add some older files
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@ -3,3 +3,10 @@ From mathcomp Require Import all_ssreflect all_algebra.
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Lemma sum_odd_3: \sum_(0 <= i < 3.*2 | odd i) i = 3^2.
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Proof. by rewrite unlock /=. Qed.
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Lemma sum_nat_const_I3 n : \sum_(i in 'I_3) n = #|'I_3| * n.
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Proof.
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rewrite big_const.
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rewrite iter_addn_0.
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by rewrite mulnC.
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Qed.
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@ -1,10 +1,14 @@
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From mathcomp Require Import all_ssreflect.
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(** * Using [exists] *)
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(* https://coq.zulipchat.com/#narrow/stream/237977-Coq-users/topic/exists.20with.20ssreflect *)
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Goal forall w:seq nat,
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(exists w1 w2: seq nat, w = (w1 ++ w2)%list /\ (length w)==0%nat) -> w = [::].
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(exists w1 w2: seq nat, w = (w1 ++ w2)%list /\ (length w)==0%nat) -> w = [::].
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Proof.
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move=> w [w1 [w2]].
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by case; case: w; case: w1; case: w2.
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move=> w [w1 [w2]].
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by case; case: w; case: w1; case: w2. (* ✓ *)
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Restart.
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move=> w [w1 [w2]] [_].
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rewrite size_eq0.
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by apply/eqP.
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Qed.
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@ -22,3 +22,6 @@ Proof.
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rewrite /fun_of_matrix /=.
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by rewrite ffunE.
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Qed.
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Check castmx.
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Print erefl.
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@ -113,6 +113,7 @@ Check \matrix_(i<3, j<7) if (i==j:>nat) then 1%:Z else 0.
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Fail Check \matrix_(i<3, j<7) if (i:nat)==j then 1 else 0.
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Fail Check \matrix_(i<3, j<7) (fun i:'I_3 j => if i==j then 1 else 0).
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Check 1%:M.
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Check const_mx (Some false): 'M_3.
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Definition M2 : 'M[int]_(2,2) := \matrix_(i,j < 2) 3%:Z.
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Example eg1: 'M[int]_ (3, 7) := \matrix_(i<3, j<7) 8%:Z.
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@ -122,6 +123,32 @@ Print Scope Z_scope.
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Print Scopes.
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Example eg3 := \matrix_(i<3, j<7)
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(fun i j => if i==j :> nat then 1 else 0%:Z).
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Locate "&&".
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Example eg6 := \matrix_(i<3, j<7)
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(if (i==0) && (j==1) then (Some true) else (Some false)).
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(* Example eg6 := \matrix_(i<3, j<7) *)
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(* (if (i==0) && (j==1) then 1 else 0%:Z). *)
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(* Example eg6 := \matrix_(i<3, j<7) *)
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(* (if ((i==0) && (j==1)) then 1 else 0%:Z). *)
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(* Example eg6 := \matrix_(i<3, j<7) *)
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(* (fun i j => if ((i=0:>nat) && (j=1):>nat) then 1 else 0%:Z). *)
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(* Example eg6 := \matrix_(i<3, j<7) *)
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(* (fun i j => if (i=0 && j=1):>nat then 1 else 0%:Z). *)
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Example eg7: 'rV_2 :=
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\row_(i<2) (if i==0 then true else false).
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Check 1%:M: 'M_2.
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Check eg7.
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Example eg8: 'cV_2 :=
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\col_(i<2) (if i!=0 then true else false).
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Check eg7 *m eg8.
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Example eg7ob: 'rV_2 :=
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\row_(i<2) (if i==0 then Some true else Some false).
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Example eg8ob: 'cV_2 :=
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\col_(i<2) (if i!=0 then Some true else Some false).
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Example eg4 := \matrix_(i<3, j<3) (i+j).
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@ -204,6 +231,26 @@ Check 'M[bool]_(3, 4).
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(* 'M_(3, 4) : predArgType *)
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Local Open Scope ring_scope.
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Lemma sum_odd_n (n: nat) : \sum_(0 <= i < n.*2 | odd i) i = n^2.
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Proof.
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elim: n => [//=|n IH]; first by rewrite double0 -mulnn muln0 big_geq.
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rewrite (@big_cat_nat _ _ _ n.*2) //=; last by rewrite -!addnn leq_add.
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rewrite IH -!mulnn mulSnr mulnSr -addnA.
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congr (_ + _).
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rewrite big_ltn_cond ?ifF ?odd_double //.
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rewrite big_ltn_cond /ifT ?oddS ?odd_double //=.
