--- /dev/null
+include "ASM/BitVector.ma".
+
+definition Identifier ≝ Word.
+
+inductive addressing_mode: Type[0] ≝
+ DIRECT: Byte → addressing_mode
+| INDIRECT: Bit → addressing_mode
+| EXT_INDIRECT: Bit → addressing_mode
+| REGISTER: BitVector 3 → addressing_mode
+| ACC_A: addressing_mode
+| ACC_B: addressing_mode
+| DPTR: addressing_mode
+| DATA: Byte → addressing_mode
+| DATA16: Word → addressing_mode
+| ACC_DPTR: addressing_mode
+| ACC_PC: addressing_mode
+| EXT_INDIRECT_DPTR: addressing_mode
+| INDIRECT_DPTR: addressing_mode
+| CARRY: addressing_mode
+| BIT_ADDR: Byte → addressing_mode
+| N_BIT_ADDR: Byte → addressing_mode
+| RELATIVE: Byte → addressing_mode
+| ADDR11: Word11 → addressing_mode
+| ADDR16: Word → addressing_mode.
+
+(* dpm: renamed register to registr to avoid clash with brian's types *)
+inductive addressing_mode_tag : Type[0] ≝
+ direct: addressing_mode_tag
+| indirect: addressing_mode_tag
+| ext_indirect: addressing_mode_tag
+| registr: addressing_mode_tag
+| acc_a: addressing_mode_tag
+| acc_b: addressing_mode_tag
+| dptr: addressing_mode_tag
+| data: addressing_mode_tag
+| data16: addressing_mode_tag
+| acc_dptr: addressing_mode_tag
+| acc_pc: addressing_mode_tag
+| ext_indirect_dptr: addressing_mode_tag
+| indirect_dptr: addressing_mode_tag
+| carry: addressing_mode_tag
+| bit_addr: addressing_mode_tag
+| n_bit_addr: addressing_mode_tag
+| relative: addressing_mode_tag
+| addr11: addressing_mode_tag
+| addr16: addressing_mode_tag.
+
+definition eq_a ≝
+ λa, b: addressing_mode_tag.
+ match a with
+ [ direct ⇒ match b with [ direct ⇒ true | _ ⇒ false ]
+ | indirect ⇒ match b with [ indirect ⇒ true | _ ⇒ false ]
+ | ext_indirect ⇒ match b with [ ext_indirect ⇒ true | _ ⇒ false ]
+ | registr ⇒ match b with [ registr ⇒ true | _ ⇒ false ]
+ | acc_a ⇒ match b with [ acc_a ⇒ true | _ ⇒ false ]
+ | acc_b ⇒ match b with [ acc_b ⇒ true | _ ⇒ false ]
+ | dptr ⇒ match b with [ dptr ⇒ true | _ ⇒ false ]
+ | data ⇒ match b with [ data ⇒ true | _ ⇒ false ]
+ | data16 ⇒ match b with [ data16 ⇒ true | _ ⇒ false ]
+ | acc_dptr ⇒ match b with [ acc_dptr ⇒ true | _ ⇒ false ]
+ | acc_pc ⇒ match b with [ acc_pc ⇒ true | _ ⇒ false ]
+ | ext_indirect_dptr ⇒ match b with [ ext_indirect_dptr ⇒ true | _ ⇒ false ]
+ | indirect_dptr ⇒ match b with [ indirect_dptr ⇒ true | _ ⇒ false ]
+ | carry ⇒ match b with [ carry ⇒ true | _ ⇒ false ]
+ | bit_addr ⇒ match b with [ bit_addr ⇒ true | _ ⇒ false ]
+ | n_bit_addr ⇒ match b with [ n_bit_addr ⇒ true | _ ⇒ false ]
+ | relative ⇒ match b with [ relative ⇒ true | _ ⇒ false ]
+ | addr11 ⇒ match b with [ addr11 ⇒ true | _ ⇒ false ]
+ | addr16 ⇒ match b with [ addr16 ⇒ true | _ ⇒ false ]
+ ].
+
+(* to avoid expansion... *)
+let rec is_a (d:addressing_mode_tag) (A:addressing_mode) on d ≝
+ match d with
+ [ direct ⇒ match A with [ DIRECT _ ⇒ true | _ ⇒ false ]
+ | indirect ⇒ match A with [ INDIRECT _ ⇒ true | _ ⇒ false ]
+ | ext_indirect ⇒ match A with [ EXT_INDIRECT _ ⇒ true | _ ⇒ false ]
+ | registr ⇒ match A with [ REGISTER _ ⇒ true | _ ⇒ false ]
+ | acc_a ⇒ match A with [ ACC_A ⇒ true | _ ⇒ false ]
+ | acc_b ⇒ match A with [ ACC_B ⇒ true | _ ⇒ false ]
+ | dptr ⇒ match A with [ DPTR ⇒ true | _ ⇒ false ]
+ | data ⇒ match A with [ DATA _ ⇒ true | _ ⇒ false ]
+ | data16 ⇒ match A with [ DATA16 _ ⇒ true | _ ⇒ false ]
+ | acc_dptr ⇒ match A with [ ACC_DPTR ⇒ true | _ ⇒ false ]
+ | acc_pc ⇒ match A with [ ACC_PC ⇒ true | _ ⇒ false ]
+ | ext_indirect_dptr ⇒ match A with [ EXT_INDIRECT_DPTR ⇒ true | _ ⇒ false ]
+ | indirect_dptr ⇒ match A with [ INDIRECT_DPTR ⇒ true | _ ⇒ false ]
+ | carry ⇒ match A with [ CARRY ⇒ true | _ ⇒ false ]
+ | bit_addr ⇒ match A with [ BIT_ADDR _ ⇒ true | _ ⇒ false ]
+ | n_bit_addr ⇒ match A with [ N_BIT_ADDR _ ⇒ true | _ ⇒ false ]
+ | relative ⇒ match A with [ RELATIVE _ ⇒ true | _ ⇒ false ]
+ | addr11 ⇒ match A with [ ADDR11 _ ⇒ true | _ ⇒ false ]
+ | addr16 ⇒ match A with [ ADDR16 _ ⇒ true | _ ⇒ false ]
+ ].
+
+
+let rec is_in n (l: Vector addressing_mode_tag n) (A:addressing_mode) on l : bool ≝
+ match l return λm.λ_:Vector addressing_mode_tag m.bool with
+ [ VEmpty ⇒ false
+ | VCons m he (tl: Vector addressing_mode_tag m) ⇒
+ is_a he A ∨ is_in ? tl A ].
+
+record subaddressing_mode (n) (l: Vector addressing_mode_tag (S n)) : Type[0] ≝
+{
+ subaddressing_modeel:> addressing_mode;
+ subaddressing_modein: bool_to_Prop (is_in ? l subaddressing_modeel)
+}.
+
+coercion subaddressing_mode : ∀n.∀l:Vector addressing_mode_tag (S n).Type[0]
+ ≝ subaddressing_mode on _l: Vector addressing_mode_tag (S ?) to Type[0].
+
+coercion mk_subaddressing_mode :
+ ∀n.∀l:Vector addressing_mode_tag (S n).∀a:addressing_mode.
+ ∀p:bool_to_Prop (is_in ? l a).subaddressing_mode n l
+ ≝ mk_subaddressing_mode on a:addressing_mode to subaddressing_mode ? ?.
+
+inductive preinstruction (A: Type[0]) : Type[0] ≝
+ ADD: [[acc_a]] → [[ registr ; direct ; indirect ; data ]] → preinstruction A
+| ADDC: [[acc_a]] → [[ registr ; direct ; indirect ; data ]] → preinstruction A
+| SUBB: [[acc_a]] → [[ registr ; direct ; indirect ; data ]] → preinstruction A
+| INC: [[ acc_a ; registr ; direct ; indirect ; dptr ]] → preinstruction A
+| DEC: [[ acc_a ; registr ; direct ; indirect ]] → preinstruction A
+| MUL: [[acc_a]] → [[acc_b]] → preinstruction A
+| DIV: [[acc_a]] → [[acc_b]] → preinstruction A
+| DA: [[acc_a]] → preinstruction A
+
+(* conditional jumps *)
+| JC: A → preinstruction A
+| JNC: A → preinstruction A
+| JB: [[bit_addr]] → A → preinstruction A
+| JNB: [[bit_addr]] → A → preinstruction A
+| JBC: [[bit_addr]] → A → preinstruction A
+| JZ: A → preinstruction A
+| JNZ: A → preinstruction A
+| CJNE:
+ [[acc_a]] × [[direct; data]] ⊎ [[registr; indirect]] × [[data]] → A → preinstruction A
+| DJNZ: [[registr ; direct]] → A → preinstruction A
+ (* logical operations *)
+| ANL:
+ [[acc_a]] × [[ registr ; direct ; indirect ; data ]] ⊎
+ [[direct]] × [[ acc_a ; data ]] ⊎
+ [[carry]] × [[ bit_addr ; n_bit_addr]] → preinstruction A
+| ORL:
+ [[acc_a]] × [[ registr ; data ; direct ; indirect ]] ⊎
+ [[direct]] × [[ acc_a ; data ]] ⊎
+ [[carry]] × [[ bit_addr ; n_bit_addr]] → preinstruction A
+| XRL:
+ [[acc_a]] × [[ data ; registr ; direct ; indirect ]] ⊎
+ [[direct]] × [[ acc_a ; data ]] → preinstruction A
+| CLR: [[ acc_a ; carry ; bit_addr ]] → preinstruction A
+| CPL: [[ acc_a ; carry ; bit_addr ]] → preinstruction A
+| RL: [[acc_a]] → preinstruction A
+| RLC: [[acc_a]] → preinstruction A
+| RR: [[acc_a]] → preinstruction A
+| RRC: [[acc_a]] → preinstruction A
+| SWAP: [[acc_a]] → preinstruction A
+
+ (* data transfer *)
+| MOV:
+ [[acc_a]] × [[ registr ; direct ; indirect ; data ]] ⊎
+ [[ registr ; indirect ]] × [[ acc_a ; direct ; data ]] ⊎
+ [[direct]] × [[ acc_a ; registr ; direct ; indirect ; data ]] ⊎
+ [[dptr]] × [[data16]] ⊎
+ [[carry]] × [[bit_addr]] ⊎
+ [[bit_addr]] × [[carry]] → preinstruction A
+| MOVX:
+ [[acc_a]] × [[ ext_indirect ; ext_indirect_dptr ]] ⊎
+ [[ ext_indirect ; ext_indirect_dptr ]] × [[acc_a]] → preinstruction A
+| SETB: [[ carry ; bit_addr ]] → preinstruction A
+| PUSH: [[direct]] → preinstruction A
+| POP: [[direct]] → preinstruction A
+| XCH: [[acc_a]] → [[ registr ; direct ; indirect ]] → preinstruction A
+| XCHD: [[acc_a]] → [[indirect]] → preinstruction A
+
+ (* program branching *)
+| RET: preinstruction A
+| RETI: preinstruction A
+| NOP: preinstruction A.
+
+inductive instruction: Type[0] ≝
+ | ACALL: [[addr11]] → instruction
+ | LCALL: [[addr16]] → instruction
+ | AJMP: [[addr11]] → instruction
+ | LJMP: [[addr16]] → instruction
+ | SJMP: [[relative]] → instruction
+ | JMP: [[indirect_dptr]] → instruction
+ | MOVC: [[acc_a]] → [[ acc_dptr ; acc_pc ]] → instruction
+ | RealInstruction: preinstruction [[ relative ]] → instruction.
+
+coercion RealInstruction: ∀p: preinstruction [[ relative ]]. instruction ≝
+ RealInstruction on _p: preinstruction ? to instruction.
+
+inductive pseudo_instruction: Type[0] ≝
+ | Instruction: preinstruction Identifier → pseudo_instruction
+ | Comment: String → pseudo_instruction
+ | Cost: Identifier → pseudo_instruction
+ | Jmp: Identifier → pseudo_instruction
+ | Call: Identifier → pseudo_instruction
+ | Mov: [[dptr]] → Identifier → pseudo_instruction.
