Nuprl Lemma : final-iterate-property
∀[A:Type]
∀f:A ⟶ (A + Top)
(SWellFounded(p-graph(A;f) y x)
⇒ (∀x:A
∃n:ℕ
((↑can-apply(f^n;x)) c∧ ((final-iterate(f;x) = do-apply(f^n;x) ∈ A) ∧ (¬↑can-apply(f;final-iterate(f;x)))))))
Proof
Definitions occuring in Statement :
final-iterate: final-iterate(f;x),
p-graph: p-graph(A;f),
p-fun-exp: f^n,
do-apply: do-apply(f;x),
can-apply: can-apply(f;x),
strongwellfounded: SWellFounded(R[x; y]),
nat: ℕ,
assert: ↑b,
uall: ∀[x:A]. B[x],
top: Top,
cand: A c∧ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
p-graph: p-graph(A;f),
strongwellfounded: SWellFounded(R[x; y]),
exists: ∃x:A. B[x],
final-iterate: final-iterate(f;x),
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
cand: A c∧ B,
so_apply: x[s],
uimplies: b supposing a,
top: Top,
and: P ∧ Q,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
ge: i ≥ j ,
le: A ≤ B,
less_than': less_than'(a;b),
assert: ↑b,
ifthenelse: if b then t else f fi ,
can-apply: can-apply(f;x),
isl: isl(x),
p-fun-exp: f^n,
primrec: primrec(n;b;c),
p-id: p-id(),
btrue: tt,
true: True,
do-apply: do-apply(f;x),
outl: outl(x),
less_than: a < b,
squash: ↓T,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff
Lemmas referenced :
le_wf,
all_wf,
subtract_wf,
exists_wf,
nat_wf,
assert_wf,
can-apply_wf,
p-fun-exp_wf,
subtype_rel_dep_function,
top_wf,
subtype_rel_union,
equal_wf,
final-iterate_wf,
do-apply_wf,
not_wf,
set_wf,
less_than_wf,
primrec-wf2,
decidable__le,
satisfiable-full-omega-tt,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
strongwellfounded_wf,
p-graph_wf2,
bool_wf,
nat_properties,
intformand_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
equal-wf-T-base,
bnot_wf,
false_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
itermAdd_wf,
int_term_value_add_lemma,
p-fun-exp-add1-sq,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
hypothesis,
addLevel,
sqequalHypSubstitution,
sqequalRule,
thin,
productElimination,
introduction,
extract_by_obid,
isectElimination,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
because_Cache,
natural_numberEquality,
rename,
setElimination,
lambdaEquality,
functionEquality,
productEquality,
unionEquality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
intEquality,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
levelHypothesis,
universeEquality,
independent_pairFormation,
applyLambdaEquality,
baseClosed,
dependent_set_memberEquality,
imageElimination,
addEquality,
equalityElimination
Latex:
\mforall{}[A:Type]
\mforall{}f:A {}\mrightarrow{} (A + Top)
(SWellFounded(p-graph(A;f) y x)
{}\mRightarrow{} (\mforall{}x:A
\mexists{}n:\mBbbN{}
((\muparrow{}can-apply(f\^{}n;x))
c\mwedge{} ((final-iterate(f;x) = do-apply(f\^{}n;x)) \mwedge{} (\mneg{}\muparrow{}can-apply(f;final-iterate(f;x)))))))
Date html generated:
2018_05_21-PM-07_36_50
Last ObjectModification:
2017_07_26-PM-05_10_49
Theory : general
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