From 6df3f662f0f23e837b70ef1f4a8b8dde4a7aa521 Mon Sep 17 00:00:00 2001
From: Hannes Schulze <haschu0103@outlook.de>
Date: Fri, 24 Oct 2025 18:20:43 +0200
Subject: use simp only everywhere except for last goal

---
 Nominal/Instances/Basic.lean | 29 ++++++++++++++++++++---------
 1 file changed, 20 insertions(+), 9 deletions(-)

(limited to 'Nominal/Instances/Basic.lean')

diff --git a/Nominal/Instances/Basic.lean b/Nominal/Instances/Basic.lean
index 577cff8..2786c86 100644
--- a/Nominal/Instances/Basic.lean
+++ b/Nominal/Instances/Basic.lean
@@ -30,12 +30,15 @@ instance : NominalSet 𝔸 𝔸 where
   · apply support_subset_supp 𝔸
     · exact NominalSet.supp_atom' 𝔸
     · exact Set.finite_singleton a
-  · suffices a ∈ supp 𝔸 a by simp; intro h; rwa [h]
+  · suffices a ∈ supp 𝔸 a by
+      simp only [Set.mem_singleton_iff]
+      intro h
+      rwa [h]
     intro S ⟨hsuppS, hfinS⟩
     by_contra ha
     -- Show that we can find a fresh c with (a c) ∈ fix S
     obtain ⟨c, hc⟩ := Infinite.exists_notMem_finset ({a} ∪ hfinS.toFinset)
-    simp at hc
+    simp only [Finset.singleton_union, Finset.mem_insert, Set.Finite.mem_toFinset, not_or] at hc
     have hfix : ⸨a, c⸩ ∈ PermSet.fix 𝔸 S := by
       intro d hd
       apply FPerm.swap_neq (ne_of_mem_of_not_mem hd ha) (ne_of_mem_of_not_mem hd hc.right)
@@ -64,7 +67,7 @@ def NominalSet.discrete (X : Type) : NominalSet.Discrete 𝔸 X where
   fin_supp := by
     intro x; use {}
     constructor
-    · intro π hπ; simp [PermSet.fix']; rfl
+    · intro π hπ; simp only [PermSet.fix', Set.mem_setOf_eq]; rfl
     · exact Set.finite_empty
 
 @[simp] lemma NominalSet.Discrete.supp {X : Type} [NominalSet.Discrete 𝔸 X] {x : X} : supp 𝔸 x = ∅ := by
@@ -118,13 +121,17 @@ instance : NominalSet 𝔸 (X × Y) where
         intro S hS
         suffices PermSet.HasSupport 𝔸 x S by apply hx; exact ⟨this, hS.out.right⟩
         intro π hπ
-        suffices π ∈ PermSet.fix' 𝔸 (x, y) by simp [PermSet.fix'] at this; exact this.left
+        suffices π ∈ PermSet.fix' 𝔸 (x, y) by
+          simp only [PermSet.fix', smul_mk, mk.injEq, Set.mem_setOf_eq] at this
+          exact this.left
         exact (hS.out.left hπ).out
     | inr hy =>
         intro S hS
         suffices PermSet.HasSupport 𝔸 y S by apply hy; exact ⟨this, hS.out.right⟩
         intro π hπ
-        suffices π ∈ PermSet.fix' 𝔸 (x, y) by simp [PermSet.fix'] at this; exact this.right
+        suffices π ∈ PermSet.fix' 𝔸 (x, y) by
+          simp only [PermSet.fix', smul_mk, mk.injEq, Set.mem_setOf_eq] at this
+          exact this.right
         exact (hS.out.left hπ).out
 
 @[simp] lemma fresh {a : 𝔸} {x : X} {y : Y} : a # (x, y) ↔ a # x ∧ a # y := by
@@ -158,12 +165,12 @@ variable {X Y : Type} [NominalSet 𝔸 X] [NominalSet 𝔸 Y]
 
 private lemma supp_inl' {x : X} : PermSet.HasSupport 𝔸 (.inl x : X ⊕ Y) (NominalSet.supp 𝔸 x) := by
   intro π hπ
-  simp [PermSet.fix']
+  simp only [PermSet.fix', smul_inl, inl.injEq, Set.mem_setOf_eq]
   exact NominalSet.supp_is_support 𝔸 hπ
 
 private lemma supp_inr' {y : Y} : PermSet.HasSupport 𝔸 (.inr y : X ⊕ Y) (NominalSet.supp 𝔸 y) := by
   intro π hπ
-  simp [PermSet.fix']
+  simp only [PermSet.fix', smul_inr, inr.injEq, Set.mem_setOf_eq]
   exact NominalSet.supp_is_support 𝔸 hπ
 
 instance : NominalSet 𝔸 (X ⊕ Y) where
@@ -181,7 +188,9 @@ instance : NominalSet 𝔸 (X ⊕ Y) where
   · intro hb S hS
     suffices PermSet.HasSupport 𝔸 x S by apply hb; exact ⟨this, hS.out.right⟩
     intro π hπ
-    suffices π ∈ PermSet.fix' 𝔸 (.inl x : X ⊕ Y) by simp [PermSet.fix'] at this; exact this
+    suffices π ∈ PermSet.fix' 𝔸 (.inl x : X ⊕ Y) by
+      simp only [PermSet.fix', smul_inl, inl.injEq, Set.mem_setOf_eq] at this
+      exact this
     exact (hS.out.left hπ).out
 
 @[simp] lemma supp_inr {y : Y} : NominalSet.supp 𝔸 (.inr y : X ⊕ Y) = NominalSet.supp 𝔸 y := by
@@ -192,7 +201,9 @@ instance : NominalSet 𝔸 (X ⊕ Y) where
   · intro hb S hS
     suffices PermSet.HasSupport 𝔸 y S by apply hb; exact ⟨this, hS.out.right⟩
     intro π hπ
-    suffices π ∈ PermSet.fix' 𝔸 (.inr y : X ⊕ Y) by simp [PermSet.fix'] at this; exact this
+    suffices π ∈ PermSet.fix' 𝔸 (.inr y : X ⊕ Y) by
+      simp only [PermSet.fix', smul_inr, inr.injEq, Set.mem_setOf_eq] at this
+      exact this
     exact (hS.out.left hπ).out
 
 @[simp] lemma fresh_inl {a : 𝔸} {x : X} : a # (.inl x : X ⊕ Y) ↔ a # x := by
-- 
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