### Nuprl Lemma : bag-combine-null

`∀[A,B:Type]. ∀[f:A ⟶ bag(B)]. ∀[b:bag(A)].  uiff(↑bag-null(⋃x∈b.f[x]);∀x:A. (x ↓∈ b `` (↑bag-null(f[x]))))`

Proof

Definitions occuring in Statement :  bag-member: `x ↓∈ bs` bag-combine: `⋃x∈bs.f[x]` bag-null: `bag-null(bs)` bag: `bag(T)` assert: `↑b` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` so_apply: `x[s]` all: `∀x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` member: `t ∈ T` all: `∀x:A. B[x]` implies: `P `` Q` uall: `∀[x:A]. B[x]` so_apply: `x[s]` rev_uimplies: `rev_uimplies(P;Q)` prop: `ℙ` so_lambda: `λ2x.t[x]` not: `¬A` false: `False` exists: `∃x:A. B[x]` cand: `A c∧ B` squash: `↓T` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q`
Lemmas referenced :  bag-member-empty-iff iff_weakening_equal true_wf squash_wf and_wf bag-member-combine empty-bag-iff-no-member bag_wf all_wf bag-combine_wf bag-null_wf assert_wf assert_witness bag-member_wf assert-bag-null
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity independent_pairFormation isect_memberFormation introduction cut lambdaFormation lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality applyEquality hypothesis productElimination independent_isectElimination sqequalRule lambdaEquality dependent_functionElimination because_Cache independent_functionElimination functionEquality universeEquality independent_pairEquality isect_memberEquality equalityTransitivity equalitySymmetry dependent_pairFormation imageMemberEquality baseClosed voidElimination imageElimination natural_numberEquality

Latex:
\mforall{}[A,B:Type].  \mforall{}[f:A  {}\mrightarrow{}  bag(B)].  \mforall{}[b:bag(A)].
uiff(\muparrow{}bag-null(\mcup{}x\mmember{}b.f[x]);\mforall{}x:A.  (x  \mdownarrow{}\mmember{}  b  {}\mRightarrow{}  (\muparrow{}bag-null(f[x]))))

Date html generated: 2016_05_15-PM-02_40_16
Last ObjectModification: 2016_01_16-AM-08_47_48

Theory : bags

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