### Nuprl Lemma : comparison-seq_wf

`∀[T:Type]. ∀[c1:comparison(T)]. ∀[c2:⋂a:T. comparison({b:T| (c1 a b) = 0 ∈ ℤ} )].  (comparison-seq(c1; c2) ∈ comparison(\000CT))`

Proof

Definitions occuring in Statement :  comparison-seq: `comparison-seq(c1; c2)` comparison: `comparison(T)` uall: `∀[x:A]. B[x]` member: `t ∈ T` set: `{x:A| B[x]} ` apply: `f a` isect: `⋂x:A. B[x]` natural_number: `\$n` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` comparison: `comparison(T)` comparison-seq: `comparison-seq(c1; c2)` and: `P ∧ Q` has-value: `(a)↓` uimplies: `b supposing a` all: `∀x:A. B[x]` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` subtype_rel: `A ⊆r B` 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` false: `False` not: `¬A` nequal: `a ≠ b ∈ T ` prop: `ℙ` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` cand: `A c∧ B` sq_stable: `SqStable(P)` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` le: `A ≤ B`
Lemmas referenced :  value-type-has-value eqtt_to_assert assert_of_eq_int int_subtype_base eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int istype-int istype-le comparison_wf equal-wf-base istype-universe decidable__equal_int full-omega-unsat intformand_wf intformnot_wf intformeq_wf itermVar_wf itermConstant_wf itermMinus_wf int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_term_value_minus_lemma int_formula_prop_wf int-value-type comparison-reflexive minus-is-int-iff false_wf eq_int_wf bool_wf sq_stable__le nequal-le-implies equal_wf squash_wf true_wf subtype_rel_self iff_weakening_equal intformle_wf int_formula_prop_le_lemma decidable__le le_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut sqequalHypSubstitution setElimination thin rename Error :dependent_set_memberEquality_alt,  productElimination Error :lambdaEquality_alt,  sqequalRule callbyvalueReduce extract_by_obid isectElimination because_Cache independent_isectElimination hypothesis Error :inhabitedIsType,  Error :lambdaFormation_alt,  unionElimination equalityElimination int_eqReduceTrueSq equalityTransitivity equalitySymmetry applyEquality hypothesisEquality Error :equalityIstype,  baseClosed sqequalBase dependent_functionElimination independent_functionElimination Error :dependent_pairFormation_alt,  promote_hyp instantiate voidElimination int_eqReduceFalseSq independent_pairFormation Error :universeIsType,  Error :productIsType,  Error :functionIsType,  minusEquality natural_numberEquality axiomEquality Error :isectIsType,  setEquality intEquality Error :isect_memberEquality_alt,  Error :isectIsTypeImplies,  universeEquality approximateComputation int_eqEquality cumulativity pointwiseFunctionality closedConclusion imageMemberEquality imageElimination

Latex:
\mforall{}[T:Type].  \mforall{}[c1:comparison(T)].  \mforall{}[c2:\mcap{}a:T.  comparison(\{b:T|  (c1  a  b)  =  0\}  )].
(comparison-seq(c1;  c2)  \mmember{}  comparison(T))

Date html generated: 2019_06_20-PM-01_42_24
Last ObjectModification: 2019_05_03-PM-02_10_37

Theory : list_1

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