+
+definition labelled_instruction ≝ option Identifier × pseudo_instruction.
+definition preamble ≝ list (Identifier × nat).
+definition assembly_program ≝ list instruction.
+definition pseudo_assembly_program ≝ preamble × (list labelled_instruction).
--- /dev/null
+(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *)
+(* BitVector.ma: Fixed length bitvectors, and common operations on them. *)
+(* Most functions are specialised versions of those found in *)
+(* Vector.ma as a courtesy, or boolean functions lifted into *)
+(* BitVector variants. *)
+(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *)
+
+include "arithmetics/nat.ma".
+
+include "ASM/FoldStuff.ma".
+include "ASM/Vector.ma".
+include "ASM/String.ma".
+
+(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *)
+(* Common synonyms. *)
+(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *)
+
+definition BitVector ≝ λn: nat. Vector bool n.
+definition Bit ≝ bool.
+definition Nibble ≝ BitVector 4.
+definition Byte7 ≝ BitVector 7.
+definition Byte ≝ BitVector 8.
+definition Word ≝ BitVector 16.
+definition Word11 ≝ BitVector 11.
+
+(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *)
+(* Inversion *)
+(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *)
+
+lemma BitVector_O: ∀v:BitVector 0. v ≃ VEmpty bool.
+ #v generalize in match (refl … 0) cases v in ⊢ (??%? → ?%%??) //
+ #n #hd #tl #abs @⊥ destruct (abs)
+qed.
+
+lemma BitVector_Sn: ∀n.∀v:BitVector (S n).
+ ∃hd.∃tl.v ≃ VCons bool n hd tl.
+ #n #v generalize in match (refl … (S n)) cases v in ⊢ (??%? → ??(λ_.??(λ_.?%%??)))
+ [ #abs @⊥ destruct (abs)
+ | #m #hd #tl #EQ <(injective_S … EQ) %[@hd] %[@tl] // ]
+qed.
+
+(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *)
+(* Lookup. *)
+(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *)
+
+(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *)
+(* Creating bitvectors from scratch. *)
+(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *)
+
+definition zero: ∀n:nat. BitVector n ≝
+ λn: nat. replicate bool n false.
+
+definition maximum: ∀n:nat. BitVector n ≝
+ λn: nat. replicate bool n true.
+
+definition pad ≝
+ λm, n: nat.
+ λb: BitVector n. pad_vector ? false m n b.
+
+(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *)
+(* Other manipulations. *)
+(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *)
+
+(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *)
+(* Logical operations. *)
+(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *)
+
+definition conjunction_bv: ∀n. ∀b, c: BitVector n. BitVector n ≝
+ λn: nat.
+ λb: BitVector n.
+ λc: BitVector n.
+ zip_with ? ? ? n (andb) b c.
+
+interpretation "BitVector conjunction" 'conjunction b c = (conjunction_bv ? b c).
+
+definition inclusive_disjunction_bv ≝
+ λn: nat.
+ λb: BitVector n.
+ λc: BitVector n.
+ zip_with ? ? ? n (orb) b c.
+
+interpretation "BitVector inclusive disjunction"
+ 'inclusive_disjunction b c = (inclusive_disjunction_bv ? b c).
+
+definition exclusive_disjunction_bv ≝
+ λn: nat.
+ λb: BitVector n.
+ λc: BitVector n.
+ zip_with ? ? ? n (exclusive_disjunction) b c.
+
+interpretation "BitVector exclusive disjunction"
+ 'exclusive_disjunction b c = (exclusive_disjunction ? b c).
+
+definition negation_bv ≝
+ λn: nat.
+ λb: BitVector n.
+ map bool bool n (notb) b.
+
+interpretation "BitVector negation" 'negation b c = (negation_bv ? b c).
+
+(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *)
+(* Rotates and shifts. *)
+(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *)
+
+(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *)
+(* Conversions to and from lists and natural numbers. *)
+(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *)
+
+definition eq_b ≝
+ λb, c: bool.
+ if b then
+ c
+ else
+ notb c.
+
+lemma eq_b_eq:
+ ∀b, c.
+ eq_b b c = true → b = c.
+ #b #c
+ cases b
+ cases c
+ normalize //
+qed.
+
+definition eq_bv ≝
+ λn: nat.
+ λb, c: BitVector n.
+ eq_v bool n eq_b b c.
+
+lemma eq_bv_elim: ∀P:bool → Type[0]. ∀n. ∀x,y.
+ (x = y → P true) →
+ (x ≠ y → P false) →
+ P (eq_bv n x y).
+#P #n #x #y #Ht #Hf whd in ⊢ (?%) @(eq_v_elim … Ht Hf)
+#Q * *; normalize /3/
+qed.
+
+lemma eq_bv_true: ∀n,v. eq_bv n v v = true.
+@eq_v_true * @refl
+qed.
+
+lemma eq_bv_false: ∀n,v,v'. v ≠ v' → eq_bv n v v' = false.
+#n #v #v' #NE @eq_v_false [ * * #H try @refl normalize in H; destruct | @NE ]
+qed.
+
+lemma eq_bv_refl:
+ ∀n,v. eq_bv n v v = true.
+ #n #v
+ elim v
+ [ //
+ | #n #hd #tl #ih
+ normalize
+ cases hd
+ [ normalize
+ @ ih
+ | normalize
+ @ ih
+ ]
+ ]
+qed.
+
+lemma eq_bv_sym: ∀n,v1,v2. eq_bv n v1 v2 = eq_bv n v2 v1.
+ #n #v1 #v2 @(eq_bv_elim … v1 v2) [// | #H >eq_bv_false /2/]
+qed.
+
+lemma eq_eq_bv:
+ ∀n, v, q.
+ v = q → eq_bv n v q = true.
+ #n #v
+ elim v
+ [ #q #h \ 5h normalize="" %="" |="" #n="" #hd="" #tl="" #ih="" #q="" #h=""\ 6h //
+ ]
+qed.
+
+lemma eq_bv_eq:
+ ∀n, v, q.
+ eq_bv n v q = true → v = q.
+ #n #v #q generalize in match v
+ elim q
+ [ #v #h @BitVector_O
+ | #n #hd #tl #ih #v' #h
+ cases (BitVector_Sn ? v')
+ #hd' * #tl' #jmeq >jmeq in h;
+ #new_h
+ change in new_h with ((andb ? ?) = ?);
+ cases(conjunction_true … new_h)
+ #eq_heads #eq_tails
+ whd in eq_heads:(??(??(%))?);
+ cases(eq_b_eq … eq_heads)
+ whd in eq_tails:(??(?????(%))?);
+ change in eq_tails with (eq_bv ??? = ?);
+ <(ih tl') //
+ ]
+qed.
+
+axiom bitvector_of_string:
+ ∀n: nat.
+ ∀s: String.
+ BitVector n.
+
+axiom string_of_bitvector:
+ ∀n: nat.
+ ∀b: BitVector n.
+ String.
+\ 5/h\ 6
\ No newline at end of file
--- /dev/null
+include "ASM/Util.ma".
+include "ASM/JMCoercions.ma".
+
+let rec foldl_strong_internal
+ (A: Type[0]) (P: list A → Type[0]) (l: list A)
+ (H: ∀prefix. ∀hd. ∀tl. l = prefix @ [hd] @ tl → P prefix → P (prefix @ [hd]))
+ (prefix: list A) (suffix: list A) (acc: P prefix) on suffix:
+ l = prefix @ suffix → P(prefix @ suffix) ≝
+ match suffix return λl'. l = prefix @ l' → P (prefix @ l') with
+ [ nil ⇒ λprf. ?
+ | cons hd tl ⇒ λprf. ?
+ ].
+ [ > (append_nil ?)
+ @ acc
+ | applyS (foldl_strong_internal A P l H (prefix @ [hd]) tl ? ?)
+ [ @ (H prefix hd tl prf acc)
+ | applyS prf
+ ]
+ ]
+qed.
+
+definition foldl_strong ≝
+ λA: Type[0].
+ λP: list A → Type[0].
+ λl: list A.
+ λH: ∀prefix. ∀hd. ∀tl. l = prefix @ [hd] @ tl → P prefix → P (prefix @ [hd]).
+ λacc: P [ ].
+ foldl_strong_internal A P l H [ ] l acc (refl …).
+
+let rec foldr_strong_internal
+ (A:Type[0])
+ (P: list A → Type[0])
+ (l: list A)
+ (H: ∀prefix,hd,tl. l = prefix @ [hd] @ tl → P tl → P (hd::tl))
+ (prefix: list A) (suffix: list A) (acc: P [ ]) on suffix : l = prefix@suffix → P suffix ≝
+ match suffix return λl'. l = prefix @ l' → P (l') with
+ [ nil ⇒ λprf. acc
+ | cons hd tl ⇒ λprf. H prefix hd tl prf (foldr_strong_internal A P l H (prefix @ [hd]) tl acc ?) ].
+ applyS prf
+qed.
+
+lemma foldr_strong:
+ ∀A:Type[0].
+ ∀P: list A → Type[0].
+ ∀l: list A.
+ ∀H: ∀prefix,hd,tl. l = prefix @ [hd] @ tl → P tl → P (hd::tl).
+ ∀acc:P [ ]. P l
+ ≝ λA,P,l,H,acc. foldr_strong_internal A P l H [ ] l acc (refl …).
+
+lemma pair_destruct: ∀A,B,a1,a2,b1,b2. pair A B a1 a2 = 〈b1,b2〉 → a1=b1 ∧ a2=b2.
+ #A #B #a1 #a2 #b1 #b2 #EQ destruct /2/
+qed.
--- /dev/null
+include "basics/jmeq.ma".
+include "basics/types.ma".
+include "basics/list.ma".
+
+definition inject : ∀A.∀P:A → Prop.∀a.∀p:P a.Σx:A.P x ≝ λA,P,a,p. dp … a p.
+definition eject : ∀A.∀P: A → Prop.(Σx:A.P x) → A ≝ λA,P,c.match c with [ dp w p ⇒ w].
+
+coercion inject nocomposites: ∀A.∀P:A → Prop.∀a.∀p:P a.Σx:A.P x ≝ inject on a:? to Σx:?.?.
+coercion eject nocomposites: ∀A.∀P:A → Prop.∀c:Σx:A.P x.A ≝ eject on _c:Σx:?.? to ?.
+
+(*axiom VOID: Type[0].
+axiom assert_false: VOID.
+definition bigbang: ∀A:Type[0].False → VOID → A.
+ #A #abs cases abs
+qed.
+
+coercion bigbang nocomposites: ∀A:Type[0].False → ∀v:VOID.A ≝ bigbang on _v:VOID to ?.*)
+
+lemma sig2: ∀A.∀P:A → Prop. ∀p:Σx:A.P x. P (eject … p).
+ #A #P #p cases p #w #q @q
+qed.
+
+(* END RUSSELL **)
--- /dev/null
+include "basics/list.ma".
+include "basics/types.ma".
+include "arithmetics/nat.ma".
+
+include "utilities/pair.ma".
+include "ASM/JMCoercions.ma".
+
+(* let's implement a daemon not used by automation *)
+inductive DAEMONXXX: Type[0] ≝ K1DAEMONXXX: DAEMONXXX | K2DAEMONXXX: DAEMONXXX.
+axiom IMPOSSIBLE: K1DAEMONXXX = K2DAEMONXXX.
+example daemon: False. generalize in match IMPOSSIBLE; #IMPOSSIBLE destruct(IMPOSSIBLE) qed.
+example not_implemented: False. cases daemon qed.
+
+notation "⊥" with precedence 90
+ for @{ match ? in False with [ ] }.
+
+definition ltb ≝
+ λm, n: nat.
+ leb (S m) n.
+
+definition geb ≝
+ λm, n: nat.
+ ltb n m.
+
+definition gtb ≝
+ λm, n: nat.
