Nuprl Lemma : map-is-permutation-on-list-of-all
∀[T:Type]
∀f:T ⟶ T. (Bij(T;T;f) ⇒ (∀as:T List. ((no_repeats(T;as) ∧ (∀t:T. (t ∈ as))) ⇒ permutation(T;as;map(f;as)))))
Proof
Definitions occuring in Statement :
permutation: permutation(T;L1;L2),
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
map: map(f;as),
list: T List,
biject: Bij(A;B;f),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
prop: ℙ,
iff: P ⇐⇒ Q,
member: t ∈ T,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
surject: Surj(A;B;f),
biject: Bij(A;B;f),
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
uimplies: b supposing a,
nat: ℕ,
cand: A c∧ B,
l_member: (x ∈ l),
true: True,
squash: ↓T,
lelt: i ≤ j < k,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
inject: Inj(A;B;f)
Lemmas referenced :
biject_wf,
list_wf,
all_wf,
no_repeats_wf,
l_member_wf,
map_wf,
permutation-when-no_repeats,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
select_wf,
equal_wf,
length_wf,
less_than_wf,
map-length,
iff_weakening_equal,
true_wf,
squash_wf,
lelt_wf,
top_wf,
subtype_rel_list,
select-map,
set_wf,
subtype_rel_dep_function,
no_repeats_map
Rules used in proof :
universeEquality,
functionEquality,
lambdaEquality,
sqequalRule,
productEquality,
because_Cache,
independent_pairFormation,
independent_functionElimination,
hypothesis,
applyEquality,
functionExtensionality,
cumulativity,
dependent_functionElimination,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
approximateComputation,
unionElimination,
natural_numberEquality,
independent_isectElimination,
rename,
setElimination,
promote_hyp,
dependent_pairFormation,
baseClosed,
imageMemberEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
dependent_set_memberEquality,
setEquality
Latex:
\mforall{}[T:Type]
\mforall{}f:T {}\mrightarrow{} T
(Bij(T;T;f)
{}\mRightarrow{} (\mforall{}as:T List. ((no\_repeats(T;as) \mwedge{} (\mforall{}t:T. (t \mmember{} as))) {}\mRightarrow{} permutation(T;as;map(f;as)))))
Date html generated:
2018_05_21-PM-00_44_09
Last ObjectModification:
2017_12_11-AM-10_52_34
Theory : list_1
Home
Index