Nuprl Lemma : urec-is-least-fixedpoint
∀[F:Type ⟶ Type]. ∀T:Type. urec(F) ⊆r T supposing F T ≡ T supposing Monotone(T.F T)
Proof
Definitions occuring in Statement :
urec: urec(F),
type-monotone: Monotone(T.F[T]),
ext-eq: A ≡ B,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀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,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
urec: urec(F),
tunion: ⋃x:A.B[x],
pi2: snd(t),
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
type-monotone: Monotone(T.F[T])
Lemmas referenced :
fun_exp_wf,
subtype_rel_weakening,
subtype_rel_transitivity,
iff_weakening_equal,
true_wf,
squash_wf,
subtype_rel_wf,
type-monotone_wf,
ext-eq_wf,
urec_wf,
subtract-add-cancel,
le_wf,
fun_exp_apply_add1,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
fun_exp0_lemma,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
productElimination,
thin,
sqequalRule,
hypothesisEquality,
applyEquality,
lemma_by_obid,
isectElimination,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
unionElimination,
instantiate,
because_Cache,
dependent_set_memberEquality,
universeEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[F:Type {}\mrightarrow{} Type]. \mforall{}T:Type. urec(F) \msubseteq{}r T supposing F T \mequiv{} T supposing Monotone(T.F T)
Date html generated:
2016_05_15-PM-06_54_53
Last ObjectModification:
2016_01_16-AM-09_48_47
Theory : general
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