### Nuprl Lemma : absval_elim

`∀[P:ℤ ⟶ ℙ]. (∀x:ℤ. P[|x|] `⇐⇒` ∀x:ℕ. P[x])`

Proof

Definitions occuring in Statement :  absval: `|i|` nat: `ℕ` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` function: `x:A ⟶ B[x]` int: `ℤ`
Definitions unfolded in proof :  so_apply: `x[s]` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` all: `∀x:A. B[x]` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` nat: `ℕ` rev_implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` uimplies: `b supposing a` less_than: `a < b` less_than': `less_than'(a;b)` top: `Top` true: `True` squash: `↓T` not: `¬A` false: `False` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` sq_stable: `SqStable(P)` le: `A ≤ B`
Lemmas referenced :  nat_wf all_wf absval_wf absval_unfold lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf less_than_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base uiff_transitivity assert_wf bnot_wf not_wf assert_of_bnot not_functionality_wrt_uiff sq_stable_from_decidable le_wf decidable__le not-lt-2 add_functionality_wrt_le add-zero le-add-cancel-alt
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep Error :isect_memberFormation_alt,  independent_pairFormation lambdaFormation cut introduction extract_by_obid hypothesis sqequalHypSubstitution isectElimination thin intEquality lambdaEquality applyEquality hypothesisEquality setElimination rename Error :functionIsType,  Error :universeIsType,  universeEquality dependent_functionElimination minusEquality natural_numberEquality because_Cache unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination independent_isectElimination lessCases axiomSqEquality Error :inhabitedIsType,  isect_memberEquality voidElimination voidEquality imageMemberEquality baseClosed imageElimination independent_functionElimination dependent_pairFormation promote_hyp instantiate cumulativity

Latex:
\mforall{}[P:\mBbbZ{}  {}\mrightarrow{}  \mBbbP{}].  (\mforall{}x:\mBbbZ{}.  P[|x|]  \mLeftarrow{}{}\mRightarrow{}  \mforall{}x:\mBbbN{}.  P[x])

Date html generated: 2019_06_20-AM-11_24_35
Last ObjectModification: 2018_09_26-AM-10_58_23

Theory : arithmetic

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