### Nuprl Lemma : strict-comparison-trans

`∀[T:Type]. ∀cmp:comparison(T). Trans(T;x,y.0 < cmp x y)`

Proof

Definitions occuring in Statement :  comparison: `comparison(T)` trans: `Trans(T;x,y.E[x; y])` less_than: `a < b` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` apply: `f a` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` trans: `Trans(T;x,y.E[x; y])` implies: `P `` Q` comparison: `comparison(T)` and: `P ∧ Q` prop: `ℙ` uimplies: `b supposing a` decidable: `Dec(P)` or: `P ∨ Q` less_than: `a < b` squash: `↓T` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` le: `A ≤ B` sq_type: `SQType(T)` guard: `{T}` cand: `A c∧ B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` true: `True` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q`
Lemmas referenced :  int_term_value_minus_lemma itermMinus_wf iff_weakening_equal le_wf equal-wf-T-base equal_wf all_wf comparison-anti true_wf squash_wf int_subtype_base subtype_base_sq int_formula_prop_eq_lemma intformeq_wf decidable__lt decidable__equal_int int_formula_prop_wf int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma intformless_wf itermVar_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le member-less_than comparison_wf less_than_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation sqequalHypSubstitution setElimination thin rename productElimination hypothesis lemma_by_obid isectElimination natural_numberEquality applyEquality hypothesisEquality dependent_functionElimination cumulativity sqequalRule lambdaEquality independent_isectElimination because_Cache universeEquality independent_functionElimination functionExtensionality unionElimination imageElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll instantiate equalityTransitivity equalitySymmetry dependent_set_memberEquality productEquality minusEquality functionEquality baseClosed imageMemberEquality

Latex:
\mforall{}[T:Type].  \mforall{}cmp:comparison(T).  Trans(T;x,y.0  <  cmp  x  y)

Date html generated: 2016_05_14-PM-02_38_18
Last ObjectModification: 2016_04_08-AM-01_26_10

Theory : list_1

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