Nuprl Lemma : list-to-set_functionality_wrt_permutation
∀[T:Type]
∀eq:EqDecider(T). ∀L1,L2:T List. (permutation(T;L1;L2) ⇒ permutation(T;list-to-set(eq;L1);list-to-set(eq;L2)))
Proof
Definitions occuring in Statement :
list-to-set: list-to-set(eq;L),
permutation: permutation(T;L1;L2),
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
le: A ≤ B,
ge: i ≥ j ,
nat: ℕ,
false: False,
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
istype: istype(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
guard: {T},
prop: ℙ,
rev_uimplies: rev_uimplies(P;Q),
rev_implies: P ⇐ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
eqof: eqof(d),
and: P ∧ Q,
iff: P ⇐⇒ Q,
deq: EqDecider(T),
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
itermConstant_wf,
intformle_wf,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__equal_int,
non_neg_length,
int_subtype_base,
le_wf,
set_subtype_base,
length_wf_nat,
istype-int,
l_member-iff-length-filter,
l_member_wf,
bool_wf,
subtype_rel_dep_function,
filter_wf5,
length_wf,
decidable__le,
member-permutation,
istype-universe,
deq_wf,
list_wf,
permutation_wf,
no-repeats-iff-count,
list-to-set-property,
list-to-set_wf,
assert_witness,
istype-assert,
safe-assert-deq,
permutation-iff-count1
Rules used in proof :
Error :isect_memberEquality_alt,
int_eqEquality,
Error :dependent_pairFormation_alt,
approximateComputation,
sqequalBase,
baseClosed,
intEquality,
voidElimination,
equalitySymmetry,
equalityTransitivity,
unionElimination,
Error :setIsType,
setEquality,
natural_numberEquality,
universeEquality,
instantiate,
Error :universeIsType,
Error :functionIsTypeImplies,
axiomEquality,
Error :lambdaEquality_alt,
independent_pairEquality,
Error :inhabitedIsType,
Error :equalityIstype,
applyEquality,
independent_isectElimination,
productElimination,
because_Cache,
sqequalRule,
independent_pairFormation,
independent_functionElimination,
hypothesis,
rename,
setElimination,
dependent_functionElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}L1,L2:T List.
(permutation(T;L1;L2) {}\mRightarrow{} permutation(T;list-to-set(eq;L1);list-to-set(eq;L2)))
Date html generated:
2019_06_20-PM-01_55_53
Last ObjectModification:
2019_06_19-AM-11_01_58
Theory : decidable!equality
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