Nuprl Lemma : permutation-iff-count
∀[T:Type]
∀eq:EqDecider(T). ∀a1,b1:T List.
(∀x:T. (||filter(eqof(eq) x;a1)|| = ||filter(eqof(eq) x;b1)|| ∈ ℤ) ⇐⇒ permutation(T;a1;b1))
Proof
Definitions occuring in Statement :
permutation: permutation(T;L1;L2),
length: ||as||,
filter: filter(P;l),
list: T List,
eqof: eqof(d),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
istype: istype(T),
nat: ℕ,
eqof: eqof(d),
top: Top,
deq: EqDecider(T),
ge: i ≥ j ,
false: False,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
bool: 𝔹,
unit: Unit,
it: ⋅,
bnot: ¬bb,
assert: ↑b,
decidable: Dec(P),
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cand: A c∧ B,
label: ...$L... t,
permutation: permutation(T;L1;L2)
Lemmas referenced :
list_induction,
list_wf,
equal-wf-base,
length_wf_nat,
filter_wf5,
eqof_wf,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
permutation_wf,
nil_wf,
permutation-nil,
cons_wf,
deq_wf,
istype-universe,
filter_nil_lemma,
istype-void,
filter_cons_lemma,
length_of_nil_lemma,
length_of_cons_lemma,
non_neg_length,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
assert_wf,
bnot_wf,
not_wf,
member_wf,
istype-assert,
bool_cases,
subtype_base_sq,
bool_subtype_base,
eqtt_to_assert,
safe-assert-deq,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
bool_cases_sqequal,
assert-bnot,
equal_wf,
permutation-cons2,
decidable__equal_int,
add-is-int-iff,
intformnot_wf,
int_formula_prop_not_lemma,
false_wf,
member-exists2,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
member_filter,
l_member_decomp,
append_wf,
istype-nat,
list_ind_cons_lemma,
list_ind_nil_lemma,
equal-wf-T-base,
uiff_transitivity,
length-append,
length_wf,
filter_append_sq,
permutation-swap-first2,
permutation_inversion,
permutation_transitivity,
permutation-rotate,
set_wf,
subtype_rel_self,
all_wf,
permutation-filter,
permutation-length,
permute_list_wf,
int_seg_wf,
inject_wf,
nat_wf,
subtype_rel_list
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
functionEquality,
hypothesis,
intEquality,
applyEquality,
because_Cache,
inhabitedIsType,
setEquality,
setIsType,
universeIsType,
independent_isectElimination,
setElimination,
rename,
natural_numberEquality,
independent_functionElimination,
functionIsType,
equalityIstype,
sqequalBase,
equalitySymmetry,
dependent_functionElimination,
instantiate,
universeEquality,
isect_memberEquality_alt,
voidElimination,
equalityTransitivity,
productElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
unionElimination,
cumulativity,
equalityElimination,
promote_hyp,
hyp_replacement,
applyLambdaEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed,
dependent_set_memberEquality_alt,
productIsType,
addEquality,
lambdaEquality,
lambdaFormation,
isect_memberFormation,
functionExtensionality,
productEquality,
dependent_set_memberEquality,
dependent_pairFormation
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}a1,b1:T List.
(\mforall{}x:T. (||filter(eqof(eq) x;a1)|| = ||filter(eqof(eq) x;b1)||) \mLeftarrow{}{}\mRightarrow{} permutation(T;a1;b1))
Date html generated:
2019_10_16-AM-11_30_19
Last ObjectModification:
2019_06_27-PM-05_26_52
Theory : bags_2
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