### Nuprl Lemma : fps-compose-atom-eq

`∀[X:Type]`
`  ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[x:X]. ∀[f:PowerSeries(X;r)].  (atom(x)(x:=f) = (f-(f[{}])*1) ∈ PowerSeries(X;r)) `
`  supposing valueall-type(X)`

Proof

Definitions occuring in Statement :  fps-compose: `g(x:=f)` fps-scalar-mul: `(c)*f` fps-sub: `(f-g)` fps-atom: `atom(x)` fps-one: `1` fps-coeff: `f[b]` power-series: `PowerSeries(X;r)` empty-bag: `{}` deq: `EqDecider(T)` valueall-type: `valueall-type(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` universe: `Type` equal: `s = t ∈ T` crng: `CRng`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` crng: `CRng` comm: `Comm(T;op)` rng: `Rng` and: `P ∧ Q` cand: `A c∧ B` prop: `ℙ` all: `∀x:A. B[x]` listp: `A List+` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` power-series: `PowerSeries(X;r)` so_apply: `x[s]` uiff: `uiff(P;Q)` fps-coeff: `f[b]` fps-atom: `atom(x)` fps-compose: `g(x:=f)` fps-single: `<c>` fps-one: `1` fps-scalar-mul: `(c)*f` fps-sub: `(f-g)` fps-neg: `-(f)` fps-add: `(f+g)` ring_p: `IsRing(T;plus;zero;neg;times;one)` group_p: `IsGroup(T;op;id;inv)` squash: `↓T` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` false: `False` not: `¬A` tlp: `tlp(L)` hdp: `hdp(L)` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` bag-member: `x ↓∈ bs` sq_stable: `SqStable(P)` nat: `ℕ` top: `Top` cons: `[a / b]` bag-rep: `bag-rep(n;x)` ge: `i ≥ j ` le: `A ≤ B` satisfiable_int_formula: `satisfiable_int_formula(fmla)` bag-union: `bag-union(bbs)` concat: `concat(ll)` empty-bag: `{}` bag-append: `as + bs` rev_uimplies: `rev_uimplies(P;Q)` length: `||as||` list_ind: list_ind nil: `[]` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` infix_ap: `x f y` bag-product: `Πx ∈ b. f[x]` single-bag: `{x}` bag-summation: `Σ(x∈b). f[x]` bag-accum: `bag-accum(v,x.f[v; x];init;bs)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]`
Lemmas referenced :  rng_plus_comm crng_properties rng_properties rng_all_properties ring_p_wf rng_car_wf rng_plus_wf rng_zero_wf rng_minus_wf rng_times_wf rng_one_wf bag-product_wf bag_wf tl_wf list-subtype-bag listp_wf fps-ext fps-compose_wf fps-atom_wf fps-sub_wf fps-scalar-mul_wf fps-coeff_wf empty-bag_wf fps-one_wf power-series_wf crng_wf deq_wf valueall-type_wf bag-summation_wf squash_wf assoc_wf comm_wf bag-eq_wf bag-append_wf hd_wf listp_properties bag-rep_wf length_wf_nat single-bag_wf bool_wf eqtt_to_assert assert-bag-eq rng_times_one eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot rng_times_zero bag-parts'_wf true_wf infix_ap_wf bag-null_wf assert-bag-null equal-wf-T-base bag-summation-filter hdp_wf tlp_wf iff_weakening_equal bag-extensionality-no-repeats bag-filter_wf ifthenelse_wf cons_wf_listp cons_wf nil_wf empty-bag-no-repeats bag-single-no-repeats bag-member_wf decidable__equal_set list_wf decidable__equal_list decidable__equal_bag decidable-equal-deq less_than_wf length_wf bag-filter-no-repeats bag-parts'-no-repeats bag-member-filter bag-append-is-single sq_stable__bag-member bag-member-parts' bag-member-single bag-size_wf nat_wf bag_size_single_lemma bag-size-rep list-cases reduce_tl_nil_lemma length_of_nil_lemma int_subtype_base product_subtype_list reduce_tl_cons_lemma reduce_hd_cons_lemma length_of_cons_lemma primrec1_lemma cons_bag_empty_lemma uiff_transitivity assert_wf iff_transitivity bnot_wf not_wf iff_weakening_uiff assert_of_bnot non_neg_length satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformeq_wf itermAdd_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_term_value_add_lemma int_formula_prop_wf reduce_cons_lemma reduce_nil_lemma bag-append-is-empty l_all_iff l_member_wf cons_member bag-union_wf subtype_rel_self list_ind_nil_lemma bag-append-empty bag-subtype-list bool_cases bag-member-empty-iff empty_bag_append_lemma l_all_cons l_all_nil equal-empty-bag bag-summation-empty rng_plus_inv list_accum_cons_lemma list_accum_nil_lemma rng_minus_zero rng_plus_zero
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality hypothesis dependent_set_memberEquality productElimination independent_pairFormation lambdaFormation cumulativity because_Cache applyEquality independent_isectElimination sqequalRule lambdaEquality isect_memberEquality axiomEquality equalityTransitivity equalitySymmetry universeEquality imageElimination productEquality functionExtensionality functionEquality dependent_functionElimination unionElimination equalityElimination dependent_pairFormation promote_hyp instantiate independent_functionElimination voidElimination imageMemberEquality baseClosed hyp_replacement natural_numberEquality applyLambdaEquality intEquality voidEquality hypothesis_subsumption impliesFunctionality int_eqEquality computeAll setEquality inlFormation equalityUniverse levelHypothesis

Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)].  \mforall{}[r:CRng].  \mforall{}[x:X].  \mforall{}[f:PowerSeries(X;r)].    (atom(x)(x:=f)  =  (f-(f[\{\}])*1))
supposing  valueall-type(X)

Date html generated: 2018_05_21-PM-10_06_07
Last ObjectModification: 2017_07_26-PM-06_34_09

Theory : power!series

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