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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