Merge pull request #31 from AeneasVerif/son_lean_backend

Improve the Lean backend

3 files changed, 907 insertions, 0 deletions

diff --git a/tests/lean/Hashmap/Funs.lean b/tests/lean/Hashmap/Funs.lean
new file mode 100644
index 00000000..870693b5
--- /dev/null
+++ b/tests/lean/Hashmap/Funs.lean
@@ -0,0 +1,449 @@
+-- THIS FILE WAS AUTOMATICALLY GENERATED BY AENEAS
+-- [hashmap]: function definitions
+import Base
+import Hashmap.Types
+open Primitives
+namespace hashmap
+
+/- [hashmap::hash_key]: forward function -/
+def hash_key (k : Usize) : Result Usize :=
+  Result.ret k
+
+/- [hashmap::HashMap::{0}::allocate_slots]: loop 0: forward function -/
+divergent def HashMap.allocate_slots_loop
+  (T : Type) (slots : Vec (List T)) (n : Usize) : Result (Vec (List T)) :=
+  if n > (Usize.ofInt 0)
+  then
+    do
+      let slots0 ← Vec.push (List T) slots List.Nil
+      let n0 ← n - (Usize.ofInt 1)
+      HashMap.allocate_slots_loop T slots0 n0
+  else Result.ret slots
+
+/- [hashmap::HashMap::{0}::allocate_slots]: forward function -/
+def HashMap.allocate_slots
+  (T : Type) (slots : Vec (List T)) (n : Usize) : Result (Vec (List T)) :=
+  HashMap.allocate_slots_loop T slots n
+
+/- [hashmap::HashMap::{0}::new_with_capacity]: forward function -/
+def HashMap.new_with_capacity
+  (T : Type) (capacity : Usize) (max_load_dividend : Usize)
+  (max_load_divisor : Usize) :
+  Result (HashMap T)
+  :=
+  do
+    let v := Vec.new (List T)
+    let slots ← HashMap.allocate_slots T v capacity
+    let i ← capacity * max_load_dividend
+    let i0 ← i / max_load_divisor
+    Result.ret
+      {
+        num_entries := (Usize.ofInt 0),
+        max_load_factor := (max_load_dividend, max_load_divisor),
+        max_load := i0,
+        slots := slots
+      }
+
+/- [hashmap::HashMap::{0}::new]: forward function -/
+def HashMap.new (T : Type) : Result (HashMap T) :=
+  HashMap.new_with_capacity T (Usize.ofInt 32) (Usize.ofInt 4) (Usize.ofInt 5)
+
+/- [hashmap::HashMap::{0}::clear]: loop 0: merged forward/backward function
+   (there is a single backward function, and the forward function returns ()) -/
+divergent def HashMap.clear_loop
+  (T : Type) (slots : Vec (List T)) (i : Usize) : Result (Vec (List T)) :=
+  let i0 := Vec.len (List T) slots
+  if i < i0
+  then
+    do
+      let i1 ← i + (Usize.ofInt 1)
+      let slots0 ← Vec.index_mut_back (List T) slots i List.Nil
+      HashMap.clear_loop T slots0 i1
+  else Result.ret slots
+
+/- [hashmap::HashMap::{0}::clear]: merged forward/backward function
+   (there is a single backward function, and the forward function returns ()) -/
+def HashMap.clear (T : Type) (self : HashMap T) : Result (HashMap T) :=
+  do
+    let v ← HashMap.clear_loop T self.slots (Usize.ofInt 0)
+    Result.ret { self with num_entries := (Usize.ofInt 0), slots := v }
+
+/- [hashmap::HashMap::{0}::len]: forward function -/
+def HashMap.len (T : Type) (self : HashMap T) : Result Usize :=
+  Result.ret self.num_entries
+
+/- [hashmap::HashMap::{0}::insert_in_list]: loop 0: forward function -/
+divergent def HashMap.insert_in_list_loop
+  (T : Type) (key : Usize) (value : T) (ls : List T) : Result Bool :=
+  match ls with
+  | List.Cons ckey cvalue tl =>
+    if ckey = key
+    then Result.ret false
+    else HashMap.insert_in_list_loop T key value tl
+  | List.Nil => Result.ret true
+
+/- [hashmap::HashMap::{0}::insert_in_list]: forward function -/
+def HashMap.insert_in_list
+  (T : Type) (key : Usize) (value : T) (ls : List T) : Result Bool :=
+  HashMap.insert_in_list_loop T key value ls
+
+/- [hashmap::HashMap::{0}::insert_in_list]: loop 0: backward function 0 -/
+divergent def HashMap.insert_in_list_loop_back
+  (T : Type) (key : Usize) (value : T) (ls : List T) : Result (List T) :=
+  match ls with
+  | List.Cons ckey cvalue tl =>
+    if ckey = key
+    then Result.ret (List.Cons ckey value tl)
+    else
+      do
+        let tl0 ← HashMap.insert_in_list_loop_back T key value tl
+        Result.ret (List.Cons ckey cvalue tl0)
+  | List.Nil => let l := List.Nil
+                Result.ret (List.Cons key value l)
+
+/- [hashmap::HashMap::{0}::insert_in_list]: backward function 0 -/
+def HashMap.insert_in_list_back
+  (T : Type) (key : Usize) (value : T) (ls : List T) : Result (List T) :=
+  HashMap.insert_in_list_loop_back T key value ls
+
+/- [hashmap::HashMap::{0}::insert_no_resize]: merged forward/backward function
+   (there is a single backward function, and the forward function returns ()) -/
+def HashMap.insert_no_resize