+ ltb n m.
+
+(* dpm: unless I'm being stupid, this isn't defined in the stdlib? *)
+let rec eq_nat (n: nat) (m: nat) on n: bool ≝
+ match n with
+ [ O ⇒ match m with [ O ⇒ true | _ ⇒ false ]
+ | S n' ⇒ match m with [ S m' ⇒ eq_nat n' m' | _ ⇒ false ]
+ ].
+
+let rec forall
+ (A: Type[0]) (f: A → bool) (l: list A)
+ on l ≝
+ match l with
+ [ nil ⇒ true
+ | cons hd tl ⇒ f hd ∧ forall A f tl
+ ].
+
+let rec prefix
+ (A: Type[0]) (k: nat) (l: list A)
+ on l ≝
+ match l with
+ [ nil ⇒ [ ]
+ | cons hd tl ⇒
+ match k with
+ [ O ⇒ [ ]
+ | S k' ⇒ hd :: prefix A k' tl
+ ]
+ ].
+
+let rec fold_left2
+ (A: Type[0]) (B: Type[0]) (C: Type[0]) (f: A → B → C → A) (accu: A)
+ (left: list B) (right: list C) (proof: |left| = |right|)
+ on left: A ≝
+ match left return λx. |x| = |right| → A with
+ [ nil ⇒ λnil_prf.
+ match right return λx. |[ ]| = |x| → A with
+ [ nil ⇒ λnil_nil_prf. accu
+ | cons hd tl ⇒ λcons_nil_absrd. ?
+ ] nil_prf
+ | cons hd tl ⇒ λcons_prf.
+ match right return λx. |hd::tl| = |x| → A with
+ [ nil ⇒ λcons_nil_absrd. ?
+ | cons hd' tl' ⇒ λcons_cons_prf.
+ fold_left2 … f (f accu hd hd') tl tl' ?
+ ] cons_prf
+ ] proof.
+ [ 1: normalize in cons_nil_absrd;
+ destruct(cons_nil_absrd)
+ | 2: normalize in cons_nil_absrd;
+ destruct(cons_nil_absrd)
+ | 3: normalize in cons_cons_prf;
+ @injective_S
+ assumption
+ ]
+qed.
+
+let rec remove_n_first_internal
+ (i: nat) (A: Type[0]) (l: list A) (n: nat)
+ on l ≝
+ match l with
+ [ nil ⇒ [ ]
+ | cons hd tl ⇒
+ match eq_nat i n with
+ [ true ⇒ l
+ | _ ⇒ remove_n_first_internal (S i) A tl n
+ ]
+ ].
+
+definition remove_n_first ≝
+ λA: Type[0].
+ λn: nat.
+ λl: list A.
+ remove_n_first_internal 0 A l n.
+
+let rec foldi_from_until_internal
+ (A: Type[0]) (i: nat) (res: ?) (rem: list A) (m: nat) (f: nat → list A → A → list A)
+ on rem ≝
+ match rem with
+ [ nil ⇒ res
+ | cons e tl ⇒
+ match geb i m with
+ [ true ⇒ res
+ | _ ⇒ foldi_from_until_internal A (S i) (f i res e) tl m f
+ ]
+ ].
+
+definition foldi_from_until ≝
+ λA: Type[0].
+ λn: nat.
+ λm: nat.
+ λf: ?.
+ λa: ?.
+ λl: ?.
+ foldi_from_until_internal A 0 a (remove_n_first A n l) m f.
+
+definition foldi_from ≝
+ λA: Type[0].
+ λn.
+ λf.
+ λa.
+ λl.
+ foldi_from_until A n (|l|) f a l.
+
+definition foldi_until ≝
+ λA: Type[0].
+ λm.
+ λf.
+ λa.
+ λl.
+ foldi_from_until A 0 m f a l.
+
+definition foldi ≝
+ λA: Type[0].
+ λf.
+ λa.
+ λl.
+ foldi_from_until A 0 (|l|) f a l.
+
+definition hd_safe ≝
+ λA: Type[0].
+ λl: list A.
+ λproof: 0 < |l|.
+ match l return λx. 0 < |x| → A with
+ [ nil ⇒ λnil_absrd. ?
+ | cons hd tl ⇒ λcons_prf. hd
+ ] proof.
+ normalize in nil_absrd;
+ cases(not_le_Sn_O 0)
+ #HYP
+ cases(HYP nil_absrd)
+qed.
+
+definition tail_safe ≝
+ λA: Type[0].
+ λl: list A.
+ λproof: 0 < |l|.
+ match l return λx. 0 < |x| → list A with
+ [ nil ⇒ λnil_absrd. ?
+ | cons hd tl ⇒ λcons_prf. tl
+ ] proof.
+ normalize in nil_absrd;
+ cases(not_le_Sn_O 0)
+ #HYP
+ cases(HYP nil_absrd)
+qed.
+
+let rec split
+ (A: Type[0]) (l: list A) (index: nat) (proof: index ≤ |l|)
+ on index ≝
+ match index return λx. x ≤ |l| → (list A) × (list A) with
+ [ O ⇒ λzero_prf. 〈[], l〉
+ | S index' ⇒ λsucc_prf.
+ match l return λx. S index' ≤ |x| → (list A) × (list A) with
+ [ nil ⇒ λnil_absrd. ?
+ | cons hd tl ⇒ λcons_prf.
+ let 〈l1, l2〉 ≝ split A tl index' ? in
+ 〈hd :: l1, l2〉
+ ] succ_prf
+ ] proof.
+ [1: normalize in nil_absrd;
+ cases(not_le_Sn_O index')
+ #HYP
+ cases(HYP nil_absrd)
+ |2: normalize in cons_prf;
+ @le_S_S_to_le
+ assumption
+ ]
+qed.
+
+let rec nth_safe
+ (elt_type: Type[0]) (index: nat) (the_list: list elt_type)
+ (proof: index < | the_list |)
+ on index ≝
+ match index return λs. s < | the_list | → elt_type with
+ [ O ⇒
+ match the_list return λt. 0 < | t | → elt_type with
+ [ nil ⇒ λnil_absurd. ?
+ | cons hd tl ⇒ λcons_proof. hd
+ ]
+ | S index' ⇒
+ match the_list return λt. S index' < | t | → elt_type with
+ [ nil ⇒ λnil_absurd. ?
+ | cons hd tl ⇒
+ λcons_proof. nth_safe elt_type index' tl ?
+ ]
+ ] proof.
+ [ normalize in nil_absurd;
+ cases (not_le_Sn_O 0)
+ #ABSURD
+ elim (ABSURD nil_absurd)
+ | normalize in nil_absurd;
+ cases (not_le_Sn_O (S index'))
+ #ABSURD
+ elim (ABSURD nil_absurd)
+ | normalize in cons_proof
+ @le_S_S_to_le
+ assumption
+ ]
+qed.
+
+definition last_safe ≝
+ λelt_type: Type[0].
+ λthe_list: list elt_type.
+ λproof : 0 < | the_list |.
+ nth_safe elt_type (|the_list| - 1) the_list ?.
+ normalize /2/
+qed.
+
+let rec reduce
+ (A: Type[0]) (B: Type[0]) (left: list A) (right: list B) on left ≝
+ match left with
+ [ nil ⇒ 〈〈[ ], [ ]〉, 〈[ ], right〉〉
+ | cons hd tl ⇒
+ match right with
+ [ nil ⇒ 〈〈[ ], left〉, 〈[ ], [ ]〉〉
+ | cons hd' tl' ⇒
+ let 〈cleft, cright〉 ≝ reduce A B tl tl' in
+ let 〈commonl, restl〉 ≝ cleft in
+ let 〈commonr, restr〉 ≝ cright in
+ 〈〈hd :: commonl, restl〉, 〈hd' :: commonr, restr〉〉
+ ]
+ ].
+
+(*
+axiom reduce_strong:
+ ∀A: Type[0].
+ ∀left: list A.
+ ∀right: list A.
+ Σret: ((list A) × (list A)) × ((list A) × (list A)). | \fst (\fst ret) | = | \fst (\snd ret) |.
+*)
+
+let rec reduce_strong
+ (A: Type[0]) (B: Type[0]) (left: list A) (right: list B)
+ on left : Σret: ((list A) × (list A)) × ((list B) × (list B)). |\fst (\fst ret)| = |\fst (\snd ret)| ≝
+ match left with
+ [ nil ⇒ 〈〈[ ], [ ]〉, 〈[ ], right〉〉
+ | cons hd tl ⇒
+ match right with
+ [ nil ⇒ 〈〈[ ], left〉, 〈[ ], [ ]〉〉
+ | cons hd' tl' ⇒
+ let 〈cleft, cright〉 ≝ reduce_strong A B tl tl' in
+ let 〈commonl, restl〉 ≝ cleft in
+ let 〈commonr, restr〉 ≝ cright in
+ 〈〈hd :: commonl, restl〉, 〈hd' :: commonr, restr〉〉
+ ]
+ ].
+ [ 1: normalize %
+ | 2: normalize %
+ | 3: normalize
+ generalize in match (sig2 … (reduce_strong A B tl tl1));
+ >p2 >p3 >p4 normalize in ⊢ (% → ?)
+ #HYP //
+ ]
+qed.
+
+let rec map2_opt
+ (A: Type[0]) (B: Type[0]) (C: Type[0]) (f: A → B → C)
+ (left: list A) (right: list B) on left ≝
+ match left with
+ [ nil ⇒
+ match right with
+ [ nil ⇒ Some ? (nil C)
+ | _ ⇒ None ?
+ ]
+ | cons hd tl ⇒
+ match right with
+ [ nil ⇒ None ?
+ | cons hd' tl' ⇒
+ match map2_opt A B C f tl tl' with
+ [ None ⇒ None ?
+ | Some tail ⇒ Some ? (f hd hd' :: tail)
+ ]
+ ]
+ ].
+
+let rec map2
+ (A: Type[0]) (B: Type[0]) (C: Type[0]) (f: A → B → C)
+ (left: list A) (right: list B) (proof: | left | = | right |) on left ≝
+ match left return λx. | x | = | right | → list C with
+ [ nil ⇒
+ match right return λy. | [] | = | y | → list C with
+ [ nil ⇒ λnil_prf. nil C
+ | _ ⇒ λcons_absrd. ?
+ ]
+ | cons hd tl ⇒
+ match right return λy. | hd::tl | = | y | → list C with
+ [ nil ⇒ λnil_absrd. ?
+ | cons hd' tl' ⇒ λcons_prf. (f hd hd') :: map2 A B C f tl tl' ?
+ ]
+ ] proof.
+ [1: normalize in cons_absrd;
+ destruct(cons_absrd)
+ |2: normalize in nil_absrd;
+ destruct(nil_absrd)
+ |3: normalize in cons_prf;
+ destruct(cons_prf)
+ assumption
+ ]
+qed.
+
+let rec map3
+ (A: Type[0]) (B: Type[0]) (C: Type[0]) (D: Type[0]) (f: A → B → C → D)
+ (left: list A) (centre: list B) (right: list C)
+ (prflc: |left| = |centre|) (prfcr: |centre| = |right|) on left ≝
+ match left return λx. |x| = |centre| → list D with
+ [ nil ⇒ λnil_prf.
+ match centre return λx. |x| = |right| → list D with
+ [ nil ⇒ λnil_nil_prf.
+ match right return λx. |nil ?| = |x| → list D with
+ [ nil ⇒ λnil_nil_nil_prf. nil D
+ | cons hd tl ⇒ λcons_nil_nil_absrd. ?
+ ] nil_nil_prf
+ | cons hd tl ⇒ λnil_cons_absrd. ?
+ ] prfcr
+ | cons hd tl ⇒ λcons_prf.
+ match centre return λx. |x| = |right| → list D with
+ [ nil ⇒ λcons_nil_absrd. ?
+ | cons hd' tl' ⇒ λcons_cons_prf.
+ match right return λx. |right| = |x| → |cons ? hd' tl'| = |x| → list D with
+ [ nil ⇒ λrefl_prf. λcons_cons_nil_absrd. ?
