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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