### Nuprl Lemma : do-apply-p-co-filter

`∀[T:Type]. ∀[P:T ⟶ ℙ]. ∀[f:∀x:T. Dec(P[x])]. ∀[x:T].`
`  do-apply(p-co-filter(f);x) = x ∈ T supposing ↑can-apply(p-co-filter(f);x)`

Proof

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

Latex:
\mforall{}[T:Type].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[f:\mforall{}x:T.  Dec(P[x])].  \mforall{}[x:T].
do-apply(p-co-filter(f);x)  =  x  supposing  \muparrow{}can-apply(p-co-filter(f);x)

Date html generated: 2016_05_15-PM-03_31_08
Last ObjectModification: 2015_12_27-PM-01_11_10

Theory : general

Home Index