Nuprl Lemma : stump-monotone
∀T:Type. ∀t:wfd-tree(T). ∀n:ℕ. ∀s:ℕn ⟶ T. ((¬↑(stump(t) n s)) ⇒ (∀x:T. (¬↑(stump(t) (n + 1) s++x))))
Proof
Definitions occuring in Statement :
stump: stump(t),
wfd-tree: wfd-tree(T),
seq-adjoin: s++t,
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
nat: ℕ,
implies: P ⇒ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
so_apply: x[s],
stump: stump(t),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
int_seg: {i..j-},
lelt: i ≤ j < k,
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
int_upper: {i...},
seq-adjoin: s++t,
seq-append: seq-append(n;m;s1;s2),
less_than: a < b
Lemmas referenced :
wfd-tree-induction,
all_wf,
nat_wf,
int_seg_wf,
not_wf,
assert_wf,
stump_wf,
decidable__le,
false_wf,
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,
le_wf,
seq-adjoin_wf,
wfd-tree_wf,
wfd_tree_rec_leaf_lemma,
wfd_tree_rec_node_lemma,
eq_int_wf,
bool_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
decidable__lt,
not-lt-2,
not-equal-2,
lelt_wf,
subtract_wf,
minus-minus,
add-member-int_seg2,
le-add-cancel2,
int_upper_subtype_nat,
nat_properties,
nequal-le-implies,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
assert_of_eq_int,
le_antisymmetry_iff,
minus-zero,
less-iff-le,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
bnot_wf,
equal-wf-T-base,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
squash_wf,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
hypothesis,
functionEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
cumulativity,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
addEquality,
unionElimination,
independent_pairFormation,
voidElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
imageMemberEquality,
baseClosed,
imageElimination,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
instantiate,
hypothesis_subsumption,
universeEquality,
lessCases,
isect_memberFormation,
sqequalAxiom,
impliesFunctionality,
hyp_replacement
Latex:
\mforall{}T:Type. \mforall{}t:wfd-tree(T). \mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} T.
((\mneg{}\muparrow{}(stump(t) n s)) {}\mRightarrow{} (\mforall{}x:T. (\mneg{}\muparrow{}(stump(t) (n + 1) s++x))))
Date html generated:
2017_04_14-AM-07_45_21
Last ObjectModification:
2017_02_27-PM-03_17_23
Theory : co-recursion
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