diff --git a/src/Iris/BI/Lib/BUpdPlain.lean b/src/Iris/BI/Lib/BUpdPlain.lean
index b525eeb8..1cdd96e2 100644
--- a/src/Iris/BI/Lib/BUpdPlain.lean
+++ b/src/Iris/BI/Lib/BUpdPlain.lean
@@ -34,8 +34,7 @@ theorem BUpdPlain_intro {P : PROP} : P ⊢ BUpdPlain P := by
   iintro Hp
   unfold BUpdPlain
   iintro _ H
-  iapply H
-  iexact Hp
+  iapply H $$ Hp
 
 theorem BUpdPlain_mono {P Q : PROP} : (P ⊢ Q) → (BUpdPlain P ⊢ BUpdPlain Q) := by
   intros H
@@ -43,27 +42,24 @@ theorem BUpdPlain_mono {P Q : PROP} : (P ⊢ Q) → (BUpdPlain P ⊢ BUpdPlain Q
   iintro R HQR
   iintro Hp
   have H1 : ⊢ iprop(Q -∗ ■ HQR) -∗ iprop(P -∗ ■ HQR) := by
-    iintro H
-    iintro Hp
+    iintro H Hp
     iapply H
     apply H
   iintro ⟨Ha, H2⟩
-  iapply Ha $! HQR
-  iapply H1
-  iexact H2
+  iapply Ha
+  iapply H1 $$ H2
 
 theorem BUpdPlain_idemp {P : PROP} : BUpdPlain (BUpdPlain P) ⊢ BUpdPlain P := by
   unfold BUpdPlain
   iintro Hp R H
-  iapply Hp $! R
+  iapply Hp
   iintro Hp
-  iapply Hp $! R
-  iassumption
+  iapply Hp $$ H
 
 theorem BUpdPlain_frame_r {P Q : PROP} : BUpdPlain P ∗ Q ⊢ (BUpdPlain iprop(P ∗ Q)) := by
   unfold BUpdPlain
   iintro ⟨Hp, Hq⟩ R H
-  iapply Hp $! R
+  iapply Hp
   iintro Hp
   iapply H
   isplitl [Hp]
@@ -73,7 +69,7 @@ theorem BUpdPlain_frame_r {P Q : PROP} : BUpdPlain P ∗ Q ⊢ (BUpdPlain iprop(
 theorem BUpdPlain_plainly {P : PROP} : BUpdPlain iprop(■ P) ⊢ (■ P) := by
   unfold BUpdPlain
   iintro H
-  iapply H $! P
+  iapply H
   exact wand_rfl
 
 /- BiBUpdPlainly entails the alternative definition -/
diff --git a/src/Iris/Examples/Proofs.lean b/src/Iris/Examples/Proofs.lean
index e61cb97c..106fac2e 100644
--- a/src/Iris/Examples/Proofs.lean
+++ b/src/Iris/Examples/Proofs.lean
@@ -13,7 +13,7 @@ theorem proof_example_1 [BI PROP] (P Q R : PROP) (Φ : α → PROP) :
   P ∗ Q ∗ □ R ⊢ □ (R -∗ ∃ x, Φ x) -∗ ∃ x, Φ x ∗ P ∗ Q
 := by
   iintro ⟨HP, HQ, □HR⟩ □HRΦ
-  ihave HΦ := HRΦ with HR
+  ihave HΦ := HRΦ $$ HR
   icases HΦ with ⟨x, _HΦ⟩
   iexists x
   isplitr
diff --git a/src/Iris/ProofMode/Goals.lean b/src/Iris/ProofMode/Goals.lean
index 80a9ba77..2ac3ce6e 100644
--- a/src/Iris/ProofMode/Goals.lean
+++ b/src/Iris/ProofMode/Goals.lean
@@ -34,10 +34,12 @@ def Goals.addPureGoal {prop : Q(Type u)} {bi : Q(BI $prop)} (g : Goals bi)
   g.goals.modify (·.push m)
 
 -- TODO: change this to replace main goal that deduplicates goals
-def Goals.getGoals {prop : Q(Type u)} {bi : Q(BI $prop)} (g : Goals bi) : MetaM (List MVarId) := do
+def Goals.getGoals {prop : Q(Type u)} {bi : Q(BI $prop)} (g : Goals bi) : TermElabM (List MVarId) := do
   let goals ← g.goals.get
   -- put the goals that depend on other goals last
   let mvars ← goals.foldlM (λ m g => do
     return m ∪ (← g.getMVarDependencies)) ∅
   let (dep, nonDep) := goals.partition (λ x => mvars.contains x)
+  -- TODO: Is this the right place to do this?
+  Term.synthesizeSyntheticMVarsNoPostponing
   return (nonDep ++ dep).toList
diff --git a/src/Iris/ProofMode/Instances.lean b/src/Iris/ProofMode/Instances.lean
index f7ea079e..c191e3c4 100644
--- a/src/Iris/ProofMode/Instances.lean
+++ b/src/Iris/ProofMode/Instances.lean
@@ -86,6 +86,13 @@ instance intoForall_intuitionistically [BI PROP] (P : PROP) (Φ : α → PROP)
     [h : IntoForall P Φ] : IntoForall iprop(□ P) (fun a => iprop(□ (Φ a))) where
   into_forall := (intuitionistically_mono h.1).trans intuitionistically_forall_1
 
+instance intoForall_wand_pure [BI PROP] (P Q : PROP) Φ
+    [h : FromPure a P Φ] : IntoForall iprop(P -∗ Q) (fun _ : Φ => Q) where
+  into_forall := forall_intro λ hΦ =>
+    emp_sep.2.trans <| (sep_mono_l <|
+      (affinelyIf_emp.mpr.trans (affinelyIf_mono (pure_intro hΦ))).trans
+        h.1).trans wand_elim_r
+
 -- FromExists
 instance (priority := default + 10) fromExists_exists [BI PROP] (Φ : α → PROP) :
     FromExists iprop(∃ a, Φ a) Φ := ⟨.rfl⟩
diff --git a/src/Iris/ProofMode/Patterns/ProofModeTerm.lean b/src/Iris/ProofMode/Patterns/ProofModeTerm.lean
index 0b45b19d..17fdc605 100644
--- a/src/Iris/ProofMode/Patterns/ProofModeTerm.lean
+++ b/src/Iris/ProofMode/Patterns/ProofModeTerm.lean
@@ -11,23 +11,17 @@ open Lean
 declare_syntax_cat pmTerm
 
 syntax term : pmTerm
-syntax term "with" specPat,+ : pmTerm
-syntax term "$!" term,+ : pmTerm
-syntax term "$!" term,+ "with" specPat,+ : pmTerm
+syntax term "$$" specPat,+ : pmTerm
 
 structure PMTerm where
   term : Term
-  terms : List Term
   spats : List SpecPat
   deriving Repr, Inhabited
 
 partial def PMTerm.parse (term : Syntax) : MacroM PMTerm := do
   match ← expandMacros term with
-  | `(pmTerm| $trm:term) => return ⟨trm, [], []⟩
-  | `(pmTerm| $trm:term with $spats,*) => return ⟨trm, [], ← parseSpats spats⟩
-  | `(pmTerm| $trm:term $! $ts,*) => return ⟨trm, ts.getElems.toList, []⟩
-  | `(pmTerm| $trm:term $! $ts,* with $spats,*) =>
-    return ⟨trm, ts.getElems.toList, ← parseSpats spats⟩
+  | `(pmTerm| $trm:term) => return ⟨trm, []⟩
+  | `(pmTerm| $trm:term $$ $spats,*) => return ⟨trm, ← parseSpats spats⟩
   | _ => Macro.throwUnsupported
 where
   parseSpats (spats : Syntax.TSepArray `specPat ",") : MacroM (List SpecPat) :=
diff --git a/src/Iris/ProofMode/Patterns/SpecPattern.lean b/src/Iris/ProofMode/Patterns/SpecPattern.lean
index c118734b..a482ca60 100644
--- a/src/Iris/ProofMode/Patterns/SpecPattern.lean
+++ b/src/Iris/ProofMode/Patterns/SpecPattern.lean
@@ -9,11 +9,13 @@ open Lean
 declare_syntax_cat specPat
 
 syntax ident : specPat
+syntax "%" term : specPat
 syntax "[" ident,* "]" optional(" as " ident) : specPat
 
 -- see https://gitlab.mpi-sws.org/iris/iris/-/blob/master/iris/proofmode/spec_patterns.v?ref_type=heads#L15
 inductive SpecPat
   | ident (name : Ident)
+  | pure (t : Term)
   | goal (names : List Ident) (goalName : Name)
   deriving Repr, Inhabited
 
@@ -24,14 +26,9 @@ partial def SpecPat.parse (pat : Syntax) : MacroM SpecPat := do
 where
   go : TSyntax `specPat → Option SpecPat
   | `(specPat| $name:ident) => some <| .ident name
+  | `(specPat| % $term:term) => some <| .pure term
   | `(specPat| [$[$names:ident],*]) => some <| .goal names.toList .anonymous
   | `(specPat| [$[$names:ident],*] as $goal:ident) => match goal.raw with
     | .ident _ _ val _ => some <| .goal names.toList val
     | _ => none
   | _ => none
-
-def headName (spats : List SpecPat) : Name :=
-  match spats.head? with
-    | some <| .ident _ => .anonymous
-    | some <| .goal _ name => name
-    | _ => .anonymous
diff --git a/src/Iris/ProofMode/Tactics/Apply.lean b/src/Iris/ProofMode/Tactics/Apply.lean
index 58c36120..105c19ec 100644
--- a/src/Iris/ProofMode/Tactics/Apply.lean
+++ b/src/Iris/ProofMode/Tactics/Apply.lean
@@ -23,7 +23,7 @@ partial def iApplyCore {prop : Q(Type u)} {bi : Q(BI $prop)} (gs : Goals bi) {e}
      let pf' ← gs.addGoal hyps' A
      return q(apply $pf $pf')
 
