### Nuprl Lemma : fps-set-to-one-one

`∀[r:CRng]. ∀[y:Atom]. ∀[n:ℕ].  ([1]_n(y:=1) = if (n =z 0) then 1 else 0 fi  ∈ PowerSeries(r))`

Proof

Definitions occuring in Statement :  fps-set-to-one: `[f]_n(y:=1)` fps-one: `1` fps-zero: `0` power-series: `PowerSeries(X;r)` nat: `ℕ` ifthenelse: `if b then t else f fi ` eq_int: `(i =z j)` uall: `∀[x:A]. B[x]` natural_number: `\$n` atom: `Atom` equal: `s = t ∈ T` crng: `CRng`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` all: `∀x:A. B[x]` fps-zero: `0` fps-one: `1` fps-coeff: `f[b]` fps-set-to-one: `[f]_n(y:=1)` subtype_rel: `A ⊆r B` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` bor: `p ∨bq` 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` le: `A ≤ B` less_than': `less_than'(a;b)` not: `¬A` ge: `i ≥ j ` int_upper: `{i...}` crng: `CRng` rng: `Rng` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` prop: `ℙ` less_than: `a < b` squash: `↓T` bag-count: `(#x in bs)` count: `count(P;L)` reduce: `reduce(f;k;as)` list_ind: list_ind empty-bag: `{}` nil: `[]` bag-size: `#(bs)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` decidable: `Dec(P)` true: `True` band: `p ∧b q`
Lemmas referenced :  fps-ext fps-set-to-one_wf fps-one_wf ifthenelse_wf eq_int_wf power-series_wf fps-zero_wf lt_int_wf bag-count_wf atom-deq_wf eqtt_to_assert assert_of_lt_int assert_of_eq_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int upper_subtype_nat istype-false nat_properties nequal-le-implies zero-add istype-le rng_zero_wf bool_wf iff_weakening_uiff assert_wf less_than_wf istype-less_than bag-size_wf bag_wf istype-nat istype-atom crng_wf bag-null_wf assert-bag-null equal-wf-T-base length_of_nil_lemma full-omega-unsat intformand_wf intformeq_wf itermVar_wf itermConstant_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_wf equal-empty-bag empty_bag_append_lemma bag_size_empty_lemma bag-null-rep subtract_wf decidable__le intformnot_wf intformle_wf itermSubtract_wf int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_subtract_lemma int_subtype_base decidable__equal_int bag_null_empty_lemma rng_one_wf equal_wf squash_wf true_wf istype-universe bag-null-append bag-rep_wf list-subtype-bag bfalse_wf subtype_rel_self iff_weakening_equal iff_imp_equal_bool bool_cases band_wf btrue_wf iff_functionality_wrt_iff false_wf iff_transitivity assert_of_band istype-assert
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin atomEquality hypothesisEquality hypothesis setElimination rename natural_numberEquality productElimination independent_isectElimination lambdaFormation_alt sqequalRule applyEquality because_Cache inhabitedIsType unionElimination equalityElimination equalityTransitivity equalitySymmetry lambdaEquality_alt dependent_pairFormation_alt equalityIstype promote_hyp dependent_functionElimination instantiate independent_functionElimination voidElimination hypothesis_subsumption independent_pairFormation dependent_set_memberEquality_alt cumulativity universeIsType isect_memberEquality_alt axiomEquality isectIsTypeImplies hyp_replacement applyLambdaEquality imageElimination baseClosed sqequalBase approximateComputation int_eqEquality intEquality universeEquality closedConclusion imageMemberEquality productEquality productIsType

Latex:
\mforall{}[r:CRng].  \mforall{}[y:Atom].  \mforall{}[n:\mBbbN{}].    ([1]\_n(y:=1)  =  if  (n  =\msubz{}  0)  then  1  else  0  fi  )

Date html generated: 2019_10_16-AM-11_36_29
Last ObjectModification: 2018_11_26-PM-03_09_16

Theory : power!series

Home Index