Nuprl Lemma : concat-lifting-member

`∀[B:Type]. ∀[n:ℕ]. ∀[A:ℕn ⟶ Type]. ∀[bags:k:ℕn ⟶ bag(A k)]. ∀[f:funtype(n;A;bag(B))]. ∀[b:B].`
`  (b ↓∈ concat-lifting(n;f;bags) `⇐⇒` ↓∃lst:k:ℕn ⟶ (A k). ((∀[k:ℕn]. lst k ↓∈ bags k) ∧ b ↓∈ uncurry-rev(n;f) lst))`

Proof

Definitions occuring in Statement :  concat-lifting: `concat-lifting(n;f;bags)` uncurry-rev: `uncurry-rev(n;f)` bag-member: `x ↓∈ bs` bag: `bag(T)` funtype: `funtype(n;A;T)` int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` squash: `↓T` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  concat-lifting: `concat-lifting(n;f;bags)` uall: `∀[x:A]. B[x]` member: `t ∈ T` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` prop: `ℙ` nat: `ℕ` ge: `i ≥ j ` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` subtype_rel: `A ⊆r B` uiff: `uiff(P;Q)` subtract: `n - m` less_than: `a < b` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` uncurry-rev: `uncurry-rev(n;f)` uncurry-gen: `uncurry-gen(n)` ifthenelse: `if b then t else f fi ` eq_int: `(i =z j)` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` bag-member: `x ↓∈ bs`
Lemmas referenced :  nat_wf concat-lifting_wf uncurry-rev_wf uall_wf exists_wf squash_wf concat-lifting-list_wf bag-member_wf add-zero int_formula_prop_eq_lemma intformeq_wf decidable__equal_int int_seg_wf add-member-int_seg2 le_wf int_term_value_subtract_lemma itermSubtract_wf decidable__le subtract_wf bag_wf funtype_wf subtype_rel-equal lelt_wf int_formula_prop_wf int_formula_prop_le_lemma int_term_value_var_lemma int_term_value_add_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma intformle_wf itermVar_wf itermAdd_wf itermConstant_wf intformless_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__lt nat_properties false_wf concat-lifting-list-member
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity cut lemma_by_obid sqequalHypSubstitution isectElimination thin because_Cache hypothesisEquality dependent_set_memberEquality natural_numberEquality independent_pairFormation sqequalRule lambdaFormation hypothesis setElimination rename dependent_functionElimination addEquality unionElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll applyEquality cumulativity productElimination imageElimination functionExtensionality imageMemberEquality baseClosed introduction independent_functionElimination functionEquality productEquality universeEquality isect_memberFormation independent_pairEquality

Latex:
\mforall{}[B:Type].  \mforall{}[n:\mBbbN{}].  \mforall{}[A:\mBbbN{}n  {}\mrightarrow{}  Type].  \mforall{}[bags:k:\mBbbN{}n  {}\mrightarrow{}  bag(A  k)].  \mforall{}[f:funtype(n;A;bag(B))].  \mforall{}[b:B].
(b  \mdownarrow{}\mmember{}  concat-lifting(n;f;bags)
\mLeftarrow{}{}\mRightarrow{}  \mdownarrow{}\mexists{}lst:k:\mBbbN{}n  {}\mrightarrow{}  (A  k).  ((\mforall{}[k:\mBbbN{}n].  lst  k  \mdownarrow{}\mmember{}  bags  k)  \mwedge{}  b  \mdownarrow{}\mmember{}  uncurry-rev(n;f)  lst))

Date html generated: 2016_05_15-PM-03_06_42
Last ObjectModification: 2016_01_16-AM-08_35_07

Theory : bags

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