-  let some ⟨_, hyps'', pf''⟩ ← try? <| iSpecializeCore gs hyps uniq [] [.goal [] .anonymous] | throwError m!"iapply: cannot apply {out} to {goal}"
+  let some ⟨_, hyps'', pf''⟩ ← try? <| iSpecializeCore gs hyps uniq [.goal [] .anonymous] | throwError m!"iapply: cannot apply {out} to {goal}"
   let pf''' ← iApplyCore gs hyps'' goal uniq
   return q($(pf'').trans $pf''')
 
diff --git a/src/Iris/ProofMode/Tactics/Have.lean b/src/Iris/ProofMode/Tactics/Have.lean
index f9d6b169..1c4e97a5 100644
--- a/src/Iris/ProofMode/Tactics/Have.lean
+++ b/src/Iris/ProofMode/Tactics/Have.lean
@@ -47,7 +47,7 @@ private def iHaveCore (gs : @Goals u prop bi) {e} (hyps : Hyps bi e)
 def iHave (gs : @Goals u prop bi) {e} (hyps : Hyps bi e)
   (pmt : PMTerm) (name : TSyntax ``binderIdent) (keep : Bool) (mayPostpone := false) : TacticM (Name × (e' : _) × Hyps bi e' × Q($e ⊢ $e')) := do
   let ⟨uniq, _, hyps', pf⟩ ← iHaveCore gs hyps pmt.term name keep mayPostpone
-  let ⟨_, hyps'', pf'⟩ ← iSpecializeCore gs hyps' uniq pmt.terms pmt.spats
+  let ⟨_, hyps'', pf'⟩ ← iSpecializeCore gs hyps' uniq pmt.spats
   return ⟨uniq, _, hyps'', q($(pf).trans $pf')⟩
 
 elab "ihave" colGt name:binderIdent " := " pmt:pmTerm  : tactic => do
diff --git a/src/Iris/ProofMode/Tactics/Specialize.lean b/src/Iris/ProofMode/Tactics/Specialize.lean
index 70c23eac..082192a7 100644
--- a/src/Iris/ProofMode/Tactics/Specialize.lean
+++ b/src/Iris/ProofMode/Tactics/Specialize.lean
@@ -38,18 +38,6 @@ theorem specialize_forall [BI PROP] {p : Bool} {A1 A2 P : PROP} {α : Sort _} {
     [inst : IntoForall P Φ] (h : A1 ⊢ A2 ∗ □?p P) (a : α) : A1 ⊢ A2 ∗ □?p (Φ a) := by
   refine h.trans <| sep_mono_r <| intuitionisticallyIf_mono <| inst.1.trans (forall_elim a)
 
-def SpecializeState.process_forall (gs : @Goals u prop bi) :
-    @SpecializeState u prop bi orig → Term → TermElabM (SpecializeState bi orig)
-  | { e, hyps, p, out, pf }, arg => do
-    let v ← mkFreshLevelMVar
-    let α : Q(Sort v) ← mkFreshExprMVarQ q(Sort v)
-    let Φ : Q($α → $prop) ← mkFreshExprMVarQ q($α → $prop)
-    let some _ ← ProofMode.trySynthInstanceQAddingGoals gs q(IntoForall $out $Φ) | throwError "ispecialize: {out} is not a forall"
-    let x ← elabTermEnsuringTypeQ (u := .succ .zero) arg α
-    have out' : Q($prop) := Expr.headBeta q($Φ $x)
-    have : $out' =Q $Φ $x := ⟨⟩
-    return { e, hyps, p, out := out', pf := q(specialize_forall $pf $x) }
-
 def SpecializeState.process_wand (gs : @Goals u prop bi) :
     @SpecializeState u prop bi orig → SpecPat → TermElabM (SpecializeState bi orig)
   | { hyps, p, out, pf, .. }, .ident i => do
@@ -64,6 +52,17 @@ def SpecializeState.process_wand (gs : @Goals u prop bi) :
       throwError m!"ispecialize: cannot instantiate {out} with {out₁'}"
     let pf := q(specialize_wand $pf $pf')
     return { e := e', hyps := hyps', p := p2, out := out₂, pf }
+  | { e, hyps, p, out, pf, .. }, .pure t => do
+    let v ← mkFreshLevelMVar
+    let α : Q(Sort v) ← mkFreshExprMVarQ q(Sort v)
+    let Φ : Q($α → $prop) ← mkFreshExprMVarQ q($α → $prop)
+    let some _ ← ProofMode.trySynthInstanceQAddingGoals gs q(IntoForall $out $Φ) | throwError "ispecialize: {out} is not a lean premise"
+    let x ← elabTermEnsuringTypeQ (u := .succ .zero) t α
+    have out' : Q($prop) := Expr.headBeta q($Φ $x)
+    have : $out' =Q $Φ $x := ⟨⟩
+    let newMVarIds ← getMVarsNoDelayed x
+    for mvar in newMVarIds do gs.addPureGoal mvar
+    return { e, hyps, p, out := out', pf := q(specialize_forall $pf $x) }
   | { hyps, p, out, pf, .. }, .goal ns g => do
     let mut uniqs : NameSet := {}
     for name in ns do
@@ -77,12 +76,11 @@ def SpecializeState.process_wand (gs : @Goals u prop bi) :
     return { e := el', hyps := hypsl', p := q(false), out := out₂, pf }
 
 def iSpecializeCore (gs : @Goals u prop bi) {e} (hyps : Hyps bi e)
-  (uniq : Name) (alls : List Term) (spats : List SpecPat) : TacticM ((e' : _) × Hyps bi e' × Q($e ⊢ $e')) := do
+  (uniq : Name) (spats : List SpecPat) : TacticM ((e' : _) × Hyps bi e' × Q($e ⊢ $e')) := do
   let some ⟨name, _, hyps, _, out, p, _, pf⟩ := Id.run <|
     hyps.removeG true λ name uniq' _ _ => if uniq == uniq' then some name else none
     | throwError "ispecialize: cannot find argument"
   let state := { hyps, out, p, pf := q(($pf).1), .. }
-  let state ← liftM <| alls.foldlM (SpecializeState.process_forall gs) state
   let state ← liftM <| spats.foldlM (SpecializeState.process_wand gs) state
 
   let hyps' := Hyps.add bi name uniq state.p state.out state.hyps
@@ -96,7 +94,7 @@ elab "ispecialize" colGt pmt:pmTerm : tactic => do
   -- hypothesis must be in the context, otherwise use pose proof
   let name := ⟨pmt.term⟩
   let some uniq ← try? <| hyps.findWithInfo name | throwError s!"{name} should be a hypothesis, use ihave instead"
-  let ⟨_, hyps', pf⟩ ← iSpecializeCore gs hyps uniq pmt.terms pmt.spats
+  let ⟨_, hyps', pf⟩ ← iSpecializeCore gs hyps uniq pmt.spats
   let pf' ← gs.addGoal hyps' goal
   mvar.assign q(($pf).trans $pf')
   replaceMainGoal (← gs.getGoals)
diff --git a/src/Iris/Tests/Tactics.lean b/src/Iris/Tests/Tactics.lean
index 800af982..669bbb20 100644
--- a/src/Iris/Tests/Tactics.lean
+++ b/src/Iris/Tests/Tactics.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2022 Lars König. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Lars König, Oliver Soeser
+Authors: Lars König, Oliver Soeser, Michael Sammler
 -/
 import Iris.BI
 import Iris.ProofMode
@@ -12,7 +12,8 @@ open Iris.BI
 /- This file contains tests with various scenarios for all available tactics. -/
 
 -- start stop
-theorem start_stop [BI PROP] (Q : PROP) (H : Q ⊢ Q) : Q ⊢ Q := by
+/-- Tests `istart` and `istop` for entering and exiting proof mode -/
+example [BI PROP] (Q : PROP) (H : Q ⊢ Q) : Q ⊢ Q := by
   istart
   iintro _HQ
   istop
@@ -21,23 +22,27 @@ theorem start_stop [BI PROP] (Q : PROP) (H : Q ⊢ Q) : Q ⊢ Q := by
 -- rename
 namespace rename
 
-theorem rename [BI PROP] (Q : PROP) : Q ⊢ Q := by
+/-- Tests basic hypothesis renaming with `irename` -/
+example [BI PROP] (Q : PROP) : Q ⊢ Q := by
   iintro HQ
   irename HQ => H
   iexact H
 
-theorem rename_by_type [BI PROP] (Q : PROP) : □ P ∗ Q ⊢ Q := by
+/-- Tests renaming a hypothesis by its type -/
+example [BI PROP] (P Q : PROP) : □ P ∗ Q ⊢ Q := by
   iintro ⟨_HP, HQ⟩
   irename: Q => H
   iexact H
 
-theorem rename_twice [BI PROP] (Q : PROP) : Q ⊢ Q := by
+/-- Tests renaming a hypothesis twice -/
+example [BI PROP] (Q : PROP) : Q ⊢ Q := by
   iintro HQ
   irename HQ => H
   irename H => HQ
   iexact HQ
 
-theorem rename_id [BI PROP] (Q : PROP) : Q ⊢ Q := by
+/-- Tests renaming a hypothesis to itself (no-op) -/
+example [BI PROP] (Q : PROP) : Q ⊢ Q := by
   iintro HQ
   irename HQ => HQ
   iexact HQ
@@ -47,13 +52,15 @@ end rename
 -- clear
 namespace clear
 
-theorem intuitionistic [BI PROP] (Q : PROP) : □ P ⊢ Q -∗ Q := by
+/-- Tests clearing an intuitionistic hypothesis with `iclear` -/
+example [BI PROP] (Q : PROP) : □ P ⊢ Q -∗ Q := by
   iintro □HP
   iintro HQ
   iclear HP
   iexact HQ
 
-theorem spatial [BI PROP] (Q : PROP) : <affine> P ⊢ Q -∗ Q := by
+/-- Tests clearing a spatial affine hypothesis with `iclear` -/
+example [BI PROP] (Q : PROP) : <affine> P ⊢ Q -∗ Q := by
   iintro HP
   iintro HQ
   iclear HP
@@ -64,52 +71,64 @@ end clear
 -- intro
 namespace intro
 
-theorem spatial [BI PROP] (Q : PROP) : Q ⊢ Q := by
+/-- Tests introducing a spatial hypothesis -/
+example [BI PROP] (Q : PROP) : Q ⊢ Q := by
   iintro HQ
   iexact HQ
 
-theorem intuitionistic [BI PROP] (Q : PROP) : □ Q ⊢ Q := by
+/-- Tests introducing an intuitionistic hypothesis with the `□` pattern -/
+example [BI PROP] (Q : PROP) : □ Q ⊢ Q := by
   iintro □HQ
   iexact HQ
 
-theorem as_intuitionistic [BI PROP] (Q : PROP) : <affine> <pers> Q ⊢ Q := by
+/-- Tests introducing an affine persistent proposition as intuitionistic -/
+example [BI PROP] (Q : PROP) : <affine> <pers> Q ⊢ Q := by
   iintro □HQ
   iexact HQ
 
-theorem as_intuitionistic_in_spatial [BI PROP] (Q : PROP) : ⊢ <pers> Q → Q := by
+/-- Tests introducing a persistent implication in the spatial context -/
+example [BI PROP] (Q : PROP) : ⊢ <pers> Q → Q := by
   iintro HQ
   iexact HQ
 
