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-rw-r--r--backends/lean/Base/Diverge/Base.lean572
-rw-r--r--backends/lean/Base/Diverge/Elab.lean1155
-rw-r--r--backends/lean/Base/Diverge/ElabBase.lean69
-rw-r--r--backends/lean/Base/Extensions.lean47
-rw-r--r--backends/lean/Base/Primitives/Scalar.lean126
-rw-r--r--backends/lean/Base/Progress/Base.lean290
-rw-r--r--backends/lean/Base/Progress/Progress.lean91
-rw-r--r--backends/lean/Base/Utils.lean114
8 files changed, 1690 insertions, 774 deletions
diff --git a/backends/lean/Base/Diverge/Base.lean b/backends/lean/Base/Diverge/Base.lean
index 6a52387d..9458c926 100644
--- a/backends/lean/Base/Diverge/Base.lean
+++ b/backends/lean/Base/Diverge/Base.lean
@@ -5,6 +5,7 @@ import Mathlib.Tactic.RunCmd
import Mathlib.Tactic.Linarith
import Base.Primitives.Base
import Base.Arith.Base
+import Base.Diverge.ElabBase
/- TODO: this is very useful, but is there more? -/
set_option profiler true
@@ -12,6 +13,78 @@ set_option profiler.threshold 100
namespace Diverge
+/- Auxiliary lemmas -/
+namespace Lemmas
+ -- TODO: not necessary anymore
+ def for_all_fin_aux {n : Nat} (f : Fin n → Prop) (m : Nat) (h : m ≤ n) : Prop :=
+ if heq: m = n then True
+ else
+ f ⟨ m, by simp_all [Nat.lt_iff_le_and_ne] ⟩ ∧
+ for_all_fin_aux f (m + 1) (by simp_all [Arith.add_one_le_iff_le_ne])
+ termination_by for_all_fin_aux n _ m h => n - m
+ decreasing_by
+ simp_wf
+ apply Nat.sub_add_lt_sub <;> try simp
+ simp_all [Arith.add_one_le_iff_le_ne]
+
+ def for_all_fin {n : Nat} (f : Fin n → Prop) := for_all_fin_aux f 0 (by simp)
+
+ theorem for_all_fin_aux_imp_forall {n : Nat} (f : Fin n → Prop) (m : Nat) :
+ (h : m ≤ n) →
+ for_all_fin_aux f m h → ∀ i, m ≤ i.val → f i
+ := by
+ generalize h: (n - m) = k
+ revert m
+ induction k -- TODO: induction h rather?
+ case zero =>
+ simp_all
+ intro m h1 h2
+ have h: n = m := by
+ linarith
+ unfold for_all_fin_aux; simp_all
+ simp_all
+ -- There is no i s.t. m ≤ i
+ intro i h3; cases i; simp_all
+ linarith
+ case succ k hi =>
+ intro m hk hmn
+ intro hf i hmi
+ have hne: m ≠ n := by
+ have hineq := Nat.lt_of_sub_eq_succ hk
+ linarith
+ -- m = i?
+ if heq: m = i then
+ -- Yes: simply use the `for_all_fin_aux` hyp
+ unfold for_all_fin_aux at hf
+ simp_all
+ else
+ -- No: use the induction hypothesis
+ have hlt: m < i := by simp_all [Nat.lt_iff_le_and_ne]
+ have hineq: m + 1 ≤ n := by
+ have hineq := Nat.lt_of_sub_eq_succ hk
+ simp [*, Nat.add_one_le_iff]
+ have heq1: n - (m + 1) = k := by
+ -- TODO: very annoying arithmetic proof
+ simp [Nat.sub_eq_iff_eq_add hineq]
+ have hineq1: m ≤ n := by linarith
+ simp [Nat.sub_eq_iff_eq_add hineq1] at hk
+ simp_arith [hk]
+ have hi := hi (m + 1) heq1 hineq
+ apply hi <;> simp_all
+ . unfold for_all_fin_aux at hf
+ simp_all
+ . simp_all [Arith.add_one_le_iff_le_ne]
+
+ -- TODO: this is not necessary anymore
+ theorem for_all_fin_imp_forall (n : Nat) (f : Fin n → Prop) :
+ for_all_fin f → ∀ i, f i
+ := by
+ intro Hf i
+ apply for_all_fin_aux_imp_forall <;> try assumption
+ simp
+
+end Lemmas
+
namespace Fix
open Primitives
@@ -436,6 +509,10 @@ namespace FixI
/- Indexed fixed-point: definitions with indexed types, convenient to use for mutually
recursive definitions. We simply port the definitions and proofs from Fix to a more
specific case.
+
+ Remark: the index designates the function in the mutually recursive group
+ (it should be a finite type). We make the return type depend on the input
+ type because we group the type parameters in the input type.
-/
open Primitives Fix
@@ -505,7 +582,6 @@ namespace FixI
kk_ty id a b → kk_ty id a b
abbrev in_out_ty : Type (imax (u + 1) (v + 1)) := (in_ty : Type u) × ((x:in_ty) → Type v)
- -- TODO: remove?
abbrev mk_in_out_ty (in_ty : Type u) (out_ty : in_ty → Type v) :
in_out_ty :=
Sigma.mk in_ty out_ty
@@ -538,73 +614,6 @@ namespace FixI
let j: Fin tys1.length := ⟨ j, jLt ⟩
Eq.mp (by simp) (get_fun tl j)
- def for_all_fin_aux {n : Nat} (f : Fin n → Prop) (m : Nat) (h : m ≤ n) : Prop :=
- if heq: m = n then True
- else
- f ⟨ m, by simp_all [Nat.lt_iff_le_and_ne] ⟩ ∧
- for_all_fin_aux f (m + 1) (by simp_all [Arith.add_one_le_iff_le_ne])
- termination_by for_all_fin_aux n _ m h => n - m
- decreasing_by
- simp_wf
- apply Nat.sub_add_lt_sub <;> try simp
- simp_all [Arith.add_one_le_iff_le_ne]
-
- def for_all_fin {n : Nat} (f : Fin n → Prop) := for_all_fin_aux f 0 (by simp)
-
- theorem for_all_fin_aux_imp_forall {n : Nat} (f : Fin n → Prop) (m : Nat) :
- (h : m ≤ n) →
- for_all_fin_aux f m h → ∀ i, m ≤ i.val → f i
- := by
- generalize h: (n - m) = k
- revert m
- induction k -- TODO: induction h rather?
- case zero =>
- simp_all
- intro m h1 h2
- have h: n = m := by
- linarith
- unfold for_all_fin_aux; simp_all
- simp_all
- -- There is no i s.t. m ≤ i
- intro i h3; cases i; simp_all
- linarith
- case succ k hi =>
- intro m hk hmn
- intro hf i hmi
- have hne: m ≠ n := by
- have hineq := Nat.lt_of_sub_eq_succ hk
- linarith
- -- m = i?
- if heq: m = i then
- -- Yes: simply use the `for_all_fin_aux` hyp
- unfold for_all_fin_aux at hf
- simp_all
- else
- -- No: use the induction hypothesis
- have hlt: m < i := by simp_all [Nat.lt_iff_le_and_ne]
- have hineq: m + 1 ≤ n := by
- have hineq := Nat.lt_of_sub_eq_succ hk
- simp [*, Nat.add_one_le_iff]
- have heq1: n - (m + 1) = k := by
- -- TODO: very annoying arithmetic proof
- simp [Nat.sub_eq_iff_eq_add hineq]
- have hineq1: m ≤ n := by linarith
- simp [Nat.sub_eq_iff_eq_add hineq1] at hk
- simp_arith [hk]
- have hi := hi (m + 1) heq1 hineq
- apply hi <;> simp_all
- . unfold for_all_fin_aux at hf
- simp_all
- . simp_all [Arith.add_one_le_iff_le_ne]
-
- -- TODO: this is not necessary anymore
- theorem for_all_fin_imp_forall (n : Nat) (f : Fin n → Prop) :
- for_all_fin f → ∀ i, f i
- := by
- intro Hf i
- apply for_all_fin_aux_imp_forall <;> try assumption
- simp
-
/- Automating the proofs -/
@[simp] theorem is_valid_p_same
(k : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)) (x : Result c) :
@@ -707,6 +716,218 @@ namespace FixI
end FixI
+namespace FixII
+ /- Similar to FixI, but we split the input arguments between the type parameters
+ and the input values.
+ -/
+ open Primitives Fix
+
+ -- The index type
+ variable {id : Type u}
+
+ -- The input/output types
+ variable {ty : id → Type v} {a : (i:id) → ty i → Type w} {b : (i:id) → ty i → Type x}
+
+ -- Monotonicity relation over monadic arrows (i.e., Kleisli arrows)
+ def karrow_rel (k1 k2 : (i:id) → (t:ty i) → (a i t) → Result (b i t)) : Prop :=
+ ∀ i t x, result_rel (k1 i t x) (k2 i t x)
+
+ def kk_to_gen (k : (i:id) → (t:ty i) → (x:a i t) → Result (b i t)) :
+ (x: (i:id) × (t:ty i) × (a i t)) → Result (b x.fst x.snd.fst) :=
+ λ ⟨ i, t, x ⟩ => k i t x
+
+ def kk_of_gen (k : (x: (i:id) × (t:ty i) × (a i t)) → Result (b x.fst x.snd.fst)) :
+ (i:id) → (t:ty i) → a i t → Result (b i t) :=
+ λ i t x => k ⟨ i, t, x ⟩
+
+ def k_to_gen (k : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t)) :
+ ((x: (i:id) × (t:ty i) × (a i t)) → Result (b x.fst x.snd.fst)) → (x: (i:id) × (t:ty i) × (a i t)) → Result (b x.fst x.snd.fst) :=
+ λ kk => kk_to_gen (k (kk_of_gen kk))
+
+ def k_of_gen (k : ((x: (i:id) × (t:ty i) × (a i t)) → Result (b x.fst x.snd.fst)) → (x: (i:id) × (t:ty i) × (a i t)) → Result (b x.fst x.snd.fst)) :
+ ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t) :=
+ λ kk => kk_of_gen (k (kk_to_gen kk))
+
+ def e_to_gen (e : ((i:id) → (t:ty i) → a i t → Result (b i t)) → Result c) :
+ ((x: (i:id) × (t:ty i) × (a i t)) → Result (b x.fst x.snd.fst)) → Result c :=
+ λ k => e (kk_of_gen k)
+
+ def is_valid_p (k : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t))
+ (e : ((i:id) → (t:ty i) → a i t → Result (b i t)) → Result c) : Prop :=
+ Fix.is_valid_p (k_to_gen k) (e_to_gen e)
+
+ def is_valid (f : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t)) : Prop :=
+ ∀ k i t x, is_valid_p k (λ k => f k i t x)
+
+ def fix
+ (f : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t)) :
+ (i:id) → (t:ty i) → a i t → Result (b i t) :=
+ kk_of_gen (Fix.fix (k_to_gen f))
+
+ theorem is_valid_fix_fixed_eq
+ {{f : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t)}}
+ (Hvalid : is_valid f) :
+ fix f = f (fix f) := by
+ have Hvalid' : Fix.is_valid (k_to_gen f) := by
+ intro k x
+ simp only [is_valid, is_valid_p] at Hvalid
+ let ⟨ i, t, x ⟩ := x
+ have Hvalid := Hvalid (k_of_gen k) i t x
+ simp only [k_to_gen, k_of_gen, kk_to_gen, kk_of_gen] at Hvalid
+ refine Hvalid
+ have Heq := Fix.is_valid_fix_fixed_eq Hvalid'
+ simp [fix]
+ conv => lhs; rw [Heq]
+
+ /- Some utilities to define the mutually recursive functions -/
+
+ -- TODO: use more
+ abbrev kk_ty (id : Type u) (ty : id → Type v) (a : (i:id) → ty i → Type w) (b : (i:id) → ty i → Type x) :=
+ (i:id) → (t:ty i) → a i t → Result (b i t)
+ abbrev k_ty (id : Type u) (ty : id → Type v) (a : (i:id) → ty i → Type w) (b : (i:id) → ty i → Type x) :=
+ kk_ty id ty a b → kk_ty id ty a b
+
+ abbrev in_out_ty : Type (imax (u + 1) (imax (v + 1) (w + 1))) :=
+ (ty : Type u) × (ty → Type v) × (ty → Type w)
+ abbrev mk_in_out_ty (ty : Type u) (in_ty : ty → Type v) (out_ty : ty → Type w) :
+ in_out_ty :=
+ Sigma.mk ty (Prod.mk in_ty out_ty)
+
+ -- Initially, we had left out the parameters id, a and b.
+ -- However, by parameterizing Funs with those parameters, we can state
+ -- and prove lemmas like Funs.is_valid_p_is_valid_p
+ inductive Funs (id : Type u) (ty : id → Type v)
+ (a : (i:id) → ty i → Type w) (b : (i:id) → ty i → Type x) :
+ List in_out_ty.{v, w, x} → Type (max (u + 1) (max (v + 1) (max (w + 1) (x + 1)))) :=
+ | Nil : Funs id ty a b []
+ | Cons {it: Type v} {ity : it → Type w} {oty : it → Type x} {tys : List in_out_ty}
+ (f : kk_ty id ty a b → (i:it) → (x:ity i) → Result (oty i)) (tl : Funs id ty a b tys) :
+ Funs id ty a b (⟨ it, ity, oty ⟩ :: tys)
+
+ def get_fun {tys : List in_out_ty} (fl : Funs id ty a b tys) :
+ (i : Fin tys.length) → kk_ty id ty a b → (t : (tys.get i).fst) →
+ ((tys.get i).snd.fst t) → Result ((tys.get i).snd.snd t) :=
+ match fl with
+ | .Nil => λ i => by have h:= i.isLt; simp at h
+ | @Funs.Cons id ty a b it ity oty tys1 f tl =>
+ λ ⟨ i, iLt ⟩ =>
+ match i with
+ | 0 =>
+ Eq.mp (by simp [List.get]) f
+ | .succ j =>
+ have jLt: j < tys1.length := by
+ simp at iLt
+ revert iLt
+ simp_arith
+ let j: Fin tys1.length := ⟨ j, jLt ⟩
+ Eq.mp (by simp) (get_fun tl j)
+
+ /- Automating the proofs -/
+ @[simp] theorem is_valid_p_same
+ (k : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t)) (x : Result c) :
+ is_valid_p k (λ _ => x) := by
+ simp [is_valid_p, k_to_gen, e_to_gen]
+
+ @[simp] theorem is_valid_p_rec
+ (k : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t)) (i : id) (t : ty i) (x : a i t) :
+ is_valid_p k (λ k => k i t x) := by
+ simp [is_valid_p, k_to_gen, e_to_gen, kk_to_gen, kk_of_gen]
+
+ theorem is_valid_p_ite
+ (k : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t))
+ (cond : Prop) [h : Decidable cond]
+ {e1 e2 : ((i:id) → (t:ty i) → a i t → Result (b i t)) → Result c}
+ (he1: is_valid_p k e1) (he2 : is_valid_p k e2) :
+ is_valid_p k (λ k => ite cond (e1 k) (e2 k)) := by
+ split <;> assumption
+
+ theorem is_valid_p_dite
+ (k : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t))
+ (cond : Prop) [h : Decidable cond]
+ {e1 : ((i:id) → (t:ty i) → a i t → Result (b i t)) → cond → Result c}
+ {e2 : ((i:id) → (t:ty i) → a i t → Result (b i t)) → Not cond → Result c}
+ (he1: ∀ x, is_valid_p k (λ k => e1 k x))
+ (he2 : ∀ x, is_valid_p k (λ k => e2 k x)) :
+ is_valid_p k (λ k => dite cond (e1 k) (e2 k)) := by
+ split <;> simp [*]
+
+ theorem is_valid_p_bind
+ {{k : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t)}}
+ {{g : ((i:id) → (t:ty i) → a i t → Result (b i t)) → Result c}}
+ {{h : c → ((i:id) → (t:ty i) → a i t → Result (b i t)) → Result d}}
+ (Hgvalid : is_valid_p k g)
+ (Hhvalid : ∀ y, is_valid_p k (h y)) :
+ is_valid_p k (λ k => do let y ← g k; h y k) := by
+ apply Fix.is_valid_p_bind
+ . apply Hgvalid
+ . apply Hhvalid
+
+ def Funs.is_valid_p
+ (k : k_ty id ty a b)
+ (fl : Funs id ty a b tys) :
+ Prop :=
+ match fl with
+ | .Nil => True
+ | .Cons f fl =>
+ (∀ i x, FixII.is_valid_p k (λ k => f k i x)) ∧ fl.is_valid_p k
+
+ theorem Funs.is_valid_p_Nil (k : k_ty id ty a b) :
+ Funs.is_valid_p k Funs.Nil := by simp [Funs.is_valid_p]
+
+ def Funs.is_valid_p_is_valid_p_aux
+ {k : k_ty id ty a b}
+ {tys : List in_out_ty}
+ (fl : Funs id ty a b tys) (Hvalid : is_valid_p k fl) :
+ ∀ (i : Fin tys.length) (t : (tys.get i).fst) (x : (tys.get i).snd.fst t),
+ FixII.is_valid_p k (fun k => get_fun fl i k t x) := by
+ -- Prepare the induction
+ have ⟨ n, Hn ⟩ : { n : Nat // tys.length = n } := ⟨ tys.length, by rfl ⟩
+ revert tys fl Hvalid
+ induction n
+ --
+ case zero =>
+ intro tys fl Hvalid Hlen;
+ have Heq: tys = [] := by cases tys <;> simp_all
+ intro i x
+ simp_all
+ have Hi := i.isLt
+ simp_all
+ case succ n Hn =>
+ intro tys fl Hvalid Hlen i x;
+ cases fl <;> simp at Hlen i x Hvalid
+ rename_i ity oty tys f fl
+ have ⟨ Hvf, Hvalid ⟩ := Hvalid
+ have Hvf1: is_valid_p k fl := by
+ simp [Hvalid, Funs.is_valid_p]
+ have Hn := @Hn tys fl Hvf1 (by simp [*])
+ -- Case disjunction on i
+ match i with
+ | ⟨ 0, _ ⟩ =>
+ simp at x
+ simp [get_fun]
+ apply (Hvf x)
+ | ⟨ .succ j, HiLt ⟩ =>
+ simp_arith at HiLt
+ simp at x
+ let j : Fin (List.length tys) := ⟨ j, by simp_arith [HiLt] ⟩
+ have Hn := Hn j x
+ apply Hn
+
+ def Funs.is_valid_p_is_valid_p
+ (tys : List in_out_ty)
+ (k : k_ty (Fin (List.length tys)) (λ i => (tys.get i).fst)
+ (fun i t => (List.get tys i).snd.fst t) (fun i t => (List.get tys i).snd.snd t))
+ (fl : Funs (Fin tys.length) (λ i => (tys.get i).fst)
+ (λ i t => (tys.get i).snd.fst t) (λ i t => (tys.get i).snd.snd t) tys) :
+ fl.is_valid_p k →
+ ∀ (i : Fin tys.length) (t : (tys.get i).fst) (x : (tys.get i).snd.fst t),
+ FixII.is_valid_p k (fun k => get_fun fl i k t x)
+ := by
+ intro Hvalid
+ apply is_valid_p_is_valid_p_aux; simp [*]
+
+end FixII
+
namespace Ex1
/- An example of use of the fixed-point -/
open Primitives Fix
@@ -1133,3 +1354,218 @@ namespace Ex6
Heqix
end Ex6
+
+namespace Ex7
+ /- `list_nth` again, but this time we use FixII -/
+ open Primitives FixII
+
+ @[simp] def tys.{u} : List in_out_ty :=
+ [mk_in_out_ty (Type u) (λ a => List a × Int) (λ a => a)]
+
+ @[simp] def ty (i : Fin 1) := (tys.get i).fst
+ @[simp] def input_ty (i : Fin 1) (t : ty i) : Type u := (tys.get i).snd.fst t
+ @[simp] def output_ty (i : Fin 1) (t : ty i) : Type u := (tys.get i).snd.snd t
+
+ def list_nth_body.{u} (k : (i:Fin 1) → (t:ty i) → input_ty i t → Result (output_ty i t))
+ (a : Type u) (x : List a × Int) : Result a :=
+ let ⟨ ls, i ⟩ := x
+ match ls with
+ | [] => .fail .panic
+ | hd :: tl =>
+ if i = 0 then .ret hd
+ else k 0 a ⟨ tl, i - 1 ⟩
+
+ @[simp] def bodies :
+ Funs (Fin 1) ty input_ty output_ty tys :=
+ Funs.Cons list_nth_body Funs.Nil
+
+ def body (k : (i : Fin 1) → (t : ty i) → (x : input_ty i t) → Result (output_ty i t)) (i: Fin 1) :
+ (t : ty i) → (x : input_ty i t) → Result (output_ty i t) := get_fun bodies i k
+
+ theorem body_is_valid: is_valid body := by
+ -- Split the proof into proofs of validity of the individual bodies
+ rw [is_valid]
+ simp only [body]
+ intro k
+ apply (Funs.is_valid_p_is_valid_p tys)
+ simp [Funs.is_valid_p]
+ (repeat (apply And.intro)); intro x; try simp at x
+ simp only [list_nth_body]
+ -- Prove the validity of the individual bodies
+ intro k x
+ split <;> try simp
+ split <;> simp
+
+ -- Writing the proof terms explicitly
+ theorem list_nth_body_is_valid' (k : k_ty (Fin 1) ty input_ty output_ty)
+ (a : Type u) (x : List a × Int) : is_valid_p k (fun k => list_nth_body k a x) :=
+ let ⟨ ls, i ⟩ := x
+ match ls with
+ | [] => is_valid_p_same k (.fail .panic)
+ | hd :: tl =>
+ is_valid_p_ite k (Eq i 0) (is_valid_p_same k (.ret hd)) (is_valid_p_rec k 0 a ⟨tl, i-1⟩)
+
+ theorem body_is_valid' : is_valid body :=
+ fun k =>
+ Funs.is_valid_p_is_valid_p tys k bodies
+ (And.intro (list_nth_body_is_valid' k) (Funs.is_valid_p_Nil k))
+
+ def list_nth {a: Type u} (ls : List a) (i : Int) : Result a :=
+ fix body 0 a ⟨ ls , i ⟩
+
+ -- The unfolding equation - diverges if `i < 0`
+ theorem list_nth_eq (ls : List a) (i : Int) :
+ list_nth ls i =
+ match ls with
+ | [] => .fail .panic
+ | hd :: tl =>
+ if i = 0 then .ret hd
+ else list_nth tl (i - 1)
+ := by
+ have Heq := is_valid_fix_fixed_eq body_is_valid
+ simp [list_nth]
+ conv => lhs; rw [Heq]
+
+ -- Write the proof term explicitly: the generation of the proof term (without tactics)
+ -- is automatable, and the proof term is actually a lot simpler and smaller when we
+ -- don't use tactics.
