Nuprl Lemma : no-weakly-safe-extensions
∀[R:ℕ ⟶ ℕ ⟶ ℙ]. ∀[n:ℕ]. ∀[s:ℕn ⟶ ℕ]. ((∀p:ℕ. (¬weakly-safe-seq(R;n + 1;s.p@n))) ⇒ (¬weakly-safe-seq(R;n;s)))
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,
all: ∀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,
prop: ℙ,
so_lambda: λ2x.t[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,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
so_apply: x[s],
top: Top,
exists: ∃x:A. B[x]
Lemmas referenced :
weakly-safe-extension,
exists_wf,
nat_wf,
weakly-safe-seq_wf,
decidable__le,
false_wf,
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,
le_wf,
seq-add_wf,
int_seg_wf,
all_wf,
not_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
independent_functionElimination,
sqequalRule,
lambdaEquality,
functionExtensionality,
applyEquality,
because_Cache,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
productElimination,
independent_isectElimination,
imageMemberEquality,
baseClosed,
imageElimination,
minusEquality,
isect_memberEquality,
voidEquality,
intEquality,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[R:\mBbbN{} {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbP{}]. \mforall{}[n:\mBbbN{}]. \mforall{}[s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}].
((\mforall{}p:\mBbbN{}. (\mneg{}weakly-safe-seq(R;n + 1;s.p@n))) {}\mRightarrow{} (\mneg{}weakly-safe-seq(R;n;s)))
Date html generated:
2018_05_21-PM-00_03_09
Last ObjectModification:
2018_05_16-AM-09_48_36
Theory : bar-induction
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