542 lines
28 KiB
Text
542 lines
28 KiB
Text
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"```lean\nTrue.intro : True\n```\n***\n`True` is true, and `True.intro` (or more commonly, `trivial`)\nis the proof. \n***\n*import Init.Prelude*",
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"```lean\nTrue.intro : True\n```\n***\n`True` is true, and `True.intro` (or more commonly, `trivial`)\nis the proof. \n***\n*import Init.Prelude*",
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"kind": "markdown"}}
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"```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`. \n***\n*import Init.Prelude*",
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"kind": "markdown"}}
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"contents": {"value": "My tactic ", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"contents": {"value": "My tactic ", "kind": "markdown"}}
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"contents":
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{"value":
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"```lean\nTrue.intro : True\n```\n***\n`True` is true, and `True.intro` (or more commonly, `trivial`)\nis the proof. \n***\n*import Init.Prelude*",
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"kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"contents":
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{"value":
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"```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\n***\n*import Init.Notation*",
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"kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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{"range":
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{"start": {"line": 28, "character": 32}, "end": {"line": 28, "character": 36}},
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"contents":
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{"value":
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"```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. \n***\n*import Init.Notation*",
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"kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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{"start": {"line": 31, "character": 6}, "end": {"line": 31, "character": 12}},
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"contents":
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{"value":
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"```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\n***\n*import Init.Notation*",
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"kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"contents":
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{"value":
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"```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. \n***\n*import Init.Notation*",
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"kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"contents": {"value": "My tactic ", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 44, "character": 2}}
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{"range":
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{"start": {"line": 44, "character": 2}, "end": {"line": 44, "character": 23}},
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"contents": {"value": "My tactic ", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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{"range":
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{"start": {"line": 49, "character": 15}, "end": {"line": 49, "character": 21}},
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"contents":
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{"value": "```lean\nmyNota : Lean.ParserDescr\n```\n***\nMy notation ",
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"kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"contents":
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{"value": "```lean\n1 : Nat\n```\n***\nMy notation ", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 59, "character": 7}}
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{"range":
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{"start": {"line": 59, "character": 7}, "end": {"line": 59, "character": 15}},
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"contents":
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{"value": "```lean\nNat\n```\n***\nMy notation ", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 69, "character": 7}}
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"contents":
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{"value": "```lean\nNat\n```\n***\nMy ultimate notation ", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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{"start": {"line": 73, "character": 18}, "end": {"line": 73, "character": 25}},
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"contents":
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{"value": "```lean\nmyInfix : Lean.TrailingParserDescr\n```",
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"kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"contents":
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"```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\n***\n*import Init.Prelude*",
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"kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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{"start": {"line": 77, "character": 7}, "end": {"line": 77, "character": 14}},
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"contents":
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{"value":
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"```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",
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"kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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{"start": {"line": 82, "character": 7}, "end": {"line": 82, "character": 8}},
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"contents":
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{"value":
