### Nuprl Lemma : l-ordered-from-upto-lt-nat

`∀[n,m:ℕ].  l-ordered(ℕ;x,y.x < y;[n, m))`

Proof

Definitions occuring in Statement :  l-ordered: `l-ordered(T;x,y.R[x; y];L)` from-upto: `[n, m)` nat: `ℕ` less_than: `a < b` uall: `∀[x:A]. B[x]`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` nat: `ℕ` decidable: `Dec(P)` or: `P ∨ Q` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` true: `True` exists: `∃x:A. B[x]` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` prop: `ℙ` l-ordered: `l-ordered(T;x,y.R[x; y];L)` subtype_rel: `A ⊆r B` le: `A ≤ B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` cand: `A c∧ B` squash: `↓T` label: `...\$L... t` guard: `{T}` sq_type: `SQType(T)`
Lemmas referenced :  decidable__le from-upto-is-nil l-ordered-nil-true less_than_wf nat_wf subtract_wf nat_properties satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_wf le_wf decidable__equal_int intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma equal_wf intformless_wf int_formula_prop_less_lemma ge_wf member-less_than l_before_wf from-upto_wf from-upto-nil l-ordered-nil subtype_rel_list subtype_rel_sets l-ordered_wf squash_wf true_wf list_wf from-upto-decomp-last decidable__lt strong-subtype-equal-lists strong-subtype-set3 strong-subtype-self append_wf cons_wf nil_wf iff_weakening_equal l-ordered-append subtype_base_sq int_subtype_base l-ordered-single member_singleton from-upto-member-nat l-ordered-cons l_member_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution dependent_functionElimination thin setElimination rename hypothesisEquality hypothesis unionElimination isectElimination because_Cache productElimination independent_pairFormation independent_isectElimination sqequalRule lambdaEquality independent_functionElimination natural_numberEquality dependent_pairFormation dependent_set_memberEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll addEquality intWeakElimination lambdaFormation applyEquality setEquality productEquality imageElimination equalityTransitivity equalitySymmetry functionEquality cumulativity universeEquality imageMemberEquality baseClosed instantiate applyLambdaEquality hyp_replacement

Latex:
\mforall{}[n,m:\mBbbN{}].    l-ordered(\mBbbN{};x,y.x  <  y;[n,  m))

Date html generated: 2018_05_21-PM-07_37_41
Last ObjectModification: 2017_07_26-PM-05_11_58

Theory : general

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