### Nuprl Lemma : fpf-join-ap-left

`∀[A:Type]. ∀[B,C:A ⟶ Type]. ∀[eq:EqDecider(A)]. ∀[f:a:A fp-> B[a]]. ∀[g:a:A fp-> C[a]]. ∀[x:A].`
`  f ⊕ g(x) = f(x) ∈ B[x] supposing ↑x ∈ dom(f)`

Proof

Definitions occuring in Statement :  fpf-join: `f ⊕ g` fpf-ap: `f(x)` fpf-dom: `x ∈ dom(f)` fpf: `a:A fp-> B[a]` deq: `EqDecider(T)` assert: `↑b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` fpf-ap: `f(x)` fpf-join: `f ⊕ g` pi2: `snd(t)` fpf-cap: `f(x)?z` prop: `ℙ` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` all: `∀x:A. B[x]` top: `Top` not: `¬A` implies: `P `` Q` false: `False` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` ifthenelse: `if b then t else f fi ` bfalse: `ff`
Lemmas referenced :  assert_wf fpf-dom_wf subtype-fpf2 top_wf fpf_wf deq_wf bool_wf fpf-ap_wf equal-wf-T-base bnot_wf not_wf eqtt_to_assert uiff_transitivity eqff_to_assert assert_of_bnot equal_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule hypothesis extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality applyEquality lambdaEquality functionExtensionality independent_isectElimination lambdaFormation isect_memberEquality voidElimination voidEquality because_Cache axiomEquality equalityTransitivity equalitySymmetry baseClosed independent_functionElimination unionElimination equalityElimination productElimination dependent_functionElimination

Latex:
\mforall{}[A:Type].  \mforall{}[B,C:A  {}\mrightarrow{}  Type].  \mforall{}[eq:EqDecider(A)].  \mforall{}[f:a:A  fp->  B[a]].  \mforall{}[g:a:A  fp->  C[a]].  \mforall{}[x:A].
f  \moplus{}  g(x)  =  f(x)  supposing  \muparrow{}x  \mmember{}  dom(f)

Date html generated: 2018_05_21-PM-09_21_48
Last ObjectModification: 2018_02_09-AM-10_18_27

Theory : finite!partial!functions

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