### Nuprl Lemma : homogeneous-extension-implies

`∀R:ℕ ⟶ ℕ ⟶ ℙ. ∀n:ℕ. ∀s:ℕn ⟶ ℕ. ∀m:ℕ.  (homogeneous(R;n + 1;s.m@n) `` homogeneous(R;n;s))`

Proof

Definitions occuring in Statement :  homogeneous: `homogeneous(R;n;s)` seq-add: `s.x@n` int_seg: `{i..j-}` nat: `ℕ` prop: `ℙ` all: `∀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` homogeneous: `homogeneous(R;n;s)` and: `P ∧ Q` strictly-increasing-seq: `strictly-increasing-seq(n;s)` member: `t ∈ T` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` nat: `ℕ` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` not: `¬A` rev_implies: `P `` Q` false: `False` prop: `ℙ` uiff: `uiff(P;Q)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` subtract: `n - m` subtype_rel: `A ⊆r B` top: `Top` less_than': `less_than'(a;b)` true: `True` squash: `↓T` seq-add: `s.x@n` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` guard: `{T}` bfalse: `ff` exists: `∃x:A. B[x]` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` sq_stable: `SqStable(P)` less_than: `a < b`
Lemmas referenced :  decidable__lt false_wf not-lt-2 less-iff-le condition-implies-le minus-add nat_wf minus-one-mul add-swap minus-one-mul-top add-commutes add_functionality_wrt_le add-associates le-add-cancel and_wf le_wf less_than_wf squash_wf true_wf eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int less_than_transitivity1 le_weakening less_than_transitivity2 le_weakening2 less_than_irreflexivity eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base iff_transitivity assert_wf bnot_wf not_wf iff_weakening_uiff assert_of_bnot int_seg_wf homogeneous_wf decidable__le not-le-2 sq_stable__le zero-add add-zero seq-add_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation sqequalHypSubstitution productElimination thin cut hypothesis dependent_functionElimination setElimination rename dependent_set_memberEquality hypothesisEquality independent_pairFormation introduction extract_by_obid addEquality natural_numberEquality unionElimination voidElimination independent_functionElimination independent_isectElimination isectElimination sqequalRule applyEquality because_Cache lambdaEquality isect_memberEquality voidEquality minusEquality hyp_replacement equalitySymmetry imageElimination equalityTransitivity intEquality equalityElimination int_eqReduceTrueSq dependent_pairFormation promote_hyp instantiate cumulativity impliesFunctionality int_eqReduceFalseSq imageMemberEquality baseClosed functionExtensionality functionEquality universeEquality

Latex:
\mforall{}R:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}  {}\mrightarrow{}  \mBbbP{}.  \mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}.  \mforall{}m:\mBbbN{}.    (homogeneous(R;n  +  1;s.m@n)  {}\mRightarrow{}  homogeneous(R;n;s))

Date html generated: 2017_04_14-AM-07_27_24
Last ObjectModification: 2017_02_27-PM-02_56_40

Theory : bar-induction

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