Nuprl Lemma : non-homogeneous-add
∀[R:ℕ ⟶ ℕ ⟶ ℙ]
∀n:ℕ. ∀s:ℕn ⟶ ℕ. ∀p,q:ℕ.
(p < q
⇒ homogeneous(R;n + 1;s.p@n)
⇒ homogeneous(R;n + 1;s.q@n)
⇒ (¬homogeneous(R;n + 2;s.p@n.q@n + 1))
⇒ {0 < n ∧ (¬(R (s (n - 1)) p ⇐⇒ R (s (n - 1)) q))})
Proof
Definitions occuring in Statement :
homogeneous: homogeneous(R;n;s),
seq-add: s.x@n,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
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,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
iff: P ⇐⇒ Q,
and: P ∧ Q,
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,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
top: Top,
strictly-increasing-seq: strictly-increasing-seq(n;s),
seq-add: s.x@n,
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
guard: {T},
lelt: i ≤ j < k,
bfalse: ff,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
homogeneous: homogeneous(R;n;s),
prop: ℙ,
cand: A c∧ B,
ge: i ≥ j
Lemmas referenced :
decidable__lt,
le2-homogeneous,
decidable__le,
istype-false,
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,
istype-le,
seq-add_wf,
not-lt-2,
istype-void,
le-add-cancel2,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
le_antisymmetry_iff,
less-iff-le,
eqff_to_assert,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
and_wf,
less_than_wf,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
equal-wf-base,
iff_weakening_uiff,
assert_of_bnot,
istype-assert,
equal_wf,
not-equal-2,
int_seg_wf,
strictly-increasing-seq-add,
istype-less_than,
subtract_wf,
minus-minus,
add-mul-special,
zero-mul,
le-add-cancel-alt,
homogeneous_wf,
nat_wf,
istype-nat,
decidable__and2,
less_than_transitivity2,
le_weakening2,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
iff_wf,
squash_wf,
true_wf,
decidable__int_equal,
istype-sqequal,
sq_stable__and,
sq_stable__less_than,
member-less_than,
two-mul,
mul-distributes-right,
one-mul,
nat_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
unionElimination,
independent_functionElimination,
isectElimination,
hypothesisEquality,
Error :dependent_set_memberEquality_alt,
addEquality,
independent_pairFormation,
voidElimination,
productElimination,
independent_isectElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
applyEquality,
minusEquality,
Error :lambdaEquality_alt,
Error :isect_memberEquality_alt,
Error :inhabitedIsType,
equalityElimination,
int_eqReduceTrueSq,
Error :dependent_pairFormation_alt,
equalityTransitivity,
equalitySymmetry,
Error :equalityIsType4,
baseApply,
closedConclusion,
intEquality,
promote_hyp,
instantiate,
cumulativity,
Error :equalityIstype,
sqequalBase,
Error :functionIsType,
int_eqReduceFalseSq,
Error :equalityIsType1,
Error :universeIsType,
functionExtensionality,
Error :productIsType,
universeEquality,
hyp_replacement,
multiplyEquality,
Error :functionIsTypeImplies
Latex:
\mforall{}[R:\mBbbN{} {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbP{}]
\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}. \mforall{}p,q:\mBbbN{}.
(p < q
{}\mRightarrow{} homogeneous(R;n + 1;s.p@n)
{}\mRightarrow{} homogeneous(R;n + 1;s.q@n)
{}\mRightarrow{} (\mneg{}homogeneous(R;n + 2;s.p@n.q@n + 1))
{}\mRightarrow{} \{0 < n \mwedge{} (\mneg{}(R (s (n - 1)) p \mLeftarrow{}{}\mRightarrow{} R (s (n - 1)) q))\})
Date html generated:
2019_06_20-AM-11_29_09
Last ObjectModification:
2018_11_22-PM-10_39_06
Theory : bar-induction
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