+  (T : Type) (self : HashMap T) (key : Usize) (value : T) :
+  Result (HashMap T)
+  :=
+  do
+    let hash ← hash_key key
+    let i := Vec.len (List T) self.slots
+    let hash_mod ← hash % i
+    let l ← Vec.index_mut (List T) self.slots hash_mod
+    let inserted ← HashMap.insert_in_list T key value l
+    if inserted
+    then
+      do
+        let i0 ← self.num_entries + (Usize.ofInt 1)
+        let l0 ← HashMap.insert_in_list_back T key value l
+        let v ← Vec.index_mut_back (List T) self.slots hash_mod l0
+        Result.ret { self with num_entries := i0, slots := v }
+    else
+      do
+        let l0 ← HashMap.insert_in_list_back T key value l
+        let v ← Vec.index_mut_back (List T) self.slots hash_mod l0
+        Result.ret { self with slots := v }
+
+/- [core::num::u32::{9}::MAX] -/
+def core_num_u32_max_body : Result U32 := Result.ret (U32.ofInt 4294967295)
+def core_num_u32_max_c : U32 := eval_global core_num_u32_max_body (by simp)
+
+/- [hashmap::HashMap::{0}::move_elements_from_list]: loop 0: merged forward/backward function
+   (there is a single backward function, and the forward function returns ()) -/
+divergent def HashMap.move_elements_from_list_loop
+  (T : Type) (ntable : HashMap T) (ls : List T) : Result (HashMap T) :=
+  match ls with
+  | List.Cons k v tl =>
+    do
+      let ntable0 ← HashMap.insert_no_resize T ntable k v
+      HashMap.move_elements_from_list_loop T ntable0 tl
+  | List.Nil => Result.ret ntable
+
+/- [hashmap::HashMap::{0}::move_elements_from_list]: merged forward/backward function
+   (there is a single backward function, and the forward function returns ()) -/
+def HashMap.move_elements_from_list
+  (T : Type) (ntable : HashMap T) (ls : List T) : Result (HashMap T) :=
+  HashMap.move_elements_from_list_loop T ntable ls
+
+/- [hashmap::HashMap::{0}::move_elements]: loop 0: merged forward/backward function
+   (there is a single backward function, and the forward function returns ()) -/
+divergent def HashMap.move_elements_loop
+  (T : Type) (ntable : HashMap T) (slots : Vec (List T)) (i : Usize) :
+  Result ((HashMap T) × (Vec (List T)))
+  :=
+  let i0 := Vec.len (List T) slots
+  if i < i0
+  then
+    do
+      let l ← Vec.index_mut (List T) slots i
+      let ls := mem.replace (List T) l List.Nil
+      let ntable0 ← HashMap.move_elements_from_list T ntable ls
+      let i1 ← i + (Usize.ofInt 1)
+      let l0 := mem.replace_back (List T) l List.Nil
+      let slots0 ← Vec.index_mut_back (List T) slots i l0
+      HashMap.move_elements_loop T ntable0 slots0 i1
+  else Result.ret (ntable, slots)
+
+/- [hashmap::HashMap::{0}::move_elements]: merged forward/backward function
+   (there is a single backward function, and the forward function returns ()) -/
+def HashMap.move_elements
+  (T : Type) (ntable : HashMap T) (slots : Vec (List T)) (i : Usize) :
+  Result ((HashMap T) × (Vec (List T)))
+  :=
+  HashMap.move_elements_loop T ntable slots i
+
+/- [hashmap::HashMap::{0}::try_resize]: merged forward/backward function
+   (there is a single backward function, and the forward function returns ()) -/
+def HashMap.try_resize (T : Type) (self : HashMap T) : Result (HashMap T) :=
+  do
+    let max_usize ← Scalar.cast .Usize core_num_u32_max_c
+    let capacity := Vec.len (List T) self.slots
+    let n1 ← max_usize / (Usize.ofInt 2)
+    let (i, i0) := self.max_load_factor
+    let i1 ← n1 / i
+    if capacity <= i1
+    then
+      do
+        let i2 ← capacity * (Usize.ofInt 2)
+        let ntable ← HashMap.new_with_capacity T i2 i i0
+        let (ntable0, _) ←
+          HashMap.move_elements T ntable self.slots (Usize.ofInt 0)
+        Result.ret
+          {
+            ntable0
+              with
+              num_entries := self.num_entries, max_load_factor := (i, i0)
+          }
+    else Result.ret { self with max_load_factor := (i, i0) }
+
+/- [hashmap::HashMap::{0}::insert]: merged forward/backward function
+   (there is a single backward function, and the forward function returns ()) -/
+def HashMap.insert
+  (T : Type) (self : HashMap T) (key : Usize) (value : T) :
+  Result (HashMap T)
+  :=
+  do
+    let self0 ← HashMap.insert_no_resize T self key value
+    let i ← HashMap.len T self0
+    if i > self0.max_load
+    then HashMap.try_resize T self0
+    else Result.ret self0
+
+/- [hashmap::HashMap::{0}::contains_key_in_list]: loop 0: forward function -/
+divergent def HashMap.contains_key_in_list_loop
+  (T : Type) (key : Usize) (ls : List T) : Result Bool :=
+  match ls with
+  | List.Cons ckey t tl =>
