### Nuprl Lemma : do-apply-p-lift

`∀[A,B:Type]. ∀[P:A ⟶ ℙ]. ∀[d:x:A ⟶ Dec(P[x])]. ∀[f:{x:A| P[x]}  ⟶ B]. ∀[x:A].`
`  do-apply(p-lift(d;f);x) = (f x) ∈ B supposing ↑can-apply(p-lift(d;f);x)`

Proof

Definitions occuring in Statement :  p-lift: `p-lift(d;f)` do-apply: `do-apply(f;x)` can-apply: `can-apply(f;x)` assert: `↑b` decidable: `Dec(P)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` set: `{x:A| B[x]} ` apply: `f a` 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` p-lift: `p-lift(d;f)` do-apply: `do-apply(f;x)` can-apply: `can-apply(f;x)` implies: `P `` Q` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` all: `∀x:A. B[x]` top: `Top` decidable: `Dec(P)` or: `P ∨ Q` isl: `isl(x)` outl: `outl(x)` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` bfalse: `ff` false: `False`
Lemmas referenced :  assert_wf can-apply_wf p-lift_wf top_wf subtype_rel_dep_function set_wf decidable_wf true_wf false_wf equal_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin sqequalRule sqequalHypSubstitution independent_functionElimination hypothesis lemma_by_obid isectElimination hypothesisEquality lambdaEquality applyEquality setEquality universeEquality because_Cache independent_isectElimination lambdaFormation isect_memberEquality voidElimination voidEquality axiomEquality equalityTransitivity equalitySymmetry functionEquality cumulativity unionElimination dependent_set_memberEquality dependent_functionElimination

Latex:
\mforall{}[A,B:Type].  \mforall{}[P:A  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[d:x:A  {}\mrightarrow{}  Dec(P[x])].  \mforall{}[f:\{x:A|  P[x]\}    {}\mrightarrow{}  B].  \mforall{}[x:A].
do-apply(p-lift(d;f);x)  =  (f  x)  supposing  \muparrow{}can-apply(p-lift(d;f);x)

Date html generated: 2016_05_15-PM-03_29_25
Last ObjectModification: 2015_12_27-PM-01_09_45

Theory : general

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