Nuprl Lemma : subtype_urec
∀[F:Type ⟶ Type]. (F urec(F)) ⊆r urec(F) supposing continuous'-monotone{i:l}(T.F T)
Proof
Definitions occuring in Statement :
continuous'-monotone: continuous'-monotone{i:l}(T.F[T]),
urec: urec(F),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
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,
continuous'-monotone: continuous'-monotone{i:l}(T.F[T]),
and: P ∧ Q,
type-continuous': semi-continuous(λT.F[T]),
all: ∀x:A. B[x],
implies: P ⇒ Q,
urec: urec(F),
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
tunion: ⋃x:A.B[x],
pi2: snd(t),
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top
Lemmas referenced :
monotone-incr-chain,
subtype_rel_transitivity,
urec_wf,
tunion_wf,
nat_wf,
fun_exp_wf,
istype-nat,
continuous'-monotone_wf,
istype-universe,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
subtype_rel-equal,
fun_exp_apply_add1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
isectElimination,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
sqequalRule,
applyEquality,
lambdaEquality_alt,
instantiate,
closedConclusion,
universeEquality,
because_Cache,
voidEquality,
independent_isectElimination,
axiomEquality,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
functionIsType,
imageElimination,
imageMemberEquality,
dependent_pairEquality_alt,
dependent_set_memberEquality_alt,
addEquality,
setElimination,
rename,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
independent_pairFormation,
baseClosed,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[F:Type {}\mrightarrow{} Type]. (F urec(F)) \msubseteq{}r urec(F) supposing continuous'-monotone\{i:l\}(T.F T)
Date html generated:
2019_10_15-AM-11_30_20
Last ObjectModification:
2018_10_31-PM-02_25_26
Theory : general
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