[coq] [mathcomp] make matrix to vector function
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Module Interval.
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Definition t (low high: nat): Type :=
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{x:nat | (x >= low) /\ (x <= high)}.
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Context {low high: nat}.
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Definition add (a b: t low high): t low high.
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Proof.
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refine(
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let (a, pfa) := a in
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let (b, pfb) := b in
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exist _ (a+b) _).
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split.
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- induction a.
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+ simpl.
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destruct pfb; assumption.
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+ simpl.
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Abort.
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End Interval.
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Definition i28: Type := Interval.t 2 8.
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Definition n1: i28.
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(* unfold i28. *)
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(* unfold Interval.t. *)
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refine (exist _ 3 _).
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split; repeat constructor.
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Qed.
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Ltac ivtac n :=
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match goal with
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| |- Interval.t _ _ =>
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refine (exist _ n _);
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split; repeat constructor
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| |- {_:nat | _ >=_ /\ _ <= _} =>
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refine (exist _ n _);
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split; repeat constructor
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end.
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(* Ltac ivtacGen low high n := *)
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(* match goal with *)
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(* | |- interval _ _ => *)
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(* refine (exist _ n _); *)
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(* split; repeat constructor *)
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(* end. *)
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Definition n2: i28.
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unfold i28.
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ivtac 5.
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Defined.
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(* Ltac ivtac n := *)
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(* match goal with *)
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(* | H: _ |- exist _ _ => idtac *)
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(* end. *)
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(* | {x:nat | x >=_ /\ x <= _} => *)
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(* refine (exist _ n _); *)
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(* split; repeat constructor. *)
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(* end *)
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From mathcomp Require Import all_ssreflect all_algebra.
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(* From mathcomp Require Import algebra.matrix. *)
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(* Check matrix. *)
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Local Open Scope type_scope.
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Local Open Scope ring_scope.
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Fail Example diagonal := \matrix_(i < 3, j < 7) if i == j then 1 else 0.
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Fail Example diagonal := \matrix_(i < 3, j < 7) (if i == j then 1 else 0).
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Check \matrix_(i<3, j<7) (fun i j => if i==j then 1 else 0).
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Fail Compute \matrix_(i<3, j<7) if (i:nat)==j then 1 else 0.
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(*
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The command has indeed failed with message:
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In environment
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i : ordinal_finType 3
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j : ordinal_finType 7
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Unable to unify "?t0" with "?t@{b:=(i : nat) == j}" (cannot satisfy
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constraint "?t0" == "?t@{b:=(i : nat) == j}").
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*)
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Fail Compute \matrix_(i<3, j<7) if (i==j :> nat) then 1 else 0.
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(*
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The command has indeed failed with message:
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In environment
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i : ordinal_finType 3
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j : ordinal_finType 7
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Unable to unify "?t0" with "?t@{b:=i == j :> nat}" (cannot satisfy constraint
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"?t0" == "?t@{b:=i == j :> nat}").
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*)
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Fail Example eg1 := \matrix_(i<3, j<7) (fun (i:'I_3) (j:'I_7) => if (i==j :> nat) then 1 else 0).
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Fail Example eg1 := \matrix_(i<3, j<7) (fun (i:'I_3) (j:'I_7) => if (i:nat) == j then 1 else 0).
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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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Definition M3 : '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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Example eg2: 'M[int]_ (3, 7) := \matrix_(i<3, j<7) 8.
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Check 3%:Z.
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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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Example eg4 := \matrix_(i<3, j<3)
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(fun i j => i+j%:Z).
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Compute eg4 0 1.
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Check eg4.
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Fail Check eg4 _ 0 _ 0.
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Compute eg4 0 0.
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Check eg4 0 0.
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Search (int_ZmodType -> nat -> int_ZmodType).
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Check eg2 _ _.
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Check eg4 _ _.
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Check matrix bool 3 4.
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(* 'M_(3, 4) : predArgType *)
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Check 'M_(3, 4).
