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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