Nuprl Lemma : fix_property
∀[T:Type]. ∀eq:EqDecider(T). ∀f:T ⟶ T. (retraction(T;f) ⇒ (∀x:T. (((f f**(x)) = f**(x) ∈ T) ∧ f**(x) is f*(x))))
Proof
Definitions occuring in Statement :
fix: f**(x),
retraction: retraction(T;f),
fun-connected: y is f*(x),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀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 :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
retraction: retraction(T;f),
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
and: P ∧ Q,
uimplies: b supposing a,
so_apply: x[s],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
cand: A c∧ B,
fix: f**(x),
ycomb: Y,
eqof: eqof(d),
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T
Lemmas referenced :
less_than_wf,
all_wf,
subtract_wf,
equal_wf,
fix_wf,
fun-connected_wf,
set_wf,
primrec-wf2,
nat_wf,
add_nat_wf,
false_wf,
le_wf,
nat_properties,
decidable__le,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
retraction_wf,
deq_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
eqof_wf,
uiff_transitivity,
eqtt_to_assert,
safe-assert-deq,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
fun-connected-test2,
itermSubtract_wf,
int_term_value_subtract_lemma,
fun-connected_transitivity,
fun-connected-step,
decidable-equal-deq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
hypothesis,
addLevel,
sqequalHypSubstitution,
productElimination,
thin,
introduction,
extract_by_obid,
isectElimination,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
because_Cache,
sqequalRule,
natural_numberEquality,
rename,
setElimination,
lambdaEquality,
functionEquality,
productEquality,
independent_isectElimination,
intEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
dependent_set_memberEquality,
addEquality,
independent_pairFormation,
applyLambdaEquality,
unionElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
levelHypothesis,
universeEquality,
equalityElimination,
impliesFunctionality,
imageElimination
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}f:T {}\mrightarrow{} T.
(retraction(T;f) {}\mRightarrow{} (\mforall{}x:T. (((f f**(x)) = f**(x)) \mwedge{} f**(x) is f*(x))))
Date html generated:
2018_05_21-PM-07_46_57
Last ObjectModification:
2017_07_26-PM-05_24_29
Theory : general
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