feat(frontends/lean): allow backticks in binder declarations

library/theories/number_theory/primes.lean

@@ -79,9 +79,9 @@ have ¬ (∀ m, m < succ n → m ∣ n → m = 1 ∨ m = n), from
   assume h, absurd (λ m hl hd, h m (lt_succ_of_le hl) hd) this,
 have {m | m < succ n ∧ ¬(m ∣ n → m = 1 ∨ m = n)}, from bsub_not_of_not_ball this,
 obtain m hlt (h₃ : ¬(m ∣ n → m = 1 ∨ m = n)), from this,
-obtain (h₄ : m ∣ n) (h₅ : ¬ (m = 1 ∨ m = n)), from iff.mp !not_implies_iff_and_not h₃,
+obtain `m ∣ n` (h₅ : ¬ (m = 1 ∨ m = n)), from iff.mp !not_implies_iff_and_not h₃,
 have ¬ m = 1 ∧ ¬ m = n,  from iff.mp !not_or_iff_not_and_not h₅,
-subtype.tag m (and.intro h₄ this)
+subtype.tag m (and.intro `m ∣ n` this)
 
 theorem ex_dvd_of_not_prime {n : nat} : n ≥ 2 → ¬ prime n → ∃ m, m ∣ n ∧ m ≠ 1 ∧ m ≠ n :=
 assume h₁ h₂, ex_of_sub (sub_dvd_of_not_prime h₁ h₂)
@@ -125,7 +125,7 @@ definition infinite_primes (n : nat) : {p | p ≥ n ∧ prime p} :=
 let m := fact (n + 1) in
 have m ≥ 1,     from le_of_lt_succ (succ_lt_succ (fact_pos _)),
 have m + 1 ≥ 2, from succ_le_succ this,
-obtain p (pp : prime p) (pd : p ∣ m + 1), from sub_prime_and_dvd this,
+obtain p `prime p` `p ∣ m + 1`, from sub_prime_and_dvd this,
 have p ≥ 2, from ge_two_of_prime `prime p`,
 have p > 0, from lt_of_succ_lt (lt_of_succ_le `p ≥ 2`),
 have p ≥ n, from by_contradiction
@@ -144,8 +144,8 @@ ex_of_sub (infinite_primes n)
 lemma odd_of_prime {p : nat} : prime p → p > 2 → odd p :=
 λ pp p_gt_2, by_contradiction (λ hn,
   have even p, from even_of_not_odd hn,
-  obtain k (hk : p = 2*k), from exists_of_even this,
-  assert 2 ∣ p, by rewrite [hk]; apply dvd_mul_right,
+  obtain k `p = 2*k`, from exists_of_even this,
+  assert 2 ∣ p, by rewrite [`p = 2*k`]; apply dvd_mul_right,
   or.elim (eq_one_or_eq_self_of_prime_of_dvd pp this)
     (suppose 2 = 1, absurd this dec_trivial)
     (suppose 2 = p, by subst this; exact absurd p_gt_2 !lt.irrefl))

src/frontends/lean/parser.cpp

@@ -934,6 +934,13 @@ void parser::parse_binders_core(buffer<expr> & r, buffer<notation_entry> * nentr
         if (curr_is_identifier()) {
             parse_binder_block(r, binder_info(), rbp);
             last_block_delimited = false;
+        } else if (curr_is_backtick()) {
+            auto p    = pos();
+            name n    = mk_fresh_name();
+            expr type = parse_backtick_expr_core();
+            expr local = save_pos(mk_local(n, type, binder_info()), p);
+            add_local(local);
+            r.push_back(local);
         } else {
             optional<binder_info> bi = parse_optional_binder_info(simple_only);
             if (bi) {
@@ -1330,8 +1337,8 @@ expr parser::parse_string_expr() {
     return from_string(v);
 }
 
-expr parser::parse_backtick_expr() {
-    auto p = pos();
+
+expr parser::parse_backtick_expr_core() {
     next();
     flet<bool> set(m_in_backtick, true);
     expr type = parse_expr();
@@ -1339,6 +1346,12 @@ expr parser::parse_backtick_expr() {
         throw parser_error("invalid expression, '`' expected", pos());
     }
     next();
+    return type;
+}
+
+expr parser::parse_backtick_expr() {
+    auto p = pos();
+    expr type   = parse_backtick_expr_core();
     expr assump = mk_by_plus(save_pos(mk_constant(get_tactic_assumption_name()), p), p);
     return save_pos(mk_typed_expr(type, assump), p);
 }

src/frontends/lean/parser.h

@@ -234,6 +234,7 @@ class parser {
     elaborator_context mk_elaborator_context(environment const & env, local_level_decls const & lls, pos_info_provider const & pp);
 
     bool m_in_backtick; // true if parser `expr` notation
+    expr parse_backtick_expr_core();
     expr parse_backtick_expr();
 
     optional<expr> is_tactic_command(name & id);
@@ -338,6 +339,8 @@ public:
     bool curr_is_token_or_id(name const & tk) const;
     /** \brief Return true iff the current token is a command, EOF, period or script block */
     bool curr_is_command_like() const;
+    /** \brief Return true iff the current token is a backtick ` */
+    bool curr_is_backtick() const { return curr() == scanner::token_kind::Backtick; }
     /** \brief Read the next token if the current one is not End-of-file. */
     void next() { if (m_curr != scanner::token_kind::Eof) scan(); }
     /** \brief Return true iff the current token is a keyword (or command keyword) named \c tk */