+    if ckey = key
+    then Result.ret true
+    else HashMap.contains_key_in_list_loop T key tl
+  | List.Nil => Result.ret false
+
+/- [hashmap::HashMap::{0}::contains_key_in_list]: forward function -/
+def HashMap.contains_key_in_list
+  (T : Type) (key : Usize) (ls : List T) : Result Bool :=
+  HashMap.contains_key_in_list_loop T key ls
+
+/- [hashmap::HashMap::{0}::contains_key]: forward function -/
+def HashMap.contains_key
+  (T : Type) (self : HashMap T) (key : Usize) : Result Bool :=
+  do
+    let hash ← hash_key key
+    let i := Vec.len (List T) self.slots
+    let hash_mod ← hash % i
+    let l ← Vec.index (List T) self.slots hash_mod
+    HashMap.contains_key_in_list T key l
+
+/- [hashmap::HashMap::{0}::get_in_list]: loop 0: forward function -/
+divergent def HashMap.get_in_list_loop
+  (T : Type) (key : Usize) (ls : List T) : Result T :=
+  match ls with
+  | List.Cons ckey cvalue tl =>
+    if ckey = key
+    then Result.ret cvalue
+    else HashMap.get_in_list_loop T key tl
+  | List.Nil => Result.fail Error.panic
+
+/- [hashmap::HashMap::{0}::get_in_list]: forward function -/
+def HashMap.get_in_list (T : Type) (key : Usize) (ls : List T) : Result T :=
+  HashMap.get_in_list_loop T key ls
+
+/- [hashmap::HashMap::{0}::get]: forward function -/
+def HashMap.get (T : Type) (self : HashMap T) (key : Usize) : Result T :=
+  do
+    let hash ← hash_key key
+    let i := Vec.len (List T) self.slots
+    let hash_mod ← hash % i
+    let l ← Vec.index (List T) self.slots hash_mod
+    HashMap.get_in_list T key l
+
+/- [hashmap::HashMap::{0}::get_mut_in_list]: loop 0: forward function -/
+divergent def HashMap.get_mut_in_list_loop
+  (T : Type) (ls : List T) (key : Usize) : Result T :=
+  match ls with
+  | List.Cons ckey cvalue tl =>
+    if ckey = key
+    then Result.ret cvalue
+    else HashMap.get_mut_in_list_loop T tl key
+  | List.Nil => Result.fail Error.panic
+
+/- [hashmap::HashMap::{0}::get_mut_in_list]: forward function -/
+def HashMap.get_mut_in_list
+  (T : Type) (ls : List T) (key : Usize) : Result T :=
+  HashMap.get_mut_in_list_loop T ls key
+
+/- [hashmap::HashMap::{0}::get_mut_in_list]: loop 0: backward function 0 -/
+divergent def HashMap.get_mut_in_list_loop_back
+  (T : Type) (ls : List T) (key : Usize) (ret0 : T) : Result (List T) :=
+  match ls with
+  | List.Cons ckey cvalue tl =>
+    if ckey = key
+    then Result.ret (List.Cons ckey ret0 tl)
+    else
+      do
+        let tl0 ← HashMap.get_mut_in_list_loop_back T tl key ret0
+        Result.ret (List.Cons ckey cvalue tl0)
+  | List.Nil => Result.fail Error.panic
+
+/- [hashmap::HashMap::{0}::get_mut_in_list]: backward function 0 -/
+def HashMap.get_mut_in_list_back
+  (T : Type) (ls : List T) (key : Usize) (ret0 : T) : Result (List T) :=
+  HashMap.get_mut_in_list_loop_back T ls key ret0
+
+/- [hashmap::HashMap::{0}::get_mut]: forward function -/
+def HashMap.get_mut (T : Type) (self : HashMap T) (key : Usize) : Result T :=
+  do
+    let hash ← hash_key key
+    let i := Vec.len (List T) self.slots
+    let hash_mod ← hash % i
+    let l ← Vec.index_mut (List T) self.slots hash_mod
+    HashMap.get_mut_in_list T l key
+
+/- [hashmap::HashMap::{0}::get_mut]: backward function 0 -/
+def HashMap.get_mut_back
+  (T : Type) (self : HashMap T) (key : Usize) (ret0 : T) :
+  Result (HashMap T)
+  :=
+  do
+    let hash ← hash_key key
+    let i := Vec.len (List T) self.slots
+    let hash_mod ← hash % i
+    let l ← Vec.index_mut (List T) self.slots hash_mod
+    let l0 ← HashMap.get_mut_in_list_back T l key ret0
+    let v ← Vec.index_mut_back (List T) self.slots hash_mod l0
+    Result.ret { self with slots := v }
+
+/- [hashmap::HashMap::{0}::remove_from_list]: loop 0: forward function -/
+divergent def HashMap.remove_from_list_loop
+  (T : Type) (key : Usize) (ls : List T) : Result (Option T) :=
+  match ls with
+  | List.Cons ckey t tl =>
+    if ckey = key
+    then
+      let mv_ls := mem.replace (List T) (List.Cons ckey t tl) List.Nil
+      match mv_ls with
+      | List.Cons i cvalue tl0 => Result.ret (Option.some cvalue)
+      | List.Nil => Result.fail Error.panic
+    else HashMap.remove_from_list_loop T key tl
+  | List.Nil => Result.ret Option.none
+
+/- [hashmap::HashMap::{0}::remove_from_list]: forward function -/
+def HashMap.remove_from_list
+  (T : Type) (key : Usize) (ls : List T) : Result (Option T) :=
+  HashMap.remove_from_list_loop T key ls
+
+/- [hashmap::HashMap::{0}::remove_from_list]: loop 0: backward function 1 -/
+divergent def HashMap.remove_from_list_loop_back
