feat(frontends/lean/elaborator): coercion from (decidable) Prop to bool

This is a hard coded extra case. It is not an instance of has_coe.
Even if we change has_coe to accomodate this case, it will not be a
satisfactory solution because this coercion depends on the element and
not the type, and the element usually contains metavariables.

We should eventually write a tactic for synthesizing coercions.

library/init/meta/expr.lean

@@ -183,14 +183,14 @@ meta def local_type : expr → expr
 | e := e
 
 meta def is_constant_of : expr → name → bool
-| (const n₁ ls) n₂ := to_bool (n₁ = n₂)
+| (const n₁ ls) n₂ := n₁ = n₂
 | e             n  := ff
 
 meta def is_app_of (e : expr) (n : name) : bool :=
 is_constant_of (get_app_fn e) n
 
 meta def is_napp_of (e : expr) (c : name) (n : nat) : bool :=
-to_bool (is_app_of e c ∧ get_app_num_args e = n)
+is_app_of e c ∧ get_app_num_args e = n
 
 meta def is_false (e : expr) : bool :=
 is_constant_of e `false

library/init/meta/fun_info.lean

@@ -32,7 +32,7 @@ group ∘ cbrace $
   when ii "inst_implicit" +++
   when p  "prop" +++
   when d  "has_fwd_deps" +++
-  when (to_bool (length ds > 0)) (to_fmt "back_deps := " ++ to_fmt ds)
+  when (length ds > 0) (to_fmt "back_deps := " ++ to_fmt ds)
 
 meta instance : has_to_format param_info :=
 has_to_format.mk param_info.to_format

library/init/meta/level.lean

@@ -58,5 +58,5 @@ meta def level.has_param : level → name → bool
 | (level.succ l)     n  := level.has_param l n
 | (level.max l₁ l₂)  n  := level.has_param l₁ n || level.has_param l₂ n
 | (level.imax l₁ l₂) n  := level.has_param l₁ n || level.has_param l₂ n
-| (level.param n₁)   n  := to_bool (n₁ = n)
+| (level.param n₁)   n  := n₁ = n
 | l                  n  := ff

library/init/meta/occurrences.lean

@@ -27,8 +27,8 @@ open occurrences
 
 def occurrences.contains : occurrences → nat → bool
 | all                  p := tt
-| (occurrences.pos ps) p := to_bool (p ∈ ps)
-| (occurrences.neg ps) p := to_bool (p ∉ ps)
+| (occurrences.pos ps) p := p ∈ ps
+| (occurrences.neg ps) p := p ∉ ps
 
 instance : inhabited occurrences :=
 ⟨all⟩

library/init/meta/simp_tactic.lean

@@ -160,7 +160,7 @@ cs^.any (λ c,
   if e^.is_app_of c then tt   /- Exact match -/
   else let f := e^.get_app_fn in
        /- f is an auxiliary constant generated when compiling c -/
-       f^.is_constant && f^.const_name^.is_internal && to_bool (f^.const_name^.get_prefix = c))
+       f^.is_constant && f^.const_name^.is_internal && (f^.const_name^.get_prefix = c))
 
 /- Delta reduce the given constant names -/
 meta def delta_core (cfg : delta_config) (cs : list name) (e : expr) : tactic expr :=

library/init/meta/smt/congruence_closure.lean

@@ -68,7 +68,7 @@ meta def eqc_of (s : cc_state) (e : expr) : list expr :=
 s^.eqc_of_core e e []
 
 meta def in_singlenton_eqc (s : cc_state) (e : expr) : bool :=
-to_bool (s^.next e = e)
+s^.next e = e
 
 meta def eqc_size (s : cc_state) (e : expr) : nat :=
 (s^.eqc_of e)^.length

library/init/meta/tactic.lean

@@ -532,7 +532,7 @@ meta def rstep {α : Type u} (line : nat) (col : nat) (t : tactic α) : tactic u
 
 meta def is_prop (e : expr) : tactic bool :=
 do t ← infer_type e,
-   return (to_bool (t = expr.prop))
+   return (t = expr.prop)
 
 /-- Return true iff n is the name of declaration that is a proposition. -/
 meta def is_prop_decl (n : name) : tactic bool :=

library/init/native/default.lean

@@ -208,7 +208,7 @@ let fst' := list.map assert_name fst,
   ])
 
 meta def is_return (n : name) : bool :=
-decidable.to_bool $ `native_compiler.return = n
+`native_compiler.return = n
 
 meta def compile_call (head : name) (arity : nat) (args : list ir.expr) : ir_compiler ir.expr := do
   trace_ir $ "compile_call: " ++ (to_string head),

library/init/native/util.lean

@@ -14,7 +14,7 @@ import init.native.internal
 import init.native.builtin
 
 meta def is_nat_cases_on (n : name) : bool :=
-  decidable.to_bool $ `nat.cases_on = n
+`nat.cases_on = n
 
 meta def is_cases_on (head : expr) : bool :=
   if is_nat_cases_on (expr.const_name head)

library/tools/debugger/cli.lean

@@ -206,7 +206,7 @@ meta def in_active_bps (s : state) : vm bool :=
 do sz ← vm.call_stack_size,
    match s^.active_bps with
    | []          := return ff
-   | (csz, _)::_ := return $ to_bool (sz = csz)
+   | (csz, _)::_ := return (sz = csz)
    end
 
 meta def run_transition (s : state) : vm state :=

library/tools/super/lpo.lean

@@ -19,7 +19,7 @@ def majo {T} (gt : T → T → bool) (s : T) : list T → bool
 
 meta def alpha (lpo : expr → expr → bool) : list expr → expr → bool
 | [] _ := ff
-| (s::ss) t := to_bool (s = t) || lpo s t || alpha ss t
+| (s::ss) t := (s = t) || lpo s t || alpha ss t
 