-theorem drop [BI PROP] (Q : PROP) : ⊢ P → Q -∗ Q := by
+/-- Tests dropping a hypothesis in an implication with the `-` pattern -/
+example [BI PROP] (Q : PROP) : ⊢ P → Q -∗ Q := by
   iintro - HQ
   iexact HQ
 
-theorem drop_after [BI PROP] (Q : PROP) : ⊢ Q -∗ P → Q := by
+/-- Tests dropping a hypothesis in an implication in a non-empty context -/
+example [BI PROP] (Q : PROP) : ⊢ Q -∗ P → Q := by
   iintro HQ -
   iexact HQ
 
-theorem «forall» [BI PROP] : ⊢ ∀ x, ⌜x = 0⌝ → (⌜x = 0⌝ : PROP) := by
+/-- Tests introducing an universally quantified variable -/
+example [BI PROP] : ⊢@{PROP} ∀ x, ⌜x = 0⌝ → ⌜x = 0⌝ := by
   iintro x
   iintro H
   iexact H
 
-theorem pure [BI PROP] [BIAffine PROP] (Q : PROP) : ⊢ ⌜φ⌝ -∗ Q -∗ Q := by
+/-- Tests introducing and extracting a pure hypothesis in affine BI -/
+example [BI PROP] [BIAffine PROP] φ (Q : PROP) : ⊢ ⌜φ⌝ -∗ Q -∗ Q := by
   iintro ⌜Hφ⌝ HQ
   iexact HQ
 
-theorem pattern [BI PROP] (Q : PROP) : □ (P1 ∨ P2) ∗ Q ⊢ Q := by
+/-- Tests introducing with disjunction pattern inside intuitionistic -/
+example [BI PROP] (P1 P2 Q : PROP) : □ (P1 ∨ P2) ∗ Q ⊢ Q := by
   iintro ⟨□(_HP1 | _HP2), HQ⟩
   <;> iexact HQ
 
-theorem multiple_spatial [BI PROP] (Q : PROP) : ⊢ <affine> P -∗ Q -∗ Q := by
+/-- Tests introducing multiple spatial hypotheses -/
+example [BI PROP] (P Q : PROP) : ⊢ <affine> P -∗ Q -∗ Q := by
   iintro _HP HQ
   iexact HQ
 
-theorem multiple_intuitionistic [BI PROP] (Q : PROP) : ⊢ □ P -∗ □ Q -∗ Q := by
+/-- Tests introducing multiple intuitionistic hypotheses -/
+example [BI PROP] (Q : PROP) : ⊢ □ P -∗ □ Q -∗ Q := by
   iintro □_HP □HQ
   iexact HQ
 
-theorem multiple_patterns [BI PROP] (Q : PROP) : ⊢ □ (P1 ∧ P2) -∗ Q ∨ Q -∗ Q := by
+/-- Tests introducing with complex nested patterns -/
+example [BI PROP] (Q : PROP) : ⊢ □ (P1 ∧ P2) -∗ Q ∨ Q -∗ Q := by
   iintro □⟨_HP1, ∗_HP2⟩ (HQ | HQ)
   <;> iexact HQ
 
@@ -118,34 +137,40 @@ end intro
 -- exists
 namespace «exists»
 
-theorem id [BI PROP] : ⊢@{PROP} ∃ x, x := by
+/-- Tests `iexists` with a BI proposition -/
+example [BI PROP] : ⊢@{PROP} ∃ x, x := by
   iexists iprop(True)
   ipure_intro
   exact True.intro
 
-theorem f [BI PROP] : ⊢@{PROP} ∃ (_x : Nat), True ∨ False := by
+/-- Tests `iexists` with a natural number -/
+example [BI PROP] : ⊢@{PROP} ∃ (_x : Nat), True ∨ False := by
   iexists 42
   ileft
   ipure_intro
   exact True.intro
 
-theorem pure [BI PROP] : ⊢@{PROP} ⌜∃ x, x ∨ False⌝ := by
+/-- Tests `iexists` with Prop -/
+example [BI PROP] : ⊢@{PROP} ⌜∃ x, x ∨ False⌝ := by
   iexists True
   ipure_intro
   exact Or.inl True.intro
 
 
-theorem mvar [BI PROP] : ⊢@{PROP} ∃ x, ⌜x = 42⌝ := by
+/-- Tests `iexists` with a named metavariable -/
+example [BI PROP] : ⊢@{PROP} ∃ x, ⌜x = 42⌝ := by
   iexists ?y
   ipure_intro
   rfl
 
-theorem mvar_anon [BI PROP] : ⊢@{PROP} ∃ x, ⌜x = 42⌝ := by
+/-- Tests `iexists` with anonymous metavariable -/
+example [BI PROP] : ⊢@{PROP} ∃ x, ⌜x = 42⌝ := by
   iexists _
   ipure_intro
   rfl
 
-theorem mvar_two [BI PROP] : ⊢@{PROP} ∃ x y : Nat, ⌜x = y⌝ := by
+/-- Tests `iexists` with two quantifiers -/
+example [BI PROP] : ⊢@{PROP} ∃ x y : Nat, ⌜x = y⌝ := by
   iexists _, 1
   ipure_intro
   rfl
@@ -156,15 +181,18 @@ end «exists»
 -- exact
 namespace exact
 
-theorem exact [BI PROP] (Q : PROP) : Q ⊢ Q := by
+/-- Tests basic `iexact` -/
+example [BI PROP] (Q : PROP) : Q ⊢ Q := by
   iintro HQ
   iexact HQ
 
-theorem def_eq [BI PROP] (Q : PROP) : <affine> <pers> Q ⊢ □ Q := by
+/-- Tests `iexact` with affine pers to intuitionistic -/
+example [BI PROP] (Q : PROP) : <affine> <pers> Q ⊢ □ Q := by
   iintro HQ
   iexact HQ
 
-theorem intuitionistic [BI PROP] (Q : PROP) : □ Q ⊢ Q := by
+/-- Tests `iexact` with intuitionistic hypothesis -/
+example [BI PROP] (Q : PROP) : □ Q ⊢ Q := by
   iintro HQ
   iexact HQ
 
@@ -173,23 +201,28 @@ end exact
 -- assumption
 namespace assumption
 
-theorem exact [BI PROP] (Q : PROP) : Q ⊢ Q := by
+/-- Tests `iassumption` for exact match -/
+example [BI PROP] (Q : PROP) : Q ⊢ Q := by
   iintro _HQ
   iassumption
 
-theorem from_assumption [BI PROP] (Q : PROP) : <affine> <pers> Q ⊢ □ Q := by
+/-- Tests `iassumption` with affine pers to intuitionistic -/
+example [BI PROP] (Q : PROP) : <affine> <pers> Q ⊢ □ Q := by
   iintro _HQ
   iassumption
 
-theorem intuitionistic [BI PROP] (Q : PROP) : □ Q ⊢ Q := by
+/-- Tests `iassumption` with intuitionistic hypothesis -/
+example [BI PROP] (Q : PROP) : □ Q ⊢ Q := by
   iintro □_HQ
   iassumption
 
-theorem lean [BI PROP] (Q : PROP) (H : ⊢ Q) : <affine> P ⊢ Q := by
+/-- Tests `iassumption` using a Lean hypothesis -/
+example [BI PROP] (Q : PROP) (H : ⊢ Q) : <affine> P ⊢ Q := by
   iintro _HP
   iassumption
 
-theorem lean_pure [BI PROP] (Q : PROP) : <affine> ⌜⊢ Q⌝ ⊢ Q := by
+/-- Tests `iassumption` with Lean hypothesis first introduced -/
+example [BI PROP] (Q : PROP) : <affine> ⌜⊢ Q⌝ ⊢ Q := by
   iintro ⌜H⌝
   iassumption
 
@@ -198,157 +231,196 @@ end assumption
 -- apply
 namespace apply
 
-theorem exact [BI PROP] (Q : PROP) : Q ⊢ Q := by
+/-- Tests `iapply` with exact match -/
+example [BI PROP] (Q : PROP) : Q ⊢ Q := by
   iintro HQ
   iapply HQ
 
-theorem apply [BI PROP] (P Q : PROP) : ⊢ P -∗ (P -∗ Q) -∗ Q := by
+/-- Tests `iapply` with wand -/
+example [BI PROP] (P Q : PROP) : ⊢ P -∗ (P -∗ Q) -∗ Q := by
   iintro HP H
-  iapply H with HP
+  iapply H $$ HP
 
-theorem multiple [BI PROP] (P Q R : PROP) : ⊢ P -∗ Q -∗ (P -∗ Q -∗ R) -∗ R := by
+/-- Tests `iapply` with multiple hypotheses -/
+example [BI PROP] (P Q R : PROP) : ⊢ P -∗ Q -∗ (P -∗ Q -∗ R) -∗ R := by
   iintro HP HQ H
-  iapply H with HP, HQ
+  iapply H $$ HP, HQ
 
-theorem multiple' [BI PROP] (P Q R S : PROP) : ⊢ (P -∗ Q) -∗ P -∗ R -∗ (Q -∗ R -∗ S) -∗ S := by
+/-- Tests `iapply` with nested wand application -/
+example [BI PROP] (P Q R S : PROP) : ⊢ (P -∗ Q) -∗ P -∗ R -∗ (Q -∗ R -∗ S) -∗ S := by
   iintro HPQ HP HR H
-  iapply H with [HPQ, HP], HR
-  iapply HPQ with HP
+  iapply H $$ [HPQ, HP], HR
+  iapply HPQ $$ HP
 
-theorem exact_intuitionistic [BI PROP] (Q : PROP) : □ Q ⊢ Q := by
+/-- Tests `iapply` with intuitionistic exact -/
+example [BI PROP] (Q : PROP) : □ Q ⊢ Q := by
   iintro □HQ
   iapply HQ
 
-theorem apply_intuitionistic [BI PROP] (P Q : PROP) : ⊢ □ P -∗ (P -∗ Q) -∗ Q := by
+/-- Tests `iapply` with intuitionistic wand argument -/
+example [BI PROP] (P Q : PROP) : ⊢ □ P -∗ (P -∗ Q) -∗ Q := by
   iintro HP H
-  iapply H with HP
+  iapply H $$ HP
 
-theorem multiple_intuitionistic [BI PROP] (P Q R : PROP) : ⊢ □ P -∗ Q -∗ □ (P -∗ Q -∗ □ R) -∗ R := by
+/-- Tests `iapply` with multiple intuitionistic hypotheses and subgoals -/
+example [BI PROP] (P Q R : PROP) : ⊢ □ P -∗ Q -∗ □ (P -∗ Q -∗ □ R) -∗ R := by
   iintro □HP HQ □H
-  iapply H with [], [HQ] as Q
+  iapply H $$ [], [HQ] as Q
   case Q => iexact HQ
   iexact HP
 