+ theorem list_nth_eq'.{u} {a : Type u} (ls : List a) (i : Int) :
+ list_nth ls i =
+ match ls with
+ | [] => .fail .panic
+ | hd :: tl =>
+ if i = 0 then .ret hd
+ else list_nth tl (i - 1)
+ :=
+ -- Use the fixed-point equation
+ have Heq := is_valid_fix_fixed_eq body_is_valid.{u}
+ -- Add the index
+ have Heqi := congr_fun Heq 0
+ -- Add the type parameter
+ have Heqia := congr_fun Heqi a
+ -- Add the input
+ have Heqix := congr_fun Heqia (ls, i)
+ -- Done
+ Heqix
+
+end Ex7
+
+namespace Ex8
+ /- Higher-order example, with FixII -/
+ open Primitives FixII
+
+ variable {id : Type u} {ty : id → Type v}
+ variable {a : (i:id) → ty i → Type w} {b : (i:id) → ty i → Type x}
+
+ /- An auxiliary function, which doesn't require the fixed-point -/
+ def map {a : Type y} {b : Type z} (f : a → Result b) (ls : List a) : Result (List b) :=
+ match ls with
+ | [] => .ret []
+ | hd :: tl =>
+ do
+ let hd ← f hd
+ let tl ← map f tl
+ .ret (hd :: tl)
+
+ /- The validity theorems for `map`, generic in `f` -/
+
+ -- This is not the most general lemma, but we keep it to test the `divergence` encoding on a simple case
+ @[divspec]
+ theorem map_is_valid_simple
+ (i : id) (t : ty i)
+ (k : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t))
+ (ls : List (a i t)) :
+ is_valid_p k (λ k => map (k i t) ls) := by
+ induction ls <;> simp [map]
+ apply is_valid_p_bind <;> try simp_all
+ intros
+ apply is_valid_p_bind <;> try simp_all
+
+ @[divspec]
+ theorem map_is_valid
+ (d : Type y)
+ {{f : ((i:id) → (t : ty i) → a i t → Result (b i t)) → d → Result c}}
+ (k : ((i:id) → (t:ty i) → a i t → Result (b i t)) → (i:id) → (t:ty i) → a i t → Result (b i t))
+ (Hfvalid : ∀ x1, is_valid_p k (fun kk1 => f kk1 x1))
+ (ls : List d) :
+ is_valid_p k (λ k => map (f k) ls) := by
+ induction ls <;> simp [map]
+ apply is_valid_p_bind <;> try simp_all
+ intros
+ apply is_valid_p_bind <;> try simp_all
+
+end Ex8
+
+namespace Ex9
+ /- An example which uses map -/
+ open Primitives FixII Ex8
+
+ inductive Tree (a : Type u) :=
+ | leaf (x : a)
+ | node (tl : List (Tree a))
+
+ @[simp] def tys.{u} : List in_out_ty :=
+ [mk_in_out_ty (Type u) (λ a => Tree a) (λ a => Tree a)]
+
+ @[simp] def ty (i : Fin 1) := (tys.get i).fst
+ @[simp] def input_ty (i : Fin 1) (t : ty i) : Type u := (tys.get i).snd.fst t
+ @[simp] def output_ty (i : Fin 1) (t : ty i) : Type u := (tys.get i).snd.snd t
+
+ def id_body.{u} (k : (i:Fin 1) → (t:ty i) → input_ty i t → Result (output_ty i t))
+ (a : Type u) (t : Tree a) : Result (Tree a) :=
+ match t with
+ | .leaf x => .ret (.leaf x)
+ | .node tl =>
+ do
+ let tl ← map (k 0 a) tl
+ .ret (.node tl)
+
+ @[simp] def bodies :
+ Funs (Fin 1) ty input_ty output_ty tys :=
+ Funs.Cons id_body Funs.Nil
+
+ theorem id_body_is_valid :
+ ∀ (k : ((i : Fin 1) → (t : ty i) → input_ty i t → Result (output_ty i t)) → (i : Fin 1) → (t : ty i) → input_ty i t → Result (output_ty i t))
+ (a : Type u) (x : Tree a),
+ @is_valid_p (Fin 1) ty input_ty output_ty (output_ty 0 a) k (λ k => id_body k a x) := by
+ intro k a x
+ simp only [id_body]
+ split <;> try simp
+ apply is_valid_p_bind <;> try simp [*]
+ -- We have to show that `map k tl` is valid
+ -- Remark: `map_is_valid` doesn't work here, we need the specialized version
+ apply map_is_valid_simple
+
+ def body (k : (i : Fin 1) → (t : ty i) → (x : input_ty i t) → Result (output_ty i t)) (i: Fin 1) :
+ (t : ty i) → (x : input_ty i t) → Result (output_ty i t) := get_fun bodies i k
+
+ theorem body_is_valid : is_valid body :=
+ fun k =>
+ Funs.is_valid_p_is_valid_p tys k bodies
+ (And.intro (id_body_is_valid k) (Funs.is_valid_p_Nil k))
+
+ def id {a: Type u} (t : Tree a) : Result (Tree a) :=
+ fix body 0 a t
+
+ -- Writing the proof term explicitly
+ theorem id_eq' {a : Type u} (t : Tree a) :
+ id t =
+ (match t with
+ | .leaf x => .ret (.leaf x)
+ | .node tl =>
+ do
+ let tl ← map id tl
+ .ret (.node tl))
+ :=
+ -- The unfolding equation
+ have Heq := is_valid_fix_fixed_eq body_is_valid.{u}
+ -- Add the index
+ have Heqi := congr_fun Heq 0
+ -- Add the type parameter
+ have Heqia := congr_fun Heqi a
+ -- Add the input
+ have Heqix := congr_fun Heqia t
+ -- Done
+ Heqix
+
+end Ex9
diff --git a/backends/lean/Base/Diverge/Elab.lean b/backends/lean/Base/Diverge/Elab.lean
index c6628486..6115b13b 100644
--- a/backends/lean/Base/Diverge/Elab.lean
+++ b/backends/lean/Base/Diverge/Elab.lean
@@ -17,15 +17,24 @@ syntax (name := divergentDef)
open Lean Elab Term Meta Primitives Lean.Meta
open Utils
+def normalize_let_bindings := true
+
/- The following was copied from the `wfRecursion` function. -/
open WF in
+-- TODO: use those
+def UnitType := Expr.const ``PUnit [Level.succ .zero]
+def UnitValue := Expr.const ``PUnit.unit [Level.succ .zero]
+
+def mkProdType (x y : Expr) : MetaM Expr :=
+ mkAppM ``Prod #[x, y]
+
def mkProd (x y : Expr) : MetaM Expr :=
mkAppM ``Prod.mk #[x, y]
-def mkInOutTy (x y : Expr) : MetaM Expr :=
- mkAppM ``FixI.mk_in_out_ty #[x, y]
+def mkInOutTy (x y z : Expr) : MetaM Expr := do
+ mkAppM ``FixII.mk_in_out_ty #[x, y, z]
-- Return the `a` in `Return a`
def getResultTy (ty : Expr) : MetaM Expr :=
@@ -47,6 +56,17 @@ def getSigmaTypes (ty : Expr) : MetaM (Expr × Expr) := do
else
pure (args.get! 0, args.get! 1)
+/- Make a sigma type.
+
+ `x` should be a variable, and `ty` and type which (might) uses `x`
+ -/
+def mkSigmaType (x : Expr) (sty : Expr) : MetaM Expr := do
+ trace[Diverge.def.sigmas] "mkSigmaType: {x} {sty}"
+ let alpha ← inferType x
+ let beta ← mkLambdaFVars #[x] sty
+ trace[Diverge.def.sigmas] "mkSigmaType: ({alpha}) ({beta})"
+ mkAppOptM ``Sigma #[some alpha, some beta]
+
/- Generate a Sigma type from a list of *variables* (all the expressions
must be variables).
@@ -60,20 +80,78 @@ def getSigmaTypes (ty : Expr) : MetaM (Expr × Expr) := do
def mkSigmasType (xl : List Expr) : MetaM Expr :=
match xl with
| [] => do
- trace[Diverge.def.sigmas] "mkSigmasOfTypes: []"
- pure (Expr.const ``PUnit.unit [])
+ trace[Diverge.def.sigmas] "mkSigmasType: []"
+ pure (Expr.const ``PUnit [Level.succ .zero])
| [x] => do
- trace[Diverge.def.sigmas] "mkSigmasOfTypes: [{x}]"
- let ty ← Lean.Meta.inferType x
+ trace[Diverge.def.sigmas] "mkSigmasType: [{x}]"
+ let ty ← inferType x
pure ty
| x :: xl => do
- trace[Diverge.def.sigmas] "mkSigmasOfTypes: [{x}::{xl}]"
- let alpha ← Lean.Meta.inferType x
+ trace[Diverge.def.sigmas] "mkSigmasType: [{x}::{xl}]"
let sty ← mkSigmasType xl
- trace[Diverge.def.sigmas] "mkSigmasOfTypes: [{x}::{xl}]: alpha={alpha}, sty={sty}"
- let beta ← mkLambdaFVars #[x] sty
- trace[Diverge.def.sigmas] "mkSigmasOfTypes: ({alpha}) ({beta})"
- mkAppOptM ``Sigma #[some alpha, some beta]
+ mkSigmaType x sty
+
+/- Generate a product type from a list of *variables* (this is similar to `mkSigmas`).
+
+ Example:
+ - xl = [(ls:List a), (i:Int)]
+
+ Generates:
+ `List a × Int`
+ -/
+def mkProdsType (xl : List Expr) : MetaM Expr :=
+ match xl with
+ | [] => do
+ trace[Diverge.def.prods] "mkProdsType: []"
+ pure (Expr.const ``PUnit [Level.succ .zero])
+ | [x] => do
+ trace[Diverge.def.prods] "mkProdsType: [{x}]"
+ let ty ← inferType x
+ pure ty
+ | x :: xl => do
+ trace[Diverge.def.prods] "mkProdsType: [{x}::{xl}]"
+ let ty ← inferType x
+ let xl_ty ← mkProdsType xl
+ mkAppM ``Prod #[ty, xl_ty]
+
+/- Split the input arguments between the types and the "regular" arguments.
+
+ We do something simple: we treat an input argument as an
+ input type iff it appears in the type of the following arguments.
+
+ Note that what really matters is that we find the arguments which appear
+ in the output type.
+
+ Also, we stop at the first input that we treat as an
+ input type.
+ -/
+def splitInputArgs (in_tys : Array Expr) (out_ty : Expr) : MetaM (Array Expr × Array Expr) := do
+ -- Look for the first parameter which appears in the subsequent parameters
+ let rec splitAux (in_tys : List Expr) : MetaM (HashSet FVarId × List Expr × List Expr) :=
+ match in_tys with
+ | [] => do
+ let fvars ← getFVarIds (← inferType out_ty)
+ pure (fvars, [], [])
+ | ty :: in_tys => do
+ let (fvars, in_tys, in_args) ← splitAux in_tys
+ -- Have we already found where to split between type variables/regular
+ -- variables?
+ if ¬ in_tys.isEmpty then
+ -- The fvars set is now useless: no need to update it anymore
+ pure (fvars, ty :: in_tys, in_args)
+ else
+ -- Check if ty appears in the set of free variables:
+ let ty_id := ty.fvarId!
+ if fvars.contains ty_id then
+ -- We must split here. Note that we don't need to update the fvars
+ -- set: it is not useful anymore
+ pure (fvars, [ty], in_args)
+ else
+ -- We must split later: update the fvars set
+ let fvars := fvars.insertMany (← getFVarIds (← inferType ty))
+ pure (fvars, [], ty :: in_args)
+ let (_, in_tys, in_args) ← splitAux in_tys.data
+ pure (Array.mk in_tys, Array.mk in_args)
/- Apply a lambda expression to some arguments, simplifying the lambdas -/
def applyLambdaToArgs (e : Expr) (xs : Array Expr) : MetaM Expr := do
@@ -105,7 +183,7 @@ def mkSigmasVal (ty : Expr) (xl : List Expr) : MetaM Expr :=
match xl with
| [] => do
trace[Diverge.def.sigmas] "mkSigmasVal: []"
- pure (Expr.const ``PUnit.unit [])
+ pure (Expr.const ``PUnit.unit [Level.succ .zero])
| [x] => do
trace[Diverge.def.sigmas] "mkSigmasVal: [{x}]"
pure x
@@ -122,6 +200,17 @@ def mkSigmasVal (ty : Expr) (xl : List Expr) : MetaM Expr :=
trace[Diverge.def.sigmas] "mkSigmasVal:\n{alpha}\n{beta}\n{fst}\n{snd}"
mkAppOptM ``Sigma.mk #[some alpha, some beta, some fst, some snd]
+/- Group a list of expressions into a (non-dependent) tuple -/
+def mkProdsVal (xl : List Expr) : MetaM Expr :=
+ match xl with
+ | [] =>
+ pure (Expr.const ``PUnit.unit [Level.succ .zero])
+ | [x] => do
+ pure x
+ | x :: xl => do
+ let xl ← mkProdsVal xl
+ mkAppM ``Prod.mk #[x, xl]
+
def mkAnonymous (s : String) (i : Nat) : Name :=
.num (.str .anonymous s) i
@@ -159,31 +248,31 @@ partial def mkSigmasMatch (xl : List Expr) (out : Expr) (index : Nat := 0) : Met
match xl with
| [] => do
-- This would be unexpected
- throwError "mkSigmasMatch: empyt list of input parameters"
+ throwError "mkSigmasMatch: empty list of input parameters"
| [x] => do
-- In the example given for the explanations: this is the inner match case
trace[Diverge.def.sigmas] "mkSigmasMatch: [{x}]"
mkLambdaFVars #[x] out
| fst :: xl => do
- -- In the example given for the explanations: this is the outer match case
- -- Remark: for the naming purposes, we use the same convention as for the
- -- fields and parameters in `Sigma.casesOn` and `Sigma.mk` (looking at
- -- those definitions might help)
- --
- -- We want to build the match expression:
- -- ```
- -- λ scrut =>
- -- match scrut with
- -- | Sigma.mk x ... -- the hole is given by a recursive call on the tail
- -- ```
+ /- In the example given for the explanations: this is the outer match case
+ Remark: for the naming purposes, we use the same convention as for the
+ fields and parameters in `Sigma.casesOn` and `Sigma.mk` (looking at
+ those definitions might help)
+
+ We want to build the match expression:
+ ```
+ λ scrut =>
+ match scrut with
+ | Sigma.mk x ... -- the hole is given by a recursive call on the tail
+ ``` -/
trace[Diverge.def.sigmas] "mkSigmasMatch: [{fst}::{xl}]"
- let alpha ← Lean.Meta.inferType fst
+ let alpha ← inferType fst
let snd_ty ← mkSigmasType xl
let beta ← mkLambdaFVars #[fst] snd_ty
let snd ← mkSigmasMatch xl out (index + 1)
let mk ← mkLambdaFVars #[fst] snd
-- Introduce the "scrut" variable
- let scrut_ty ← mkSigmasType (fst :: xl)
+ let scrut_ty ← mkSigmaType fst snd_ty
withLocalDeclD (mkAnonymous "scrut" index) scrut_ty fun scrut => do
trace[Diverge.def.sigmas] "mkSigmasMatch: scrut: ({scrut}) : ({← inferType scrut})"
-- TODO: make the computation of the motive more efficient
@@ -206,6 +295,67 @@ partial def mkSigmasMatch (xl : List Expr) (out : Expr) (index : Nat := 0) : Met
trace[Diverge.def.sigmas] "mkSigmasMatch: sm: {sm}"
pure sm
+/- This is similar to `mkSigmasMatch`, but with non-dependent tuples
+
+ Remark: factor out with `mkSigmasMatch`? This is extremely similar.
+-/
+partial def mkProdsMatch (xl : List Expr) (out : Expr) (index : Nat := 0) : MetaM Expr :=
+ match xl with
+ | [] => do
+ -- This would be unexpected
+ throwError "mkProdsMatch: empty list of input parameters"
+ | [x] => do
+ -- In the example given for the explanations: this is the inner match case
+ trace[Diverge.def.prods] "mkProdsMatch: [{x}]"
+ mkLambdaFVars #[x] out
+ | fst :: xl => do
+ trace[Diverge.def.prods] "mkProdsMatch: [{fst}::{xl}]"
+ let alpha ← inferType fst
+ let beta ← mkProdsType xl
+ let snd ← mkProdsMatch xl out (index + 1)
+ let mk ← mkLambdaFVars #[fst] snd
+ -- Introduce the "scrut" variable
+ let scrut_ty ← mkProdType alpha beta
+ withLocalDeclD (mkAnonymous "scrut" index) scrut_ty fun scrut => do
+ trace[Diverge.def.prods] "mkProdsMatch: scrut: ({scrut}) : ({← inferType scrut})"
+ -- TODO: make the computation of the motive more efficient
+ let motive ← do
+ let out_ty ← inferType out
+ mkLambdaFVars #[scrut] out_ty
+ -- The final expression: putting everything together
+ trace[Diverge.def.prods] "mkProdsMatch:\n ({alpha})\n ({beta})\n ({motive})\n ({scrut})\n ({mk})"
+ let sm ← mkAppOptM ``Prod.casesOn #[some alpha, some beta, some motive, some scrut, some mk]
+ -- Abstracting the "scrut" variable
+ let sm ← mkLambdaFVars #[scrut] sm
+ trace[Diverge.def.prods] "mkProdsMatch: sm: {sm}"
+ pure sm
+
+/- Same as `mkSigmasMatch` but also accepts an empty list of inputs, in which case
+ it generates the expression:
+ ```
+ λ () => e
+ ``` -/
+def mkSigmasMatchOrUnit (xl : List Expr) (out : Expr) : MetaM Expr :=
+ if xl.isEmpty then do
+ let scrut_ty := Expr.const ``PUnit [Level.succ .zero]
+ withLocalDeclD (mkAnonymous "scrut" 0) scrut_ty fun scrut => do
+ mkLambdaFVars #[scrut] out
+ else
+ mkSigmasMatch xl out
+
+/- Same as `mkProdsMatch` but also accepts an empty list of inputs, in which case
+ it generates the expression:
+ ```
+ λ () => e
+ ``` -/
+def mkProdsMatchOrUnit (xl : List Expr) (out : Expr) : MetaM Expr :=
+ if xl.isEmpty then do
+ let scrut_ty := Expr.const ``PUnit [Level.succ .zero]
+ withLocalDeclD (mkAnonymous "scrut" 0) scrut_ty fun scrut => do
+ mkLambdaFVars #[scrut] out
+ else
+ mkProdsMatch xl out
+
/- Small tests for list_nth: give a model of what `mkSigmasMatch` should generate -/
private def list_nth_out_ty_inner (a :Type) (scrut1: @Sigma (List a) (fun (_ls : List a) => Int)) :=
@Sigma.casesOn (List a)
@@ -238,6 +388,52 @@ def mkFinVal (n i : Nat) : MetaM Expr := do
let ofNat ← mkAppOptM ``Fin.instOfNatFinHAddNatInstHAddInstAddNatOfNat #[n_lit, i_lit]
mkAppOptM ``OfNat.ofNat #[none, none, ofNat]
+/- Information about the type of a function in a declaration group.
