`∀[n:ℕ]. ∀[p,q:polyform(n)]. ∀[rmz:𝔹].  (add-polynom(n;rmz;p;q) ∈ polyform(n))`

Proof

Definitions occuring in Statement :  add-polynom: `add-polynom(n;rmz;p;q)` polyform: `polyform(n)` nat: `ℕ` bool: `𝔹` uall: `∀[x:A]. B[x]` member: `t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` decidable: `Dec(P)` or: `P ∨ Q` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` polyform: `polyform(n)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b` less_than': `less_than'(a;b)` int_upper: `{i...}` add-polynom: `add-polynom(n;rmz;p;q)` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` nil: `[]` less_than: `a < b` squash: `↓T` callbyvalueall: callbyvalueall evalall: `evalall(t)` length: `||as||` list_ind: list_ind has-value: `(a)↓` has-valueall: `has-valueall(a)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` rm-zeros: `rm-zeros(n;p)` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` true: `True`
Lemmas referenced :  nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf bool_wf polyform_wf le_wf int_seg_wf int_seg_properties decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma decidable__equal_int subtype_base_sq set_subtype_base int_subtype_base intformeq_wf int_formula_prop_eq_lemma decidable__lt lelt_wf subtype_rel_self eq_int_wf eqtt_to_assert assert_of_eq_int eqff_to_assert equal_wf bool_cases_sqequal bool_subtype_base assert-bnot neg_assert_of_eq_int upper_subtype_nat false_wf nequal-le-implies zero-add itermAdd_wf int_term_value_add_lemma nat_wf int_upper_properties equal-wf-T-base colength_wf_list list-cases product_subtype_list spread_cons_lemma list_wf assert_wf bnot_wf not_wf null_nil_lemma length_of_nil_lemma valueall-type-has-valueall list-valueall-type valueall-type-polyform evalall-reduce uiff_transitivity iff_transitivity iff_weakening_uiff assert_of_bnot cons_wf null_cons_lemma length_of_cons_lemma value-type-has-value int-value-type length_wf bfalse_wf btrue_wf list_ind_cons_lemma poly-zero_wf list_ind_wf nil_wf top_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation axiomEquality equalityTransitivity equalitySymmetry dependent_set_memberEquality productElimination because_Cache unionElimination applyEquality instantiate applyLambdaEquality hypothesis_subsumption equalityElimination promote_hyp cumulativity addEquality baseClosed imageElimination callbyvalueReduce sqleReflexivity impliesFunctionality lessCases imageMemberEquality axiomSqEquality

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[p,q:polyform(n)].  \mforall{}[rmz:\mBbbB{}].    (add-polynom(n;rmz;p;q)  \mmember{}  polyform(n))

Date html generated: 2019_06_20-PM-01_52_11
Last ObjectModification: 2018_08_20-PM-09_32_27

Theory : integer!polynomials

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