+  (T : Type) (key : Usize) (ls : List T) : Result (List T) :=
+  match ls with
+  | List.Cons ckey t tl =>
+    if ckey = key
+    then
+      let mv_ls := mem.replace (List T) (List.Cons ckey t tl) List.Nil
+      match mv_ls with
+      | List.Cons i cvalue tl0 => Result.ret tl0
+      | List.Nil => Result.fail Error.panic
+    else
+      do
+        let tl0 ← HashMap.remove_from_list_loop_back T key tl
+        Result.ret (List.Cons ckey t tl0)
+  | List.Nil => Result.ret List.Nil
+
+/- [hashmap::HashMap::{0}::remove_from_list]: backward function 1 -/
+def HashMap.remove_from_list_back
+  (T : Type) (key : Usize) (ls : List T) : Result (List T) :=
+  HashMap.remove_from_list_loop_back T key ls
+
+/- [hashmap::HashMap::{0}::remove]: forward function -/
+def HashMap.remove
+  (T : Type) (self : HashMap T) (key : Usize) : Result (Option T) :=
+  do
+    let hash ← hash_key key
+    let i := Vec.len (List T) self.slots
+    let hash_mod ← hash % i
+    let l ← Vec.index_mut (List T) self.slots hash_mod
+    let x ← HashMap.remove_from_list T key l
+    match x with
+    | Option.none => Result.ret Option.none
+    | Option.some x0 =>
+      do
+        let _ ← self.num_entries - (Usize.ofInt 1)
+        Result.ret (Option.some x0)
+
+/- [hashmap::HashMap::{0}::remove]: backward function 0 -/
+def HashMap.remove_back
+  (T : Type) (self : HashMap T) (key : Usize) : Result (HashMap T) :=
+  do
+    let hash ← hash_key key
+    let i := Vec.len (List T) self.slots
+    let hash_mod ← hash % i
+    let l ← Vec.index_mut (List T) self.slots hash_mod
+    let x ← HashMap.remove_from_list T key l
+    match x with
+    | Option.none =>
+      do
+        let l0 ← HashMap.remove_from_list_back T key l
+        let v ← Vec.index_mut_back (List T) self.slots hash_mod l0
+        Result.ret { self with slots := v }
+    | Option.some x0 =>
+      do
+        let i0 ← self.num_entries - (Usize.ofInt 1)
+        let l0 ← HashMap.remove_from_list_back T key l
+        let v ← Vec.index_mut_back (List T) self.slots hash_mod l0
+        Result.ret { self with num_entries := i0, slots := v }
+
+/- [hashmap::test1]: forward function -/
+def test1 : Result Unit :=
+  do
+    let hm ← HashMap.new U64
+    let hm0 ← HashMap.insert U64 hm (Usize.ofInt 0) (U64.ofInt 42)
+    let hm1 ← HashMap.insert U64 hm0 (Usize.ofInt 128) (U64.ofInt 18)
+    let hm2 ← HashMap.insert U64 hm1 (Usize.ofInt 1024) (U64.ofInt 138)
+    let hm3 ← HashMap.insert U64 hm2 (Usize.ofInt 1056) (U64.ofInt 256)
+    let i ← HashMap.get U64 hm3 (Usize.ofInt 128)
+    if not (i = (U64.ofInt 18))
+    then Result.fail Error.panic
+    else
+      do
+        let hm4 ←
+          HashMap.get_mut_back U64 hm3 (Usize.ofInt 1024) (U64.ofInt 56)
+        let i0 ← HashMap.get U64 hm4 (Usize.ofInt 1024)
+        if not (i0 = (U64.ofInt 56))
+        then Result.fail Error.panic
+        else
+          do
+            let x ← HashMap.remove U64 hm4 (Usize.ofInt 1024)
+            match x with
+            | Option.none => Result.fail Error.panic
+            | Option.some x0 =>
+              if not (x0 = (U64.ofInt 56))
+              then Result.fail Error.panic
+              else
+                do
+                  let hm5 ← HashMap.remove_back U64 hm4 (Usize.ofInt 1024)
+                  let i1 ← HashMap.get U64 hm5 (Usize.ofInt 0)
+                  if not (i1 = (U64.ofInt 42))
+                  then Result.fail Error.panic
+                  else
+                    do
+                      let i2 ← HashMap.get U64 hm5 (Usize.ofInt 128)
+                      if not (i2 = (U64.ofInt 18))
+                      then Result.fail Error.panic
+                      else
+                        do
+                          let i3 ← HashMap.get U64 hm5 (Usize.ofInt 1056)
+                          if not (i3 = (U64.ofInt 256))
+                          then Result.fail Error.panic
+                          else Result.ret ()
+
+/- Unit test for [hashmap::test1] -/
+#assert (test1 == .ret ())
+
+end hashmap
diff --git a/tests/lean/Hashmap/Properties.lean b/tests/lean/Hashmap/Properties.lean
new file mode 100644
index 00000000..3652f608
--- /dev/null
+++ b/tests/lean/Hashmap/Properties.lean
@@ -0,0 +1,439 @@
+import Hashmap.Funs
+
+open Primitives
+open Result
+
+namespace List
+
+-- TODO: we don't want to use the original List.lookup because it uses BEq
+-- TODO: rewrite rule: match x == y with ... -> if x = y then ... else ... ? (actually doesn't work because of sugar)
+-- TODO: move?
