Nuprl Lemma : monotone-bar-induction3-2
∀B,Q:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ ℙ.
((∀n:ℕ. ∀s:ℕn ⟶ ℕ. (B[n;s] ⇒ (∀m:ℕ. B[n + 1;s.m@n])))
⇒ (∀n:ℕ. ∀s:ℕn ⟶ ℕ. (B[n;s] ⇒ Q[n;s]))
⇒ (∀n:ℕ. ∀s:ℕn ⟶ ℕ. ((∀m:ℕ. Q[n + 1;s.m@n]) ⇒ Q[n;s]))
⇒ (∀alpha:ℕ ⟶ ℕ. ⇃(∃m:ℕ. B[m;alpha]))
⇒ ⇃(Q[0;λx.⊥]))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
seq-add: s.x@n,
int_seg: {i..j-},
nat: ℕ,
bottom: ⊥,
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
true: True,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
so_apply: x[s1;s2],
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
so_lambda: λ2x y.t[x; y],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
true: True,
ext2Baire: ext2Baire(n;f;d),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
cand: A c∧ B,
outl: outl(x),
isl: isl(x),
less_than: a < b,
seq-add: s.x@n,
nequal: a ≠ b ∈ T ,
subtract: n - m,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
strong-continuity-rel,
subtype_rel_dep_function,
nat_wf,
int_seg_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
implies-quotient-true,
exists_wf,
unit_wf2,
all_wf,
equal_wf,
subtype_rel_union,
assert_wf,
isl_wf,
le_wf,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
quotient_wf,
true_wf,
equiv_rel_true,
nat_properties,
decidable__le,
intformnot_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
seq-add_wf,
basic_bar_induction,
decidable__assert,
seq-adjoin_wf,
ext2Baire_wf,
squash_wf,
bool_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
decidable__equal_int,
lelt_wf,
bfalse_wf,
and_wf,
btrue_wf,
btrue_neq_bfalse,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__lt,
equal-wf-base-T,
int_subtype_base,
int_seg_subtype,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
set_wf,
primrec-wf2,
add-zero,
add-member-int_seg2,
add-associates,
subtract-add-cancel,
eq_int_wf,
assert_of_eq_int,
decidable__equal_nat,
neg_assert_of_eq_int,
minus-one-mul,
add-commutes,
add-mul-special,
zero-mul,
zero-add,
seq-adjoin-is-seq-add,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
rename,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
because_Cache,
isectElimination,
hypothesis,
sqequalRule,
natural_numberEquality,
setElimination,
independent_isectElimination,
independent_pairFormation,
functionEquality,
unionEquality,
productEquality,
inlEquality,
dependent_set_memberEquality,
productElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
addEquality,
unionElimination,
universeEquality,
cumulativity,
imageElimination,
imageMemberEquality,
baseClosed,
addLevel,
hyp_replacement,
equalitySymmetry,
equalityTransitivity,
levelHypothesis,
equalityElimination,
promote_hyp,
instantiate,
applyLambdaEquality,
baseApply,
closedConclusion,
inrEquality,
int_eqReduceTrueSq,
int_eqReduceFalseSq
Latex:
\mforall{}B,Q:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbP{}.
((\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}. (B[n;s] {}\mRightarrow{} (\mforall{}m:\mBbbN{}. B[n + 1;s.m@n])))
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}. (B[n;s] {}\mRightarrow{} Q[n;s]))
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}. ((\mforall{}m:\mBbbN{}. Q[n + 1;s.m@n]) {}\mRightarrow{} Q[n;s]))
{}\mRightarrow{} (\mforall{}alpha:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \00D9(\mexists{}m:\mBbbN{}. B[m;alpha]))
{}\mRightarrow{} \00D9(Q[0;\mlambda{}x.\mbot{}]))
Date html generated:
2017_04_17-AM-10_03_45
Last ObjectModification:
2017_02_27-PM-05_55_58
Theory : continuity
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