+ | cons hd'' tl'' ⇒ λrefl_prf. λcons_cons_cons_prf.
+ (f hd hd' hd'') :: (map3 A B C D f tl tl' tl'' ? ?)
+ ] (refl ? (|right|)) cons_cons_prf
+ ] prfcr
+ ] prflc.
+ [ 1: normalize in cons_nil_nil_absrd;
+ destruct(cons_nil_nil_absrd)
+ | 2: generalize in match nil_cons_absrd;
+ \ 5prfcr\ 6\ 5nil_prf #hyp="" normalize="" hyp;="" destruct(hyp)="" |="" 3:="" generalize="" in="" match="" cons_nil_absrd;=""\ 6\ 5prfcr\ 6\ 5cons_prf #hyp="" hyp;="" destruct(hyp)="" 4:="" cons_cons_nil_absrd;="" destruct(cons_cons_nil_absrd)="" 5:="" normalize="" destruct(cons_cons_cons_prf)="" assumption="" |="" 6:="" generalize="" in="" match="" cons_cons_cons_prf;=""\ 6\ 5refl_prf\ 6\ 5prfcr\ 6\ 5cons_prf #hyp="" normalize="" hyp;="" destruct(hyp)="" @sym_eq="" assumption="" ]="" lemma="" eq_rect_type0_r="" :="" ∀a:="" ∀a:a.="" ∀p:="" ∀x:a.="" eq="" type[0].="" (refl="" a="" →="" ∀x:="" a.∀p:eq="" ?="" a.="" x="" p.="" #a="" #h="" #x="" #p="" h="" generalize="" in="" match="" cases="" p="" qed.="" let="" rec="" safe_nth="" (a:="" type[0])="" (n:="" nat)="" (l:="" list="" a)="" (p:="" n=""\ 6< length A l) on n: A ≝
+ match n return λo. o < length A l → A with
+ [ O ⇒
+ match l return λm. 0 < length A m → A with
+ [ nil ⇒ λabsd1. ?
+ | cons hd tl ⇒ λprf1. hd
+ ]
+ | S n' ⇒
+ match l return λm. S n' < length A m → A with
+ [ nil ⇒ λabsd2. ?
+ | cons hd tl ⇒ λprf2. safe_nth A n' tl ?
+ ]
+ ] ?.
+ [ 1:
+ @ p
+ | 4:
+ normalize in prf2
+ normalize
+ @ le_S_S_to_le
+ assumption
+ | 2:
+ normalize in absd1;
+ cases (not_le_Sn_O O)
+ # H
+ elim (H absd1)
+ | 3:
+ normalize in absd2;
+ cases (not_le_Sn_O (S n'))
+ # H
+ elim (H absd2)
+ ]
+qed.
+
+let rec nub_by_internal (A: Type[0]) (f: A → A → bool) (l: list A) (n: nat) on n ≝
+ match n with
+ [ O ⇒
+ match l with
+ [ nil ⇒ [ ]
+ | cons hd tl ⇒ l
+ ]
+ | S n ⇒
+ match l with
+ [ nil ⇒ [ ]
+ | cons hd tl ⇒
+ hd :: nub_by_internal A f (filter ? (λy. notb (f y hd)) tl) n
+ ]
+ ].
+
+definition nub_by ≝
+ λA: Type[0].
+ λf: A → A → bool.
+ λl: list A.
+ nub_by_internal A f l (length ? l).
+
+let rec member (A: Type[0]) (eq: A → A → bool) (a: A) (l: list A) on l ≝
+ match l with
+ [ nil ⇒ false
+ | cons hd tl ⇒ orb (eq a hd) (member A eq a tl)
+ ].
+
+let rec take (A: Type[0]) (n: nat) (l: list A) on n: list A ≝
+ match n with
+ [ O ⇒ [ ]
+ | S n ⇒
+ match l with
+ [ nil ⇒ [ ]
+ | cons hd tl ⇒ hd :: take A n tl
+ ]
+ ].
+
+let rec drop (A: Type[0]) (n: nat) (l: list A) on n ≝
+ match n with
+ [ O ⇒ l
+ | S n ⇒
+ match l with
+ [ nil ⇒ [ ]
+ | cons hd tl ⇒ drop A n tl
+ ]
+ ].
+
+definition list_split ≝
+ λA: Type[0].
+ λn: nat.
+ λl: list A.
+ 〈take A n l, drop A n l〉.
+
+let rec mapi_internal (A: Type[0]) (B: Type[0]) (n: nat) (f: nat → A → B)
+ (l: list A) on l: list B ≝
+ match l with
+ [ nil ⇒ nil ?
+ | cons hd tl ⇒ (f n hd) :: (mapi_internal A B (n + 1) f tl)
+ ].
+
+definition mapi ≝
+ λA, B: Type[0].
+ λf: nat → A → B.
+ λl: list A.
+ mapi_internal A B 0 f l.
+
+let rec zip_pottier
+ (A: Type[0]) (B: Type[0]) (left: list A) (right: list B)
+ on left ≝
+ match left with
+ [ nil ⇒ [ ]
+ | cons hd tl ⇒
+ match right with
+ [ nil ⇒ [ ]
+ | cons hd' tl' ⇒ 〈hd, hd'〉 :: zip_pottier A B tl tl'
+ ]
+ ].
+
+let rec zip_safe
+ (A: Type[0]) (B: Type[0]) (left: list A) (right: list B) (prf: |left| = |right|)
+ on left ≝
+ match left return λx. |x| = |right| → list (A × B) with
+ [ nil ⇒ λnil_prf.
+ match right return λx. |[ ]| = |x| → list (A × B) with
+ [ nil ⇒ λnil_nil_prf. [ ]
+ | cons hd tl ⇒ λnil_cons_absrd. ?
+ ] nil_prf
+ | cons hd tl ⇒ λcons_prf.
+ match right return λx. |hd::tl| = |x| → list (A × B) with
+ [ nil ⇒ λcons_nil_absrd. ?
+ | cons hd' tl' ⇒ λcons_cons_prf. 〈hd, hd'〉 :: zip_safe A B tl tl' ?
+ ] cons_prf
+ ] prf.
+ [ 1: normalize in nil_cons_absrd;
+ destruct(nil_cons_absrd)
+ | 2: normalize in cons_nil_absrd;
+ destruct(cons_nil_absrd)
+ | 3: normalize in cons_cons_prf;
+ @injective_S
+ assumption
+ ]
+qed.
+
+let rec zip (A: Type[0]) (B: Type[0]) (l: list A) (r: list B) on l: option (list (A × B)) ≝
+ match l with
+ [ nil ⇒ Some ? (nil (A × B))
+ | cons hd tl ⇒
+ match r with
+ [ nil ⇒ None ?
+ | cons hd' tl' ⇒
+ match zip ? ? tl tl' with
+ [ None ⇒ None ?
+ | Some tail ⇒ Some ? (〈hd, hd'〉 :: tail)
+ ]
+ ]
+ ].
+
+let rec foldl (A: Type[0]) (B: Type[0]) (f: A → B → A) (a: A) (l: list B) on l ≝
+ match l with
+ [ nil ⇒ a
+ | cons hd tl ⇒ foldl A B f (f a hd) tl
+ ].
+
+lemma foldl_step:
+ ∀A:Type[0].
+ ∀B: Type[0].
+ ∀H: A → B → A.
+ ∀acc: A.
+ ∀pre: list B.
+ ∀hd:B.
+ foldl A B H acc (pre@[hd]) = (H (foldl A B H acc pre) hd).
+ #A #B #H #acc #pre generalize in match acc; -acc; elim pre
+ [ normalize; //
+ | #hd #tl #IH #acc #X normalize; @IH ]
+qed.
+
+lemma foldl_append:
+ ∀A:Type[0].
+ ∀B: Type[0].
+ ∀H: A → B → A.
+ ∀acc: A.
+ ∀suff,pre: list B.
+ foldl A B H acc (pre@suff) = (foldl A B H (foldl A B H acc pre) suff).
+ #A #B #H #acc #suff elim suff
+ [ #pre >append_nil %
+ | #hd #tl #IH #pre whd in ⊢ (???%) <(foldl_step … H ??) applyS (IH (pre@[hd])) ]
+qed.
+
+definition flatten ≝
+ λA: Type[0].
+ λl: list (list A).
+ foldr ? ? (append ?) [ ] l.
+
+let rec rev (A: Type[0]) (l: list A) on l ≝
+ match l with
+ [ nil ⇒ nil A
+ | cons hd tl ⇒ (rev A tl) @ [ hd ]
+ ].
+
+lemma append_length:
+ ∀A: Type[0].
+ ∀l, r: list A.
+ |(l @ r)| = |l| + |r|.
+ #A #L #R
+ elim L
+ [ %
+ | #HD #TL #IH
+ normalize >IH %
+ ]
+qed.
+
+lemma append_nil:
+ ∀A: Type[0].
+ ∀l: list A.
+ l @ [ ] = l.
+ #A #L
+ elim L //
+qed.
+
+lemma rev_append:
+ ∀A: Type[0].
+ ∀l, r: list A.
+ rev A (l @ r) = rev A r @ rev A l.
+ #A #L #R
+ elim L
+ [ normalize >append_nil %
+ | #HD #TL #IH
+ normalize >IH
+ @associative_append
+ ]
+qed.
+
+lemma rev_length:
+ ∀A: Type[0].
+ ∀l: list A.
+ |rev A l| = |l|.
+ #A #L
+ elim L
+ [ %
+ | #HD #TL #IH
+ normalize
+ >(append_length A (rev A TL) [HD])
+ normalize /2/
+ ]
+qed.
+
+lemma nth_append_first:
+ ∀A:Type[0].
+ ∀n:nat.∀l1,l2:list A.∀d:A.
+ n < |l1| → nth n A (l1@l2) d = nth n A l1 d.
+ #A #n #l1 #l2 #d
+ generalize in match n; -n; elim l1
+ [ normalize #k #Hk @⊥ @(absurd … Hk) @not_le_Sn_O
+ | #h #t #Hind #k normalize
+ cases k -k
+ [ #Hk normalize @refl
+ | #k #Hk normalize @(Hind k) @le_S_S_to_le @Hk
+ ]
+ ]
+qed.
+
+lemma nth_append_second:
+ ∀A: Type[0].∀n.∀l1,l2:list A.∀d.n ≥ length A l1 ->
+ nth n A (l1@l2) d = nth (n - length A l1) A l2 d.
+ #A #n #l1 #l2 #d
+ generalize in match n; -n; elim l1
+ [ normalize #k #Hk <(minus_n_O) @refl
+ | #h #t #Hind #k normalize
+ cases k -k;
+ [ #Hk @⊥ @(absurd (S (|t|) ≤ 0)) [ @Hk | @not_le_Sn_O ]
+ | #k #Hk normalize @(Hind k) @le_S_S_to_le @Hk
+ ]
+ ]
+qed.
+
+
+notation > "'if' term 19 e 'then' term 19 t 'else' term 48 f" non associative with precedence 19
+ for @{ match $e in bool with [ true ⇒ $t | false ⇒ $f] }.
+notation < "hvbox('if' \nbsp term 19 e \nbsp break 'then' \nbsp term 19 t \nbsp break 'else' \nbsp term 48 f \nbsp)" non associative with precedence 19
+ for @{ match $e with [ true ⇒ $t | false ⇒ $f] }.
+
+let rec fold_left_i_aux (A: Type[0]) (B: Type[0])
+ (f: nat → A → B → A) (x: A) (i: nat) (l: list B) on l ≝
+ match l with
+ [ nil ⇒ x
+ | cons hd tl ⇒ fold_left_i_aux A B f (f i x hd) (S i) tl
+ ].
+
+definition fold_left_i ≝ λA,B,f,x. fold_left_i_aux A B f x O.
+
+notation "hvbox(t⌈o ↦ h⌉)"
+ with precedence 45
+ for @{ match (? : $o=$h) with [ refl ⇒ $t ] }.
