### Nuprl Lemma : l-union-subset

`∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[as,bs:T List].  as ⋃ bs ~ as supposing bs ⊆ as`

Proof

Definitions occuring in Statement :  l-union: `as ⋃ bs` l_contains: `A ⊆ B` list: `T List` deq: `EqDecider(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` universe: `Type` sqequal: `s ~ t`
Definitions unfolded in proof :  l-union: `as ⋃ bs` member: `t ∈ T` uall: `∀[x:A]. B[x]` uimplies: `b supposing a` prop: `ℙ` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` subtype_rel: `A ⊆r B` guard: `{T}` or: `P ∨ Q` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` nil: `[]` it: `⋅` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` iff: `P `⇐⇒` Q` insert: `insert(a;L)` has-value: `(a)↓` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b` rev_implies: `P `` Q`
Lemmas referenced :  l_contains_wf list_wf deq_wf nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases reduce_nil_lemma nil_wf product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf equal_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int reduce_cons_lemma l_contains_cons eval_list_sq subtype_rel_list top_wf value-type-has-value list-value-type deq-member_wf bool_wf eqtt_to_assert assert-deq-member eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot l_member_wf cons_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality hypothesis because_Cache universeEquality isect_memberFormation sqequalAxiom sqequalRule isect_memberEquality equalityTransitivity equalitySymmetry lambdaFormation setElimination rename intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination applyEquality unionElimination promote_hyp hypothesis_subsumption productElimination applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination callbyvalueReduce equalityElimination

Latex:
\mforall{}[T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[as,bs:T  List].    as  \mcup{}  bs  \msim{}  as  supposing  bs  \msubseteq{}  as

Date html generated: 2017_04_17-AM-09_09_53
Last ObjectModification: 2017_02_27-PM-05_18_01

Theory : decidable!equality

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