+
+ In the comments about the fields, we take as example the
+ `list_nth (α : Type) (ls : List α) (i : Int) : Result α` function.
+ -/
+structure TypeInfo where
+ /- The total number of input arguments.
+
+ For list_nth: 3
+ -/
+ total_num_args : ℕ
+ /- The number of type parameters (they should be a prefix of the input arguments).
+
+ For `list_nth`: 1
+ -/
+ num_params : ℕ
+ /- The type of the dependent tuple grouping the type parameters.
+
+ For `list_nth`: `Type`
+ -/
+ params_ty : Expr
+ /- The type of the tuple grouping the input values. This is a function taking
+ as input a value of type `params_ty`.
+
+ For `list_nth`: `λ a => List a × Int`
+ -/
+ in_ty : Expr
+ /- The output type, without the `Return`. This is a function taking
+ as input a value of type `params_ty`.
+
+ For `list_nth`: `λ a => a`
+ -/
+ out_ty : Expr
+
+def mkInOutTyFromTypeInfo (info : TypeInfo) : MetaM Expr := do
+ mkInOutTy info.params_ty info.in_ty info.out_ty
+
+instance : Inhabited TypeInfo :=
+ { default := { total_num_args := 0, num_params := 0, params_ty := UnitType,
+ in_ty := UnitType, out_ty := UnitType } }
+
+instance : ToMessageData TypeInfo :=
+ ⟨ λ ⟨ total_num_args, num_params, params_ty, in_ty, out_ty ⟩ =>
+ f!"\{ total_num_args: {total_num_args}, num_params: {num_params}, params_ty: {params_ty}, in_ty: {in_ty}, out_ty: {out_ty} }}"
+ ⟩
+
/- Generate and declare as individual definitions the bodies for the individual funcions:
- replace the recursive calls with calls to the continutation `k`
- make those bodies take one single dependent tuple as input
@@ -246,15 +442,17 @@ def mkFinVal (n i : Nat) : MetaM Expr := do
We return the new declarations.
-/
def mkDeclareUnaryBodies (grLvlParams : List Name) (kk_var : Expr)
- (inOutTys : Array (Expr × Expr)) (preDefs : Array PreDefinition) :
+ (paramInOutTys : Array TypeInfo) (preDefs : Array PreDefinition) :
MetaM (Array Expr) := do
let grSize := preDefs.size
- -- Compute the map from name to (index × input type).
- -- Remark: the continuation has an indexed type; we use the index (a finite number of
- -- type `Fin`) to control which function we call at the recursive call site.
- let nameToInfo : HashMap Name (Nat × Expr) :=
- let bl := preDefs.mapIdx fun i d => (d.declName, (i.val, (inOutTys.get! i.val).fst))
+ /- Compute the map from name to (index, type info).
+
+ Remark: the continuation has an indexed type; we use the index (a finite number of
+ type `Fin`) to control which function we call at the recursive call site. -/
+ let nameToInfo : HashMap Name (Nat × TypeInfo) :=
+ let bl := preDefs.mapIdx fun i d =>
+ (d.declName, (i.val, paramInOutTys.get! i.val))
HashMap.ofList bl.toList
trace[Diverge.def.genBody] "nameToId: {nameToInfo.toList}"
@@ -262,35 +460,65 @@ def mkDeclareUnaryBodies (grLvlParams : List Name) (kk_var : Expr)
-- Auxiliary function to explore the function bodies and replace the
-- recursive calls
let visit_e (i : Nat) (e : Expr) : MetaM Expr := do
- trace[Diverge.def.genBody] "visiting expression (dept: {i}): {e}"
+ trace[Diverge.def.genBody.visit] "visiting expression (dept: {i}): {e}"
let ne ← do
match e with
| .app .. => do
e.withApp fun f args => do
- trace[Diverge.def.genBody] "this is an app: {f} {args}"
+ trace[Diverge.def.genBody.visit] "this is an app: {f} {args}"
-- Check if this is a recursive call
if f.isConst then
let name := f.constName!
match nameToInfo.find? name with
| none => pure e
- | some (id, in_ty) =>
- trace[Diverge.def.genBody] "this is a recursive call"
+ | some (id, type_info) =>
+ trace[Diverge.def.genBody.visit] "this is a recursive call"
-- This is a recursive call: replace it
-- Compute the index
let i ← mkFinVal grSize id
- -- Put the arguments in one big dependent tuple
- let args ← mkSigmasVal in_ty args.toList
- mkAppM' kk_var #[i, args]
+ -- It can happen that there are no input values given to the
+ -- recursive calls, and only type parameters.
+ let num_args := args.size
+ if num_args ≠ type_info.total_num_args ∧ num_args ≠ type_info.num_params then
+ throwError "Invalid number of arguments for the recursive call: {e}"
+ -- Split the arguments, and put them in two tuples (the first
+ -- one is a dependent tuple)
+ let (param_args, args) := args.toList.splitAt type_info.num_params
+ trace[Diverge.def.genBody.visit] "param_args: {param_args}, args: {args}"
+ let param_args ← mkSigmasVal type_info.params_ty param_args
+ -- Check if there are input values
+ if num_args = type_info.total_num_args then do
+ trace[Diverge.def.genBody.visit] "Recursive call with input values"
+ let args ← mkProdsVal args
+ mkAppM' kk_var #[i, param_args, args]
+ else do
+ trace[Diverge.def.genBody.visit] "Recursive call without input values"
+ mkAppM' kk_var #[i, param_args]
else
-- Not a recursive call: do nothing
pure e
| .const name _ =>
- -- Sanity check: we eliminated all the recursive calls
- if (nameToInfo.find? name).isSome then
- throwError "mkUnaryBodies: a recursive call was not eliminated"
+ /- This might refer to the one of the top-level functions if we use
+ it without arguments (if we give it to a higher-order
+ function for instance) and there are actually no type parameters.
+ -/
+ if (nameToInfo.find? name).isSome then do
+ -- Checking the type information
+ match nameToInfo.find? name with
+ | none => pure e
+ | some (id, type_info) =>
+ trace[Diverge.def.genBody.visit] "this is a recursive call"
+ -- This is a recursive call: replace it
+ -- Compute the index
+ let i ← mkFinVal grSize id
+ -- Check that there are no type parameters
+ if type_info.num_params ≠ 0 then throwError "mkUnaryBodies: a recursive call was not eliminated"
+ -- Introduce the call to the continuation
+ let param_args ← mkSigmasVal type_info.params_ty []
+ mkAppM' kk_var #[i, param_args]
else pure e
| _ => pure e
- trace[Diverge.def.genBody] "done with expression (depth: {i}): {e}"
+ trace[Diverge.def.genBody.visit] "done with expression (depth: {i}): {e}"
pure ne
-- Explore the bodies
@@ -300,13 +528,20 @@ def mkDeclareUnaryBodies (grLvlParams : List Name) (kk_var : Expr)
let body ← mapVisit visit_e preDef.value
trace[Diverge.def.genBody] "Body after replacement of the recursive calls: {body}"
- -- Currify the function by grouping the arguments into a dependent tuple
+ -- Currify the function by grouping the arguments into dependent tuples
-- (over which we match to retrieve the individual arguments).
lambdaTelescope body fun args body => do
- let body ← mkSigmasMatch args.toList body 0
+ -- Split the arguments between the type parameters and the "regular" inputs
+ let (_, type_info) := nameToInfo.find! preDef.declName
+ let (param_args, args) := args.toList.splitAt type_info.num_params
+ let body ← mkProdsMatchOrUnit args body
+ trace[Diverge.def.genBody] "Body after mkProdsMatchOrUnit: {body}"
+ let body ← mkSigmasMatchOrUnit param_args body
+ trace[Diverge.def.genBody] "Body after mkSigmasMatchOrUnit: {body}"
-- Add the declaration
let value ← mkLambdaFVars #[kk_var] body
+ trace[Diverge.def.genBody] "Body after abstracting kk: {value}"
let name := preDef.declName.append "body"
let levelParams := grLvlParams
let decl := Declaration.defnDecl {
@@ -318,41 +553,46 @@ def mkDeclareUnaryBodies (grLvlParams : List Name) (kk_var : Expr)
safety := .safe
all := [name]
}
+ trace[Diverge.def.genBody] "About to add decl"
addDecl decl
trace[Diverge.def] "individual body of {preDef.declName}: {body}"
-- Return the constant
let body := Lean.mkConst name (levelParams.map .param)
- -- let body ← mkAppM' body #[kk_var]
trace[Diverge.def] "individual body (after decl): {body}"
pure body
--- Generate a unique function body from the bodies of the mutually recursive group,
--- and add it as a declaration in the context.
--- We return the list of bodies (of type `FixI.Funs ...`) and the mutually recursive body.
+/- Generate a unique function body from the bodies of the mutually recursive group,
+ and add it as a declaration in the context.
+ We return the list of bodies (of type `FixI.Funs ...`) and the mutually recursive body.
+ -/
def mkDeclareMutRecBody (grName : Name) (grLvlParams : List Name)
(kk_var i_var : Expr)
- (in_ty out_ty : Expr) (inOutTys : List (Expr × Expr))
+ (param_ty in_ty out_ty : Expr) (paramInOutTys : Array TypeInfo)
(bodies : Array Expr) : MetaM (Expr × Expr) := do
-- Generate the body
let grSize := bodies.size
let finTypeExpr := mkFin grSize
-- TODO: not very clean
- let inOutTyType ← do
- let (x, y) := inOutTys.get! 0
- inferType (← mkInOutTy x y)
- let rec mkFuns (inOutTys : List (Expr × Expr)) (bl : List Expr) : MetaM Expr :=
- match inOutTys, bl with
+ let paramInOutTyType ← do
+ let info := paramInOutTys.get! 0
+ inferType (← mkInOutTyFromTypeInfo info)
+ let rec mkFuns (paramInOutTys : List TypeInfo) (bl : List Expr) : MetaM Expr :=
+ match paramInOutTys, bl with
| [], [] =>
- mkAppOptM ``FixI.Funs.Nil #[finTypeExpr, in_ty, out_ty]
- | (ity, oty) :: inOutTys, b :: bl => do
+ mkAppOptM ``FixII.Funs.Nil #[finTypeExpr, param_ty, in_ty, out_ty]
+ | info :: paramInOutTys, b :: bl => do
+ let pty := info.params_ty
+ let ity := info.in_ty
+ let oty := info.out_ty
-- Retrieving ity and oty - this is not very clean
- let inOutTysExpr ← mkListLit inOutTyType (← inOutTys.mapM (λ (x, y) => mkInOutTy x y))
- let fl ← mkFuns inOutTys bl
- mkAppOptM ``FixI.Funs.Cons #[finTypeExpr, in_ty, out_ty, ity, oty, inOutTysExpr, b, fl]
+ let paramInOutTysExpr ← mkListLit paramInOutTyType
+ (← paramInOutTys.mapM mkInOutTyFromTypeInfo)
+ let fl ← mkFuns paramInOutTys bl
+ mkAppOptM ``FixII.Funs.Cons #[finTypeExpr, param_ty, in_ty, out_ty, pty, ity, oty, paramInOutTysExpr, b, fl]
| _, _ => throwError "mkDeclareMutRecBody: `tys` and `bodies` don't have the same length"
- let bodyFuns ← mkFuns inOutTys bodies.toList
+ let bodyFuns ← mkFuns paramInOutTys.toList bodies.toList
-- Wrap in `get_fun`
- let body ← mkAppM ``FixI.get_fun #[bodyFuns, i_var, kk_var]
+ let body ← mkAppM ``FixII.get_fun #[bodyFuns, i_var, kk_var]
-- Add the index `i` and the continuation `k` as a variables
let body ← mkLambdaFVars #[kk_var, i_var] body
trace[Diverge.def] "mkDeclareMutRecBody: body: {body}"
@@ -391,11 +631,11 @@ instance : ToMessageData MatchInfo where
-- This is not a very clean formatting, but we don't need more
toMessageData := fun me => m!"\n- matcherName: {me.matcherName}\n- params: {me.params}\n- motive: {me.motive}\n- scruts: {me.scruts}\n- branchesNumParams: {me.branchesNumParams}\n- branches: {me.branches}"
--- Small helper: prove that an expression which doesn't use the continuation `kk`
--- is valid, and return the proof.
+/- Small helper: prove that an expression which doesn't use the continuation `kk`
+ is valid, and return the proof. -/
def proveNoKExprIsValid (k_var : Expr) (e : Expr) : MetaM Expr := do
trace[Diverge.def.valid] "proveNoKExprIsValid: {e}"
- let eIsValid ← mkAppM ``FixI.is_valid_p_same #[k_var, e]
+ let eIsValid ← mkAppM ``FixII.is_valid_p_same #[k_var, e]
trace[Diverge.def.valid] "proveNoKExprIsValid: result:\n{eIsValid}:\n{← inferType eIsValid}"
pure eIsValid
@@ -410,7 +650,16 @@ mutual
```
-/
partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
- trace[Diverge.def.valid] "proveValid: {e}"
+ trace[Diverge.def.valid] "proveExprIsValid: {e}"
+ -- Normalize to eliminate the lambdas - TODO: this is slightly dangerous.
+ let e ← do
+ if e.isLet ∧ normalize_let_bindings then do
+ let updt_config config :=
+ { config with transparency := .reducible, zetaNonDep := false }
+ let e ← withConfig updt_config (whnf e)
+ trace[Diverge.def.valid] "e (after normalization): {e}"
+ pure e
+ else pure e
match e with
| .const _ _ => throwError "Unimplemented" -- Shouldn't get there?
| .bvar _
@@ -418,9 +667,10 @@ partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
| .lit _
| .mvar _
| .sort _ => throwError "Unreachable"
- | .lam .. => throwError "Unimplemented"
+ | .lam .. => throwError "Unimplemented" -- TODO
| .forallE .. => throwError "Unreachable" -- Shouldn't get there
| .letE .. => do
+ -- Remark: this branch is not taken if we normalize the expressions (above)
-- Telescope all the let-bindings (remark: this also telescopes the lambdas)
lambdaLetTelescope e fun xs body => do
-- Note that we don't visit the bound values: there shouldn't be
@@ -438,164 +688,268 @@ partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
proveNoKExprIsValid k_var e
| .app .. =>
e.withApp fun f args => do
- -- There are several cases: first, check if this is a match/if
- -- Check if the expression is a (dependent) if then else.
- -- We treat the if then else expressions differently from the other matches,
- -- and have dedicated theorems for them.
- let isIte := e.isIte
- if isIte || e.isDIte then do
- e.withApp fun f args => do
- trace[Diverge.def.valid] "ite/dite: {f}:\n{args}"
- if args.size ≠ 5 then
- throwError "Wrong number of parameters for {f}: {args}"
- let cond := args.get! 1
- let dec := args.get! 2
- -- Prove that the branches are valid
- let br0 := args.get! 3
- let br1 := args.get! 4
- let proveBranchValid (br : Expr) : MetaM Expr :=
- if isIte then proveExprIsValid k_var kk_var br
- else do
- -- There is a lambda
- lambdaOne br fun x br => do
- let brValid ← proveExprIsValid k_var kk_var br
- mkLambdaFVars #[x] brValid
- let br0Valid ← proveBranchValid br0
- let br1Valid ← proveBranchValid br1
- let const := if isIte then ``FixI.is_valid_p_ite else ``FixI.is_valid_p_dite
- let eIsValid ← mkAppOptM const #[none, none, none, none, some k_var, some cond, some dec, none, none, some br0Valid, some br1Valid]
- trace[Diverge.def.valid] "ite/dite: result:\n{eIsValid}:\n{← inferType eIsValid}"
- pure eIsValid
- -- Check if the expression is a match (this case is for when the elaborator
- -- introduces auxiliary definitions to hide the match behind syntactic
- -- sugar):
- else if let some me := ← matchMatcherApp? e then do
- trace[Diverge.def.valid]
- "matcherApp:
- - params: {me.params}
- - motive: {me.motive}
- - discrs: {me.discrs}
- - altNumParams: {me.altNumParams}
- - alts: {me.alts}
- - remaining: {me.remaining}"
- -- matchMatcherApp does all the work for us: we simply need to gather
- -- the information and call the auxiliary helper `proveMatchIsValid`
- if me.remaining.size ≠ 0 then
- throwError "MatcherApp: non empty remaining array: {me.remaining}"
- let me : MatchInfo := {
- matcherName := me.matcherName
- matcherLevels := me.matcherLevels
- params := me.params
- motive := me.motive
- scruts := me.discrs
- branchesNumParams := me.altNumParams
- branches := me.alts
- }
- proveMatchIsValid k_var kk_var me
- -- Check if the expression is a raw match (this case is for when the expression
- -- is a direct call to the primitive `casesOn` function, without syntactic sugar).
- -- We have to check this case because functions like `mkSigmasMatch`, which we
- -- use to currify function bodies, introduce such raw matches.
- else if ← isCasesExpr f then do
- trace[Diverge.def.valid] "rawMatch: {e}"
- -- Deconstruct the match, and call the auxiliary helper `proveMatchIsValid`.
- --
- -- The casesOn definition is always of the following shape:
- -- - input parameters (implicit parameters)
- -- - motive (implicit), -- the motive gives the return type of the match
- -- - scrutinee (explicit)
- -- - branches (explicit).
- -- In particular, we notice that the scrutinee is the first *explicit*
- -- parameter - this is how we spot it.
- let matcherName := f.constName!
- let matcherLevels := f.constLevels!.toArray
- -- Find the first explicit parameter: this is the scrutinee
- forallTelescope (← inferType f) fun xs _ => do
- let rec findFirstExplicit (i : Nat) : MetaM Nat := do
- if i ≥ xs.size then throwError "Unexpected: could not find an explicit parameter"
- else
- let x := xs.get! i
- let xFVarId := x.fvarId!
- let localDecl ← xFVarId.getDecl
- match localDecl.binderInfo with
- | .default => pure i
- | _ => findFirstExplicit (i + 1)
- let scrutIdx ← findFirstExplicit 0
- -- Split the arguments
- let params := args.extract 0 (scrutIdx - 1)
- let motive := args.get! (scrutIdx - 1)
- let scrut := args.get! scrutIdx
- let branches := args.extract (scrutIdx + 1) args.size
- -- Compute the number of parameters for the branches: for this we use
- -- the type of the uninstantiated casesOn constant (we can't just
- -- destruct the lambdas in the branch expressions because the result
- -- of a match might be a lambda expression).
- let branchesNumParams : Array Nat ← do
- let env ← getEnv
- let decl := env.constants.find! matcherName
- let ty := decl.type
- forallTelescope ty fun xs _ => do
- let xs := xs.extract (scrutIdx + 1) xs.size
- xs.mapM fun x => do
- let xty ← inferType x
- forallTelescope xty fun ys _ => do
- pure ys.size
- let me : MatchInfo := {
- matcherName,
- matcherLevels,
- params,
- motive,
- scruts := #[scrut],
- branchesNumParams,
- branches,
- }
- proveMatchIsValid k_var kk_var me
- -- Check if this is a monadic let-binding
- else if f.isConstOf ``Bind.bind then do
- trace[Diverge.def.valid] "bind:\n{args}"
- -- We simply need to prove that the subexpressions are valid, and call
- -- the appropriate lemma.
