### Nuprl Lemma : mapfilter-no-rep-fun

`∀[T,U,V:Type]. ∀[eq:EqDecider(U)]. ∀[L:T List]. ∀[u:U]. ∀[f:T ⟶ U]. ∀[g:{x:{x:T| (x ∈ L)} | ↑(eq f[x] u)}  ⟶ V].`
`  ||mapfilter(g;λx.(eq f[x] u);L)|| ≤ 1 supposing no_repeats(U;map(f;L))`

Proof

Definitions occuring in Statement :  mapfilter: `mapfilter(f;P;L)` no_repeats: `no_repeats(T;l)` l_member: `(x ∈ l)` length: `||as||` map: `map(f;as)` list: `T List` deq: `EqDecider(T)` assert: `↑b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` le: `A ≤ B` set: `{x:A| B[x]} ` apply: `f a` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` deq: `EqDecider(T)` so_apply: `x[s]` subtype_rel: `A ⊆r B` prop: `ℙ` so_lambda: `λ2x.t[x]` and: `P ∧ Q` uimplies: `b supposing a` all: `∀x:A. B[x]` implies: `P `` Q` cand: `A c∧ B` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` le: `A ≤ B` mapfilter: `mapfilter(f;P;L)` true: `True` squash: `↓T` guard: `{T}` iff: `P `⇐⇒` Q` eqof: `eqof(d)` uiff: `uiff(P;Q)` less_than': `less_than'(a;b)` int_seg: `{i..j-}` lelt: `i ≤ j < k` less_than: `a < b` l_before: `x before y ∈ l` sublist: `L1 ⊆ L2` select: `L[n]` cons: `[a / b]` subtract: `n - m` no_repeats: `no_repeats(T;l)` ge: `i ≥ j ` nat: `ℕ` increasing: `increasing(f;k)`
Lemmas referenced :  length_wf mapfilter-wf subtype_rel_dep_function l_member_wf assert_wf subtype_rel_sets set_wf decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf less_than'_wf no_repeats_wf map_wf list_wf deq_wf length-map filter_wf5 filter_is_sublist sublist_wf equal_wf squash_wf true_wf iff_weakening_equal member_filter_2 safe-assert-deq select_wf false_wf decidable__lt intformless_wf int_formula_prop_less_lemma select_member lelt_wf l_before_sublist l_before_select le_wf length_of_cons_lemma length_of_nil_lemma int_seg_wf map-length non_neg_length length_wf_nat nat_properties int_seg_properties nat_wf subtype_rel_list top_wf select-map intformeq_wf int_formula_prop_eq_lemma
Rules used in proof :  cut introduction extract_by_obid sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isectElimination thin hypothesisEquality because_Cache lambdaEquality applyEquality setElimination rename hypothesis functionExtensionality cumulativity sqequalRule setEquality productEquality independent_isectElimination lambdaFormation productElimination dependent_functionElimination equalityTransitivity equalitySymmetry natural_numberEquality unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll functionEquality universeEquality isect_memberFormation independent_pairEquality axiomEquality imageElimination imageMemberEquality baseClosed independent_functionElimination dependent_set_memberEquality hyp_replacement applyLambdaEquality

Latex:
\mforall{}[T,U,V:Type].  \mforall{}[eq:EqDecider(U)].  \mforall{}[L:T  List].  \mforall{}[u:U].  \mforall{}[f:T  {}\mrightarrow{}  U].  \mforall{}[g:\{x:\{x:T|  (x  \mmember{}  L)\}  |
\muparrow{}(eq  f[x]  u)\}    {}\mrightarrow{}  V].
||mapfilter(g;\mlambda{}x.(eq  f[x]  u);L)||  \mleq{}  1  supposing  no\_repeats(U;map(f;L))

Date html generated: 2017_09_29-PM-06_04_32
Last ObjectModification: 2017_07_26-PM-02_53_05

Theory : decidable!equality

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