;====================================================================
;
;       examples for testing the intuitionistic prover
;
;===================================================================

(define ex
  '((frm (-> (- (& (a c) (b c))) 
	     (v (- (a c)) (- (b c))) ))))

(define ex1
  '((frm (-> (- (- (a c))) (a c)))) )


(define lo1
  '((frm
      (-> (& 
	     (all x y (- (& (g x y) (g y x))))
	     (all x y z (-> (g x y) (v (g x z) (g z y))))
	     )
	  (all x (- (g x x))) ))))


(define lo2
  '((frm
      (-> (& 
	     (all x y (- (& (g x y) (g y x))))
	     (all x y z (-> (g x y) (v (g x z) (g z y))))
	     )
	  (all x y (-> (g x y) (- (g y x)))) ))))

(define lo3
  '((frm
      (-> (& 
	     (all x y (- (& (g x y) (g y x))))
	     (all x y z (-> (g x y) (v (g x z) (g z y))))
	     )
	  (all x y z (-> (& (g x y) (g y z)) (g x z))) ))))


(define lo4
  '((frm
      (-> (& 
	     (all x y (- (& (g x y) (g y x))))
	     (all x y z (-> (g x y) (v (g x z) (g z y))))
	     )
	  (all x (v (- (g x x)) (g x x))) ))))

(define lo5
  '((frm
      (-> (& 
	     (all x y (- (& (g x y) (g y x))))
	     (all x y z (-> (g x y) (v (g x z) (g z y))))
	     )
	  (all x y z (->  (v (g x y) (g y x))
			  (v (v (g x z) (g z x))
			     (v (g y z) (g z y))))) ))))

(define lo6
  '((frm
      (-> (& 
	     (all x y (- (& (g x y) (g y x))))
	     (all x y z (-> (g x y) (v (g x z) (g z y))))
	     )
	  (all x y (-> (g x y) (- (g y x)))) ))))


(define lo7
  '((frm
      (-> (& 
	     (all x y (- (& (g x y) (g y x))))
	     (all x y z (-> (g x y) (v (g x z) (g z y))))
	     )
	  (all x y z (-> (& (g x y) (- (g z y))) (g x z)))))))


;(define lo8
;  '((frm
;      (-> (& 
;	     (all x y (- (& (g x y) (g y x))))
;	     (all x y z (-> (g x y) (v (g x z) (g z y))))
;	     )
;	  (-> (- (- (g a b))) (g a b)) ))))


;(define lo8
;  '((frm
;      (-> (& 
;	     (all x y (- (& (g x y) (g y x))))
;	     (all x y z (-> (g x y) (v (g x z) (g z y))))
;	     (all x (- (g 0 x)))
;	     (all x (- (g x 1))) 
;	     (
;	     )
;	  (-> (- (- (g a b))) (g a b)) ))))


(define lom1
  '((frm
      (->
        (&
	 (all x y z (->  (v (g x y) (g y x))
			 (v (v (g x z) (g z x))
			    (v (g y z) (g z y)))))
	 (all x y z (-> (& (g x y) (g y z)) (g x z))) )
	(all x y z (-> (g x y) (v (g x z) (g z y)))) ))))


(define ax1
  '( (all x (- (dipt x x)))
     (all x (- (diln x x)))
     (all x (- (con x x)))
     (all x y z
	  (-> (dipt x y) (v (dipt x z) (dipt y z))) ) 
     (all x y z
	  (-> (diln x y) (v (diln x z) (diln y z))) )
     (all x y z
	  (-> (con x y) (v (con x z) (con y z))) )))

(define ax2
  '(  (all x y
	   (-> (dipt x y) (- (apt x (ln x y)))) )
      (all x y
	   (-> (dipt x y) (- (apt y (ln x y)))) )
      (all x y
	   (-> (con x y) (- (apt (pt x y) x))) )
      (all x y
	   (-> (con x y) (- (apt (pt x y) y))) )))

(define ax3
  '(    (all x y u v
	     (-> (& (dipt x y) (diln u v))
		 (v (v (apt x u) (apt x v))
		    (v (apt y u) (apt y v)) )))))