- let x := args.get! 4
- let y := args.get! 5
- -- Prove that the subexpressions are valid
- let xValid ← proveExprIsValid k_var kk_var x
- trace[Diverge.def.valid] "bind: xValid:\n{xValid}:\n{← inferType xValid}"
- let yValid ← do
- -- This is a lambda expression
- lambdaOne y fun x y => do
- trace[Diverge.def.valid] "bind: y: {y}"
- let yValid ← proveExprIsValid k_var kk_var y
- trace[Diverge.def.valid] "bind: yValid (no forall): {yValid}"
- trace[Diverge.def.valid] "bind: yValid: x: {x}"
- let yValid ← mkLambdaFVars #[x] yValid
- trace[Diverge.def.valid] "bind: yValid (forall): {yValid}: {← inferType yValid}"
- pure yValid
- -- Put everything together
- trace[Diverge.def.valid] "bind:\n- xValid: {xValid}: {← inferType xValid}\n- yValid: {yValid}: {← inferType yValid}"
- mkAppM ``FixI.is_valid_p_bind #[xValid, yValid]
- -- Check if this is a recursive call, i.e., a call to the continuation `kk`
- else if f.isFVarOf kk_var.fvarId! then do
- trace[Diverge.def.valid] "rec: args: \n{args}"
- if args.size ≠ 2 then throwError "Recursive call with invalid number of parameters: {args}"
- let i_arg := args.get! 0
- let x_arg := args.get! 1
- let eIsValid ← mkAppM ``FixI.is_valid_p_rec #[k_var, i_arg, x_arg]
- trace[Diverge.def.valid] "rec: result: \n{eIsValid}"
- pure eIsValid
+ proveAppIsValid k_var kk_var e f args
+
+partial def proveAppIsValid (k_var kk_var : Expr) (e : Expr) (f : Expr) (args : Array Expr): MetaM Expr := do
+ trace[Diverge.def.valid] "proveAppIsValid: {e}\nDecomposed: {f} {args}"
+ /- There are several cases: first, check if this is a match/if
+ Check if the expression is a (dependent) if then else.
+ We treat the if then else expressions differently from the other matches,
+ and have dedicated theorems for them. -/
+ let isIte := e.isIte
+ if isIte || e.isDIte then do
+ e.withApp fun f args => do
+ trace[Diverge.def.valid] "ite/dite: {f}:\n{args}"
+ if args.size ≠ 5 then
+ throwError "Wrong number of parameters for {f}: {args}"
+ let cond := args.get! 1
+ let dec := args.get! 2
+ -- Prove that the branches are valid
+ let br0 := args.get! 3
+ let br1 := args.get! 4
+ let proveBranchValid (br : Expr) : MetaM Expr :=
+ if isIte then proveExprIsValid k_var kk_var br
else do
- -- Remaining case: normal application.
- -- It shouldn't use the continuation.
- proveNoKExprIsValid k_var e
+ -- There is a lambda
+ lambdaOne br fun x br => do
+ let brValid ← proveExprIsValid k_var kk_var br
+ mkLambdaFVars #[x] brValid
+ let br0Valid ← proveBranchValid br0
+ let br1Valid ← proveBranchValid br1
+ let const := if isIte then ``FixII.is_valid_p_ite else ``FixII.is_valid_p_dite
+ let eIsValid ←
+ mkAppOptM const #[none, none, none, none, none,
+ some k_var, some cond, some dec, none, none,
+ some br0Valid, some br1Valid]
+ trace[Diverge.def.valid] "ite/dite: result:\n{eIsValid}:\n{← inferType eIsValid}"
+ pure eIsValid
+ /- Check if the expression is a match (this case is for when the elaborator
+ introduces auxiliary definitions to hide the match behind syntactic
+ sugar): -/
+ else if let some me := ← matchMatcherApp? e then do
+ trace[Diverge.def.valid]
+ "matcherApp:
+ - params: {me.params}
+ - motive: {me.motive}
+ - discrs: {me.discrs}
+ - altNumParams: {me.altNumParams}
+ - alts: {me.alts}
+ - remaining: {me.remaining}"
+ -- matchMatcherApp does all the work for us: we simply need to gather
+ -- the information and call the auxiliary helper `proveMatchIsValid`
+ if me.remaining.size ≠ 0 then
+ throwError "MatcherApp: non empty remaining array: {me.remaining}"
+ let me : MatchInfo := {
+ matcherName := me.matcherName
+ matcherLevels := me.matcherLevels
+ params := me.params
+ motive := me.motive
+ scruts := me.discrs
+ branchesNumParams := me.altNumParams
+ branches := me.alts
+ }
+ proveMatchIsValid k_var kk_var me
+ /- Check if the expression is a raw match (this case is for when the expression
+ is a direct call to the primitive `casesOn` function, without syntactic sugar).
+ We have to check this case because functions like `mkSigmasMatch`, which we
+ use to currify function bodies, introduce such raw matches. -/
+ else if ← isCasesExpr f then do
+ trace[Diverge.def.valid] "rawMatch: {e}"
+ /- Deconstruct the match, and call the auxiliary helper `proveMatchIsValid`.
+
+ The casesOn definition is always of the following shape:
+ - input parameters (implicit parameters)
+ - motive (implicit), -- the motive gives the return type of the match
+ - scrutinee (explicit)
+ - branches (explicit).
+ In particular, we notice that the scrutinee is the first *explicit*
+ parameter - this is how we spot it.
+ -/
+ let matcherName := f.constName!
+ let matcherLevels := f.constLevels!.toArray
+ -- Find the first explicit parameter: this is the scrutinee
+ forallTelescope (← inferType f) fun xs _ => do
+ let rec findFirstExplicit (i : Nat) : MetaM Nat := do
+ if i ≥ xs.size then throwError "Unexpected: could not find an explicit parameter"
+ else
+ let x := xs.get! i
+ let xFVarId := x.fvarId!
+ let localDecl ← xFVarId.getDecl
+ match localDecl.binderInfo with
+ | .default => pure i
+ | _ => findFirstExplicit (i + 1)
+ let scrutIdx ← findFirstExplicit 0
+ -- Split the arguments
+ let params := args.extract 0 (scrutIdx - 1)
+ let motive := args.get! (scrutIdx - 1)
+ let scrut := args.get! scrutIdx
+ let branches := args.extract (scrutIdx + 1) args.size
+ /- Compute the number of parameters for the branches: for this we use
+ the type of the uninstantiated casesOn constant (we can't just
+ destruct the lambdas in the branch expressions because the result
+ of a match might be a lambda expression). -/
+ let branchesNumParams : Array Nat ← do
+ let env ← getEnv
+ let decl := env.constants.find! matcherName
+ let ty := decl.type
+ forallTelescope ty fun xs _ => do
+ let xs := xs.extract (scrutIdx + 1) xs.size
+ xs.mapM fun x => do
+ let xty ← inferType x
+ forallTelescope xty fun ys _ => do
+ pure ys.size
+ let me : MatchInfo := {
+ matcherName,
+ matcherLevels,
+ params,
+ motive,
+ scruts := #[scrut],
+ branchesNumParams,
+ branches,
+ }
+ proveMatchIsValid k_var kk_var me
+ -- Check if this is a monadic let-binding
+ else if f.isConstOf ``Bind.bind then do
+ trace[Diverge.def.valid] "bind:\n{args}"
+ -- We simply need to prove that the subexpressions are valid, and call
+ -- the appropriate lemma.
+ let x := args.get! 4
+ let y := args.get! 5
+ -- Prove that the subexpressions are valid
+ let xValid ← proveExprIsValid k_var kk_var x
+ trace[Diverge.def.valid] "bind: xValid:\n{xValid}:\n{← inferType xValid}"
+ let yValid ← do
+ -- This is a lambda expression
+ lambdaOne y fun x y => do
+ trace[Diverge.def.valid] "bind: y: {y}"
+ let yValid ← proveExprIsValid k_var kk_var y
+ trace[Diverge.def.valid] "bind: yValid (no forall): {yValid}"
+ trace[Diverge.def.valid] "bind: yValid: x: {x}"
+ let yValid ← mkLambdaFVars #[x] yValid
+ trace[Diverge.def.valid] "bind: yValid (forall): {yValid}: {← inferType yValid}"
+ pure yValid
+ -- Put everything together
+ trace[Diverge.def.valid] "bind:\n- xValid: {xValid}: {← inferType xValid}\n- yValid: {yValid}: {← inferType yValid}"
+ mkAppM ``FixII.is_valid_p_bind #[xValid, yValid]
+ -- Check if this is a recursive call, i.e., a call to the continuation `kk`
+ else if f.isFVarOf kk_var.fvarId! then do
+ trace[Diverge.def.valid] "rec: args: \n{args}"
+ if args.size ≠ 3 then throwError "Recursive call with invalid number of parameters: {args}"
+ let i_arg := args.get! 0
+ let t_arg := args.get! 1
+ let x_arg := args.get! 2
+ let eIsValid ← mkAppM ``FixII.is_valid_p_rec #[k_var, i_arg, t_arg, x_arg]
+ trace[Diverge.def.valid] "rec: result: \n{eIsValid}"
+ pure eIsValid
+ else do
+ /- Remaining case: normal application.
+ Check if the arguments use the continuation:
+ - if no: this is simple
+ - if yes: we have to lookup theorems in div spec database and continue -/
+ trace[Diverge.def.valid] "regular app: {e}, f: {f}, args: {args}"
+ let argsFVars ← args.mapM getFVarIds
+ let allArgsFVars := argsFVars.foldl (fun hs fvars => hs.insertMany fvars) HashSet.empty
+ trace[Diverge.def.valid] "allArgsFVars: {allArgsFVars.toList.map mkFVar}"
+ if ¬ allArgsFVars.contains kk_var.fvarId! then do
+ -- Simple case
+ trace[Diverge.def.valid] "kk doesn't appear in the arguments"
+ proveNoKExprIsValid k_var e
+ else do
+ -- Lookup in the database for suitable theorems
+ trace[Diverge.def.valid] "kk appears in the arguments"
+ let thms ← divspecAttr.find? e
+ trace[Diverge.def.valid] "Looked up theorems: {thms}"
+ -- Try the theorems one by one
+ proveAppIsValidApplyThms k_var kk_var e f args thms.toList
+
+partial def proveAppIsValidApplyThms (k_var kk_var : Expr) (e : Expr)
+ (f : Expr) (args : Array Expr) (thms : List Name) : MetaM Expr := do
+ match thms with
+ | [] => throwError "Could not prove that the following expression is valid: {e}"
+ | thName :: thms =>
+ -- Lookup the theorem itself
+ let env ← getEnv
+ let thDecl := env.constants.find! thName
+ -- Introduce fresh meta-variables for the universes
+ let ul : List (Name × Level) ←
+ thDecl.levelParams.mapM (λ x => do pure (x, ← mkFreshLevelMVar))
+ let ulMap : HashMap Name Level := HashMap.ofList ul
+ let thTy := thDecl.type.instantiateLevelParamsCore (λ x => ulMap.find! x)
+ trace[Diverge.def.valid] "Trying with theorem {thName}: {thTy}"
+ -- Introduce meta variables for the universally quantified variables
+ let (mvars, _binders, thTyBody) ← forallMetaTelescope thTy
+ let thTermToMatch := thTyBody
+ trace[Diverge.def.valid] "thTermToMatch: {thTermToMatch}"
+ -- Create the term: `is_valid_p k (λ kk => e)`
+ let termToMatch ← mkLambdaFVars #[kk_var] e
+ let termToMatch ← mkAppM ``FixII.is_valid_p #[k_var, termToMatch]
+ trace[Diverge.def.valid] "termToMatch: {termToMatch}"
+ -- Attempt to match
+ trace[Diverge.def.valid] "Matching terms:\n\n{termToMatch}\n\n{thTermToMatch}"
+ let ok ← isDefEq termToMatch thTermToMatch
+ if ¬ ok then
+ -- Failure: attempt with the other theorems
+ proveAppIsValidApplyThms k_var kk_var e f args thms
+ else do
+ /- Success: continue with this theorem
+
+ Instantiate the meta variables (some of them will not be instantiated:
+ they are new subgoals)
+ -/
+ let mvars ← mvars.mapM instantiateMVars
+ let th ← mkAppOptM thName (Array.map some mvars)
+ trace[Diverge.def.valid] "Instantiated theorem: {th}\n{← inferType th}"
+ -- Filter the instantiated meta variables
+ let mvars := mvars.filter (fun v => v.isMVar)
+ let mvars := mvars.map (fun v => v.mvarId!)
+ trace[Diverge.def.valid] "Remaining subgoals: {mvars}"
+ for mvarId in mvars do
+ -- Prove the subgoal (i.e., the precondition of the theorem)
+ let mvarDecl ← mvarId.getDecl
+ let declType ← instantiateMVars mvarDecl.type
+ -- Reduce the subgoal before diving in, if necessary
+ trace[Diverge.def.valid] "Subgoal: {declType}"
+ -- Dive in the type
+ forallTelescope declType fun forall_vars mvar_e => do
+ trace[Diverge.def.valid] "forall_vars: {forall_vars}"
+ -- `mvar_e` should have the shape `is_valid_p k (λ kk => ...)`
+ -- We need to retrieve the new `k` variable, and dive into the
+ -- `λ kk => ...`
+ mvar_e.consumeMData.withApp fun is_valid args => do
+ if is_valid.constName? ≠ ``FixII.is_valid_p ∨ args.size ≠ 7 then
+ throwError "Invalid precondition: {mvar_e}"
+ else do
+ let k_var := args.get! 5
+ let e_lam := args.get! 6
+ trace[Diverge.def.valid] "k_var: {k_var}\ne_lam: {e_lam}"
+ -- The outer lambda should be for the kk_var
+ lambdaOne e_lam.consumeMData fun kk_var e => do
+ -- Continue
+ trace[Diverge.def.valid] "kk_var: {kk_var}\ne: {e}"
+ -- We sometimes need to reduce the term - TODO: this is really dangerous
+ let e ← do
+ let updt_config config :=
+ { config with transparency := .reducible, zetaNonDep := false }
+ withConfig updt_config (whnf e)
+ trace[Diverge.def.valid] "e (after normalization): {e}"
+ let e_valid ← proveExprIsValid k_var kk_var e
+ trace[Diverge.def.valid] "e_valid (for e): {e_valid}"
+ let e_valid ← mkLambdaFVars forall_vars e_valid
+ trace[Diverge.def.valid] "e_valid (with foralls): {e_valid}"
+ let _ ← inferType e_valid -- Sanity check
+ -- Assign the meta variable
+ mvarId.assign e_valid
+ pure th
-- Prove that a match expression is valid.
partial def proveMatchIsValid (k_var kk_var : Expr) (me : MatchInfo) : MetaM Expr := do
trace[Diverge.def.valid] "proveMatchIsValid: {me}"
-- Prove the validity of the branch expressions
let branchesValid:Array Expr ← me.branches.mapIdxM fun idx br => do
- -- Go inside the lambdas - note that we have to be careful: some of the
- -- binders might come from the match, and some of the binders might come
- -- from the fact that the expression in the match is a lambda expression:
- -- we use the branchesNumParams field for this reason
+ /- Go inside the lambdas - note that we have to be careful: some of the
+ binders might come from the match, and some of the binders might come
+ from the fact that the expression in the match is a lambda expression:
+ we use the branchesNumParams field for this reason. -/
let numParams := me.branchesNumParams.get! idx
lambdaTelescopeN br numParams fun xs br => do
-- Prove that the branch expression is valid
@@ -603,13 +957,14 @@ partial def proveMatchIsValid (k_var kk_var : Expr) (me : MatchInfo) : MetaM Exp
-- Reconstruct the lambda expression
mkLambdaFVars xs brValid
trace[Diverge.def.valid] "branchesValid:\n{branchesValid}"
- -- Compute the motive, which has the following shape:
- -- ```
- -- λ scrut => is_valid_p k (λ k => match scrut with ...)
- -- ^^^^^^^^^^^^^^^^^^^^
- -- this is the original match expression, with the
- -- the difference that the scrutinee(s) is a variable
- -- ```
+ /- Compute the motive, which has the following shape:
+ ```
+ λ scrut => is_valid_p k (λ k => match scrut with ...)
+ ^^^^^^^^^^^^^^^^^^^^
+ this is the original match expression, with the
+ the difference that the scrutinee(s) is a variable
+ ```
+ -/
let validMotive : Expr ← do
-- The motive is a function of the scrutinees (i.e., a lambda expression):
-- introduce binders for the scrutinees
@@ -628,7 +983,7 @@ partial def proveMatchIsValid (k_var kk_var : Expr) (me : MatchInfo) : MetaM Exp
let matchE ← mkAppOptM me.matcherName args
-- Wrap in the `is_valid_p` predicate
let matchE ← mkLambdaFVars #[kk_var] matchE
- let validMotive ← mkAppM ``FixI.is_valid_p #[k_var, matchE]
+ let validMotive ← mkAppM ``FixII.is_valid_p #[k_var, matchE]
-- Abstract away the scrutinee variables
mkLambdaFVars scrutVars validMotive
trace[Diverge.def.valid] "valid motive: {validMotive}"
@@ -646,10 +1001,10 @@ partial def proveMatchIsValid (k_var kk_var : Expr) (me : MatchInfo) : MetaM Exp
end
--- Prove that a single body (in the mutually recursive group) is valid.
---
--- For instance, if we define the mutually recursive group [`is_even`, `is_odd`],
--- we prove that `is_even.body` and `is_odd.body` are valid.
+/- Prove that a single body (in the mutually recursive group) is valid.
+
+ For instance, if we define the mutually recursive group [`is_even`, `is_odd`],
+ we prove that `is_even.body` and `is_odd.body` are valid. -/
partial def proveSingleBodyIsValid
(k_var : Expr) (preDef : PreDefinition) (bodyConst : Expr) :
MetaM Expr := do
@@ -661,24 +1016,29 @@ partial def proveSingleBodyIsValid
let body := (env.constants.find! name).value!
trace[Diverge.def.valid] "body: {body}"
lambdaTelescope body fun xs body => do
- assert! xs.size = 2
+ trace[Diverge.def.valid] "xs: {xs}"
+ if xs.size ≠ 3 then throwError "Invalid number of lambdas: {xs} (expected 3)"
let kk_var := xs.get! 0
- let x_var := xs.get! 1
+ let t_var := xs.get! 1
+ let x_var := xs.get! 2
-- State the type of the theorem to prove
- let thmTy ← mkAppM ``FixI.is_valid_p
- #[k_var, ← mkLambdaFVars #[kk_var] (← mkAppM' bodyConst #[kk_var, x_var])]
+ trace[Diverge.def.valid] "bodyConst: {bodyConst} : {← inferType bodyConst}"
+ let bodyApp ← mkAppOptM' bodyConst #[.some kk_var, .some t_var, .some x_var]
+ trace[Diverge.def.valid] "bodyApp: {bodyApp}"
+ let bodyApp ← mkLambdaFVars #[kk_var] bodyApp
+ trace[Diverge.def.valid] "bodyApp: {bodyApp}"
+ let thmTy ← mkAppM ``FixII.is_valid_p #[k_var, bodyApp]
trace[Diverge.def.valid] "thmTy: {thmTy}"
-- Prove that the body is valid
+ trace[Diverge.def.valid] "body: {body}"
let proof ← proveExprIsValid k_var kk_var body
- let proof ← mkLambdaFVars #[k_var, x_var] proof
+ let proof ← mkLambdaFVars #[k_var, t_var, x_var] proof
trace[Diverge.def.valid] "proveSingleBodyIsValid: proof:\n{proof}:\n{← inferType proof}"
-- The target type (we don't have to do this: this is simply a sanity check,
-- and this allows a nicer debugging output)
let thmTy ← do
- let body ← mkAppM' bodyConst #[kk_var, x_var]
- let body ← mkLambdaFVars #[kk_var] body
- let ty ← mkAppM ``FixI.is_valid_p #[k_var, body]
- mkForallFVars #[k_var, x_var] ty
+ let ty ← mkAppM ``FixII.is_valid_p #[k_var, bodyApp]
+ mkForallFVars #[k_var, t_var, x_var] ty
trace[Diverge.def.valid] "proveSingleBodyIsValid: thmTy\n{thmTy}:\n{← inferType thmTy}"
-- Save the theorem
let name := preDef.declName ++ "body_is_valid"
@@ -694,18 +1054,18 @@ partial def proveSingleBodyIsValid
-- Return the theorem
pure (Expr.const name (preDef.levelParams.map .param))
--- Prove that the list of bodies are valid.
