Nuprl Lemma : fun-connected-iff-fun_exp
∀[T:Type]. ∀f:T ⟶ T. ((∀x:T. Dec((f x) = x ∈ T)) ⇒ (∀x,y:T. (x is f*(y) ⇐⇒ ∃n:ℕ. (x = (f^n y) ∈ T))))
Proof
Definitions occuring in Statement :
fun-connected: y is f*(x),
fun_exp: f^n,
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: 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,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
fun-connected: y is f*(x),
exists: ∃x:A. B[x],
fun-path: y=f*(x) via L,
select: L[n],
uimplies: b supposing a,
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
subtract: n - m,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
uiff: uiff(P;Q),
guard: {T},
not: ¬A,
decidable: Dec(P),
or: P ∨ Q,
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
le: A ≤ B
Lemmas referenced :
fun-connected_wf,
exists_wf,
nat_wf,
equal_wf,
fun_exp_wf,
all_wf,
decidable_wf,
list_induction,
fun-path_wf,
list_wf,
length_of_nil_lemma,
stuck-spread,
base_wf,
nil_wf,
fun-path-cons,
isect_wf,
less_than_wf,
length_wf,
not_wf,
decidable__lt,
cons_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
fun_exp_add1,
false_wf,
fun_exp0_lemma,
fun-connected-test2,
subtract_wf,
itermSubtract_wf,
intformless_wf,
int_term_value_subtract_lemma,
int_formula_prop_less_lemma,
set_wf,
primrec-wf2,
subtract-add-cancel,
fun-connected-step,
fun-connected_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
functionExtensionality,
applyEquality,
hypothesis,
sqequalRule,
lambdaEquality,
functionEquality,
universeEquality,
productElimination,
independent_functionElimination,
baseClosed,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
imageElimination,
because_Cache,
rename,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
natural_numberEquality,
productEquality,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
dependent_set_memberEquality,
addEquality,
setElimination,
int_eqEquality,
intEquality,
computeAll
Latex:
\mforall{}[T:Type]. \mforall{}f:T {}\mrightarrow{} T. ((\mforall{}x:T. Dec((f x) = x)) {}\mRightarrow{} (\mforall{}x,y:T. (x is f*(y) \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}. (x = (f\^{}n y)))))
Date html generated:
2018_05_21-PM-07_45_39
Last ObjectModification:
2017_07_26-PM-05_23_12
Theory : general
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