--- /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
projects / helm.git / blobdiff