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by rewrite big_geq ?addn0 -addnn addnS // -addnn addSn.
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Qed.
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Fail Check MatrixFormula.seq_of_rV eg4.
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From CoqEAL Require Import hrel param refinements trivial_seq seqmx.
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35
coq/matrix.v
35
coq/matrix.v
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@ -5,8 +5,17 @@ Require Import Bvector.
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Definition matrix (rows cols: nat): Type :=
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Vector.t (Bvector cols) rows.
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Axiom zip: forall {A B: Type} {n:nat},
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Vector.t A n -> Vector.t B n -> Vector.t (A*B) n.
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(* Uses /convoy pattern/.
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See: https://stackoverflow.com/questions/59290356/dependent-pattern-matching-zip-two-vectors
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*)
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Fixpoint zip {A B: Type} {n:nat}
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(av: Vector.t A n) (bv: Vector.t B n): Vector.t (A*B) n :=
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match av in Vector.t _ n
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return Vector.t B n -> Vector.t (A*B) n with
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| [] => fun _ => []
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| a::av' => fun bv => (a, Vector.hd bv) :: zip av' (Vector.tl bv)
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end bv.
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Section Matrix.
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Context {A: Type}.
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@ -20,10 +29,20 @@ Section Matrix.
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Definition vecMatrixProd {rows cols: nat}
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(vec: Bvector rows) (mat: matrix cols rows): Bvector cols :=
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Vector.map (fun mrow => dotProduct vec mrow) mat.
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Definition vecMatrixProd {rows cols: nat}
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(vec: Bvector rows) (mat: matrix cols rows): Bvector cols :=
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Vector.map (fun mrow => dotProduct vec mrow) mat.
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End Matrix.
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Definition isAllZero {n:nat} (v: Vector.t bool n): bool :=
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negb (Vector.fold_right (fun elem acc => orb elem acc) v false).
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(* Make vector of size [sz] with its [i]-th bit set, where indexing
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starts from 0. *)
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Fixpoint seti (i sz:nat): Vector.t bool sz :=
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match sz with
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| O => []
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| S sz' =>
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match i with
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| O => true :: (repeat false sz')
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| S i' => false :: (seti i' sz')
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end
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end.
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173
ocaml/ex1.ml
173
ocaml/ex1.ml
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@ -40,14 +40,179 @@ let rec flatten = function
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(* # flatten ([[1;2];[3;4]]);; *)
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(* - : int list = [1; 2; 3; 4] *)
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let remove_stutter l =
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List.fold_right
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let remove_stutter l = snd (List.fold_right
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(fun x (prev, res) ->
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match prev with
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| None -> (Some x, x::res)
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| Some prev' ->
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if x=prev' then (prev, res)
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else (Some x, x::res))
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l (None, [])
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l (None, []))
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(* # remove_stutter [1;2;2;3;1;1;1;4;4;2;2];; *)
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(* - : int option * int list = (Some 1, [1; 2; 3; 1; 4; 2]) *)
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(* - : int list = [1; 2; 3; 1; 4; 2] *)
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(*********************************************************************)
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(* https://github.com/kayceesrk/cs3100_f19/blob/cc71e59a081543257adc72a2428685e891bcd307/assignments/assignment2.zip *)
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type expr
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= Var of string
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| App of expr * expr
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| Lam of string * expr
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(* prettify: expr -> string *)
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let rec prettify e =
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match e with
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| Var v -> v
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| App (e1, e2) -> "(" ^ (prettify e1) ^ " " ^ (prettify e2) ^ ")"
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| Lam (v, e') -> "λ" ^ v ^ "." ^ (prettify e')
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let mem v vars = List.exists (fun x -> x=v) vars
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(* # mem "b" ["a";"b";"c"];; *)
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(* - : bool = true *)
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(* *)
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(* # mem "x" ["a";"b";"c"];; *)
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(* - : bool = false *)
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let remove v vars = List.filter (fun x -> not (v=x)) vars
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(* # remove "b" ["a";"b";"c"];; *)
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(* - : string list = ["a"; "c"] *)
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(* *)
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(* # remove "x" ["a";"b";"c"];; *)
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(* - : string list = ["a"; "b"; "c"] *)
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(* union: string list -> string list -> string list *)
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let union s1 s2 =
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remove_stutter (List.sort String.compare (s1 @ s2))
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(* # union ["a"; "c"; "b"] ["d"; "b"; "x"; "a"];; *)
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(* - : String.t list = ["a"; "b"; "c"; "d"; "x"] *)
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let add a l =
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let l' =
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(if List.exists (fun x -> x=a) l then l
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else a::l) in
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List.sort
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(fun x y ->
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if x < y then -1
