Theory "defCNF"

Parents rich_list

Signature

Constant	Type
DEF	:(num -> bool) -> num -> (bool reln, (num + bool) # (num + bool)) alist -> bool
OK	:num -> bool reln # (num + bool) # (num + bool) -> bool
OKDEF	:num -> (bool reln, (num + bool) # (num + bool)) alist -> bool
OK_tupled	:num # bool reln # (num + bool) # (num + bool) -> bool
UNIQUE	:(num -> bool) -> num -> bool reln # (num + bool) # (num + bool) -> bool
UNIQUE_tupled	:(num -> bool) # num # bool reln # (num + bool) # (num + bool) -> bool

Definitions

UNIQUE_tupled_primitive_def

|- UNIQUE_tupled =
   WFREC (@R. WF R)
     (λUNIQUE_tupled a'.
        case a' of
          (v,n,conn,INL i,INL j) => I (v n ⇔ conn (v i) (v j))
        | (v,n,conn,INL i,INR b) => I (v n ⇔ conn (v i) b)
        | (v,n,conn,INR a,INL j') => I (v n ⇔ conn a (v j'))
        | (v,n,conn,INR a,INR b') => I (v n ⇔ conn a b'))

UNIQUE_curried_def

|- ∀x x1 x2. defCNF$UNIQUE x x1 x2 ⇔ UNIQUE_tupled (x,x1,x2)

DEF_def

|- (∀v n. defCNF$DEF v n [] ⇔ T) ∧
   ∀v n x xs.
     defCNF$DEF v n (x::xs) ⇔ defCNF$UNIQUE v n x ∧ defCNF$DEF v (SUC n) xs

OK_tupled_primitive_def

|- OK_tupled =
   WFREC (@R. WF R)
     (λOK_tupled a'.
        case a' of
          (n,conn,INL i,INL j) => I (i < n ∧ j < n)
        | (n,conn,INL i,INR b) => I (i < n)
        | (n,conn,INR a,INL j') => I (j' < n)
        | (n,conn,INR a,INR b') => I T)

OK_curried_def

|- ∀x x1. defCNF$OK x x1 ⇔ OK_tupled (x,x1)

OKDEF_def

|- (∀n. defCNF$OKDEF n [] ⇔ T) ∧
   ∀n x xs. defCNF$OKDEF n (x::xs) ⇔ defCNF$OK n x ∧ defCNF$OKDEF (SUC n) xs

Theorems

UNIQUE_ind

|- ∀P.
     (∀v n conn i j. P v n (conn,INL i,INL j)) ∧
     (∀v n conn i b. P v n (conn,INL i,INR b)) ∧
     (∀v n conn a j. P v n (conn,INR a,INL j)) ∧
     (∀v n conn a b. P v n (conn,INR a,INR b)) ⇒
     ∀v v1 v2 v3 v4. P v v1 (v2,v3,v4)

UNIQUE_def

|- (defCNF$UNIQUE v n (conn,INL i,INL j) ⇔ (v n ⇔ conn (v i) (v j))) ∧
   (defCNF$UNIQUE v n (conn,INL i,INR b) ⇔ (v n ⇔ conn (v i) b)) ∧
   (defCNF$UNIQUE v n (conn,INR a,INL j) ⇔ (v n ⇔ conn a (v j))) ∧
   (defCNF$UNIQUE v n (conn,INR a,INR b) ⇔ (v n ⇔ conn a b))

OK_ind

|- ∀P.
     (∀n conn i j. P n (conn,INL i,INL j)) ∧
     (∀n conn i b. P n (conn,INL i,INR b)) ∧
     (∀n conn a j. P n (conn,INR a,INL j)) ∧
     (∀n conn a b. P n (conn,INR a,INR b)) ⇒
     ∀v v1 v2 v3. P v (v1,v2,v3)

OK_def

|- (defCNF$OK n (conn,INL i,INL j) ⇔ i < n ∧ j < n) ∧
   (defCNF$OK n (conn,INL i,INR b) ⇔ i < n) ∧
   (defCNF$OK n (conn,INR a,INL j) ⇔ j < n) ∧
   (defCNF$OK n (conn,INR a,INR b) ⇔ T)

DEF_SNOC

|- ∀n x l v.
     defCNF$DEF v n (SNOC x l) ⇔
     defCNF$DEF v n l ∧ defCNF$UNIQUE v (n + LENGTH l) x

OKDEF_SNOC

|- ∀n x l.
     defCNF$OKDEF n (SNOC x l) ⇔ defCNF$OKDEF n l ∧ defCNF$OK (n + LENGTH l) x

CONSISTENCY

|- ∀n l. defCNF$OKDEF n l ⇒ ∃v. defCNF$DEF v n l

BIGSTEP

|- ∀P Q R. (∀v. P v ⇒ (Q ⇔ R v)) ⇒ ((∃v. P v) ∧ Q ⇔ ∃v. P v ∧ R v)

FINAL_DEF

|- ∀v n x. (v n ⇔ x) ⇔ (v n ⇔ x) ∧ defCNF$DEF v (SUC n) []