Nuprl Lemma : continuous'-monotone-product
∀[F,G:Type ⟶ Type].
(continuous'-monotone{i:l}(T.F[T] × G[T])) supposing
(continuous'-monotone{i:l}(T.G[T]) and
continuous'-monotone{i:l}(T.F[T]))
Proof
Definitions occuring in Statement :
continuous'-monotone: continuous'-monotone{i:l}(T.F[T]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
so_apply: x[s],
continuous'-monotone: continuous'-monotone{i:l}(T.F[T]),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
type-monotone: Monotone(T.F[T]),
subtype_rel: A ⊆r B,
type-continuous': semi-continuous(λT.F[T]),
tunion: ⋃x:A.B[x],
pi2: snd(t),
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
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,
prop: ℙ,
type-incr-chain: type-incr-chain{i:l}(),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x]
Lemmas referenced :
subtype_rel_simple_product,
subtype_rel_wf,
imax_wf,
imax_nat,
nat_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
equal_wf,
le_wf,
type-incr-chain-subtype,
imax_ub,
tunion_wf,
type-incr-chain_wf,
type-monotone_wf,
type-continuous'_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
applyEquality,
hypothesisEquality,
independent_isectElimination,
hypothesis,
axiomEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
independent_pairFormation,
lambdaEquality,
independent_pairEquality,
imageElimination,
imageMemberEquality,
dependent_pairEquality,
dependent_set_memberEquality,
setElimination,
rename,
lambdaFormation,
applyLambdaEquality,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
inlFormation,
inrFormation,
productEquality,
baseClosed,
functionEquality
Latex:
\mforall{}[F,G:Type {}\mrightarrow{} Type].
(continuous'-monotone\{i:l\}(T.F[T] \mtimes{} G[T])) supposing
(continuous'-monotone\{i:l\}(T.G[T]) and
continuous'-monotone\{i:l\}(T.F[T]))
Date html generated:
2018_05_21-PM-08_44_12
Last ObjectModification:
2018_05_19-PM-05_06_24
Theory : general
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