`∀[R:ℕ ⟶ ℕ ⟶ ℙ]`
`  ∀n:ℕ. ∀s:ℕn ⟶ ℕ. ∀p,q:ℕ.`
`    (p < q`
`    `` homogeneous(R;n + 1;s.p@n)`
`    `` homogeneous(R;n + 1;s.q@n)`
`    `` (¬homogeneous(R;n + 2;s.p@n.q@n + 1))`
`    `` {0 < n ∧ (¬(R (s (n - 1)) p `⇐⇒` R (s (n - 1)) q))})`

Proof

Definitions occuring in Statement :  homogeneous: `homogeneous(R;n;s)` seq-add: `s.x@n` int_seg: `{i..j-}` nat: `ℕ` less_than: `a < b` uall: `∀[x:A]. B[x]` prop: `ℙ` guard: `{T}` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` not: `¬A` implies: `P `` Q` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` subtract: `n - m` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` nat: `ℕ` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` false: `False` 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` top: `Top` strictly-increasing-seq: `strictly-increasing-seq(n;s)` seq-add: `s.x@n` int_seg: `{i..j-}` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` guard: `{T}` lelt: `i ≤ j < k` bfalse: `ff` exists: `∃x:A. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` homogeneous: `homogeneous(R;n;s)` prop: `ℙ` cand: `A c∧ B` ge: `i ≥ j `
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  Error :lambdaFormation_alt,  cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin natural_numberEquality setElimination rename because_Cache hypothesis unionElimination independent_functionElimination isectElimination hypothesisEquality Error :dependent_set_memberEquality_alt,  addEquality independent_pairFormation voidElimination productElimination independent_isectElimination sqequalRule imageMemberEquality baseClosed imageElimination applyEquality minusEquality Error :lambdaEquality_alt,  Error :isect_memberEquality_alt,  Error :inhabitedIsType,  equalityElimination int_eqReduceTrueSq Error :dependent_pairFormation_alt,  equalityTransitivity equalitySymmetry Error :equalityIsType4,  baseApply closedConclusion intEquality promote_hyp instantiate cumulativity Error :equalityIstype,  sqequalBase Error :functionIsType,  int_eqReduceFalseSq Error :equalityIsType1,  Error :universeIsType,  functionExtensionality Error :productIsType,  universeEquality hyp_replacement multiplyEquality Error :functionIsTypeImplies

Latex:
\mforall{}[R:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}  {}\mrightarrow{}  \mBbbP{}]
\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}.  \mforall{}p,q:\mBbbN{}.
(p  <  q
{}\mRightarrow{}  homogeneous(R;n  +  1;s.p@n)
{}\mRightarrow{}  homogeneous(R;n  +  1;s.q@n)
{}\mRightarrow{}  (\mneg{}homogeneous(R;n  +  2;s.p@n.q@n  +  1))
{}\mRightarrow{}  \{0  <  n  \mwedge{}  (\mneg{}(R  (s  (n  -  1))  p  \mLeftarrow{}{}\mRightarrow{}  R  (s  (n  -  1))  q))\})

Date html generated: 2019_06_20-AM-11_29_09
Last ObjectModification: 2018_11_22-PM-10_39_06

Theory : bar-induction

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