`∀r:Rng. ∀p,q:iMonomial() List.  ipolynomial-term(add-ipoly(p;q)) ≡ ipolynomial-term(p) (+) ipolynomial-term(q)`

Proof

Definitions occuring in Statement :  ringeq_int_terms: `t1 ≡ t2` rng: `Rng` add-ipoly: `add-ipoly(p;q)` ipolynomial-term: `ipolynomial-term(p)` iMonomial: `iMonomial()` itermAdd: `left (+) right` list: `T List` all: `∀x:A. B[x]`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` 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]` and: `P ∧ Q` prop: `ℙ` ringeq_int_terms: `t1 ≡ t2` le: `A ≤ B` less_than': `less_than'(a;b)` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` uiff: `uiff(P;Q)` add-ipoly: `add-ipoly(p;q)` has-value: `(a)↓` ifthenelse: `if b then t else f fi ` btrue: `tt` cons: `[a / b]` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` bfalse: `ff` ipolynomial-term: `ipolynomial-term(p)` ring_term_value: `ring_term_value(f;t)` itermAdd: `left (+) right` int_term_ind: int_term_ind itermConstant: `"const"` rng: `Rng` bool: `𝔹` unit: `Unit` it: `⋅` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` iMonomial: `iMonomial()` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` so_apply: `x[s]` int_nzero: `ℤ-o` callbyvalueall: callbyvalueall has-valueall: `has-valueall(a)` pi1: `fst(t)` nequal: `a ≠ b ∈ T ` rev_uimplies: `rev_uimplies(P;Q)` imonomial-le: `imonomial-le(m1;m2)` pi2: `snd(t)` squash: `↓T` label: `...\$L... t` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` infix_ap: `x f y`
Lemmas referenced :  nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_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_less_lemma int_formula_prop_wf ge_wf istype-less_than length_wf iMonomial_wf subtract-1-ge-0 istype-nat add_nat_wf length_wf_nat istype-void istype-le decidable__le add-is-int-iff intformnot_wf itermAdd_wf intformeq_wf int_formula_prop_not_lemma int_term_value_add_lemma int_formula_prop_eq_lemma false_wf decidable__lt list_wf rng_wf non_neg_length list-cases value-type-has-value list-value-type nil_wf null_nil_lemma product_subtype_list cons_wf null_cons_lemma spread_cons_lemma ipolynomial-term_wf int-to-ring-zero rng_plus_zero ring_term_value_wf ring_term_value_add_lemma ring_term_value_const_lemma rng_car_wf imonomial-le_wf eqtt_to_assert eqff_to_assert bool_cases_sqequal subtype_base_sq bool_wf bool_subtype_base assert-bnot add-ipoly_wf1 valueall-type-has-valueall list-valueall-type product-valueall-type int_nzero_wf sorted_wf set-valueall-type nequal_wf int-valueall-type evalall-reduce ipolynomial-term-cons-ringeq int-value-type eq_int_wf assert_of_eq_int neg_assert_of_eq_int imonomial-term_wf length_of_cons_lemma int_nzero_properties subtract_wf itermSubtract_wf int_term_value_subtract_lemma ringeq_int_terms_functionality itermAdd_functionality_wrt_ringeq ringeq_int_terms_transitivity ringeq_int_terms_weakening intlex_wf intlex-antisym subtype_rel_universe1 set_subtype_base list_subtype_base int_subtype_base equal_wf squash_wf true_wf istype-universe set_wf member_wf subtype_rel_self iff_weakening_equal imonomial-term-add-ringeq rng_zero_wf imonomial-term-linear-ringeq rng_times_zero rng_plus_wf rng_properties rng_plus_assoc rng_plus_ac_1 rng_plus_comm
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt int_eqEquality dependent_functionElimination Error :memTop,  sqequalRule independent_pairFormation universeIsType voidElimination axiomEquality functionIsTypeImplies inhabitedIsType addEquality because_Cache dependent_set_memberEquality_alt equalityTransitivity equalitySymmetry applyLambdaEquality unionElimination pointwiseFunctionality promote_hyp baseApply closedConclusion baseClosed productElimination equalityIstype callbyvalueReduce voidEquality hypothesis_subsumption functionIsType equalityElimination instantiate cumulativity setEquality intEquality int_eqReduceTrueSq int_eqReduceFalseSq independent_pairEquality applyEquality imageElimination universeEquality hyp_replacement imageMemberEquality

Latex:
\mforall{}r:Rng.  \mforall{}p,q:iMonomial()  List.