Nuprl Lemma : from-upto-singleton

`∀[n,m,k:ℤ].  uiff([n, m) = [k] ∈ (ℤ List);(m = (n + 1) ∈ ℤ) ∧ (k = n ∈ ℤ))`

Proof

Definitions occuring in Statement :  from-upto: `[n, m)` cons: `[a / b]` nil: `[]` list: `T List` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` and: `P ∧ Q` add: `n + m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` prop: `ℙ` subtype_rel: `A ⊆r B` from-upto: `[n, m)` has-value: `(a)↓` all: `∀x:A. B[x]` top: `Top` not: `¬A` implies: `P `` Q` false: `False` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` sq_type: `SQType(T)` guard: `{T}` ifthenelse: `if b then t else f fi ` btrue: `tt` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` bfalse: `ff` squash: `↓T` ge: `i ≥ j ` bool: `𝔹` unit: `Unit` it: `⋅` bnot: `¬bb` assert: `↑b`
Lemmas referenced :  equal-wf-base list_wf int_subtype_base lt_int_wf value-type-has-value int-value-type null_nil_lemma btrue_wf reduce_tl_cons_lemma nil_wf and_wf equal_wf tl_wf cons_wf from-upto_wf subtype_rel_list le_wf less_than_wf null_wf null_cons_lemma bfalse_wf btrue_neq_bfalse assert_wf bnot_wf not_wf decidable__equal_int satisfiable-full-omega-tt intformand_wf intformnot_wf intformeq_wf itermVar_wf itermAdd_wf itermConstant_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_add_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_wf bool_cases subtype_base_sq bool_wf bool_subtype_base eqtt_to_assert assert_of_lt_int eqff_to_assert iff_transitivity iff_weakening_uiff assert_of_bnot reduce_hd_cons_lemma hd_wf squash_wf ge_wf length_wf length_cons_ge_one top_wf from-upto-nil decidable__le intformle_wf int_formula_prop_le_lemma bool_cases_sqequal assert-bnot
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation hypothesis sqequalRule sqequalHypSubstitution productElimination thin independent_pairEquality axiomEquality extract_by_obid isectElimination intEquality baseApply closedConclusion baseClosed hypothesisEquality applyEquality because_Cache productEquality isect_memberEquality equalityTransitivity equalitySymmetry callbyvalueReduce independent_isectElimination addEquality natural_numberEquality dependent_functionElimination voidElimination voidEquality dependent_set_memberEquality applyLambdaEquality setElimination rename setEquality lambdaEquality independent_functionElimination unionElimination dependent_pairFormation int_eqEquality computeAll instantiate cumulativity lambdaFormation impliesFunctionality imageElimination universeEquality imageMemberEquality equalityElimination promote_hyp

Latex:
\mforall{}[n,m,k:\mBbbZ{}].    uiff([n,  m)  =  [k];(m  =  (n  +  1))  \mwedge{}  (k  =  n))

Date html generated: 2017_04_17-AM-07_56_04
Last ObjectModification: 2017_02_27-PM-04_28_26

Theory : list_1

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