Nuprl Lemma : urec_subtype
∀[F:Type ⟶ Type]. urec(F) ⊆r (F urec(F)) supposing Monotone(T.F[T])
Proof
Definitions occuring in Statement :
urec: urec(F),
type-monotone: Monotone(T.F[T]),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
apply: f a,
function: x:A ⟶ B[x],
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],
subtype_rel: A ⊆r B,
prop: ℙ,
urec: urec(F),
tunion: ⋃x:A.B[x],
pi2: snd(t),
all: ∀x:A. B[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
top: Top,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
nat_plus: ℕ+,
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
less_than': less_than'(a;b),
true: True,
subtract: n - m,
union-continuous: union-continuous{i:l}(T.F[T])
Lemmas referenced :
subtype_rel_transitivity,
urec_wf,
tunion_wf,
nat_wf,
fun_exp_wf,
istype-nat,
type-monotone_wf,
istype-universe,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
fun_exp0_lemma,
istype-void,
subtract_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformeq_wf,
istype-int,
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_eq_lemma,
int_formula_prop_wf,
istype-le,
subtype_rel-equal,
fun_exp_add1_sub,
decidable__lt,
istype-false,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
istype-less_than,
type-monotone-union-continuous
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality_alt,
applyEquality,
instantiate,
closedConclusion,
universeEquality,
because_Cache,
voidEquality,
independent_isectElimination,
axiomEquality,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
functionIsType,
imageElimination,
productElimination,
dependent_functionElimination,
setElimination,
rename,
natural_numberEquality,
unionElimination,
cumulativity,
intEquality,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
voidElimination,
imageMemberEquality,
dependent_pairEquality_alt,
dependent_set_memberEquality_alt,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
lambdaFormation_alt,
addEquality,
minusEquality,
baseClosed,
lambdaEquality,
lemma_by_obid
Latex:
\mforall{}[F:Type {}\mrightarrow{} Type]. urec(F) \msubseteq{}r (F urec(F)) supposing Monotone(T.F[T])
Date html generated:
2019_10_15-AM-11_30_03
Last ObjectModification:
2018_10_31-PM-02_25_38
Theory : general
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