### Nuprl Lemma : cWO-induction_1

`∀[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 :  uall: `∀[x:A]. B[x]` uimplies: `b supposing a` member: `t ∈ T` cWO: `cWO(T;x,y.R[x; y])` all: `∀x:A. B[x]` squash: `↓T` TI: `TI(T;x,y.R[x; y];t.Q[t])` implies: `P `` Q` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` prop: `ℙ` so_apply: `x[s]` so_lambda: `λ2x y.t[x; y]` so_lambda: `so_lambda(x,y,z.t[x; y; z])` bfalse: `ff` ifthenelse: `if b then t else f fi ` assert: `↑b` outl: `outl(x)` true: `True` less_than': `less_than'(a;b)` le: `A ≤ B` top: `Top` subtract: `n - m` uiff: `uiff(P;Q)` false: `False` rev_implies: `P `` Q` not: `¬A` iff: `P `⇐⇒` Q` or: `P ∨ Q` decidable: `Dec(P)` lelt: `i ≤ j < k` int_seg: `{i..j-}` isl: `isl(x)` nat: `ℕ` and: `P ∧ Q` isr: `isr(x)` consistent-seq: `R-consistent-seq(n)` so_lambda: `λ2x.t[x]` btrue: `tt` cand: `A c∧ B` bnot: `¬bb` guard: `{T}` sq_type: `SQType(T)` exists: `∃x:A. B[x]` it: `⋅` unit: `Unit` bool: `𝔹` seq-add: `s.x@n` sq_stable: `SqStable(P)` less_than: `a < b`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  cut introduction sqequalRule sqequalHypSubstitution Error :lambdaEquality_alt,  dependent_functionElimination thin hypothesisEquality imageElimination hypothesis imageMemberEquality baseClosed Error :functionIsTypeImplies,  Error :inhabitedIsType,  rename Error :lambdaFormation_alt,  extract_by_obid isectElimination Error :functionIsType,  Error :setIsType,  Error :universeIsType,  applyEquality instantiate universeEquality setElimination because_Cache independent_functionElimination unionEquality Error :unionIsType,  Error :equalityIsType1,  Error :productIsType,  equalitySymmetry equalityTransitivity minusEquality Error :isect_memberEquality_alt,  addEquality independent_isectElimination voidElimination independent_pairFormation Error :dependent_set_memberEquality_alt,  productElimination unionElimination natural_numberEquality productEquality functionEquality setEquality closedConclusion voidEquality Error :inlEquality_alt,  multiplyEquality int_eqReduceFalseSq baseApply Error :equalityIsType4,  intEquality cumulativity promote_hyp Error :dependent_pairFormation_alt,  int_eqReduceTrueSq equalityElimination applyLambdaEquality hyp_replacement

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: 2019_06_20-AM-11_29_48
Last ObjectModification: 2018_10_12-AM-11_32_44

Theory : bar-induction

Home Index