Nuprl Lemma : sqle-list_ind

        F[rec-case(as) of
          [] => b1
          h::t =>
           r.H[h;t;r]] ≤ G[rec-case(as) of
                           [] => b2
                           h::t =>
        supposing F[b1] ≤ G[b2] 
      supposing ∀x,y,r1,r2:Base.  ((F[r1] ≤ G[r2])  (F[H[x;y;r1]] ≤ G[J[x;y;r2]])) 
    supposing strict1(λx.G[x]) 
  supposing strict1(λx.F[x])


Definitions occuring in Statement :  list_ind: list_ind strict1: strict1(F) uimplies: supposing a uall: [x:A]. B[x] so_apply: x[s1;s2;s3] so_apply: x[s] all: x:A. B[x] implies:  Q lambda: λx.A[x] base: Base sqle: s ≤ t
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a all: x:A. B[x] prop: so_lambda: λ2x.t[x] implies:  Q so_apply: x[s] nat: false: False ge: i ≥  guard: {T} top: Top strict1: strict1(F) and: P ∧ Q cand: c∧ B not: ¬A squash: T or: P ∨ Q decidable: Dec(P) iff: ⇐⇒ Q rev_implies:  Q uiff: uiff(P;Q) subtract: m subtype_rel: A ⊆B le: A ≤ B less_than': less_than'(a;b) true: True nat_plus: + has-value: (a)↓ pi1: fst(t) pi2: snd(t) so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]) so_apply: x[s1;s2;s3;s4] list_ind: list_ind
Lemmas referenced :  le_wf le_weakening2 fixpoint-upper-bound bottom-sqle lifting-strict-isaxiom lifting-strict-ispair lifting-strict-callbyvalue fun_exp_unroll_1 has-value-implies-dec-isaxiom-2 top_wf has-value-implies-dec-ispair-2 int_subtype_base cbv_sqle strict1-strict4 nat_wf le-add-cancel add-zero add_functionality_wrt_le add-commutes add-swap add-associates minus-minus minus-add minus-one-mul-top zero-add minus-one-mul condition-implies-le less-iff-le not-ge-2 false_wf subtract_wf decidable__le is-exception_wf has-value_wf_base exception-not-bottom bottom_diverge strictness-apply fun_exp0_lemma less_than_wf ge_wf less_than_irreflexivity less_than_transitivity1 nat_properties strict1_wf all_wf base_wf sqle_wf_base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation axiomSqleEquality hypothesis lemma_by_obid sqequalHypSubstitution isectElimination thin sqequalRule baseApply closedConclusion baseClosed hypothesisEquality lambdaEquality dependent_functionElimination isect_memberEquality because_Cache equalityTransitivity equalitySymmetry functionEquality setElimination rename intWeakElimination natural_numberEquality independent_isectElimination independent_functionElimination voidElimination voidEquality productElimination divergentSqle imageElimination unionElimination independent_pairFormation addEquality applyEquality intEquality minusEquality comment dependent_set_memberEquality callbyvalueIspair ispairExceptionCases exceptionSqequal sqleReflexivity sqleTransitivity fixpointLeast sqleRule

                F[rec-case(as)  of
                    []  =>  b1
                    h::t  =>
                      r.H[h;t;r]]  \mleq{}  G[rec-case(as)  of
                                                      []  =>  b2
                                                      h::t  =>
                supposing  F[b1]  \mleq{}  G[b2] 
            supposing  \mforall{}x,y,r1,r2:Base.    ((F[r1]  \mleq{}  G[r2])  {}\mRightarrow{}  (F[H[x;y;r1]]  \mleq{}  G[J[x;y;r2]])) 
        supposing  strict1(\mlambda{}x.G[x]) 
    supposing  strict1(\mlambda{}x.F[x])

Date html generated: 2016_05_14-AM-06_27_22
Last ObjectModification: 2016_01_14-PM-08_27_12

Theory : list_0

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