Nuprl Lemma : min-increasing-sequence-prop2
∀b,a:ℕ ⟶ ℕ. ∀n,x,k:ℕ.
((b = a ∈ (ℕx ⟶ ℕ))
⇒ increasing-sequence(a)
⇒ (min-increasing-sequence(b;n;(a x) + 1) = (inl k) ∈ (ℕ?))
⇒ (x ≤ k))
Proof
Definitions occuring in Statement :
min-increasing-sequence: min-increasing-sequence(a;n;k),
increasing-sequence: increasing-sequence(a),
int_seg: {i..j-},
nat: ℕ,
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
inl: inl x,
union: left + right,
add: n + m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
iff: P ⇐⇒ Q,
true: True,
squash: ↓T,
lelt: i ≤ j < k,
int_seg: {i..j-},
so_apply: x[s],
so_lambda: λ2x.t[x],
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
uiff: uiff(P;Q),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
guard: {T},
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
subtype_rel: A ⊆r B,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
int_seg_properties,
increasing-sequence-prop1,
iff_weakening_equal,
true_wf,
squash_wf,
and_wf,
lelt_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
min-increasing-sequence-prop1,
subtype_rel_self,
int_seg_subtype_nat,
subtype_rel_dep_function,
int_seg_wf,
increasing-sequence_wf,
int_formula_prop_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
add-is-int-iff,
decidable__le,
nat_properties,
le_wf,
false_wf,
add_nat_wf,
min-increasing-sequence_wf,
unit_wf2,
nat_wf,
equal_wf
Rules used in proof :
universeEquality,
imageMemberEquality,
imageElimination,
functionEquality,
inlEquality,
independent_functionElimination,
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_isectElimination,
productElimination,
baseClosed,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality,
unionElimination,
dependent_functionElimination,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
independent_pairFormation,
natural_numberEquality,
sqequalRule,
rename,
setElimination,
lambdaEquality,
addEquality,
dependent_set_memberEquality,
because_Cache,
hypothesisEquality,
applyEquality,
functionExtensionality,
hypothesis,
unionEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}b,a:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mforall{}n,x,k:\mBbbN{}.
((b = a)
{}\mRightarrow{} increasing-sequence(a)
{}\mRightarrow{} (min-increasing-sequence(b;n;(a x) + 1) = (inl k))
{}\mRightarrow{} (x \mleq{} k))
Date html generated:
2017_04_21-AM-11_20_53
Last ObjectModification:
2017_04_20-PM-05_34_59
Theory : continuity
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