Nuprl Lemma : homogeneous-extension-implies
∀R:ℕ ⟶ ℕ ⟶ ℙ. ∀n:ℕ. ∀s:ℕn ⟶ ℕ. ∀m:ℕ. (homogeneous(R;n + 1;s.m@n) ⇒ homogeneous(R;n;s))
Proof
Definitions occuring in Statement :
homogeneous: homogeneous(R;n;s),
seq-add: s.x@n,
int_seg: {i..j-},
nat: ℕ,
prop: ℙ,
all: ∀x:A. B[x],
implies: 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,
homogeneous: homogeneous(R;n;s),
and: P ∧ Q,
strictly-increasing-seq: strictly-increasing-seq(n;s),
member: t ∈ T,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
seq-add: s.x@n,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
guard: {T},
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
sq_stable: SqStable(P),
less_than: a < b
Lemmas referenced :
decidable__lt,
false_wf,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-add,
nat_wf,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
le-add-cancel,
and_wf,
le_wf,
less_than_wf,
squash_wf,
true_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
less_than_transitivity1,
le_weakening,
less_than_transitivity2,
le_weakening2,
less_than_irreflexivity,
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,
int_seg_wf,
homogeneous_wf,
decidable__le,
not-le-2,
sq_stable__le,
zero-add,
add-zero,
seq-add_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
hypothesis,
dependent_functionElimination,
setElimination,
rename,
dependent_set_memberEquality,
hypothesisEquality,
independent_pairFormation,
introduction,
extract_by_obid,
addEquality,
natural_numberEquality,
unionElimination,
voidElimination,
independent_functionElimination,
independent_isectElimination,
isectElimination,
sqequalRule,
applyEquality,
because_Cache,
lambdaEquality,
isect_memberEquality,
voidEquality,
minusEquality,
hyp_replacement,
equalitySymmetry,
imageElimination,
equalityTransitivity,
intEquality,
equalityElimination,
int_eqReduceTrueSq,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
impliesFunctionality,
int_eqReduceFalseSq,
imageMemberEquality,
baseClosed,
functionExtensionality,
functionEquality,
universeEquality
Latex:
\mforall{}R:\mBbbN{} {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbP{}. \mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}. \mforall{}m:\mBbbN{}. (homogeneous(R;n + 1;s.m@n) {}\mRightarrow{} homogeneous(R;n;s))
Date html generated:
2017_04_14-AM-07_27_24
Last ObjectModification:
2017_02_27-PM-02_56_40
Theory : bar-induction
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