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"```lean\nNat : Type\n```\n***\nThe type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n induction n with\n | zero =>\n show 0 < 1\n decide\n | succ i ih => -- ih : i < succ i\n show succ i < succ (succ i)\n exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n linked list with `n` links, which is horribly inefficient. Instead, the\n runtime itself has a special representation for `Nat` which stores numbers up\n to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n library (usually [GMP](https://gmplib.org/)).\n\n***\n*import Init.Prelude*",
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"kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 87, "character": 14}}
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{"start": {"line": 87, "character": 14}, "end": {"line": 87, "character": 36}},
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"contents":
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{"value":
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"```lean\nLean.Parser.Term.doSeq : Lean.Parser.Parser\n```\n***\n*import Lean.Parser.Do*",
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"kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 89, "character": 30}}
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{"start": {"line": 89, "character": 29}, "end": {"line": 89, "character": 34}},
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"contents":
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{"value":
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"```lean\nLean.Parser.Term.doSeq : Lean.Parser.Parser\n```\n***\n*import Lean.Parser.Do*",
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"kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 92, "character": 2}}
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{"range":
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{"start": {"line": 92, "character": 0}, "end": {"line": 92, "character": 7}},
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"contents": {"value": "My command ", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 99, "character": 2}}
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{"range":
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{"start": {"line": 99, "character": 0}, "end": {"line": 99, "character": 7}},
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"contents": {"value": "My command ", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 102, "character": 16}}
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{"range":
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{"start": {"line": 102, "character": 16},
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"end": {"line": 102, "character": 23}},
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"contents":
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{"value":
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"```lean\nLean.Parser.ppSpace : Lean.Parser.Parser\n```\n***\nNo-op parser that advises the pretty printer to emit a space/soft line break. \n***\n*import Lean.Parser.Extra*",
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"kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 102, "character": 24}}
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{"range":
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{"start": {"line": 102, "character": 24},
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"end": {"line": 102, "character": 31}},
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"contents":
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{"value":
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"```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`. \n***\n*import Init.Prelude*",
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"kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 102, "character": 31}}
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{"range":
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{"start": {"line": 102, "character": 31},
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"end": {"line": 102, "character": 35}},
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"contents":
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{"value":
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"```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. \n***\n*import Init.Notation*",
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"kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 111, "character": 2}}
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{"range":
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{"start": {"line": 111, "character": 0}, "end": {"line": 111, "character": 8}},
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"contents": {"value": "My ultimate command ", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 115, "character": 10}}
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{"range":
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{"start": {"line": 115, "character": 8},
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"end": {"line": 115, "character": 14}},
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"contents":
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{"value":
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"Structure instance. `{ x := e, ... }` assigns `e` to field `x`, which may be\ninherited. If `e` is itself a variable called `x`, it can be elided:\n`fun y => { x := 1, y }`.\nA *structure update* of an existing value can be given via `with`:\n`{ point with x := 1 }`.\nThe structure type can be specified if not inferable:\n`{ x := 1, y := 2 : Point }`.\n",
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"kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 119, "character": 8}}
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{"range":
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{"start": {"line": 119, "character": 8},
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"end": {"line": 119, "character": 10}},
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"contents":
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{"value":
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"```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\n***\n*import Init.Prelude*",
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"kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 119, "character": 10}}
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{"range":
