Nuprl Lemma : dependent-choice
∀[T:ℕ ⟶ Type]. ∀[R:n:ℕ ⟶ T[n] ⟶ T[n + 1] ⟶ ℙ].
((∀n:ℕ. ∀x:T[n]. ∃y:T[n + 1]. R[n;x;y])
⇒ (∀x0:T[0]. ∃f:n:ℕ ⟶ T[n]. (((f 0) = x0 ∈ T[0]) ∧ (∀n:ℕ. R[n;f n;f (n + 1)]))))
Proof
Definitions occuring in Statement :
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2;s3],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
cand: A c∧ B,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
guard: {T},
ge: i ≥ j ,
pi1: fst(t),
so_apply: x[s1;s2;s3],
true: True,
top: Top,
subtype_rel: A ⊆r B,
subtract: n - m,
squash: ↓T,
sq_stable: SqStable(P),
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
so_lambda: λ2x.t[x],
prop: ℙ,
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
so_apply: x[s],
member: t ∈ T,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
add-subtract-cancel,
primrec-wf2,
set_wf,
primrec1_lemma,
le_weakening2,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
le-add-cancel2,
subtype_rel-equal,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
primrec-unroll,
minus-minus,
less-iff-le,
not-ge-2,
subtract_wf,
primrec0_lemma,
less_than_wf,
ge_wf,
less_than_irreflexivity,
less_than_transitivity1,
nat_properties,
equal_wf,
subtype_rel_self,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
sq_stable__le,
not-le-2,
decidable__le,
exists_wf,
all_wf,
le_wf,
false_wf,
nat_wf
Rules used in proof :
productEquality,
instantiate,
equalityElimination,
axiomEquality,
intWeakElimination,
equalitySymmetry,
equalityTransitivity,
dependent_pairFormation,
universeEquality,
cumulativity,
functionEquality,
minusEquality,
intEquality,
voidEquality,
isect_memberEquality,
imageElimination,
baseClosed,
imageMemberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
unionElimination,
dependent_functionElimination,
rename,
setElimination,
addEquality,
because_Cache,
lambdaEquality,
isectElimination,
independent_pairFormation,
sqequalRule,
natural_numberEquality,
dependent_set_memberEquality,
extract_by_obid,
introduction,
hypothesisEquality,
functionExtensionality,
applyEquality,
productElimination,
sqequalHypSubstitution,
thin,
promote_hyp,
hypothesis,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:\mBbbN{} {}\mrightarrow{} Type]. \mforall{}[R:n:\mBbbN{} {}\mrightarrow{} T[n] {}\mrightarrow{} T[n + 1] {}\mrightarrow{} \mBbbP{}].
((\mforall{}n:\mBbbN{}. \mforall{}x:T[n]. \mexists{}y:T[n + 1]. R[n;x;y])
{}\mRightarrow{} (\mforall{}x0:T[0]. \mexists{}f:n:\mBbbN{} {}\mrightarrow{} T[n]. (((f 0) = x0) \mwedge{} (\mforall{}n:\mBbbN{}. R[n;f n;f (n + 1)]))))
Date html generated:
2018_07_25-PM-02_09_07
Last ObjectModification:
2018_07_25-PM-01_31_32
Theory : fun_1
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