Nuprl Lemma : fix-connected
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[f:T ⟶ T].
∀[x,y:T]. f**(x) = f**(y) ∈ T supposing x is f*(y) supposing retraction(T;f)
Proof
Definitions occuring in Statement :
fix: f**(x),
retraction: retraction(T;f),
fun-connected: y is f*(x),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
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,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
fun-connected_wf,
retraction_wf,
fun-connected-induction,
equal_wf,
fix_wf,
not_wf,
squash_wf,
true_wf,
fix-step,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
extract_by_obid,
isectElimination,
cumulativity,
functionExtensionality,
applyEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
independent_isectElimination,
lambdaFormation,
hyp_replacement,
applyLambdaEquality,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[f:T {}\mrightarrow{} T].
\mforall{}[x,y:T]. f**(x) = f**(y) supposing x is f*(y) supposing retraction(T;f)
Date html generated:
2018_05_21-PM-07_47_16
Last ObjectModification:
2017_07_26-PM-05_24_49
Theory : general
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