### Nuprl Lemma : seq-append-assoc

`∀[n,m,k:ℕ]. ∀[s1,s2,s3:Top].`
`  (seq-append(n;m + k;s1;seq-append(m;k;s2;s3)) ~ seq-append(n + m;k;seq-append(n;m;s1;s2);s3))`

Proof

Definitions occuring in Statement :  seq-append: `seq-append(n;m;s1;s2)` nat: `ℕ` uall: `∀[x:A]. B[x]` top: `Top` add: `n + m` sqequal: `s ~ t`
Definitions unfolded in proof :  seq-append: `seq-append(n;m;s1;s2)` uall: `∀[x:A]. B[x]` member: `t ∈ T` implies: `P `` Q` top: `Top` nat: `ℕ` all: `∀x:A. B[x]` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` less_than: `a < b` less_than': `less_than'(a;b)` true: `True` squash: `↓T` not: `¬A` false: `False` bfalse: `ff` exists: `∃x:A. B[x]` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` prop: `ℙ` ge: `i ≥ j ` subtract: `n - m` nat_plus: `ℕ+` le: `A ≤ B` decidable: `Dec(P)` has-value: `(a)↓`
Lemmas referenced :  istype-int istype-top istype-nat istype-void lt_int_wf eqtt_to_assert assert_of_lt_int eqff_to_assert set_subtype_base int_subtype_base bool_subtype_base bool_cases_sqequal subtype_base_sq iff_transitivity assert_wf bnot_wf not_wf less_than_wf iff_weakening_uiff assert_of_bnot istype-less_than istype-assert le_wf bool_wf add_functionality_wrt_le subtract_wf le_reflexive minus-one-mul zero-add one-mul add-mul-special add-associates two-mul add-commutes mul-distributes-right zero-mul less-iff-le add-zero not-lt-2 minus-one-mul-top add-swap omega-shadow mul-distributes minus-add mul-associates mul-swap mul-commutes le-add-cancel nat_properties decidable__lt has-value_wf_base is-exception_wf bottom-sqle
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep Error :isect_memberFormation_alt,  introduction cut thin Error :lambdaFormation_alt,  Error :equalityIsType4,  extract_by_obid hypothesis hypothesisEquality because_Cache sqequalHypSubstitution axiomSqEquality Error :inhabitedIsType,  Error :isect_memberEquality_alt,  isectElimination Error :isectIsTypeImplies,  voidElimination setElimination rename multiplyEquality minusEquality natural_numberEquality unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination independent_isectElimination lessCases independent_pairFormation imageMemberEquality baseClosed imageElimination independent_functionElimination addEquality Error :dependent_pairFormation_alt,  baseApply closedConclusion applyEquality intEquality Error :lambdaEquality_alt,  promote_hyp dependent_functionElimination instantiate Error :functionIsType,  Error :universeIsType,  Error :equalityIsType1,  cumulativity Error :dependent_set_memberEquality_alt,  sqequalSqle divergentSqle callbyvalueLess sqleReflexivity lessExceptionCases axiomSqleEquality exceptionSqequal

Latex:
\mforall{}[n,m,k:\mBbbN{}].  \mforall{}[s1,s2,s3:Top].
(seq-append(n;m  +  k;s1;seq-append(m;k;s2;s3))  \msim{}  seq-append(n  +  m;k;seq-append(n;m;s1;s2);s3))

Date html generated: 2019_06_20-AM-11_28_36
Last ObjectModification: 2018_10_27-AM-11_38_11

Theory : bar-induction

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