ac600853c0
This PR improves the `congr` tactic so that it can handle function applications with fewer arguments than the arity of the head function. This also fixes a bug where `congr` could not make progress with `Set`-valued functions in Mathlib, since `Set` was being unfolded and making such functions have an apparently higher arity. This addresses issue #2128 for the `congr` tactic, but not `simp` and others.
73 lines
2 KiB
Text
73 lines
2 KiB
Text
/-!
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# Tests of the `congr` tactic
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-/
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example (h : a = b) : Nat.succ (a + 1) = Nat.succ (b + 1) := by
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congr
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example (h : a = b) : Nat.succ (a + 1) = Nat.succ (b + 1) := by
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congr 1
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show a + 1 = b + 1
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rw [h]
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def f (p : Prop) (a : Nat) (h : a > 0) [Decidable p] : Nat :=
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if p then
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a - 1
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else
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a + 1
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example (h : a = b) : f True (a + 1) (by simp +arith) = f (0 = 0) (b + 1) (by simp +arith) := by
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congr
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decide
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example (h : a = b) : f True (a + 1) (by simp +arith) = f (0 = 0) (b + 1) (by simp +arith) := by
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congr 1
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· decide
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· show a + 1 = b + 1
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rw [h]
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example (h₁ : α = β) (h₂ : α = γ) (a : α) : cast h₁ a ≍ cast h₂ a := by
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congr
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· subst h₁ h₂; rfl
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· subst h₁ h₂; apply heq_of_eq; rfl
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example (f : Nat → Nat) (g : Nat → Nat) : f (g (x + y)) = f (g (y + x)) := by
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congr 2
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rw [Nat.add_comm]
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example (p q r : Prop) (h : q = r) : (p → q) = (p → r) := by
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congr
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example (p q r s : Prop) (h₁ : q = r) (h₂ : r = s) : (p → q) = (p → s) := by
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congr
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rw [h₁, h₂]
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namespace Tao1
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/-!
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Reported on Zulip: https://leanprover.zulipchat.com/#narrow/channel/287929-mathlib4/topic/congr.20unexpectedly.20fails.20with.20Set.2Eimage/near/527382064
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The `congr` tactic used to make no progress here because `congr` was computing the arity from the head function
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(`Set.image`) rather than from the actual number of supplied arguments.
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-/
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def Set (α : Type _) := α → Prop
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def Set.image {α β : Type _} (f : α → β) (s : Set α) : Set β :=
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fun y => ∃ x, s x ∧ f x = y
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infixl:80 " '' " => Set.image
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example {X Y : Type} (f : X → Y) (E F : Set X) (h : E = F) : f '' E = f '' F := by
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congr
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/-!
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This also didn't work, for the same reason.
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-/
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example {X Y : Type} (f g : X → Y) (h : f = g) : Set.image f = Set.image g := by
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congr
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/-!
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The `HEq` version also did not work.
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-/
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example {X Y Y' : Type} (h : Y = Y') (f : X → Y) (f' : X → Y') (hf : f ≍ f') (E : Set X) : f '' E ≍ f' '' E := by
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congr
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end Tao1
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