`∀[p,q:iMonomial() List].  (add-ipoly(p;q) ∈ iMonomial() List)`

Proof

Definitions occuring in Statement :  add-ipoly: `add-ipoly(p;q)` iMonomial: `iMonomial()` list: `T List` 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 ` guard: `{T}` uimplies: `b supposing a` prop: `ℙ` subtype_rel: `A ⊆r B` or: `P ∨ Q` add-ipoly: `add-ipoly(p;q)` has-value: `(a)↓` ifthenelse: `if b then t else f fi ` btrue: `tt` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` top: `Top` so_apply: `x[s1;s2]` squash: `↓T` sq_stable: `SqStable(P)` uiff: `uiff(P;Q)` and: `P ∧ Q` le: `A ≤ B` not: `¬A` less_than': `less_than'(a;b)` true: `True` nil: `[]` it: `⋅` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` subtract: `n - m` bfalse: `ff` bool: `𝔹` unit: `Unit` exists: `∃x:A. B[x]` bnot: `¬bb` assert: `↑b` has-valueall: `has-valueall(a)` callbyvalueall: callbyvalueall int_nzero: `ℤ-o` iMonomial: `iMonomial()` nequal: `a ≠ b ∈ T ` pi1: `fst(t)` pi2: `snd(t)` imonomial-le: `imonomial-le(m1;m2)` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` decidable: `Dec(P)` istype: `istype(T)`
Lemmas referenced :  nat_properties less_than_transitivity1 less_than_irreflexivity ge_wf less_than_wf nat_wf colength_wf_list iMonomial_wf list_wf list-cases value-type-has-value list-value-type nil_wf null_nil_lemma product_subtype_list spread_cons_lemma istype-void sq_stable__le le_antisymmetry_iff add_functionality_wrt_le add-associates istype-int add-zero zero-add le-add-cancel int_subtype_base subtract-1-ge-0 subtype_base_sq set_subtype_base le_wf add-commutes add-swap cons_wf null_cons_lemma imonomial-le_wf eqtt_to_assert eqff_to_assert bool_cases_sqequal bool_wf bool_subtype_base assert-bnot evalall-reduce int-valueall-type nequal_wf set-valueall-type subtype_rel_self sorted_wf int_nzero_wf product-valueall-type list-valueall-type valueall-type-has-valueall equal_wf int-value-type minus-minus less-iff-le not-ge-2 subtract_wf minus-one-mul-top minus-one-mul minus-add condition-implies-le not-le-2 false_wf decidable__le equal-wf-T-base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut thin Error :lambdaFormation_alt,  extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination natural_numberEquality independent_isectElimination independent_functionElimination voidElimination Error :universeIsType,  Error :lambdaEquality_alt,  dependent_functionElimination Error :isect_memberEquality_alt,  axiomEquality equalityTransitivity equalitySymmetry Error :inhabitedIsType,  Error :equalityIsType3,  applyEquality unionElimination callbyvalueReduce voidEquality because_Cache promote_hyp hypothesis_subsumption productElimination applyLambdaEquality imageMemberEquality baseClosed imageElimination addEquality Error :equalityIsType4,  Error :equalityIsType1,  instantiate cumulativity intEquality Error :dependent_set_memberEquality_alt,  minusEquality equalityElimination Error :dependent_pairFormation_alt,  lambdaFormation setEquality lambdaEquality dependent_pairFormation dependent_set_memberEquality independent_pairEquality int_eqEquality independent_pairFormation isect_memberEquality

Latex:
\mforall{}[p,q:iMonomial()  List].    (add-ipoly(p;q)  \mmember{}  iMonomial()  List)

Date html generated: 2019_06_20-PM-00_45_10
Last ObjectModification: 2018_10_02-PM-11_35_52

Theory : omega

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