Nuprl Lemma : list_accum-mapfilter

`∀[T,U,A:Type]. ∀[f:A ⟶ U ⟶ A]. ∀[L:T List]. ∀[p:{a:T| (a ∈ L)}  ⟶ 𝔹]. ∀[g:{a:T| (a ∈ L) ∧ (↑(p a))}  ⟶ U]. ∀[x:A].`
`  (accumulate (with value a and list item x):`
`    f[a;x]`
`   over list:`
`     mapfilter(g;p;L)`
`   with starting value:`
`    x) ~ accumulate (with value a and list item x):`
`          if p x then f[a;g x] else a fi `
`         over list:`
`           L`
`         with starting value:`
`          x))`

Proof

Definitions occuring in Statement :  mapfilter: `mapfilter(f;P;L)` l_member: `(x ∈ l)` list_accum: list_accum list: `T List` assert: `↑b` ifthenelse: `if b then t else f fi ` bool: `𝔹` uall: `∀[x:A]. B[x]` so_apply: `x[s1;s2]` and: `P ∧ Q` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type` sqequal: `s ~ t`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` subtype_rel: `A ⊆r B` guard: `{T}` or: `P ∨ Q` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` cons: `[a / b]` colength: `colength(L)` decidable: `Dec(P)` nil: `[]` it: `⋅` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` mapfilter: `mapfilter(f;P;L)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` cand: `A c∧ B` bfalse: `ff` bnot: `¬bb` assert: `↑b`
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf l_member_wf assert_wf bool_wf equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases mapfilter_nil_lemma list_accum_nil_lemma nil_wf product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf equal_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int list_accum_cons_lemma filter_cons_lemma cons_member cons_wf eqtt_to_assert map_cons_lemma subtype_rel_dep_function subtype_rel_sets subtype_rel_self set_wf eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot list_wf
Rules used in proof :  cut thin sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination sqequalAxiom functionEquality setEquality cumulativity productEquality applyEquality functionExtensionality dependent_set_memberEquality because_Cache unionElimination promote_hyp hypothesis_subsumption productElimination equalityTransitivity equalitySymmetry applyLambdaEquality addEquality baseClosed instantiate imageElimination inlFormation equalityElimination inrFormation universeEquality isect_memberFormation

Latex:
\mforall{}[T,U,A:Type].  \mforall{}[f:A  {}\mrightarrow{}  U  {}\mrightarrow{}  A].  \mforall{}[L:T  List].  \mforall{}[p:\{a:T|  (a  \mmember{}  L)\}    {}\mrightarrow{}  \mBbbB{}].  \mforall{}[g:\{a:T|
(a  \mmember{}  L)  \mwedge{}  (\muparrow{}(p  a))\}
{}\mrightarrow{}  U].  \mforall{}[x:A].
(accumulate  (with  value  a  and  list  item  x):
f[a;x]
over  list:
mapfilter(g;p;L)
with  starting  value:
x)  \msim{}  accumulate  (with  value  a  and  list  item  x):
if  p  x  then  f[a;g  x]  else  a  fi
over  list:
L
with  starting  value:
x))

Date html generated: 2017_04_17-AM-07_37_24
Last ObjectModification: 2017_02_27-PM-04_12_30

Theory : list_1

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