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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