Nuprl Lemma : polymorphic-choice-sq
∀f:⋂A:Type. (A ⟶ A ⟶ A). ((f ~ λx.if f x is lambda then λy.x otherwise ⊥) ∨ (f ~ λx,y. y))
Proof
Definitions occuring in Statement :
bottom: ⊥,
islambda: if z is lambda then a otherwise b,
all: ∀x:A. B[x],
or: P ∨ Q,
apply: f a,
lambda: λx.A[x],
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
subtype_rel: A ⊆r B,
guard: {T},
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
top: Top,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
has-value: (a)↓,
btrue: tt,
it: ⋅,
bfalse: ff,
ge: i ≥ j ,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
sq_type: SQType(T)
Lemmas referenced :
istype-universe,
polymorphic-choice-base-sq,
nat_wf,
false_wf,
le_wf,
top_wf,
value-type-has-value,
set-value-type,
int-value-type,
has-value_wf_base,
is-exception_wf,
set_subtype_base,
int_subtype_base,
equal_wf,
strictness-apply,
bottom_diverge,
int_term_value_var_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformand_wf,
nat_properties,
istype-le,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
istype-void,
int_formula_prop_not_lemma,
istype-int,
itermConstant_wf,
intformle_wf,
intformnot_wf,
full-omega-unsat,
decidable__le,
equal_functionality_wrt_subtype_rel2,
istype-top,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
true_wf,
squash_wf,
member_wf,
istype-sqequal,
not_zero_sqequal_one,
iff_weakening_equal,
subtype_rel_self,
istype-base,
subtype_base_sq,
equal-wf-base,
or_wf,
sqequal-wf-base
Rules used in proof :
because_Cache,
hypothesisEquality,
Error :universeIsType,
Error :functionIsType,
hypothesis,
universeEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
instantiate,
cut,
Error :isectIsType,
Error :lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
pointwiseFunctionality,
dependent_functionElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
unionElimination,
sqequalRule,
applyEquality,
lambdaEquality,
isectEquality,
cumulativity,
functionEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
lambdaFormation,
islambdaCases,
isect_memberFormation,
axiomSqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
intEquality,
functionExtensionality,
callbyvalueApply,
divergentSqle,
baseClosed,
baseApply,
closedConclusion,
inlEquality,
sqequalIntensionalEquality,
inrEquality,
Error :equalityIstype,
int_eqEquality,
rename,
setElimination,
applyLambdaEquality,
Error :isect_memberEquality_alt,
Error :dependent_pairFormation_alt,
approximateComputation,
Error :dependent_set_memberEquality_alt,
Error :inhabitedIsType,
Error :lambdaEquality_alt,
Error :isectIsTypeImplies,
Error :isect_memberFormation_alt,
int_eqReduceTrueSq,
imageMemberEquality,
imageElimination,
unionEquality,
sqequalExtensionalEquality,
Error :inlEquality_alt,
productElimination,
callbyvalueIslambda
Latex:
\mforall{}f:\mcap{}A:Type. (A {}\mrightarrow{} A {}\mrightarrow{} A). ((f \msim{} \mlambda{}x.if f x is lambda then \mlambda{}y.x otherwise \mbot{}) \mvee{} (f \msim{} \mlambda{}x,y. y))
Date html generated:
2019_06_20-PM-02_45_22
Last ObjectModification:
2019_01_09-PM-03_39_35
Theory : num_thy_1
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