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{"start": {"line": 119, "character": 8},
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"end": {"line": 119, "character": 21}},
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"contents": {"value": "```lean\nTrue\n```", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 126, "character": 2}}
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{"range":
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{"start": {"line": 126, "character": 2}, "end": {"line": 126, "character": 3}},
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"contents": {"value": "```lean\nn : Id ℕ\n```", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"end": {"line": 133, "character": 17}},
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"contents": {"value": "```lean\nfoo : ℕ\n```", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 138, "character": 7}}
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{"range":
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{"start": {"line": 138, "character": 7},
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"end": {"line": 138, "character": 10}},
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"contents": {"value": "```lean\nBar.foo : ℕ\n```", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 141, "character": 9}}
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{"range":
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{"start": {"line": 141, "character": 7},
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"end": {"line": 141, "character": 17}},
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"contents": {"value": "```lean\n_root_.foo : ℕ\n```", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 144, "character": 4}}
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||
{"range":
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{"start": {"line": 144, "character": 4}, "end": {"line": 144, "character": 7}},
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"contents": {"value": "```lean\nBar.bar : ℕ\n```", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 147, "character": 10}}
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{"range":
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{"start": {"line": 147, "character": 10},
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"end": {"line": 147, "character": 13}},
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"contents": {"value": "```lean\nBar.Foo : Type\n```", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 147, "character": 17}}
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{"range":
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{"start": {"line": 147, "character": 17},
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"end": {"line": 147, "character": 19}},
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"contents":
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{"value": "```lean\nBar.Foo.mk : ℕ → Foo\n```", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 150, "character": 2}}
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{"range":
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{"start": {"line": 150, "character": 2}, "end": {"line": 150, "character": 4}},
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"contents":
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{"value": "```lean\nBar.Foo.hi : Foo → ℕ\n```", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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||
"position": {"line": 153, "character": 10}}
|
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{"range":
|
||
{"start": {"line": 153, "character": 10},
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"end": {"line": 153, "character": 13}},
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"contents": {"value": "```lean\nBar.Bar : Type\n```", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 155, "character": 4}}
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{"range":
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{"start": {"line": 155, "character": 4}, "end": {"line": 155, "character": 6}},
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"contents": {"value": "```lean\nBar.Bar.mk : Bar\n```", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 158, "character": 2}}
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{"range":
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{"start": {"line": 158, "character": 0}, "end": {"line": 158, "character": 8}},
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"contents":
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{"value": "```lean\nBar.instToStringNat : ToString ℕ\n```",
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"kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 160, "character": 9}}
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{"range":
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{"start": {"line": 160, "character": 9},
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"end": {"line": 160, "character": 10}},
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"contents": {"value": "```lean\nBar.f : ToString ℕ\n```", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
|
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"position": {"line": 163, "character": 10}}
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{"range":
|
||
{"start": {"line": 163, "character": 10},
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"end": {"line": 163, "character": 16}},
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"contents":
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{"value": "A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. ",
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"kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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||
"position": {"line": 166, "character": 4}}
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{"range":
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{"start": {"line": 166, "character": 4},
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||
"end": {"line": 166, "character": 11}},
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"contents": {"value": "```lean\nBar.foo.bar : ℕ\n```", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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||
"position": {"line": 166, "character": 8}}
