### Nuprl Lemma : assert-bag_all

`∀[T:Type]. ∀[f:T ⟶ 𝔹]. ∀[b:bag(T)].  (∀x:T. (x ↓∈ b `` (↑f[x])) `⇐⇒` ↑bag_all(b;f))`

Proof

Definitions occuring in Statement :  bag-member: `x ↓∈ bs` bag_all: `bag_all(b;f)` bag: `bag(T)` assert: `↑b` bool: `𝔹` uall: `∀[x:A]. B[x]` so_apply: `x[s]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rev_implies: `P `` Q` all: `∀x:A. B[x]` squash: `↓T` sq_stable: `SqStable(P)` exists: `∃x:A. B[x]` subtype_rel: `A ⊆r B` uimplies: `b supposing a` assert: `↑b` ifthenelse: `if b then t else f fi ` bag_all: `bag_all(b;f)` bag-accum: `bag-accum(v,x.f[v; x];init;bs)` list_accum: list_accum nil: `[]` it: `⋅` btrue: `tt` true: `True` cons-bag: `x.b` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` cand: `A c∧ B` sq_or: `a ↓∨ b` or: `P ∨ Q` guard: `{T}` empty-bag: `{}` top: `Top` false: `False` band: `p ∧b q` bfalse: `ff` sq_type: `SQType(T)`
Lemmas referenced :  all_wf bag-member_wf assert_wf bag_all_wf assert_witness bag_wf bool_wf bag_to_squash_list sq_stable__all sq_stable_from_decidable decidable__assert squash_wf list_induction list-subtype-bag list_wf nil_wf bag_all-cons assert_of_band bag-member-cons equal_wf cons_wf bag_all-empty bag-member-empty-iff empty-bag_wf true_wf bool_cases_sqequal cons-bag_wf band_wf and_wf assert_elim subtype_base_sq bool_subtype_base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality sqequalRule lambdaEquality functionEquality hypothesis applyEquality functionExtensionality productElimination independent_pairEquality dependent_functionElimination independent_functionElimination isect_memberEquality because_Cache universeEquality imageElimination promote_hyp rename independent_isectElimination natural_numberEquality voidEquality voidElimination inlFormation imageMemberEquality baseClosed inrFormation hyp_replacement equalitySymmetry Error :applyLambdaEquality,  unionElimination comment dependent_set_memberEquality setElimination setEquality equalityTransitivity instantiate

Latex:
\mforall{}[T:Type].  \mforall{}[f:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[b:bag(T)].    (\mforall{}x:T.  (x  \mdownarrow{}\mmember{}  b  {}\mRightarrow{}  (\muparrow{}f[x]))  \mLeftarrow{}{}\mRightarrow{}  \muparrow{}bag\_all(b;f))

Date html generated: 2016_10_25-AM-10_28_45
Last ObjectModification: 2016_07_12-AM-06_45_14

Theory : bags

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