+@[simp]
+def lookup' {α : Type} (ls: _root_.List (Usize × α)) (key: Usize) : Option α :=
+  match ls with
+  | [] => none
+  | (k, x) :: tl => if k = key then some x else lookup' tl key
+
+end List
+
+namespace hashmap
+
+namespace List
+
+def v {α : Type} (ls: List α) : _root_.List (Usize × α) :=
+  match ls with
+  | Nil => []
+  | Cons k x tl => (k, x) :: v tl
+
+@[simp] theorem v_nil (α : Type) : (Nil : List α).v = [] := by rfl
+@[simp] theorem v_cons {α : Type} k x (tl: List α) : (Cons k x tl).v = (k, x) :: v tl := by rfl
+
+@[simp]
+abbrev lookup {α : Type} (ls: List α) (key: Usize) : Option α :=
+  ls.v.lookup' key
+
+@[simp]
+abbrev len {α : Type} (ls : List α) : Int := ls.v.len
+
+end List
+
+namespace HashMap
+
+abbrev Core.List := _root_.List
+
+namespace List
+
+end List
+
+-- TODO: move
+@[simp] theorem neq_imp_nbeq [BEq α] [LawfulBEq α] (x y : α) (heq : ¬ x = y) : ¬ x == y := by simp [*]
+@[simp] theorem neq_imp_nbeq_rev [BEq α] [LawfulBEq α] (x y : α) (heq : ¬ x = y) : ¬ y == x := by simp [*]
+
+-- TODO: move
+-- TODO: this doesn't work because of sugar
+theorem match_lawful_beq [BEq α] [LawfulBEq α] [DecidableEq α] (x y : α) :
+  (x == y) = (if x = y then true else false) := by
+  split <;> simp_all
+
+@[pspec]
+theorem insert_in_list_spec0 {α : Type} (key: Usize) (value: α) (ls: List α) :
+  ∃ b,
+    insert_in_list α key value ls = ret b ∧
+    (b ↔ ls.lookup key = none)
+  := match ls with
+  | .Nil => by simp [insert_in_list, insert_in_list_loop]
+  | .Cons k v tl =>
+    if h: k = key then -- TODO: The order of k/key matters
+      by
+        simp [insert_in_list]
+        rw [insert_in_list_loop]
+        simp [h]
+    else
+      have ⟨ b, hi ⟩ := insert_in_list_spec0 key value tl
+      by
+        exists b
+        simp [insert_in_list]
+        rw [insert_in_list_loop] -- TODO: Using simp leads to infinite recursion
+        simp only [insert_in_list] at hi
+        simp [*]
+
+-- Variation: use progress
+theorem insert_in_list_spec1 {α : Type} (key: Usize) (value: α) (ls: List α) :
+  ∃ b,
+    insert_in_list α key value ls = ret b ∧
+    (b ↔ ls.lookup key = none)
+  := match ls with
+  | .Nil => by simp [insert_in_list, insert_in_list_loop]
+  | .Cons k v tl =>
+    if h: k = key then -- TODO: The order of k/key matters
+      by
+        simp [insert_in_list]
+        rw [insert_in_list_loop]
+        simp [h]
+    else
+      by
+        simp only [insert_in_list]
+        rw [insert_in_list_loop]
+        conv => rhs; ext; simp [*]
+        progress keep heq as ⟨ b, hi ⟩
+        simp only [insert_in_list] at heq
+        exists b
+
+-- Variation: use tactics from the beginning
+theorem insert_in_list_spec2 {α : Type} (key: Usize) (value: α) (ls: List α) :
+  ∃ b,
+    insert_in_list α key value ls = ret b ∧
+    (b ↔ (ls.lookup key = none))
+  := by
+  induction ls
+  case Nil => simp [insert_in_list, insert_in_list_loop]
+  case Cons k v tl ih =>
+    simp only [insert_in_list]
+    rw [insert_in_list_loop]
+    simp only
+    if h: k = key then
+     simp [h]
+    else
+     conv => rhs; ext; left; simp [h] -- TODO: Simplify
+     simp only [insert_in_list] at ih;
+     -- TODO: give the possibility of using underscores
+     progress as ⟨ b, h ⟩
+     simp [*]
+
+def distinct_keys (ls : Core.List (Usize × α)) := ls.pairwise_rel (λ x y => x.fst ≠ y.fst)
+
+def hash_mod_key (k : Usize) (l : Int) : Int :=
+  match hash_key k with
+  | .ret k => k.val % l
+  | _ => 0
+
+@[simp]
+theorem hash_mod_key_eq : hash_mod_key k l = k.val % l := by
