Nuprl Lemma : fix-step
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[f:T ⟶ T]. ∀[x:T]. f**(f x) = f**(x) ∈ T supposing retraction(T;f)
Proof
Definitions occuring in Statement :
fix: f**(x),
retraction: retraction(T;f),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
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,
prop: ℙ,
implies: P ⇒ Q,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
fix: f**(x),
ycomb: Y,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
eqof: eqof(d),
deq: EqDecider(T),
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A
Lemmas referenced :
retraction_wf,
equal_wf,
squash_wf,
true_wf,
fix_wf,
deq_wf,
iff_weakening_equal,
eqof_wf,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
assert_wf,
bnot_wf,
not_wf,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
functionExtensionality,
applyEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
lambdaEquality,
imageElimination,
universeEquality,
independent_isectElimination,
functionEquality,
imageMemberEquality,
baseClosed,
natural_numberEquality,
productElimination,
independent_functionElimination,
unionElimination,
equalityElimination,
setElimination,
rename,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
voidElimination,
independent_pairFormation,
impliesFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[f:T {}\mrightarrow{} T]. \mforall{}[x:T]. f**(f x) = f**(x) supposing retraction(T;f)
Date html generated:
2018_05_21-PM-07_47_07
Last ObjectModification:
2017_07_26-PM-05_24_39
Theory : general
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