Nuprl Lemma : prior_wf
∀[T:Type]. ∀[f:ℕ ⟶ (T + Top)]. ∀[n:ℕ]. (prior(n;f) ∈ ℕn × T?)
Proof
Definitions occuring in Statement :
prior: prior(n;f),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
top: Top,
unit: Unit,
member: t ∈ T,
function: x:A ⟶ B[x],
product: x:A × B[x],
union: left + right,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
prior: prior(n;f),
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
all: ∀x:A. B[x],
implies: P ⇒ Q,
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
ge: i ≥ j ,
int_upper: {i...},
has-value: (a)↓,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
subtype_rel: A ⊆r B,
so_apply: x[s1;s2]
Lemmas referenced :
natrec_wf,
int_seg_wf,
unit_wf2,
nat_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
it_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_upper_subtype_nat,
false_wf,
le_wf,
nat_properties,
nequal-le-implies,
zero-add,
value-type-has-value,
int-value-type,
subtract_wf,
int_upper_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_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_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
top_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
subtype_rel_union,
subtype_rel_product,
int_seg_subtype
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
unionEquality,
productEquality,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
cumulativity,
because_Cache,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
inrEquality,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
independent_functionElimination,
voidElimination,
hypothesis_subsumption,
dependent_set_memberEquality,
independent_pairFormation,
callbyvalueReduce,
intEquality,
applyEquality,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
inlEquality,
independent_pairEquality,
functionExtensionality,
functionEquality,
axiomEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[f:\mBbbN{} {}\mrightarrow{} (T + Top)]. \mforall{}[n:\mBbbN{}]. (prior(n;f) \mmember{} \mBbbN{}n \mtimes{} T?)
Date html generated:
2017_10_01-AM-09_12_01
Last ObjectModification:
2017_07_26-PM-04_47_50
Theory : general
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