Nuprl Lemma : remove-repeats-fun-map2
∀[A,B:Type]. ∀[eq:EqDecider(B)]. ∀[L:A List]. ∀[f:{a:A| (a ∈ L)} ⟶ B].
(map(f;remove-repeats-fun(eq;f;L)) = remove-repeats(eq;map(f;L)) ∈ (B List))
Proof
Definitions occuring in Statement :
remove-repeats-fun: remove-repeats-fun(eq;f;L),
remove-repeats: remove-repeats(eq;L),
l_member: (x ∈ l),
map: map(f;as),
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
or: P ∨ Q,
remove-repeats-fun: remove-repeats-fun(eq;f;L),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
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),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
compose: f o g,
deq: EqDecider(T),
true: True
Lemmas referenced :
nat_properties,
full-omega-unsat,
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,
equal-wf-T-base,
nat_wf,
colength_wf_list,
int_subtype_base,
list-cases,
list_ind_nil_lemma,
map_nil_lemma,
remove_repeats_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,
decidable__equal_int,
list_ind_cons_lemma,
map_cons_lemma,
remove_repeats_cons_lemma,
cons_wf,
cons_member,
filter-map,
map_wf,
filter_wf5,
remove-repeats-fun_wf,
list-set-type,
subtype_rel_list_set,
bnot_wf,
list_wf,
deq_wf,
subtype_rel_dep_function,
subtype_rel_sets,
set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
axiomEquality,
functionEquality,
setEquality,
applyEquality,
because_Cache,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
cumulativity,
imageElimination,
functionExtensionality,
inlFormation,
inrFormation,
imageMemberEquality,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[eq:EqDecider(B)]. \mforall{}[L:A List]. \mforall{}[f:\{a:A| (a \mmember{} L)\} {}\mrightarrow{} B].
(map(f;remove-repeats-fun(eq;f;L)) = remove-repeats(eq;map(f;L)))
Date html generated:
2018_05_21-PM-00_51_19
Last ObjectModification:
2018_05_19-AM-06_41_07
Theory : decidable!equality
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