Nuprl Lemma : bar-induction
∀[T:Type]. ∀[R,A:(T List) ⟶ ℙ].
((∀s:T List. Dec(R[s]))
⇒ (∀s:T List. (R[s] ⇒ A[s]))
⇒ (∀s:T List. ((∀t:T. A[s @ [t]]) ⇒ A[s]))
⇒ (∀s:T List. ((∀alpha:ℕ ⟶ T. (↓∃n:ℕ. R[s @ map(alpha;upto(n))])) ⇒ A[s])))
Proof
Definitions occuring in Statement :
upto: upto(n),
map: map(f;as),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
so_apply: x[s],
nat: ℕ,
so_apply: x[s1;s2],
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
guard: {T},
top: Top,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
seq-adjoin: s++t,
seq-append: seq-append(n;m;s1;s2),
less_than: a < b,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
label: ...$L... t,
rev_implies: P ⇐ Q,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
cand: A c∧ B
Lemmas referenced :
bar_induction,
list_wf,
map_wf,
int_seg_wf,
upto_wf,
nat_wf,
all_wf,
seq-adjoin_wf,
squash_wf,
exists_wf,
append_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
cons_wf,
nil_wf,
decidable_wf,
list_extensionality,
length-append,
map-length,
length_of_cons_lemma,
length_of_nil_lemma,
length_upto,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
select-map,
subtype_rel_list,
top_wf,
less_than_wf,
true_wf,
length_append,
iff_weakening_equal,
add_functionality_wrt_eq,
length_wf,
map_length_nat,
length-singleton,
lelt_wf,
length-map,
intformless_wf,
int_formula_prop_less_lemma,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
equal_wf,
select_upto,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__lt,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
select-cons-hd,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
select-append,
select-upto,
length_wf_nat,
select_wf,
int_seg_properties,
seq-append_wf,
add_nat_wf,
add-is-int-iff,
non_neg_length
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
hypothesisEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
cumulativity,
hypothesis,
natural_numberEquality,
setElimination,
rename,
because_Cache,
functionEquality,
independent_functionElimination,
dependent_functionElimination,
addEquality,
independent_isectElimination,
independent_pairFormation,
universeEquality,
hyp_replacement,
equalitySymmetry,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
imageElimination,
equalityTransitivity,
productElimination,
imageMemberEquality,
baseClosed,
lessCases,
sqequalAxiom,
equalityElimination,
promote_hyp,
instantiate,
applyLambdaEquality,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}[T:Type]. \mforall{}[R,A:(T List) {}\mrightarrow{} \mBbbP{}].
((\mforall{}s:T List. Dec(R[s]))
{}\mRightarrow{} (\mforall{}s:T List. (R[s] {}\mRightarrow{} A[s]))
{}\mRightarrow{} (\mforall{}s:T List. ((\mforall{}t:T. A[s @ [t]]) {}\mRightarrow{} A[s]))
{}\mRightarrow{} (\mforall{}s:T List. ((\mforall{}alpha:\mBbbN{} {}\mrightarrow{} T. (\mdownarrow{}\mexists{}n:\mBbbN{}. R[s @ map(alpha;upto(n))])) {}\mRightarrow{} A[s])))
Date html generated:
2017_04_17-AM-08_59_46
Last ObjectModification:
2017_02_27-PM-05_16_39
Theory : list_1
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