Nuprl Lemma : strong-continuity-test-sp_wf
∀[M:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ (ℕ?)]. ∀[n:ℕ]. ∀[f:ℕn ⟶ ℕ]. ∀[k:ℕ]. (strong-continuity-test-sp(M;n;f;k) ∈ ℕ?)
Proof
Definitions occuring in Statement :
strong-continuity-test-sp: strong-continuity-test-sp(M;n;f;k),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
unit: Unit,
member: t ∈ T,
function: x:A ⟶ B[x],
union: left + right,
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
strong-continuity-test-sp: strong-continuity-test-sp(M;n;f;k),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
exposed-it: exposed-it,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
int_seg: {i..j-},
nequal: a ≠ b ∈ T ,
lelt: i ≤ j < k
Lemmas referenced :
nat_wf,
int_seg_wf,
unit_wf2,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
primrec0_lemma,
primrec-unroll-1,
le_wf,
subtype_rel_dep_function,
int_seg_subtype,
false_wf,
subtype_rel_self,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
primrec_wf,
int_seg_subtype_nat,
int_seg_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
hypothesis,
because_Cache,
functionEquality,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
unionEquality,
isect_memberFormation,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
intWeakElimination,
lambdaFormation,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
unionElimination,
inlEquality,
dependent_set_memberEquality,
applyEquality,
functionExtensionality,
equalityElimination,
productElimination,
inrEquality,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}[M:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} (\mBbbN{}?)]. \mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}]. \mforall{}[k:\mBbbN{}].
(strong-continuity-test-sp(M;n;f;k) \mmember{} \mBbbN{}?)
Date html generated:
2017_04_17-AM-10_01_49
Last ObjectModification:
2017_02_27-PM-05_53_27
Theory : continuity
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