Nuprl Lemma : stump'_property
∀T:Type. ∀t:wfd-tree(T). ∀n:ℕ. ∀s:ℕn ⟶ T.
(↑(stump'(t) n s) ⇐⇒ (¬↑(stump(t) n s)) ∧ (0 < n ⇒ (↑(stump(t) (n - 1) s))))
Proof
Definitions occuring in Statement :
stump': stump'(t),
stump: stump(t),
wfd-tree: wfd-tree(T),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
less_than: a < b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
stump': stump'(t),
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
ge: i ≥ j ,
int_upper: {i...},
subtype_rel: A ⊆r B,
subtract: n - m,
iff: P ⇐⇒ Q,
less_than: a < b,
squash: ↓T,
rev_implies: P ⇐ Q,
true: True,
decidable: Dec(P),
sq_stable: SqStable(P),
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
band: p ∧b q
Lemmas referenced :
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_upper_subtype_nat,
false_wf,
le_wf,
nat_properties,
nequal-le-implies,
zero-add,
int_seg_wf,
nat_wf,
wfd-tree_wf,
int_subtype_base,
stump-nil,
subtype_rel_self,
subtype_rel_wf,
empty-wfd-tree_wf,
less_than_wf,
true_wf,
not_wf,
assert_wf,
stump_wf,
less_than_irreflexivity,
subtract_wf,
decidable__le,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-one-mul,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
le-add-cancel,
subtype_rel_dep_function,
int_seg_subtype,
int_upper_wf,
add-mul-special,
zero-mul,
add-zero,
le-add-cancel2,
decidable__lt,
not-lt-2,
bnot_wf,
iff_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
natural_numberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
dependent_pairFormation,
hypothesisEquality,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
hypothesis_subsumption,
dependent_set_memberEquality,
independent_pairFormation,
functionEquality,
universeEquality,
intEquality,
applyEquality,
hyp_replacement,
applyLambdaEquality,
imageElimination,
productEquality,
functionExtensionality,
minusEquality,
imageMemberEquality,
baseClosed,
addEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
multiplyEquality,
addLevel,
impliesFunctionality
Latex:
\mforall{}T:Type. \mforall{}t:wfd-tree(T). \mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} T.
(\muparrow{}(stump'(t) n s) \mLeftarrow{}{}\mRightarrow{} (\mneg{}\muparrow{}(stump(t) n s)) \mwedge{} (0 < n {}\mRightarrow{} (\muparrow{}(stump(t) (n - 1) s))))
Date html generated:
2017_04_14-AM-07_45_32
Last ObjectModification:
2017_02_27-PM-03_16_48
Theory : co-recursion
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