Nuprl Lemma : distinct_iff_counts_le_one
∀s:DSet. ∀ps:|s| List. (↑distinct{s}(ps) ⇐⇒ ∀x:|s|. ((x #∈ ps) ≤ 1))
Proof
Definitions occuring in Statement :
count: a #∈ as,
distinct: distinct{s}(ps),
list: T List,
assert: ↑b,
le: A ≤ B,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
dset: DSet,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
and: P ∧ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
true: True,
infix_ap: x f y,
ball: ball,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
uimplies: b supposing a,
band: p ∧b q,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
gt: i > j,
b2i: b2i(b),
squash: ↓T,
subtype_rel: A ⊆r B,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
less_than: a < b
Lemmas referenced :
list_induction,
set_car_wf,
iff_wf,
assert_wf,
distinct_wf,
all_wf,
le_wf,
count_wf,
distinct_nil_lemma,
istype-void,
count_nil_lemma,
distinct_cons_lemma,
count_cons_lemma,
list_wf,
dset_wf,
istype-false,
true_wf,
iff_weakening_uiff,
set_eq_wf,
equal_wf,
assert_of_dset_eq,
mem_wf,
not_wf,
b2i_wf,
bnot_wf,
assert_of_bnot,
ball_char,
infix_ap_wf,
bool_wf,
ball_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bfalse_wf,
assert_of_band,
mem_iff_count_nzero,
gt_wf,
equal-wf-T-base,
uiff_transitivity,
iff_transitivity,
squash_wf,
istype-int,
add_functionality_wrt_eq,
subtype_rel_self,
iff_weakening_equal,
decidable__equal_int,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermAdd_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_add_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
non_neg_length,
count_bounds,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
zero-add,
b2i_bounds
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
setElimination,
rename,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
hypothesisEquality,
natural_numberEquality,
universeIsType,
independent_functionElimination,
isect_memberEquality_alt,
voidElimination,
productIsType,
functionIsType,
inhabitedIsType,
independent_pairFormation,
productElimination,
applyEquality,
equalityIsType1,
promote_hyp,
addEquality,
unionElimination,
equalityElimination,
independent_isectElimination,
dependent_pairFormation_alt,
equalityTransitivity,
equalitySymmetry,
instantiate,
productEquality,
baseClosed,
imageElimination,
imageMemberEquality,
universeEquality,
approximateComputation,
int_eqEquality
Latex:
\mforall{}s:DSet. \mforall{}ps:|s| List. (\muparrow{}distinct\{s\}(ps) \mLeftarrow{}{}\mRightarrow{} \mforall{}x:|s|. ((x \#\mmember{} ps) \mleq{} 1))
Date html generated:
2019_10_16-PM-01_05_28
Last ObjectModification:
2018_10_08-AM-10_19_15
Theory : list_2
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