Nuprl Lemma : fixpoint-upper-bound
∀j:ℕ. ∀F:Top. (F^j ⊥ ≤ fix(F))
Proof
Definitions occuring in Statement :
fun_exp: f^n,
nat: ℕ,
bottom: ⊥,
top: Top,
all: ∀x:A. B[x],
apply: f a,
fix: fix(F),
sqle: s ≤ t
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
nat_plus: ℕ+
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
top_wf,
fun_exp0_lemma,
bottom-sqle,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
fun_exp_unroll_1,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
introduction,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
axiomSqleEquality,
isect_memberEquality,
voidEquality,
unionElimination,
independent_pairFormation,
productElimination,
addEquality,
applyEquality,
intEquality,
minusEquality,
because_Cache,
dependent_set_memberEquality,
sqleRule,
sqleReflexivity
Latex:
\mforall{}j:\mBbbN{}. \mforall{}F:Top. (F\^{}j \mbot{} \mleq{} fix(F))
Date html generated:
2016_05_13-PM-04_07_22
Last ObjectModification:
2015_12_26-AM-11_03_59
Theory : fun_1
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