Nuprl Lemma : monotone-bar-induction-strict2
∀B,Q:n:ℕ ⟶ {s:ℕn ⟶ ℕ| strictly-increasing-seq(n;s)} ⟶ ℙ.
((∀n:ℕ. ∀s:{s:ℕn ⟶ ℕ| strictly-increasing-seq(n;s)} .
(B[n;s] ⇒ (∀m:ℕ. (strictly-increasing-seq(n + 1;s.m@n) ⇒ B[n + 1;s.m@n]))))
⇒ (∀n:ℕ. ∀s:{s:ℕn ⟶ ℕ| strictly-increasing-seq(n;s)} . (B[n;s] ⇒ ⇃(Q[n;s])))
⇒ (∀n:ℕ. ∀s:{s:ℕn ⟶ ℕ| strictly-increasing-seq(n;s)} .
((∀m:ℕ. (strictly-increasing-seq(n + 1;s.m@n) ⇒ ⇃(Q[n + 1;s.m@n]))) ⇒ ⇃(Q[n;s])))
⇒ (∀alpha:StrictInc. ∃m:ℕ. B[m;alpha])
⇒ ⇃(Q[0;λx.⊥]))
Proof
Definitions occuring in Statement :
strict-inc: StrictInc,
quotient: x,y:A//B[x; y],
strictly-increasing-seq: strictly-increasing-seq(n;s),
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,
set: {x:A| B[x]} ,
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,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
so_apply: x[s],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
strictly-increasing-seq: strictly-increasing-seq(n;s),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
seq-add: s.x@n,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
guard: {T},
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
true: True,
quotient: x,y:A//B[x; y],
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
strict-inc: StrictInc
Lemmas referenced :
all_wf,
strict-inc_wf,
exists_wf,
nat_wf,
strict-inc-subtype,
int_seg_wf,
strictly-increasing-seq_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
seq-add_wf,
quotient_wf,
true_wf,
equiv_rel_true,
monotone-bar-induction2,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
less_than_wf,
squash_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
int_seg_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
decidable__strictly-increasing-seq,
quotient-member-eq,
equal-wf-base,
it_wf,
top_wf,
subtype_rel_dep_function,
unit_wf2,
make-strict_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
make-strict-agrees,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
dependent_functionElimination,
setEquality,
functionEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
dependent_set_memberEquality,
addEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
universeEquality,
cumulativity,
independent_functionElimination,
productElimination,
hyp_replacement,
equalitySymmetry,
imageElimination,
equalityTransitivity,
equalityElimination,
int_eqReduceTrueSq,
promote_hyp,
instantiate,
int_eqReduceFalseSq,
imageMemberEquality,
baseClosed,
pointwiseFunctionalityForEquality,
pertypeElimination,
productEquality,
applyLambdaEquality
Latex:
\mforall{}B,Q:n:\mBbbN{} {}\mrightarrow{} \{s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}| strictly-increasing-seq(n;s)\} {}\mrightarrow{} \mBbbP{}.
((\mforall{}n:\mBbbN{}. \mforall{}s:\{s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}| strictly-increasing-seq(n;s)\} .
(B[n;s] {}\mRightarrow{} (\mforall{}m:\mBbbN{}. (strictly-increasing-seq(n + 1;s.m@n) {}\mRightarrow{} B[n + 1;s.m@n]))))
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}s:\{s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}| strictly-increasing-seq(n;s)\} . (B[n;s] {}\mRightarrow{} \00D9(Q[n;s])))
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}s:\{s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}| strictly-increasing-seq(n;s)\} .
((\mforall{}m:\mBbbN{}. (strictly-increasing-seq(n + 1;s.m@n) {}\mRightarrow{} \00D9(Q[n + 1;s.m@n]))) {}\mRightarrow{} \00D9(Q[n;s])))
{}\mRightarrow{} (\mforall{}alpha:StrictInc. \mexists{}m:\mBbbN{}. B[m;alpha])
{}\mRightarrow{} \00D9(Q[0;\mlambda{}x.\mbot{}]))
Date html generated:
2017_04_20-AM-07_23_43
Last ObjectModification:
2017_02_27-PM-05_59_45
Theory : continuity
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