+axiom daemon : ∀P:Prop.P.
+
+lemma wsem_copy0 : WRealize ? copy0 R_copy0.
+#intape #k #outc #Hloop
+lapply (sem_while … sem_copy_step intape k outc Hloop) [%] -Hloop
+* #ta * #Hstar @(star_ind_l ??????? Hstar)
+[ #tb whd in ⊢ (%→?); #Hleft
+ #ls #c #c0 #rs #l1 #l3 #l4 #Htb #Hl1nomarks #Hl3l4nomarks #Hlen #l1' #bv
+ #Hl1 #Hl1bits #l4' #bg #Hl4 #Hl4bits
+ cases (Hleft … Htb) -Hleft #Hc #Houtc % %
+ [ generalize in match Hl1bits; -Hl1bits cases l1' in Hl1;
+ [ normalize #Hl1 #c1 destruct (Hl1) %
+ | * #c' #b' #l0 #Heq normalize in Heq; destruct (Heq)
+ #Hl1bits lapply (Hl1bits 〈c',false〉 ?) [ @memb_hd ]
+ >Hc #Hfalse destruct ]
+ | @Houtc ]
+| #tb #tc #td whd in ⊢ (%→?→(?→%)→%→?); #Htc #Hstar1 #Hind #Htd
+ lapply (Hind Htd) -Hind #Hind
+ #ls #c #c0 #rs #l1 #l3 #l4 #Htb #Hl1nomarks #Hl3l4nomarks #Hlen #l1' #bv
+ #Hl1 #Hl1bits #l4' #bg #Hl4 #Hl4bits %2
+ cases (Htc … Htb) -Htc #Hcbitnull #Htc
+ % [ % #Hc' >Hc' in Hcbitnull; normalize #Hfalse destruct (Hfalse) ]
+ cut (|l1| = |reverse ? l4|) [>length_reverse @Hlen] #Hlen1
+ @(list_cases2 … Hlen1)
+ [ (* case l1 = [] is discriminated because l1 contains at least comma *)
+ #Hl1nil @False_ind >Hl1nil in Hl1; cases l1' normalize
+ [ #Hl1 destruct normalize in Hcbitnull; destruct (Hcbitnull)
+ | #p0 #l0 normalize #Hfalse destruct (Hfalse) cases l0 in e0;
+ [ normalize #Hfalse1 destruct (Hfalse1)
+ | #p0' #l0' normalize #Hfalse1 destruct (Hfalse1) ] ]
+ | (* case c::l1 = c::a::l1'' *)
+ * #a #ba * #a0 #ba0 #l1'' #l4'' #Hl1cons #Hl4cons
+ lapply (eq_f ?? (reverse ?) ?? Hl4cons) >reverse_reverse >reverse_cons -Hl4cons #Hl4cons
+ cut (ba = false)
+ [ >Hl1cons in Hl1nomarks; #Hl1nomarks lapply (Hl1nomarks 〈a,ba〉 ?)
+ [ @memb_hd | normalize // ] ] #Hba
+ cut (ba0 = false)
+ [ >Hl4cons in Hl3l4nomarks; #Hl3l4nomarks lapply (Hl3l4nomarks 〈a0,ba0〉 ?)
+ [ @memb_append_l2 @memb_append_l2 @memb_hd | normalize // ] ] #Hba0
+ >Hba0 in Hl4cons; >Hba in Hl1cons; -Hba0 -Hba #Hl1cons #Hl4cons
+ >Hl4cons in Htc; >Hl1cons #Htc
+ lapply (Htc a (l3@reverse ? l4'') c0 a0 ls (l1''@rs) ? (refl ??) ?)
+ [ #x #Hx @Hl3l4nomarks >Hl4cons <associative_append
+ @memb_append_l1 @Hx
+ | >associative_append >associative_append %
+ | -Htc
+ cut (∃la.l1' = 〈c,false〉::la)
+ [ >Hl1cons in Hl1; cases l1'
+ [normalize #Hfalse destruct (Hfalse)
+ | #p #la normalize #Hla destruct (Hla) @(ex_intro ?? la) % ] ]
+ * #la #Hla
+ cut (∃lb.l4' = lb@[〈c0,false〉])
+ [ >Hl4cons in Hl4;
+ @(list_elim_left … l4')
+ [ #Heq lapply (eq_f … (length ?) … Heq)
+ >length_append >length_append
+ >commutative_plus normalize >commutative_plus normalize
+ #Hfalse destruct
+ | #a1 #tl #_ #Heq
+ >(inj_append_singleton_l2 ? (reverse ? l4''@[〈a0,false〉]) (〈grid,bg〉::tl) 〈c0,false〉 a1 Heq)
+ @ex_intro //
+ ] ] * #lb #Hlb
+ cut (|lb| = |reverse ? la|)
+ [ >Hla in Hl1; >Hlb in Hl4; #Hl4 #Hl1
+ >(?:l1 = la@[〈comma,bv〉]) in Hlen;
+ [|normalize in Hl1; destruct (Hl1) %]
+ >(?:l4 = 〈grid,bg〉::lb)
+ [|@(inj_append_singleton_l1 ?? (〈grid,bg〉::lb) ?? Hl4) ]
+ >length_append >commutative_plus >length_reverse
+ normalize #Hlalb destruct (Hlalb) //
+ ] #Hlen2
+ *
+ (* by hyp on the first iteration step,
+ we consider whether c = bit x or c = null *)
+ (* c = bit x *)
+ [ * #x * #Hx #Htc
+ lapply (Hind (〈bit x,false〉::ls) a a0 rs l1''
+ (〈bit x,false〉::l3) (reverse ? l4'') ????)
