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

`∀[X:Type]`
`  ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[x,y:X].`
`    ∀[f:PowerSeries(X;r)]. (atom(y)(x:=f) = atom(y) ∈ PowerSeries(X;r)) supposing ¬(x = y ∈ X) `
`  supposing valueall-type(X)`

Proof

Definitions occuring in Statement :  fps-compose: `g(x:=f)` fps-atom: `atom(x)` power-series: `PowerSeries(X;r)` deq: `EqDecider(T)` valueall-type: `valueall-type(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` not: `¬A` 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-atom: `atom(x)` fps-coeff: `f[b]` fps-compose: `g(x:=f)` fps-single: `<c>` not: `¬A` implies: `P `` Q` false: `False` ring_p: `IsRing(T;plus;zero;neg;times;one)` group_p: `IsGroup(T;op;id;inv)` squash: `↓T` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` guard: `{T}` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` tlp: `tlp(L)` hdp: `hdp(L)` true: `True` bag-member: `x ↓∈ bs` sq_stable: `SqStable(P)` nat: `ℕ` ge: `i ≥ j ` le: `A ≤ B` less_than': `less_than'(a;b)` length: `||as||` list_ind: list_ind nil: `[]` cons: `[a / b]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` single-bag: `{x}` bag-null: `bag-null(bs)` null: `null(as)` bag-union: `bag-union(bbs)` concat: `concat(ll)` reduce: `reduce(f;k;as)` append: `as @ bs` istype: `istype(T)` rev_uimplies: `rev_uimplies(P;Q)` bag-rep: `bag-rep(n;x)` bag-product: `Πx ∈ b. f[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]` tl: `tl(l)` pi2: `snd(t)` infix_ap: `x f y` list_accum: list_accum rng_one: `1` pi1: `fst(t)` empty-bag: `{}` rng_zero: `0`
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 power-series_wf istype-void crng_wf deq_wf valueall-type_wf istype-universe 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 eqtt_to_assert assert-bag-eq rng_times_one eqff_to_assert bool_cases_sqequal subtype_base_sq bool_wf bool_subtype_base assert-bnot iff_weakening_uiff assert_wf equal_wf rng_times_zero bag-parts'_wf crng_all_properties true_wf ifthenelse_wf bag-summation-filter hdp_wf tlp_wf subtype_rel_self iff_weakening_equal bag-extensionality-no-repeats bag-filter_wf subtype_rel_bag istype-assert bag-null_wf empty-bag_wf cons-listp nil_wf assert-bag-null empty-bag-no-repeats equal-wf-T-base 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-size_wf istype-nat bag_size_empty_lemma bag-size-rep list-cases product_subtype_list reduce_tl_cons_lemma reduce_hd_cons_lemma length_of_cons_lemma bag-member-parts' non_neg_length full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformeq_wf itermAdd_wf istype-int 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 bag-member-single cons_wf istype-less_than bag-union_wf uiff_transitivity iff_transitivity bnot_wf not_wf assert_of_bnot cons_wf_listp subtype_rel_set bag_qinc bag-member-empty-iff bag-subtype-list bag-union-single bag_size_single_lemma reduce_tl_nil_lemma length_of_nil_lemma int_subtype_base primrec1_lemma cons_bag_empty_lemma single-bags-equal bool_cases primrec0_lemma bag-append-empty l_all_nil bag-summation-empty list_accum_cons_lemma list_accum_nil_lemma rng_plus_zero list_accum_wf bag_null_empty_lemma btrue_neq_bfalse
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality hypothesis dependent_set_memberEquality_alt universeIsType productElimination independent_pairFormation lambdaFormation_alt because_Cache applyEquality independent_isectElimination sqequalRule lambdaEquality_alt isect_memberEquality_alt axiomEquality isectIsTypeImplies inhabitedIsType functionIsType equalityIstype instantiate universeEquality imageElimination equalityTransitivity equalitySymmetry productEquality dependent_functionElimination unionElimination equalityElimination dependent_pairFormation_alt promote_hyp cumulativity independent_functionElimination voidElimination imageMemberEquality baseClosed hyp_replacement natural_numberEquality applyLambdaEquality setEquality setIsType intEquality Error :memTop,  hypothesis_subsumption approximateComputation int_eqEquality sqequalBase

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

Date html generated: 2020_05_20-AM-09_05_59
Last ObjectModification: 2019_12_31-PM-04_59_20

Theory : power!series

Home Index