473 lines
11 KiB
Text
473 lines
11 KiB
Text
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set_option pp.mvars.levels false
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/-!
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This file tests the `rfl` tactic:
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* Extensibility
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* Error messages
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* Effect of `with_reducible`
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-/
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-- First, let's see what `rfl` does:
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/--
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error: tactic 'rfl' failed, the left-hand side
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false
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is not definitionally equal to the right-hand side
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true
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⊢ false = true
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-/
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#guard_msgs in
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example : false = true := by rfl
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-- Now to `apply_rfl`.
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-- A plain reflexive predicate
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inductive P : α → α → Prop where | refl : P a a
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attribute [refl] P.refl
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-- Plain error
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/--
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error: tactic 'rfl' failed, the left-hand side
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42
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is not definitionally equal to the right-hand side
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23
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⊢ P 42 23
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-/
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#guard_msgs in
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example : P 42 23 := by apply_rfl
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-- Revealing implicit arguments
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opaque withImplicitNat {n : Nat} : Nat
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/--
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error: tactic 'rfl' failed, the left-hand side
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@withImplicitNat 42
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is not definitionally equal to the right-hand side
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@withImplicitNat 23
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⊢ P withImplicitNat withImplicitNat
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-/
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#guard_msgs in
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example : P (@withImplicitNat 42) (@withImplicitNat 23) := by apply_rfl
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-- Exhaustive testing of various combinations:
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-- In addition to Eq, HEq and Iff we test four relations:
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-- Defeq to relation `P` at reducible transparency
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abbrev Q : α → α → Prop := P
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-- Defeq to relation `P` at default transparency
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def Q' : α → α → Prop := P
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-- No refl attribute
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inductive R : α → α → Prop where | refl : R a a
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/-
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Now we systematically test all relations with
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* syntactic equal arguments
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* reducibly equal arguments
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* semireducibly equal arguments
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* nonequal arguments
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and all using `apply_rfl` and `with_reducible apply_rfl`
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-/
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-- Syntactic equal
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example : true = true := by apply_rfl
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example : HEq true true := by apply_rfl
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example : True ↔ True := by apply_rfl
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example : P true true := by apply_rfl
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example : Q true true := by apply_rfl
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/--
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error: tactic 'rfl' failed, no @[refl] lemma registered for relation
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Q'
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⊢ Q' true true
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-/
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#guard_msgs in example : Q' true true := by apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, no @[refl] lemma registered for relation
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R
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⊢ R true true
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-/
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#guard_msgs in example : R true true := by apply_rfl -- Error
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example : true = true := by with_reducible apply_rfl
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example : HEq true true := by with_reducible apply_rfl
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example : True ↔ True := by with_reducible apply_rfl
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example : P true true := by with_reducible apply_rfl
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example : Q true true := by with_reducible apply_rfl
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/--
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error: tactic 'rfl' failed, no @[refl] lemma registered for relation
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Q'
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⊢ Q' true true
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-/
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#guard_msgs in
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example : Q' true true := by with_reducible apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, no @[refl] lemma registered for relation
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R
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⊢ R true true
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-/
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#guard_msgs in
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example : R true true := by with_reducible apply_rfl -- Error
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-- Reducibly equal
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abbrev true' := true
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abbrev True' := True
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example : true' = true := by apply_rfl
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example : HEq true' true := by apply_rfl
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example : True' ↔ True := by apply_rfl
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example : P true' true := by apply_rfl
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example : Q true' true := by apply_rfl
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/--
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error: tactic 'rfl' failed, no @[refl] lemma registered for relation
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Q'
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⊢ Q' true' true'
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-/
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#guard_msgs in
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example : Q' true' true := by apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, no @[refl] lemma registered for relation
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R
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⊢ R true' true'
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-/
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#guard_msgs in
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example : R true' true := by apply_rfl -- Error
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example : true' = true := by with_reducible apply_rfl
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example : HEq true' true := by with_reducible apply_rfl
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example : True' ↔ True := by with_reducible apply_rfl
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example : P true' true := by with_reducible apply_rfl
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example : Q true' true := by with_reducible apply_rfl -- NB: No error, Q and true' reducible
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/--
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error: tactic 'rfl' failed, no @[refl] lemma registered for relation
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Q'
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⊢ Q' true' true'
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-/
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#guard_msgs in
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example : Q' true' true := by with_reducible apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, no @[refl] lemma registered for relation
