Nuprl Lemma : trivial_map
∀[T:Type]. ∀[a:T List]. ∀[f:T ⟶ T]. map(f;a) = a ∈ (T List) supposing ∀x:T. ((x ∈ a) ⇒ ((f x) = x ∈ T))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
map: map(f;as),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
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,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
true: True,
subtype_rel: A ⊆r B
Lemmas referenced :
list_induction,
all_wf,
l_member_wf,
equal_wf,
list_wf,
map_wf,
map_nil_lemma,
nil_wf,
map_cons_lemma,
squash_wf,
true_wf,
cons_wf,
cons_member,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
hypothesis,
applyEquality,
functionExtensionality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
rename,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
productElimination,
inrFormation,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
universeEquality,
inlFormation,
axiomEquality
Latex:
\mforall{}[T:Type]. \mforall{}[a:T List]. \mforall{}[f:T {}\mrightarrow{} T]. map(f;a) = a supposing \mforall{}x:T. ((x \mmember{} a) {}\mRightarrow{} ((f x) = x))
Date html generated:
2017_04_14-AM-08_54_22
Last ObjectModification:
2017_02_27-PM-03_38_47
Theory : list_0
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