Nuprl Lemma : permutation-iff-count1
∀[T:Type]
∀eq:T ⟶ T ⟶ 𝔹
((∀x,y:T. (↑(eq x y) ⇐⇒ x = y ∈ T))
⇒ (∀a1,b1:T List. (∀x:T. (||filter(eq x;a1)|| = ||filter(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,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
nat: ℕ,
istype: istype(T),
uimplies: b supposing a,
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
top: Top,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
ge: i ≥ j ,
false: False,
le: A ≤ B,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
bfalse: ff,
it: ⋅,
unit: Unit,
bool: 𝔹,
decidable: Dec(P),
squash: ↓T,
true: True,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
deq: EqDecider(T),
cand: A c∧ B,
permutation: permutation(T;L1;L2)
Lemmas referenced :
istype-universe,
istype-assert,
cons_wf,
permutation-nil,
nil_wf,
permutation_wf,
int_subtype_base,
istype-int,
le_wf,
set_subtype_base,
l_member_wf,
bool_wf,
subtype_rel_dep_function,
filter_wf5,
length_wf_nat,
equal-wf-base,
list_wf,
list_induction,
filter_nil_lemma,
istype-void,
filter_cons_lemma,
length_of_nil_lemma,
bool_cases,
subtype_base_sq,
bool_subtype_base,
eqtt_to_assert,
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,
eqff_to_assert,
assert_of_bnot,
not_wf,
bnot_wf,
assert_wf,
equal-wf-T-base,
uiff_transitivity,
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,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
l_member_decomp,
append_wf,
istype-nat,
list_ind_cons_lemma,
list_ind_nil_lemma,
length-append,
add-associates,
length_wf,
filter_append_sq,
equal_wf,
add_functionality_wrt_eq,
ifthenelse_wf,
ite_rw_false,
iff_weakening_uiff,
assert-deq,
permutation-swap-first2,
permutation_inversion,
permutation_transitivity,
permutation-rotate,
iff_wf,
set_wf,
all_wf,
permute_list_wf,
int_seg_wf,
inject_wf,
nat_wf,
subtype_rel_list,
permutation-filter,
permutation-length
Rules used in proof :
cut,
universeEquality,
instantiate,
productIsType,
dependent_functionElimination,
equalitySymmetry,
sqequalBase,
equalityIstype,
functionIsType,
independent_functionElimination,
natural_numberEquality,
rename,
setElimination,
independent_isectElimination,
universeIsType,
setIsType,
setEquality,
inhabitedIsType,
because_Cache,
applyEquality,
intEquality,
hypothesis,
functionEquality,
lambdaEquality_alt,
sqequalRule,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
thin,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
isect_memberEquality_alt,
voidElimination,
equalityTransitivity,
unionElimination,
cumulativity,
productElimination,
closedConclusion,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
baseClosed,
equalityElimination,
hyp_replacement,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
equalityIsType1,
imageElimination,
imageMemberEquality,
dependent_set_memberEquality_alt,
addEquality,
functionExtensionality,
lambdaEquality,
lambdaFormation,
isect_memberFormation,
productEquality,
dependent_set_memberEquality,
dependent_pairFormation
Latex:
\mforall{}[T:Type]
\mforall{}eq:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbB{}
((\mforall{}x,y:T. (\muparrow{}(eq x y) \mLeftarrow{}{}\mRightarrow{} x = y))
{}\mRightarrow{} (\mforall{}a1,b1:T List.
(\mforall{}x:T. (||filter(eq x;a1)|| = ||filter(eq x;b1)||) \mLeftarrow{}{}\mRightarrow{} permutation(T;a1;b1))))
Date html generated:
2019_10_15-AM-10_24_11
Last ObjectModification:
2019_08_05-PM-02_09_03
Theory : decidable!equality
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