Nuprl Lemma : poly-choice-eta-2
∀f:Base. ((∀x,y:Base. ((f x y) = x ∈ Base)) ⇒ (f ~ λx.if f x is lambda then λy.x otherwise ⊥))
Proof
Definitions occuring in Statement :
bottom: ⊥,
islambda: if z is lambda then a otherwise b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
lambda: λx.A[x],
base: Base,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
false: False,
not: ¬A,
top: Top,
so_apply: x[s],
so_lambda: λ2x.t[x],
or: P ∨ Q,
has-value: (a)↓,
sq_type: SQType(T),
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
true: True,
prop: ℙ,
squash: ↓T,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
sq_stable: SqStable(P),
is-exception: is-exception(t),
label: ...$L... t
Lemmas referenced :
sqle_wf_base,
bottom-sqle,
not_id_sqeq_bottom,
strictness-apply,
equal-wf-base,
all_wf,
is-exception_wf,
has-value_wf_base,
int-value-type,
value-type-has-value,
iff_weakening_equal,
true_wf,
squash_wf,
equal_wf,
subtype_rel_self,
base_wf,
subtype_base_sq,
has-value-implies-dec-islambda-2,
sqequal-wf-base,
sq_stable__sqle,
exception-not-value,
not-id-sqle-bottom,
exception-not-bottom,
bottom_diverge,
and_wf
Rules used in proof :
sqequalSqle,
voidEquality,
voidElimination,
isect_memberEquality,
closedConclusion,
baseApply,
sqleRule,
sqleReflexivity,
divergentSqle,
unionElimination,
callbyvalueApplyCases,
callbyvalueApply,
intEquality,
independent_functionElimination,
productElimination,
imageMemberEquality,
sqequalRule,
natural_numberEquality,
because_Cache,
baseClosed,
dependent_functionElimination,
universeEquality,
equalitySymmetry,
equalityTransitivity,
hypothesisEquality,
imageElimination,
lambdaEquality,
applyEquality,
independent_isectElimination,
hypothesis,
cumulativity,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
instantiate,
thin,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
axiomSqleEquality,
applyExceptionCases,
exceptionApplyCases,
sqequalIntensionalEquality,
sqleLambda,
levelHypothesis,
rename,
setElimination,
applyLambdaEquality,
dependent_set_memberEquality,
hyp_replacement,
addLevel,
independent_pairFormation,
exceptionSqequal,
callbyvalueIslambda,
islambdaExceptionCases
Latex:
\mforall{}f:Base. ((\mforall{}x,y:Base. ((f x y) = x)) {}\mRightarrow{} (f \msim{} \mlambda{}x.if f x is lambda then \mlambda{}y.x otherwise \mbot{}))
Date html generated:
2019_06_20-PM-02_45_08
Last ObjectModification:
2019_01_12-AM-10_05_30
Theory : num_thy_1
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