(define ax4
  '(   (all x y z 
	    (-> (apt x y) (v (dipt x z) (apt z y))) )
       (all x y z 
	    (-> (apt x y) (v (diln y z) (apt x z))) )
       (all x y z 
	    (-> (con x y) (v (diln y z) (con x z))) )))

; 3.1 proved in 158 sec

(define thm3.1
  `((frm
      (-> (& 
	   ,@ax1 
	   ,@ax2 
	   ,@ax3 
	   ,@ax4) 
	  (all x y u
	     (-> (& (dipt x y) (& (- (apt x u)) (- (apt y u))))
		 (- (diln u (ln x y))) )) ))))


; 3.1 proved in 158 sec

(define thm3.1.1
  `((frm
      (-> (& 
	   ;,@ax1 
	   ,@ax2 
	   ,@ax3 )
	   ;,@ax4) 
	  (all x y u
	     (-> (& (dipt x y) (& (- (apt x u)) (- (apt y u))))
		 (- (diln u (ln x y))) )) ))))

; 3.2 proved in 188 sec

(define thm3.2
  `((frm
      (-> (& 
	   ,@ax1 
	   ,@ax2 
	   ,@ax3 
	   ,@ax4) 
	  (all x y z
	       (-> (& (con x y) (- (apt z x)) (- (apt z y)))
		   (- (dipt z (pt x y))) )) ))))
 
; 3.2.1 proved in 1.5 sec

(define thm3.2.1
  `((frm
      (-> (& 
	   ,@ax1 
	   ,@ax2 
	   ,@ax3 
	   ,@ax4
	   (all x y (-> (con x y) (diln x y)))
	   ) 
	  (all x y z
	       (-> (& (con x y) (- (apt z x)) (- (apt z y)))
		   (- (dipt z (pt x y))) )) ))))
 

(define thm3.2.2
  `((frm
      (-> (& 
	   ,@ax1 
	   ;,@ax2 
	   ;,@ax3 
	   ,@ax4
	   (all x y (-> (con x y) (diln x y)))
	   ) 
	  (all x y z
	       (-> (& (con x y) (- (apt z x)) (- (apt z y)))
		   (- (dipt z (pt x y))) )) ))))
 

; 3.3 proved in 1.3 sec

(define thm3.3
  `((frm
      (-> (& 
	   ,@ax1 
	   ,@ax2 
	   ,@ax3 
	   ,@ax4) 
	  (all x y (-> (con x y) (diln x y))) ))))

; thm4.1.i.1 proved in 1.3 sec

(define thm4.1.i.1
  `((frm
      (-> (& 
	   ; ,@ax1 
	   ,@ax2 
	   ,@ax3 
	   ; ,@ax4
           (dipt a b)
           (con l m) 
           (diln l (ln a b)) ) 
          (v (apt a l) (apt b l)) ))))
      
; thm4.1.i.1-full not proved in ca 18 hours.

(define thm4.1.i.1-full
  `((frm
      (-> (& 
	   ,@ax1 
	   ,@ax2 
	   ,@ax3 
	   ,@ax4
           (dipt a b)
           (con l m) 
           (diln l (ln a b)) ) 
          (v (apt a l) (apt b l)) ))))


; thm4.1.i-full proved in ...

(define thm4.1.i-full
  `((frm
      (-> (& 
           ,@ax1 
           ,@ax2 
           ,@ax3 
           ,@ax4
           (dipt a b)
           (con l m) 
           (v (apt a l) (apt b l)) )
          (diln l (ln a b)) ))))    


; thm4.1.i.2 proved in 1.58 sec

(define thm4.1.i.2
  `((frm
      (-> (& 
	   ; ,@ax1 
	   ,@ax2 
	   ; ,@ax3 
	   ,@ax4
           (dipt a b)
           (con l m) 
	   (v (apt a l) (apt b l)) )
          (diln l (ln a b)) ))))    


; thm4.1.ii.1 proved in 1.8 seconds

(define thm4.1.ii.1   
  `((frm
      (-> (& 
	   ; ,@ax1 
	   ,@ax2 
	   ,@ax3 
	   ; ,@ax4
           (dipt a b)
           (con l m)
	   (diln l m)
           (dipt a (pt l m)) ) 
          (v (apt a l) (apt a m)) ))))
      
