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
Lemmas referenced :
istype-universe,
subtype_rel_self,
cWO_wf,
basic_strong_bar_induction,
unit_wf2,
int_seg_wf,
outl_wf,
le_wf,
le-add-cancel-alt,
zero-mul,
add-mul-special,
not-lt-2,
decidable__lt,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-minus,
istype-int,
minus-add,
nat_wf,
istype-void,
minus-one-mul-top,
zero-add,
minus-one-mul,
condition-implies-le,
less-iff-le,
not-le-2,
istype-false,
decidable__le,
subtract_wf,
bfalse_wf,
btrue_wf,
assert_wf,
less_than_wf,
consistent-seq_wf,
all_wf,
decidable__assert,
decidable__and2,
true_wf,
istype-less_than,
isl_wf,
not-equal-2,
int_subtype_base,
set_subtype_base,
assert_of_bnot,
iff_weakening_uiff,
equal_wf,
not_wf,
bnot_wf,
iff_transitivity,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
eq_int_wf,
istype-assert,
seq-add_wf,
minus-zero,
le-add-cancel2,
sq_stable__le,
add-subtract-cancel,
squash_wf,
iff_weakening_equal,
isr_wf
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
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