Nuprl Lemma : l-last-is-last

`∀[l:Top]. (l-last(l) ~ last(l))`

Proof

Definitions occuring in Statement :  l-last: `l-last(l)` last: `last(L)` uall: `∀[x:A]. B[x]` top: `Top` sqequal: `s ~ 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` or: `P ∨ Q` cons: `[a / b]` le: `A ≤ B` less_than': `less_than'(a;b)` colength: `colength(L)` nil: `[]` it: `⋅` guard: `{T}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` subtype_rel: `A ⊆r B` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` list_ind: list_ind l-last-default: `l-last-default(l;d)` l-last: `l-last(l)` cand: `A c∧ B` is-exception: `is-exception(t)` has-value: `(a)↓` strict4: `strict4(F)` so_apply: `x[s1;s2;s3;s4]` so_lambda: `so_lambda(x,y,z,w.t[x; y; z; w])` nat_plus: `ℕ+` select: `L[n]` last: `last(L)` true: `True` label: `...\$L... t` subtract: `n - m` pi2: `snd(t)` pi1: `fst(t)` length: `||as||` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` null: `null(as)`
Lemmas referenced :  nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void 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 top_wf list-cases l-last-nil last_nil product_subtype_list colength-cons-not-zero colength_wf_list istype-false le_wf subtract-1-ge-0 subtype_base_sq intformeq_wf int_formula_prop_eq_lemma set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf intformnot_wf itermSubtract_wf itermAdd_wf int_formula_prop_not_lemma int_term_value_subtract_lemma int_term_value_add_lemma decidable__le l-last-cons last_cons2 null_wf eqtt_to_assert assert_of_null eqff_to_assert bool_subtype_base bool_cases_sqequal bool_wf assert-bnot iff_weakening_uiff assert_wf equal-wf-T-base list_wf nat_wf has-value-l-last-list has-value_wf_base is-exception_wf has-value-last-list istype-top base_wf sqle_wf_base has-value-implies-dec-isaxiom-2 has-value-implies-dec-ispair-2 exception-not-bottom bottom_diverge lifting-strict-callbyvalue fun_exp_unroll_1 exception-not-bottom_1 strictness-apply fun_exp0_lemma less_than_wf satisfiable-full-omega-tt equal-wf-base equal_wf assert_of_lt_int lt_int_wf length_of_cons_lemma int-value-type value-type-has-value length-nat-if-has-value false_wf pair-eta set-value-type exception-not-value length_of_nil_lemma istype-base decidable__lt add-associates add-swap add-commutes zero-add length-zero-implies-nil list_ind_cons_lemma list_ind_nil_lemma bfalse_wf exception-not-value_1
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut sqequalRule thin Error :lambdaFormation_alt,  extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality dependent_functionElimination Error :isect_memberEquality_alt,  voidElimination independent_pairFormation Error :universeIsType,  axiomSqEquality Error :functionIsTypeImplies,  Error :inhabitedIsType,  unionElimination promote_hyp hypothesis_subsumption productElimination Error :equalityIsType1,  because_Cache Error :dependent_set_memberEquality_alt,  instantiate equalityTransitivity equalitySymmetry applyLambdaEquality imageElimination Error :equalityIsType4,  baseApply closedConclusion baseClosed applyEquality intEquality equalityElimination Error :equalityIsType3,  cumulativity sqequalSqle divergentSqle sqleReflexivity lambdaFormation axiomSqleEquality exceptionSqequal exceptionCompactness callbyvalueReduce callbyvalueExceptionCases sqleRule sqleTransitivity inlFormation inrFormation applyExceptionCases callbyvalueApply dependent_set_memberEquality imageMemberEquality computeAll voidEquality isect_memberEquality lambdaEquality dependent_pairFormation sqequalIntensionalEquality productEquality spreadExceptionCases isect_memberFormation lessCases addEquality callbyvalueAdd lessExceptionCases addExceptionCases minusEquality levelHypothesis addLevel Error :isectIsTypeImplies,  callbyvalueCallbyvalue

Latex:
\mforall{}[l:Top].  (l-last(l)  \msim{}  last(l))

Date html generated: 2019_06_20-PM-01_20_47
Last ObjectModification: 2018_10_17-AM-11_51_01

Theory : list_1

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