Nuprl Lemma : permutation-when-no_repeats
∀[T:Type]
∀sa,sb:T List. ((∀x:T. ((x ∈ sa) ⇐⇒ (x ∈ sb))) ⇒ no_repeats(T;sb) ⇒ no_repeats(T;sa) ⇒ permutation(T;sb;sa))
Proof
Definitions occuring in Statement :
permutation: permutation(T;L1;L2),
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
int_seg: {i..j-},
uimplies: b supposing a,
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
l_member: (x ∈ l),
cand: A c∧ B,
nat: ℕ,
ge: i ≥ j ,
subtype_rel: A ⊆r B,
guard: {T},
pi1: fst(t),
inject: Inj(A;B;f),
so_lambda: λ2x.t[x],
so_apply: x[s],
no_repeats: no_repeats(T;l),
less_than': less_than'(a;b),
sq_type: SQType(T),
true: True,
permutation: permutation(T;L1;L2)
Lemmas referenced :
select_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
length_wf,
intformless_wf,
int_formula_prop_less_lemma,
select_member,
nat_properties,
istype-le,
istype-less_than,
int_seg_wf,
no_repeats_wf,
l_member_wf,
list_wf,
istype-universe,
non_neg_length,
length_wf_nat,
pigeon-hole,
decidable__equal_int_seg,
set_subtype_base,
lelt_wf,
int_subtype_base,
int_seg_subtype_nat,
istype-false,
intformeq_wf,
int_formula_prop_eq_lemma,
subtype_base_sq,
istype-nat,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
le_wf,
less_than_wf,
decidable__equal_int,
subtype_rel_function,
int_seg_subtype,
le_weakening,
inject_wf,
permute_list_wf,
list_extensionality,
permute_list_length,
permute_list_select
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
productElimination,
imageElimination,
sqequalRule,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
universeIsType,
voidElimination,
dependent_set_memberEquality_alt,
productIsType,
equalityIstype,
functionIsType,
inhabitedIsType,
instantiate,
universeEquality,
promote_hyp,
applyEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
functionExtensionality,
intEquality,
sqequalBase,
cumulativity,
imageMemberEquality,
baseClosed,
productEquality
Latex:
\mforall{}[T:Type]
\mforall{}sa,sb:T List.
((\mforall{}x:T. ((x \mmember{} sa) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} sb)))
{}\mRightarrow{} no\_repeats(T;sb)
{}\mRightarrow{} no\_repeats(T;sa)
{}\mRightarrow{} permutation(T;sb;sa))
Date html generated:
2020_05_19-PM-09_45_17
Last ObjectModification:
2019_12_31-PM-00_12_11
Theory : list_1
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