### Nuprl Lemma : cWO-induction_1-ext

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  ∀[Q:T ⟶ ℙ]. TI(T;x,y.R[x;y];t.Q[t]) supposing cWO(T;x,y.R[x;y])`

Proof

Definitions occuring in Statement :  cWO: `cWO(T;x,y.R[x; y])` TI: `TI(T;x,y.R[x; y];t.Q[t])` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` so_apply: `x[s]` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  member: `t ∈ T` cWO-induction_1 basic_strong_bar_induction decidable__and2 decidable__lt decidable__squash uall: `∀[x:A]. B[x]` so_lambda: `so_lambda(x,y,z,w.t[x; y; z; w])` so_apply: `x[s1;s2;s3;s4]` top: `Top` uimplies: `b supposing a` strict4: `strict4(F)` and: `P ∧ Q` all: `∀x:A. B[x]` implies: `P `` Q` has-value: `(a)↓` prop: `ℙ` guard: `{T}` or: `P ∨ Q` squash: `↓T` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` decidable_functionality iff_preserves_decidability decidable__and decidable__less_than' decidable__assert isr: `isr(x)` subtract: `n - m` bfalse: `ff` any: `any x` btrue: `tt` squash_elim it: `⋅` seq-normalize: `seq-normalize(n;s)` ifthenelse: `if b then t else f fi `
Lemmas referenced :  cWO-induction_1 lifting-strict-spread has-value_wf_base base_wf is-exception_wf top_wf equal_wf lifting-strict-decide lifting-strict-less strict4-spread basic_strong_bar_induction decidable__and2 decidable__lt decidable__squash decidable_functionality iff_preserves_decidability decidable__and decidable__less_than' decidable__assert squash_elim
Rules used in proof :  introduction sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity cut instantiate extract_by_obid hypothesis sqequalRule thin sqequalHypSubstitution isectElimination baseClosed isect_memberEquality voidElimination voidEquality independent_isectElimination independent_pairFormation lambdaFormation callbyvalueApply baseApply closedConclusion hypothesisEquality applyExceptionCases inrFormation imageMemberEquality imageElimination exceptionSqequal inlFormation callbyvalueDecide equalityTransitivity equalitySymmetry unionEquality unionElimination sqleReflexivity dependent_functionElimination independent_functionElimination decideExceptionCases because_Cache

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].    \mforall{}[Q:T  {}\mrightarrow{}  \mBbbP{}].  TI(T;x,y.R[x;y];t.Q[t])  supposing  cWO(T;x,y.R[x;y])

Date html generated: 2017_04_14-AM-07_29_02
Last ObjectModification: 2017_02_27-PM-02_56_56

Theory : bar-induction

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