Nuprl Lemma : AF-induction-iff
∀T:Type. ∀R:T ⟶ T ⟶ ℙ.
((∀x,y:T. Dec(R+[x;y]))
⇒ (∃R':T ⟶ T ⟶ ℙ. (AFx,y:T.R'[x;y] ∧ (∀x,y:T. (R+[x;y] ⇒ (¬R'[x;y])))) ⇐⇒ ∀Q:T ⟶ ℙ. TI(T;x,y.R[x;y];t.Q[t])))
Proof
Definitions occuring in Statement :
rel_plus: R+,
almost-full: AFx,y:T.R[x; y],
TI: TI(T;x,y.R[x; y];t.Q[t]),
decidable: Dec(P),
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
not: ¬A,
subtype_rel: A ⊆r B,
false: False,
rev_implies: P ⇐ Q,
cand: A c∧ B,
almost-full: AFx,y:T.R[x; y],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
ge: i ≥ j ,
TI: TI(T;x,y.R[x; y];t.Q[t]),
le: A ≤ B,
less_than': less_than'(a;b),
squash: ↓T,
infix_ap: x f y,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
eq_int: (i =z j),
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
true: True
Lemmas referenced :
AF-induction4,
almost-full_wf,
rel_plus_wf,
subtype_rel_self,
istype-void,
not_wf,
TI_wf,
decidable_wf,
istype-universe,
istype-nat,
all_wf,
nat_wf,
equal_wf,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
squash_wf,
exists_wf,
nat_properties,
intformand_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
istype-false,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-less_than,
rel_plus_iff2,
eq_int_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
assert_wf,
bnot_wf,
equal-wf-base,
set_subtype_base,
le_wf,
int_subtype_base,
istype-assert,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
rel-star-iff-rel-plus,
true_wf,
iff_weakening_equal,
zero-add
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_pairFormation,
independent_isectElimination,
sqequalRule,
Error :lambdaEquality_alt,
applyEquality,
Error :universeIsType,
Error :functionIsType,
universeEquality,
Error :productIsType,
Error :inhabitedIsType,
isectElimination,
because_Cache,
instantiate,
Error :dependent_pairFormation_alt,
independent_functionElimination,
voidElimination,
functionExtensionality,
functionEquality,
cumulativity,
Error :dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
approximateComputation,
Error :isect_memberEquality_alt,
addEquality,
setElimination,
rename,
int_eqEquality,
imageElimination,
productElimination,
imageMemberEquality,
baseClosed,
Error :equalityIstype,
Error :setIsType,
applyLambdaEquality,
equalityTransitivity,
equalitySymmetry,
closedConclusion,
equalityElimination,
hyp_replacement,
intEquality,
baseApply,
sqequalBase
Latex:
\mforall{}T:Type. \mforall{}R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}.
((\mforall{}x,y:T. Dec(R\msupplus{}[x;y]))
{}\mRightarrow{} (\mexists{}R':T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}. (AFx,y:T.R'[x;y] \mwedge{} (\mforall{}x,y:T. (R\msupplus{}[x;y] {}\mRightarrow{} (\mneg{}R'[x;y]))))
\mLeftarrow{}{}\mRightarrow{} \mforall{}Q:T {}\mrightarrow{} \mBbbP{}. TI(T;x,y.R[x;y];t.Q[t])))
Date html generated:
2019_06_20-PM-02_02_20
Last ObjectModification:
2018_12_07-PM-06_37_22
Theory : relations2
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