Nuprl Lemma : fun-connected-to-same
∀[T:Type]
∀f:T ⟶ T
(retraction(T;f)
⇒ (∀x,y:T. Dec(x = y ∈ T))
⇒ (∀x,z:T. (x is f*(z) ⇒ (∀y:T. (y is f*(z) ⇒ (x is f*(y) ∨ y is f*(x)))))))
Proof
Definitions occuring in Statement :
retraction: retraction(T;f),
fun-connected: y is f*(x),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
so_apply: x[s1;s2],
guard: {T},
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
false: False,
decidable: Dec(P)
Lemmas referenced :
fun-connected-induction,
all_wf,
fun-connected_wf,
or_wf,
equal_wf,
not_wf,
decidable_wf,
retraction_wf,
fun-connected_transitivity,
fun-connected-step,
fun-connected-step-back
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
cumulativity,
functionEquality,
functionExtensionality,
applyEquality,
hypothesis,
independent_functionElimination,
inrFormation,
because_Cache,
axiomEquality,
rename,
voidElimination,
universeEquality,
unionElimination,
inlFormation,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
independent_isectElimination
Latex:
\mforall{}[T:Type]
\mforall{}f:T {}\mrightarrow{} T
(retraction(T;f)
{}\mRightarrow{} (\mforall{}x,y:T. Dec(x = y))
{}\mRightarrow{} (\mforall{}x,z:T. (x is f*(z) {}\mRightarrow{} (\mforall{}y:T. (y is f*(z) {}\mRightarrow{} (x is f*(y) \mvee{} y is f*(x)))))))
Date html generated:
2018_05_21-PM-07_48_44
Last ObjectModification:
2017_07_26-PM-05_26_29
Theory : general
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