### Nuprl Lemma : decidable__cmp-le

`∀[T:Type]. ∀cmp:comparison(T). ∀x,y:cmp-type(T;cmp).  Dec(cmp-le(cmp;x;y))`

Proof

Definitions occuring in Statement :  cmp-le: `cmp-le(cmp;x;y)` cmp-type: `cmp-type(T;cmp)` comparison: `comparison(T)` decidable: `Dec(P)` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` member: `t ∈ T` cmp-type: `cmp-type(T;cmp)` prop: `ℙ` quotient: `x,y:A//B[x; y]` and: `P ∧ Q` subtype_rel: `A ⊆r B` guard: `{T}` comparison: `comparison(T)` uimplies: `b supposing a` sq_type: `SQType(T)` implies: `P `` Q` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` cmp-le: `cmp-le(cmp;x;y)` decidable: `Dec(P)` not: `¬A` or: `P ∨ Q` exposed-it: `exposed-it` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` le: `A ≤ B` less_than': `less_than'(a;b)` less_than: `a < b` top: `Top` false: `False` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` bfalse: `ff` bnot: `¬bb` assert: `↑b`
Lemmas referenced :  decidable_wf cmp-le_wf subtype_rel_self cmp-type_wf equal-wf-base equal-wf-T-base comparison_wf subtype_base_sq int_subtype_base equal_wf squash_wf true_wf iff_weakening_equal minus_functionality_wrt_eq le_int_wf bool_wf eqtt_to_assert assert_of_le_int subtype_rel-equal top_wf less_than_wf false_wf lt_int_wf assert_of_lt_int full-omega-unsat intformand_wf intformless_wf itermVar_wf itermConstant_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_wf eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot less_than'_wf le_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation introduction sqequalHypSubstitution pointwiseFunctionalityForEquality cut extract_by_obid isectElimination thin hypothesisEquality sqequalRule hypothesis pertypeElimination productElimination cumulativity equalityTransitivity equalitySymmetry applyEquality dependent_functionElimination productEquality because_Cache intEquality setElimination rename baseClosed universeEquality instantiate independent_isectElimination independent_functionElimination lambdaEquality imageElimination natural_numberEquality imageMemberEquality minusEquality unionElimination equalityElimination inlEquality independent_pairEquality lessCases independent_pairFormation axiomSqEquality isect_memberEquality voidElimination voidEquality approximateComputation dependent_pairFormation int_eqEquality promote_hyp axiomEquality functionEquality inrEquality

Latex:
\mforall{}[T:Type].  \mforall{}cmp:comparison(T).  \mforall{}x,y:cmp-type(T;cmp).    Dec(cmp-le(cmp;x;y))

Date html generated: 2019_06_20-PM-01_41_56
Last ObjectModification: 2018_08_20-PM-09_32_17

Theory : list_1

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