+
+definition function_apply ≝
+ λA, B: Type[0].
+ λf: A → B.
+ λa: A.
+ f a.
+
+notation "f break $ x"
+ left associative with precedence 99
+ for @{ 'function_apply $f $x }.
+
+interpretation "Function application" 'function_apply f x = (function_apply ? ? f x).
+
+let rec iterate (A: Type[0]) (f: A → A) (a: A) (n: nat) on n ≝
+ match n with
+ [ O ⇒ a
+ | S o ⇒ f (iterate A f a o)
+ ].
+
+(* Yeah, I probably ought to do something more general... *)
+notation "hvbox(\langle term 19 a, break term 19 b, break term 19 c\rangle)"
+with precedence 90 for @{ 'triple $a $b $c}.
+interpretation "Triple construction" 'triple x y z = (pair ? ? (pair ? ? x y) z).
+
+notation "hvbox(\langle term 19 a, break term 19 b, break term 19 c, break term 19 d\rangle)"
+with precedence 90 for @{ 'quadruple $a $b $c $d}.
+interpretation "Quadruple construction" 'quadruple w x y z = (pair ? ? (pair ? ? w x) (pair ? ? y z)).
+
+notation > "hvbox('let' 〈ident w,ident x,ident y,ident z〉 ≝ t 'in' s)"
+ with precedence 10
+for @{ match $t with [ pair ${fresh wx} ${fresh yz} ⇒ match ${fresh wx} with [ pair ${ident w} ${ident x} ⇒ match ${fresh yz} with [ pair ${ident y} ${ident z} ⇒ $s ] ] ] }.
+
+notation > "hvbox('let' 〈ident x,ident y,ident z〉 ≝ t 'in' s)"
+ with precedence 10
+for @{ match $t with [ pair ${fresh xy} ${ident z} ⇒ match ${fresh xy} with [ pair ${ident x} ${ident y} ⇒ $s ] ] }.
+
+notation < "hvbox('let' \nbsp hvbox(〈ident x,ident y〉\nbsp ≝ break t \nbsp 'in' \nbsp) break s)"
+ with precedence 10
+for @{ match $t with [ pair (${ident x}:$ignore) (${ident y}:$ignora) ⇒ $s ] }.
+
+axiom pair_elim':
+ ∀A,B,C: Type[0].
+ ∀T: A → B → C.
+ ∀p.
+ ∀P: A×B → C → Prop.
+ (∀lft, rgt. p = 〈lft,rgt〉 → P 〈lft,rgt〉 (T lft rgt)) →
+ P p (let 〈lft, rgt〉 ≝ p in T lft rgt).
+
+axiom pair_elim'':
+ ∀A,B,C,C': Type[0].
+ ∀T: A → B → C.
+ ∀T': A → B → C'.
+ ∀p.
+ ∀P: A×B → C → C' → Prop.
+ (∀lft, rgt. p = 〈lft,rgt〉 → P 〈lft,rgt〉 (T lft rgt) (T' lft rgt)) →
+ P p (let 〈lft, rgt〉 ≝ p in T lft rgt) (let 〈lft, rgt〉 ≝ p in T' lft rgt).
+
+lemma pair_destruct_1:
+ ∀A,B.∀a:A.∀b:B.∀c. 〈a,b〉 = c → a = \fst c.
+ #A #B #a #b *; /2/
+qed.
+
+lemma pair_destruct_2:
+ ∀A,B.∀a:A.∀b:B.∀c. 〈a,b〉 = c → b = \snd c.
+ #A #B #a #b *; /2/
+qed.
+
+
+let rec exclusive_disjunction (b: bool) (c: bool) on b ≝
+ match b with
+ [ true ⇒
+ match c with
+ [ false ⇒ true
+ | true ⇒ false
+ ]
+ | false ⇒
+ match c with
+ [ false ⇒ false
+ | true ⇒ true
+ ]
+ ].
+
+(* dpm: conflicts with library definitions
+interpretation "Nat less than" 'lt m n = (ltb m n).
+interpretation "Nat greater than" 'gt m n = (gtb m n).
+interpretation "Nat greater than eq" 'geq m n = (geb m n).
+*)
+
+let rec division_aux (m: nat) (n : nat) (p: nat) ≝
+ match ltb n (S p) with
+ [ true ⇒ O
+ | false ⇒
+ match m with
+ [ O ⇒ O
+ | (S q) ⇒ S (division_aux q (n - (S p)) p)
+ ]
+ ].
+
+definition division ≝
+ λm, n: nat.
+ match n with
+ [ O ⇒ S m
+ | S o ⇒ division_aux m m o
+ ].
+
+notation "hvbox(n break ÷ m)"
+ right associative with precedence 47
+ for @{ 'division $n $m }.
+
+interpretation "Nat division" 'division n m = (division n m).
+
+let rec modulus_aux (m: nat) (n: nat) (p: nat) ≝
+ match leb n p with
+ [ true ⇒ n
+ | false ⇒
+ match m with
+ [ O ⇒ n
+ | S o ⇒ modulus_aux o (n - (S p)) p
+ ]
+ ].
+
+definition modulus ≝
+ λm, n: nat.
+ match n with
+ [ O ⇒ m
+ | S o ⇒ modulus_aux m m o
+ ].
+
+notation "hvbox(n break 'mod' m)"
+ right associative with precedence 47
+ for @{ 'modulus $n $m }.
+
+interpretation "Nat modulus" 'modulus m n = (modulus m n).
+
+definition divide_with_remainder ≝
+ λm, n: nat.
+ pair ? ? (m ÷ n) (modulus m n).
+
+let rec exponential (m: nat) (n: nat) on n ≝
+ match n with
+ [ O ⇒ S O
+ | S o ⇒ m * exponential m o
+ ].
+
+interpretation "Nat exponential" 'exp n m = (exponential n m).
+
+notation "hvbox(a break ⊎ b)"
+ left associative with precedence 50
+for @{ 'disjoint_union $a $b }.
+interpretation "sum" 'disjoint_union A B = (Sum A B).
+
+theorem less_than_or_equal_monotone:
+ ∀m, n: nat.
+ m ≤ n → (S m) ≤ (S n).
+ #m #n #H
+ elim H
+ /2/
+qed.
+
+theorem less_than_or_equal_b_complete:
+ ∀m, n: nat.
+ leb m n = false → ¬(m ≤ n).
+ #m;
+ elim m;
+ normalize
+ [ #n #H
+ destruct
+ | #y #H1 #z
+ cases z
+ normalize
+ [ #H
+ /2/
+ | /3/
+ ]
+ ]
+qed.
+
+theorem less_than_or_equal_b_correct:
+ ∀m, n: nat.
+ leb m n = true → m ≤ n.
+ #m
+ elim m
+ //
+ #y #H1 #z
+ cases z
+ normalize
+ [ #H
+ destruct
+ | #n #H lapply (H1 … H) /2/
+ ]
+qed.
+
+definition less_than_or_equal_b_elim:
+ ∀m, n: nat.
+ ∀P: bool → Type[0].
+ (m ≤ n → P true) → (¬(m ≤ n) → P false) → P (leb m n).
+ #m #n #P #H1 #H2;
+ lapply (less_than_or_equal_b_correct m n)
+ lapply (less_than_or_equal_b_complete m n)
+ cases (leb m n)
+ /3/
+qed.
+
+lemma inclusive_disjunction_true:
+ ∀b, c: bool.
+ (orb b c) = true → b = true ∨ c = true.
+ # b
+ # c
+ elim b
+ [ normalize
+ # H
+ @ or_introl
+ %
+ | normalize
+ /2/
+ ]
+qed.
+
+lemma conjunction_true:
+ ∀b, c: bool.
+ andb b c = true → b = true ∧ c = true.
+ # b
+ # c
+ elim b
+ normalize
+ [ /2/
+ | # K
+ destruct
+ ]
+qed.
+
+lemma eq_true_false: false=true → False.
+ # K
+ destruct
+qed.
+
+lemma inclusive_disjunction_b_true: ∀b. orb b true = true.
+ # b
+ cases b
+ %
+qed.
+
+definition bool_to_Prop ≝
+ λb. match b with [ true ⇒ True | false ⇒ False ].
+
+coercion bool_to_Prop: ∀b:bool. Prop ≝ bool_to_Prop on _b:bool to Type[0].
+
+lemma eq_false_to_notb: ∀b. b = false → ¬ b.
+ *; /2/
+qed.
+
+lemma length_append:
+ ∀A.∀l1,l2:list A.
+ |l1 @ l2| = |l1| + |l2|.
+ #A #l1 elim l1
+ [ //
+ | #hd #tl #IH #l2 normalize \ 5ih ]="" qed.=""\ 6\ 5/ih\ 6\ 5/cons_prf\ 6\ 5/prfcr\ 6\ 5/refl_prf\ 6\ 5/cons_prf\ 6\ 5/prfcr\ 6\ 5/nil_prf\ 6\ 5/prfcr\ 6
\ No newline at end of file
(* sigma *)
inductive Sig (A:Type[0]) (f:A→Type[0]) : Type[0] ≝
- dp: ∀a:A.(f a)→Sig A f.
\ No newline at end of file
+ dp: ∀a:A.(f a)→Sig A f.
+
+interpretation "Sigma" 'sigma x = (Sig ? x).
-
include "tutorial/chapter2.ma".
include "basics/bool.ma".
interpretation "in_prl" 'in_l E = (in_prl ? E).
lemma not_epsilon_lp :∀S.∀pi:\ 5a href="cic:/matita/tutorial/chapter4/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S. \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6 ((\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6ℓ\ 5/a\ 6 pi) \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]).
-#S #pi (elim pi) normalize /2/
- [#pi1 #pi2 #H1 #H2 % * /2/ * #w1 * #w2 * * #appnil
- cases (\ 5a href="cic:/matita/tutorial/chapter3/nil_to_nil.def(5)"\ 6nil_to_nil\ 5/a\ 6 … appnil) /2/
- |#p11 #p12 #H1 #H2 % * /2/
- |#pi #H % * #w1 * #w2 * * #appnil (cases (\ 5a href="cic:/matita/tutorial/chapter3/nil_to_nil.def(5)"\ 6nil_to_nil\ 5/a\ 6 … appnil)) /2/
+#S #pi (elim pi) normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Not.con(0,1,1)"\ 6nmk\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+ [#pi1 #pi2 #H1 #H2 % * /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ * #w1 * #w2 * * #appnil
+ cases (\ 5a href="cic:/matita/tutorial/chapter3/nil_to_nil.def(5)"\ 6nil_to_nil\ 5/a\ 6 … appnil) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+ |#p11 #p12 #H1 #H2 % * /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+ |#pi #H % * #w1 * #w2 * * #appnil (cases (\ 5a href="cic:/matita/tutorial/chapter3/nil_to_nil.def(5)"\ 6nil_to_nil\ 5/a\ 6 … appnil)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
]
qed.
#S #e #H %2 >H // qed.
lemma if_epsilon_true : ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter4/pre.def(1)"\ 6pre\ 5/a\ 6 S. \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 ] \ 5a title="in_prl mem" href="cic:/fakeuri.def(1)"\ 6∈\ 5/a\ 6 e → \ 5a title="snd" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 e \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
-#S * #pi #b * [#abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /2/] cases b normalize // @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6
+#S * #pi #b * [normalize #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /2/] cases b normalize // @False_ind
qed.
-
+definition lor ≝ λS:Alpha.λa,b:pre S.〈\fst a + \fst b,\snd a ∨ \snd b〉.
+
+notation "a ⊕ b" left associative with precedence 60 for @{'oplus $a $b}.
+interpretation "oplus" 'oplus a b = (lo ? a b).
+
+ndefinition lc ≝ λS:Alpha.λbcast:∀S:Alpha.∀E:pitem S.pre S.λa,b:pre S.
