### Nuprl Lemma : fps-product-upto

`∀[X:Type]`
`  ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[k:ℕ+]. ∀[f:ℕk ⟶ PowerSeries(X;r)].`
`    (Π(x∈upto(k)).f[x] = (f[0]*Π(x∈upto(k - 1)).f[x + 1]) ∈ PowerSeries(X;r)) `
`  supposing valueall-type(X)`

Proof

Definitions occuring in Statement :  fps-product: `Π(x∈b).f[x]` fps-mul: `(f*g)` power-series: `PowerSeries(X;r)` upto: `upto(n)` deq: `EqDecider(T)` int_seg: `{i..j-}` nat_plus: `ℕ+` valueall-type: `valueall-type(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` function: `x:A ⟶ B[x]` subtract: `n - m` add: `n + m` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T` crng: `CRng`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` nat_plus: `ℕ+` single-bag: `{x}` bag-append: `as + bs` upto: `upto(n)` append: `as @ bs` all: `∀x:A. B[x]` so_lambda: `so_lambda(x,y,z.t[x; y; z])` top: `Top` so_apply: `x[s1;s2;s3]` from-upto: `[n, m)` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` ifthenelse: `if b then t else f fi ` bfalse: `ff` exists: `∃x:A. B[x]` prop: `ℙ` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` false: `False` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` cand: `A c∧ B` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` lelt: `i ≤ j < k` le: `A ≤ B` decidable: `Dec(P)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtract: `n - m` less_than': `less_than'(a;b)` true: `True` squash: `↓T` less_than: `a < b` bag-map: `bag-map(f;bs)`
Lemmas referenced :  int_seg_wf power-series_wf nat_plus_wf crng_wf deq_wf valueall-type_wf list_ind_cons_lemma list_ind_nil_lemma lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot less_than_wf nat_plus_properties satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf from-upto_wf list-subtype-bag subtype_rel_sets le_wf lelt_wf decidable__le false_wf not-le-2 condition-implies-le minus-add minus-one-mul zero-add minus-one-mul-top add-commutes add_functionality_wrt_le add-associates le-add-cancel single-bag_wf decidable__lt upto_wf subtract_wf subtype_rel_self bag-append_wf fps-product-append squash_wf true_wf fps-mul_wf fps-product_wf add-member-int_seg2 intformle_wf itermSubtract_wf int_formula_prop_le_lemma int_term_value_subtract_lemma iff_weakening_equal fps-product-single fps-product-reindex int_seg_properties decidable__equal_int intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma subtype_rel_dep_function int_seg_subtype from-upto-shift list_wf list_subtype_base set_subtype_base int_subtype_base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut hypothesis functionEquality extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality setElimination rename hypothesisEquality cumulativity sqequalRule isect_memberEquality axiomEquality because_Cache equalityTransitivity equalitySymmetry universeEquality dependent_functionElimination voidElimination voidEquality callbyvalueReduce sqleReflexivity lambdaFormation unionElimination equalityElimination productElimination independent_isectElimination dependent_pairFormation promote_hyp instantiate independent_functionElimination lambdaEquality int_eqEquality intEquality independent_pairFormation computeAll applyEquality productEquality addEquality minusEquality dependent_set_memberEquality imageElimination functionExtensionality imageMemberEquality baseClosed setEquality

Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)].  \mforall{}[r:CRng].  \mforall{}[k:\mBbbN{}\msupplus{}].  \mforall{}[f:\mBbbN{}k  {}\mrightarrow{}  PowerSeries(X;r)].
(\mPi{}(x\mmember{}upto(k)).f[x]  =  (f[0]*\mPi{}(x\mmember{}upto(k  -  1)).f[x  +  1]))
supposing  valueall-type(X)

Date html generated: 2018_05_21-PM-09_57_22
Last ObjectModification: 2017_07_26-PM-06_33_19

Theory : power!series

Home Index