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{"range":
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{"start": {"line": 166, "character": 4},
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"end": {"line": 166, "character": 11}},
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"contents": {"value": "```lean\nBar.foo.bar : ℕ\n```", "kind": "markdown"}}
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"contents": {"value": "```lean\nx : ℕ\n```", "kind": "markdown"}}
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"position": {"line": 174, "character": 4}}
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{"start": {"line": 174, "character": 4}, "end": {"line": 174, "character": 5}},
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"contents": {"value": "```lean\nx : ℕ\n```", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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{"range":
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{"start": {"line": 178, "character": 11},
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"end": {"line": 178, "character": 33}},
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"contents":
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{"value": "enable the 'unused variables' linter", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 181, "character": 8}}
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{"range":
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"contents": {"value": "```lean\nx : ℕ\n```", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 184, "character": 8}}
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[{"targetUri": "file://hover.lean",
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"position": {"line": 184, "character": 8}}
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"contents": {"value": "```lean\nx : ℕ\n```", "kind": "markdown"}}
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"end": {"line": 189, "character": 26}},
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"contents": {"value": "```lean\nn : ℕ\n```", "kind": "markdown"}}
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"position": {"line": 192, "character": 2}}
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null
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 198, "character": 2}}
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"end": {"line": 198, "character": 15}},
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"contents":
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{"value":
|
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"`· tac` focuses on the main goal and tries to solve it using `tac`, or else fails. ",
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"kind": "markdown"}}
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"end": {"line": 201, "character": 32}},
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"contents":
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{"value":
|
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"```lean\nId ℕ\n```\n***\nParentheses, used for grouping expressions (e.g., `a * (b + c)`).\nCan also be used for 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 - also applies to other parentheses-like notations such as `(·, 1)`\n",
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"kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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{"range":
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{"start": {"line": 204, "character": 8}, "end": {"line": 204, "character": 9}},
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"contents": {"value": "```lean\n?m\n```", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 204, "character": 10}}
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{"range":
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{"start": {"line": 204, "character": 8},
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"end": {"line": 204, "character": 13}},
|
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"contents":
|
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{"value":
|
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"```lean\n?m x✝¹ x✝\n```\n***\n`a + b` computes the sum of `a` and `b`.\nThe meaning of this notation is type-dependent. ",
|
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"kind": "markdown"}}
|
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{"textDocument": {"uri": "file://hover.lean"},
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"position": {"line": 212, "character": 28}}
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"contents": {"value": "```lean\nx : α\n```", "kind": "markdown"}}
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{"textDocument": {"uri": "file://hover.lean"},
|
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{"range":
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|
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"end": {"line": 214, "character": 29}},
|
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"contents": {"value": "```lean\nα\n```", "kind": "markdown"}}
|
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"position": {"line": 216, "character": 31}}
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|
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"end": {"line": 216, "character": 32}},
|
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"contents": {"value": "```lean\nx : α\n```", "kind": "markdown"}}
|
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{"textDocument": {"uri": "file://hover.lean"},
|
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"position": {"line": 218, "character": 31}}
|
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{"range":
|
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{"start": {"line": 218, "character": 31},
|
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"end": {"line": 218, "character": 32}},
|
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"contents": {"value": "```lean\nα\n```", "kind": "markdown"}}
|
||
{"textDocument": {"uri": "file://hover.lean"},
|
||
"position": {"line": 222, "character": 8}}
|
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{"range":
|
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{"start": {"line": 222, "character": 8}, "end": {"line": 222, "character": 9}},
|
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"contents": {"value": "```lean\nx : ℕ\n```", "kind": "markdown"}}
