Nuprl Lemma : simple-dependent-choice
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
((∀x:T. ∃y:T. R[x;y]) ⇒ (∀x0:T. ∃f:ℕ ⟶ T. (((f 0) = x0 ∈ T) ∧ (∀i:ℕ. R[f i;f (i + 1)]))))
Proof
Definitions occuring in Statement :
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
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 :
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
guard: {T},
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
bfalse: ff,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
compose: f o g,
pi1: fst(t),
so_apply: x[s],
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_apply: x[s1;s2],
so_lambda: λ2x.t[x],
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
prop: ℙ,
primrec: primrec(n;b;c),
fun_exp: f^n,
cand: A c∧ B,
and: P ∧ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
le_weakening2,
primrec-wf2,
less_than_wf,
set_wf,
minus-minus,
subtract_wf,
add-subtract-cancel,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
le_antisymmetry_iff,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
less-iff-le,
fun_exp_unroll,
exists_wf,
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,
all_wf,
le_wf,
false_wf,
equal_wf,
nat_wf,
fun_exp_wf
Rules used in proof :
instantiate,
equalityElimination,
equalitySymmetry,
equalityTransitivity,
universeEquality,
functionEquality,
minusEquality,
intEquality,
voidEquality,
isect_memberEquality,
imageElimination,
baseClosed,
imageMemberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
unionElimination,
dependent_functionElimination,
rename,
setElimination,
addEquality,
because_Cache,
natural_numberEquality,
dependent_set_memberEquality,
functionExtensionality,
cumulativity,
productEquality,
independent_pairFormation,
sqequalRule,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
applyEquality,
lambdaEquality,
dependent_pairFormation,
productElimination,
sqequalHypSubstitution,
thin,
promote_hyp,
hypothesis,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}x:T. \mexists{}y:T. R[x;y]) {}\mRightarrow{} (\mforall{}x0:T. \mexists{}f:\mBbbN{} {}\mrightarrow{} T. (((f 0) = x0) \mwedge{} (\mforall{}i:\mBbbN{}. R[f i;f (i + 1)]))))
Date html generated:
2018_07_25-PM-02_09_04
Last ObjectModification:
2018_07_25-PM-01_14_39
Theory : fun_1
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