{"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 3, "character": 8}} {"range": {"start": {"line": 3, "character": 8}, "end": {"line": 3, "character": 18}}, "contents": {"value": "```lean\nTrue.intro : True\n```\n***\n`True` is true, and `True.intro` (or more commonly, `trivial`)\nis the proof. ", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 7, "character": 8}} {"range": {"start": {"line": 7, "character": 8}, "end": {"line": 7, "character": 18}}, "contents": {"value": "```lean\nTrue.intro : True\n```\n***\n`True` is true, and `True.intro` (or more commonly, `trivial`)\nis the proof. ", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 12, "character": 4}} {"range": {"start": {"line": 12, "character": 4}, "end": {"line": 12, "character": 12}}, "contents": {"value": "```lean\nNat.zero : Nat\n```\n***\n`Nat.zero`, normally written `0 : Nat`, is the smallest natural number.\nThis is one of the two constructors of `Nat`. ", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 21, "character": 2}} {"range": {"start": {"line": 21, "character": 2}, "end": {"line": 21, "character": 23}}, "contents": {"value": "My tactic ", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 21, "character": 8}} {"range": {"start": {"line": 21, "character": 2}, "end": {"line": 21, "character": 23}}, "contents": {"value": "My tactic ", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 21, "character": 13}} {"range": {"start": {"line": 21, "character": 13}, "end": {"line": 21, "character": 23}}, "contents": {"value": "```lean\nTrue.intro : True\n```\n***\n`True` is true, and `True.intro` (or more commonly, `trivial`)\nis the proof. ", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 28, "character": 6}} {"range": {"start": {"line": 28, "character": 6}, "end": {"line": 28, "character": 12}}, "contents": {"value": "```lean\nLean.Parser.Category.tactic : Lean.Parser.Category\n```\n***\n`tactic` is the builtin syntax category for tactics. These appear after\n`by` in proofs, and they are programs that take in the proof context\n(the hypotheses in scope plus the type of the term to synthesize) and construct\na term of the expected type. For example, `simp` is a tactic, used in:\n```\nexample : 2 + 2 = 4 := by simp\n```\n", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 28, "character": 32}} {"range": {"start": {"line": 28, "character": 32}, "end": {"line": 28, "character": 36}}, "contents": {"value": "```lean\nLean.Parser.Category.term : Lean.Parser.Category\n```\n***\n`term` is the builtin syntax category for terms. A term denotes an expression\nin lean's type theory, for example `2 + 2` is a term. The difference between\n`Term` and `Expr` is that the former is a kind of syntax, while the latter is\nthe result of elaboration. For example `by simp` is also a `Term`, but it elaborates\nto different `Expr`s depending on the context. ", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 31, "character": 6}} {"range": {"start": {"line": 31, "character": 6}, "end": {"line": 31, "character": 12}}, "contents": {"value": "```lean\nLean.Parser.Category.tactic : Lean.Parser.Category\n```\n***\n`tactic` is the builtin syntax category for tactics. These appear after\n`by` in proofs, and they are programs that take in the proof context\n(the hypotheses in scope plus the type of the term to synthesize) and construct\na term of the expected type. For example, `simp` is a tactic, used in:\n```\nexample : 2 + 2 = 4 := by simp\n```\n", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 31, "character": 24}} {"range": {"start": {"line": 31, "character": 23}, "end": {"line": 31, "character": 27}}, "contents": {"value": "```lean\nLean.Parser.Category.term : Lean.Parser.Category\n```\n***\n`term` is the builtin syntax category for terms. A term denotes an expression\nin lean's type theory, for example `2 + 2` is a term. The difference between\n`Term` and `Expr` is that the former is a kind of syntax, while the latter is\nthe result of elaboration. For example `by simp` is also a `Term`, but it elaborates\nto different `Expr`s depending on the context. ", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 36, "character": 2}} {"range": {"start": {"line": 36, "character": 2}, "end": {"line": 36, "character": 23}}, "contents": {"value": "My tactic ", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 44, "character": 2}} {"range": {"start": {"line": 44, "character": 2}, "end": {"line": 44, "character": 23}}, "contents": {"value": "My tactic ", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 49, "character": 16}} {"range": {"start": {"line": 49, "character": 15}, "end": {"line": 49, "character": 21}}, "contents": {"value": "```lean\nmyNota : Lean.ParserDescr\n```\n***\nMy notation ", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 52, "character": 7}} {"range": {"start": {"line": 52, "character": 7}, "end": {"line": 52, "character": 15}}, "contents": {"value": "```lean\n1 : Nat\n```\n***\nMy notation ", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 59, "character": 7}} {"range": {"start": {"line": 59, "character": 7}, "end": {"line": 59, "character": 15}}, "contents": {"value": "```lean\nNat\n```\n***\nMy notation ", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 69, "character": 7}} {"range": {"start": {"line": 69, "character": 7}, "end": {"line": 69, "character": 16}}, "contents": {"value": "```lean\nNat\n```\n***\nMy ultimate notation ", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 73, "character": 21}} {"range": {"start": {"line": 73, "character": 18}, "end": {"line": 73, "character": 25}}, "contents": {"value": "```lean\nmyInfix : Lean.TrailingParserDescr\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 73, "character": 39}} {"range": {"start": {"line": 73, "character": 38}, "end": {"line": 73, "character": 45}}, "contents": {"value": "```lean\nNat.add : Nat → Nat → Nat\n```\n***\nAddition of natural numbers.