Nuprl Lemma : stump-bars
∀T:Type. ∀t:wfd-tree(T). ∀alpha:ℕ ⟶ T. ∃n:ℕ. (¬↑(stump(t) n alpha))
Proof
Definitions occuring in Statement :
stump: stump(t),
wfd-tree: wfd-tree(T),
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
stump: stump(t),
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
guard: {T},
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
sq_type: SQType(T),
bnot: ¬bb,
ge: i ≥ j ,
int_upper: {i...},
int_seg: {i..j-},
lelt: i ≤ j < k
Lemmas referenced :
wfd-tree-induction,
all_wf,
nat_wf,
exists_wf,
not_wf,
assert_wf,
stump_wf,
subtype_rel_dep_function,
int_seg_wf,
int_seg_subtype_nat,
false_wf,
wfd-tree_wf,
wfd_tree_rec_leaf_lemma,
wfd_tree_rec_node_lemma,
le_wf,
decidable__le,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
le_antisymmetry_iff,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
add-subtract-cancel,
int_upper_subtype_nat,
nat_properties,
nequal-le-implies,
subtract_wf,
minus-minus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
functionEquality,
hypothesis,
cumulativity,
because_Cache,
applyEquality,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
functionExtensionality,
universeEquality,
dependent_pairFormation,
dependent_set_memberEquality,
addEquality,
unionElimination,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
intEquality,
minusEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
hypothesis_subsumption
Latex:
\mforall{}T:Type. \mforall{}t:wfd-tree(T). \mforall{}alpha:\mBbbN{} {}\mrightarrow{} T. \mexists{}n:\mBbbN{}. (\mneg{}\muparrow{}(stump(t) n alpha))
Date html generated:
2017_04_14-AM-07_45_24
Last ObjectModification:
2017_02_27-PM-03_16_23
Theory : co-recursion
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