-theorem later [BI PROP] (P Q : PROP) : ⊢ (▷ P -∗ Q) -∗ P -∗ Q := by
+/-- Tests `iapply` with later modality -/
+example [BI PROP] (P Q : PROP) : ⊢ (▷ P -∗ Q) -∗ P -∗ Q := by
   iintro H HP
-  iapply H with HP
+  iapply H $$ HP
 
-theorem test_affine [BI PROP] [BIAffine PROP] (P Q : PROP) : ⊢ (P → Q) -∗ <pers> P -∗ Q := by
+/-- Tests `iapply` with implication -/
+example [BI PROP] [BIAffine PROP] (P Q : PROP) : ⊢ (P → Q) -∗ <pers> P -∗ Q := by
   iintro H HP
-  iapply H with HP
+  iapply H $$ HP
 
-theorem later_affine [BI PROP] [BIAffine PROP] (P Q : PROP) : ⊢ (▷ P → Q) -∗ P -∗ Q := by
+/-- Tests `iapply` with later and implication -/
+example [BI PROP] [BIAffine PROP] (P Q : PROP) : ⊢ (▷ P → Q) -∗ P -∗ Q := by
   iintro H HP
-  iapply H with HP
+  iapply H $$ HP
 
-theorem exact_lean [BI PROP] (Q : PROP) (H : ⊢ Q) : ⊢ Q := by
+/-- Tests `iapply` with Lean hypothesis -/
+example [BI PROP] (Q : PROP) (H : ⊢ Q) : ⊢ Q := by
   iapply H
 
-theorem exact_lean' [BI PROP] (Q : PROP) : Q ⊢ (emp ∗ Q) ∗ emp := by
+/-- Tests `iapply` with lemma -/
+example [BI PROP] (Q : PROP) : Q ⊢ (emp ∗ Q) ∗ emp := by
   iapply (wand_intro sep_emp.mpr)
   iemp_intro
 
-theorem exact_lean'' [BI PROP] (Q : PROP) (H : 0 = 0 → ⊢ Q) : ⊢ Q := by
+/-- Tests `iapply` with pure sidecondition -/
+example [BI PROP] (Q : PROP) (H : 0 = 0 → ⊢ Q) : ⊢ Q := by
   iapply H
   rfl
 
-theorem exact_lean''' [BI PROP] : ⊢@{PROP} ⌜1 = 1⌝ := by
+/-- Tests `iapply` with lemma with sidecondition -/
+example [BI PROP] : ⊢@{PROP} ⌜1 = 1⌝ := by
   istart
   iapply (pure_intro (P:=emp))
   . rfl
   iemp_intro
 
-theorem apply_lean [BI PROP] (P Q : PROP) (H : P ⊢ Q) (HP : ⊢ P) : ⊢ Q := by
+/-- Tests `iapply` with entailment as Lean hypothesis -/
+example [BI PROP] (P Q : PROP) (H : P ⊢ Q) (HP : ⊢ P) : ⊢ Q := by
   iapply H
   iapply HP
 
-theorem apply_lean' [BI PROP] (P Q : PROP) (H : ⊢ P -∗ Q) (HP : ⊢ P) : ⊢ Q := by
-  iapply H with []
+/-- Tests `iapply` with wand entailment as Lean hypothesis -/
+example [BI PROP] (P Q : PROP) (H : ⊢ P -∗ Q) (HP : ⊢ P) : ⊢ Q := by
+  iapply H $$ []
   iapply HP
 
-theorem apply_lean'' [BI PROP] (P Q : PROP) (H1 : P ⊢ Q) (H2 : Q ⊢ R) : P ⊢ R := by
+/-- Tests `iapply` with constructed term -/
+example [BI PROP] (P Q : PROP) (H1 : P ⊢ Q) (H2 : Q ⊢ R) : P ⊢ R := by
   iintro HP
   iapply (wand_intro (emp_sep.mp.trans H2))
   . ipure_intro; trivial
-  iapply H1 with HP
+  iapply H1 $$ HP
 
-theorem multiple_lean [BI PROP] (P Q R : PROP) (H : P ⊢ Q -∗ R) (HP : ⊢ P) : ⊢ Q -∗ R := by
+/-- Tests `iapply` with Lean wand entailment and subgoal -/
+example [BI PROP] (P Q R : PROP) (H : P ⊢ Q -∗ R) (HP : ⊢ P) : ⊢ Q -∗ R := by
   iintro HQ
-  iapply H with [], HQ
+  iapply H $$ [], HQ
   iapply HP
 
-theorem multiple_lean' [BI PROP] (P Q R : PROP) (H : P ∗ Q ⊢ R) (HP : ⊢ P) : ⊢ Q -∗ R := by
+/-- Tests `iapply` with lemma and subgoal -/
+example [BI PROP] (P Q R : PROP) (H : P ∗ Q ⊢ R) (HP : ⊢ P) : ⊢ Q -∗ R := by
   iintro HQ
-  iapply (wand_intro H) with [], HQ
+  iapply (wand_intro H) $$ [], HQ
   iapply HP
 
-theorem exact_forall [BI PROP] (P : α → PROP) (a : α) (H : ⊢ ∀ x, P x) : ⊢ P a := by
+/-- Tests `iapply` with forall -/
+example [BI PROP] (P : α → PROP) (a : α) (H : ⊢ ∀ x, P x) : ⊢ P a := by
   istart
   iapply H
 
-theorem exact_forall' [BI PROP] (P : α → PROP) (a : α) (H : ∀ x, ⊢ P x) : ⊢ P a := by
+/-- Tests `iapply` with Lean forall -/
+example [BI PROP] (P : α → PROP) (a : α) (H : ∀ x, ⊢ P x) : ⊢ P a := by
   iapply H
 
-theorem apply_forall [BI PROP] (P Q : α → PROP) (a b : α) (H : ⊢ ∀ x, ∀ y, P x -∗ Q y) : P a ⊢ Q b := by
+/-- Tests `iapply` with forall specialization -/
+example [BI PROP] (P Q : α → PROP) (a b : α) (H : ⊢ ∀ x, ∀ y, P x -∗ Q y) : P a ⊢ Q b := by
   iintro HP
-  iapply H $! a, b with HP
+  iapply H $$ %a, %b, HP
 
-theorem apply_forall' [BI PROP] (P Q : α → PROP) (a b : α) : (∀ x, ∀ y, P x -∗ Q y) ⊢ P a -∗ Q b := by
+/-- Tests `iapply` with forall specialization from hypothesis -/
+example [BI PROP] (P Q : α → PROP) (a b : α) : (∀ x, ∀ y, P x -∗ Q y) ⊢ P a -∗ Q b := by
   iintro H HP
-  iapply H $! a, b with HP
+  iapply H $$ %a, %b, HP
 
-/-- error: ispecialize: iprop(P a -∗ Q b) is not a forall -/
+/-- Tests `iapply` with tactic -/
+example [BI PROP] (P Q : α → PROP) (a b : α) : (∀ x, ∀ y, P x -∗ Q y) ⊢ P a -∗ Q b := by
+  iintro H HP
+  iapply H $$ %by exact a, %b, [HP]
+  iapply HP
+
+/-- error: ispecialize: iprop(P a -∗ Q b) is not a lean premise -/
 #guard_msgs in
-theorem apply_forall_too_many [BI PROP] (P Q : α → PROP) (a b : α) : (∀ x, ∀ y, P x -∗ Q y) ⊢ P a -∗ Q b := by
+example [BI PROP] (P Q : α → PROP) (a b : α) : (∀ x, ∀ y, P x -∗ Q y) ⊢ P a -∗ Q b := by
   iintro H HP
-  iapply H $! a, b, _ with HP
+  iapply H $$ %a, %b, %_, HP
 
-theorem apply_forall2 [BI PROP] (P Q : α → PROP) (a b : α) : (∀ x, ∀ y, P x -∗ Q y) ⊢ P a -∗ Q b := by
+/-- Tests `iapply` using unification for foralls -/
+example [BI PROP] (P Q : α → PROP) (a b : α) : (∀ x, ∀ y, P x -∗ Q y) ⊢ P a -∗ Q b := by
   iintro H HP
-  iapply H with HP
+  iapply H $$ HP
 
-theorem apply_forall3 [BI PROP] (P Q : α → PROP) (a b : α) : (∀ x, ∀ y, P x -∗ Q y) ⊢ P a -∗ Q b := by
+/-- Tests `iapply` using manually created metavariables -/
+example [BI PROP] (P Q : α → PROP) (a b : α) : (∀ x, ∀ y, P x -∗ Q y) ⊢ P a -∗ Q b := by
   iintro H HP
-  iapply H $! ?_, ?_ with HP
+  iapply H $$ %?_, %?_, HP
 
-theorem apply_forall4 [BI PROP] (P Q : α → PROP) (a b : α) : (∀ x, ∀ y, P x -∗ Q y) ⊢ P a -∗ Q b := by
+/-- Tests `iapply` using unification in two steps, instantiating metavars  -/
+example [BI PROP] (P Q : α → PROP) (a b : α) : (∀ x, ∀ y, P x -∗ Q y) ⊢ P a -∗ Q b := by
   iintro H HP
   iapply H
   iapply HP
 
-theorem apply_forall_intuitionistic [BI PROP] (P Q : α → PROP) (a b : α) (H : ⊢ □ ∀ x, ∀ y, P x -∗ Q y) : P a ⊢ Q b := by
+/-- Tests `iapply` with intuitionistic forall from Lean -/
+example [BI PROP] (P Q : α → PROP) (a b : α) (H : ⊢ □ ∀ x, ∀ y, P x -∗ Q y) : P a ⊢ Q b := by
   iintro HP
-  iapply H $! a with HP
+  iapply H $$ %a, HP
 
-theorem apply_forall_intuitionistic' [BI PROP] (P Q : α → PROP) (a b : α) : (□ ∀ x, ∀ y, P x -∗ Q y) ⊢ P a -∗ Q b := by
+/-- Tests `iapply` with intuitionistic forall from hypothesis -/
+example [BI PROP] (P Q : α → PROP) (a b : α) : (□ ∀ x, ∀ y, P x -∗ Q y) ⊢ P a -∗ Q b := by
   iintro H HP
-  iapply H $! a, b with HP
+  iapply H $$ %a, %b, HP
 
-theorem apply_two_wands [BI PROP] (P Q : Nat → PROP) :
+/-- Tests `iapply` with two wands and subgoals -/
+example [BI PROP] (P Q : Nat → PROP) :
   (P 1 -∗ P 2 -∗ Q 1) ⊢ □ P 1 -∗ P 2 -∗ Q 1 := by
   iintro H □HP1 HP2
   iapply H
   . iexact HP1
   . iexact HP2
 