---
--- For instance, if we define the mutually recursive group [`is_even`, `is_odd`],
--- we prove that `Funs.Cons is_even.body (Funs.Cons is_odd.body Funs.Nil)` is
--- valid.
-partial def proveFunsBodyIsValid (inOutTys: Expr) (bodyFuns : Expr)
+/- Prove that the list of bodies are valid.
+
+ For instance, if we define the mutually recursive group [`is_even`, `is_odd`],
+ we prove that `Funs.Cons is_even.body (Funs.Cons is_odd.body Funs.Nil)` is
+ valid. -/
+partial def proveFunsBodyIsValid (paramInOutTys: Expr) (bodyFuns : Expr)
(k_var : Expr) (bodiesValid : Array Expr) : MetaM Expr := do
-- Create the big "and" expression, which groups the validity proof of the individual bodies
let rec mkValidConj (i : Nat) : MetaM Expr := do
if i = bodiesValid.size then
-- We reached the end
- mkAppM ``FixI.Funs.is_valid_p_Nil #[k_var]
+ mkAppM ``FixII.Funs.is_valid_p_Nil #[k_var]
else do
-- We haven't reached the end: introduce a conjunction
let valid := bodiesValid.get! i
@@ -713,20 +1073,20 @@ partial def proveFunsBodyIsValid (inOutTys: Expr) (bodyFuns : Expr)
mkAppM ``And.intro #[valid, ← mkValidConj (i + 1)]
let andExpr ← mkValidConj 0
-- Wrap in the `is_valid_p_is_valid_p` theorem, and abstract the continuation
- let isValid ← mkAppM ``FixI.Funs.is_valid_p_is_valid_p #[inOutTys, k_var, bodyFuns, andExpr]
+ let isValid ← mkAppM ``FixII.Funs.is_valid_p_is_valid_p #[paramInOutTys, k_var, bodyFuns, andExpr]
mkLambdaFVars #[k_var] isValid
--- Prove that the mut rec body (i.e., the unary body which groups the bodies
--- of all the functions in the mutually recursive group and on which we will
--- apply the fixed-point operator) is valid.
---
--- We save the proof in the theorem "[GROUP_NAME]."mut_rec_body_is_valid",
--- which we return.
---
--- TODO: maybe this function should introduce k_var itself
+/- Prove that the mut rec body (i.e., the unary body which groups the bodies
+ of all the functions in the mutually recursive group and on which we will
+ apply the fixed-point operator) is valid.
+
+ We save the proof in the theorem "[GROUP_NAME]."mut_rec_body_is_valid",
+ which we return.
+
+ TODO: maybe this function should introduce k_var itself -/
def proveMutRecIsValid
(grName : Name) (grLvlParams : List Name)
- (inOutTys : Expr) (bodyFuns mutRecBodyConst : Expr)
+ (paramInOutTys : Expr) (bodyFuns mutRecBodyConst : Expr)
(k_var : Expr) (preDefs : Array PreDefinition)
(bodies : Array Expr) : MetaM Expr := do
-- First prove that the individual bodies are valid
@@ -737,9 +1097,10 @@ def proveMutRecIsValid
proveSingleBodyIsValid k_var preDef body
-- Then prove that the mut rec body is valid
trace[Diverge.def.valid] "## Proving that the 'Funs' body is valid"
- let isValid ← proveFunsBodyIsValid inOutTys bodyFuns k_var bodiesValid
+ let isValid ← proveFunsBodyIsValid paramInOutTys bodyFuns k_var bodiesValid
+ trace[Diverge.def.valid] "Generated the term: {isValid}"
-- Save the theorem
- let thmTy ← mkAppM ``FixI.is_valid #[mutRecBodyConst]
+ let thmTy ← mkAppM ``FixII.is_valid #[mutRecBodyConst]
let name := grName ++ "mut_rec_body_is_valid"
let decl := Declaration.thmDecl {
name
@@ -753,26 +1114,29 @@ def proveMutRecIsValid
-- Return the theorem
pure (Expr.const name (grLvlParams.map .param))
--- Generate the final definions by using the mutual body and the fixed point operator.
---
--- For instance:
--- ```
--- def is_even (i : Int) : Result Bool := mut_rec_body 0 i
--- def is_odd (i : Int) : Result Bool := mut_rec_body 1 i
--- ```
-def mkDeclareFixDefs (mutRecBody : Expr) (inOutTys : Array (Expr × Expr)) (preDefs : Array PreDefinition) :
+/- Generate the final definions by using the mutual body and the fixed point operator.
+
+ For instance:
+ ```
+ def is_even (i : Int) : Result Bool := mut_rec_body 0 i
+ def is_odd (i : Int) : Result Bool := mut_rec_body 1 i
+ ```
+ -/
+def mkDeclareFixDefs (mutRecBody : Expr) (paramInOutTys : Array TypeInfo) (preDefs : Array PreDefinition) :
TermElabM (Array Name) := do
let grSize := preDefs.size
let defs ← preDefs.mapIdxM fun idx preDef => do
lambdaTelescope preDef.value fun xs _ => do
- -- Retrieve the input type
- let in_ty := (inOutTys.get! idx.val).fst
+ -- Retrieve the parameters info
+ let type_info := paramInOutTys.get! idx.val
-- Create the index
let idx ← mkFinVal grSize idx.val
- -- Group the inputs into a dependent tuple
- let input ← mkSigmasVal in_ty xs.toList
+ -- Group the inputs into two tuples
+ let (params_args, input_args) := xs.toList.splitAt type_info.num_params
+ let params ← mkSigmasVal type_info.params_ty params_args
+ let input ← mkProdsVal input_args
-- Apply the fixed point
- let fixedBody ← mkAppM ``FixI.fix #[mutRecBody, idx, input]
+ let fixedBody ← mkAppM ``FixII.fix #[mutRecBody, idx, params, input]
let fixedBody ← mkLambdaFVars xs fixedBody
-- Create the declaration
let name := preDef.declName
@@ -790,7 +1154,8 @@ def mkDeclareFixDefs (mutRecBody : Expr) (inOutTys : Array (Expr × Expr)) (preD
pure defs
-- Prove the equations that we will use as unfolding theorems
-partial def proveUnfoldingThms (isValidThm : Expr) (inOutTys : Array (Expr × Expr))
+partial def proveUnfoldingThms (isValidThm : Expr)
+ (paramInOutTys : Array TypeInfo)
(preDefs : Array PreDefinition) (decls : Array Name) : MetaM Unit := do
let grSize := preDefs.size
let proveIdx (i : Nat) : MetaM Unit := do
@@ -810,14 +1175,18 @@ partial def proveUnfoldingThms (isValidThm : Expr) (inOutTys : Array (Expr × Ex
trace[Diverge.def.unfold] "proveUnfoldingThms: thm statement: {thmTy}"
-- The proof
-- Use the fixed-point equation
- let proof ← mkAppM ``FixI.is_valid_fix_fixed_eq #[isValidThm]
+ let proof ← mkAppM ``FixII.is_valid_fix_fixed_eq #[isValidThm]
-- Add the index
let idx ← mkFinVal grSize i
let proof ← mkAppM ``congr_fun #[proof, idx]
- -- Add the input argument
- let arg ← mkSigmasVal (inOutTys.get! i).fst xs.toList
- let proof ← mkAppM ``congr_fun #[proof, arg]
- -- Abstract the arguments away
+ -- Add the input arguments
+ let type_info := paramInOutTys.get! i
+ let (params, args) := xs.toList.splitAt type_info.num_params
+ let params ← mkSigmasVal type_info.params_ty params
+ let args ← mkProdsVal args
+ let proof ← mkAppM ``congr_fun #[proof, params]
+ let proof ← mkAppM ``congr_fun #[proof, args]
+ -- Abstract all the arguments away
let proof ← mkLambdaFVars xs proof
trace[Diverge.def.unfold] "proveUnfoldingThms: proof: {proof}:\n{← inferType proof}"
-- Declare the theorem
@@ -845,7 +1214,9 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
let msg := toMessageData <| preDefs.map fun pd => (pd.declName, pd.levelParams, pd.type, pd.value)
trace[Diverge.def] ("divRecursion: defs:\n" ++ msg)
- -- TODO: what is this?
+ -- Apply all the "attribute" functions (for instance, the function which
+ -- registers the theorem in the simp database if there is the `simp` attribute,
+ -- etc.)
for preDef in preDefs do
applyAttributesOf #[preDef] AttributeApplicationTime.afterCompilation
@@ -859,40 +1230,53 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
let grLvlParams := def0.levelParams
trace[Diverge.def] "def0 universe levels: {def0.levelParams}"
- -- We first compute the list of pairs: (input type × output type)
- let inOutTys : Array (Expr × Expr) ←
- preDefs.mapM (fun preDef => do
- withRef preDef.ref do -- is the withRef useful?
- -- Check the universe parameters - TODO: I'm not sure what the best thing
- -- to do is. In practice, all the type parameters should be in Type 0, so
- -- we shouldn't have universe issues.
- if preDef.levelParams ≠ grLvlParams then
- throwError "Non-uniform polymorphism in the universes"
- forallTelescope preDef.type (fun in_tys out_ty => do
- let in_ty ← liftM (mkSigmasType in_tys.toList)
- -- Retrieve the type in the "Result"
- let out_ty ← getResultTy out_ty
- let out_ty ← liftM (mkSigmasMatch in_tys.toList out_ty)
- pure (in_ty, out_ty)
- )
- )
- trace[Diverge.def] "inOutTys: {inOutTys}"
- -- Turn the list of input/output type pairs into an expresion
- let inOutTysExpr ← inOutTys.mapM (λ (x, y) => mkInOutTy x y)
- let inOutTysExpr ← mkListLit (← inferType (inOutTysExpr.get! 0)) inOutTysExpr.toList
-
- -- From the list of pairs of input/output types, actually compute the
- -- type of the continuation `k`.
- -- We first introduce the index `i : Fin n` where `n` is the number of
- -- functions in the group.
+ /- We first compute the tuples: (type parameters × input type × output type)
+ - type parameters: this is a sigma type
+ - input type: λ params_type => product type
+ - output type: λ params_type => output type
+ For instance, on the function:
+ `list_nth (α : Type) (ls : List α) (i : Int) : Result α`:
+ we generate:
+ `(Type, λ α => List α × i, λ α => Result α)`
+ -/
+ let paramInOutTys : Array TypeInfo ←
+ preDefs.mapM (fun preDef => do
+ -- Check the universe parameters - TODO: I'm not sure what the best thing
+ -- to do is. In practice, all the type parameters should be in Type 0, so
+ -- we shouldn't have universe issues.
+ if preDef.levelParams ≠ grLvlParams then
+ throwError "Non-uniform polymorphism in the universes"
+ forallTelescope preDef.type (fun in_tys out_ty => do
+ let total_num_args := in_tys.size
+ let (params, in_tys) ← splitInputArgs in_tys out_ty
+ trace[Diverge.def] "Decomposed arguments: {preDef.declName}: {params}, {in_tys}, {out_ty}"
+ let num_params := params.size
+ let params_ty ← mkSigmasType params.data
+ let in_ty ← mkSigmasMatchOrUnit params.data (← mkProdsType in_tys.data)
+ -- Retrieve the type in the "Result"
+ let out_ty ← getResultTy out_ty
+ let out_ty ← mkSigmasMatchOrUnit params.data out_ty
+ trace[Diverge.def] "inOutTy: {preDef.declName}: {params_ty}, {in_tys}, {out_ty}"
+ pure ⟨ total_num_args, num_params, params_ty, in_ty, out_ty ⟩))
+ trace[Diverge.def] "paramInOutTys: {paramInOutTys}"
+ -- Turn the list of input types/input args/output type tuples into expressions
+ let paramInOutTysExpr ← liftM (paramInOutTys.mapM mkInOutTyFromTypeInfo)
+ let paramInOutTysExpr ← mkListLit (← inferType (paramInOutTysExpr.get! 0)) paramInOutTysExpr.toList
+ trace[Diverge.def] "paramInOutTys: {paramInOutTys}"
+
+ /- From the list of pairs of input/output types, actually compute the
+ type of the continuation `k`.
+ We first introduce the index `i : Fin n` where `n` is the number of
+ functions in the group.
+ -/
let i_var_ty := mkFin preDefs.size
withLocalDeclD (mkAnonymous "i" 0) i_var_ty fun i_var => do
- let in_out_ty ← mkAppM ``List.get #[inOutTysExpr, i_var]
- trace[Diverge.def] "in_out_ty := {in_out_ty} : {← inferType in_out_ty}"
- -- Add an auxiliary definition for `in_out_ty`
- let in_out_ty ← do
- let value ← mkLambdaFVars #[i_var] in_out_ty
- let name := grName.append "in_out_ty"
+ let param_in_out_ty ← mkAppM ``List.get #[paramInOutTysExpr, i_var]
+ trace[Diverge.def] "param_in_out_ty := {param_in_out_ty} : {← inferType param_in_out_ty}"
+ -- Add an auxiliary definition for `param_in_out_ty` (this is a potentially big term)
+ let param_in_out_ty ← do
+ let value ← mkLambdaFVars #[i_var] param_in_out_ty
+ let name := grName.append "param_in_out_ty"
let levelParams := grLvlParams
let decl := Declaration.defnDecl {
name := name
@@ -905,19 +1289,28 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
}
addDecl decl
-- Return the constant
- let in_out_ty := Lean.mkConst name (levelParams.map .param)
- mkAppM' in_out_ty #[i_var]
- trace[Diverge.def] "in_out_ty (after decl) := {in_out_ty} : {← inferType in_out_ty}"
- let in_ty ← mkAppM ``Sigma.fst #[in_out_ty]
+ let param_in_out_ty := Lean.mkConst name (levelParams.map .param)
+ mkAppM' param_in_out_ty #[i_var]
+ trace[Diverge.def] "param_in_out_ty (after decl) := {param_in_out_ty} : {← inferType param_in_out_ty}"
+ -- Decompose between: param_ty, in_ty, out_ty
+ let param_ty ← mkAppM ``Sigma.fst #[param_in_out_ty]
+ let in_out_ty ← mkAppM ``Sigma.snd #[param_in_out_ty]
+ let in_ty ← mkAppM ``Prod.fst #[in_out_ty]
+ let out_ty ← mkAppM ``Prod.snd #[in_out_ty]
+ trace[Diverge.def] "param_ty: {param_ty}"
+ trace[Diverge.def] "in_ty: {in_ty}"
+ trace[Diverge.def] "out_ty: {out_ty}"
+ withLocalDeclD (mkAnonymous "t" 1) param_ty fun param => do
+ let in_ty ← mkAppM' in_ty #[param]
+ let out_ty ← mkAppM' out_ty #[param]
trace[Diverge.def] "in_ty: {in_ty}"
- withLocalDeclD (mkAnonymous "x" 1) in_ty fun input => do
- let out_ty ← mkAppM' (← mkAppM ``Sigma.snd #[in_out_ty]) #[input]
trace[Diverge.def] "out_ty: {out_ty}"
-- Introduce the continuation `k`
- let in_ty ← mkLambdaFVars #[i_var] in_ty
- let out_ty ← mkLambdaFVars #[i_var, input] out_ty
- let kk_var_ty ← mkAppM ``FixI.kk_ty #[i_var_ty, in_ty, out_ty]
+ let param_ty ← mkLambdaFVars #[i_var] param_ty
+ let in_ty ← mkLambdaFVars #[i_var, param] in_ty
+ let out_ty ← mkLambdaFVars #[i_var, param] out_ty
+ let kk_var_ty ← mkAppM ``FixII.kk_ty #[i_var_ty, param_ty, in_ty, out_ty]
trace[Diverge.def] "kk_var_ty: {kk_var_ty}"
withLocalDeclD (mkAnonymous "kk" 2) kk_var_ty fun kk_var => do
trace[Diverge.def] "kk_var: {kk_var}"
@@ -925,29 +1318,30 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
-- Replace the recursive calls in all the function bodies by calls to the
-- continuation `k` and and generate for those bodies declarations
trace[Diverge.def] "# Generating the unary bodies"
- let bodies ← mkDeclareUnaryBodies grLvlParams kk_var inOutTys preDefs
+ let bodies ← mkDeclareUnaryBodies grLvlParams kk_var paramInOutTys preDefs
trace[Diverge.def] "Unary bodies (after decl): {bodies}"
+
-- Generate the mutually recursive body
trace[Diverge.def] "# Generating the mut rec body"
- let (bodyFuns, mutRecBody) ← mkDeclareMutRecBody grName grLvlParams kk_var i_var in_ty out_ty inOutTys.toList bodies
+ let (bodyFuns, mutRecBody) ← mkDeclareMutRecBody grName grLvlParams kk_var i_var param_ty in_ty out_ty paramInOutTys bodies
trace[Diverge.def] "mut rec body (after decl): {mutRecBody}"
-- Prove that the mut rec body satisfies the validity criteria required by
-- our fixed-point
- let k_var_ty ← mkAppM ``FixI.k_ty #[i_var_ty, in_ty, out_ty]
+ let k_var_ty ← mkAppM ``FixII.k_ty #[i_var_ty, param_ty, in_ty, out_ty]
withLocalDeclD (mkAnonymous "k" 3) k_var_ty fun k_var => do
trace[Diverge.def] "# Proving that the mut rec body is valid"
- let isValidThm ← proveMutRecIsValid grName grLvlParams inOutTysExpr bodyFuns mutRecBody k_var preDefs bodies
+ let isValidThm ← proveMutRecIsValid grName grLvlParams paramInOutTysExpr bodyFuns mutRecBody k_var preDefs bodies
-- Generate the final definitions
trace[Diverge.def] "# Generating the final definitions"
- let decls ← mkDeclareFixDefs mutRecBody inOutTys preDefs
+ let decls ← mkDeclareFixDefs mutRecBody paramInOutTys preDefs
-- Prove the unfolding theorems
trace[Diverge.def] "# Proving the unfolding theorems"
- proveUnfoldingThms isValidThm inOutTys preDefs decls
+ proveUnfoldingThms isValidThm paramInOutTys preDefs decls
- -- Generating code -- TODO
+ -- Generating code
addAndCompilePartialRec preDefs
-- The following function is copy&pasted from Lean.Elab.PreDefinition.Main
@@ -1068,15 +1462,23 @@ elab_rules : command
Command.elabCommand <| ← `(namespace $(mkIdentFrom id ns) $cmd end $(mkIdentFrom id ns))
namespace Tests
+
/- Some examples of partial functions -/
- divergent def list_nth {a: Type} (ls : List a) (i : Int) : Result a :=
+ --set_option trace.Diverge.def true
+ --set_option trace.Diverge.def.genBody true
+ --set_option trace.Diverge.def.valid true
+ --set_option trace.Diverge.def.genBody.visit true
+
+ divergent def list_nth {a: Type u} (ls : List a) (i : Int) : Result a :=
match ls with
| [] => .fail .panic
| x :: ls =>
if i = 0 then return x
else return (← list_nth ls (i - 1))
+ --set_option trace.Diverge false
+
#check list_nth.unfold
example {a: Type} (ls : List a) :
@@ -1087,17 +1489,33 @@ namespace Tests
. intro i hpos h; simp at h; linarith
. rename_i hd tl ih
intro i hpos h
- -- We can directly use `rw [list_nth]`!
