Nuprl Lemma : gen-bar-rec
∀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.⊥]))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
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 :
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
lelt: i ≤ j < k,
guard: {T},
int_seg: {i..j-},
prop: ℙ,
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
uimplies: b supposing a,
int_upper: {i...},
so_apply: x[s],
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
implies: P ⇒ Q,
all: ∀x:A. B[x],
squash: ↓T,
isl: isl(x),
true: True,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
bfalse: ff,
sq_type: SQType(T),
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
spector-bar-rec: spector-bar-rec(Y;G;H;n;s),
bnot: ¬bb,
it: ⋅,
unit: Unit,
bool: 𝔹,
ext2Baire: ext2Baire(n;f;d),
seq-add: s.x@n
Lemmas referenced :
seq-add_wf,
int_term_value_add_lemma,
int_formula_prop_not_lemma,
itermAdd_wf,
intformnot_wf,
decidable__le,
nat_properties,
equiv_rel_true,
true_wf,
quotient_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
satisfiable-full-omega-tt,
int_seg_properties,
isl_wf,
assert_wf,
equal_wf,
unit_wf2,
le_wf,
ext2Baire_wf,
exists_wf,
implies-quotient-true,
subtype_rel_self,
false_wf,
int_seg_subtype_nat,
int_seg_wf,
nat_wf,
subtype_rel_dep_function,
int_upper_subtype_nat,
int_upper_wf,
all_wf,
strong-continuity-rel-unique-pair,
decidable__equal_int,
btrue_wf,
bfalse_wf,
seq-normalize_wf,
seq-normalize-equal,
iff_wf,
assert_of_le_int,
le_int_wf,
iff_imp_equal_bool,
bool_subtype_base,
bool_wf,
subtype_base_sq,
less_than_wf,
assert-bnot,
bool_cases_sqequal,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf
Rules used in proof :
cumulativity,
universeEquality,
unionElimination,
addEquality,
independent_functionElimination,
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
productElimination,
dependent_pairEquality,
inlEquality,
dependent_set_memberEquality,
productEquality,
unionEquality,
functionEquality,
independent_pairFormation,
independent_isectElimination,
natural_numberEquality,
hypothesisEquality,
functionExtensionality,
applyEquality,
hypothesis,
because_Cache,
setElimination,
isectElimination,
lambdaEquality,
sqequalRule,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
rename,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
applyLambdaEquality,
hyp_replacement,
equalitySymmetry,
equalityTransitivity,
imageElimination,
SquashedBarInduction,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
Error :equalityIstype,
baseClosed,
imageMemberEquality,
impliesFunctionality,
addLevel,
instantiate,
promote_hyp,
equalityElimination
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{}]))
Date html generated:
2019_06_20-PM-03_07_08
Last ObjectModification:
2019_01_15-PM-03_07_44
Theory : continuity
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