Nuprl Lemma : polymorphic-choice-int
∀f:⋂A:Type. (A ⟶ A ⟶ A). ((∀x,y:ℤ. ((f x y) = x ∈ ℤ)) ∨ (∀x,y:ℤ. ((f x y) = y ∈ ℤ)))
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],
int: ℤ,
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,
implies: P ⇒ Q,
squash: ↓T,
decidable: Dec(P),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
label: ...$L... t,
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
int_mod: ℤ_n,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
nat_plus: ℕ+,
less_than: a < b,
less_than': less_than'(a;b),
modulus: a mod n,
int_seg: {i..j-},
quotient: x,y:A//B[x; y],
cand: A c∧ B,
absval: |i|,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
istype-universe,
base_wf,
or_wf,
equal-wf-base,
istype-base,
decidable__equal_int,
int_subtype_base,
subtype_base_sq,
subtype_rel_self,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
istype-int,
int-subtype-int_mod,
int_mod_wf,
quotient-member-eq,
eqmod_wf,
eqmod_equiv_rel,
eqmod-zero,
modulus-equal-iff-eqmod,
istype-less_than,
modulus_wf_int_mod,
full-omega-unsat,
intformeq_wf,
itermConstant_wf,
int_formula_prop_eq_lemma,
istype-void,
int_term_value_constant_lemma,
int_formula_prop_wf,
intformand_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
exists-type-equating-ints,
equal-wf-T-base,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
Error :isectIsType,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
universeEquality,
hypothesis,
Error :functionIsType,
Error :universeIsType,
hypothesisEquality,
because_Cache,
setEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
Error :inlFormation_alt,
Error :equalityIstype,
Error :inhabitedIsType,
Error :dependent_set_memberEquality_alt,
sqequalRule,
Error :unionIsType,
Error :inrFormation_alt,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_functionElimination,
independent_functionElimination,
closedConclusion,
intEquality,
natural_numberEquality,
unionElimination,
cumulativity,
independent_isectElimination,
Error :lambdaEquality_alt,
productElimination,
sqequalBase,
voidElimination,
independent_pairFormation,
pertypeElimination,
promote_hyp,
Error :productIsType,
callbyvalueReduce,
sqleReflexivity,
Error :dependent_pairFormation_alt,
approximateComputation,
Error :isect_memberEquality_alt,
int_eqEquality,
hyp_replacement,
applyLambdaEquality,
functionEquality
Latex:
\mforall{}f:\mcap{}A:Type. (A {}\mrightarrow{} A {}\mrightarrow{} A). ((\mforall{}x,y:\mBbbZ{}. ((f x y) = x)) \mvee{} (\mforall{}x,y:\mBbbZ{}. ((f x y) = y)))
Date html generated:
2019_06_20-PM-02_44_31
Last ObjectModification:
2018_11_25-PM-01_33_30
Theory : num_thy_1
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