; thm4.1.ii.2 proved in 1.3 seconds

(define thm4.1.ii.2
  `((frm
      (-> (& 
	   ; ,@ax1 
	   ,@ax2 
	   ; ,@ax3 
	   ,@ax4
           (dipt a b)
           (con l m) 
	   (v (apt a l) (apt a m)) )
          (dipt a (pt l m)) ))))    
          
; thm4.2 proof not found (too hard)

(define thm4.2
  `((frm
      (-> (& 
	   ,@ax1 
	   ,@ax2 
	   ,@ax3 
	   ,@ax4
           (dipt a b)
           (dipt c d) )
        (-> (v (apt a (ln c d)) (apt b (ln c d)))
	    (v (apt c (ln a b)) (apt d (ln a b))) )))))

; thm4.2.a shown not provable in 1.2 sec

(define thm4.2.a
  `((frm
      (-> (& 
	  ; ,@ax1 
	  ; ,@ax2 
	  ; ,@ax3 
	  ; ,@ax4
	  (all x (-> (diln x (ln a b)) (v (apt a x) (apt b x))))
	  (all x (-> (v (apt a x) (apt b x)) (diln x (ln a b))))
	  (-> (diln (ln c d) (ln a b)) (diln (ln a b) (ln c d))) )
	  ; (dipt a b)
          ; (dipt c d)  
        (-> (v (apt a (ln c d)) (apt b (ln c d)))
	    (v (apt c (ln a b)) (apt d (ln a b))) )))))


; thm4.2.b proved in 64 sec
;  uses lemma 4.1.i

(define thm4.2.b
  `((frm
      (-> (& 
	   ,@ax1 
	   ;,@ax2 
	   ;,@ax3 
	   ;,@ax4
	  (all y u
	     (-> (dipt y u)  
		 (&  
		  (all x (-> (diln x (ln y u)) (v (apt y x) (apt u x))))
		  (all x (-> (v (apt y x) (apt u x)) (diln x (ln y u)))) )))
          ;(all x y u v
	  ;     (-> (diln (ln x y) (ln u v)) (diln (ln u v) (ln x y))) )
	  (dipt a b)
          (dipt c d)  )
        (-> (v (apt a (ln c d)) (apt b (ln c d)))
	    (v (apt c (ln a b)) (apt d (ln a b))) )))))

; thm4.3.i proved in 37 sec. 

(define thm4.3.i 
  `((frm
      (-> (& 
	   ,@ax1 
	   ,@ax2 
	   ,@ax3 
	   ,@ax4
           (dipt a b)
           (apt c (ln a b)) )
          (& (dipt c a) (dipt c b)) ))))


; thm4.3.i.b proved in 1 sec. 

(define thm4.3.i.b ; ax1 and ax3 unnecessary
  `((frm
      (-> (& 
	   ; ,@ax1 
	   ,@ax2 
	   ; ,@ax3 
	   ,@ax4
           (dipt a b)
           (apt c (ln a b)) )
          (& (dipt c a) (dipt c b)) ))))


;  thm4.3.ii proved in ????

(define thm4.3.ii 
  `((frm
      (-> (& 
	   ,@ax1 
	   ,@ax2 
	   ,@ax3 
	   ,@ax4
           (dipt a b)
           (apt c (ln a b)) )
          (& (diln (ln a b) (ln c a)) 
	     (diln (ln a b) (ln c b)) )))))

;  thm4.3.ii.b proved in 1.4 sec

(define thm4.3.ii.b ; ax1 and ax3 unnecessary (important!)
  `((frm
      (-> (& 
	  ; ,@ax1 
	   ,@ax2 
	  ; ,@ax3 
	   ,@ax4
           (dipt a b)
           (apt c (ln a b)) )
          (& (diln (ln a b) (ln c a)) 
	     (diln (ln a b) (ln c b)) )))))