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R
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⊢ R true' true'
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-/
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#guard_msgs in
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example : R true' true := by with_reducible apply_rfl -- Error
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-- Equal at default transparency only
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def true'' := true
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def True'' := True
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example : true'' = true := by apply_rfl
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example : HEq true'' true := by apply_rfl
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example : True'' ↔ True := by apply_rfl
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example : P true'' true := by apply_rfl
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example : Q true'' true := by apply_rfl
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/--
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error: tactic 'rfl' failed, no @[refl] lemma registered for relation
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Q'
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⊢ Q' true'' true''
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-/
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#guard_msgs in
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example : Q' true'' true := by apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, no @[refl] lemma registered for relation
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R
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⊢ R true'' true''
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-/
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#guard_msgs in
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example : R true'' true := by apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, the left-hand side
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true''
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is not definitionally equal to the right-hand side
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true
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⊢ true'' = true
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-/
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#guard_msgs in
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example : true'' = true := by with_reducible apply_rfl -- Error
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/--
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error: tactic 'apply' failed, could not unify the conclusion of 'HEq.refl'
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@HEq ?α ?a ?α ?a
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with the goal
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@HEq Bool true'' Bool true
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Note: The full type of 'HEq.refl' is
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∀ {α : Sort _} (a : α), a ≍ a
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⊢ true'' ≍ true
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-/
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#guard_msgs in
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example : HEq true'' true := by with_reducible apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, the left-hand side
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True''
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is not definitionally equal to the right-hand side
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True
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⊢ True'' ↔ True
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-/
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#guard_msgs in
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example : True'' ↔ True := by with_reducible apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, the left-hand side
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true''
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is not definitionally equal to the right-hand side
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true
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⊢ P true'' true
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-/
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#guard_msgs in
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example : P true'' true := by with_reducible apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, the left-hand side
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true''
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is not definitionally equal to the right-hand side
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true
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⊢ Q true'' true
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-/
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#guard_msgs in
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example : Q true'' true := by with_reducible apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, the left-hand side
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true''
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is not definitionally equal to the right-hand side
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true
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⊢ Q' true'' true
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-/
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#guard_msgs in
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example : Q' true'' true := by with_reducible apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, the left-hand side
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true''
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is not definitionally equal to the right-hand side
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true
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⊢ R true'' true
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-/
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#guard_msgs in
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example : R true'' true := by with_reducible apply_rfl -- Error
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-- Unequal
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/--
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error: tactic 'rfl' failed, the left-hand side
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false
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is not definitionally equal to the right-hand side
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true
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⊢ false = true
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-/
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#guard_msgs in
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example : false = true := by apply_rfl -- Error
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/--
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error: tactic 'apply' failed, could not unify the conclusion of 'HEq.refl'
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?a ≍ ?a
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with the goal
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false ≍ true
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Note: The full type of 'HEq.refl' is
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∀ {α : Sort _} (a : α), a ≍ a
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⊢ false ≍ true
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-/
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#guard_msgs in
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example : HEq false true := by apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, the left-hand side
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False
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is not definitionally equal to the right-hand side
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True
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⊢ False ↔ True
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-/
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#guard_msgs in
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example : False ↔ True := by apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, the left-hand side
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false
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is not definitionally equal to the right-hand side
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true
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⊢ P false true
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-/
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#guard_msgs in
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example : P false true := by apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, the left-hand side
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false
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is not definitionally equal to the right-hand side
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true
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⊢ Q false true
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-/
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#guard_msgs in
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example : Q false true := by apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, the left-hand side
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false
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is not definitionally equal to the right-hand side
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true
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⊢ Q' false true
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-/
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#guard_msgs in
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example : Q' false true := by apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, the left-hand side
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false
