Nuprl Lemma : strictly-increasing-seq-add2-implies
∀n:ℕ. ∀s:ℕn ⟶ ℕ. ∀x,y:ℕ.
(strictly-increasing-seq(n + 2;s.x@n.y@n + 1)
⇒ {x < y ∧ strictly-increasing-seq(n + 1;s.x@n) ∧ strictly-increasing-seq(n + 1;s.y@n)})
Proof
Definitions occuring in Statement :
strictly-increasing-seq: strictly-increasing-seq(n;s),
seq-add: s.x@n,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
iff: 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_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
strictly-increasing-seq: strictly-increasing-seq(n;s),
seq-add: s.x@n,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
less_than: a < b,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
strictly-increasing-seq_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-add_wf,
nat_wf,
int_seg_wf,
subtype_rel_dep_function,
subtype_rel_sets,
and_wf,
less_than_wf,
decidable__lt,
not-lt-2,
add-mul-special,
zero-mul,
le-add-cancel2,
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,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
less-iff-le,
less_than_transitivity1,
le_weakening,
less_than_transitivity2,
le_weakening2,
less_than_irreflexivity,
not-equal-2,
or_wf,
decidable__int_equal,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
because_Cache,
hypothesis,
natural_numberEquality,
dependent_functionElimination,
hypothesisEquality,
unionElimination,
voidElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
functionExtensionality,
setEquality,
functionEquality,
multiplyEquality,
int_eqReduceTrueSq,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
impliesFunctionality,
int_eqReduceFalseSq,
inlFormation,
inrFormation,
addLevel,
orFunctionality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}. \mforall{}x,y:\mBbbN{}.
(strictly-increasing-seq(n + 2;s.x@n.y@n + 1)
{}\mRightarrow{} \{x < y \mwedge{} strictly-increasing-seq(n + 1;s.x@n) \mwedge{} strictly-increasing-seq(n + 1;s.y@n)\})
Date html generated:
2017_04_14-AM-07_26_27
Last ObjectModification:
2017_02_27-PM-02_56_12
Theory : bar-induction
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