style: correct typos in comments and documentation (#13181)

This PR fixes 15 spelling errors across 8 files in source comments,
docstrings,
and Lake CLI help text. No functional code changes.

Typos corrected: `auxillary` → `auxiliary`, `occurence`/`occuring` →
`occurrence`/`occurring`, `paramaters` → `parameters`, `precendence` →
`precedence`, `similiar` → `similar`, `contianing` → `containing`,
`specifed` → `specified`, `sythesized` → `synthesized`, `enviroment` →
`environment`. Also fixes grammar in `lake cache put` help text
(`used via be specified` → `used can be specified`, `Lake will used` →
`Lake will use`).

Files changed:
- `src/Lean/Elab/Coinductive.lean` — auxillary, occurence (×2),
paramaters
- `src/Lean/Server/FileWorker/SemanticHighlighting.lean` — precendence
(×2)
- `src/Lean/Compiler/LCNF/CoalesceRC.lean` — similiar
- `src/Lean/Compiler/LCNF/ExpandResetReuse.lean` — occuring, contianing
- `src/Lean/LibrarySuggestions/Basic.lean` — occuring (×2)
- `src/Lean/Meta/Tactic/Simp/Rewrite.lean` — sythesized
- `src/lake/Lake/CLI/Help.lean` — specifed (×2), grammar fixes
- `src/lake/Lake/CLI/Main.lean` — enviroment

Closes #13182

src/Lean/Compiler/LCNF/CoalesceRC.lean

@@ -21,7 +21,7 @@ Within a basic block, it is always safe to:
   until the later inc) and thus doing all relevant `inc` in the beginning doesn't change
   semantics.
 - Move all decrements on a variable to the last `dec` location (summing the counts). Because the
-  value is guaranteed to stay alive until at least the last `dec` anyway so a similiar argument to
+  value is guaranteed to stay alive until at least the last `dec` anyway so a similar argument to
   `inc` holds.
 
 Crucially this pass must be placed after `expandResetReuse` as that one relies on `inc`s still being

src/Lean/Compiler/LCNF/ExpandResetReuse.lean

@@ -69,8 +69,8 @@ open ImpureType
 abbrev Mask := Array (Option FVarId)
 
 /--
-Try to erase `inc` instructions on projections of `targetId` occuring in the tail of `ds`.
-Return the updated `ds` and mask contianing the `FVarId`s whose `inc` was removed.
+Try to erase `inc` instructions on projections of `targetId` occurring in the tail of `ds`.
+Return the updated `ds` and mask containing the `FVarId`s whose `inc` was removed.
 -/
 partial def eraseProjIncFor (nFields : Nat) (targetId : FVarId) (ds : Array (CodeDecl .impure)) :
     CompilerM (Array (CodeDecl .impure) × Mask) := do

src/Lean/Elab/Coinductive.lean

@@ -43,7 +43,7 @@ builtin_initialize
 
   Upon such rewrite, the code for adding flat inductives does not diverge much from the usual
   way its done for inductive declarations, but we omit applying attributes/modifiers and
-  we do not set the syntax references to track those declarations (as this is auxillary piece of
+  we do not set the syntax references to track those declarations (as this is auxiliary piece of
   data hidden from the user).
 
   Then, upon adding such flat inductives for each definition in the mutual block to the environment,
@@ -345,7 +345,7 @@ private def mkCasesOnCoinductive (infos : Array InductiveVal) : MetaM Unit := do
         | throwError "expected to be quantifier"
       let motiveMVar ← mkFreshExprMVar type
       /-
-        We intro all the indices and the occurence of the coinductive predicate
+        We intro all the indices and the occurrence of the coinductive predicate
       -/
       let (fvars, subgoal) ← motiveMVar.mvarId!.introN (info.numIndices + 1)
       subgoal.withContext do
@@ -373,7 +373,7 @@ private def mkCasesOnCoinductive (infos : Array InductiveVal) : MetaM Unit := do
           -/
           let originalCasesOn := mkAppN originalCasesOn indices
           /-
-            The next argument is the occurence of the coinductive predicate.
+            The next argument is the occurrence of the coinductive predicate.
             The original `casesOn` of the flat inductive mentions it in
             unrolled form, so we need to rewrite it.
           -/
@@ -447,7 +447,7 @@ public def elabCoinductive (coinductiveElabData : Array CoinductiveElabData) : T
   let consts := namesAndTypes.map fun (name, _) => (mkConst name levelParams)
   /-
     We create values of each of PreDefinitions, by taking existential (see `Meta.SumOfProducts`)
-    form of the associated flat inductives and applying paramaters, as well as recursive calls
+    form of the associated flat inductives and applying parameters, as well as recursive calls
     (with their parameters passed).
   -/
   let preDefVals ← forallBoundedTelescope infos[0]!.type originalNumParams fun params _ => do

