Nuprl Lemma : do-apply-p-co-restrict

`∀[A,B:Type]. ∀[f:A ⟶ (B + Top)]. ∀[P:A ⟶ ℙ]. ∀[p:∀x:A. Dec(P[x])]. ∀[x:A].`
`  do-apply(p-co-restrict(f;p);x) = do-apply(f;x) ∈ B supposing ↑can-apply(p-co-restrict(f;p);x)`

Proof

Definitions occuring in Statement :  p-co-restrict: `p-co-restrict(f;p)` 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]` top: `Top` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` function: `x:A ⟶ B[x]` union: `left + right` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  p-co-restrict: `p-co-restrict(f;p)` uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` and: `P ∧ Q` squash: `↓T` true: `True` guard: `{T}` uiff: `uiff(P;Q)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q`
Lemmas referenced :  assert_wf can-apply_wf p-compose_wf top_wf p-co-filter_wf all_wf decidable_wf equal_wf do-apply_wf do-apply-p-co-filter assert_functionality_wrt_uiff squash_wf true_wf iff_weakening_equal do-apply-compose can-apply-compose-iff
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation introduction cut hypothesis extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality lambdaEquality applyEquality functionExtensionality because_Cache isect_memberEquality axiomEquality equalityTransitivity equalitySymmetry functionEquality universeEquality productElimination imageElimination independent_isectElimination imageMemberEquality baseClosed unionEquality natural_numberEquality independent_functionElimination

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

Date html generated: 2017_10_01-AM-09_14_21
Last ObjectModification: 2017_07_26-PM-04_49_28

Theory : general

Home Index