 meta def lex_ma (lpo : expr → expr → bool) (s t : expr) : list expr → list expr → bool
 | (si::ss) (ti::ts) :=
@@ -36,7 +36,7 @@ else alpha lpo (get_app_args s) t
 meta def prec_gt_of_name_list (ns : list name) : expr → expr → bool :=
 let nis := rb_map.of_list ns^.zip_with_index in
 λs t, match (nis^.find (name_of_funsym s), nis^.find (name_of_funsym t)) with
-| (some si, some ti) := to_bool (si > ti)
+| (some si, some ti) := si > ti
 | _ := ff
 end
 

library/tools/super/prover_state.lean

@@ -419,7 +419,7 @@ meta def empty (local_false : expr) : prover_state :=
 
 meta def initial (local_false : expr) (clauses : list clause) : tactic prover_state := do
 after_setup ← for' clauses (λc,
-  let in_sos := decidable.to_bool $ ((contained_lconsts c^.proof)^.erase local_false^.local_uniq_name)^.size = 0 in
+  let in_sos := ((contained_lconsts c^.proof)^.erase local_false^.local_uniq_name)^.size = 0 in
   do mk_derived c { priority := score.prio.immediate, in_sos := in_sos,
                     age := 0, cost := 0 } >>= add_inferred
 ) $ empty local_false,

src/frontends/lean/elaborator.cpp

@@ -510,8 +510,17 @@ void elaborator::trace_coercion_failure(expr const & e_type, expr const & type,
         });
 }
 
+optional<expr> elaborator::mk_Prop_to_bool_coercion(expr const & e, expr const & ref) {
+    expr dec    = mk_app(mk_constant(get_decidable_name()), e);
+    expr inst   = mk_instance(dec, ref);
+    expr r      = mk_app(mk_constant(get_decidable_to_bool_name()), e, inst);
+    return some_expr(r);
+}
+
 optional<expr> elaborator::mk_coercion_core(expr const & e, expr const & e_type, expr const & type, expr const & ref) {
-    if (!has_expr_metavar(e_type) && !has_expr_metavar(type)) {
+    if (e_type == mk_Prop() && m_ctx.is_def_eq(type, mk_bool())) {
+        return mk_Prop_to_bool_coercion(e, ref);
+    } else if (!has_expr_metavar(e_type) && !has_expr_metavar(type)) {
         expr has_coe_t;
         try {
             has_coe_t = mk_app(m_ctx, get_has_coe_t_name(), e_type, type);

src/frontends/lean/elaborator.h

@@ -133,6 +133,7 @@ private:
     level replace_univ_placeholder(level const & l);
 
     void trace_coercion_failure(expr const & e_type, expr const & type, expr const & ref, char const * error_msg);
+    optional<expr> mk_Prop_to_bool_coercion(expr const & e, expr const & ref);
     optional<expr> mk_coercion_core(expr const & e, expr const & e_type, expr const & type, expr const & ref);
     bool is_monad(expr const & e);
     bool is_monad_fail(expr const & e);

src/library/constants.cpp

@@ -48,6 +48,7 @@ name const * g_cyclic_numerals = nullptr;
 name const * g_cyclic_numerals_bound = nullptr;
 name const * g_decidable = nullptr;
 name const * g_decidable_by_contradiction = nullptr;
+name const * g_decidable_to_bool = nullptr;
 name const * g_discrete_field = nullptr;
 name const * g_distinct = nullptr;
 name const * g_distrib = nullptr;
@@ -535,6 +536,7 @@ void initialize_constants() {
     g_cyclic_numerals_bound = new name{"cyclic_numerals", "bound"};
     g_decidable = new name{"decidable"};
     g_decidable_by_contradiction = new name{"decidable", "by_contradiction"};
+    g_decidable_to_bool = new name{"decidable", "to_bool"};
     g_discrete_field = new name{"discrete_field"};
     g_distinct = new name{"distinct"};
     g_distrib = new name{"distrib"};
@@ -1023,6 +1025,7 @@ void finalize_constants() {
     delete g_cyclic_numerals_bound;
     delete g_decidable;
     delete g_decidable_by_contradiction;
+    delete g_decidable_to_bool;
     delete g_discrete_field;
     delete g_distinct;
     delete g_distrib;
@@ -1510,6 +1513,7 @@ name const & get_cyclic_numerals_name() { return *g_cyclic_numerals; }
 name const & get_cyclic_numerals_bound_name() { return *g_cyclic_numerals_bound; }
 name const & get_decidable_name() { return *g_decidable; }
 name const & get_decidable_by_contradiction_name() { return *g_decidable_by_contradiction; }
+name const & get_decidable_to_bool_name() { return *g_decidable_to_bool; }
 name const & get_discrete_field_name() { return *g_discrete_field; }
 name const & get_distinct_name() { return *g_distinct; }
 name const & get_distrib_name() { return *g_distrib; }

src/library/constants.h

@@ -50,6 +50,7 @@ name const & get_cyclic_numerals_name();
 name const & get_cyclic_numerals_bound_name();
 name const & get_decidable_name();
 name const & get_decidable_by_contradiction_name();
+name const & get_decidable_to_bool_name();
 name const & get_discrete_field_name();
 name const & get_distinct_name();
 name const & get_distrib_name();

src/library/constants.txt

@@ -43,6 +43,7 @@ cyclic_numerals
 cyclic_numerals.bound
 decidable
 decidable.by_contradiction
+decidable.to_bool
 discrete_field
 distinct
 distrib