Nuprl Lemma : polymorphic-choice-base
∀f:⋂A:Type. (A ⟶ A ⟶ A). ((∀x,y:Base. ((f x y) = x ∈ Base)) ∨ (∀x,y:Base. ((f x y) = y ∈ Base)))
Proof
Definitions occuring in Statement :
all: ∀x:A. B[x],
or: P ∨ Q,
apply: f a,
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
base: Base,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
or: P ∨ Q,
prop: ℙ,
squash: ↓T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
false: False,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
rev_implies: P ⇐ Q
Lemmas referenced :
istype-universe,
base_wf,
or_wf,
equal-wf-base,
istype-base,
set_subtype_base,
equal_wf,
subtype_rel_self,
subtype_base_sq,
polymorphic-choice-int,
squash_wf,
true_wf,
iff_weakening_equal,
int_subtype_base,
type-with-y=n,
full-omega-unsat,
intformeq_wf,
itermConstant_wf,
istype-int,
int_formula_prop_eq_lemma,
istype-void,
int_term_value_constant_lemma,
int_formula_prop_wf,
type-with-x=0-y=1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
Error :isectIsType,
cut,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
universeEquality,
hypothesis,
Error :functionIsType,
Error :universeIsType,
hypothesisEquality,
Error :inhabitedIsType,
setEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
imageElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
applyEquality,
Error :inlFormation_alt,
Error :equalityIsType4,
Error :dependent_set_memberEquality_alt,
Error :unionIsType,
Error :inrFormation_alt,
applyLambdaEquality,
setElimination,
rename,
baseApply,
closedConclusion,
Error :lambdaEquality_alt,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
cumulativity,
independent_functionElimination,
Error :inlEquality_alt,
axiomEquality,
Error :inrEquality_alt,
natural_numberEquality,
productElimination,
independent_pairFormation,
intEquality,
voidElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :isect_memberEquality_alt,
hyp_replacement,
Error :equalityIsType1
Latex:
\mforall{}f:\mcap{}A:Type. (A {}\mrightarrow{} A {}\mrightarrow{} A). ((\mforall{}x,y:Base. ((f x y) = x)) \mvee{} (\mforall{}x,y:Base. ((f x y) = y)))
Date html generated:
2019_06_20-PM-02_44_48
Last ObjectModification:
2018_10_17-AM-10_42_24
Theory : num_thy_1
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