Nuprl Lemma : is-above-pair
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[a:A]. ∀[b:B[a]].
∀z:Base. (is-above(a:A × B[a];<a, b>;z) ⇒ (∃c,d:Base. ((z ~ <c, d>) ∧ is-above(A;a;c) ∧ is-above(B[a];b;d))))
Proof
Definitions occuring in Statement :
is-above: is-above(T;a;z),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
base: Base,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
is-above: is-above(T;a;z),
and: P ∧ Q,
subtype_rel: A ⊆r B,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
top: Top,
pi2: snd(t),
pi1: fst(t),
has-value: (a)↓,
or: P ∨ Q,
prop: ℙ,
cand: A c∧ B,
istype: istype(T),
false: False,
not: ¬A
Lemmas referenced :
pair-eta,
subtype_rel_product,
top_wf,
istype-top,
istype-void,
has-value-monotonic,
has-value_wf_base,
is-exception_wf,
has-value-implies-dec-ispair-2,
is-above_wf,
base_wf,
sqle_wf_base,
subtype_rel-equal,
equal_functionality_wrt_subtype_rel2,
not-btrue-sqle-bfalse
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
Error :dependent_pairFormation_alt,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
applyEquality,
Error :lambdaEquality_alt,
functionExtensionality,
cumulativity,
Error :inhabitedIsType,
independent_isectElimination,
Error :isect_memberEquality_alt,
voidElimination,
Error :universeIsType,
because_Cache,
applyLambdaEquality,
divergentSqle,
sqleReflexivity,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
Error :productIsType,
sqequalIntensionalEquality,
productEquality,
Error :dependent_pairEquality_alt,
Error :functionIsType,
universeEquality,
independent_pairFormation,
Error :equalityIsType2,
sqleRule
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[a:A]. \mforall{}[b:B[a]].
\mforall{}z:Base. (is-above(a:A \mtimes{} B[a];<a, b>z) {}\mRightarrow{} (\mexists{}c,d:Base. ((z \msim{} <c, d>) \mwedge{} is-above(A;a;c) \mwedge{} is-above(\000CB[a];b;d))))
Date html generated:
2019_06_20-PM-00_28_20
Last ObjectModification:
2018_09_29-PM-11_16_01
Theory : subtype_1
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