Nuprl Lemma : no_repeats_map
∀[T:Type]. ∀[L:T List]. ∀[A:Type]. ∀[f:{x:T| (x ∈ L)} ⟶ A]. no_repeats(A;map(f;L)) supposing Inj({x:T| (x ∈ L)} ;A;f\000C) supposing no_repeats(T;L)
Proof
Definitions occuring in Statement :
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
map: map(f;as),
list: T List,
inject: Inj(A;B;f),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
mapl: mapl(f;l),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
guard: {T},
or: P ∨ Q,
uiff: uiff(P;Q),
cand: A c∧ B,
inject: Inj(A;B;f),
squash: ↓T,
not: ¬A,
exists: ∃x:A. B[x],
false: False
Lemmas referenced :
list_induction,
no_repeats_wf,
all_wf,
l_member_wf,
inject_wf,
mapl_wf,
list_wf,
map_nil_lemma,
no_repeats_nil,
nil_wf,
map_cons_lemma,
no_repeats_cons,
subtype_rel_dep_function,
cons_wf,
subtype_rel_sets,
cons_member,
equal_wf,
set_wf,
no_repeats_witness,
member_wf,
member-mapl,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
lambdaEquality,
functionEquality,
cumulativity,
hypothesis,
setEquality,
functionExtensionality,
applyEquality,
independent_functionElimination,
lambdaFormation,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
rename,
independent_isectElimination,
because_Cache,
setElimination,
productElimination,
inrFormation,
inlFormation,
dependent_set_memberEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
universeEquality,
hyp_replacement,
Error :applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List].
\mforall{}[A:Type]. \mforall{}[f:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} A]. no\_repeats(A;map(f;L)) supposing Inj(\{x:T| (x \mmember{} L)\} ;A;f) s\000Cupposing no\_repeats(T;L)
Date html generated:
2016_10_21-AM-10_30_08
Last ObjectModification:
2016_07_12-AM-05_43_24
Theory : list_1
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