Nuprl Lemma : final-iterate_wf
∀[A:Type]. ∀[f:A ⟶ (A + Top)]. ∀[x:A]. (final-iterate(f;x) ∈ A) supposing SWellFounded(p-graph(A;f) y x)
Proof
Definitions occuring in Statement :
final-iterate: final-iterate(f;x),
p-graph: p-graph(A;f),
strongwellfounded: SWellFounded(R[x; y]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
p-graph: p-graph(A;f),
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
prop: ℙ,
so_lambda: λ2x y.t[x; y],
cand: A c∧ B,
subtype_rel: A ⊆r B,
top: Top,
so_apply: x[s1;s2],
strongwellfounded: SWellFounded(R[x; y]),
exists: ∃x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
and: P ∧ Q,
final-iterate: final-iterate(f;x),
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
less_than: a < b,
squash: ↓T,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
strongwellfounded_wf,
assert_wf,
can-apply_wf,
subtype_rel_union,
top_wf,
equal_wf,
do-apply_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,
le_wf,
nat_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
bool_wf,
equal-wf-T-base,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
thin,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
axiomEquality,
extract_by_obid,
isectElimination,
cumulativity,
lambdaEquality,
productEquality,
functionExtensionality,
applyEquality,
because_Cache,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
functionEquality,
unionEquality,
universeEquality,
productElimination,
lambdaFormation,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
unionElimination,
applyLambdaEquality,
baseClosed,
imageElimination,
equalityElimination
Latex:
\mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} (A + Top)].
\mforall{}[x:A]. (final-iterate(f;x) \mmember{} A) supposing SWellFounded(p-graph(A;f) y x)
Date html generated:
2018_05_21-PM-07_36_42
Last ObjectModification:
2017_07_26-PM-05_10_42
Theory : general
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