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is not definitionally equal to the right-hand side
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true
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⊢ R false true
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-/
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#guard_msgs in
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example : R false true := by apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, the left-hand side
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false
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is not definitionally equal to the right-hand side
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true
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⊢ false = true
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-/
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#guard_msgs in
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example : false = true := by with_reducible apply_rfl -- Error
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/--
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error: tactic 'apply' failed, could not unify the conclusion of 'HEq.refl'
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?a ≍ ?a
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with the goal
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false ≍ true
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Note: The full type of 'HEq.refl' is
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∀ {α : Sort _} (a : α), a ≍ a
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⊢ false ≍ true
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-/
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#guard_msgs in
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example : HEq false true := by with_reducible apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, the left-hand side
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False
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is not definitionally equal to the right-hand side
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True
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⊢ False ↔ True
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-/
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#guard_msgs in
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example : False ↔ True := by with_reducible apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, the left-hand side
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false
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is not definitionally equal to the right-hand side
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true
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⊢ P false true
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-/
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#guard_msgs in
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example : P false true := by with_reducible apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, the left-hand side
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false
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is not definitionally equal to the right-hand side
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true
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⊢ Q false true
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-/
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#guard_msgs in
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example : Q false true := by with_reducible apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, the left-hand side
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false
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is not definitionally equal to the right-hand side
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true
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⊢ Q' false true
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-/
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#guard_msgs in
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example : Q' false true := by with_reducible apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, the left-hand side
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false
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is not definitionally equal to the right-hand side
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true
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⊢ R false true
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-/
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#guard_msgs in
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example : R false true := by with_reducible apply_rfl -- Error
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-- Inheterogeneous unequal
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/--
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error: tactic 'apply' failed, could not unify the conclusion of 'HEq.refl'
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?a ≍ ?a
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with the goal
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true ≍ 1
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Note: The full type of 'HEq.refl' is
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∀ {α : Sort _} (a : α), a ≍ a
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⊢ true ≍ 1
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-/
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#guard_msgs in
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example : HEq true 1 := by apply_rfl -- Error
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/--
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error: tactic 'apply' failed, could not unify the conclusion of 'HEq.refl'
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?a ≍ ?a
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with the goal
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true ≍ 1
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Note: The full type of 'HEq.refl' is
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∀ {α : Sort _} (a : α), a ≍ a
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⊢ true ≍ 1
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-/
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#guard_msgs in
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example : HEq true 1 := by with_reducible apply_rfl -- Error
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-- Rfl lemma with side condition:
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-- Error message should show left-over goal
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inductive S : Bool → Bool → Prop where | refl : a = true → S a a
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attribute [refl] S.refl
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/--
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error: tactic 'rfl' failed, the left-hand side
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true
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is not definitionally equal to the right-hand side
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false
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⊢ S true false
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-/
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#guard_msgs in
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example : S true false := by apply_rfl -- Error
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/--
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error: tactic 'rfl' failed, the left-hand side
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true
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is not definitionally equal to the right-hand side
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false
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⊢ S true false
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-/
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#guard_msgs in
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example : S true false := by with_reducible apply_rfl -- Error
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/--
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error: unsolved goals
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case a
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⊢ true = true
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-/
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#guard_msgs in
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example : S true true := by apply_rfl -- Error (left-over goal)
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/--
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error: unsolved goals
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case a
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⊢ true = true
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-/
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#guard_msgs in
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example : S true true := by with_reducible apply_rfl -- Error (left-over goal)
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/--
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error: unsolved goals
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case a
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⊢ false = true
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-/
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#guard_msgs in
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example : S false false := by apply_rfl -- Error (left-over goal)
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/--
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error: unsolved goals
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case a
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⊢ false = true
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-/
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#guard_msgs in
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example : S false false := by with_reducible apply_rfl -- Error (left-over goal)
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