+  simp [hash_mod_key, hash_key]
+
+def slot_s_inv_hash (l i : Int) (ls : Core.List (Usize × α)) : Prop :=
+  ls.allP (λ (k, _) => hash_mod_key k l = i)
+
+@[simp]
+def slot_s_inv (l i : Int) (ls : Core.List (Usize × α)) : Prop :=
+  distinct_keys ls ∧
+  slot_s_inv_hash l i ls
+
+def slot_t_inv (l i : Int) (s : List α) : Prop := slot_s_inv l i s.v
+
+-- Interpret the hashmap as a list of lists
+def v (hm : HashMap α) : Core.List (Core.List (Usize × α)) :=
+  hm.slots.val.map List.v
+
+-- Interpret the hashmap as an associative list
+def al_v (hm : HashMap α) : Core.List (Usize × α) :=
+  hm.v.flatten
+
+-- TODO: automatic derivation
+instance : Inhabited (List α) where
+  default := .Nil
+
+@[simp]
+def slots_s_inv (s : Core.List (List α)) : Prop :=
+  ∀ (i : Int), 0 ≤ i → i < s.len → slot_t_inv s.len i (s.index i)
+
+def slots_t_inv (s : Vec (List α)) : Prop :=
+  slots_s_inv s.v
+
+@[simp]
+def base_inv (hm : HashMap α) : Prop :=
+  -- [num_entries] correctly tracks the number of entries
+  hm.num_entries.val = hm.al_v.len ∧
+  -- Slots invariant
+  slots_t_inv hm.slots ∧
+  -- The capacity must be > 0 (otherwise we can't resize)
+  0 < hm.slots.length
+  -- TODO: load computation
+
+def inv (hm : HashMap α) : Prop :=
+  -- Base invariant
+  base_inv hm
+  -- TODO: either the hashmap is not overloaded, or we can't resize it
+
+theorem insert_in_list_back_spec_aux {α : Type} (l : Int) (key: Usize) (value: α) (l0: List α)
+  (hinv : slot_s_inv_hash l (hash_mod_key key l) l0.v)
+  (hdk : distinct_keys l0.v) :
+  ∃ l1,
+    insert_in_list_back α key value l0 = ret l1 ∧
+    -- We update the binding
+    l1.lookup key = value ∧
+    (∀ k, k ≠ key → l1.lookup k = l0.lookup k) ∧
+    -- We preserve part of the key invariant
+    slot_s_inv_hash l (hash_mod_key key l) l1.v ∧
+    -- Reasoning about the length
+    (match l0.lookup key with
+     | none => l1.len = l0.len + 1
+     | some _ => l1.len = l0.len) ∧
+    -- The keys are distinct
+    distinct_keys l1.v ∧
+    -- We need this auxiliary property to prove that the keys distinct properties is preserved
+    (∀ k, k ≠ key → l0.v.allP (λ (k1, _) => k ≠ k1) → l1.v.allP (λ (k1, _) => k ≠ k1))
+  := match l0 with
+  | .Nil => by checkpoint
+    simp (config := {contextual := true})
+      [insert_in_list_back, insert_in_list_loop_back,
+       List.v, slot_s_inv_hash, distinct_keys, List.pairwise_rel]
+  | .Cons k v tl0 =>
+     if h: k = key then by checkpoint
+       simp [insert_in_list_back]
+       rw [insert_in_list_loop_back]
+       simp [h]
+       split_conjs
+       . intros; simp [*]
+       . simp [List.v, slot_s_inv_hash] at *
+         simp [*]
+       . simp [*, distinct_keys] at *
+         apply hdk
+       . tauto
+     else by checkpoint
+       simp [insert_in_list_back]
+       rw [insert_in_list_loop_back]
+       simp [h]
+       have : slot_s_inv_hash l (hash_mod_key key l) (List.v tl0) := by checkpoint
+         simp_all [List.v, slot_s_inv_hash]
+       have : distinct_keys (List.v tl0) := by checkpoint
+         simp [distinct_keys] at hdk
+         simp [hdk, distinct_keys]
+       progress keep heq as ⟨ tl1 .. ⟩
+       simp only [insert_in_list_back] at heq
+       have : slot_s_inv_hash l (hash_mod_key key l) (List.v (List.Cons k v tl1)) := by checkpoint
+         simp [List.v, slot_s_inv_hash] at *
+         simp [*]
+       have : distinct_keys ((k, v) :: List.v tl1) := by checkpoint
+         simp [distinct_keys] at *
+         simp [*]
+       -- TODO: canonize addition by default?