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else if x = y then 0
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else 1) l'
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(* # add "b" ["a";"c"];; *)
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(* - : string list = ["a"; "b"; "c"] *)
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(* *)
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(* # add "a" ["c"; "a"];; *)
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(* - : string list = ["a"; "c"] *)
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let varcount = ref 0
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let fresh s =
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let v = !varcount in
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s ^ (string_of_int v)
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(* free_variables: expr -> string list *)
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let rec free_variables = function
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| Var v -> [v]
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| App (e1, e2) ->
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let fv1 = free_variables e1 in
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let fv2 = free_variables e2 in
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union fv1 fv2
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| Lam (v, e') ->
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let fv = free_variables e' in
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remove v fv
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(* # free_variables (Lam ("x", Var "x"));; *)
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(* - : String.t list = [] *)
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(* *)
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(* # free_variables (Lam ("x", Var "y"));; *)
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(* - : String.t list = ["y"] *)
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(* substitute : expr -> string -> expr -> expr *)
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(* substitute e1 var e2 means e1[e2/var] *)
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let rec substitute e var repl =
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match e with
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| Var v -> repl
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| App (e1, e2) -> App (substitute e1 var repl, substitute e2 var repl)
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| Lam (v, body) ->
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if v=var then
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(* A bound variable. Can be interfered with only on a beta-reduction *)
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e
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else
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let v' =
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(* Avoid free variable capture *)
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if mem v (free_variables repl) then fresh v
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else v in
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let body' = substitute body var repl in
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Lam (v', body')
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(* (λy.x)[(λz.z w)/x] is (λy.λz.z w) *)
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(* # substitute (Lam ("y", Var "x")) "x" (Lam ("z", App (Var "z", (Var "w"))));; *)
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(* - : expr = Lam ("y", Lam ("z", App (Var "z", Var "w"))) *)
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(* *)
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(* (λx.x)[y/x] is (λx.x) *)
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(* # substitute (Lam ("x", Var "x")) "x" (Var "y");; *)
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(* - : expr = Lam ("x", Var "x") *)
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(* *)
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(* (λx.y)[x/y] is (λx0. x) *)
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(* # substitute (Lam ("x", Var "y")) "y" (Var "x");; *)
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(* - : expr = Lam ("x0", Var "x") *)
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(* reduce_cbv : expr -> expr * bool *)
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let reduce_cbv e =
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match e with
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| App (e1, e2) ->
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(match e1 with
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| Lam (v, e') -> (substitute e' v e2, true)
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| _ ->
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(* Still got space to reduce *)
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let (e1', flag) = reduce_cbv e1 in
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(App (e1', e2), flag))
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| _ -> (e, false)
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(* # let e, flag = *)
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(* reduce_cbv ( *)
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(* App ( *)
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(* App ( *)
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(* Lam ("x", Var "x"), *)
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(* Lam ("x", Var "x")), *)
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(* Lam ("z", *)
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(* App (Lam ("x", Var "x"), *)
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(* Var "z")))) in (prettify e, flag);; *)
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(* - : string * bool = ("(λx.x λz.(λx.x z))", true) *)
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(* prettify ( *)
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(* App ( *)
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(* Lam ("x", Var "x), *)
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(* Lam ("z", *)
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(* App ( *)
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(* Lam ("x", Var "x), *)
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(* Var "z")))) *)
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(* App (Lam ("z", *)
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(* Lam ("x", Var "x")), *)
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(* Var "z" *)
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(* let expr, reduced = reduce_cbv (parse_string "(λx.x) (λz.(λx.x) z)") in *)
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(* assert (reduced = true && *)
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(* alpha_equiv expr (parse_string "λz.(λx.x) z")); *)
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(* let expr, reduced = reduce_cbv (parse_string "λz.(λx.x) z") in *)
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(* assert (reduced = false && *)
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(* alpha_equiv expr (parse_string "λz.(λx.x) z")); *)
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(* let expr, reduced = reduce_cbv (parse_string "(λx.y) ((λx.x x) (λx.x x))") in *)
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(* assert (reduced = true && *)
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(* alpha_equiv expr (parse_string "(λx.y) ((λx.x x) (λx.x x))")); *)
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(* let expr, reduced = reduce_cbv (parse_string "x y z") in *)
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(* assert (reduced = false && *)
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(* alpha_equiv expr (parse_string "x y z")) *)
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let rec reduce_normal e =
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let e', flag = reduce_cbv e in
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if flag then reduce_normal e'
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else e'
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