Nuprl Lemma : AF-uniform-induction
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(∀[Q:T ⟶ ℙ]. uniform-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],
uniform-TI: uniform-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,
uniform-TI: uniform-TI(T;x,y.R[x; y];t.Q[t]),
nat: ℕ,
decidable: Dec(P),
consistent-seq: R-consistent-seq(n),
so_lambda: λ2x.t[x],
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
so_lambda: λ2x y.t[x; y],
AFbar: AFbar(),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
top: Top,
true: True,
isr: isr(x),
assert: ↑b,
bfalse: ff,
exposed-it: exposed-it,
bool: 𝔹,
unit: Unit,
it: ⋅,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
ge: i ≥ j ,
nequal: a ≠ b ∈ T ,
int_upper: {i...},
sq_stable: SqStable(P),
AF-spread-law: AF-spread-law(x,y.R[x; y]),
cand: A c∧ B,
outl: outl(x),
isl: isl(x),
less_than: a < b
Lemmas referenced :
not_wf,
nat_wf,
AF-spread-law_wf,
AFbar_wf,
unit_wf2,
int_seg_wf,
decidable__AFbar,
all_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_dep_function,
subtype_rel_sets,
and_wf,
le_wf,
less_than_wf,
less_than_transitivity2,
le_weakening2,
subtype_rel_self,
AF-path-barred,
uall_wf,
almost-full_wf,
subtract_wf,
decidable__le,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
decidable__lt,
not-lt-2,
add-mul-special,
zero-mul,
le-add-cancel-alt,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
equal-wf-T-base,
iff_weakening_uiff,
assert_of_bnot,
int_upper_subtype_nat,
nat_properties,
nequal-le-implies,
true_wf,
sq_stable__le,
le_antisymmetry_iff,
not-equal-2,
set_wf,
isl_wf,
int_upper_wf,
le-add-cancel2,
outl_wf,
int_subtype_base,
set_subtype_base,
decidable__int_equal
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,
isect_memberEquality,
because_Cache,
strong_bar_Induction,
unionEquality,
natural_numberEquality,
setElimination,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
intEquality,
setEquality,
productElimination,
independent_functionElimination,
unionElimination,
addEquality,
voidEquality,
minusEquality,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
impliesFunctionality,
hypothesis_subsumption,
axiomEquality,
int_eqReduceTrueSq,
int_eqReduceFalseSq,
multiplyEquality,
inlEquality,
productEquality,
isectEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(\mforall{}[Q:T {}\mrightarrow{} \mBbbP{}]. uniform-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_28_07
Last ObjectModification:
2017_02_27-PM-02_58_25
Theory : bar-induction
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