Nuprl Lemma : polymorphic-choice
∀f:⋂A:Type. (A ⟶ A ⟶ A). ((f = (λx,y. x) ∈ (⋂A:Type. (A ⟶ A ⟶ A))) ∨ (f = (λx,y. y) ∈ (⋂A:Type. (A ⟶ A ⟶ A))))
Proof
Definitions occuring in Statement :
all: ∀x:A. B[x],
or: P ∨ Q,
lambda: λx.A[x],
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
or: P ∨ Q,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
label: ...$L... t,
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
polymorphic-choice-base,
equal_wf,
squash_wf,
true_wf,
istype-universe,
subtype_base_sq,
base_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
unionElimination,
Error :inlFormation_alt,
Error :isect_memberEquality_alt,
functionExtensionality,
sqequalRule,
pointwiseFunctionalityForEquality,
because_Cache,
applyEquality,
Error :lambdaEquality_alt,
imageElimination,
isectElimination,
equalityTransitivity,
equalitySymmetry,
Error :universeIsType,
Error :inhabitedIsType,
universeEquality,
instantiate,
cumulativity,
independent_isectElimination,
independent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
Error :equalityIsType3,
Error :inrFormation_alt,
Error :isectIsType,
Error :functionIsType
Latex:
\mforall{}f:\mcap{}A:Type. (A {}\mrightarrow{} A {}\mrightarrow{} A). ((f = (\mlambda{}x,y. x)) \mvee{} (f = (\mlambda{}x,y. y)))
Date html generated:
2019_06_20-PM-02_44_55
Last ObjectModification:
2018_10_06-AM-11_24_31
Theory : num_thy_1
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