### Nuprl Lemma : seq-append-bar

`∀k:ℕ. ∀s:ℕk ⟶ ℕ. ∀x:ℕ. ∀Q:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ ℙ.`
`  ((∀f:ℕ ⟶ ℕ. ∃n:ℕ. ∀m:{n...}. Q[m + k;seq-append(k;m;s;f)])`
`  `` (∀f:ℕ ⟶ ℕ. ∃n:ℕ. ∀m:{n...}. Q[m + k;seq-append(k + 1;m;s.x@k;f)]))`

Proof

Definitions occuring in Statement :  seq-add: `s.x@n` seq-append: `seq-append(n;m;s1;s2)` int_upper: `{i...}` int_seg: `{i..j-}` nat: `ℕ` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` nat: `ℕ` false: `False` not: `¬A` uall: `∀[x:A]. B[x]` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s1;s2]` int_upper: `{i...}` guard: `{T}` subtype_rel: `A ⊆r B` sq_stable: `SqStable(P)` squash: `↓T` le: `A ≤ B` less_than': `less_than'(a;b)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m` true: `True` so_apply: `x[s]` seq-append: `seq-append(n;m;s1;s2)` seq-add: `s.x@n` int_seg: `{i..j-}` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` less_than: `a < b` lelt: `i ≤ j < k` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` nequal: `a ≠ b ∈ T `
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut hypothesis sqequalHypSubstitution dependent_functionElimination thin lambdaEquality int_eqEquality setElimination rename because_Cache natural_numberEquality hypothesisEquality applyEquality functionExtensionality dependent_set_memberEquality introduction extract_by_obid isectElimination unionElimination independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation intEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation productElimination addEquality imageMemberEquality baseClosed imageElimination minusEquality multiplyEquality functionEquality cumulativity universeEquality hyp_replacement equalitySymmetry equalityTransitivity equalityElimination lessCases isect_memberFormation axiomSqEquality int_eqReduceTrueSq promote_hyp instantiate int_eqReduceFalseSq

Latex:
\mforall{}k:\mBbbN{}.  \mforall{}s:\mBbbN{}k  {}\mrightarrow{}  \mBbbN{}.  \mforall{}x:\mBbbN{}.  \mforall{}Q:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbP{}.
((\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  \mexists{}n:\mBbbN{}.  \mforall{}m:\{n...\}.  Q[m  +  k;seq-append(k;m;s;f)])
{}\mRightarrow{}  (\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  \mexists{}n:\mBbbN{}.  \mforall{}m:\{n...\}.  Q[m  +  k;seq-append(k  +  1;m;s.x@k;f)]))

Date html generated: 2019_06_20-PM-02_54_52
Last ObjectModification: 2018_08_20-PM-09_36_09

Theory : continuity

Home Index