Nuprl Lemma : urec-level_wf
∀[F:Type ⟶ Type]
∀[f:destructor{i:l}(T.F[T])]. ∀[x:urec(F)]. (urec-level(f;x) ∈ ℕ) supposing ∀T:Type. ((T ⊆r Base) ⇒ (F[T] ⊆r Base))
Proof
Definitions occuring in Statement :
urec-level: urec-level(f;x),
destructor: destructor{i:l}(T.F[T]),
urec: urec(F),
nat: ℕ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
function: x:A ⟶ B[x],
base: Base,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
urec: urec(F),
tunion: ⋃x:A.B[x],
pi2: snd(t),
destructor: destructor{i:l}(T.F[T]),
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
urec-level: urec-level(f;x),
constructor: Constr(T.F[T]),
ap-con: ap-con(con;L),
decomp: decomp{i:l}(S.F[S];T;x),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
respects-equality: respects-equality(S;T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
le: A ≤ B,
less_than': less_than'(a;b),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
cons: [a / b],
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
subtract: n - m,
l_exists: (∃x∈L. P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
select: L[n]
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-less_than,
fun_exp0_lemma,
subtract-1-ge-0,
subtype_rel-equal,
fun_exp_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
fun_exp_add1_sub,
decidable__lt,
urec_wf,
destructor_wf,
istype-universe,
subtype_rel_wf,
base_wf,
subtype_rel_transitivity,
istype-nat,
le_wf,
decomp_wf,
constructor_wf,
list_wf,
ap-con_wf,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
equal_functionality_wrt_subtype_rel2,
istype-base,
subtype-respects-equality,
null_wf3,
subtype_rel_list,
top_wf,
eqtt_to_assert,
assert_of_null,
istype-false,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal-wf-T-base,
map_wf,
nat_wf,
list-cases,
product_subtype_list,
nil_wf,
map_cons_lemma,
list_ind_nil_lemma,
list_ind_cons_lemma,
int_subtype_base,
list_subtype_base,
set_subtype_base,
imax-list-ub,
length_of_nil_lemma,
length_of_cons_lemma,
length_wf_nat,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
add_nat_plus,
less_than_wf,
nat_plus_properties,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
length_wf,
select_wf,
cons_wf,
int_seg_properties,
imax-list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
imageElimination,
productElimination,
thin,
sqequalRule,
rename,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
applyEquality,
instantiate,
universeEquality,
dependent_set_memberEquality_alt,
unionElimination,
voidEquality,
because_Cache,
equalityIstype,
isectIsTypeImplies,
functionIsType,
isectIsType,
setIsType,
dependent_pairEquality_alt,
productIsType,
imageMemberEquality,
baseClosed,
hyp_replacement,
closedConclusion,
equalityElimination,
equalityIsType1,
promote_hyp,
cumulativity,
equalityIsType3,
hypothesis_subsumption,
intEquality,
applyLambdaEquality,
equalityIsType4,
baseApply,
addEquality,
minusEquality,
pointwiseFunctionality
Latex:
\mforall{}[F:Type {}\mrightarrow{} Type]
\mforall{}[f:destructor\{i:l\}(T.F[T])]. \mforall{}[x:urec(F)]. (urec-level(f;x) \mmember{} \mBbbN{})
supposing \mforall{}T:Type. ((T \msubseteq{}r Base) {}\mRightarrow{} (F[T] \msubseteq{}r Base))
Date html generated:
2020_05_20-AM-08_18_07
Last ObjectModification:
2019_11_27-PM-03_30_33
Theory : general
Home
Index