+ match a with [ mk_pair e1 b1 ⇒
+ match b1 with
+ [ false ⇒ 〈e1 · \fst b, \snd b〉
+ | true ⇒ 〈e1 · \fst (bcast ? (\fst b)),\snd b || \snd (bcast ? (\fst b))〉]].
+
+notation < "a ⊙ b" left associative with precedence 60 for @{'lc $op $a $b}.
+interpretation "lc" 'lc op a b = (lc ? op a b).
+notation > "a ⊙ b" left associative with precedence 60 for @{'lc eclose $a $b}.
+
+ndefinition lk ≝ λS:Alpha.λbcast:∀S:Alpha.∀E:pitem S.pre S.λa:pre S.
+ match a with [ mk_pair e1 b1 ⇒
+ match b1 with
+ [ false ⇒ 〈e1^*, false〉
+ | true ⇒ 〈(\fst (bcast ? e1))^*, true〉]].
+
+notation < "a \sup ⊛" non associative with precedence 90 for @{'lk $op $a}.
+interpretation "lk" 'lk op a = (lk ? op a).
+notation > "a^⊛" non associative with precedence 90 for @{'lk eclose $a}.
+
+notation > "•" non associative with precedence 60 for @{eclose ?}.
+nlet rec eclose (S: Alpha) (E: pitem S) on E : pre S ≝
+ match E with
+ [ pz ⇒ 〈 ∅, false 〉
+ | pe ⇒ 〈 ϵ, true 〉
+ | ps x ⇒ 〈 `.x, false 〉
+ | pp x ⇒ 〈 `.x, false 〉
+ | po E1 E2 ⇒ •E1 ⊕ •E2
+ | pc E1 E2 ⇒ •E1 ⊙ 〈 E2, false 〉
+ | pk E ⇒ 〈(\fst (•E))^*,true〉].
+notation < "• x" non associative with precedence 60 for @{'eclose $x}.
+interpretation "eclose" 'eclose x = (eclose ? x).
+notation > "• x" non associative with precedence 60 for @{'eclose $x}.
+
+ndefinition reclose ≝ λS:Alpha.λp:pre S.let p' ≝ •\fst p in 〈\fst p',\snd p || \snd p'〉.
+interpretation "reclose" 'eclose x = (reclose ? x).
+
+ndefinition eq_f1 ≝ λS.λa,b:word S → Prop.∀w.a w ↔ b w.
+notation > "A =1 B" non associative with precedence 45 for @{'eq_f1 $A $B}.
+notation "A =\sub 1 B" non associative with precedence 45 for @{'eq_f1 $A $B}.
+interpretation "eq f1" 'eq_f1 a b = (eq_f1 ? a b).
+
+naxiom extP : ∀S.∀p,q:word S → Prop.(p =1 q) → p = q.
+
+nlemma epsilon_or : ∀S:Alpha.∀b1,b2. ϵ(b1 || b2) = ϵ b1 ∪ ϵ b2. ##[##2: napply S]
+#S b1 b2; ncases b1; ncases b2; napply extP; #w; nnormalize; @; /2/; *; //; *;
+nqed.
+
+nlemma cupA : ∀S.∀a,b,c:word S → Prop.a ∪ b ∪ c = a ∪ (b ∪ c).
+#S a b c; napply extP; #w; nnormalize; @; *; /3/; *; /3/; nqed.
+
+nlemma cupC : ∀S. ∀a,b:word S → Prop.a ∪ b = b ∪ a.
+#S a b; napply extP; #w; @; *; nnormalize; /2/; nqed.
+
+(* theorem 16: 2 *)
+nlemma oplus_cup : ∀S:Alpha.∀e1,e2:pre S.𝐋\p (e1 ⊕ e2) = 𝐋\p e1 ∪ 𝐋\p e2.
+#S r1; ncases r1; #e1 b1 r2; ncases r2; #e2 b2;
+nwhd in ⊢ (??(??%)?);
+nchange in ⊢(??%?) with (𝐋\p (e1 + e2) ∪ ϵ (b1 || b2));
+nchange in ⊢(??(??%?)?) with (𝐋\p (e1) ∪ 𝐋\p (e2));
+nrewrite > (epsilon_or S …); nrewrite > (cupA S (𝐋\p e1) …);
+nrewrite > (cupC ? (ϵ b1) …); nrewrite < (cupA S (𝐋\p e2) …);
+nrewrite > (cupC ? ? (ϵ b1) …); nrewrite < (cupA …); //;
+nqed.
+
+nlemma odotEt :
+ ∀S.∀e1,e2:pitem S.∀b2. 〈e1,true〉 ⊙ 〈e2,b2〉 = 〈e1 · \fst (•e2),b2 || \snd (•e2)〉.
+#S e1 e2 b2; nnormalize; ncases (•e2); //; nqed.
+
+nlemma LcatE : ∀S.∀e1,e2:pitem S.𝐋\p (e1 · e2) = 𝐋\p e1 · 𝐋 |e2| ∪ 𝐋\p e2. //; nqed.
+
+nlemma cup_dotD : ∀S.∀p,q,r:word S → Prop.(p ∪ q) · r = (p · r) ∪ (q · r).
+#S p q r; napply extP; #w; nnormalize; @;
+##[ *; #x; *; #y; *; *; #defw; *; /7/ by or_introl, or_intror, ex_intro, conj;
+##| *; *; #x; *; #y; *; *; /7/ by or_introl, or_intror, ex_intro, conj; ##]
+nqed.
+
+nlemma cup0 :∀S.∀p:word S → Prop.p ∪ {} = p.
+#S p; napply extP; #w; nnormalize; @; /2/; *; //; *; nqed.
+
+nlemma erase_dot : ∀S.∀e1,e2:pitem S.𝐋 |e1 · e2| = 𝐋 |e1| · 𝐋 |e2|.
+#S e1 e2; napply extP; nnormalize; #w; @; *; #w1; *; #w2; *; *; /7/ by ex_intro, conj;
+nqed.
+
+nlemma erase_plus : ∀S.∀e1,e2:pitem S.𝐋 |e1 + e2| = 𝐋 |e1| ∪ 𝐋 |e2|.
+#S e1 e2; napply extP; nnormalize; #w; @; *; /4/ by or_introl, or_intror; nqed.
+
+nlemma erase_star : ∀S.∀e1:pitem S.𝐋 |e1|^* = 𝐋 |e1^*|. //; nqed.
+
+ndefinition substract := λS.λp,q:word S → Prop.λw.p w ∧ ¬ q w.
+interpretation "substract" 'minus a b = (substract ? a b).
+
+nlemma cup_sub: ∀S.∀a,b:word S → Prop. ¬ (a []) → a ∪ (b - {[]}) = (a ∪ b) - {[]}.
+#S a b c; napply extP; #w; nnormalize; @; *; /4/; *; /4/; nqed.
+
+nlemma sub0 : ∀S.∀a:word S → Prop. a - {} = a.
+#S a; napply extP; #w; nnormalize; @; /3/; *; //; nqed.
+
+nlemma subK : ∀S.∀a:word S → Prop. a - a = {}.
+#S a; napply extP; #w; nnormalize; @; *; /2/; nqed.
+
+nlemma subW : ∀S.∀a,b:word S → Prop.∀w.(a - b) w → a w.
+#S a b w; nnormalize; *; //; nqed.
+
+nlemma erase_bull : ∀S.∀a:pitem S. |\fst (•a)| = |a|.
+#S a; nelim a; // by {};
+##[ #e1 e2 IH1 IH2; nchange in ⊢ (???%) with (|e1| · |e2|);
+ nrewrite < IH1; nrewrite < IH2;
+ nchange in ⊢ (??(??%)?) with (\fst (•e1 ⊙ 〈e2,false〉));
+ ncases (•e1); #e3 b; ncases b; nnormalize;
+ ##[ ncases (•e2); //; ##| nrewrite > IH2; //]
+##| #e1 e2 IH1 IH2; nchange in ⊢ (???%) with (|e1| + |e2|);
+ nrewrite < IH2; nrewrite < IH1;
+ nchange in ⊢ (??(??%)?) with (\fst (•e1 ⊕ •e2));
+ ncases (•e1); ncases (•e2); //;
+##| #e IH; nchange in ⊢ (???%) with (|e|^* ); nrewrite < IH;
+ nchange in ⊢ (??(??%)?) with (\fst (•e))^*; //; ##]
+nqed.
+
+nlemma eta_lp : ∀S.∀p:pre S.𝐋\p p = 𝐋\p 〈\fst p, \snd p〉.
+#S p; ncases p; //; nqed.
+
+nlemma epsilon_dot: ∀S.∀p:word S → Prop. {[]} · p = p.
+#S e; napply extP; #w; nnormalize; @; ##[##2: #Hw; @[]; @w; /3/; ##]
+*; #w1; *; #w2; *; *; #defw defw1 Hw2; nrewrite < defw; nrewrite < defw1;
+napply Hw2; nqed.
+
+(* theorem 16: 1 → 3 *)
+nlemma odot_dot_aux : ∀S.∀e1,e2: pre S.
+ 𝐋\p (•(\fst e2)) = 𝐋\p (\fst e2) ∪ 𝐋 |\fst e2| →
+ 𝐋\p (e1 ⊙ e2) = 𝐋\p e1 · 𝐋 |\fst e2| ∪ 𝐋\p e2.
+#S e1 e2 th1; ncases e1; #e1' b1'; ncases b1';
+##[ nwhd in ⊢ (??(??%)?); nletin e2' ≝ (\fst e2); nletin b2' ≝ (\snd e2);
+ nletin e2'' ≝ (\fst (•(\fst e2))); nletin b2'' ≝ (\snd (•(\fst e2)));
+ nchange in ⊢ (??%?) with (?∪?);
+ nchange in ⊢ (??(??%?)?) with (?∪?);
+ nchange in match (𝐋\p 〈?,?〉) with (?∪?);
+ nrewrite > (epsilon_or …); nrewrite > (cupC ? (ϵ ?)…);
+ nrewrite > (cupA …);nrewrite < (cupA ?? (ϵ?)…);
+ nrewrite > (?: 𝐋\p e2'' ∪ ϵ b2'' = 𝐋\p e2' ∪ 𝐋 |e2'|); ##[##2:
+ nchange with (𝐋\p 〈e2'',b2''〉 = 𝐋\p e2' ∪ 𝐋 |e2'|);
+ ngeneralize in match th1;
+ nrewrite > (eta_lp…); #th1; nrewrite > th1; //;##]
+ nrewrite > (eta_lp ? e2);
+ nchange in match (𝐋\p 〈\fst e2,?〉) with (𝐋\p e2'∪ ϵ b2');
+ nrewrite > (cup_dotD …); nrewrite > (epsilon_dot…);
+ nrewrite > (cupC ? (𝐋\p e2')…); nrewrite > (cupA…);nrewrite > (cupA…);
+ nrewrite < (erase_bull S e2') in ⊢ (???(??%?)); //;
+##| ncases e2; #e2' b2'; nchange in match (〈e1',false〉⊙?) with 〈?,?〉;
+ nchange in match (𝐋\p ?) with (?∪?);
+ nchange in match (𝐋\p (e1'·?)) with (?∪?);
+ nchange in match (𝐋\p 〈e1',?〉) with (?∪?);
+ nrewrite > (cup0…);
+ nrewrite > (cupA…); //;##]
+nqed.
+
+nlemma sub_dot_star :
+ ∀S.∀X:word S → Prop.∀b. (X - ϵ b) · X^* ∪ {[]} = X^*.
