### Nuprl Lemma : bag-maximal?-cons

`∀[T:Type]. ∀[b:bag(T)]. ∀[R:T ⟶ T ⟶ 𝔹]. ∀[x,v:T].  uiff(↑bag-maximal?(v.b;x;R);(↑bag-maximal?(b;x;R)) ∧ (↑(R x v)))`

Proof

Definitions occuring in Statement :  bag-maximal?: `bag-maximal?(bg;x;R)` cons-bag: `x.b` bag: `bag(T)` assert: `↑b` bool: `𝔹` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  single-bag: `{x}` bag-append: `as + bs` cons-bag: `x.b` append: `as @ bs` all: `∀x:A. B[x]` so_lambda: `so_lambda(x,y,z.t[x; y; z])` member: `t ∈ T` top: `Top` so_apply: `x[s1;s2;s3]` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` implies: `P `` Q` prop: `ℙ` rev_uimplies: `rev_uimplies(P;Q)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  list_ind_cons_lemma list_ind_nil_lemma bag-maximal?-single assert_witness bag-maximal?_wf single-bag_wf and_wf assert_wf iff_weakening_uiff bag-append_wf bag-maximal?-append uiff_wf cons-bag_wf bool_wf bag_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity sqequalRule cut lemma_by_obid sqequalHypSubstitution dependent_functionElimination thin isect_memberEquality voidElimination voidEquality hypothesis independent_pairFormation isect_memberFormation introduction productElimination isectElimination hypothesisEquality independent_isectElimination independent_pairEquality independent_functionElimination applyEquality because_Cache addLevel cumulativity functionEquality universeEquality equalityTransitivity equalitySymmetry

Latex:
\mforall{}[T:Type].  \mforall{}[b:bag(T)].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[x,v:T].
uiff(\muparrow{}bag-maximal?(v.b;x;R);(\muparrow{}bag-maximal?(b;x;R))  \mwedge{}  (\muparrow{}(R  x  v)))

Date html generated: 2016_05_15-PM-02_30_36
Last ObjectModification: 2015_12_27-AM-09_48_48

Theory : bags

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