-theorem apply_and_l [BI PROP] (P Q : Nat → PROP) :
+/-- Tests `iapply` selecting left conjunct -/
+example [BI PROP] (P Q : Nat → PROP) :
   ((P 1 -∗ P 2) ∧ (Q 1 -∗ Q 2)) ⊢ P 1 -∗ P 2 := by
   iintro H HP1
   iapply H
   iexact HP1
 
-theorem apply_and_r [BI PROP] (P Q : Nat → PROP) :
+/-- Tests `iapply` selecting right conjunct -/
+example [BI PROP] (P Q : Nat → PROP) :
   ((P 1 -∗ P 2) ∧ (Q 1 -∗ Q 2)) ⊢ Q 1 -∗ Q 2 := by
   iintro H HQ1
   iapply H
   iexact HQ1
 
-theorem apply_and_l_exact [BI PROP] (P Q : Nat → PROP) :
+/-- Tests `iapply` selecting left conjunct (exact match) -/
+example [BI PROP] (P Q : Nat → PROP) :
   (P 1 ∧ Q 1) ⊢ P 1 := by
   iintro H
   iapply H
 
-theorem apply_and_r_exact [BI PROP] (P Q : Nat → PROP) :
+/-- Tests `iapply` selecting right conjunct (exact match) -/
+example [BI PROP] (P Q : Nat → PROP) :
   (P 1 ∧ Q 1) ⊢ Q 1 := by
   iintro H
   iapply H
@@ -358,67 +430,80 @@ end apply
 -- have
 namespace ihave
 
-theorem exact_lean [BI PROP] (Q : PROP) (H : ⊢ Q) : ⊢ Q := by
+/-- Tests `ihave` with Lean hypothesis -/
+example [BI PROP] (Q : PROP) (H : ⊢ Q) : ⊢ Q := by
   ihave HQ := H
   iexact HQ
 
-theorem exact_lean_forall [BI PROP] (Q : Nat → PROP) (H : ∀ x, ⊢ Q x) : ⊢ Q 1 := by
+/-- Tests `ihave` with forall specialization via case -/
+example [BI PROP] (Q : Nat → PROP) (H : ∀ x, ⊢ Q x) : ⊢ Q 1 := by
   ihave HQ := H
   case x => exact 1
   iapply HQ
 
-theorem exact_lean_forall2 [BI PROP] (Q : Nat → PROP) (H : ∀ x, ⊢ Q x) : ⊢ Q 1 := by
+/-- Tests `ihave` with forall specialization via named hole -/
+example [BI PROP] (Q : Nat → PROP) (H : ∀ x, ⊢ Q x) : ⊢ Q 1 := by
   ihave HQ := H ?res
   case res => exact 1
   iexact HQ
 
-theorem exact_lean_forall3 [BI PROP] (Q : Nat → Nat → PROP) (H : ∀ x y, ⊢ Q x y) : ⊢ Q 1 1 := by
+/-- Tests `ihave` with two named holes -/
+example [BI PROP] (Q : Nat → Nat → PROP) (H : ∀ x y, ⊢ Q x y) : ⊢ Q 1 1 := by
   ihave HQ := H ?res ?res
   case res => exact 1
   iexact HQ
 
-theorem exact_lean_forall4 [BI PROP] (Q : Nat → PROP) (H : ∀ x, ⊢ Q x) : ⊢ Q 1 := by
+/-- Tests `ihave` creating metavars -/
+example [BI PROP] (Q : Nat → PROP) (H : ∀ x, ⊢ Q x) : ⊢ Q 1 := by
   ihave HQ := H
   iexact HQ
 
-theorem exact_lean_tc [BI PROP] (Q : Nat → PROP) (H : ∀ (P : PROP) [Persistent P], ⊢ P) : ⊢ Q 1 := by
+/-- Tests `ihave` with typeclass argument (failing search) -/
+example [BI PROP] (Q : Nat → PROP) (H : ∀ (P : PROP) [Persistent P], ⊢ P) : ⊢ Q 1 := by
   ihave HQ := H
   rotate_right 1; exact iprop(□ Q 1)
   . apply inferInstance
   iexact HQ
 
-theorem exact_lean_tc2 [BI PROP] (Q : Nat → PROP) (H : ∀ (P : PROP) [Persistent P], ⊢ P) : ⊢ Q 1 := by
+/-- Tests `ihave` with typeclass argument (successful search) -/
+example [BI PROP] (Q : Nat → PROP) (H : ∀ (P : PROP) [Persistent P], ⊢ P) : ⊢ Q 1 := by
   ihave HQ := H iprop(□ Q _)
   rotate_right 1; exact 1
   iexact HQ
 
-theorem exact_spatial [BI PROP] (Q : PROP) : Q ⊢ Q := by
+/-- Tests `ihave` from spatial hypothesis -/
+example [BI PROP] (Q : PROP) : Q ⊢ Q := by
   iintro H
   ihave HQ := H
   iexact HQ
 
-theorem apply_lean [BI PROP] (P Q : PROP) (H : P ⊢ Q) : ⊢ P -∗ Q := by
+/-- Tests `ihave` with Lean entailment -/
+example [BI PROP] (P Q : PROP) (H : P ⊢ Q) : ⊢ P -∗ Q := by
   ihave HPQ := H
   iexact HPQ
 
-theorem apply_forall [BI PROP] (P Q : α → PROP) (a b : α) (H : ⊢ ∀ x, ∀ y, P x -∗ Q y) : P a ⊢ Q b := by
+/-- Tests `ihave` with forall specialization from Lean -/
+example [BI PROP] (P Q : α → PROP) (a b : α) (H : ⊢ ∀ x, ∀ y, P x -∗ Q y) : P a ⊢ Q b := by
   iintro HP
-  ihave H' := H $! a, b
-  iapply H' with HP
+  ihave H' := H $$ %a, %b
+  iapply H' $$ HP
 
-theorem apply_forall_spatial [BI PROP] (P Q : α → PROP) (a b : α) : (∀ x, ∀ y, P x -∗ Q y) ⊢ P a -∗ Q b := by
+/-- Tests `ihave` with forall specialization from hypothesis -/
+example [BI PROP] (P Q : α → PROP) (a b : α) : (∀ x, ∀ y, P x -∗ Q y) ⊢ P a -∗ Q b := by
   iintro H HP
-  ihave H' := H $! a, b with HP
+  ihave H' := H $$ %a, %b, HP
   iexact H'
 
-theorem apply_forall_intuitionistic [BI PROP] (P Q : α → PROP) (a b : α) (H : ⊢ □ ∀ x, ∀ y, P x -∗ Q y) : P a ⊢ Q b := by
+/-- Tests `ihave` with intuitionistic forall specialization from Lean -/
+example [BI PROP] (P Q : α → PROP) (a b : α) (H : ⊢ □ ∀ x, ∀ y, P x -∗ Q y) : P a ⊢ Q b := by
   iintro HP
-  ihave H' := H $! a, b
-  iapply H' with HP
+  ihave H' := H $$ %a, %b
+  iapply H' $$ HP
 
-theorem apply_forall_intuitionistic_iris [BI PROP] (P Q : α → PROP) (a b : α) : (□ ∀ x, ∀ y, P x -∗ Q y) ⊢ P a -∗ Q b := by
+/-- Tests `ihave` with intuitionistic forall specialization and subgoal -/
+example [BI PROP] (P Q : α → PROP) (a b : α) : (□ ∀ x, ∀ y, P x -∗ Q y) ⊢ P a -∗ Q b := by
   iintro H HP
-  ihave H' := H $! a, b with [HP]
+  ihave H' := H $$ %a, %b, [HP]
   . iexact HP
   iexact H'
 
@@ -427,15 +512,18 @@ end ihave
 -- ex falso
 namespace exfalso
 
-theorem false_intro [BI PROP] (Q : PROP) : False ⊢ Q := by
+/-- Tests false elimination via empty pattern -/
+example [BI PROP] (Q : PROP) : False ⊢ Q := by
   iintro ⟨⟩
 
-theorem false [BI PROP] (P : PROP) : □ P ⊢ False -∗ Q := by
+/-- Tests `iexfalso` with false hypothesis -/
+example [BI PROP] (P : PROP) : □ P ⊢ False -∗ Q := by
   iintro _HP HF
   iexfalso
   iexact HF
 
-theorem pure [BI PROP] (P : PROP) (HF : False) : ⊢ P := by
+/-- Tests `iexfalso` with pure false from Lean -/
+example [BI PROP] (P : PROP) (HF : False) : ⊢ P := by
   istart
   iexfalso
   ipure_intro
@@ -446,13 +534,15 @@ end exfalso
 -- pure
 namespace pure
 
-theorem move [BI PROP] (Q : PROP) : <affine> ⌜φ⌝ ⊢ Q -∗ Q := by
+/-- Tests `ipure` to move pure hypothesis to Lean context -/
+example [BI PROP] (Q : PROP) : <affine> ⌜φ⌝ ⊢ Q -∗ Q := by
   iintro Hφ
   iintro HQ
   ipure Hφ
   iexact HQ
 
-theorem move_multiple [BI PROP] (Q : PROP) : <affine> ⌜φ1⌝ ⊢ <affine> ⌜φ2⌝ -∗ Q -∗ Q := by
+/-- Tests `ipure` with multiple pure hypotheses -/
+example [BI PROP] (Q : PROP) : <affine> ⌜φ1⌝ ⊢ <affine> ⌜φ2⌝ -∗ Q -∗ Q := by
   iintro Hφ1
   iintro Hφ2
   iintro HQ
@@ -460,7 +550,8 @@ theorem move_multiple [BI PROP] (Q : PROP) : <affine> ⌜φ1⌝ ⊢ <affine> ⌜
   ipure Hφ2
   iexact HQ
 
-theorem move_conjunction [BI PROP] (Q : PROP) : (⌜φ1⌝ ∧ <affine> ⌜φ2⌝) ⊢ Q -∗ Q := by
+/-- Tests `ipure` with conjunction containing pure -/
+example [BI PROP] (Q : PROP) : (⌜φ1⌝ ∧ <affine> ⌜φ2⌝) ⊢ Q -∗ Q := by
   iintro Hφ
   iintro HQ
   ipure Hφ
@@ -471,20 +562,23 @@ end pure
 -- intuitionistic
 namespace intuitionistic
 
-theorem move [BI PROP] (P : PROP) : □ P ⊢ Q -∗ Q := by
+/-- Tests `iintuitionistic` to move hypothesis to intuitionistic context -/
+example [BI PROP] (P : PROP) : □ P ⊢ Q -∗ Q := by
   iintro HP
   iintro HQ
   iintuitionistic HP
   iexact HQ
 
