Nuprl Lemma : cons_succ

∀[T:Type]
  ∀l:T List
    ∀[P:T ⟶ ℙ]
      ∀a,x:T.
        (y = succ(x) in [a / l]
        ⇒ P[y]
        ⇒ ((P[hd(l)]) supposing (0 < ||l|| and (x = a ∈ T)) ∧ y = succ(x) in l⇒ P[y] supposing ¬(x = a ∈ T)))


Proof



Definitions occuring in Statement :  l_succ: l_succ,  length: ||as||,  hd: hd(l),  cons: [a / b],  list: T List,  less_than: a < b,  uimplies: b supposing a,  uall: ∀[x:A]. B[x],  prop: ℙ,  so_apply: x[s],  all: ∀x:A. B[x],  not: ¬A,  implies: P ⇒ Q,  and: P ∧ Q,  function: x:A ⟶ B[x],  natural_number: $n,  universe: Type,  equal: s = t ∈ T
Definitions unfolded in proof :  l_succ: l_succ,  uall: ∀[x:A]. B[x],  all: ∀x:A. B[x],  implies: P ⇒ Q,  and: P ∧ Q,  cand: A c∧ B,  uimplies: b supposing a,  member: t ∈ T,  prop: ℙ,  not: ¬A,  false: False,  nat: ℕ,  top: Top,  ge: i ≥ j ,  decidable: Dec(P),  or: P ∨ Q,  satisfiable_int_formula: satisfiable_int_formula(fmla),  exists: ∃x:A. B[x],  less_than: a < b,  squash: ↓T,  uiff: uiff(P;Q),  so_apply: x[s],  subtype_rel: A ⊆r B,  select: L[n],  cons: [a / b],  subtract: n - m,  le: A ≤ B,  less_than': less_than'(a;b),  nil: [],  it: ⋅,  so_lambda: λ2x y.t[x; y],  so_apply: x[s1;s2],  guard: {T},  iff: P ⇐⇒ Q
Lemmas referenced :  member-less_than,  length_wf,  less_than_wf,  select_wf,  length_of_cons_lemma,  istype-void,  nat_properties,  decidable__le,  full-omega-unsat,  intformand_wf,  intformnot_wf,  intformle_wf,  itermConstant_wf,  itermVar_wf,  istype-int,  int_formula_prop_and_lemma,  int_formula_prop_not_lemma,  int_formula_prop_le_lemma,  int_term_value_constant_lemma,  int_term_value_var_lemma,  int_formula_prop_wf,  decidable__lt,  intformless_wf,  itermAdd_wf,  int_formula_prop_less_lemma,  int_term_value_add_lemma,  nat_wf,  not_wf,  equal_wf,  cons_wf,  add-is-int-iff,  false_wf,  istype-universe,  list_wf,  list-cases,  length_of_nil_lemma,  istype-false,  product_subtype_list,  le_wf,  stuck-spread,  istype-base,  reduce_hd_cons_lemma,  select-cons-tl,  add-subtract-cancel,  select_cons_tl,  subtype_rel_self,  iff_weakening_equal
Rules used in proof :  sqequalSubstitution,  sqequalRule,  sqequalReflexivity,  sqequalTransitivity,  computationStep,  isect_memberFormation_alt,  lambdaFormation_alt,  cut,  introduction,  axiomEquality,  hypothesis,  thin,  rename,  extract_by_obid,  sqequalHypSubstitution,  isectElimination,  natural_numberEquality,  hypothesisEquality,  independent_isectElimination,  universeIsType,  equalityIsType1,  inhabitedIsType,  independent_pairFormation,  lambdaEquality_alt,  dependent_functionElimination,  voidElimination,  functionIsTypeImplies,  setElimination,  because_Cache,  isect_memberEquality_alt,  unionElimination,  approximateComputation,  independent_functionElimination,  dependent_pairFormation_alt,  int_eqEquality,  imageElimination,  productElimination,  addEquality,  functionIsType,  pointwiseFunctionality,  equalityTransitivity,  equalitySymmetry,  promote_hyp,  baseApply,  closedConclusion,  baseClosed,  applyEquality,  universeEquality,  dependent_set_memberEquality_alt,  hypothesis_subsumption,  instantiate
Latex:
\mforall{}[T:Type]
    \mforall{}l:T  List
        \mforall{}[P:T  {}\mrightarrow{}  \mBbbP{}]
            \mforall{}a,x:T.
                (y  =  succ(x)  in  [a  /  l]
                {}\mRightarrow{}  P[y]
                {}\mRightarrow{}  ((P[hd(l)])  supposing  (0  <  ||l||  and  (x  =  a))  \mwedge{}  y  =  succ(x)  in  l{}\mRightarrow{}  P[y]  supposing  \mneg{}(x  =  a\000C)))


Date html generated: 2019_10_15-AM-10_53_20
Last ObjectModification: 2018_10_09-AM-09_54_24

Theory : list!


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