### 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]`
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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