+ -- We can directly use `rw [list_nth]`
rw [list_nth]; simp
split <;> try simp [*]
. tauto
- . -- TODO: we shouldn't have to do that
+ . -- We don't have to do this if we use scalar_tac
have hneq : 0 < i := by cases i <;> rename_i a _ <;> simp_all; cases a <;> simp_all
simp at h
have ⟨ x, ih ⟩ := ih (i - 1) (by linarith) (by linarith)
simp [ih]
tauto
+ -- Return a continuation
+ divergent def list_nth_with_back {a: Type} (ls : List a) (i : Int) :
+ Result (a × (a → Result (List a))) :=
+ match ls with
+ | [] => .fail .panic
+ | x :: ls =>
+ if i = 0 then return (x, (λ ret => return (ret :: ls)))
+ else do
+ let (x, back) ← list_nth_with_back ls (i - 1)
+ return (x,
+ (λ ret => do
+ let ls ← back ret
+ return (x :: ls)))
+
+ #check list_nth_with_back.unfold
+
mutual
divergent def is_even (i : Int) : Result Bool :=
if i = 0 then return true else return (← is_odd (i - 1))
@@ -1121,7 +1539,6 @@ namespace Tests
#check bar.unfold
-- Testing dependent branching and let-bindings
- -- TODO: why the linter warning?
divergent def isNonZero (i : Int) : Result Bool :=
if _h:i = 0 then return false
else
@@ -1157,6 +1574,82 @@ namespace Tests
#check test1.unfold
+ /- Tests with higher-order functions -/
+ section HigherOrder
+ open Ex8
+
+ inductive Tree (a : Type u) :=
+ | leaf (x : a)
+ | node (tl : List (Tree a))
+
+ divergent def id {a : Type u} (t : Tree a) : Result (Tree a) :=
+ match t with
+ | .leaf x => .ret (.leaf x)
+ | .node tl =>
+ do
+ let tl ← map id tl
+ .ret (.node tl)
+
+ #check id.unfold
+
+ divergent def id1 {a : Type u} (t : Tree a) : Result (Tree a) :=
+ match t with
+ | .leaf x => .ret (.leaf x)
+ | .node tl =>
+ do
+ let tl ← map (fun x => id1 x) tl
+ .ret (.node tl)
+
+ #check id1.unfold
+
+ divergent def id2 {a : Type u} (t : Tree a) : Result (Tree a) :=
+ match t with
+ | .leaf x => .ret (.leaf x)
+ | .node tl =>
+ do
+ let tl ← map (fun x => do let _ ← id2 x; id2 x) tl
+ .ret (.node tl)
+
+ #check id2.unfold
+
+ divergent def incr (t : Tree Nat) : Result (Tree Nat) :=
+ match t with
+ | .leaf x => .ret (.leaf (x + 1))
+ | .node tl =>
+ do
+ let tl ← map incr tl
+ .ret (.node tl)
+
+ -- We handle this by inlining the let-binding
+ divergent def id3 (t : Tree Nat) : Result (Tree Nat) :=
+ match t with
+ | .leaf x => .ret (.leaf (x + 1))
+ | .node tl =>
+ do
+ let f := id3
+ let tl ← map f tl
+ .ret (.node tl)
+
+ #check id3.unfold
+
+ /-
+ -- This is not handled yet: we can only do it if we introduce "general"
+ -- relations for the input types and output types (result_rel should
+ -- be parameterized by something).
+ divergent def id4 (t : Tree Nat) : Result (Tree Nat) :=
+ match t with
+ | .leaf x => .ret (.leaf (x + 1))
+ | .node tl =>
+ do
+ let f ← .ret id4
+ let tl ← map f tl
+ .ret (.node tl)
+
+ #check id4.unfold
+ -/
+
+ end HigherOrder
+
end Tests
end Diverge
diff --git a/backends/lean/Base/Diverge/ElabBase.lean b/backends/lean/Base/Diverge/ElabBase.lean
index fedb1c74..0d33e9d2 100644
--- a/backends/lean/Base/Diverge/ElabBase.lean
+++ b/backends/lean/Base/Diverge/ElabBase.lean
@@ -1,15 +1,84 @@
import Lean
+import Base.Utils
+import Base.Primitives.Base
+import Base.Extensions
namespace Diverge
open Lean Elab Term Meta
+open Utils Extensions
-- We can't define and use trace classes in the same file
+initialize registerTraceClass `Diverge
initialize registerTraceClass `Diverge.elab
initialize registerTraceClass `Diverge.def
initialize registerTraceClass `Diverge.def.sigmas
+initialize registerTraceClass `Diverge.def.prods
initialize registerTraceClass `Diverge.def.genBody
+initialize registerTraceClass `Diverge.def.genBody.visit
initialize registerTraceClass `Diverge.def.valid
initialize registerTraceClass `Diverge.def.unfold
+-- For the attribute (for higher-order functions)
+initialize registerTraceClass `Diverge.attr
+
+-- Attribute
+
+-- divspec attribute
+structure DivSpecAttr where
+ attr : AttributeImpl
+ ext : DiscrTreeExtension Name true
+ deriving Inhabited
+
+/- The persistent map from expressions to divspec theorems. -/
+initialize divspecAttr : DivSpecAttr ← do
+ let ext ← mkDiscrTreeExtention `divspecMap true
+ let attrImpl : AttributeImpl := {
+ name := `divspec
+ descr := "Marks theorems to use with the `divergent` encoding"
+ add := fun thName stx attrKind => do
+ Attribute.Builtin.ensureNoArgs stx
+ -- TODO: use the attribute kind
+ unless attrKind == AttributeKind.global do
+ throwError "invalid attribute divspec, must be global"
+ -- Lookup the theorem
+ let env ← getEnv
+ let thDecl := env.constants.find! thName
+ let fKey : Array (DiscrTree.Key true) ← MetaM.run' (do
+ /- The theorem should have the shape:
+ `∀ ..., is_valid_p k (λ k => ...)`
+
+ Dive into the ∀:
+ -/
+ let (_, _, fExpr) ← forallMetaTelescope thDecl.type.consumeMData
+ /- Dive into the argument of `is_valid_p`: -/
+ fExpr.consumeMData.withApp fun _ args => do
+ if args.size ≠ 7 then throwError "Invalid number of arguments to is_valid_p"
+ let fExpr := args.get! 6
+ /- Dive into the lambda: -/
+ let (_, _, fExpr) ← lambdaMetaTelescope fExpr.consumeMData
+ trace[Diverge] "Registering divspec theorem for {fExpr}"
+ -- Convert the function expression to a discrimination tree key
+ DiscrTree.mkPath fExpr)
+ let env := ext.addEntry env (fKey, thName)
+ setEnv env
+ trace[Diverge] "Saved the environment"
+ pure ()
+ }
+ registerBuiltinAttribute attrImpl
+ pure { attr := attrImpl, ext := ext }
+
+def DivSpecAttr.find? (s : DivSpecAttr) (e : Expr) : MetaM (Array Name) := do
+ (s.ext.getState (← getEnv)).getMatch e
+
+def DivSpecAttr.getState (s : DivSpecAttr) : MetaM (DiscrTree Name true) := do
+ pure (s.ext.getState (← getEnv))
+
+def showStoredDivSpec : MetaM Unit := do
+ let st ← divspecAttr.getState
+ -- TODO: how can we iterate over (at least) the values stored in the tree?
+ --let s := st.toList.foldl (fun s (f, th) => f!"{s}\n{f} → {th}") f!""
+ let s := f!"{st}"
+ IO.println s
+
end Diverge
diff --git a/backends/lean/Base/Extensions.lean b/backends/lean/Base/Extensions.lean
new file mode 100644
index 00000000..b34f41dc
--- /dev/null
+++ b/backends/lean/Base/Extensions.lean
@@ -0,0 +1,47 @@
+import Lean
+import Std.Lean.HashSet
+import Base.Utils
+import Base.Primitives.Base
+
+import Lean.Meta.DiscrTree
+import Lean.Meta.Tactic.Simp
+
+/-! Various state extensions used in the library -/
+namespace Extensions
+
+open Lean Elab Term Meta
+open Utils
+
+-- This is not used anymore but we keep it here.
+-- TODO: the original function doesn't define correctly the `addImportedFn`. Do a PR?
+def mkMapDeclarationExtension [Inhabited α] (name : Name := by exact decl_name%) :
+ IO (MapDeclarationExtension α) :=
+ registerSimplePersistentEnvExtension {
+ name := name,
+ addImportedFn := fun a => a.foldl (fun s a => a.foldl (fun s (k, v) => s.insert k v) s) RBMap.empty,
+ addEntryFn := fun s n => s.insert n.1 n.2 ,
+ toArrayFn := fun es => es.toArray.qsort (fun a b => Name.quickLt a.1 b.1)
+ }
+
+/- Discrimination trees map expressions to values. When storing an expression
+ in a discrimination tree, the expression is first converted to an array
+ of `DiscrTree.Key`, which are the keys actually used by the discrimination
+ trees. The conversion operation is monadic, however, and extensions require
+ all the operations to be pure. For this reason, in the state extension, we
+ store the keys from *after* the transformation (i.e., the `DiscrTreeKey`
+ below). The transformation itself can be done elsewhere.
+ -/
+abbrev DiscrTreeKey (simpleReduce : Bool) := Array (DiscrTree.Key simpleReduce)
+
+abbrev DiscrTreeExtension (α : Type) (simpleReduce : Bool) :=
+ SimplePersistentEnvExtension (DiscrTreeKey simpleReduce × α) (DiscrTree α simpleReduce)
+
+def mkDiscrTreeExtention [Inhabited α] [BEq α] (name : Name := by exact decl_name%) (simpleReduce : Bool) :
+ IO (DiscrTreeExtension α simpleReduce) :=
+ registerSimplePersistentEnvExtension {
+ name := name,
+ addImportedFn := fun a => a.foldl (fun s a => a.foldl (fun s (k, v) => s.insertCore k v) s) DiscrTree.empty,
+ addEntryFn := fun s n => s.insertCore n.1 n.2 ,
+ }
+
+end Extensions
diff --git a/backends/lean/Base/Primitives/Scalar.lean b/backends/lean/Base/Primitives/Scalar.lean
index f74fecd4..db522df2 100644
--- a/backends/lean/Base/Primitives/Scalar.lean
+++ b/backends/lean/Base/Primitives/Scalar.lean
@@ -528,7 +528,7 @@ instance {ty} : HAnd (Scalar ty) (Scalar ty) (Scalar ty) where
hAnd x y := Scalar.and x y
-- Generic theorem - shouldn't be used much
-@[cpspec]
+@[pspec]
theorem Scalar.add_spec {ty} {x y : Scalar ty}
(hmin : Scalar.min ty ≤ x.val + y.val)
(hmax : x.val + y.val ≤ Scalar.max ty) :
@@ -550,62 +550,62 @@ theorem Scalar.add_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty}
apply add_spec <;> assumption
/- Fine-grained theorems -/
-@[cepspec] theorem Usize.add_spec {x y : Usize} (hmax : x.val + y.val ≤ Usize.max) :
+@[pspec] theorem Usize.add_spec {x y : Usize} (hmax : x.val + y.val ≤ Usize.max) :
∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
-@[cepspec] theorem U8.add_spec {x y : U8} (hmax : x.val + y.val ≤ U8.max) :
+@[pspec] theorem U8.add_spec {x y : U8} (hmax : x.val + y.val ≤ U8.max) :
∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
-@[cepspec] theorem U16.add_spec {x y : U16} (hmax : x.val + y.val ≤ U16.max) :
+@[pspec] theorem U16.add_spec {x y : U16} (hmax : x.val + y.val ≤ U16.max) :
∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
-@[cepspec] theorem U32.add_spec {x y : U32} (hmax : x.val + y.val ≤ U32.max) :
+@[pspec] theorem U32.add_spec {x y : U32} (hmax : x.val + y.val ≤ U32.max) :
∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
-@[cepspec] theorem U64.add_spec {x y : U64} (hmax : x.val + y.val ≤ U64.max) :
+@[pspec] theorem U64.add_spec {x y : U64} (hmax : x.val + y.val ≤ U64.max) :
∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
-@[cepspec] theorem U128.add_spec {x y : U128} (hmax : x.val + y.val ≤ U128.max) :
+@[pspec] theorem U128.add_spec {x y : U128} (hmax : x.val + y.val ≤ U128.max) :
∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
-@[cepspec] theorem Isize.add_spec {x y : Isize}
+@[pspec] theorem Isize.add_spec {x y : Isize}
(hmin : Isize.min ≤ x.val + y.val) (hmax : x.val + y.val ≤ Isize.max) :
∃ z, x + y = ret z ∧ z.val = x.val + y.val :=
Scalar.add_spec hmin hmax
-@[cepspec] theorem I8.add_spec {x y : I8}
+@[pspec] theorem I8.add_spec {x y : I8}
(hmin : I8.min ≤ x.val + y.val) (hmax : x.val + y.val ≤ I8.max) :
∃ z, x + y = ret z ∧ z.val = x.val + y.val :=
Scalar.add_spec hmin hmax
-@[cepspec] theorem I16.add_spec {x y : I16}
+@[pspec] theorem I16.add_spec {x y : I16}
(hmin : I16.min ≤ x.val + y.val) (hmax : x.val + y.val ≤ I16.max) :
∃ z, x + y = ret z ∧ z.val = x.val + y.val :=
Scalar.add_spec hmin hmax
-@[cepspec] theorem I32.add_spec {x y : I32}
+@[pspec] theorem I32.add_spec {x y : I32}
(hmin : I32.min ≤ x.val + y.val) (hmax : x.val + y.val ≤ I32.max) :
∃ z, x + y = ret z ∧ z.val = x.val + y.val :=
Scalar.add_spec hmin hmax
-@[cepspec] theorem I64.add_spec {x y : I64}
+@[pspec] theorem I64.add_spec {x y : I64}
(hmin : I64.min ≤ x.val + y.val) (hmax : x.val + y.val ≤ I64.max) :
∃ z, x + y = ret z ∧ z.val = x.val + y.val :=
Scalar.add_spec hmin hmax
-@[cepspec] theorem I128.add_spec {x y : I128}
+@[pspec] theorem I128.add_spec {x y : I128}
(hmin : I128.min ≤ x.val + y.val) (hmax : x.val + y.val ≤ I128.max) :
∃ z, x + y = ret z ∧ z.val = x.val + y.val :=
Scalar.add_spec hmin hmax
-- Generic theorem - shouldn't be used much
-@[cpspec]
+@[pspec]
theorem Scalar.sub_spec {ty} {x y : Scalar ty}
(hmin : Scalar.min ty ≤ x.val - y.val)
(hmax : x.val - y.val ≤ Scalar.max ty) :
@@ -629,56 +629,56 @@ theorem Scalar.sub_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty}
apply sub_spec <;> assumption
/- Fine-grained theorems -/
-@[cepspec] theorem Usize.sub_spec {x y : Usize} (hmin : Usize.min ≤ x.val - y.val) :
+@[pspec] theorem Usize.sub_spec {x y : Usize} (hmin : Usize.min ≤ x.val - y.val) :
∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
-@[cepspec] theorem U8.sub_spec {x y : U8} (hmin : U8.min ≤ x.val - y.val) :
+@[pspec] theorem U8.sub_spec {x y : U8} (hmin : U8.min ≤ x.val - y.val) :
∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
-@[cepspec] theorem U16.sub_spec {x y : U16} (hmin : U16.min ≤ x.val - y.val) :
+@[pspec] theorem U16.sub_spec {x y : U16} (hmin : U16.min ≤ x.val - y.val) :
∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
-@[cepspec] theorem U32.sub_spec {x y : U32} (hmin : U32.min ≤ x.val - y.val) :
+@[pspec] theorem U32.sub_spec {x y : U32} (hmin : U32.min ≤ x.val - y.val) :
∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
-@[cepspec] theorem U64.sub_spec {x y : U64} (hmin : U64.min ≤ x.val - y.val) :
+@[pspec] theorem U64.sub_spec {x y : U64} (hmin : U64.min ≤ x.val - y.val) :
∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
-@[cepspec] theorem U128.sub_spec {x y : U128} (hmin : U128.min ≤ x.val - y.val) :
+@[pspec] theorem U128.sub_spec {x y : U128} (hmin : U128.min ≤ x.val - y.val) :
∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
-@[cepspec] theorem Isize.sub_spec {x y : Isize} (hmin : Isize.min ≤ x.val - y.val)
+@[pspec] theorem Isize.sub_spec {x y : Isize} (hmin : Isize.min ≤ x.val - y.val)
(hmax : x.val - y.val ≤ Isize.max) :
∃ z, x - y = ret z ∧ z.val = x.val - y.val :=
Scalar.sub_spec hmin hmax
-@[cepspec] theorem I8.sub_spec {x y : I8} (hmin : I8.min ≤ x.val - y.val)
+@[pspec] theorem I8.sub_spec {x y : I8} (hmin : I8.min ≤ x.val - y.val)
(hmax : x.val - y.val ≤ I8.max) :
∃ z, x - y = ret z ∧ z.val = x.val - y.val :=
Scalar.sub_spec hmin hmax
-@[cepspec] theorem I16.sub_spec {x y : I16} (hmin : I16.min ≤ x.val - y.val)
+@[pspec] theorem I16.sub_spec {x y : I16} (hmin : I16.min ≤ x.val - y.val)
(hmax : x.val - y.val ≤ I16.max) :
∃ z, x - y = ret z ∧ z.val = x.val - y.val :=
Scalar.sub_spec hmin hmax
-@[cepspec] theorem I32.sub_spec {x y : I32} (hmin : I32.min ≤ x.val - y.val)
+@[pspec] theorem I32.sub_spec {x y : I32} (hmin : I32.min ≤ x.val - y.val)
(hmax : x.val - y.val ≤ I32.max) :
∃ z, x - y = ret z ∧ z.val = x.val - y.val :=
Scalar.sub_spec hmin hmax
-@[cepspec] theorem I64.sub_spec {x y : I64} (hmin : I64.min ≤ x.val - y.val)
+@[pspec] theorem I64.sub_spec {x y : I64} (hmin : I64.min ≤ x.val - y.val)
(hmax : x.val - y.val ≤ I64.max) :
∃ z, x - y = ret z ∧ z.val = x.val - y.val :=
Scalar.sub_spec hmin hmax
-@[cepspec] theorem I128.sub_spec {x y : I128} (hmin : I128.min ≤ x.val - y.val)
+@[pspec] theorem I128.sub_spec {x y : I128} (hmin : I128.min ≤ x.val - y.val)
(hmax : x.val - y.val ≤ I128.max) :
∃ z, x - y = ret z ∧ z.val = x.val - y.val :=
Scalar.sub_spec hmin hmax
@@ -705,62 +705,62 @@ theorem Scalar.mul_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty}
apply mul_spec <;> assumption
/- Fine-grained theorems -/
-@[cepspec] theorem Usize.mul_spec {x y : Usize} (hmax : x.val * y.val ≤ Usize.max) :
+@[pspec] theorem Usize.mul_spec {x y : Usize} (hmax : x.val * y.val ≤ Usize.max) :
∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
-@[cepspec] theorem U8.mul_spec {x y : U8} (hmax : x.val * y.val ≤ U8.max) :
+@[pspec] theorem U8.mul_spec {x y : U8} (hmax : x.val * y.val ≤ U8.max) :
∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
-@[cepspec] theorem U16.mul_spec {x y : U16} (hmax : x.val * y.val ≤ U16.max) :
+@[pspec] theorem U16.mul_spec {x y : U16} (hmax : x.val * y.val ≤ U16.max) :
∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
-@[cepspec] theorem U32.mul_spec {x y : U32} (hmax : x.val * y.val ≤ U32.max) :
+@[pspec] theorem U32.mul_spec {x y : U32} (hmax : x.val * y.val ≤ U32.max) :
∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
-@[cepspec] theorem U64.mul_spec {x y : U64} (hmax : x.val * y.val ≤ U64.max) :
+@[pspec] theorem U64.mul_spec {x y : U64} (hmax : x.val * y.val ≤ U64.max) :
∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
-@[cepspec] theorem U128.mul_spec {x y : U128} (hmax : x.val * y.val ≤ U128.max) :
+@[pspec] theorem U128.mul_spec {x y : U128} (hmax : x.val * y.val ≤ U128.max) :
∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
