Nuprl Lemma : no-repeats-iff-count
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[L:T List].
uiff(no_repeats(T;L);∀[x:T]. uiff(1 ≤ ||filter(eq x;L)||;||filter(eq x;L)|| = 1 ∈ ℤ))
Proof
Definitions occuring in Statement :
no_repeats: no_repeats(T;l),
length: ||as||,
filter: filter(P;l),
list: T List,
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
apply: f a,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
deq: EqDecider(T),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
set-equal: set-equal(T;x;y),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
l_member: (x ∈ l),
nat: ℕ,
cand: A c∧ B,
ge: i ≥ j ,
eqof: eqof(d),
int_seg: {i..j-},
lelt: i ≤ j < k,
no_repeats: no_repeats(T;l),
rev_uimplies: rev_uimplies(P;Q),
guard: {T},
length: ||as||,
list_ind: list_ind,
cons: [a / b],
nil: [],
it: ⋅
Lemmas referenced :
le_wf,
length_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
subtype_rel_self,
set_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_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_eq_lemma,
int_formula_prop_wf,
less_than'_wf,
equal-wf-T-base,
no_repeats_wf,
no_repeats_witness,
uall_wf,
uiff_wf,
list_wf,
deq_wf,
set-equal-no_repeats-length,
cons_wf,
select_wf,
false_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
nil_wf,
no_repeats_singleton,
no_repeats_filter,
length_of_cons_lemma,
length_of_nil_lemma,
member_singleton,
member_filter,
iff_wf,
equal_wf,
assert_wf,
less_than_wf,
nat_properties,
safe-assert-deq,
select_member,
lelt_wf,
not_wf,
nat_wf,
sublist_filter,
l_all_cons,
assert_functionality_wrt_uiff,
l_all_nil,
length_sublist,
sublist_pair,
decidable__equal_int,
equal-wf-base,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
cumulativity,
hypothesisEquality,
applyEquality,
setElimination,
rename,
sqequalRule,
lambdaEquality,
setEquality,
independent_isectElimination,
because_Cache,
lambdaFormation,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
productElimination,
independent_pairEquality,
axiomEquality,
baseClosed,
independent_functionElimination,
universeEquality,
addLevel,
impliesFunctionality,
productEquality,
dependent_set_memberEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[L:T List].
uiff(no\_repeats(T;L);\mforall{}[x:T]. uiff(1 \mleq{} ||filter(eq x;L)||;||filter(eq x;L)|| = 1))
Date html generated:
2017_09_29-PM-06_04_09
Last ObjectModification:
2017_07_26-PM-02_53_02
Theory : decidable!equality
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