### Nuprl Lemma : less-iff-le

`∀x,y:ℤ.  uiff(x < y;(1 + x) ≤ y)`

Proof

Definitions occuring in Statement :  less_than: `a < b` uiff: `uiff(P;Q)` le: `A ≤ B` all: `∀x:A. B[x]` add: `n + m` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  prop: `ℙ` uall: `∀[x:A]. B[x]` false: `False` implies: `P `` Q` not: `¬A` le: `A ≤ B` member: `t ∈ T` uimplies: `b supposing a` and: `P ∧ Q` uiff: `uiff(P;Q)` all: `∀x:A. B[x]` rev_uimplies: `rev_uimplies(P;Q)` subtype_rel: `A ⊆r B` squash: `↓T` cand: `A c∧ B` less_than: `a < b` or: `P ∨ Q` true: `True` less_than': `less_than'(a;b)` guard: `{T}` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` top: `Top` bfalse: `ff` exists: `∃x:A. B[x]` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  le_wf less_than_wf less_than'_wf le-iff-less-or-equal int_subtype_base equal-wf-base lt_int_wf eqtt_to_assert assert_of_lt_int istype-top istype-void eqff_to_assert bool_subtype_base bool_cases_sqequal subtype_base_sq bool_wf iff_transitivity assert_wf bnot_wf not_wf iff_weakening_uiff assert_of_bnot istype-assert zero-add le_reflexive add_functionality_wrt_lt less_than_transitivity1
Rules used in proof :  intEquality equalitySymmetry equalityTransitivity axiomEquality hypothesis natural_numberEquality addEquality isectElimination extract_by_obid voidElimination hypothesisEquality dependent_functionElimination lambdaEquality independent_pairEquality thin productElimination sqequalHypSubstitution sqequalRule cut introduction isect_memberFormation independent_pairFormation lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution independent_isectElimination lessCases applyEquality closedConclusion baseApply baseClosed imageMemberEquality inlFormation inrFormation lessDiscrete Error :inhabitedIsType,  Error :lambdaFormation_alt,  unionElimination equalityElimination because_Cache Error :isect_memberFormation_alt,  axiomSqEquality Error :isect_memberEquality_alt,  Error :isectIsTypeImplies,  imageElimination independent_functionElimination Error :dependent_pairFormation_alt,  Error :equalityIsType4,  promote_hyp instantiate cumulativity Error :functionIsType,  Error :equalityIsType1

Latex:
\mforall{}x,y:\mBbbZ{}.    uiff(x  <  y;(1  +  x)  \mleq{}  y)

Date html generated: 2019_06_20-AM-11_23_03
Last ObjectModification: 2018_10_16-PM-02_47_37

Theory : arithmetic

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