-@[cepspec] theorem Isize.mul_spec {x y : Isize} (hmin : Isize.min ≤ x.val * y.val)
+@[pspec] theorem Isize.mul_spec {x y : Isize} (hmin : Isize.min ≤ x.val * y.val)
(hmax : x.val * y.val ≤ Isize.max) :
∃ z, x * y = ret z ∧ z.val = x.val * y.val :=
Scalar.mul_spec hmin hmax
-@[cepspec] theorem I8.mul_spec {x y : I8} (hmin : I8.min ≤ x.val * y.val)
+@[pspec] theorem I8.mul_spec {x y : I8} (hmin : I8.min ≤ x.val * y.val)
(hmax : x.val * y.val ≤ I8.max) :
∃ z, x * y = ret z ∧ z.val = x.val * y.val :=
Scalar.mul_spec hmin hmax
-@[cepspec] theorem I16.mul_spec {x y : I16} (hmin : I16.min ≤ x.val * y.val)
+@[pspec] theorem I16.mul_spec {x y : I16} (hmin : I16.min ≤ x.val * y.val)
(hmax : x.val * y.val ≤ I16.max) :
∃ z, x * y = ret z ∧ z.val = x.val * y.val :=
Scalar.mul_spec hmin hmax
-@[cepspec] theorem I32.mul_spec {x y : I32} (hmin : I32.min ≤ x.val * y.val)
+@[pspec] theorem I32.mul_spec {x y : I32} (hmin : I32.min ≤ x.val * y.val)
(hmax : x.val * y.val ≤ I32.max) :
∃ z, x * y = ret z ∧ z.val = x.val * y.val :=
Scalar.mul_spec hmin hmax
-@[cepspec] theorem I64.mul_spec {x y : I64} (hmin : I64.min ≤ x.val * y.val)
+@[pspec] theorem I64.mul_spec {x y : I64} (hmin : I64.min ≤ x.val * y.val)
(hmax : x.val * y.val ≤ I64.max) :
∃ z, x * y = ret z ∧ z.val = x.val * y.val :=
Scalar.mul_spec hmin hmax
-@[cepspec] theorem I128.mul_spec {x y : I128} (hmin : I128.min ≤ x.val * y.val)
+@[pspec] theorem I128.mul_spec {x y : I128} (hmin : I128.min ≤ x.val * y.val)
(hmax : x.val * y.val ≤ I128.max) :
∃ z, x * y = ret z ∧ z.val = x.val * y.val :=
Scalar.mul_spec hmin hmax
-- Generic theorem - shouldn't be used much
-@[cpspec]
+@[pspec]
theorem Scalar.div_spec {ty} {x y : Scalar ty}
(hnz : y.val ≠ 0)
(hmin : Scalar.min ty ≤ scalar_div x.val y.val)
@@ -788,66 +788,66 @@ theorem Scalar.div_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : S
apply hs
/- Fine-grained theorems -/
-@[cepspec] theorem Usize.div_spec (x : Usize) {y : Usize} (hnz : y.val ≠ 0) :
+@[pspec] theorem Usize.div_spec (x : Usize) {y : Usize} (hnz : y.val ≠ 0) :
∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
apply Scalar.div_unsigned_spec <;> simp [*]
-@[cepspec] theorem U8.div_spec (x : U8) {y : U8} (hnz : y.val ≠ 0) :
+@[pspec] theorem U8.div_spec (x : U8) {y : U8} (hnz : y.val ≠ 0) :
∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
-@[cepspec] theorem U16.div_spec (x : U16) {y : U16} (hnz : y.val ≠ 0) :
+@[pspec] theorem U16.div_spec (x : U16) {y : U16} (hnz : y.val ≠ 0) :
∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
-@[cepspec] theorem U32.div_spec (x : U32) {y : U32} (hnz : y.val ≠ 0) :
+@[pspec] theorem U32.div_spec (x : U32) {y : U32} (hnz : y.val ≠ 0) :
∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
-@[cepspec] theorem U64.div_spec (x : U64) {y : U64} (hnz : y.val ≠ 0) :
+@[pspec] theorem U64.div_spec (x : U64) {y : U64} (hnz : y.val ≠ 0) :
∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
-@[cepspec] theorem U128.div_spec (x : U128) {y : U128} (hnz : y.val ≠ 0) :
+@[pspec] theorem U128.div_spec (x : U128) {y : U128} (hnz : y.val ≠ 0) :
∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
-@[cepspec] theorem Isize.div_spec (x : Isize) {y : Isize}
+@[pspec] theorem Isize.div_spec (x : Isize) {y : Isize}
(hnz : y.val ≠ 0)
(hmin : Isize.min ≤ scalar_div x.val y.val)
(hmax : scalar_div x.val y.val ≤ Isize.max):
∃ z, x / y = ret z ∧ z.val = scalar_div x.val y.val :=
Scalar.div_spec hnz hmin hmax
-@[cepspec] theorem I8.div_spec (x : I8) {y : I8}
+@[pspec] theorem I8.div_spec (x : I8) {y : I8}
(hnz : y.val ≠ 0)
(hmin : I8.min ≤ scalar_div x.val y.val)
(hmax : scalar_div x.val y.val ≤ I8.max):
∃ z, x / y = ret z ∧ z.val = scalar_div x.val y.val :=
Scalar.div_spec hnz hmin hmax
-@[cepspec] theorem I16.div_spec (x : I16) {y : I16}
+@[pspec] theorem I16.div_spec (x : I16) {y : I16}
(hnz : y.val ≠ 0)
(hmin : I16.min ≤ scalar_div x.val y.val)
(hmax : scalar_div x.val y.val ≤ I16.max):
∃ z, x / y = ret z ∧ z.val = scalar_div x.val y.val :=
Scalar.div_spec hnz hmin hmax
-@[cepspec] theorem I32.div_spec (x : I32) {y : I32}
+@[pspec] theorem I32.div_spec (x : I32) {y : I32}
(hnz : y.val ≠ 0)
(hmin : I32.min ≤ scalar_div x.val y.val)
(hmax : scalar_div x.val y.val ≤ I32.max):
∃ z, x / y = ret z ∧ z.val = scalar_div x.val y.val :=
Scalar.div_spec hnz hmin hmax
-@[cepspec] theorem I64.div_spec (x : I64) {y : I64}
+@[pspec] theorem I64.div_spec (x : I64) {y : I64}
(hnz : y.val ≠ 0)
(hmin : I64.min ≤ scalar_div x.val y.val)
(hmax : scalar_div x.val y.val ≤ I64.max):
∃ z, x / y = ret z ∧ z.val = scalar_div x.val y.val :=
Scalar.div_spec hnz hmin hmax
-@[cepspec] theorem I128.div_spec (x : I128) {y : I128}
+@[pspec] theorem I128.div_spec (x : I128) {y : I128}
(hnz : y.val ≠ 0)
(hmin : I128.min ≤ scalar_div x.val y.val)
(hmax : scalar_div x.val y.val ≤ I128.max):
@@ -855,7 +855,7 @@ theorem Scalar.div_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : S
Scalar.div_spec hnz hmin hmax
-- Generic theorem - shouldn't be used much
-@[cpspec]
+@[pspec]
theorem Scalar.rem_spec {ty} {x y : Scalar ty}
(hnz : y.val ≠ 0)
(hmin : Scalar.min ty ≤ scalar_rem x.val y.val)
@@ -883,59 +883,59 @@ theorem Scalar.rem_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : S
simp [*] at hs
simp [*]
-@[cepspec] theorem Usize.rem_spec (x : Usize) {y : Usize} (hnz : y.val ≠ 0) :
+@[pspec] theorem Usize.rem_spec (x : Usize) {y : Usize} (hnz : y.val ≠ 0) :
∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
apply Scalar.rem_unsigned_spec <;> simp [*]
-@[cepspec] theorem U8.rem_spec (x : U8) {y : U8} (hnz : y.val ≠ 0) :
+@[pspec] theorem U8.rem_spec (x : U8) {y : U8} (hnz : y.val ≠ 0) :
∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
-@[cepspec] theorem U16.rem_spec (x : U16) {y : U16} (hnz : y.val ≠ 0) :
+@[pspec] theorem U16.rem_spec (x : U16) {y : U16} (hnz : y.val ≠ 0) :
∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
-@[cepspec] theorem U32.rem_spec (x : U32) {y : U32} (hnz : y.val ≠ 0) :
+@[pspec] theorem U32.rem_spec (x : U32) {y : U32} (hnz : y.val ≠ 0) :
∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
-@[cepspec] theorem U64.rem_spec (x : U64) {y : U64} (hnz : y.val ≠ 0) :
+@[pspec] theorem U64.rem_spec (x : U64) {y : U64} (hnz : y.val ≠ 0) :
∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
-@[cepspec] theorem U128.rem_spec (x : U128) {y : U128} (hnz : y.val ≠ 0) :
+@[pspec] theorem U128.rem_spec (x : U128) {y : U128} (hnz : y.val ≠ 0) :
∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
-@[cepspec] theorem I8.rem_spec (x : I8) {y : I8}
+@[pspec] theorem I8.rem_spec (x : I8) {y : I8}
(hnz : y.val ≠ 0)
(hmin : I8.min ≤ scalar_rem x.val y.val)
(hmax : scalar_rem x.val y.val ≤ I8.max):
∃ z, x % y = ret z ∧ z.val = scalar_rem x.val y.val :=
Scalar.rem_spec hnz hmin hmax
-@[cepspec] theorem I16.rem_spec (x : I16) {y : I16}
+@[pspec] theorem I16.rem_spec (x : I16) {y : I16}
(hnz : y.val ≠ 0)
(hmin : I16.min ≤ scalar_rem x.val y.val)
(hmax : scalar_rem x.val y.val ≤ I16.max):
∃ z, x % y = ret z ∧ z.val = scalar_rem x.val y.val :=
Scalar.rem_spec hnz hmin hmax
-@[cepspec] theorem I32.rem_spec (x : I32) {y : I32}
+@[pspec] theorem I32.rem_spec (x : I32) {y : I32}
(hnz : y.val ≠ 0)
(hmin : I32.min ≤ scalar_rem x.val y.val)
(hmax : scalar_rem x.val y.val ≤ I32.max):
∃ z, x % y = ret z ∧ z.val = scalar_rem x.val y.val :=
Scalar.rem_spec hnz hmin hmax
-@[cepspec] theorem I64.rem_spec (x : I64) {y : I64}
+@[pspec] theorem I64.rem_spec (x : I64) {y : I64}
(hnz : y.val ≠ 0)
(hmin : I64.min ≤ scalar_rem x.val y.val)
(hmax : scalar_rem x.val y.val ≤ I64.max):
∃ z, x % y = ret z ∧ z.val = scalar_rem x.val y.val :=
Scalar.rem_spec hnz hmin hmax
-@[cepspec] theorem I128.rem_spec (x : I128) {y : I128}
+@[pspec] theorem I128.rem_spec (x : I128) {y : I128}
(hnz : y.val ≠ 0)
(hmin : I128.min ≤ scalar_rem x.val y.val)
(hmax : scalar_rem x.val y.val ≤ I128.max):
diff --git a/backends/lean/Base/Progress/Base.lean b/backends/lean/Base/Progress/Base.lean
index 76a92795..0ad16ab6 100644
--- a/backends/lean/Base/Progress/Base.lean
+++ b/backends/lean/Base/Progress/Base.lean
@@ -2,11 +2,12 @@ import Lean
import Std.Lean.HashSet
import Base.Utils
import Base.Primitives.Base
+import Base.Extensions
namespace Progress
open Lean Elab Term Meta
-open Utils
+open Utils Extensions
-- We can't define and use trace classes in the same file
initialize registerTraceClass `Progress
@@ -15,17 +16,17 @@ initialize registerTraceClass `Progress
structure PSpecDesc where
-- The universally quantified variables
+ -- Can be fvars or mvars
fvars : Array Expr
-- The existentially quantified variables
evars : Array Expr
+ -- The function applied to its arguments
+ fArgsExpr : Expr
-- The function
- fExpr : Expr
fName : Name
-- The function arguments
fLevels : List Level
args : Array Expr
- -- The universally quantified variables which appear in the function arguments
- argsFVars : Array FVarId
-- The returned value
ret : Expr
-- The postcondition (if there is)
@@ -37,7 +38,7 @@ section Methods
variable [MonadError m]
variable {a : Type}
- /- Analyze a pspec theorem to decompose its arguments.
+ /- Analyze a goal or a pspec theorem to decompose its arguments.
PSpec theorems should be of the following shape:
```
@@ -56,12 +57,20 @@ section Methods
TODO: generalize for when we do inductive proofs
-/
partial
- def withPSpec [Inhabited (m a)] [Nonempty (m a)] (th : Expr) (k : PSpecDesc → m a)
- (sanityChecks : Bool := false) :
+ def withPSpec [Inhabited (m a)] [Nonempty (m a)]
+ (isGoal : Bool) (th : Expr) (k : PSpecDesc → m a) :
m a := do
trace[Progress] "Proposition: {th}"
-- Dive into the quantified variables and the assumptions
- forallTelescope th.consumeMData fun fvars th => do
+ -- Note that if we analyze a pspec theorem to register it in a database (i.e.
+ -- a discrimination tree), we need to introduce *meta-variables* for the
+ -- quantified variables.
+ let telescope (k : Array Expr → Expr → m a) : m a :=
+ if isGoal then forallTelescope th.consumeMData (fun fvars th => k fvars th)
+ else do
+ let (fvars, _, th) ← forallMetaTelescope th.consumeMData
+ k fvars th
+ telescope fun fvars th => do
trace[Progress] "Universally quantified arguments and assumptions: {fvars}"
-- Dive into the existentials
existsTelescope th.consumeMData fun evars th => do
@@ -78,7 +87,7 @@ section Methods
-- destruct the application to get the function name
mExpr.consumeMData.withApp fun mf margs => do
trace[Progress] "After stripping the arguments of the monad expression:\n- mf: {mf}\n- margs: {margs}"
- let (fExpr, f, args) ← do
+ let (fArgsExpr, f, args) ← do
if mf.isConst ∧ mf.constName = ``Bind.bind then do
-- Dive into the bind
let fExpr := (margs.get! 4).consumeMData
@@ -86,29 +95,27 @@ section Methods
else pure (mExpr, mf, margs)
trace[Progress] "After stripping the arguments of the function call:\n- f: {f}\n- args: {args}"
if ¬ f.isConst then throwError "Not a constant: {f}"
- -- Compute the set of universally quantified variables which appear in the function arguments
- let allArgsFVars ← args.foldlM (fun hs arg => getFVarIds arg hs) HashSet.empty
- -- Sanity check
- if sanityChecks then
- -- All the variables which appear in the inputs given to the function are
- -- universally quantified (in particular, they are not *existentially* quantified)
- let fvarsSet : HashSet FVarId := HashSet.ofArray (fvars.map (fun x => x.fvarId!))
- let filtArgsFVars := allArgsFVars.toArray.filter (fun fvar => ¬ fvarsSet.contains fvar)
- if ¬ filtArgsFVars.isEmpty then
+ -- *Sanity check* (activated if we are analyzing a theorem to register it in a DB)
+ -- Check if some existentially quantified variables
+ let _ := do
+ -- Collect all the free variables in the arguments
+ let allArgsFVars ← args.foldlM (fun hs arg => getFVarIds arg hs) HashSet.empty
+ -- Check if they intersect the fvars we introduced for the existentially quantified variables
+ let evarsSet : HashSet FVarId := HashSet.ofArray (evars.map (fun (x : Expr) => x.fvarId!))
+ let filtArgsFVars := allArgsFVars.toArray.filter (fun var => evarsSet.contains var)
+ if filtArgsFVars.isEmpty then pure ()
+ else
let filtArgsFVars := filtArgsFVars.map (fun fvarId => Expr.fvar fvarId)
throwError "Some of the function inputs are not universally quantified: {filtArgsFVars}"
- let argsFVars := fvars.map (fun x => x.fvarId!)
- let argsFVars := argsFVars.filter (fun fvar => allArgsFVars.contains fvar)
-- Return
- trace[Progress] "Function: {f.constName!}";
+ trace[Progress] "Function with arguments: {fArgsExpr}";
let thDesc := {
fvars := fvars
evars := evars
- fExpr
+ fArgsExpr
fName := f.constName!
fLevels := f.constLevels!
args := args
- argsFVars
ret := ret
post := post
}
@@ -116,117 +123,18 @@ section Methods
end Methods
-def getPSpecFunName (th : Expr) : MetaM Name :=
- withPSpec th (fun d => do pure d.fName) true
+def getPSpecFunArgsExpr (isGoal : Bool) (th : Expr) : MetaM Expr :=
+ withPSpec isGoal th (fun d => do pure d.fArgsExpr)
-def getPSpecClassFunNames (th : Expr) : MetaM (Name × Name) :=
- withPSpec th (fun d => do
- let arg0 := d.args.get! 0
- arg0.withApp fun f _ => do
- if ¬ f.isConst then throwError "Not a constant: {f}"
- pure (d.fName, f.constName)
- ) true
-
-def getPSpecClassFunNameArg (th : Expr) : MetaM (Name × Expr) :=
- withPSpec th (fun d => do
- let arg0 := d.args.get! 0
- pure (d.fName, arg0)
- ) true
-
--- "Regular" pspec attribute
+-- pspec attribute
structure PSpecAttr where
attr : AttributeImpl
- ext : MapDeclarationExtension Name
- deriving Inhabited
-
-/- pspec attribute for type classes: we use the name of the type class to
- lookup another map. We use the *first* argument of the type class to lookup
- into this second map.
-
- Example:
- ========
- We use type classes for addition. For instance, the addition between two
- U32 is written (without syntactic sugar) as `HAdd.add (Scalar ty) x y`. As a consequence,
- we store the theorem through the bindings: HAdd.add → Scalar → ...
-
- SH: TODO: this (and `PSpecClassExprAttr`) is a bit ad-hoc. For now it works for the
- specs of the scalar operations, which is what I really need, but I'm not sure it
- applies well to other situations. A better way would probably to use type classes, but
- I couldn't get them to work on those cases. It is worth retrying.
--/
-structure PSpecClassAttr where
- attr : AttributeImpl
- ext : MapDeclarationExtension (NameMap Name)
- deriving Inhabited
-
-/- Same as `PSpecClassAttr` but we use the full first argument (it works when it
- is a constant). -/
-structure PSpecClassExprAttr where
- attr : AttributeImpl
- ext : MapDeclarationExtension (HashMap Expr Name)
+ ext : DiscrTreeExtension Name true
deriving Inhabited
--- TODO: the original function doesn't define correctly the `addImportedFn`. Do a PR?
-def mkMapDeclarationExtension [Inhabited α] (name : Name := by exact decl_name%) :
- IO (MapDeclarationExtension α) :=
- registerSimplePersistentEnvExtension {
- name := name,
- addImportedFn := fun a => a.foldl (fun s a => a.foldl (fun s (k, v) => s.insert k v) s) RBMap.empty,
- addEntryFn := fun s n => s.insert n.1 n.2 ,
- toArrayFn := fun es => es.toArray.qsort (fun a b => Name.quickLt a.1 b.1)
- }
-
--- Declare an extension of maps of maps (using [RBMap]).
--- The important point is that we need to merge the bound values (which are maps).
-def mkMapMapDeclarationExtension [Inhabited β] (ord : α → α → Ordering)
- (name : Name := by exact decl_name%) :
- IO (MapDeclarationExtension (RBMap α β ord)) :=
- registerSimplePersistentEnvExtension {
- name := name,
- addImportedFn := fun a =>
- a.foldl (fun s a => a.foldl (
- -- We need to merge the maps
- fun s (k0, k1_to_v) =>
- match s.find? k0 with
- | none =>
- -- No binding: insert one
- s.insert k0 k1_to_v
- | some m =>
- -- There is already a binding: merge
- let m := RBMap.fold (fun m k v => m.insert k v) m k1_to_v
- s.insert k0 m)
- s) RBMap.empty,
- addEntryFn := fun s n => s.insert n.1 n.2 ,
- toArrayFn := fun es => es.toArray.qsort (fun a b => Name.quickLt a.1 b.1)
- }
-
--- Declare an extension of maps of maps (using [HashMap]).
--- The important point is that we need to merge the bound values (which are maps).