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(*
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'M_(3, 4)
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: predArgType
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where
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?R : [ |- Type]
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*)
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Check 'M[bool]_(3, 4).
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(* 'M_(3, 4) : predArgType *)
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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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Check mkseqmx_ord.
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Check matrix_of_fun.
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Check fun_of_matrix.
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Check matrix_of_fun tt.
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Check matrix_of_fun _ _ _ _ eg4.
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Check fun_of_matrix eg4.
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(* Example smx4 := mkseqmx_ord (fun_of_matrix eg4). *)
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(* Check mkseqmx_ord (fun_of_matrix eg4). *)
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(* Compute List.tl smx4. *)
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(* Check smx4. *)
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Locate "'I_".
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Print ordinal.
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Compute [seq x <- ord_enum 2 | false ].
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Compute [seq x <- ord_enum 2 | true ].
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Check filter.
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Compute List.hd 1 (ord_enum 2).
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Check List.hd (Ordinal _) (ord_enum 2).
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Compute List.hd (Ordinal _) (ord_enum 2).
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Check ord_enum 2.
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Check ord_enum.
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Require Import Bvector.
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Print Ordinal.
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Check Vector.of_list.
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Lemma length_size {A:Type} (l: seq A): List.length l = size l.
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Proof. by []. Qed.
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Lemma size_ord_enum n : size (ord_enum n) = n.
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Proof.
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by rewrite -{3}(size_iota 0 n) -val_ord_enum size_map.
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Restart.
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rewrite -{3}(size_iota 0 n).
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rewrite -val_ord_enum.
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by rewrite size_map.
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Qed.
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Goal forall n:nat,
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size (ord_enum n) = n.
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Proof.
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move=> n.
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Search size.
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rewrite -{3}(size_iota 0 n).
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Search ord_enum.
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rewrite -val_ord_enum.
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Search size.
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by rewrite size_map.
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Search ord_enum.
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Qed.
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Definition mkRow {A:Type} {rows cols:nat}
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(r: 'I_rows) (f: 'I_rows -> 'I_cols -> A): Vector.t A cols.
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refine(
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match r with
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| @Ordinal _ r' _ =>
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let cids := Vector.of_list (ord_enum cols) in (* list 'I_cols *)
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let mapped := Vector.map (fun cid => f r cid) cids in
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_
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end).
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assert ((size (ord_enum cols)) = cols).
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by rewrite size_ord_enum.
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move: mapped.
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by rewrite (length_size (ord_enum cols)) H.
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Defined.
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Check Vector.fold_right.
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Definition mkMat {A:Type} {rows cols:nat}
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(f: 'I_rows -> 'I_cols -> A): Vector.t (Vector.t A cols) rows.
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Proof.
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refine(
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let rids := Vector.of_list (ord_enum rows) in (* list 'I_rows *)
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let rws := Vector.map (fun rid => mkRow rid f) rids in
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(* let res := Vector.fold_right (fun rw acc => rw ++ acc) rws [] in *)
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_).
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move: rws.
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by rewrite (length_size (ord_enum rows)) (size_ord_enum).
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Defined.
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Compute mkMat (fun_of_matrix eg4).
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Definition mkseqmx_ord {A: Type} m n (f : 'I_m -> 'I_n -> A): nat.
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Abort.
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Require Import Extraction.
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Example e1 := (ord_enum 2).
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Recursive Extraction e1. (* Ocaml ran this! *)
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Recursive Extraction smx4. (* Ocaml couldn't run this.. *)
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Goal ((0 : 'M[int]_(2,2)) == 0).
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by coqeal.
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Abort.
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Search (matrix _ _ _ -> seq _).
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Search (seqmx).
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Require Import Bvector.
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Compute \matrix_(i<3, j<7) if i==j then 1 else 0.
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Check \matrix_(i<3, j<7) if i==j then 1 else 0.
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Check \matrix_(i, j) (fun i j => i = j).
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