### Nuprl Lemma : fpf-compatible-join

`∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[f,g,h:a:A fp-> B[a]].  (h || f ⊕ g) supposing (h || g and h || f)`

Proof

Definitions occuring in Statement :  fpf-join: `f ⊕ g` fpf-compatible: `f || g` fpf: `a:A fp-> B[a]` deq: `EqDecider(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` fpf-compatible: `f || g` all: `∀x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` top: `Top` prop: `ℙ` cand: `A c∧ B` squash: `↓T` true: `True` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` or: `P ∨ Q` not: `¬A` false: `False`
Lemmas referenced :  assert_wf fpf-dom_wf subtype-fpf2 top_wf fpf-join_wf fpf-compatible_wf fpf_wf deq_wf bool_wf equal-wf-T-base bnot_wf not_wf equal_wf squash_wf true_wf fpf-ap_wf fpf-join-ap iff_weakening_equal eqtt_to_assert uiff_transitivity eqff_to_assert assert_of_bnot fpf-join-dom
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution lambdaFormation hypothesisEquality sqequalRule lambdaEquality dependent_functionElimination thin axiomEquality hypothesis productEquality extract_by_obid isectElimination cumulativity applyEquality functionExtensionality independent_isectElimination isect_memberEquality voidElimination voidEquality because_Cache equalityTransitivity equalitySymmetry functionEquality universeEquality independent_functionElimination independent_pairFormation baseClosed productElimination imageElimination natural_numberEquality imageMemberEquality unionElimination equalityElimination

Latex:
\mforall{}[A:Type].  \mforall{}[eq:EqDecider(A)].  \mforall{}[B:A  {}\mrightarrow{}  Type].  \mforall{}[f,g,h:a:A  fp->  B[a]].
(h  ||  f  \moplus{}  g)  supposing  (h  ||  g  and  h  ||  f)

Date html generated: 2018_05_21-PM-09_28_24
Last ObjectModification: 2018_02_09-AM-10_23_43

Theory : finite!partial!functions

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