Nuprl Lemma : retraction-fixedpoint
∀[T:Type]. ∀f:T ⟶ T. (retraction(T;f) ⇒ (∀x:T. ∃y:T. (((f y) = y ∈ T) ∧ y is f*(x))))
Proof
Definitions occuring in Statement :
retraction: retraction(T;f),
fun-connected: y is f*(x),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
retraction: retraction(T;f),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
nat: ℕ,
guard: {T},
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
less_than: a < b,
squash: ↓T,
fun-connected: y is f*(x),
fun-path: y=f*(x) via L,
subtract: n - m,
last: last(L),
select: L[n],
cons: [a / b],
true: True,
int_seg: {i..j-},
sq_type: SQType(T),
lelt: i ≤ j < k
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
fun-connected-test2,
equal_wf,
fun-connected_wf,
less_than_wf,
all_wf,
subtract_wf,
exists_wf,
set_wf,
primrec-wf2,
nat_wf,
add_nat_wf,
false_wf,
le_wf,
decidable__le,
add-is-int-iff,
intformnot_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
decidable__lt,
or_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
fun-connected_transitivity,
cons_wf,
nil_wf,
fun-path_wf,
length_of_cons_lemma,
length_of_nil_lemma,
reduce_hd_cons_lemma,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
select_wf,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
dependent_pairFormation,
because_Cache,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
introduction,
extract_by_obid,
isectElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
applyLambdaEquality,
setElimination,
rename,
natural_numberEquality,
independent_isectElimination,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
productEquality,
functionEquality,
dependent_set_memberEquality,
addEquality,
unionElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
independent_functionElimination,
universeEquality,
imageElimination,
imageMemberEquality,
instantiate,
independent_pairEquality,
axiomEquality,
hyp_replacement
Latex:
\mforall{}[T:Type]. \mforall{}f:T {}\mrightarrow{} T. (retraction(T;f) {}\mRightarrow{} (\mforall{}x:T. \mexists{}y:T. (((f y) = y) \mwedge{} y is f*(x))))
Date html generated:
2018_05_21-PM-07_47_52
Last ObjectModification:
2017_07_26-PM-05_25_46
Theory : general
Home
Index