-def mkMapHashMapDeclarationExtension [BEq α] [Hashable α] [Inhabited β]
- (name : Name := by exact decl_name%) :
- IO (MapDeclarationExtension (HashMap α β)) :=
- registerSimplePersistentEnvExtension {
- name := name,
- addImportedFn := fun a =>
- a.foldl (fun s a => a.foldl (
- -- We need to merge the maps
- fun s (k0, k1_to_v) =>
- match s.find? k0 with
- | none =>
- -- No binding: insert one
- s.insert k0 k1_to_v
- | some m =>
- -- There is already a binding: merge
- let m := HashMap.fold (fun m k v => m.insert k v) m k1_to_v
- s.insert k0 m)
- s) RBMap.empty,
- addEntryFn := fun s n => s.insert n.1 n.2 ,
- toArrayFn := fun es => es.toArray.qsort (fun a b => Name.quickLt a.1 b.1)
- }
-
-/- The persistent map from function to pspec theorems. -/
+/- The persistent map from expressions to pspec theorems. -/
initialize pspecAttr : PSpecAttr ← do
- let ext ← mkMapDeclarationExtension `pspecMap
+ let ext ← mkDiscrTreeExtention `pspecMap true
let attrImpl : AttributeImpl := {
name := `pspec
descr := "Marks theorems to use with the `progress` tactic"
@@ -238,130 +146,30 @@ initialize pspecAttr : PSpecAttr ← do
-- Lookup the theorem
let env ← getEnv
let thDecl := env.constants.find! thName
- let fName ← MetaM.run' (getPSpecFunName thDecl.type)
- trace[Progress] "Registering spec theorem for {fName}"
- let env := ext.addEntry env (fName, thName)
- setEnv env
- pure ()
- }
- registerBuiltinAttribute attrImpl
- pure { attr := attrImpl, ext := ext }
-
-/- The persistent map from type classes to pspec theorems -/
-initialize pspecClassAttr : PSpecClassAttr ← do
- let ext : MapDeclarationExtension (NameMap Name) ←
- mkMapMapDeclarationExtension Name.quickCmp `pspecClassMap
- let attrImpl : AttributeImpl := {
- name := `cpspec
- descr := "Marks theorems to use for type classes with the `progress` tactic"
- add := fun thName stx attrKind => do
- Attribute.Builtin.ensureNoArgs stx
- -- TODO: use the attribute kind
- unless attrKind == AttributeKind.global do
- throwError "invalid attribute 'cpspec', must be global"
- -- Lookup the theorem
- let env ← getEnv
- let thDecl := env.constants.find! thName
- let (fName, argName) ← MetaM.run' (getPSpecClassFunNames thDecl.type)
- trace[Progress] "Registering class spec theorem for ({fName}, {argName})"
- -- Update the entry if there is one, add an entry if there is none
- let env :=
- match (ext.getState (← getEnv)).find? fName with
- | none =>
- let m := RBMap.ofList [(argName, thName)]
- ext.addEntry env (fName, m)
- | some m =>
- let m := m.insert argName thName
- ext.addEntry env (fName, m)
+ let fKey ← MetaM.run' (do
+ let fExpr ← getPSpecFunArgsExpr false thDecl.type
+ trace[Progress] "Registering spec theorem for {fExpr}"
+ -- Convert the function expression to a discrimination tree key
+ DiscrTree.mkPath fExpr)
+ let env := ext.addEntry env (fKey, thName)
setEnv env
+ trace[Progress] "Saved the environment"
pure ()
}
registerBuiltinAttribute attrImpl
pure { attr := attrImpl, ext := ext }
-/- The 2nd persistent map from type classes to pspec theorems -/
-initialize pspecClassExprAttr : PSpecClassExprAttr ← do
- let ext : MapDeclarationExtension (HashMap Expr Name) ←
- mkMapHashMapDeclarationExtension `pspecClassExprMap
- let attrImpl : AttributeImpl := {
- name := `cepspec
- descr := "Marks theorems to use for type classes with the `progress` tactic"
- add := fun thName stx attrKind => do
- Attribute.Builtin.ensureNoArgs stx
- -- TODO: use the attribute kind
- unless attrKind == AttributeKind.global do
- throwError "invalid attribute 'cpspec', must be global"
- -- Lookup the theorem
- let env ← getEnv
- let thDecl := env.constants.find! thName
- let (fName, arg) ← MetaM.run' (getPSpecClassFunNameArg thDecl.type)
- -- Sanity check: no variables appear in the argument
- MetaM.run' do
- let fvars ← getFVarIds arg
- if ¬ fvars.isEmpty then throwError "The first argument ({arg}) contains variables"
- -- We store two bindings:
- -- - arg to theorem name
- -- - reduced arg to theorem name
- let rarg ← MetaM.run' (reduceAll arg)
- trace[Progress] "Registering class spec theorem for ({fName}, {arg}) and ({fName}, {rarg})"
- -- Update the entry if there is one, add an entry if there is none
- let env :=
- match (ext.getState (← getEnv)).find? fName with
- | none =>
- let m := HashMap.ofList [(arg, thName), (rarg, thName)]
- ext.addEntry env (fName, m)
- | some m =>
- let m := m.insert arg thName
- let m := m.insert rarg thName
- ext.addEntry env (fName, m)
- setEnv env
- pure ()
- }
- registerBuiltinAttribute attrImpl
- pure { attr := attrImpl, ext := ext }
-
-
-def PSpecAttr.find? (s : PSpecAttr) (name : Name) : MetaM (Option Name) := do
- return (s.ext.getState (← getEnv)).find? name
-
-def PSpecClassAttr.find? (s : PSpecClassAttr) (className argName : Name) : MetaM (Option Name) := do
- match (s.ext.getState (← getEnv)).find? className with
- | none => return none
- | some map => return map.find? argName
-
-def PSpecClassExprAttr.find? (s : PSpecClassExprAttr) (className : Name) (arg : Expr) : MetaM (Option Name) := do
- match (s.ext.getState (← getEnv)).find? className with
- | none => return none
- | some map => return map.find? arg
-
-def PSpecAttr.getState (s : PSpecAttr) : MetaM (NameMap Name) := do
- pure (s.ext.getState (← getEnv))
+def PSpecAttr.find? (s : PSpecAttr) (e : Expr) : MetaM (Array Name) := do
+ (s.ext.getState (← getEnv)).getMatch e
-def PSpecClassAttr.getState (s : PSpecClassAttr) : MetaM (NameMap (NameMap Name)) := do
- pure (s.ext.getState (← getEnv))
-
-def PSpecClassExprAttr.getState (s : PSpecClassExprAttr) : MetaM (NameMap (HashMap Expr Name)) := do
+def PSpecAttr.getState (s : PSpecAttr) : MetaM (DiscrTree Name true) := do
pure (s.ext.getState (← getEnv))
def showStoredPSpec : MetaM Unit := do
let st ← pspecAttr.getState
- let s := st.toList.foldl (fun s (f, th) => f!"{s}\n{f} → {th}") f!""
- IO.println s
-
-def showStoredPSpecClass : MetaM Unit := do
- let st ← pspecClassAttr.getState
- let s := st.toList.foldl (fun s (f, m) =>
- let ms := m.toList.foldl (fun s (f, th) =>
- f!"{s}\n {f} → {th}") f!""
- f!"{s}\n{f} → [{ms}]") f!""
- IO.println s
-
-def showStoredPSpecExprClass : MetaM Unit := do
- let st ← pspecClassExprAttr.getState
- let s := st.toList.foldl (fun s (f, m) =>
- let ms := m.toList.foldl (fun s (f, th) =>
- f!"{s}\n {f} → {th}") f!""
- f!"{s}\n{f} → [{ms}]") f!""
+ -- TODO: how can we iterate over (at least) the values stored in the tree?
+ --let s := st.toList.foldl (fun s (f, th) => f!"{s}\n{f} → {th}") f!""
+ let s := f!"{st}"
IO.println s
end Progress
diff --git a/backends/lean/Base/Progress/Progress.lean b/backends/lean/Base/Progress/Progress.lean
index ba63f09d..a6a4e82a 100644
--- a/backends/lean/Base/Progress/Progress.lean
+++ b/backends/lean/Base/Progress/Progress.lean
@@ -204,11 +204,11 @@ def getFirstArg (args : Array Expr) : Option Expr := do
if args.size = 0 then none
else some (args.get! 0)
-/- Helper: try to lookup a theorem and apply it, or continue with another tactic
- if it fails -/
+/- Helper: try to lookup a theorem and apply it.
+ Return true if it succeeded. -/
def tryLookupApply (keep : Option Name) (ids : Array (Option Name)) (splitPost : Bool)
(asmTac : TacticM Unit) (fExpr : Expr)
- (kind : String) (th : Option TheoremOrLocal) (x : TacticM Unit) : TacticM Unit := do
+ (kind : String) (th : Option TheoremOrLocal) : TacticM Bool := do
let res ← do
match th with
| none =>
@@ -223,9 +223,9 @@ def tryLookupApply (keep : Option Name) (ids : Array (Option Name)) (splitPost :
pure (some res)
catch _ => none
match res with
- | some .Ok => return ()
+ | some .Ok => return true
| some (.Error msg) => throwError msg
- | none => x
+ | none => return false
-- The array of ids are identifiers to use when introducing fresh variables
def progressAsmsOrLookupTheorem (keep : Option Name) (withTh : Option TheoremOrLocal)
@@ -236,11 +236,19 @@ def progressAsmsOrLookupTheorem (keep : Option Name) (withTh : Option TheoremOrL
let goalTy ← mgoal.getType
trace[Progress] "goal: {goalTy}"
-- Dive into the goal to lookup the theorem
- let (fExpr, fName, args) ← do
- withPSpec goalTy fun desc =>
- -- TODO: check that no quantified variables in the arguments
- pure (desc.fExpr, desc.fName, desc.args)
- trace[Progress] "Function: {fName}"
+ -- Remark: if we don't isolate the call to `withPSpec` to immediately "close"
+ -- the terms immediately, we may end up with the error:
+ -- "(kernel) declaration has free variables"
+ -- I'm not sure I understand why.
+ -- TODO: we should also check that no quantified variable appears in fExpr.
+ -- If such variables appear, we should just fail because the goal doesn't
+ -- have the proper shape.
+ let fExpr ← do
+ let isGoal := true
+ withPSpec isGoal goalTy fun desc => do
+ let fExpr := desc.fArgsExpr
+ trace[Progress] "Expression to match: {fExpr}"
+ pure fExpr
-- If the user provided a theorem/assumption: use it.
-- Otherwise, lookup one.
match withTh with
@@ -258,36 +266,24 @@ def progressAsmsOrLookupTheorem (keep : Option Name) (withTh : Option TheoremOrL
match res with
| .Ok => return ()
| .Error msg => throwError msg
- -- It failed: try to lookup a theorem
- -- TODO: use a list of theorems, and try them one by one?
- trace[Progress] "No assumption succeeded: trying to lookup a theorem"
- let pspec ← do
- let thName ← pspecAttr.find? fName
- pure (thName.map fun th => .Theorem th)
- tryLookupApply keep ids splitPost asmTac fExpr "pspec theorem" pspec do
- -- It failed: try to lookup a *class* expr spec theorem (those are more
- -- specific than class spec theorems)
- trace[Progress] "Failed using a pspec theorem: trying to lookup a pspec class expr theorem"
- let pspecClassExpr ← do
- match getFirstArg args with
- | none => pure none
- | some arg => do
- trace[Progress] "Using: f:{fName}, arg: {arg}"
- let thName ← pspecClassExprAttr.find? fName arg
- pure (thName.map fun th => .Theorem th)
- tryLookupApply keep ids splitPost asmTac fExpr "pspec class expr theorem" pspecClassExpr do
- -- It failed: try to lookup a *class* spec theorem
- trace[Progress] "Failed using a pspec class expr theorem: trying to lookup a pspec class theorem"
- let pspecClass ← do
- match ← getFirstArgAppName args with
- | none => pure none
- | some argName => do
- trace[Progress] "Using: f: {fName}, arg: {argName}"
- let thName ← pspecClassAttr.find? fName argName
- pure (thName.map fun th => .Theorem th)
- tryLookupApply keep ids splitPost asmTac fExpr "pspec class theorem" pspecClass do
- trace[Progress] "Failed using a pspec class theorem: trying to use a recursive assumption"
- -- Try a recursive call - we try the assumptions of kind "auxDecl"
+ -- It failed: lookup the pspec theorems which match the expression
+ trace[Progress] "No assumption succeeded: trying to lookup a pspec theorem"
+ let pspecs : Array TheoremOrLocal ← do
+ let thNames ← pspecAttr.find? fExpr
+ -- TODO: because of reduction, there may be several valid theorems (for
+ -- instance for the scalars). We need to sort them from most specific to
+ -- least specific. For now, we assume the most specific theorems are at
+ -- the end.
+ let thNames := thNames.reverse
+ trace[Progress] "Looked up pspec theorems: {thNames}"
+ pure (thNames.map fun th => TheoremOrLocal.Theorem th)
+ -- Try the theorems one by one
+ for pspec in pspecs do
+ if ← tryLookupApply keep ids splitPost asmTac fExpr "pspec theorem" pspec then return ()
+ else pure ()
+ -- It failed: try to use the recursive assumptions
+ trace[Progress] "Failed using a pspec theorem: trying to use a recursive assumption"
+ -- We try to apply the assumptions of kind "auxDecl"
let ctx ← Lean.MonadLCtx.getLCtx
let decls ← ctx.getAllDecls
let decls := decls.filter (λ decl => match decl.kind with
@@ -381,8 +377,6 @@ namespace Test
-- The following commands display the databases of theorems
-- #eval showStoredPSpec
- -- #eval showStoredPSpecClass
- -- #eval showStoredPSpecExprClass
open alloc.vec
example {ty} {x y : Scalar ty}
@@ -402,6 +396,19 @@ namespace Test
example {x y : U32}
(hmax : x.val + y.val ≤ U32.max) :
∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+ -- This spec theorem is suboptimal, but it is good to check that it works
+ progress with Scalar.add_spec as ⟨ z, h1 .. ⟩
+ simp [*, h1]
+
+ example {x y : U32}
+ (hmax : x.val + y.val ≤ U32.max) :
+ ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+ progress with U32.add_spec as ⟨ z, h1 .. ⟩
+ simp [*, h1]
+
+ example {x y : U32}
+ (hmax : x.val + y.val ≤ U32.max) :
+ ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
progress keep _ as ⟨ z, h1 .. ⟩
simp [*, h1]
diff --git a/backends/lean/Base/Utils.lean b/backends/lean/Base/Utils.lean
index b917a789..b0032281 100644
--- a/backends/lean/Base/Utils.lean
+++ b/backends/lean/Base/Utils.lean
@@ -159,47 +159,96 @@ elab "print_ctx_decls" : tactic => do
let decls ← ctx.getDecls
printDecls decls
--- A map visitor function for expressions (adapted from `AbstractNestedProofs.visit`)
+-- A map-reduce visitor function for expressions (adapted from `AbstractNestedProofs.visit`)
-- The continuation takes as parameters:
-- - the current depth of the expression (useful for printing/debugging)
-- - the expression to explore
-partial def mapVisit (k : Nat → Expr → MetaM Expr) (e : Expr) : MetaM Expr := do
- let mapVisitBinders (xs : Array Expr) (k2 : MetaM Expr) : MetaM Expr := do
+partial def mapreduceVisit {a : Type} (k : Nat → a → Expr → MetaM (a × Expr))
+ (state : a) (e : Expr) : MetaM (a × Expr) := do
+ let mapreduceVisitBinders (state : a) (xs : Array Expr) (k2 : a → MetaM (a × Expr)) :
+ MetaM (a × Expr) := do
let localInstances ← getLocalInstances
- let mut lctx ← getLCtx
- for x in xs do
- let xFVarId := x.fvarId!
- let localDecl ← xFVarId.getDecl
- let type ← mapVisit k localDecl.type
- let localDecl := localDecl.setType type
- let localDecl ← match localDecl.value? with
- | some value => let value ← mapVisit k value; pure <| localDecl.setValue value
- | none => pure localDecl
- lctx :=lctx.modifyLocalDecl xFVarId fun _ => localDecl
- withLCtx lctx localInstances k2
+ -- Update the local declarations for the bindings in context `lctx`
+ let rec visit_xs (lctx : LocalContext) (state : a) (xs : List Expr) : MetaM (LocalContext × a) := do
+ match xs with
+ | [] => pure (lctx, state)
+ | x :: xs => do
+ let xFVarId := x.fvarId!
+ let localDecl ← xFVarId.getDecl
+ let (state, type) ← mapreduceVisit k state localDecl.type
+ let localDecl := localDecl.setType type
+ let (state, localDecl) ← match localDecl.value? with
+ | some value =>
+ let (state, value) ← mapreduceVisit k state value
+ pure (state, localDecl.setValue value)
+ | none => pure (state, localDecl)
+ let lctx := lctx.modifyLocalDecl xFVarId fun _ => localDecl
+ -- Recursive call
+ visit_xs lctx state xs
+ let (lctx, state) ← visit_xs (← getLCtx) state xs.toList
+ -- Call the continuation with the updated context
+ withLCtx lctx localInstances (k2 state)
-- TODO: use a cache? (Lean.checkCache)
- let rec visit (i : Nat) (e : Expr) : MetaM Expr := do
+ let rec visit (i : Nat) (state : a) (e : Expr) : MetaM (a × Expr) := do
-- Explore
- let e ← k i e
+ let (state, e) ← k i state e
match e with
| .bvar _
| .fvar _
| .mvar _
| .sort _
| .lit _
- | .const _ _ => pure e
- | .app .. => do e.withApp fun f args => return mkAppN f (← args.mapM (visit (i + 1)))
+ | .const _ _ => pure (state, e)
+ | .app .. => do e.withApp fun f args => do
+ let (state, args) ← args.foldlM (fun (state, args) arg => do let (state, arg) ← visit (i + 1) state arg; pure (state, arg :: args)) (state, [])
+ let args := args.reverse
+ let (state, f) ← visit (i + 1) state f
+ let e' := mkAppN f (Array.mk args)
+ return (state, e')
| .lam .. =>
lambdaLetTelescope e fun xs b =>
- mapVisitBinders xs do mkLambdaFVars xs (← visit (i + 1) b) (usedLetOnly := false)
+ mapreduceVisitBinders state xs fun state => do
+ let (state, b) ← visit (i + 1) state b
+ let e' ← mkLambdaFVars xs b (usedLetOnly := false)
+ return (state, e')
| .forallE .. => do
- forallTelescope e fun xs b => mapVisitBinders xs do mkForallFVars xs (← visit (i + 1) b)
+ forallTelescope e fun xs b =>
+ mapreduceVisitBinders state xs fun state => do
+ let (state, b) ← visit (i + 1) state b
+ let e' ← mkForallFVars xs b
+ return (state, e')
| .letE .. => do
- lambdaLetTelescope e fun xs b => mapVisitBinders xs do
- mkLambdaFVars xs (← visit (i + 1) b) (usedLetOnly := false)
- | .mdata _ b => return e.updateMData! (← visit (i + 1) b)
- | .proj _ _ b => return e.updateProj! (← visit (i + 1) b)
- visit 0 e
+ lambdaLetTelescope e fun xs b =>
+ mapreduceVisitBinders state xs fun state => do
+ let (state, b) ← visit (i + 1) state b
+ let e' ← mkLambdaFVars xs b (usedLetOnly := false)
+ return (state, e')
+ | .mdata _ b => do
+ let (state, b) ← visit (i + 1) state b
+ return (state, e.updateMData! b)
+ | .proj _ _ b => do
+ let (state, b) ← visit (i + 1) state b
+ return (state, e.updateProj! b)
+ visit 0 state e
+
+-- A map visitor function for expressions (adapted from `AbstractNestedProofs.visit`)
+-- The continuation takes as parameters:
+-- - the current depth of the expression (useful for printing/debugging)
+-- - the expression to explore
+partial def mapVisit (k : Nat → Expr → MetaM Expr) (e : Expr) : MetaM Expr := do
+ let k' i (_ : Unit) e := do
+ let e ← k i e
+ pure ((), e)
+ let (_, e) ← mapreduceVisit k' () e
+ pure e
+
+-- A reduce visitor
+partial def reduceVisit {a : Type} (k : Nat → a → Expr → MetaM a) (s : a) (e : Expr) : MetaM a := do
+ let k' i (s : a) e := do
+ let s ← k i s e
+ pure (s, e)
+ let (s, _) ← mapreduceVisit k' s e
+ pure s
-- Generate a fresh user name for an anonymous proposition to introduce in the
-- assumptions
@@ -371,15 +420,22 @@ def splitConjTarget : TacticM Unit := do
-- Destruct an equaliy and return the two sides
def destEq (e : Expr) : MetaM (Expr × Expr) := do
- e.withApp fun f args =>
+ e.consumeMData.withApp fun f args =>
if f.isConstOf ``Eq ∧ args.size = 3 then pure (args.get! 1, args.get! 2)
else throwError "Not an equality: {e}"
-- Return the set of FVarIds in the expression
+-- TODO: this collects fvars introduced in the inner bindings
partial def getFVarIds (e : Expr) (hs : HashSet FVarId := HashSet.empty) : MetaM (HashSet FVarId) := do
- e.withApp fun body args => do
- let hs := if body.isFVar then hs.insert body.fvarId! else hs
- args.foldlM (fun hs arg => getFVarIds arg hs) hs
+ reduceVisit (fun _ (hs : HashSet FVarId) e =>
+ if e.isFVar then pure (hs.insert e.fvarId!) else pure hs)
+ hs e
+
+-- Return the set of MVarIds in the expression
+partial def getMVarIds (e : Expr) (hs : HashSet MVarId := HashSet.empty) : MetaM (HashSet MVarId) := do
+ reduceVisit (fun _ (hs : HashSet MVarId) e =>
+ if e.isMVar then pure (hs.insert e.mvarId!) else pure hs)
+ hs e
-- Tactic to split on a disjunction.
-- The expression `h` should be an fvar.