### Nuprl Lemma : int-bag-product-append

`∀[b1,b2:bag(ℤ)].  (Π(b1 + b2) ~ Π(b1) * Π(b2))`

Proof

Definitions occuring in Statement :  int-bag-product: `Π(b)` bag-append: `as + bs` bag: `bag(T)` uall: `∀[x:A]. B[x]` multiply: `n * m` int: `ℤ` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` int-bag-product: `Π(b)` bag-product: `Πx ∈ b. f[x]` sq_type: `SQType(T)` all: `∀x:A. B[x]` implies: `P `` Q` guard: `{T}` monoid_p: `IsMonoid(T;op;id)` and: `P ∧ Q` assoc: `Assoc(T;op)` infix_ap: `x f y` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` prop: `ℙ` ident: `Ident(T;op;id)` cand: `A c∧ B` comm: `Comm(T;op)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` true: `True` squash: `↓T` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  subtype_base_sq int_subtype_base bag_wf decidable__equal_int satisfiable-full-omega-tt intformnot_wf intformeq_wf itermMultiply_wf itermVar_wf int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_mul_lemma int_term_value_var_lemma int_formula_prop_wf itermConstant_wf int_term_value_constant_lemma bag-summation_wf equal_wf squash_wf true_wf bag-summation-append iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin instantiate extract_by_obid sqequalHypSubstitution isectElimination cumulativity intEquality independent_isectElimination hypothesis dependent_functionElimination equalityTransitivity equalitySymmetry independent_functionElimination sqequalAxiom sqequalRule isect_memberEquality hypothesisEquality because_Cache independent_pairFormation unionElimination natural_numberEquality dependent_pairFormation lambdaEquality int_eqEquality voidElimination voidEquality computeAll axiomEquality productElimination independent_pairEquality multiplyEquality applyEquality imageElimination universeEquality imageMemberEquality baseClosed

Latex:
\mforall{}[b1,b2:bag(\mBbbZ{})].    (\mPi{}(b1  +  b2)  \msim{}  \mPi{}(b1)  *  \mPi{}(b2))

Date html generated: 2017_10_01-AM-08_51_39
Last ObjectModification: 2017_07_26-PM-04_33_25

Theory : bags

Home Index