Nuprl Lemma : gen-bar-ind-implies-monotone
(∀P:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ ℙ
((∀n:ℕ. ∀s:ℕn ⟶ ℕ. ((∀m:ℕ. P[n + 1;s.m@n]) ⇒ P[n;s])) ⇒ (∀f:ℕ ⟶ ℕ. ⇃(∃n:ℕ. ∀m:{n...}. P[m;f])) ⇒ ⇃(P[0;λx.⊥])))
⇒ (∀B,Q:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ ℙ. ∀bar:∀s:ℕ ⟶ ℕ. ⇃(∃n:ℕ. B[n;s]). ∀mon:∀n:ℕ. ∀m:ℕn. ∀s:ℕn ⟶ ℕ. (B[m;s] ⇒ B[n;s]).
∀base:∀n:ℕ. ∀s:ℕn ⟶ ℕ. (B[n;s] ⇒ Q[n;s]). ∀ind:∀n:ℕ. ∀s:ℕn ⟶ ℕ. ((∀m:ℕ. Q[n + 1;s.m@n]) ⇒ Q[n;s]).
⇃(Q[0;seq-normalize(0;⊥)]))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
seq-normalize: seq-normalize(n;s),
seq-add: s.x@n,
int_upper: {i...},
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: $n
Definitions unfolded in proof :
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s1;s2],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
so_apply: x[s],
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
so_lambda: λ2x y.t[x; y],
int_upper: {i...},
squash: ↓T,
label: ...$L... t,
sq_type: SQType(T),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
nat_wf,
all_wf,
int_seg_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,
int_seg_subtype_nat,
false_wf,
subtype_rel_dep_function,
int_seg_subtype,
int_seg_properties,
intformless_wf,
int_formula_prop_less_lemma,
subtype_rel_self,
quotient_wf,
exists_wf,
true_wf,
equiv_rel_true,
int_upper_wf,
int_upper_subtype_nat,
implies-quotient-true,
subtract_wf,
int_upper_properties,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
equal_wf,
squash_wf,
subtype_base_sq,
int_subtype_base,
iff_weakening_equal,
add-zero,
set_wf,
less_than_wf,
primrec-wf2,
decidable__lt,
lelt_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
functionEquality,
introduction,
extract_by_obid,
isectElimination,
sqequalRule,
lambdaEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
addEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
universeEquality,
productElimination,
cumulativity,
instantiate,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
hyp_replacement,
applyLambdaEquality
Latex:
(\mforall{}P:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbP{}
((\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}. ((\mforall{}m:\mBbbN{}. P[n + 1;s.m@n]) {}\mRightarrow{} P[n;s]))
{}\mRightarrow{} (\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \00D9(\mexists{}n:\mBbbN{}. \mforall{}m:\{n...\}. P[m;f]))
{}\mRightarrow{} \00D9(P[0;\mlambda{}x.\mbot{}])))
{}\mRightarrow{} (\mforall{}B,Q:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbP{}. \mforall{}bar:\mforall{}s:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \00D9(\mexists{}n:\mBbbN{}. B[n;s]). \mforall{}mon:\mforall{}n:\mBbbN{}. \mforall{}m:\mBbbN{}n. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}.
(B[m;s] {}\mRightarrow{} B[n;s]).
\mforall{}base:\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}. (B[n;s] {}\mRightarrow{} Q[n;s]). \mforall{}ind:\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}.
((\mforall{}m:\mBbbN{}. Q[n + 1;s.m@n]) {}\mRightarrow{} Q[n;s]).
\00D9(Q[0;seq-normalize(0;\mbot{})]))
Date html generated:
2017_04_20-AM-07_35_16
Last ObjectModification:
2017_02_27-PM-06_03_39
Theory : continuity
Home
Index