(define thm4.4.i ; no success without extra assumptions
  `((frm
      (-> (& 
	   ,@ax1 
	 ;  ,@ax2 
	   ,@ax3 
	 ;  ,@ax4
	   (dipt a b)
	   ; success with the following three added:
	   ;(- (apt a (ln a b)))
	   ;(- (apt b (ln a b)))
	   ;(- (apt b (ln c b)))
           ; following two come from ax2:
	   (all x y
		(-> (dipt x y) (- (apt x (ln x y)))) )
	   (all x y
		(-> (dipt x y) (- (apt y (ln x y)))) )
	   ; (dipt c a) ; following four axioms: lemma 4.3, one MP applied.
	   (dipt c b)
	   ; (diln (ln a b) (ln c a)) 
           (diln (ln a b) (ln c b)) 

	   (apt c (ln a b)) )
	  (apt a (ln c b)) ))))

(define thm4.4.i.test
  `((frm
      (-> (& 
	 ;  ,@ax1 
	 ;  (all x (- (dipt x x)))
	 ;  ,@ax2 
	   ,@ax3 
	 ;  ,@ax4
	   (dipt a b)
	   ; success with the following three added:
	   ;(- (apt a (ln a b)))
	   ;(- (apt b (ln a b)))
	   ;(- (apt b (ln c b)))
           ; following two come from ax2:
	   (all x y
	   	(-> (dipt x y) (- (apt x (ln x y)))) )
	   (all x y
	   	(-> (dipt x y) (- (apt y (ln x y)))) )
	   ; following is thm4.3.i:
           (all x y z
	   	(-> (& (dipt x y)
	   	       (apt z (ln x y)) )
	   	    (& (dipt z x) (dipt z y)) ))
	   ; following is thm4.3.ii:
           (all x y z
	   	(-> (& (dipt x y)
	   	       (apt z (ln x y)) )
	   	    (& (diln (ln x y) (ln z x))
	   	       (diln (ln x y) (ln z y)) )))
	   
	   ;(dipt c a) ; following four axioms: lemma 4.3, one MP applied.
	   ;(dipt c b)
	   (diln (ln a b) (ln c a)) 
           (diln (ln a b) (ln c b)) 

	   (apt c (ln a b)) )
	  (apt a (ln c b)) ))))


(define thm4.4.i.hyper
  `(
    (sos ((dipt a b))
    	 ((apt c (ln a b)))
    ;	 ; ((diln (ln a b) (ln c a))) 
    ;	 ; ((diln (ln a b) (ln c b)))
    	 ((-dipt x y)(-apt z (ln x y)) (dipt z x))
    	 ((-dipt x y)(-apt z (ln x y)) (dipt z y))
   	 ((-dipt x y)(-apt z (ln x y)) (diln (ln x y) (ln z x)))
   	 ((-dipt x y)(-apt z (ln x y)) (diln (ln x y) (ln z y))) )
    (frm
      (-> (& ,@ax3
	     
	     (dipt a b)
	     (apt c (ln a b))

	  ;   (dipt c a)
	  ;   (dipt c b)
	  ;   (diln (ln a b) (ln c a)) 
	  ;   (diln (ln a b) (ln c b))

	 ;  ,@ax1 
	 ;  (all x (- (dipt x x)))
	 ;  ,@ax2 
	 ;  ,@ax3 
	 ;  ,@ax4
	   ; (dipt a b)
	   ; success with the following three added:
	   ;(- (apt a (ln a b)))
	   ;(- (apt b (ln a b)))
	   ;(- (apt b (ln c b)))
           ; following two come from ax2:

	   (all x y
	   	(-> (dipt x y) (- (apt x (ln x y)))) )
	   (all x y
	   	(-> (dipt x y) (- (apt y (ln x y)))) ) )

	   ; following is thm4.3.i:
           ;(all x y z
	   ;	(-> (& (dipt x y)
	   ;	       (apt z (ln x y)) )
	   ;	    (& (dipt z x) (dipt z y)) ))
	   ; following is thm4.3.ii:
           ;(all x y z
	   ;	(-> (& (dipt x y)
	   ;	       (apt z (ln x y)) )
	   ;	    (& (diln (ln x y) (ln z x))
	   ;	       (diln (ln x y) (ln z y)) )))
	   
	   ;(dipt c a) ; following four axioms: lemma 4.3, one MP applied.
	   ;(dipt c b)
	   ;(diln (ln a b) (ln c a)) 
           ;(diln (ln a b) (ln c b)) 