-theorem move_multiple [BI PROP] (P : PROP) : □ P ⊢ □ Q -∗ Q := by
+/-- Tests `iintuitionistic` with multiple hypotheses -/
+example [BI PROP] (P : PROP) : □ P ⊢ □ Q -∗ Q := by
   iintro HP
   iintro HQ
   iintuitionistic HP
   iintuitionistic HQ
   iexact HQ
 
-theorem move_twice [BI PROP] (P : PROP) : □ P ⊢ Q -∗ Q := by
+/-- Tests `iintuitionistic` applied twice to same hypothesis -/
+example [BI PROP] (P : PROP) : □ P ⊢ Q -∗ Q := by
   iintro HP
   iintro HQ
   iintuitionistic HP
@@ -496,20 +590,23 @@ end intuitionistic
 -- spatial
 namespace spatial
 
-theorem move [BI PROP] (P : PROP) : □ P ⊢ □ Q -∗ Q := by
+/-- Tests `ispatial` to move hypothesis to spatial context -/
+example [BI PROP] (P : PROP) : □ P ⊢ □ Q -∗ Q := by
   iintro □HP
   iintro □HQ
   ispatial HP
   iexact HQ
 
-theorem move_multiple [BI PROP] (P : PROP) : □ P ⊢ □ Q -∗ Q := by
+/-- Tests `ispatial` with multiple hypotheses -/
+example [BI PROP] (P : PROP) : □ P ⊢ □ Q -∗ Q := by
   iintro □HP
   iintro □HQ
   ispatial HP
   ispatial HQ
   iexact HQ
 
-theorem move_twice [BI PROP] (P : PROP) : □ P ⊢ □ Q -∗ Q := by
+/-- Tests `ispatial` applied twice to same hypothesis -/
+example [BI PROP] (P : PROP) : □ P ⊢ □ Q -∗ Q := by
   iintro □HP
   iintro □HQ
   ispatial HP
@@ -521,10 +618,12 @@ end spatial
 -- emp intro
 namespace empintro
 
-theorem simple [BI PROP] : ⊢ (emp : PROP) := by
+/-- Tests `iemp_intro` for proving emp -/
+example [BI PROP] : ⊢@{PROP} emp := by
   iemp_intro
 
-theorem affine_env [BI PROP] (P : PROP) : <affine> P ⊢ emp := by
+/-- Tests `iemp_intro` with affine environment -/
+example [BI PROP] (P : PROP) : <affine> P ⊢ emp := by
   iintro _HP
   iemp_intro
 
@@ -533,15 +632,18 @@ end empintro
 -- pure intro
 namespace pureintro
 
-theorem simple [BI PROP] : ⊢ (⌜True⌝ : PROP) := by
+/-- Tests `ipure_intro` for True -/
+example [BI PROP] : ⊢@{PROP} ⌜True⌝ := by
   ipure_intro
   exact True.intro
 
-theorem or [BI PROP] : ⊢ True ∨ (False : PROP) := by
+/-- Tests `ipure_intro` for disjunction -/
+example [BI PROP] : ⊢@{PROP} True ∨ False := by
   ipure_intro
   apply Or.inl True.intro
 
-theorem with_proof [BI PROP] (H : A → B) (P Q : PROP) : <affine> P ⊢ <pers> Q → ⌜A⌝ → ⌜B⌝ := by
+/-- Tests `ipure_intro` with context -/
+example [BI PROP] (H : A → B) (P Q : PROP) : <affine> P ⊢ <pers> Q → ⌜A⌝ → ⌜B⌝ := by
   iintro _HP □_HQ
   ipure_intro
   exact H
@@ -551,128 +653,178 @@ end pureintro
 -- specialize
 namespace specialize
 
-theorem wand_spatial [BI PROP] (Q : PROP) : P ⊢ (P -∗ Q) -∗ Q := by
+/-- Tests `ispecialize` with spatial wand -/
+example [BI PROP] (Q : PROP) : P ⊢ (P -∗ Q) -∗ Q := by
   iintro HP HPQ
-  ispecialize HPQ with HP
+  ispecialize HPQ $$ HP
   iexact HPQ
 
-theorem wand_spatial_subgoal [BI PROP] (Q : PROP) : P ⊢ (P -∗ Q) -∗ Q := by
+/-- Tests `ispecialize` with subgoal -/
+example [BI PROP] (Q : PROP) : P ⊢ (P -∗ Q) -∗ Q := by
   iintro HP HPQ
-  ispecialize HPQ with [HP]
+  ispecialize HPQ $$ [HP]
   . iexact HP
   iexact HPQ
 
-theorem wand_spatial_subgoal_named [BI PROP] (Q : PROP) : P ⊢ (P -∗ Q) -∗ Q := by
+/-- Tests `ispecialize` with named subgoal -/
+example [BI PROP] (Q : PROP) : P ⊢ (P -∗ Q) -∗ Q := by
   iintro HP HPQ
-  ispecialize HPQ with [HP] as G
+  ispecialize HPQ $$ [HP] as G
   case G => iexact HP
   iexact HPQ
 
-theorem wand_intuitionistic [BI PROP] (Q : PROP) : □ P ⊢ □ (P -∗ Q) -∗ □ Q := by
+/-- Tests `ispecialize` with intuitionistic wand -/
+example [BI PROP] (Q : PROP) : □ P ⊢ □ (P -∗ Q) -∗ □ Q := by
   iintro □HP □HPQ
-  ispecialize HPQ with HP
+  ispecialize HPQ $$ HP
   iexact HPQ
 
-theorem wand_intuitionistic_subgoal [BI PROP] (Q : PROP) : □ P ⊢ □ (P -∗ Q) -∗ Q := by
+/-- Tests `ispecialize` with intuitionistic wand and subgoal -/
+example [BI PROP] (Q : PROP) : □ P ⊢ □ (P -∗ Q) -∗ Q := by
   iintro □HP □HPQ
-  ispecialize HPQ with []
+  ispecialize HPQ $$ []
   . iexact HP
   iexact HPQ
 
-theorem wand_intuitionistic_required [BI PROP] (Q : PROP) : □ P ⊢ □ (□ P -∗ Q) -∗ □ Q := by
+/-- Tests `ispecialize` with intuitionistic wand requiring intuitionistic argument -/
+example [BI PROP] (Q : PROP) : □ P ⊢ □ (□ P -∗ Q) -∗ □ Q := by
   iintro □HP □HPQ
-  ispecialize HPQ with HP
+  ispecialize HPQ $$ HP
   iexact HPQ
 
-theorem wand_intuitionistic_spatial [BI PROP] (Q : PROP) : □ P ⊢ (P -∗ Q) -∗ Q := by
+/-- Tests `ispecialize` with intuitionistic premise and spatial wand -/
+example [BI PROP] (Q : PROP) : □ P ⊢ (P -∗ Q) -∗ Q := by
   iintro □HP HPQ
-  ispecialize HPQ with HP
+  ispecialize HPQ $$ HP
   iexact HPQ
 
-theorem wand_intuitionistic_required_spatial [BI PROP] (Q : PROP) : □ P ⊢ (□ P -∗ Q) -∗ Q := by
+/-- Tests `ispecialize` with intuitionistic premise required by spatial wand -/
+example [BI PROP] (Q : PROP) : □ P ⊢ (□ P -∗ Q) -∗ Q := by
   iintro □HP HPQ
-  ispecialize HPQ with HP
+  ispecialize HPQ $$ HP
   iexact HPQ
 
-theorem wand_spatial_intuitionistic [BI PROP] (Q : PROP) : P ⊢ □ (P -∗ Q) -∗ Q := by
+/-- Tests `ispecialize` with spatial premise and intuitionistic wand -/
+example [BI PROP] (Q : PROP) : P ⊢ □ (P -∗ Q) -∗ Q := by
   iintro HP □HPQ
-  ispecialize HPQ with HP
+  ispecialize HPQ $$ HP
   iexact HPQ
 
-theorem wand_spatial_multiple [BI PROP] (Q : PROP) : ⊢ P1 -∗ P2 -∗ (P1 -∗ P2 -∗ Q) -∗ Q := by
+/-- Tests `ispecialize` with multiple spatial arguments -/
+example [BI PROP] (Q : PROP) : ⊢ P1 -∗ P2 -∗ (P1 -∗ P2 -∗ Q) -∗ Q := by
   iintro HP1 HP2 HPQ
-  ispecialize HPQ with HP1, HP2
+  ispecialize HPQ $$ HP1, HP2
   iexact HPQ
 
-theorem wand_spatial_multiple_subgoal [BI PROP] (Q : PROP) : ⊢ P1 -∗ P2 -∗ (P1 -∗ P2 -∗ Q) -∗ Q := by
+/-- Tests `ispecialize` with multiple subgoals -/
+example [BI PROP] (Q : PROP) : ⊢ P1 -∗ P2 -∗ (P1 -∗ P2 -∗ Q) -∗ Q := by
   iintro HP1 HP2 HPQ
-  ispecialize HPQ with [HP1], [HP2]
+  ispecialize HPQ $$ [HP1], [HP2]
   . iexact HP1
   . iexact HP2
   iexact HPQ
 
-theorem wand_intuitionistic_multiple [BI PROP] (Q : PROP) :
+/-- Tests `ispecialize` with multiple intuitionistic arguments -/
+example [BI PROP] (Q : PROP) :
     ⊢ □ P1 -∗ □ P2 -∗ □ (P1 -∗ □ P2 -∗ Q) -∗ □ Q := by
   iintro □HP1 □HP2 □HPQ
-  ispecialize HPQ with HP1, HP2
+  ispecialize HPQ $$ HP1, HP2
   iexact HPQ
 
-theorem wand_multiple [BI PROP] (Q : PROP) :
+/-- Tests `ispecialize` with mixed spatial and intuitionistic arguments -/
+example [BI PROP] (Q : PROP) :
     ⊢ P1 -∗ □ P2 -∗ P3 -∗ □ (P1 -∗ P2 -∗ P3 -∗ Q) -∗ Q := by
   iintro HP1 □HP2 HP3 HPQ
-  ispecialize HPQ with HP1, HP2, HP3
+  ispecialize HPQ $$ HP1, HP2, HP3
   iexact HPQ
 
-theorem forall_spatial [BI PROP] (Q : Nat → PROP) : ⊢ (∀ x, Q x) -∗ Q (y + 1) := by
+/-- Tests `ispecialize` with forall in spatial context -/
+example [BI PROP] (Q : Nat → PROP) : ⊢ (∀ x, Q x) -∗ Q (y + 1) := by
   iintro HQ
-  ispecialize HQ $! (y + 1)
+  ispecialize HQ $$ %(y + 1)
   iexact HQ
 