+ [ >Hl1cons in Hlen; >Hl4cons >length_append >commutative_plus
+ normalize #Hlen destruct (Hlen) //
+ | #x0 #Hx0 cases (orb_true_l … Hx0)
+ [ #Hx0eq >(\P Hx0eq) %
+ | -Hx0 #Hx0 @Hl3l4nomarks >Hl4cons
+ <associative_append @memb_append_l1 // ]
+ | #x0 #Hx0 @Hl1nomarks >Hl1cons @memb_cons //
+ | >Htc >associative_append %
+ | -Hind
+ <Hl1cons <Hl4cons #Hind lapply (Hind la bv ?? lb bg ??)
+ [ #x0 #Hx0 @Hl4bits >Hlb @memb_append_l1 //
+ | >Hlb in Hl4; normalize in ⊢ (%→?); #Hl4
+ @(inj_append_singleton_l1 ? l4 (〈grid,bg〉::lb) … Hl4)
+ | #x0 #Hx0 @Hl1bits >Hla @memb_cons //
+ | >Hla in Hl1; normalize in ⊢ (%→?); #Hl1
+ destruct (Hl1) // ] -Hind
+ (* by IH, we proceed by cases, whether a = comma
+ (consequently several lists = []) or not *)
+ *
+ [ * #Ha #Houtc1
+ cut (la = [] ∧ lb = [] ∧ l1'' = [] ∧ l4'' = [])
+ [ @daemon ] * * * #Hla1 #Hlb1 #Hl1nil #Hl4nil
+ >Hl1cons in Hl1; >Hla
+ >Houtc1 >Htc #Hl1
+ >Hl4cons in Hl4; >Hlb #Hl4
+ >Hla1 >Hlb1 >Hl1nil >Hl4nil >Hx
+ cut (a0 = grid) [ @daemon ] #Ha0 <Ha <Ha0
+ normalize in ⊢ (??(??%?%)(??%?%)); >associative_append %
+ | * #Ha #Houtc1 >Houtc1 @eq_f3 //
+ >Hla >reverse_cons >associative_append @eq_f
+ >Hx whd in ⊢ (??%?); @eq_f whd in ⊢ (???%); @eq_f @eq_f
+ >Hlb >append_cons @eq_f2 // >(merge_config_append … Hlen2) %
+ ]
+ ]
+ | (* c = null *)
+ * #Hc #Htc
+ lapply (Hind (〈c0,false〉::ls) a a0 rs l1'' (〈null,false〉::l3) (reverse ? l4'') ????)
+ [ >Hl1cons in Hlen; >Hl4cons >length_append >commutative_plus normalize
+ #Hlen destruct (Hlen) @e0
+ | #x0 #Hx0 cases (memb_append STape ? [〈null,false〉] (l3@reverse ? l4'') … Hx0) -Hx0 #Hx0
+ [ >(memb_single … Hx0) %
+ | @Hl3l4nomarks cases (memb_append … Hx0) -Hx0 #Hx0
+ [ @memb_append_l1 //
+ | @memb_append_l2 >Hl4cons @memb_append_l1 // ]
+ ]
+ | >Hl1cons #x' #Hx0 @Hl1nomarks >Hl1cons @memb_cons //
+ | >Htc @eq_f3 // >associative_append % ] -Hind <Hl1cons <Hl4cons #Hind
+ lapply (Hind la bv ?? lb bg ??)
+ [ #x0 #Hx0 @Hl4bits >Hlb @memb_append_l1 //
+ | >Hlb in Hl4; normalize in ⊢ (%→?); #Hl4
+ @(inj_append_singleton_l1 ? l4 (〈grid,bg〉::lb) … Hl4)
+ | #x0 #Hx0 @Hl1bits >Hla @memb_cons //
+ | >Hla in Hl1; normalize in ⊢ (%→?); #Hl1
+ destruct (Hl1) // ] -Hind *
+ (* by IH, we proceed by cases, whether a = comma
+ (consequently several lists = []) or not *)
+ [ * #Ha #Houtc1 >Hl1cons in Hl1; >Hla
+ >Houtc1 >Htc #Hl1
+ >Hl4cons in Hl4; >Hlb #Hl4
+ cut (la = [] ∧ lb = [] ∧ l1'' = [] ∧ l4'' = [])
+ [@daemon] * * * #Hla1 #Hlb1 #Hl1nil #Hl4nil
+ >Hla1 >Hlb1 >Hl1nil >Hl4nil >Hc
+ cut (a0 = grid) [ @daemon ] #Ha0 <Ha <Ha0
+ normalize in ⊢ (??(??%?%)(??%?%)); >associative_append %
+ | * #Ha #Houtc1 >Houtc1 @eq_f3 //
+ >Hla >reverse_cons >associative_append @eq_f
+ >Hc whd in ⊢ (??%?); @eq_f whd in ⊢ (???%); @eq_f @eq_f
+ >Hlb >append_cons @eq_f2 // >(merge_config_append … Hlen2) %
+ ]
+ ]
+]]]
+qed.