Nuprl Lemma : urec-level-property
∀[F:Type ⟶ Type]
(∀[f:destructor{i:l}(T.F[T])]. ∀[x:urec(F)]. (x ∈ F^urec-level(f;x) Void)) supposing
((∀T:Type. ((T ⊆r Base) ⇒ (F[T] ⊆r Base))) and
Monotone(T.F[T]))
Proof
Definitions occuring in Statement :
urec-level: urec-level(f;x),
destructor: destructor{i:l}(T.F[T]),
urec: urec(F),
type-monotone: Monotone(T.F[T]),
fun_exp: f^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,
apply: f a,
function: x:A ⟶ B[x],
base: Base,
void: Void,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
nat: ℕ,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
guard: {T},
le: A ≤ B,
subtype_rel: A ⊆r B,
urec: urec(F),
tunion: ⋃x:A.B[x],
pi2: snd(t),
destructor: destructor{i:l}(T.F[T]),
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
urec-level: urec-level(f;x),
subtract: n - m,
decomp: decomp{i:l}(S.F[S];T;x),
ap-con: ap-con(con;L),
cons: [a / b],
ifthenelse: if b then t else f fi ,
btrue: tt,
compose: f o g,
constructor: Constr(T.F[T]),
bfalse: ff,
listp: A List+,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than': less_than'(a;b),
true: True,
l_exists: (∃x∈L. P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
uiff: uiff(P;Q),
select: L[n],
less_than: a < b,
squash: ↓T,
l_member: (x ∈ l),
cand: A c∧ B,
colength: colength(L),
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
urec_wf,
destructor_wf,
subtype_rel_wf,
base_wf,
type-monotone_wf,
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,
less_than_wf,
urec-level_wf,
subtract-1-ge-0,
nat_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
fun_exp0_lemma,
fun_exp_wf,
subtract_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
le_wf,
fun_exp_add1,
type-monotone-fun_exp,
subtract-add-cancel,
subtype_rel_transitivity,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
add-associates,
add-swap,
add-commutes,
zero-add,
decomp_wf,
subtype_rel_self,
list-cases,
product_subtype_list,
null_nil_lemma,
map_nil_lemma,
fun_exp1_lemma,
nil_wf,
list_wf,
null_cons_lemma,
cons-listp,
map_wf_listp,
subtype_rel_dep_function,
void_wf,
listp_properties,
imax-list_wf,
subtype_rel_list,
decidable__lt,
length_wf,
imax-list-ub,
length_of_nil_lemma,
length_of_cons_lemma,
istype-false,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
false_wf,
select_wf,
cons_wf,
int_seg_properties,
map_cons_lemma,
l_all_iff,
l_member_wf,
member_wf,
map_wf,
map-length,
length-map,
select-map,
top_wf,
colength-cons-not-zero,
colength_wf_list,
spread_cons_lemma,
l_all_wf2,
l_all_cons,
equal-wf-base,
equal_wf,
satisfiable-full-omega-tt,
add_nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
isect_memberEquality_alt,
because_Cache,
lambdaEquality_alt,
applyEquality,
universeEquality,
functionIsType,
inhabitedIsType,
lambdaFormation_alt,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
dependent_functionElimination,
voidElimination,
independent_pairFormation,
functionIsTypeImplies,
applyLambdaEquality,
productElimination,
imageElimination,
unionElimination,
instantiate,
cumulativity,
intEquality,
closedConclusion,
dependent_set_memberEquality_alt,
voidEquality,
addEquality,
isectIsType,
setIsType,
equalityIsType1,
promote_hyp,
hypothesis_subsumption,
imageMemberEquality,
dependent_pairEquality_alt,
baseClosed,
pointwiseFunctionality,
baseApply,
productIsType,
equalityIsType4,
hyp_replacement,
computeAll,
isect_memberEquality,
dependent_pairFormation,
lambdaFormation,
lambdaEquality,
functionExtensionality,
dependent_set_memberEquality
Latex:
\mforall{}[F:Type {}\mrightarrow{} Type]
(\mforall{}[f:destructor\{i:l\}(T.F[T])]. \mforall{}[x:urec(F)]. (x \mmember{} F\^{}urec-level(f;x) Void)) supposing
((\mforall{}T:Type. ((T \msubseteq{}r Base) {}\mRightarrow{} (F[T] \msubseteq{}r Base))) and
Monotone(T.F[T]))
Date html generated:
2019_10_15-AM-11_31_30
Last ObjectModification:
2018_10_11-PM-11_01_57
Theory : general
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