	   ; (apt c (ln a b)) )

	  (apt a (ln c b)) ))))

(define thm4.4.ii ; no success
  `((frm
      (-> (& 
	   ,@ax1 
	   ,@ax2 
	   ,@ax3 
	   ,@ax4
	   (dipt a b)
	   (dipt c a) ; following four axioms: lemma 4.3, one MP applied.
	   (dipt c b)
           (diln (ln a b) (ln c a)) 
           (diln (ln a b) (ln c b)) 
	   (apt c (ln a b)) )
	  (apt b (ln a c)) ))))

(define thm4.4.iii ; no success
  `((frm
      (-> (& 
	   ,@ax1 
	   ,@ax2 
	   ,@ax3 
	   ,@ax4
	   (dipt a b)
	   (dipt c a) ; following four axioms: lemma 4.3, one MP applied.
	   (dipt c b)
           (diln (ln a b) (ln c a)) 
           (diln (ln a b) (ln c b)) 
	   (apt c (ln a b)) )
	  (apt c (ln b a)) ))))

; lemma 4.1:
;(all y u
;	(-> (dipt y u)  
;	    (&  
;	     (all x (-> (diln x (ln y u)) (v (apt y x) (apt u x))))
;	     (all x (-> (v (apt y x) (apt u x)) (diln x (ln y u)))) )))
;  (all y u
;	(-> (con y u)  
;	    (&  
;	     (all x (-> (dipt x (pt y u)) (v (apt x y) (apt x u))))
;	     (all x (-> (v (apt x y) (apt x u)) (dipt x (pt y u)))) )))


; thm4.5.i1 proved in ca 12.3 sec

(define thm4.5.i1 
  `((frm
      (-> (& 
	   ,@ax1 
	   ,@ax2 
	   ,@ax3 
	   ,@ax4
	   (dipt a b)
           (con l m)
	   (- (apt a l))
	   (- (apt b l))
	 
	   )
	 (- (diln l (ln a b))) ))))  

; thm thm4.5.i2 proved in 0.6 sec	

(define thm4.5.i2  
  `((frm
      (-> (& 
	   ,@ax1 
	   ,@ax2 
	   ,@ax3 
	   ,@ax4
	   (dipt a b)
           (con l m)
           (- (diln l (ln a b)))
	   )
	  (& (- (apt a l))
	     (- (apt b l)) )))))


; thm4.5.ii1 proved in ca 12.6 sec

(define thm4.5.ii1 
  `((frm
      (-> (& 
	   ,@ax1 
	   ,@ax2 
	   ,@ax3 
	   ,@ax4
	   (dipt a b)
           (con l m)
	   (- (apt a l))
	   (- (apt a m))
	   )
	 (- (dipt a (pt l m))) ))))  

; thm thm4.5.ii2 proved in ca 1 second

(define thm4.5.ii2  
  `((frm
      (-> (& 
	   ,@ax1 
	   ,@ax2 
	   ,@ax3 
	   ,@ax4
	   (dipt a b)
           (con l m)
           (- (dipt a (pt l m)))
	   )
	  (& (- (apt a l))
	     (- (apt a m)) )))))

; thm thm4.6 proof not found in 10 minutes

(define thm4.6
  `((frm
      (-> (& 
	   ,@ax1 
	   ,@ax2 
	   ,@ax3 
	   ,@ax4
	   (dipt a b)
           (dipt c d)
	   (- (apt a (ln c d)))
	   (- (apt b (ln c d))) )
	  (& (- (apt c (ln a b)))
	     (- (apt d (ln a b))) )))))


; thm thm4.6b proved in 5 sec

(define thm4.6b
  `((frm
      (-> (& 
	   ,@ax1 
	   ,@ax2 
	   ,@ax3 
	   ,@ax4
	   (dipt a b)
           (dipt c d)
	   (- (apt a (ln c d)))
	   (- (apt b (ln c d))) 
	   ; lemma 4.2:
	   (all xa xb xc xd
            (-> (& (dipt xa xb)
		   (dipt xc xd) )
		(-> (v (apt xa (ln xc xd)) (apt xb (ln xc xd)))
		    (v (apt xc (ln xa xb)) (apt xd (ln xa xb))) )))
	   )
	  (& (- (apt c (ln a b)))
	     (- (apt d (ln a b))) )))))