\n\nThis definition is overridden in both the kernel and the compiler to efficiently\nevaluate using the \"bignum\" representation (see `Nat`). The definition provided\nhere is the logical model (and it is soundness-critical that they coincide).\n", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 77, "character": 10}} {"range": {"start": {"line": 77, "character": 7}, "end": {"line": 77, "character": 14}}, "contents": {"value": "```lean\nNat\n```\n***\nAddition of natural numbers.\n\nThis definition is overridden in both the kernel and the compiler to efficiently\nevaluate using the \"bignum\" representation (see `Nat`). The definition provided\nhere is the logical model (and it is soundness-critical that they coincide).\n", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 82, "character": 14}} {"range": {"start": {"line": 82, "character": 14}, "end": {"line": 82, "character": 36}}, "contents": {"value": "```lean\nLean.Parser.Term.doSeq : Lean.Parser.Parser\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 84, "character": 30}} {"range": {"start": {"line": 84, "character": 29}, "end": {"line": 84, "character": 34}}, "contents": {"value": "```lean\nLean.Parser.Term.doSeq : Lean.Parser.Parser\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 87, "character": 2}} {"range": {"start": {"line": 87, "character": 0}, "end": {"line": 87, "character": 7}}, "contents": {"value": "My command ", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 94, "character": 2}} {"range": {"start": {"line": 94, "character": 0}, "end": {"line": 94, "character": 7}}, "contents": {"value": "My command ", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 97, "character": 16}} {"range": {"start": {"line": 97, "character": 16}, "end": {"line": 97, "character": 23}}, "contents": {"value": "```lean\nLean.Parser.ppSpace : Lean.Parser.Parser\n```\n***\nNo-op parser that advises the pretty printer to emit a space/soft line break. ", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 97, "character": 24}} {"range": {"start": {"line": 97, "character": 24}, "end": {"line": 97, "character": 31}}, "contents": {"value": "```lean\nLean.ParserDescr.sepBy1 : Lean.ParserDescr → String → Lean.ParserDescr → optParam Bool false → Lean.ParserDescr\n```\n***\n`sepBy1` is just like `sepBy`, except it takes 1 or more instead of\n0 or more occurrences of `p`. ", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 97, "character": 31}} {"range": {"start": {"line": 97, "character": 31}, "end": {"line": 97, "character": 35}}, "contents": {"value": "```lean\nLean.Parser.Category.term : Lean.Parser.Category\n```\n***\n`term` is the builtin syntax category for terms. A term denotes an expression\nin lean's type theory, for example `2 + 2` is a term. The difference between\n`Term` and `Expr` is that the former is a kind of syntax, while the latter is\nthe result of elaboration. For example `by simp` is also a `Term`, but it elaborates\nto different `Expr`s depending on the context. ", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 106, "character": 2}} {"range": {"start": {"line": 106, "character": 0}, "end": {"line": 106, "character": 8}}, "contents": {"value": "My ultimate command ", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 110, "character": 10}} null {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 114, "character": 8}} {"range": {"start": {"line": 114, "character": 8}, "end": {"line": 114, "character": 10}}, "contents": {"value": "```lean\nid.{0} : ∀ {α : Prop}, α → α\n```\n***\nThe identity function. `id` takes an implicit argument `α : Sort u`\n(a type in any universe), and an argument `a : α`, and returns `a`.\n\nAlthough this may look like a useless function, one application of the identity\nfunction is to explicitly put a type on an expression. If `e` has type `T`,\nand `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and lean\nknows that this expression has type `T'` rather than `T`. This can make a\ndifference for typeclass inference, since `T` and `T'` may have different\ntypeclass instances on them. `show T' from e` is sugar for an `@id T' e`\nexpression.\n", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 114, "character": 10}} {"range": {"start": {"line": 114, "character": 8}, "end": {"line": 114, "character": 21}}, "contents": {"value": "```lean\nTrue\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 121, "character": 2}} {"range": {"start": {"line": 121, "character": 2}, "end": {"line": 121, "character": 3}}, "contents": {"value": "```lean\nn : Id Nat\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 128, "character": 9}} {"range": {"start": {"line": 128, "character": 7}, "end": {"line": 128, "character": 17}}, "contents": {"value": "```lean\nfoo : Nat\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 133, "character": 7}} {"range": {"start": {"line": 133, "character": 7}, "end": {"line": 133, "character": 10}}, "contents": {"value": "```lean\nBar.foo : Nat\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 136, "character": 9}} {"range": {"start": {"line": 136, "character": 7}, "end": {"line": 136, "character": 17}}, "contents": {"value": "```lean\n_root_.foo : Nat\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 139, "character": 4}} {"range": {"start": {"line": 139, "character": 4}, "end": {"line": 139, "character": 7}}, "contents": {"value": "```lean\nBar.bar : Nat\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 142, "character": 10}} {"range": {"start": {"line": 142, "character": 10}, "end": {"line": 142, "character": 13}}, "contents": {"value": "```lean\nBar.Foo : Type\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 142, "character": 17}} {"range": {"start": {"line": 142, "character": 17}, "end": {"line": 142, "character": 19}}, "contents": {"value": "```lean\nBar.Foo.mk : Nat → Foo\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 145, "character": 2}} {"range": {"start": {"line": 145, "character": 2}, "end": {"line": 145, "character": 4}}, "contents": {"value": "```lean\nBar.Foo.hi : Foo → Nat\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 148, "character": 10}} {"range": {"start": {"line": 148, "character": 10}, "end": {"line": 148, "character": 13}}, "contents": {"value": "```lean\nBar.Bar : Type\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 150, "character": 4}} {"range": {"start": {"line": 150, "character": 4}, "end": {"line": 150, "character": 6}}, "contents": {"value": "```lean\nBar.Bar.mk : Bar\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 153, "character": 2}} {"range": {"start": {"line": 153, "character": 0}, "end": {"line": 153, "character": 8}}, "contents": {"value": "```lean\nBar.instToStringNat : ToString Nat\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 155, "character": 9}} {"range": {"start": {"line": 155, "character": 9}, "end": {"line": 155, "character": 10}}, "contents": {"value": "```lean\nBar.f : ToString Nat\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 158, "character": 10}} {"range": {"start": {"line": 158, "character": 10}, "end": {"line": 158, "character": 16}}, "contents": {"value": "A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. ", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 161, "character": 4}} {"range": {"start": {"line": 161, "character": 4}, "end": {"line": 161, "character": 11}}, "contents": {"value": "```lean\nBar.foo.bar : Nat\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 161, "character": 8}} {"range": {"start": {"line": 161, "character": 4}, "end": {"line": 161, "character": 11}}, "contents": {"value": "```lean\nBar.foo.bar : Nat\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 166, "character": 6}} {"range": {"start": {"line": 166, "character": 6}, "end": {"line": 166, "character": 7}}, "contents": {"value": "```lean\nx : Nat\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 169, "character": 4}} [{"targetUri": "file://hover.lean", "targetSelectionRange": {"start": {"line": 166, "character": 6}, "end": {"line": 166, "character": 7}}, "targetRange": {"start": {"line": 166, "character": 6}, "end": {"line": 166, "character": 7}}, "originSelectionRange": {"start": {"line": 169, "character": 4}, "end": {"line": 169, "character": 5}}}] {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 169, "character": 4}} {"range": {"start": {"line": 169, "character": 4}, "end": {"line": 169, "character": 5}}, "contents": {"value": "```lean\nx : Nat\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 173, "character": 8}} {"range": {"start": {"line": 173, "character": 8}, "end": {"line": 173, "character": 9}}, "contents": {"value": "```lean\nx : Nat\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 176, "character": 8}} [{"targetUri": "file://hover.lean", "targetSelectionRange": {"start": {"line": 173, "character": 8}, "end": {"line": 173, "character": 9}}, "targetRange": {"start": {"line": 173, "character": 8}, "end": {"line": 173, "character": 9}}, "originSelectionRange": {"start": {"line": 176, "character": 8}, "end": {"line": 176, "character": 9}}}] {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 176, "character": 8}} {"range": {"start": {"line": 176, "character": 8}, "end": {"line": 176, "character": 9}}, "contents": {"value": "```lean\nx : Nat\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 181, "character": 25}} {"range": {"start": {"line": 181, "character": 25}, "end": {"line": 181, "character": 26}}, "contents": {"value": "```lean\nn : Nat\n```", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 184, "character": 2}} null {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 190, "character": 2}} {"range": {"start": {"line": 190, "character": 2}, "end": {"line": 190, "character": 15}}, "contents": {"value": "`· tac` focuses on the main goal and tries to solve it using `tac`, or else fails. ", "kind": "markdown"}} {"textDocument": {"uri": "file://hover.lean"}, "position": {"line": 193, "character": 28}} {"range": {"start": {"line": 193, "character": 27}, "end": {"line": 193, "character": 32}}, "contents": {"value": "```lean\nId Nat\n```\n***\nYou can use parentheses for\n- Grouping expressions, e.g., `a * (b + c)`.\n- Creating tuples, e.g., `(a, b, c)` is notation for `Prod.mk a (Prod.mk b c)`.\n- Performing type ascription, e.g., `(0 : Int)` instructs Lean to process `0` as a value of type `Int`.\n- Creating `Unit.unit`, `()` is just a shorthand for `Unit.unit`.\n- Creating simple functions when combined with `·`. Here are some examples:\n - `(· + 1)` is shorthand for `fun x => x + 1`\n - `(· + ·)` is shorthand for `fun x y => x + y`\n - `(f · a b)` is shorthand for `fun x => f x a b`\n - `(h (· + 1) ·)` is shorthand for `fun x => h (fun y => y + 1) x`\n", "kind": "markdown"}}