Nuprl Lemma : permr_iff_eq_counts
∀s:DSet. ∀as,bs:|s| List. (as ≡(|s|) bs ⇐⇒ ∀x:|s|. ((x #∈ as) = (x #∈ bs) ∈ ℤ))
Proof
Definitions occuring in Statement :
count: a #∈ as,
permr: as ≡(T) bs,
list: T List,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
int: ℤ,
equal: s = t ∈ T,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
dset: DSet,
implies: P ⇒ Q,
prop: ℙ,
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
rev_implies: P ⇐ Q,
nat: ℕ,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
or: P ∨ Q,
bpermr: as ≡b bs,
ycomb: Y,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
infix_ap: x f y,
b2i: b2i(b),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
gt: i > j
Lemmas referenced :
list_wf,
set_car_wf,
dset_wf,
permr_wf,
count_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
count_functionality,
subtype_rel_self,
iff_weakening_equal,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
assert_witness,
intformeq_wf,
int_formula_prop_eq_lemma,
list-cases,
count_nil_lemma,
list_ind_nil_lemma,
int_subtype_base,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
le_wf,
bpermr_wf,
subtract-1-ge-0,
subtype_base_sq,
set_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
count_cons_lemma,
list_ind_cons_lemma,
b2i_wf,
infix_ap_wf,
bool_wf,
set_eq_wf,
nat_wf,
assert_of_bpermr,
null_nil_lemma,
null_cons_lemma,
add_functionality_wrt_eq,
btrue_wf,
dset_eq_refl,
non_neg_length,
count_bounds,
assert_of_band,
mem_wf,
remove1_wf,
assert_wf,
mem_iff_count_nzero,
decidable__lt,
cons_remove1_permr,
cons_wf,
permr_inversion,
add-is-int-iff,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
inhabitedIsType,
hypothesisEquality,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesis,
dependent_functionElimination,
intEquality,
because_Cache,
natural_numberEquality,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
instantiate,
independent_isectElimination,
productElimination,
intWeakElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
applyLambdaEquality,
functionIsTypeImplies,
unionElimination,
functionIsType,
equalityIsType4,
promote_hyp,
hypothesis_subsumption,
equalityIsType1,
dependent_set_memberEquality_alt,
baseApply,
closedConclusion,
addEquality,
pointwiseFunctionality
Latex:
\mforall{}s:DSet. \mforall{}as,bs:|s| List. (as \mequiv{}(|s|) bs \mLeftarrow{}{}\mRightarrow{} \mforall{}x:|s|. ((x \#\mmember{} as) = (x \#\mmember{} bs)))
Date html generated:
2019_10_16-PM-01_04_06
Last ObjectModification:
2018_10_08-AM-11_39_44
Theory : list_2
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