src/Lean/LibrarySuggestions/Basic.lean

@@ -91,10 +91,10 @@ end FoldRelevantConstantsImpl
 @[implemented_by FoldRelevantConstantsImpl.foldUnsafe]
 public opaque foldRelevantConstants {α : Type} (e : Expr) (init : α) (f : Name → α → MetaM α) : MetaM α := pure init
 
-/-- Collect the constants occuring in `e` (once each), skipping instance arguments and proofs. -/
+/-- Collect the constants occurring in `e` (once each), skipping instance arguments and proofs. -/
 public def relevantConstants (e : Expr) : MetaM (Array Name) := foldRelevantConstants e #[] (fun n ns => return ns.push n)
 
-/-- Collect the constants occuring in `e` (once each), skipping instance arguments and proofs. -/
+/-- Collect the constants occurring in `e` (once each), skipping instance arguments and proofs. -/
 public def relevantConstantsAsSet (e : Expr) : MetaM NameSet := foldRelevantConstants e ∅ (fun n ns => return ns.insert n)
 
 end Lean.Expr

src/Lean/Meta/Tactic/Simp/Rewrite.lean

@@ -99,7 +99,7 @@ where
       if (← withReducibleAndInstances <| isDefEq x val) then
         return true
       else
-        trace[Meta.Tactic.simp.discharge] "{← ppOrigin thmId}, failed to assign instance{indentExpr type}\nsythesized value{indentExpr val}\nis not definitionally equal to{indentExpr x}"
+        trace[Meta.Tactic.simp.discharge] "{← ppOrigin thmId}, failed to assign instance{indentExpr type}\nsynthesized value{indentExpr val}\nis not definitionally equal to{indentExpr x}"
         return false
     | _ =>
       trace[Meta.Tactic.simp.discharge] "{← ppOrigin thmId}, failed to synthesize instance{indentExpr type}"

src/Lean/Server/FileWorker/SemanticHighlighting.lean

@@ -46,7 +46,7 @@ structure LeanSemanticToken where
   stx  : Syntax
   /-- Type of the semantic token. -/
   type : SemanticTokenType
-  /-- In case of overlap, higher-priority tokens will take precendence -/
+  /-- In case of overlap, higher-priority tokens will take precedence -/
   priority : Nat := 5
 
 /-- Semantic token information with absolute LSP positions. -/
@@ -57,7 +57,7 @@ structure AbsoluteLspSemanticToken where
   tailPos : Lsp.Position
   /-- Start position of the semantic token. -/
   type    : SemanticTokenType
-  /-- In case of overlap, higher-priority tokens will take precendence -/
+  /-- In case of overlap, higher-priority tokens will take precedence -/
   priority : Nat := 5
   deriving BEq, Hashable, FromJson, ToJson
 

src/lake/Lake/CLI/Help.lean

@@ -386,7 +386,7 @@ OPTIONS:
   --force-download                redownload existing files
 
 Downloads build outputs for packages in the workspace from a remote cache
-service. The cache service used can be specifed via the `--service` option.
+service. The cache service used can be specified via the `--service` option.
 Otherwise, Lake will the system default, or, if none is configured, Reservoir.
 See `lake cache services` for more information on how to configure services.
 
@@ -429,7 +429,7 @@ USAGE:
 
 Uploads the input-to-output mappings contained in the specified file along
 with the corresponding output artifacts to a remote cache. The cache service
-used via be specified via `--service` option. If not specifed, Lake will used
+used can be specified via the `--service` option. If not specified, Lake will use
 the system default, or error if none is configured. See the help page of
 `lake cache services` for more information on how to configure services.
 

src/lake/Lake/CLI/Main.lean

@@ -446,7 +446,7 @@ protected def get : CliM PUnit := do
           logWarning endpointDeprecation
           if opts.mappingsOnly then
             error "`--mappings-only` requires services to be configured
-              via the Lake system configuration (not enviroment variables)"
+              via the Lake system configuration (not environment variables)"
           return .downloadService artifactEndpoint revisionEndpoint ws.lakeEnv.cacheService?
         | none, none =>
             return ws.defaultCacheService