|
||
{"textDocument": {"uri": "file://hover.lean"},
|
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"position": {"line": 225, "character": 4}}
|
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{"range":
|
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{"start": {"line": 225, "character": 4}, "end": {"line": 225, "character": 8}},
|
||
"contents":
|
||
{"value":
|
||
"```lean\nNat.zero : ℕ\n```\n***\n`Nat.zero`, normally written `0 : Nat`, is the smallest natural number.\nThis is one of the two constructors of `Nat`. \n***\n*import Init.Prelude*",
|
||
"kind": "markdown"}}
|
||
{"textDocument": {"uri": "file://hover.lean"},
|
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"position": {"line": 228, "character": 4}}
|
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{"range":
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{"start": {"line": 228, "character": 4}, "end": {"line": 228, "character": 8}},
|
||
"contents":
|
||
{"value":
|
||
"```lean\nNat.succ : ℕ → ℕ\n```\n***\nThe successor function on natural numbers, `succ n = n + 1`.\nThis is one of the two constructors of `Nat`. \n***\n*import Init.Prelude*",
|
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"kind": "markdown"}}
|
||
{"textDocument": {"uri": "file://hover.lean"},
|
||
"position": {"line": 228, "character": 9}}
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{"range":
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|
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"end": {"line": 228, "character": 10}},
|
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"contents": {"value": "```lean\nx : ℕ\n```", "kind": "markdown"}}
|
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{"textDocument": {"uri": "file://hover.lean"},
|
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"position": {"line": 232, "character": 8}}
|
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{"range":
|
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{"start": {"line": 232, "character": 8}, "end": {"line": 232, "character": 9}},
|
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"contents": {"value": "```lean\nx : ℕ\n```", "kind": "markdown"}}
|
||
{"textDocument": {"uri": "file://hover.lean"},
|
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"position": {"line": 234, "character": 12}}
|
||
{"range":
|
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{"start": {"line": 234, "character": 12},
|
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"end": {"line": 234, "character": 13}},
|
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"contents": {"value": "```lean\nx : ℕ\n```", "kind": "markdown"}}
|
||
{"textDocument": {"uri": "file://hover.lean"},
|
||
"position": {"line": 236, "character": 4}}
|
||
{"range":
|
||
{"start": {"line": 236, "character": 4}, "end": {"line": 236, "character": 8}},
|
||
"contents":
|
||
{"value":
|
||
"```lean\nNat.zero : ℕ\n```\n***\n`Nat.zero`, normally written `0 : Nat`, is the smallest natural number.\nThis is one of the two constructors of `Nat`. \n***\n*import Init.Prelude*",
|
||
"kind": "markdown"}}
|
||
{"textDocument": {"uri": "file://hover.lean"},
|
||
"position": {"line": 239, "character": 9}}
|
||
{"range":
|
||
{"start": {"line": 239, "character": 9},
|
||
"end": {"line": 239, "character": 10}},
|
||
"contents":
|
||
{"value":
|
||
"```lean\nℕ\n```\n***\nA placeholder term, to be synthesized by unification. ",
|
||
"kind": "markdown"}}
|
||
{"textDocument": {"uri": "file://hover.lean"},
|
||
"position": {"line": 239, "character": 11}}
|
||
{"range":
|
||
{"start": {"line": 239, "character": 11},
|
||
"end": {"line": 239, "character": 13}},
|
||
"contents": {"value": "```lean\nih : True\n```", "kind": "markdown"}}
|
||
{"textDocument": {"uri": "file://hover.lean"},
|
||
"position": {"line": 244, "character": 6}}
|
||
{"range":
|
||
{"start": {"line": 244, "character": 4}, "end": {"line": 244, "character": 9}},
|
||
"contents":
|
||
{"value":
|
||
"```lean\nNat.zero : ℕ\n```\n***\n`Nat.zero`, normally written `0 : Nat`, is the smallest natural number.\nThis is one of the two constructors of `Nat`. \n***\n*import Init.Prelude*",
|
||
"kind": "markdown"}}
|
||
{"textDocument": {"uri": "file://hover.lean"},
|
||
"position": {"line": 244, "character": 15}}
|
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{"range":
|
||
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|
||
"end": {"line": 244, "character": 18}},
|
||
"contents":
|
||
{"value":
|
||
"```lean\nNat.zero : ℕ\n```\n***\n`Nat.zero`, normally written `0 : Nat`, is the smallest natural number.\nThis is one of the two constructors of `Nat`. \n***\n*import Init.Prelude*",
|
||
"kind": "markdown"}}
|
||
{"textDocument": {"uri": "file://hover.lean"},
|
||
"position": {"line": 247, "character": 6}}
|
||
{"range":
|
||
{"start": {"line": 247, "character": 4}, "end": {"line": 247, "character": 9}},
|
||
"contents":
|
||
{"value":
|
||
"```lean\nNat.succ : ℕ → ℕ\n```\n***\nThe successor function on natural numbers, `succ n = n + 1`.\nThis is one of the two constructors of `Nat`. \n***\n*import Init.Prelude*",
|
||
"kind": "markdown"}}
|
||
{"textDocument": {"uri": "file://hover.lean"},
|
||
"position": {"line": 247, "character": 17}}
|
||
{"range":
|
||
{"start": {"line": 247, "character": 15},
|
||
"end": {"line": 247, "character": 20}},
|
||
"contents":
|
||
{"value":
|
||
"```lean\nNat.succ : ℕ → ℕ\n```\n***\nThe successor function on natural numbers, `succ n = n + 1`.\nThis is one of the two constructors of `Nat`. \n***\n*import Init.Prelude*",
|
||
"kind": "markdown"}}
|
||
{"textDocument": {"uri": "file://hover.lean"},
|
||
"position": {"line": 250, "character": 27}}
|
||
{"range":
|
||
{"start": {"line": 250, "character": 27},
|
||
"end": {"line": 250, "character": 37}},
|
||
"contents":
|
||
{"value":
|
||
"```lean\nInhabited.mk.{1} : {α : Type} → α → Inhabited α\n```\n***\n*import Init.Prelude*",
|
||
"kind": "markdown"}}
|
||
{"textDocument": {"uri": "file://hover.lean"},
|
||
"position": {"line": 250, "character": 28}}
|
||
{"range":
|
||
{"start": {"line": 250, "character": 28},
|
||
"end": {"line": 250, "character": 36}},
|
||
"contents":
|
||
{"value":
|
||
"```lean\nNat.zero : ℕ\n```\n***\n`Nat.zero`, normally written `0 : Nat`, is the smallest natural number.\nThis is one of the two constructors of `Nat`. \n***\n*import Init.Prelude*",
|
||
"kind": "markdown"}}
|