+#S X b; napply extP; #w; @;
+##[ *; ##[##2: nnormalize; #defw; nrewrite < defw; @[]; @; //]
+ *; #w1; *; #w2; *; *; #defw sube; *; #lw; *; #flx cj;
+ @ (w1 :: lw); nrewrite < defw; nrewrite < flx; @; //;
+ @; //; napply (subW … sube);
+##| *; #wl; *; #defw Pwl; nrewrite < defw; nelim wl in Pwl; ##[ #_; @2; //]
+ #w' wl' IH; *; #Pw' IHp; nlapply (IH IHp); *;
+ ##[ *; #w1; *; #w2; *; *; #defwl' H1 H2;
+ @; ncases b in H1; #H1;
+ ##[##2: nrewrite > (sub0…); @w'; @(w1@w2);
+ nrewrite > (associative_append ? w' w1 w2);
+ nrewrite > defwl'; @; ##[@;//] @(wl'); @; //;
+ ##| ncases w' in Pw';
+ ##[ #ne; @w1; @w2; nrewrite > defwl'; @; //; @; //;
+ ##| #x xs Px; @(x::xs); @(w1@w2);
+ nrewrite > (defwl'); @; ##[@; //; @; //; @; nnormalize; #; ndestruct]
+ @wl'; @; //; ##] ##]
+ ##| #wlnil; nchange in match (flatten ? (w'::wl')) with (w' @ flatten ? wl');
+ nrewrite < (wlnil); nrewrite > (append_nil…); ncases b;
+ ##[ ncases w' in Pw'; /2/; #x xs Pxs; @; @(x::xs); @([]);
+ nrewrite > (append_nil…); @; ##[ @; //;@; //; nnormalize; @; #; ndestruct]
+ @[]; @; //;
+ ##| @; @w'; @[]; nrewrite > (append_nil…); @; ##[##2: @[]; @; //]
+ @; //; @; //; @; *;##]##]##]
+nqed.
+
+(* theorem 16: 1 *)
+alias symbol "pc" (instance 13) = "cat lang".
+alias symbol "in_pl" (instance 23) = "in_pl".
+alias symbol "in_pl" (instance 5) = "in_pl".
+alias symbol "eclose" (instance 21) = "eclose".
+ntheorem bull_cup : ∀S:Alpha. ∀e:pitem S. 𝐋\p (•e) = 𝐋\p e ∪ 𝐋 |e|.
+#S e; nelim e; //;
+ ##[ #a; napply extP; #w; nnormalize; @; *; /3/ by or_introl, or_intror;
+ ##| #a; napply extP; #w; nnormalize; @; *; /3/ by or_introl; *;
+ ##| #e1 e2 IH1 IH2;
+ nchange in ⊢ (??(??(%))?) with (•e1 ⊙ 〈e2,false〉);
+ nrewrite > (odot_dot_aux S (•e1) 〈e2,false〉 IH2);
+ nrewrite > (IH1 …); nrewrite > (cup_dotD …);
+ nrewrite > (cupA …); nrewrite > (cupC ?? (𝐋\p ?) …);
+ nchange in match (𝐋\p 〈?,?〉) with (𝐋\p e2 ∪ {}); nrewrite > (cup0 …);
+ nrewrite < (erase_dot …); nrewrite < (cupA …); //;
+ ##| #e1 e2 IH1 IH2;
+ nchange in match (•(?+?)) with (•e1 ⊕ •e2); nrewrite > (oplus_cup …);
+ nrewrite > (IH1 …); nrewrite > (IH2 …); nrewrite > (cupA …);
+ nrewrite > (cupC ? (𝐋\p e2)…);nrewrite < (cupA ??? (𝐋\p e2)…);
+ nrewrite > (cupC ?? (𝐋\p e2)…); nrewrite < (cupA …);
+ nrewrite < (erase_plus …); //.
+ ##| #e; nletin e' ≝ (\fst (•e)); nletin b' ≝ (\snd (•e)); #IH;
+ nchange in match (𝐋\p ?) with (𝐋\p 〈e'^*,true〉);
+ nchange in match (𝐋\p ?) with (𝐋\p (e'^* ) ∪ {[ ]});
+ nchange in ⊢ (??(??%?)?) with (𝐋\p e' · 𝐋 |e'|^* );
+ nrewrite > (erase_bull…e);
+ nrewrite > (erase_star …);
+ nrewrite > (?: 𝐋\p e' = 𝐋\p e ∪ (𝐋 |e| - ϵ b')); ##[##2:
+ nchange in IH : (??%?) with (𝐋\p e' ∪ ϵ b'); ncases b' in IH;
+ ##[ #IH; nrewrite > (cup_sub…); //; nrewrite < IH;
+ nrewrite < (cup_sub…); //; nrewrite > (subK…); nrewrite > (cup0…);//;
+ ##| nrewrite > (sub0 …); #IH; nrewrite < IH; nrewrite > (cup0 …);//; ##]##]
+ nrewrite > (cup_dotD…); nrewrite > (cupA…);
+ nrewrite > (?: ((?·?)∪{[]} = 𝐋 |e^*|)); //;
+ nchange in match (𝐋 |e^*|) with ((𝐋 |e|)^* ); napply sub_dot_star;##]
+ nqed.
+
+(* theorem 16: 3 *)
+nlemma odot_dot:
+ ∀S.∀e1,e2: pre S. 𝐋\p (e1 ⊙ e2) = 𝐋\p e1 · 𝐋 |\fst e2| ∪ 𝐋\p e2.
+#S e1 e2; napply odot_dot_aux; napply (bull_cup S (\fst e2)); nqed.
+
+nlemma dot_star_epsilon : ∀S.∀e:re S.𝐋 e · 𝐋 e^* ∪ {[]} = 𝐋 e^*.
+#S e; napply extP; #w; nnormalize; @;
+##[ *; ##[##2: #H; nrewrite < H; @[]; /3/] *; #w1; *; #w2;
+ *; *; #defw Hw1; *; #wl; *; #defw2 Hwl; @(w1 :: wl);
+ nrewrite < defw; nrewrite < defw2; @; //; @;//;
+##| *; #wl; *; #defw Hwl; ncases wl in defw Hwl; ##[#defw; #; @2; nrewrite < defw; //]
+ #x xs defw; *; #Hx Hxs; @; @x; @(flatten ? xs); nrewrite < defw;
+ @; /2/; @xs; /2/;##]
+ nqed.
+
+nlemma nil_star : ∀S.∀e:re S. [ ] ∈ e^*.
+#S e; @[]; /2/; nqed.
+
+nlemma cupID : ∀S.∀l:word S → Prop.l ∪ l = l.
+#S l; napply extP; #w; @; ##[*]//; #; @; //; nqed.
+
+nlemma cup_star_nil : ∀S.∀l:word S → Prop. l^* ∪ {[]} = l^*.
+#S a; napply extP; #w; @; ##[*; //; #H; nrewrite < H; @[]; @; //] #;@; //;nqed.
+
+nlemma rcanc_sing : ∀S.∀A,C:word S → Prop.∀b:word S .
+ ¬ (A b) → A ∪ { (b) } = C → A = C - { (b) }.
+#S A C b nbA defC; nrewrite < defC; napply extP; #w; @;
+##[ #Aw; /3/| *; *; //; #H nH; ncases nH; #abs; nlapply (abs H); *]
+nqed.
+
+(* theorem 16: 4 *)
+nlemma star_dot: ∀S.∀e:pre S. 𝐋\p (e^⊛) = 𝐋\p e · (𝐋 |\fst e|)^*.
+#S p; ncases p; #e b; ncases b;
+##[ nchange in match (〈e,true〉^⊛) with 〈?,?〉;
+ nletin e' ≝ (\fst (•e)); nletin b' ≝ (\snd (•e));
+ nchange in ⊢ (??%?) with (?∪?);
+ nchange in ⊢ (??(??%?)?) with (𝐋\p e' · 𝐋 |e'|^* );
+ nrewrite > (?: 𝐋\p e' = 𝐋\p e ∪ (𝐋 |e| - ϵ b' )); ##[##2:
+ nlapply (bull_cup ? e); #bc;
+ nchange in match (𝐋\p (•e)) in bc with (?∪?);
+ nchange in match b' in bc with b';
+ ncases b' in bc; ##[##2: nrewrite > (cup0…); nrewrite > (sub0…); //]
+ nrewrite > (cup_sub…); ##[napply rcanc_sing] //;##]
+ nrewrite > (cup_dotD…); nrewrite > (cupA…);nrewrite > (erase_bull…);
+ nrewrite > (sub_dot_star…);
+ nchange in match (𝐋\p 〈?,?〉) with (?∪?);
+ nrewrite > (cup_dotD…); nrewrite > (epsilon_dot…); //;
+##| nwhd in match (〈e,false〉^⊛); nchange in match (𝐋\p 〈?,?〉) with (?∪?);
+ nrewrite > (cup0…);
+ nchange in ⊢ (??%?) with (𝐋\p e · 𝐋 |e|^* );
+ nrewrite < (cup0 ? (𝐋\p e)); //;##]
+nqed.
+
+nlet rec pre_of_re (S : Alpha) (e : re S) on e : pitem S ≝
+ match e with
+ [ z ⇒ pz ?
+ | e ⇒ pe ?
+ | s x ⇒ ps ? x
+ | c e1 e2 ⇒ pc ? (pre_of_re ? e1) (pre_of_re ? e2)
+ | o e1 e2 ⇒ po ? (pre_of_re ? e1) (pre_of_re ? e2)
+ | k e1 ⇒ pk ? (pre_of_re ? e1)].
+
+nlemma notFalse : ¬False. @; //; nqed.
+
+nlemma dot0 : ∀S.∀A:word S → Prop. {} · A = {}.
+#S A; nnormalize; napply extP; #w; @; ##[##2: *]
+*; #w1; *; #w2; *; *; //; nqed.
+
+nlemma Lp_pre_of_re : ∀S.∀e:re S. 𝐋\p (pre_of_re ? e) = {}.
+#S e; nelim e; ##[##1,2,3: //]
+##[ #e1 e2 H1 H2; nchange in match (𝐋\p (pre_of_re S (e1 e2))) with (?∪?);
+ nrewrite > H1; nrewrite > H2; nrewrite > (dot0…); nrewrite > (cupID…);//
+##| #e1 e2 H1 H2; nchange in match (𝐋\p (pre_of_re S (e1+e2))) with (?∪?);
+ nrewrite > H1; nrewrite > H2; nrewrite > (cupID…); //
+##| #e1 H1; nchange in match (𝐋\p (pre_of_re S (e1^* ))) with (𝐋\p (pre_of_re ??) · ?);
+ nrewrite > H1; napply dot0; ##]
+nqed.
+
+nlemma erase_pre_of_reK : ∀S.∀e. 𝐋 |pre_of_re S e| = 𝐋 e.
+#S A; nelim A; //;
+##[ #e1 e2 H1 H2; nchange in match (𝐋 (e1 · e2)) with (𝐋 e1·?);
+ nrewrite < H1; nrewrite < H2; //
+##| #e1 e2 H1 H2; nchange in match (𝐋 (e1 + e2)) with (𝐋 e1 ∪ ?);
+ nrewrite < H1; nrewrite < H2; //
+##| #e1 H1; nchange in match (𝐋 (e1^* )) with ((𝐋 e1)^* );
+ nrewrite < H1; //]
+nqed.
+
+(* corollary 17 *)
+nlemma L_Lp_bull : ∀S.∀e:re S.𝐋 e = 𝐋\p (•pre_of_re ? e).
+#S e; nrewrite > (bull_cup…); nrewrite > (Lp_pre_of_re…);
+nrewrite > (cupC…); nrewrite > (cup0…); nrewrite > (erase_pre_of_reK…); //;
+nqed.
+
+nlemma Pext : ∀S.∀f,g:word S → Prop. f = g → ∀w.f w → g w.
+#S f g H; nrewrite > H; //; nqed.
+
+(* corollary 18 *)
+ntheorem bull_true_epsilon : ∀S.∀e:pitem S. \snd (•e) = true ↔ [ ] ∈ |e|.
+#S e; @;
+##[ #defsnde; nlapply (bull_cup ? e); nchange in match (𝐋\p (•e)) with (?∪?);
+ nrewrite > defsnde; #H;
+ nlapply (Pext ??? H [ ] ?); ##[ @2; //] *; //;
+
+STOP
+
+notation > "\move term 90 x term 90 E"
+non associative with precedence 60 for @{move ? $x $E}.
