Nuprl Lemma : cyclic-map-conjugate
∀[T:Type]. ∀[g,h:T ⟶ T].
(∀[f:cyclic-map(T)]. (g o (f o h) ∈ cyclic-map(T))) supposing
((∀a:T. ((g (h a)) = a ∈ T)) and
(∀b:T. ((h (g b)) = b ∈ T)))
Proof
Definitions occuring in Statement :
cyclic-map: cyclic-map(T),
compose: f o g,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
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,
cyclic-map: cyclic-map(T),
injection: A →⟶ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
prop: ℙ,
inject: Inj(A;B;f),
compose: f o g,
implies: P ⇒ Q,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
all_wf,
exists_wf,
nat_wf,
equal_wf,
fun_exp_wf,
cyclic-map_wf,
compose_wf,
squash_wf,
true_wf,
iff_weakening_equal,
inject_wf,
iterated-conjugate2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
hypothesis,
applyEquality,
functionExtensionality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
because_Cache,
functionEquality,
universeEquality,
lambdaFormation,
applyLambdaEquality,
imageElimination,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
dependent_pairFormation,
hyp_replacement
Latex:
\mforall{}[T:Type]. \mforall{}[g,h:T {}\mrightarrow{} T].
(\mforall{}[f:cyclic-map(T)]. (g o (f o h) \mmember{} cyclic-map(T))) supposing
((\mforall{}a:T. ((g (h a)) = a)) and
(\mforall{}b:T. ((h (g b)) = b)))
Date html generated:
2018_05_21-PM-08_25_43
Last ObjectModification:
2017_07_26-PM-05_54_07
Theory : general
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