Nuprl Lemma : fix_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[f:T ⟶ T].  ∀[x:T]. (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]
, 
member: t ∈ T
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
retraction: retraction(T;f)
, 
exists: ∃x:A. B[x]
, 
all: ∀x:A. B[x]
, 
prop: ℙ
, 
nat: ℕ
, 
implies: P 
⇒ Q
, 
false: False
, 
ge: i ≥ j 
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
not: ¬A
, 
top: Top
, 
and: P ∧ Q
, 
subtype_rel: A ⊆r B
, 
guard: {T}
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
le: A ≤ B
, 
less_than': less_than'(a;b)
, 
uiff: uiff(P;Q)
, 
fix: f**(x)
, 
ycomb: Y
, 
less_than: a < b
, 
squash: ↓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
Lemmas referenced : 
retraction_wf, 
deq_wf, 
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, 
ge_wf, 
less_than_wf, 
decidable__le, 
subtract_wf, 
intformnot_wf, 
itermSubtract_wf, 
int_formula_prop_not_lemma, 
int_term_value_subtract_lemma, 
add_nat_wf, 
false_wf, 
le_wf, 
nat_wf, 
add-is-int-iff, 
itermAdd_wf, 
intformeq_wf, 
int_term_value_add_lemma, 
int_formula_prop_eq_lemma, 
equal_wf, 
decidable__lt, 
eqof_wf, 
bool_wf, 
equal-wf-T-base, 
assert_wf, 
bnot_wf, 
not_wf, 
uiff_transitivity, 
eqtt_to_assert, 
safe-assert-deq, 
iff_transitivity, 
iff_weakening_uiff, 
eqff_to_assert, 
assert_of_bnot
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
thin, 
sqequalHypSubstitution, 
productElimination, 
dependent_functionElimination, 
hypothesisEquality, 
equalityTransitivity, 
hypothesis, 
equalitySymmetry, 
sqequalRule, 
axiomEquality, 
isect_memberEquality, 
isectElimination, 
because_Cache, 
extract_by_obid, 
cumulativity, 
functionExtensionality, 
applyEquality, 
functionEquality, 
universeEquality, 
lambdaFormation, 
setElimination, 
rename, 
intWeakElimination, 
natural_numberEquality, 
independent_isectElimination, 
dependent_pairFormation, 
lambdaEquality, 
int_eqEquality, 
intEquality, 
voidElimination, 
voidEquality, 
independent_pairFormation, 
computeAll, 
independent_functionElimination, 
applyLambdaEquality, 
unionElimination, 
dependent_set_memberEquality, 
addEquality, 
pointwiseFunctionality, 
promote_hyp, 
baseApply, 
closedConclusion, 
baseClosed, 
imageElimination, 
equalityElimination, 
impliesFunctionality
Latex:
\mforall{}[T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[f:T  {}\mrightarrow{}  T].    \mforall{}[x:T].  (f**(x)  \mmember{}  T)  supposing  retraction(T;f)
Date html generated:
2018_05_21-PM-07_46_45
Last ObjectModification:
2017_07_26-PM-05_24_20
Theory : general
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