Nuprl Lemma : omega_step_measure
∀p:IntConstraints
(0 < dim(p)
⇒ (dim((λp.omega_step(p)) p) < dim(p)
∨ ((dim((λp.omega_step(p)) p) = dim(p) ∈ ℤ) ∧ num-eq-constraints((λp.omega_step(p)) p) < num-eq-constraints(p))))
Proof
Definitions occuring in Statement :
omega_step: omega_step(p),
num-eq-constraints: num-eq-constraints(p),
int-problem-dimension: dim(p),
int-constraint-problem: IntConstraints,
less_than: a < b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
lambda: λx.A[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
omega_step: omega_step(p),
int-problem-dimension: dim(p),
and: P ∧ Q,
prop: ℙ,
false: False,
not: ¬A,
uiff: uiff(P;Q),
uimplies: b supposing a,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
int-constraint-problem: IntConstraints,
tunion: ⋃x:A.B[x],
pi2: snd(t),
sq_type: SQType(T),
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
nil: [],
it: ⋅,
less_than: a < b,
squash: ↓T,
cons: [a / b],
subtract: n - m,
int_seg: {i..j-},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P),
lelt: i ≤ j < k,
unit: Unit,
num-eq-constraints: num-eq-constraints(p),
pi1: fst(t),
exists: ∃x:A. B[x],
ge: i ≥ j ,
nat_plus: ℕ+,
bool: 𝔹,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
int_upper: {i...},
length: ||as||,
list_ind: list_ind,
hd: hd(l),
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cand: A c∧ B
Lemmas referenced :
decidable__lt,
int-problem-dimension_wf,
omega_step_wf,
istype-int,
less_than_wf,
num-eq-constraints_wf,
decidable__and2,
equal_wf,
decidable__equal_int,
less-iff-le,
add_functionality_wrt_le,
add-associates,
istype-void,
add-zero,
add-commutes,
le-add-cancel2,
subtype_base_sq,
int_subtype_base,
set_wf,
list_wf,
length_wf,
list-cases,
product_subtype_list,
reduce_hd_cons_lemma,
subtract_wf,
first-success_wf,
equal-wf-base-T,
list_subtype_base,
equal-wf-base,
int_seg_wf,
equal-wf-T-base,
absval_wf,
select_wf,
decidable__le,
istype-false,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
le-add-cancel,
find-exact-eq-constraint_wf,
not-lt-2,
le_antisymmetry_iff,
add-swap,
istype-top,
exact-reduce-constraints_wf2,
set_subtype_base,
lelt_wf,
gcd-reduce-eq-constraints_wf2,
not-equal-2,
minus-zero,
minus-minus,
le_wf,
nil_wf,
gcd-reduce-ineq-constraints_wf2,
length_of_nil_lemma,
length_of_cons_lemma,
not_wf,
non_neg_length,
length_wf_nat,
not-equal-implies-less,
less_than_transitivity1,
less_than_irreflexivity,
le_reflexive,
one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
omega-shadow,
int_seg_properties,
nat_properties,
le-add-cancel-alt,
null_wf,
eqtt_to_assert,
assert_of_null,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
shadow_inequalities_wf,
subtype_rel_list_set,
null_nil_lemma,
btrue_wf,
null_cons_lemma,
bfalse_wf,
btrue_neq_bfalse,
nat_wf,
eager-map-is-map,
list-value-type,
eager-map_wf,
int-value-type,
append_wf,
map_wf,
map-length,
map_cons_lemma,
list_ind_cons_lemma,
cons_one_one,
cons_wf,
length-map,
member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
applyEquality,
Error :lambdaEquality_alt,
setElimination,
rename,
Error :inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
because_Cache,
unionElimination,
Error :inlFormation_alt,
Error :productIsType,
Error :equalityIsType1,
Error :universeIsType,
Error :inrFormation_alt,
intEquality,
Error :isect_memberEquality_alt,
independent_functionElimination,
voidElimination,
natural_numberEquality,
productElimination,
independent_isectElimination,
addEquality,
imageElimination,
instantiate,
cumulativity,
independent_pairFormation,
imageMemberEquality,
baseClosed,
promote_hyp,
hypothesis_subsumption,
setEquality,
baseApply,
closedConclusion,
productEquality,
minusEquality,
Error :setIsType,
Error :equalityIsType4,
equalityElimination,
lessCases,
Error :isect_memberFormation_alt,
axiomSqEquality,
Error :dependent_set_memberEquality_alt,
applyLambdaEquality,
Error :dependent_pairFormation_alt,
sqequalIntensionalEquality,
multiplyEquality,
Error :inlEquality_alt,
Error :dependent_pairEquality_alt,
independent_pairEquality
Latex:
\mforall{}p:IntConstraints
(0 < dim(p)
{}\mRightarrow{} (dim((\mlambda{}p.omega\_step(p)) p) < dim(p)
\mvee{} ((dim((\mlambda{}p.omega\_step(p)) p) = dim(p))
\mwedge{} num-eq-constraints((\mlambda{}p.omega\_step(p)) p) < num-eq-constraints(p))))
Date html generated:
2019_06_20-PM-00_51_30
Last ObjectModification:
2018_10_03-AM-00_13_25
Theory : omega
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