; thm4.7i proved in 7 sec

(define thm4.7i
  `((frm
      (-> (& 
	   ,@ax1 
	   ,@ax2 
	   ,@ax3 
	   ,@ax4
	   (dipt a b)
           (dipt a c)
	   (dipt b c)
	   (- (apt c (ln a b)))
	   ; lemma 4.6:
	   (all xa xb xc xd
            (-> (& (dipt xa xb)
		   (dipt xc xd) )
		(-> (& (- (apt xa (ln xc xd))) (- (apt xb (ln xc xd))))
		    (& (- (apt xc (ln xa xb))) (- (apt xd (ln xa xb)))) )))
	   ; lemma 4.2:
	   ;(all xa xb xc xd
           ; (-> (& (dipt xa xb)
	   ;	   (dipt xc xd) )
	   ;	(-> (v (apt xa (ln xc xd)) (apt xb (ln xc xd)))
	   ;	    (v (apt xc (ln xa xb)) (apt xd (ln xa xb))) )))
	   )
	  (- (apt a (ln c b))) ))))


; thm4.7ii proved in 7 seconds

(define thm4.7ii
  `((frm
      (-> (& 
	   ,@ax1 
	   ,@ax2 
	   ,@ax3 
	   ,@ax4
	   (dipt a b)
           (dipt a c)
	   (dipt b c)
	   (- (apt c (ln a b)))
	   ; lemma 4.6:
	   (all xa xb xc xd
            (-> (& (dipt xa xb)
		   (dipt xc xd) )
		(-> (& (- (apt xa (ln xc xd))) (- (apt xb (ln xc xd))))
		    (& (- (apt xc (ln xa xb))) (- (apt xd (ln xa xb)))) )))
	   )
	  (- (apt b (ln a c))) ))))


; thm4.7iii proved in 6.3 sec

(define thm4.7iii
  `((frm
      (-> (& 
	   ,@ax1 
	   ,@ax2 
	   ,@ax3 
	   ,@ax4
	   (dipt a b)
           (dipt a c)
	   (dipt b c)
	   (- (apt c (ln a b)))

	   ; 4.7i:

	   (- (apt a (ln c b)))

	   ; 4.7ii:

	   (- (apt b (ln a c)))

	   ; lemma 4.6:
	   (all xa xb xc xd
            (-> (& (dipt xa xb)
		   (dipt xc xd) )
		(-> (& (- (apt xa (ln xc xd))) (- (apt xb (ln xc xd))))
		    (& (- (apt xc (ln xa xb))) (- (apt xd (ln xa xb)))) )))
	   )
	  (- (apt c (ln b a))) ))))



;-------------------------------------------------------------

(define thm5.1 ; no success
  `((frm
      (-> (& 
	   ,@ax1 
	   ,@ax2 
	   ,@ax3 
	   ,@ax4
	   (dipt a b)
	   ; following two: lemma 4.5.i
      	   (-> (- (diln l (ln a b))) (& (- (apt a l)) (- (apt b l))))
	   (-> (& (- (apt a l)) (- (apt b l))) (- (diln l (ln a b)))) ) 
	  (- (diln (ln a b) (ln b a))) ))))

      

;---------------------------------------------------------------------

(define ax1p
  '(& (all x (- (dipt x x)))
     (all x (- (diln x x)))
     (all x (- (con x x)))
     (all x y z
	  (-> (dipt x y) (v (dipt x z) (dipt y z))) ) 
     (all x y z
	  (-> (diln x y) (v (diln x z) (diln y z))) )
     (all x y z
	  (-> (con x y) (v (con x z) (con y z))) )))

(define ax2p
  '(&  (all x y
	   (-> (dipt x y) (- (apt x (ln x y)))) )
      (all x y
	   (-> (dipt x y) (- (apt y (ln x y)))) )
      (all x y
	   (-> (con x y) (- (apt (pt x y) x))) )
      (all x y
	   (-> (con x y) (- (apt (pt x y) y))) )))

(define ax3p
  '    (all x y u v
	     (-> (& (dipt x y) (diln u v))
		 (v (v (apt x u) (apt x v))
		    (v (apt y u) (apt y v)) ))))

