Nuprl Lemma : type-monotone_fun_exp_top
∀[F:Type ⟶ Type]. ∀[n,m:ℕ]. (F^n Top) ⊆r (F^m Top) supposing m ≤ n supposing Monotone(T.F[T])
Proof
Definitions occuring in Statement :
type-monotone: Monotone(T.F[T]),
fun_exp: f^n,
nat: ℕ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
le: A ≤ B,
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,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
and: P ∧ Q,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
prop: ℙ,
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
nat_plus: ℕ+,
less_than: a < b,
less_than': less_than'(a;b),
not: ¬A,
false: False,
decidable: Dec(P),
or: P ∨ Q,
ge: i ≥ j ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
bnot: ¬bb,
assert: ↑b,
compose: f o g,
type-monotone: Monotone(T.F[T]),
nequal: a ≠ b ∈ T
Lemmas referenced :
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
minus-one-mul,
add-commutes,
minus-one-mul-top,
add-associates,
add-mul-special,
zero-mul,
zero-add,
subtract_wf,
subtype_rel_wf,
squash_wf,
true_wf,
fun_exp_add_apply,
top_wf,
fun_exp_wf,
iff_weakening_equal,
equal_wf,
type-monotone_wf,
add_functionality_wrt_le,
le_reflexive,
one-mul,
two-mul,
mul-distributes-right,
add-zero,
not-le-2,
minus-zero,
add-swap,
omega-shadow,
less_than_wf,
mul-distributes,
mul-associates,
mul-commutes,
le-add-cancel-alt,
nat_properties,
decidable__le,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
fun_exp0_lemma,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-minus,
le-add-cancel,
le_weakening2,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
not-equal-2,
fun_exp_unroll
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesis,
independent_isectElimination,
sqequalRule,
intEquality,
lambdaEquality,
natural_numberEquality,
hypothesisEquality,
setElimination,
rename,
dependent_set_memberEquality,
productElimination,
applyEquality,
isect_memberEquality,
voidElimination,
voidEquality,
minusEquality,
because_Cache,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
lambdaFormation,
imageElimination,
universeEquality,
functionExtensionality,
imageMemberEquality,
baseClosed,
axiomEquality,
functionEquality,
multiplyEquality,
addEquality,
independent_pairFormation,
unionElimination,
intWeakElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp
Latex:
\mforall{}[F:Type {}\mrightarrow{} Type]. \mforall{}[n,m:\mBbbN{}]. (F\^{}n Top) \msubseteq{}r (F\^{}m Top) supposing m \mleq{} n supposing Monotone(T.F[T])
Date html generated:
2017_04_14-AM-07_37_25
Last ObjectModification:
2017_02_27-PM-03_09_44
Theory : subtype_1
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