Nuprl Lemma : le2-homogeneous
∀[R:ℕ ⟶ ℕ ⟶ ℙ]. ∀[n:ℕ]. ∀[s:ℕn ⟶ ℕ]. ((n ≤ 2) ⇒ strictly-increasing-seq(n;s) ⇒ homogeneous(R;n;s))
Proof
Definitions occuring in Statement :
homogeneous: homogeneous(R;n;s),
strictly-increasing-seq: strictly-increasing-seq(n;s),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
le: A ≤ B,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
homogeneous: homogeneous(R;n;s),
and: P ∧ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
int_seg: {i..j-},
nat: ℕ,
le: A ≤ B,
member: t ∈ T,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
lelt: i ≤ j < k,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
decidable__int_equal,
false_wf,
not-equal-2,
less-iff-le,
add_functionality_wrt_le,
add-associates,
add-zero,
add-swap,
add-commutes,
le-add-cancel2,
sq_stable__le,
and_wf,
le_wf,
less_than_wf,
equal_wf,
nat_wf,
int_seg_wf,
strictly-increasing-seq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
hypothesis,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
unionElimination,
voidElimination,
independent_functionElimination,
independent_isectElimination,
isectElimination,
addEquality,
natural_numberEquality,
sqequalRule,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
intEquality,
because_Cache,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry,
setEquality,
hyp_replacement,
Error :applyLambdaEquality,
functionExtensionality,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[R:\mBbbN{} {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbP{}]. \mforall{}[n:\mBbbN{}]. \mforall{}[s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}].
((n \mleq{} 2) {}\mRightarrow{} strictly-increasing-seq(n;s) {}\mRightarrow{} homogeneous(R;n;s))
Date html generated:
2016_10_21-AM-09_37_56
Last ObjectModification:
2016_07_12-AM-05_01_09
Theory : bar-induction
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