+nlet rec move (S: Alpha) (x:S) (E: pitem S) on E : pre S ≝
+ match E with
+ [ pz ⇒ 〈 ∅, false 〉
+ | pe ⇒ 〈 ϵ, false 〉
+ | ps y ⇒ 〈 `y, false 〉
+ | pp y ⇒ 〈 `y, x == y 〉
+ | po e1 e2 ⇒ \move x e1 ⊕ \move x e2
+ | pc e1 e2 ⇒ \move x e1 ⊙ \move x e2
+ | pk e ⇒ (\move x e)^⊛ ].
+notation < "\move\shy x\shy E" non associative with precedence 60 for @{'move $x $E}.
+notation > "\move term 90 x term 90 E" non associative with precedence 60 for @{'move $x $E}.
+interpretation "move" 'move x E = (move ? x E).
+
+ndefinition rmove ≝ λS:Alpha.λx:S.λe:pre S. \move x (\fst e).
+interpretation "rmove" 'move x E = (rmove ? x E).
+
+nlemma XXz : ∀S:Alpha.∀w:word S. w ∈ ∅ → False.
+#S w abs; ninversion abs; #; ndestruct;
+nqed.
+
+
+nlemma XXe : ∀S:Alpha.∀w:word S. w .∈ ϵ → False.
+#S w abs; ninversion abs; #; ndestruct;
+nqed.
+
+nlemma XXze : ∀S:Alpha.∀w:word S. w .∈ (∅ · ϵ) → False.
+#S w abs; ninversion abs; #; ndestruct; /2/ by XXz,XXe;
+nqed.
+
+
+naxiom in_move_cat:
+ ∀S.∀w:word S.∀x.∀E1,E2:pitem S. w .∈ \move x (E1 · E2) →
+ (∃w1.∃w2. w = w1@w2 ∧ w1 .∈ \move x E1 ∧ w2 ∈ .|E2|) ∨ w .∈ \move x E2.
+#S w x e1 e2 H; nchange in H with (w .∈ \move x e1 ⊙ \move x e2);
+ncases e1 in H; ncases e2;
+##[##1: *; ##[*; nnormalize; #; ndestruct]
+ #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
+ nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz,XXze;
+##|##2: *; ##[*; nnormalize; #; ndestruct]
+ #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
+ nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz,XXze;
+##| #r; *; ##[ *; nnormalize; #; ndestruct]
+ #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
+ ##[##2: nnormalize; #; ndestruct; @2; @2; //.##]
+ nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz;
+##| #y; *; ##[ *; nnormalize; #defw defx; ndestruct; @2; @1; /2/ by conj;##]
+ #H; ninversion H; nnormalize; #; ndestruct;
+ ##[ncases (?:False); /2/ by XXz] /3/ by or_intror;
+##| #r1 r2; *; ##[ *; #defw]
+ ...
+nqed.
+
+ntheorem move_ok:
+ ∀S:Alpha.∀E:pre S.∀a,w.w .∈ \move a E ↔ (a :: w) .∈ E.
+#S E; ncases E; #r b; nelim r;
+##[##1,2: #a w; @;
+ ##[##1,3: nnormalize; *; ##[##1,3: *; #; ndestruct; ##| #abs; ncases (XXz … abs); ##]
+ #H; ninversion H; #; ndestruct;
+ ##|##*:nnormalize; *; ##[##1,3: *; #; ndestruct; ##| #H1; ncases (XXz … H1); ##]
+ #H; ninversion H; #; ndestruct;##]
+##|#a c w; @; nnormalize; ##[*; ##[*; #; ndestruct; ##] #abs; ninversion abs; #; ndestruct;##]
+ *; ##[##2: #abs; ninversion abs; #; ndestruct; ##] *; #; ndestruct;
+##|#a c w; @; nnormalize;
+ ##[ *; ##[ *; #defw; nrewrite > defw; #ca; @2; nrewrite > (eqb_t … ca); @; ##]
+ #H; ninversion H; #; ndestruct;
+ ##| *; ##[ *; #; ndestruct; ##] #H; ninversion H; ##[##2,3,4,5,6: #; ndestruct]
+ #d defw defa; ndestruct; @1; @; //; nrewrite > (eqb_true S d d); //. ##]
+##|#r1 r2 H1 H2 a w; @;
+ ##[ #H; ncases (in_move_cat … H);
+ ##[ *; #w1; *; #w2; *; *; #defw w1m w2m;
+ ncases (H1 a w1); #H1w1; #_; nlapply (H1w1 w1m); #good;
+ nrewrite > defw; @2; @2 (a::w1); //; ncases good; ##[ *; #; ndestruct] //.
+ ##|
+ ...
+##|
+##|
+##]
+nqed.
+
+
+notation > "x ↦* E" non associative with precedence 60 for @{move_star ? $x $E}.
+nlet rec move_star (S : decidable) w E on w : bool × (pre S) ≝
+ match w with
+ [ nil ⇒ E
+ | cons x w' ⇒ w' ↦* (x ↦ \snd E)].
+
+ndefinition in_moves ≝ λS:decidable.λw.λE:bool × (pre S). \fst(w ↦* E).
+
+ncoinductive equiv (S:decidable) : bool × (pre S) → bool × (pre S) → Prop ≝
+ mk_equiv:
+ ∀E1,E2: bool × (pre S).
+ \fst E1 = \fst E2 →
+ (∀x. equiv S (x ↦ \snd E1) (x ↦ \snd E2)) →
+ equiv S E1 E2.
+
+ndefinition NAT: decidable.
+ @ nat eqb; /2/.
+nqed.
+
+include "hints_declaration.ma".
+
+alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
+unification hint 0 ≔ ; X ≟ NAT ⊢ carr X ≡ nat.
+
+ninductive unit: Type[0] ≝ I: unit.
+
+nlet corec foo_nop (b: bool):
+ equiv ?
+ 〈 b, pc ? (ps ? 0) (pk ? (pc ? (ps ? 1) (ps ? 0))) 〉
+ 〈 b, pc ? (pk ? (pc ? (ps ? 0) (ps ? 1))) (ps ? 0) 〉 ≝ ?.
+ @; //; #x; ncases x
+ [ nnormalize in ⊢ (??%%); napply (foo_nop false)
+ | #y; ncases y
+ [ nnormalize in ⊢ (??%%); napply (foo_nop false)
+ | #w; nnormalize in ⊢ (??%%); napply (foo_nop false) ]##]
+nqed.
+
+(*
+nlet corec foo (a: unit):
+ equiv NAT
+ (eclose NAT (pc ? (ps ? 0) (pk ? (pc ? (ps ? 1) (ps ? 0)))))
+ (eclose NAT (pc ? (pk ? (pc ? (ps ? 0) (ps ? 1))) (ps ? 0)))
+≝ ?.
+ @;
+ ##[ nnormalize; //
+ ##| #x; ncases x
+ [ nnormalize in ⊢ (??%%);
+ nnormalize in foo: (? → ??%%);
+ @; //; #y; ncases y
+ [ nnormalize in ⊢ (??%%); napply foo_nop
+ | #y; ncases y
+ [ nnormalize in ⊢ (??%%);
+
+ ##| #z; nnormalize in ⊢ (??%%); napply foo_nop ]##]
+ ##| #y; nnormalize in ⊢ (??%%); napply foo_nop
+ ##]
+nqed.
+*)
+
+ndefinition test1 : pre ? ≝ ❨ `0 | `1 ❩^* `0.
+ndefinition test2 : pre ? ≝ ❨ (`0`1)^* `0 | (`0`1)^* `1 ❩.
+ndefinition test3 : pre ? ≝ (`0 (`0`1)^* `1)^*.
+
+
+nlemma foo: in_moves ? [0;0;1;0;1;1] (ɛ test3) = true.
+ nnormalize in match test3;
+ nnormalize;
+//;
+nqed.
+
+(**********************************************************)
+
+ninductive der (S: Type[0]) (a: S) : re S → re S → CProp[0] ≝
+ der_z: der S a (z S) (z S)
+ | der_e: der S a (e S) (z S)
+ | der_s1: der S a (s S a) (e ?)
+ | der_s2: ∀b. a ≠ b → der S a (s S b) (z S)
+ | der_c1: ∀e1,e2,e1',e2'. in_l S [] e1 → der S a e1 e1' → der S a e2 e2' →
+ der S a (c ? e1 e2) (o ? (c ? e1' e2) e2')
+ | der_c2: ∀e1,e2,e1'. Not (in_l S [] e1) → der S a e1 e1' →
+ der S a (c ? e1 e2) (c ? e1' e2)
+ | der_o: ∀e1,e2,e1',e2'. der S a e1 e1' → der S a e2 e2' →
+ der S a (o ? e1 e2) (o ? e1' e2').
+
+nlemma eq_rect_CProp0_r:
+ ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (refl A a) → P x p.
+ #A; #a; #x; #p; ncases p; #P; #H; nassumption.
+nqed.
+
+nlemma append1: ∀A.∀a:A.∀l. [a] @ l = a::l. //. nqed.
+
+naxiom in_l1: ∀S,r1,r2,w. in_l S [ ] r1 → in_l S w r2 → in_l S w (c S r1 r2).
+(* #S; #r1; #r2; #w; nelim r1
+ [ #K; ninversion K
+ | #H1; #H2; napply (in_c ? []); //
+ | (* tutti casi assurdi *) *)
+
+ninductive in_l' (S: Type[0]) : word S → re S → CProp[0] ≝
+ in_l_empty1: ∀E.in_l S [] E → in_l' S [] E
+ | in_l_cons: ∀a,w,e,e'. in_l' S w e' → der S a e e' → in_l' S (a::w) e.
+
+ncoinductive eq_re (S: Type[0]) : re S → re S → CProp[0] ≝
+ mk_eq_re: ∀E1,E2.
+ (in_l S [] E1 → in_l S [] E2) →
+ (in_l S [] E2 → in_l S [] E1) →
+ (∀a,E1',E2'. der S a E1 E1' → der S a E2 E2' → eq_re S E1' E2') →
+ eq_re S E1 E2.
+
+(* serve il lemma dopo? *)
+ntheorem eq_re_is_eq: ∀S.∀E1,E2. eq_re S E1 E2 → ∀w. in_l ? w E1 → in_l ? w E2.
+ #S; #E1; #E2; #H1; #w; #H2; nelim H2 in E2 H1 ⊢ %
+ [ #r; #K (* ok *)
+ | #a; #w; #R1; #R2; #K1; #K2; #K3; #R3; #K4; @2 R2; //; ncases K4;
+
+(* IL VICEVERSA NON VALE *)
+naxiom in_l_to_in_l: ∀S,w,E. in_l' S w E → in_l S w E.
+(* #S; #w; #E; #H; nelim H
+ [ //
+ | #a; #w'; #r; #r'; #H1; (* e si cade qua sotto! *)
+ ]
+nqed. *)
+
+ntheorem der1: ∀S,a,e,e',w. der S a e e' → in_l S w e' → in_l S (a::w) e.
+ #S; #a; #E; #E'; #w; #H; nelim H
+ [##1,2: #H1; ninversion H1
+ [##1,8: #_; #K; (* non va ndestruct K; *) ncases (?:False); (* perche' due goal?*) /2/
+ |##2,9: #X; #Y; #K; ncases (?:False); /2/
+ |##3,10: #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
+ |##4,11: #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+ |##5,12: #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+ |##6,13: #x; #y; #K; ncases (?:False); /2/
+ |##7,14: #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/]
+##| #H1; ninversion H1
+ [ //
+ | #X; #Y; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+ | #x; #y; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/ ]
+##| #H1; #H2; #H3; ninversion H3
+ [ #_; #K; ncases (?:False); /2/
+ | #X; #Y; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+ | #x; #y; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/ ]
+##| #r1; #r2; #r1'; #r2'; #H1; #H2; #H3; #H4; #H5; #H6;
\ No newline at end of file
projects / helm.git / commitdiff