}  fp  \moplus{}  fnp  \mwedge{}  fp  \moplus{}  fnp  \msubseteq{}  f)
\mwedge{}  ((\mforall{}a:A.  P[a]  supposing  \muparrow{}a  \mmember{}  dom(fp))  \mwedge{}  (\mforall{}a:A.  \mneg{}P[a]  supposing  \muparrow{}a  \mmember{}  dom(fnp)))
\mwedge{}  fpf-domain(fp)  \msubseteq{}  fpf-domain(f)
\mwedge{}  fpf-domain(fnp)  \msubseteq{}  fpf-domain(f))))

Date html generated: 2018_05_21-PM-09_24_58
Last ObjectModification: 2018_05_19-PM-04_38_11

Theory : finite!partial!functions

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