-theorem forall_intuitionistic [BI PROP] (Q : Nat → PROP) : ⊢ □ (∀ x, Q x) -∗ □ Q y := by
+/-- Tests `ispecialize` with forall in intuitionistic context -/
+example [BI PROP] (Q : Nat → PROP) : ⊢ □ (∀ x, Q x) -∗ □ Q y := by
   iintro □HQ
-  ispecialize HQ $! y
+  ispecialize HQ $$ %y
   iexact HQ
 
-theorem forall_spatial_intuitionistic [BI PROP] (Q : Nat → PROP) : ⊢ (∀ x, □ Q x) -∗ □ Q y := by
+/-- Tests `ispecialize` with forall returning intuitionistic proposition -/
+example [BI PROP] (Q : Nat → PROP) : ⊢ (∀ x, □ Q x) -∗ □ Q y := by
   iintro HQ
-  ispecialize HQ $! y
+  ispecialize HQ $$ %y
   iexact HQ
 
-theorem forall_spatial_multiple [BI PROP] (Q : Nat → Nat → PROP) :
+/-- Tests `ispecialize` with multiple forall in spatial context -/
+example [BI PROP] (Q : Nat → Nat → PROP) :
     ⊢ (∀ x, ∀ y, Q x y) -∗ Q x y := by
   iintro HQ
-  ispecialize HQ $! x, y
+  ispecialize HQ $$ %x, %y
   iexact HQ
 
-theorem forall_intuitionistic_multiple [BI PROP] (Q : Nat → Nat → PROP) :
+/-- Tests `ispecialize` with multiple forall in intuitionistic context -/
+example [BI PROP] (Q : Nat → Nat → PROP) :
     ⊢ □ (∀ x, ∀ y, Q x y) -∗ □ Q x y := by
   iintro □HQ
-  ispecialize HQ $! x, y
+  ispecialize HQ $$ %x, %y
   iexact HQ
 
-theorem forall_multiple [BI PROP] (Q : Nat → Nat → PROP) : ⊢ (∀ x, □ (∀ y, Q x y)) -∗ □ Q x y := by
+/-- Tests `ispecialize` with nested forall and intuitionistic -/
+example [BI PROP] (Q : Nat → Nat → PROP) : ⊢ (∀ x, □ (∀ y, Q x y)) -∗ □ Q x y := by
   iintro HQ
-  ispecialize HQ $! x, y
+  ispecialize HQ $$ %x, %y
   iexact HQ
 
-theorem multiple [BI PROP] (Q : Nat → PROP) :
+/-- Tests `ispecialize` with mixed forall and wand specialization -/
+example [BI PROP] (Q : Nat → PROP) :
     ⊢ □ P1 -∗ P2 -∗ (□ P1 -∗ (∀ x, P2 -∗ Q x)) -∗ Q y := by
   iintro □HP1 HP2 HPQ
-  ispecialize HPQ with HP1
-  ispecialize HPQ $! y with HP2
+  ispecialize HPQ $$ HP1, %y, HP2
   iexact HPQ
 
+/-- Tests `ispecialize` with pure True wand using `.intro` -/
+example [BI PROP] (P : PROP) :
+    ⊢ (True -∗ P) -∗ P := by
+  iintro H
+  ispecialize H $$ %.intro
+  iexact H
+
+/-- Tests `ispecialize` with pure wand using tactic -/
+example [BI PROP] (P : PROP) :
+    ⊢ (True -∗ P) -∗ P := by
+  iintro H
+  ispecialize H $$ %(by grind)
+  iexact H
+
+/-- Tests `ispecialize` alternating pure and spatial arguments -/
+example [BI PROP] (P Q : PROP) :
+    ⊢ (∀ x, P -∗ ⌜x = 1⌝ -∗ Q) -∗ P -∗ Q := by
+  iintro H HP
+  ispecialize H $$ %_, HP, %rfl
+  iexact H
+
+/-- Tests `ispecialize` with pure subgoal -/
+example [BI PROP] (P Q : PROP) :
+    ⊢ (∀ x, P -∗ ⌜x = 1⌝ -∗ Q) -∗ P -∗ Q := by
+  iintro H HP
+  ispecialize H $$ %_, HP, %_
+  · rfl
+  iexact H
+
 end specialize
 
 -- split
 namespace split
 
-theorem and [BI PROP] (Q : PROP) : Q ⊢ Q ∧ Q := by
+/-- Tests `isplit` for conjunction -/
+example [BI PROP] (Q : PROP) : Q ⊢ Q ∧ Q := by
   iintro HQ
   isplit
   <;> iexact HQ
 
-theorem sep_left [BI PROP] [BIAffine PROP] (Q : PROP) : ⊢ P -∗ Q -∗ R -∗ P ∗ Q := by
+/-- Tests `isplitl` with explicit left hypotheses -/
+example [BI PROP] [BIAffine PROP] (Q : PROP) : ⊢ P -∗ Q -∗ R -∗ P ∗ Q := by
   iintro HP
   iintro HQ
   iintro _HR
@@ -680,7 +832,8 @@ theorem sep_left [BI PROP] [BIAffine PROP] (Q : PROP) : ⊢ P -∗ Q -∗ R -∗
   · iexact HP
   · iexact HQ
 
-theorem sep_right [BI PROP] [BIAffine PROP] (Q : PROP) : ⊢ P -∗ Q -∗ R -∗ P ∗ Q := by
+/-- Tests `isplitr` with explicit right hypotheses -/
+example [BI PROP] [BIAffine PROP] (Q : PROP) : ⊢ P -∗ Q -∗ R -∗ P ∗ Q := by
   iintro HP
   iintro HQ
   iintro _HR
@@ -688,7 +841,8 @@ theorem sep_right [BI PROP] [BIAffine PROP] (Q : PROP) : ⊢ P -∗ Q -∗ R -
   · iexact HP
   · iexact HQ
 
-theorem sep_left_all [BI PROP] [BIAffine PROP] (Q : PROP) : ⊢ P -∗ □ Q -∗ R -∗ P ∗ Q := by
+/-- Tests `isplitl` without argument -/
+example [BI PROP] [BIAffine PROP] (Q : PROP) : ⊢ P -∗ □ Q -∗ R -∗ P ∗ Q := by
   iintro HP
   iintro □HQ
   iintro _HR
@@ -696,7 +850,8 @@ theorem sep_left_all [BI PROP] [BIAffine PROP] (Q : PROP) : ⊢ P -∗ □ Q -
   · iexact HP
   · iexact HQ
 
-theorem sep_right_all [BI PROP] [BIAffine PROP] (Q : PROP) : ⊢ □ P -∗ Q -∗ R -∗ P ∗ Q := by
+/-- Tests `isplitr` without argument -/
+example [BI PROP] [BIAffine PROP] (Q : PROP) : ⊢ □ P -∗ Q -∗ R -∗ P ∗ Q := by
   iintro □HP
   iintro HQ
   iintro _HR
@@ -709,17 +864,20 @@ end split
 -- left / right
 namespace leftright
 
-theorem left [BI PROP] (P : PROP) : P ⊢ P ∨ Q := by
+/-- Tests `ileft` -/
+example [BI PROP] (P : PROP) : P ⊢ P ∨ Q := by
   iintro HP
   ileft
   iexact HP
 
-theorem right [BI PROP] (Q : PROP) : Q ⊢ P ∨ Q := by
+/-- Tests `iright` -/
+example [BI PROP] (Q : PROP) : Q ⊢ P ∨ Q := by
   iintro HQ
   iright
   iexact HQ
 
-theorem complex [BI PROP] (P Q : PROP) : ⊢ P -∗ Q -∗ P ∗ (R ∨ Q ∨ R) := by
+/-- Tests nested disjunction with left and right -/
+example [BI PROP] (P Q : PROP) : ⊢ P -∗ Q -∗ P ∗ (R ∨ Q ∨ R) := by
   iintro HP HQ
   isplitl [HP]
   · iassumption
@@ -732,177 +890,212 @@ end leftright
 -- cases
 namespace cases
 
-theorem rename [BI PROP] (P : PROP) : P ⊢ P := by
+/-- Tests `icases` for simple renaming -/
+example [BI PROP] (P : PROP) : P ⊢ P := by
   iintro HP
   icases HP with H
   iexact H
 
-theorem clear [BI PROP] (P Q : PROP) : ⊢ P -∗ <affine> Q -∗ P := by
+/-- Tests `icases` to clear hypothesis -/
+example [BI PROP] (P Q : PROP) : ⊢ P -∗ <affine> Q -∗ P := by
   iintro HP
   iintro HQ
   icases HQ with -
   iexact HP
 
-theorem and [BI PROP] (Q : PROP) : □ (P1 ∧ P2 ∧ Q) ⊢ Q := by
+/-- Tests `icases` with nested conjunction -/
+example [BI PROP] (Q : PROP) : □ (P1 ∧ P2 ∧ Q) ⊢ Q := by
   iintro □HP
   icases HP with ⟨_HP1, _HP2, HQ⟩
   iexact HQ
 
-theorem and_intuitionistic [BI PROP] (Q : PROP) : □ P ∧ Q ⊢ Q := by
+/-- Tests `icases` with intuitionistic conjunction -/
+example [BI PROP] (Q : PROP) : □ P ∧ Q ⊢ Q := by
   iintro HPQ
   icases HPQ with ⟨_HP, HQ⟩
   iexact HQ
 
-theorem and_persistent_left [BI PROP] (Q : PROP) : <pers> Q ∧ <affine> P ⊢ Q := by
+/-- Tests `icases` on conjunction with persistent left -/
+example [BI PROP] (Q : PROP) : <pers> Q ∧ <affine> P ⊢ Q := by
   iintro HQP
   icases HQP with ⟨□HQ, _HP⟩
   iexact HQ
 
-theorem and_persistent_right [BI PROP] (Q : PROP) : Q ∧ <pers> P ⊢ Q := by
+/-- Tests `icases` on conjunction with persistent right -/
+example [BI PROP] (Q : PROP) : Q ∧ <pers> P ⊢ Q := by
   iintro HQP
   icases HQP with ⟨HQ, _HP⟩
   iexact HQ
 
-theorem sep [BI PROP] [BIAffine PROP] (Q : PROP) : P1 ∗ P2 ∗ Q ⊢ Q := by
+/-- Tests `icases` with nested separating conjunction -/
+example [BI PROP] [BIAffine PROP] (Q : PROP) : P1 ∗ P2 ∗ Q ⊢ Q := by
   iintro HPQ
   icases HPQ with ⟨_HP1, _HP2, HQ⟩
   iexact HQ
 
-theorem disjunction [BI PROP] (Q : PROP) : Q ⊢ <affine> (P1 ∨ P2 ∨ P3) -∗ Q := by
+/-- Tests `icases` with nested disjunction -/
+example [BI PROP] (Q : PROP) : Q ⊢ <affine> (P1 ∨ P2 ∨ P3) -∗ Q := by
   iintro HQ
   iintro HP
   icases HP with (_HP1 | _HP2 | _HP3)
   <;> iexact HQ
 