+       simp_all [Int.add_assoc, Int.add_comm, Int.add_left_comm]
+
+@[pspec]
+theorem insert_in_list_back_spec {α : Type} (l : Int) (key: Usize) (value: α) (l0: List α)
+  (hinv : slot_s_inv_hash l (hash_mod_key key l) l0.v)
+  (hdk : distinct_keys l0.v) :
+  ∃ l1,
+    insert_in_list_back α key value l0 = ret l1 ∧
+    -- We update the binding
+    l1.lookup key = value ∧
+    (∀ k, k ≠ key → l1.lookup k = l0.lookup k) ∧
+    -- We preserve part of the key invariant
+    slot_s_inv_hash l (hash_mod_key key l) l1.v ∧
+    -- Reasoning about the length
+    (match l0.lookup key with
+     | none => l1.len = l0.len + 1
+     | some _ => l1.len = l0.len) ∧
+    -- The keys are distinct
+    distinct_keys l1.v
+  := by
+  progress with insert_in_list_back_spec_aux as ⟨ l1 .. ⟩
+  exists l1
+
+@[simp]
+def slots_t_lookup (s : Core.List (List α)) (k : Usize) : Option α :=
+  let i := hash_mod_key k s.len
+  let slot := s.index i
+  slot.lookup k
+
+def lookup (hm : HashMap α) (k : Usize) : Option α :=
+  slots_t_lookup hm.slots.val k
+
+@[simp]
+abbrev len_s (hm : HashMap α) : Int := hm.al_v.len
+
+-- Remark: α and β must live in the same universe, otherwise the
+-- bind doesn't work
+theorem if_update_eq
+  {α β : Type u} (b : Bool) (y : α) (e : Result α) (f : α → Result β) :
+  (if b then Bind.bind e f else f y) = Bind.bind (if b then e else pure y) f
+  := by
+  split <;> simp [Pure.pure]
+
+-- Small helper
+-- TODO: move, and introduce a better solution with nice syntax
+def mk_opaque {α : Sort u} (x : α) : { y : α // y = x}  :=
+  ⟨ x, by simp ⟩
+
+--set_option profiler true
+--set_option profiler.threshold 10
+--set_option trace.profiler true
+
+-- For pretty printing (useful when copy-pasting goals)
+attribute [pp_dot] List.length -- use the dot notation when printing
+set_option pp.coercions false -- do not print coercions with ↑ (this doesn't parse)
+
+theorem insert_no_resize_spec {α : Type} (hm : HashMap α) (key : Usize) (value : α)
+  (hinv : hm.inv) (hnsat : hm.lookup key = none → hm.len_s < Usize.max) :
+  ∃ nhm, hm.insert_no_resize α key value = ret nhm  ∧
+  -- We preserve the invariant
+  nhm.inv ∧
+  -- We updated the binding for key
+  nhm.lookup key = some value ∧
+  -- We left the other bindings unchanged
+  (∀ k, ¬ k = key → nhm.lookup k = hm.lookup k) ∧
+  -- Reasoning about the length
+  (match hm.lookup key with
+   | none => nhm.len_s = hm.len_s + 1
+   | some _ => nhm.len_s = hm.len_s) := by
+  rw [insert_no_resize]
+  simp only [hash_key, bind_tc_ret] -- TODO: annoying
+  have _ : (Vec.len (List α) hm.slots).val ≠ 0 := by checkpoint
+   intro
+   simp_all [inv]
+  progress keep _ as ⟨ hash_mod, hhm ⟩
+  have _ : 0 ≤ hash_mod.val := by checkpoint scalar_tac
+  have _ : hash_mod.val < Vec.length hm.slots := by
+    have : 0 < hm.slots.val.len := by
+      simp [inv] at hinv
+      simp [hinv]
+    -- TODO: we want to automate that
+    simp [*, Int.emod_lt_of_pos]
+  -- TODO: change the spec of Vec.index_mut to introduce a let-binding.
+  -- or: make progress introduce the let-binding by itself (this is clearer)
+  progress as ⟨ l, h_leq ⟩
+  -- TODO: make progress use the names written in the goal
+  progress as ⟨ inserted ⟩
+  rw [if_update_eq] -- TODO: necessary because we don't have a join
+  -- TODO: progress to ...
+  have hipost :
+    ∃ i0, (if inserted = true then hm.num_entries + Usize.ofInt 1 else pure hm.num_entries) = ret i0 ∧
+    i0.val = if inserted then hm.num_entries.val + 1 else hm.num_entries.val
+    := by
+    if inserted then
+      simp [*]
+      have hbounds : hm.num_entries.val + (Usize.ofInt 1).val ≤ Usize.max := by
+        simp [lookup] at hnsat
+        simp_all
+        simp [inv] at hinv
+        int_tac
+      -- TODO: progress fails in command line mode with "index out of bounds"
+      -- and I have no idea how to fix this. The error happens after progress
+      -- introduced the new goals. It must be when we exit the "withApp", etc.
+      -- helpers.
