Nuprl Lemma : AF-induction
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(∀[Q:T ⟶ ℙ]. TI(T;x,y.¬R[x;y];t.Q[t])) supposing
(AFx,y:T.R[x;y] and
(∀x,y,z:T. ((¬R[x;y]) ⇒ (¬R[y;z]) ⇒ (¬R[x;z]))))
Proof
Definitions occuring in Statement :
almost-full: AFx,y:T.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],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
prop: ℙ,
almost-full: AFx,y:T.R[x; y],
squash: ↓T,
TI: TI(T;x,y.R[x; y];t.Q[t]),
so_lambda: λ2x y.t[x; y],
consistent-seq: R-consistent-seq(n),
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
seq-add: s.x@n,
AF-spread-law: AF-spread-law(x,y.R[x; y]),
isl: isl(x),
outl: outl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
and: P ∧ Q,
cand: A c∧ B,
true: True,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
lelt: i ≤ j < k,
le: A ≤ B,
decidable: Dec(P),
less_than': less_than'(a;b),
subtract: n - m,
sq_stable: SqStable(P),
top: Top,
less_than: a < b
Lemmas referenced :
not_wf,
nat_wf,
AF-spread-law_wf,
AFbar_wf,
basic_strong_bar_induction,
unit_wf2,
all_wf,
consistent-seq_wf,
decidable__AFbar,
at_AFbar,
AF-path-barred,
almost-full_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
iff_weakening_uiff,
assert_of_bnot,
outl_wf,
decidable__lt,
false_wf,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-swap,
add-commutes,
le-add-cancel,
less-iff-le,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
minus-one-mul-top,
zero-add,
le-add-cancel2,
int_seg_wf,
true_wf,
decidable__le,
not-le-2,
sq_stable__le,
add-zero,
le_wf,
seq-add_wf,
set_wf,
and_wf,
less_than_wf,
less_than_transitivity1,
less_than_irreflexivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
applyEquality,
functionExtensionality,
cumulativity,
hypothesis,
universeEquality,
extract_by_obid,
isectElimination,
rename,
imageElimination,
imageMemberEquality,
baseClosed,
functionEquality,
lambdaFormation,
because_Cache,
unionEquality,
equalityTransitivity,
equalitySymmetry,
setEquality,
setElimination,
inlEquality,
independent_functionElimination,
dependent_set_memberEquality,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
int_eqReduceTrueSq,
natural_numberEquality,
independent_pairFormation,
dependent_pairFormation,
promote_hyp,
instantiate,
intEquality,
impliesFunctionality,
int_eqReduceFalseSq,
addEquality,
minusEquality,
isect_memberEquality,
voidEquality,
productEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(\mforall{}[Q:T {}\mrightarrow{} \mBbbP{}]. TI(T;x,y.\mneg{}R[x;y];t.Q[t])) supposing
(AFx,y:T.R[x;y] and
(\mforall{}x,y,z:T. ((\mneg{}R[x;y]) {}\mRightarrow{} (\mneg{}R[y;z]) {}\mRightarrow{} (\mneg{}R[x;z]))))
Date html generated:
2017_04_14-AM-07_27_57
Last ObjectModification:
2017_02_27-PM-02_57_29
Theory : bar-induction
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