(define ax4p
  '(&   (all x y z 
	    (-> (apt x y) (v (dipt x z) (apt z y))) )
       (all x y z 
	    (-> (apt x y) (v (diln y z) (apt x z))) )
       (all x y z 
	    (-> (con x y) (v (diln y z) (con x z))) )))


; 3.1 proved in 158 sec

(define thm3.1p
  `((frm
	  (all x y u
	     (-> (& (dipt x y) (& (- (apt x u)) (- (apt y u))))
		 (- (diln u (ln x y))) )) )))


; 3.2 proved in 188 sec

(define thm3.2p
  `((frm
	  (all x y z
	       (-> (& (con x y) (- (apt z x)) (- (apt z y)))
		   (- (dipt z (pt x y))) )) )))

 
; 3.3 proved in 1.3 sec

(define thm3.3p
  `((frm
	  (all x y (-> (con x y) (diln x y))) )))

; thm4.1.i.1 proved in 1.3 sec

(define thm4.1.i.1p
  `((frm
      (-> (&
           (dipt a b)
           (con l m) 
           (diln l (ln a b)) )
          (v (apt a l) (apt b l)) ))))
      
; thm4.1.i.1-full not proved in ca 18 hours.

; thm4.1.i.2 proved in 1.58 sec

(define thm4.1.i.2p
  `((frm
      (-> (& 
           (dipt a b)
           (con l m) 
	   (v (apt a l) (apt b l)) )
          (diln l (ln a b)) ))))    


; thm4.1.ii.1 proved in 1.8 seconds

(define thm4.1.ii.1p   
  `((frm
      (-> (& 
           (dipt a b)
           (con l m)
	   (diln l m)
           (dipt a (pt l m)) ) 
          (v (apt a l) (apt a m)) ))))
      
; thm4.1.ii.2 proved in 1.3 seconds

(define thm4.1.ii.2p
  `((frm
      (-> (& 
           (dipt a b)
           (con l m) 
	   (v (apt a l) (apt a m)) )
          (dipt a (pt l m)) ))))    
          


(define thm4.2p
  `((frm
      (-> (& 
	   (dipt a b)
           (dipt c d) )
        (-> (v (apt a (ln c d)) (apt b (ln c d)))
	    (v (apt c (ln a b)) (apt d (ln a b))) )))))

(define thm4.3.i.p 
  `((frm
      (-> (& 
           (dipt a b)
           (apt c (ln a b)) )
          (& (dipt c a) (dipt c b)) ))))

(define thm4.3.i.b.p ; ax1 and ax3 unnecessary
  `((frm
      (-> (& 
	   (dipt a b)
           (apt c (ln a b)) )
          (& (dipt c a) (dipt c b)) ))))


(define thm4.3.ii.b.p ; ax1 and ax3 unnecessary (important!)
  `((frm
      (-> (& 
           (dipt a b)
           (apt c (ln a b)) )
          (& (diln (ln a b) (ln c a)) 
	     (diln (ln a b) (ln c b)) )))))


;----------------------------------------------------------
(define (i100 x) (int-prove x 9999999 100))

(define (i60 x) (int-prove x 9999999 60))

(define (i50 x) (int-prove x 9999999 50))


(define (i30 x) (int-prove x 9999999 30))


(define (i0 x) (int-prove x 9999999 0))
				  
(define (prbup) (load "resintprb.scm"))

(verbose 2)

(require 'pretty-print)

;(define *make-int-frm-axioms-strat* #t) ; nonatomic axioms to be created?? ;#f
(set! *make-int-frm-axioms-strat* #f)
;(define *make-int-instantiated-axioms-strat* #t) ; only instances created??
;(define *int-prepare-sos-strat* #t) ; optimizing the initial sos
;(define *int-label-atoms-strat* #f) ; label atoms?
;(define *int-inversion-stratflag* #t) ; use invertibility? ;
;(define *int-reduction-strat* #t) ; use reductions?
;(define *int-use-unique-atoms-flag* #f) ; analyse uniqueness of atoms
;(define *int-prohibited-pr-strat* #t) ; test nesting, etc?