+      -- progress as ⟨ z, hp ⟩
+      have ⟨ z, hp ⟩ := Usize.add_spec hbounds
+      simp [hp]
+    else
+      simp [*, Pure.pure]
+  progress as ⟨ i0 ⟩
+  have h_slot : slot_s_inv_hash hm.slots.length (hash_mod_key key hm.slots.length) l.v
+    := by
+    simp [inv] at hinv
+    have h := (hinv.right.left hash_mod.val (by assumption) (by assumption)).right
+    simp [slot_t_inv, hhm] at h
+    simp [h, hhm, h_leq]
+  have hd : distinct_keys l.v := by checkpoint
+    simp [inv, slots_t_inv, slot_t_inv] at hinv
+    have h := hinv.right.left hash_mod.val (by assumption) (by assumption)
+    simp [h, h_leq]
+  -- TODO: hide the variables and only keep the props
+  -- TODO: allow providing terms to progress to instantiate the meta variables
+  -- which are not propositions
+  progress as ⟨ l0, _, _, _, hlen .. ⟩
+  progress keep hv as ⟨ v, h_veq ⟩
+  -- TODO: update progress to automate that
+  -- TODO: later I don't want to inline nhm - we need to control simp: deactivate
+  -- zeta reduction? For now I have to do this peculiar manipulation
+  have ⟨ nhm, nhm_eq ⟩ := @mk_opaque (HashMap α) { num_entries := i0, max_load_factor := hm.max_load_factor, max_load := hm.max_load, slots := v }
+  exists nhm
+  have hupdt : lookup nhm key = some value := by checkpoint
+    simp [lookup, List.lookup] at *
+    simp_all
+  have hlkp : ∀ k, ¬ k = key → nhm.lookup k = hm.lookup k := by
+    simp [lookup, List.lookup] at *
+    intro k hk
+    -- We have to make a case disjunction: either the hashes are different,
+    -- in which case we don't even lookup the same slots, or the hashes
+    -- are the same, in which case we have to reason about what happens
+    -- in one slot
+    let k_hash_mod := k.val % v.val.len
+    have : 0 < hm.slots.val.len := by simp_all [inv]
+    have hvpos : 0 < v.val.len := by simp_all
+    have hvnz: v.val.len ≠ 0 := by
+      simp_all
+    have _ : 0 ≤ k_hash_mod := by
+      -- TODO: we want to automate this
+      simp
+      apply Int.emod_nonneg k.val hvnz
+    have _ : k_hash_mod < Vec.length hm.slots := by
+      -- TODO: we want to automate this
+      simp
+      have h := Int.emod_lt_of_pos k.val hvpos
+      simp_all
+    if h_hm : k_hash_mod = hash_mod.val then
+      simp_all
+    else
+      simp_all
+  have _ :
+    match hm.lookup key with
+    | none => nhm.len_s = hm.len_s + 1
+    | some _ => nhm.len_s = hm.len_s := by checkpoint
+    simp only [lookup, List.lookup, len_s, al_v, HashMap.v, slots_t_lookup] at *
+    -- We have to do a case disjunction
+    simp_all
+    simp [_root_.List.update_map_eq]
+    -- TODO: dependent rewrites
+    have _ : key.val % hm.slots.val.len < (List.map List.v hm.slots.val).len := by
+      simp [*]
+    simp [_root_.List.len_flatten_update_eq, *]
+    split <;>
+    rename_i heq <;>
+    simp [heq] at hlen <;>
+    -- TODO: canonize addition by default? We need a tactic to simplify arithmetic equalities
+    -- with addition and substractions ((ℤ, +) is a group or something - there should exist a tactic
+    -- somewhere in mathlib?)
+    simp [Int.add_assoc, Int.add_comm, Int.add_left_comm] <;>
+    int_tac
+  have hinv : inv nhm := by
+    simp [inv] at *
+    split_conjs
+    . match h: lookup hm key with
+      | none =>
+        simp [h, lookup] at *
+        simp_all
+      | some _ =>
+        simp_all [lookup]
+    . simp [slots_t_inv, slot_t_inv] at *
+      intro i hipos _
+      have _ := hinv.right.left i hipos (by simp_all)
+      simp [hhm, h_veq, nhm_eq] at * -- TODO: annoying, we do that because simp_all fails below
+      -- We need a case disjunction
+      if h_ieq : i = key.val % _root_.List.len hm.slots.val then
+        -- TODO: simp_all fails: "(deterministic) timeout at 'whnf'"
+        -- Also, it is annoying to do this kind
+        -- of rewritings by hand. We could have a different simp
+        -- which safely substitutes variables when we have an
+        -- equality `x = ...` and `x` doesn't appear in the rhs
+        simp [h_ieq] at *
+        simp [*]
+      else
+        simp [*]
+    . -- TODO: simp[*] fails: "(deterministic) timeout at 'whnf'"
+      simp [hinv, h_veq, nhm_eq]
+  simp_all
+
+end HashMap
+
+end hashmap
diff --git a/tests/lean/Hashmap/Types.lean b/tests/lean/Hashmap/Types.lean
new file mode 100644
index 00000000..6606cf9e
--- /dev/null
+++ b/tests/lean/Hashmap/Types.lean
@@ -0,0 +1,19 @@
+-- THIS FILE WAS AUTOMATICALLY GENERATED BY AENEAS
+-- [hashmap]: type definitions
+import Base
+open Primitives
+namespace hashmap
+
+/- [hashmap::List] -/
+inductive List (T : Type) :=
+| Cons : Usize → T → List T → List T
+| Nil : List T
+
+/- [hashmap::HashMap] -/
+structure HashMap (T : Type) where
+  num_entries : Usize
+  max_load_factor : (Usize × Usize)
+  max_load : Usize
+  slots : Vec (List T)
+
+end hashmap