Nuprl Lemma : baf-bar-monotone
∀R,T:ℕ ⟶ ℕ ⟶ ℙ. ∀n:ℕ. ∀s:{s:ℕn ⟶ ℕ| strictly-increasing-seq(n;s)} .
(baf-bar(n,m.R[n;m];n,m.T[n;m];n;s)
⇒ (∀m:ℕ. (strictly-increasing-seq(n + 1;s.m@n) ⇒ baf-bar(n,m.R[n;m];n,m.T[n;m];n + 1;s.m@n))))
Proof
Definitions occuring in Statement :
baf-bar: baf-bar(n,m.R[n; m];n,m.T[n; m];l;a),
strictly-increasing-seq: strictly-increasing-seq(n;s),
seq-add: s.x@n,
int_seg: {i..j-},
nat: ℕ,
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
baf-bar: baf-bar(n,m.R[n; m];n,m.T[n; m];l;a),
and: P ∧ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
int_seg: {i..j-},
lelt: i ≤ j < k,
uall: ∀[x:A]. B[x],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
prop: ℙ,
le: A ≤ B,
less_than: a < b,
guard: {T},
so_apply: x[s1;s2],
seq-add: s.x@n,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y]
Lemmas referenced :
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
lelt_wf,
int_seg_properties,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
intformle_wf,
int_formula_prop_eq_lemma,
int_formula_prop_le_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
nat_wf,
decidable__equal_int,
le_wf,
decidable__le,
seq-add_wf,
exists_wf,
int_seg_wf,
strictly-increasing-seq_wf,
baf-bar_wf,
set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
independent_pairFormation,
productElimination,
thin,
promote_hyp,
hypothesis,
dependent_pairFormation,
setElimination,
rename,
dependent_set_memberEquality,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
dependent_functionElimination,
addEquality,
natural_numberEquality,
unionElimination,
independent_isectElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll,
because_Cache,
addLevel,
hyp_replacement,
equalitySymmetry,
applyEquality,
equalityElimination,
equalityTransitivity,
int_eqReduceTrueSq,
instantiate,
cumulativity,
independent_functionElimination,
int_eqReduceFalseSq,
functionExtensionality,
levelHypothesis,
productEquality,
universeEquality,
functionEquality
Latex:
\mforall{}R,T:\mBbbN{} {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbP{}. \mforall{}n:\mBbbN{}. \mforall{}s:\{s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}| strictly-increasing-seq(n;s)\} .
(baf-bar(n,m.R[n;m];n,m.T[n;m];n;s)
{}\mRightarrow{} (\mforall{}m:\mBbbN{}. (strictly-increasing-seq(n + 1;s.m@n) {}\mRightarrow{} baf-bar(n,m.R[n;m];n,m.T[n;m];n + 1;s.m@n))))
Date html generated:
2017_04_20-AM-07_26_28
Last ObjectModification:
2017_02_27-PM-05_59_51
Theory : continuity
Home
Index