-theorem conjunction_and_disjunction [BI PROP] [BIAffine PROP] (Q : PROP) :
+/-- Tests `icases` with complex mixed conjunction and disjunction -/
+example [BI PROP] [BIAffine PROP] (Q : PROP) :
     (P11 ∨ P12 ∨ P13) ∗ P2 ∗ (P31 ∨ P32 ∨ P33) ∗ Q ⊢ Q := by
   iintro HP
   icases HP with ⟨_HP11 | _HP12 | _HP13, HP2, HP31 | HP32 | HP33, HQ⟩
   <;> iexact HQ
 
-theorem move_to_pure [BI PROP] (Q : PROP) : ⊢ <affine> ⌜⊢ Q⌝ -∗ Q := by
+/-- Tests `icases` moving pure to Lean context with ⌜⌝ -/
+example [BI PROP] (Q : PROP) : ⊢ <affine> ⌜⊢ Q⌝ -∗ Q := by
   iintro HQ
   icases HQ with ⌜HQ⌝
   istop
   exact HQ
 
-theorem move_to_pure_ascii [BI PROP] (Q : PROP) : ⊢ <affine> ⌜⊢ Q⌝ -∗ Q := by
+/-- Tests `icases` moving pure to Lean context with % -/
+example [BI PROP] (Q : PROP) : ⊢ <affine> ⌜⊢ Q⌝ -∗ Q := by
   iintro HQ
   icases HQ with %HQ
   istop
   exact HQ
 
-theorem move_to_intuitionistic [BI PROP] (Q : PROP) : ⊢ □ Q -∗ Q := by
+/-- Tests `icases` moving to intuitionistic with □ -/
+example [BI PROP] (Q : PROP) : ⊢ □ Q -∗ Q := by
   iintro HQ
   icases HQ with □HQ
   iexact HQ
 
-theorem move_to_intuitionistic_ascii [BI PROP] (Q : PROP) : ⊢ □ Q -∗ Q := by
+/-- Tests `icases` moving to intuitionistic with # -/
+example [BI PROP] (Q : PROP) : ⊢ □ Q -∗ Q := by
   iintro HQ
   icases HQ with #HQ
   iexact HQ
 
-theorem move_to_spatial [BI PROP] (Q : PROP) : ⊢ □ Q -∗ Q := by
+/-- Tests `icases` moving to spatial with ∗ -/
+example [BI PROP] (Q : PROP) : ⊢ □ Q -∗ Q := by
   iintro □HQ
   icases HQ with ∗HQ
   iexact HQ
 
-theorem move_to_spatial_ascii [BI PROP] (Q : PROP) : ⊢ □ Q -∗ Q := by
+/-- Tests `icases` moving to spatial with * -/
+example [BI PROP] (Q : PROP) : ⊢ □ Q -∗ Q := by
   iintro □HQ
   icases HQ with *HQ
   iexact HQ
 
-theorem move_to_pure_conjunction [BI PROP] (Q : PROP) : ⊢ <affine> ⌜φ⌝ ∗ Q -∗ Q := by
+/-- Tests `icases` with pure in conjunction -/
+example [BI PROP] (Q : PROP) : ⊢ <affine> ⌜φ⌝ ∗ Q -∗ Q := by
   iintro HφQ
   icases HφQ with ⟨⌜Hφ⌝, HQ⟩
   iexact HQ
 
-theorem move_to_pure_disjunction [BI PROP] (Q : PROP) :
+/-- Tests `icases` with pure in disjunction -/
+example [BI PROP] (Q : PROP) :
     ⊢ <affine> ⌜φ1⌝ ∨ <affine> ⌜φ2⌝ -∗ Q -∗ Q := by
   iintro Hφ
   iintro HQ
   icases Hφ with (⌜Hφ1⌝ | ⌜Hφ2⌝)
   <;> iexact HQ
 
-theorem move_to_intuitionistic_conjunction [BI PROP] (Q : PROP) : ⊢ □ P ∗ Q -∗ Q := by
+/-- Tests `icases` with intuitionistic in conjunction -/
+example [BI PROP] (Q : PROP) : ⊢ □ P ∗ Q -∗ Q := by
   iintro HPQ
   icases HPQ with ⟨□_HP, HQ⟩
   iexact HQ
 
-theorem move_to_intuitionistic_disjunction [BI PROP] (Q : PROP) : ⊢ □ Q ∨ Q -∗ Q := by
+/-- Tests `icases` with intuitionistic in disjunction -/
+example [BI PROP] (Q : PROP) : ⊢ □ Q ∨ Q -∗ Q := by
   iintro HQQ
   icases HQQ with (□HQ | HQ)
   <;> iexact HQ
 
-theorem move_to_spatial_conjunction [BI PROP] (Q : PROP) : ⊢ □ (P ∧ Q) -∗ Q := by
+/-- Tests `icases` moving to spatial inside intuitionistic conjunction -/
+example [BI PROP] (Q : PROP) : ⊢ □ (P ∧ Q) -∗ Q := by
   iintro □HPQ
   icases HPQ with ⟨_HP, ∗HQ⟩
   iexact HQ
 
-theorem move_to_spatial_disjunction [BI PROP] (Q : PROP) : ⊢ □ (Q ∨ Q) -∗ Q := by
+/-- Tests `icases` with or inside intuitionistic, moving one to spatial -/
+example [BI PROP] (Q : PROP) : ⊢ □ (Q ∨ Q) -∗ Q := by
   iintro □HPQ
   icases HPQ with (HQ | ∗HQ)
   <;> iexact HQ
 
-theorem move_to_intuitionistic_and_back_conjunction [BI PROP] (Q : PROP) : ⊢ □ (P ∧ Q) -∗ Q := by
+/-- Tests `icases` moving whole hypothesis to intuitionistic then destructing -/
+example [BI PROP] (Q : PROP) : ⊢ □ (P ∧ Q) -∗ Q := by
   iintro HPQ
   icases HPQ with □⟨_HP, ∗HQ⟩
   iexact HQ
 
-theorem move_to_intuitionistic_and_back_disjunction [BI PROP] (Q : PROP) : ⊢ □ (Q ∨ Q) -∗ Q := by
+/-- Tests `icases` with or, moving whole to intuitionistic -/
+example [BI PROP] (Q : PROP) : ⊢ □ (Q ∨ Q) -∗ Q := by
   iintro HPQ
   icases HPQ with □(HQ | ∗HQ)
   <;> iexact HQ
 
-theorem conjunction_clear [BI PROP] [BIAffine PROP] (Q : PROP) : Q ∗ P ⊢ Q := by
+/-- Tests `icases` clearing in conjunction -/
+example [BI PROP] [BIAffine PROP] (Q : PROP) : Q ∗ P ⊢ Q := by
   iintro HQP
   icases HQP with ⟨HQ, -⟩
   iexact HQ
 
-theorem disjunction_clear [BI PROP] [BIAffine PROP] (Q : PROP) : Q ⊢ P1 ∨ P2 -∗ Q := by
+/-- Tests `icases` clearing in disjunction -/
+example [BI PROP] [BIAffine PROP] (Q : PROP) : Q ⊢ P1 ∨ P2 -∗ Q := by
   iintro HQ
   iintro HP
   icases HP with (- | _HP2)
   <;> iexact HQ
 
-theorem and_destruct_spatial_right [BI PROP] (Q : PROP) : P ∧ Q ⊢ Q := by
+/-- Tests `icases` destructing conjunction left -/
+example [BI PROP] (Q : PROP) : P ∧ Q ⊢ Q := by
   iintro HPQ
   icases HPQ with ⟨-, HQ⟩
   iexact HQ
 
-theorem and_destruct_spatial_left [BI PROP] (Q : PROP) : Q ∧ P ⊢ Q := by
+/-- Tests `icases` destructing conjunction right -/
+example [BI PROP] (Q : PROP) : Q ∧ P ⊢ Q := by
   iintro HQP
   icases HQP with ⟨HQ, -⟩
   iexact HQ
 
-theorem and_clear_spatial_multiple [BI PROP] (Q : PROP) : P1 ∧ P2 ∧ Q ∧ P3 ⊢ Q := by
+/-- Tests `icases` destructing multiple conjunctions  -/
+example [BI PROP] (Q : PROP) : P1 ∧ P2 ∧ Q ∧ P3 ⊢ Q := by
   iintro HPQ
   icases HPQ with ⟨-, -, HQ, -⟩
   iexact HQ
 
-theorem and_destruct_intuitionistic_right [BI PROP] (Q : PROP) : □ (P ∧ Q) ⊢ Q := by
+/-- Tests `icases` destructing intuitionistic conjunction, clearing left -/
+example [BI PROP] (Q : PROP) : □ (P ∧ Q) ⊢ Q := by
   iintro □HPQ
   icases HPQ with ⟨-, HQ⟩
   iexact HQ
 
-theorem and_destruct_intuitionistic_left [BI PROP] (Q : PROP) : □ (Q ∧ P) ⊢ Q := by
+/-- Tests `icases` destructing intuitionistic conjunction, clearing right -/
+example [BI PROP] (Q : PROP) : □ (Q ∧ P) ⊢ Q := by
   iintro □HQP
   icases HQP with ⟨HQ, -⟩
   iexact HQ
 
-theorem and_clear_intuitionistic_multiple [BI PROP] (Q : PROP) : □ (P1 ∧ P2 ∧ Q ∧ P3) ⊢ Q := by
+/-- Tests `icases` destructing multiple intuitionistic conjunctions -/
+example [BI PROP] (Q : PROP) : □ (P1 ∧ P2 ∧ Q ∧ P3) ⊢ Q := by
   iintro □HPQ
   icases HPQ with ⟨-, -, HQ, -⟩
   iexact HQ
 
-theorem «exists» [BI PROP] (Q : Nat → PROP) : (∃ x, Q x) ⊢ ∃ x, Q x ∨ False := by
+/-- Tests `icases` with existential -/
+example [BI PROP] (Q : Nat → PROP) : (∃ x, Q x) ⊢ ∃ x, Q x ∨ False := by
   iintro ⟨x, H⟩
   iexists x
   ileft
   iexact H
 
-theorem exists_intuitionistic [BI PROP] (Q : Nat → PROP) : □ (∃ x, Q x) ⊢ ∃ x, □ Q x ∨ False := by
+/-- Tests `icases` with intuitionistic existential -/
+example [BI PROP] (Q : Nat → PROP) : □ (∃ x, Q x) ⊢ ∃ x, □ Q x ∨ False := by
   iintro ⟨x, □H⟩
   iexists x
   ileft
   iexact H
+
+end cases