### Nuprl Lemma : simple_more_general_fan_theorem

`∀[T:ℕ ⟶ Type]`
`  (∀i:ℕ. Bounded(T[i]))`
`  `` (∀[X:n:ℕ ⟶ (i:ℕn ⟶ T[i]) ⟶ ℙ]`
`        (∀n:ℕ. ∀s:i:ℕn ⟶ T[i].  Dec(X[n;s])) `` (∃k:ℕ [(∀f:i:ℕ ⟶ T[i]. ∃n:ℕk. X[n;f])]) `
`        supposing ∀f:i:ℕ ⟶ T[i]. (↓∃n:ℕ. X[n;f])) `
`  supposing ∀i:ℕ. T[i]`

Proof

Definitions occuring in Statement :  bounded-type: `Bounded(T)` int_seg: `{i..j-}` nat: `ℕ` decidable: `Dec(P)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` so_apply: `x[s]` all: `∀x:A. B[x]` sq_exists: `∃x:A [B[x]]` exists: `∃x:A. B[x]` squash: `↓T` implies: `P `` Q` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  nequal: `a ≠ b ∈ T ` nat_plus: `ℕ+` ge: `i ≥ j ` seq-adjoin: `s++t` pi2: `snd(t)` project-seq: `project-seq(s)` pi1: `fst(t)` istype: `istype(T)` decidable: `Dec(P)` sq_exists: `∃x:A [B[x]]` subtract: `n - m` gt: `i > j` assert: `↑b` ifthenelse: `if b then t else f fi ` bnot: `¬bb` sq_type: `SQType(T)` or: `P ∨ Q` exists: `∃x:A. B[x]` bfalse: `ff` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` cand: `A c∧ B` true: `True` top: `Top` uiff: `uiff(P;Q)` btrue: `tt` it: `⋅` unit: `Unit` bool: `𝔹` seq-append: `seq-append(n;m;s1;s2)` not: `¬A` false: `False` less_than': `less_than'(a;b)` sq_stable: `SqStable(P)` guard: `{T}` so_apply: `x[s1;s2]` less_than: `a < b` le: `A ≤ B` and: `P ∧ Q` lelt: `i ≤ j < k` int_seg: `{i..j-}` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` nat: `ℕ` prop: `ℙ` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s]` squash: `↓T` all: `∀x:A. B[x]` member: `t ∈ T` implies: `P `` Q` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` bounded-type: `Bounded(T)`
Rules used in proof :  functionExtensionality dependent_pairEquality_alt functionExtensionality_alt hyp_replacement dependent_set_memberFormation_alt inrFormation_alt inlFormation_alt multiplyEquality setIsType isectIsType axiomEquality sqequalBase minusEquality baseApply cumulativity promote_hyp dependent_pairFormation_alt universeEquality instantiate voidElimination isectIsTypeImplies isect_memberEquality_alt axiomSqEquality lessCases equalityElimination unionElimination independent_pairFormation equalitySymmetry equalityTransitivity equalityIstype independent_functionElimination addEquality dependent_set_memberEquality_alt closedConclusion productIsType universeIsType functionIsType productElimination because_Cache independent_isectElimination intEquality setElimination natural_numberEquality functionEquality isectEquality applyEquality productEquality isectElimination extract_by_obid rename inhabitedIsType functionIsTypeImplies baseClosed imageMemberEquality hypothesis imageElimination hypothesisEquality thin dependent_functionElimination lambdaEquality_alt sqequalHypSubstitution introduction cut lambdaFormation_alt isect_memberFormation_alt computationStep sqequalTransitivity sqequalReflexivity sqequalRule sqequalSubstitution

Latex:
\mforall{}[T:\mBbbN{}  {}\mrightarrow{}  Type]
(\mforall{}i:\mBbbN{}.  Bounded(T[i]))
{}\mRightarrow{}  (\mforall{}[X:n:\mBbbN{}  {}\mrightarrow{}  (i:\mBbbN{}n  {}\mrightarrow{}  T[i])  {}\mrightarrow{}  \mBbbP{}]
(\mforall{}n:\mBbbN{}.  \mforall{}s:i:\mBbbN{}n  {}\mrightarrow{}  T[i].    Dec(X[n;s]))  {}\mRightarrow{}  (\mexists{}k:\mBbbN{}  [(\mforall{}f:i:\mBbbN{}  {}\mrightarrow{}  T[i].  \mexists{}n:\mBbbN{}k.  X[n;f])])
supposing  \mforall{}f:i:\mBbbN{}  {}\mrightarrow{}  T[i].  (\mdownarrow{}\mexists{}n:\mBbbN{}.  X[n;f]))
supposing  \mforall{}i:\mBbbN{}.  T[i]

Date html generated: 2019_10_15-AM-10_20_16
Last ObjectModification: 2019_10_07-PM-04_40_21

Theory : bar-induction

Home Index