Nuprl Lemma : weakly-safe-extension
∀[R:ℕ ⟶ ℕ ⟶ ℙ]. ∀[n:ℕ]. ∀[s:ℕn ⟶ ℕ].  (weakly-safe-seq(R;n;s) 
⇒ (¬¬(∃p:ℕ. weakly-safe-seq(R;n + 1;s.p@n))))
Proof
Definitions occuring in Statement : 
weakly-safe-seq: weakly-safe-seq(R;n;s)
, 
seq-add: s.x@n
, 
int_seg: {i..j-}
, 
nat: ℕ
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
exists: ∃x:A. B[x]
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
function: x:A ⟶ B[x]
, 
add: n + m
, 
natural_number: $n
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
implies: P 
⇒ Q
, 
not: ¬A
, 
false: False
, 
exists: ∃x:A. B[x]
, 
nat: ℕ
, 
all: ∀x:A. B[x]
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
rev_implies: P 
⇐ Q
, 
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
, 
prop: ℙ
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
weakly-safe-seq: weakly-safe-seq(R;n;s)
, 
weakly-infinite: w∃∞p.S[p]
, 
cand: A c∧ B
, 
guard: {T}
, 
istype: istype(T)
, 
sq_type: SQType(T)
Lemmas referenced : 
weakly-safe-seq_wf, 
decidable__le, 
istype-false, 
not-le-2, 
sq_stable__le, 
condition-implies-le, 
minus-add, 
istype-void, 
istype-int, 
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, 
nat_wf, 
int_seg_wf, 
istype-nat, 
false_wf, 
or_wf, 
all_wf, 
homogeneous_wf, 
less_than_wf, 
not_wf, 
subtype_rel_dep_function, 
subtype_rel_sets, 
and_wf, 
le_wf, 
istype-less_than, 
exists_wf, 
minimal-double-negation-hyp-elim, 
minimal-not-not-excluded-middle, 
double-negation-hyp-elim, 
le_reflexive, 
less_than_transitivity2, 
int_subtype_base, 
subtype_base_sq, 
weakly-infinite-cases, 
zero-mul, 
add-mul-special, 
not-lt-2, 
decidable__lt, 
less-iff-le, 
le_weakening2, 
non-homogeneous-add, 
iff_wf, 
subtype_rel_self, 
le-add-cancel-alt, 
minus-minus, 
subtract_wf, 
weakly-infinite_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
Error :isect_memberFormation_alt, 
introduction, 
cut, 
Error :lambdaFormation_alt, 
thin, 
hypothesis, 
sqequalHypSubstitution, 
independent_functionElimination, 
voidElimination, 
sqequalRule, 
Error :functionIsType, 
Error :productIsType, 
Error :inhabitedIsType, 
hypothesisEquality, 
Error :universeIsType, 
extract_by_obid, 
isectElimination, 
Error :dependent_set_memberEquality_alt, 
addEquality, 
setElimination, 
rename, 
natural_numberEquality, 
dependent_functionElimination, 
unionElimination, 
independent_pairFormation, 
productElimination, 
independent_isectElimination, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
applyEquality, 
Error :lambdaEquality_alt, 
Error :isect_memberEquality_alt, 
because_Cache, 
minusEquality, 
Error :functionIsTypeImplies, 
Error :isectIsTypeImplies, 
universeEquality, 
closedConclusion, 
functionEquality, 
intEquality, 
Error :setIsType, 
Error :unionIsType, 
Error :dependent_pairFormation_alt, 
equalityTransitivity, 
equalitySymmetry, 
productEquality, 
cumulativity, 
instantiate, 
Error :inlFormation_alt, 
multiplyEquality, 
promote_hyp, 
Error :inrFormation_alt
Latex:
\mforall{}[R:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[n:\mBbbN{}].  \mforall{}[s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}].
    (weakly-safe-seq(R;n;s)  {}\mRightarrow{}  (\mneg{}\mneg{}(\mexists{}p:\mBbbN{}.  weakly-safe-seq(R;n  +  1;s.p@n))))
Date html generated:
2019_06_20-AM-11_29_19
Last ObjectModification:
2018_10_18-PM-03_54_59
Theory : bar-induction
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