Nuprl Lemma : weakly-safe-extension

[R:ℕ ⟶ ℕ ⟶ ℙ]. ∀[n:ℕ]. ∀[s:ℕn ⟶ ℕ].  (weakly-safe-seq(R;n;s)  (¬¬(∃p:ℕweakly-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: uall: [x:A]. B[x] prop: exists: x:A. B[x] not: ¬A implies:  Q function: x:A ⟶ B[x] add: m natural_number: $n
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T implies:  Q not: ¬A false: False exists: x:A. B[x] nat: all: x:A. B[x] decidable: Dec(P) or: P ∨ Q iff: ⇐⇒ Q and: P ∧ Q rev_implies:  Q uiff: uiff(P;Q) uimplies: supposing a sq_stable: SqStable(P) squash: T subtract: m subtype_rel: A ⊆B top: Top le: A ≤ B less_than': less_than'(a;b) true: True prop: 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: 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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