use simp only everywhere except for last goal

7 files changed, 40 insertions, 24 deletions

diff --git a/Nominal/Category.lean b/Nominal/Category.lean
index 29a0b73..721dfb0 100644
--- a/Nominal/Category.lean
+++ b/Nominal/Category.lean
@@ -66,6 +66,10 @@ instance : ConcreteCategory (Nom 𝔸) (· →ₑ[𝔸] ·) where
 instance : HasForget₂ (Nom 𝔸) (Action Type (FPerm 𝔸)) where
   forget₂ :=
     { obj X := Action.ofMulAction (FPerm 𝔸) X.carrier
-      map f := ⟨f.toFun, by intro π; ext x; simp; apply f.equivariant⟩ }
+      map f := ⟨f.toFun, by
+        intro π
+        ext x
+        simp only [EquivariantMap.toFun_eq_coe, types_comp_apply, Action.ofMulAction_apply]
+        apply f.equivariant⟩ }
 
 end Forget
diff --git a/Nominal/Equivariance.lean b/Nominal/Equivariance.lean
index 368d266..50cf7c2 100644
--- a/Nominal/Equivariance.lean
+++ b/Nominal/Equivariance.lean
@@ -38,7 +38,7 @@ lemma equivariant_set_iff {S : Set X} : EquivariantSet 𝔸 S ↔ ∀ (x : X) (�
     · intro hx; exact h x π hx
     · intro hx
       apply (h (π • x) π⁻¹) at hx
-      simp at hx
+      simp only [inv_smul_smul] at hx
       exact hx
   · intro h x π hx
     exact (h x π).mp hx
@@ -50,7 +50,7 @@ lemma equivariant_relation_iff {R : Rel X Y} : EquivariantRelation 𝔸 R ↔ �
     · intro hxy; exact h x y π hxy
     · intro hxy
       apply (h (π • x) (π • y) π⁻¹) at hxy
-      simp at hxy
+      simp only [inv_smul_smul] at hxy
       exact hxy
   · intro h x y π hxy
     exact (h x y π).mp hxy
@@ -201,7 +201,7 @@ private lemma for_fresh_iff₁ : EquivariantRelation 𝔸 R → (И a, R a x) �
   intro hequi h
   have hinf : ({a | R a x} \ NominalSet.supp 𝔸 x).Infinite := Set.Infinite.diff h (NominalSet.supp_finite 𝔸)
   obtain ⟨a, ha⟩ := hinf.nonempty
-  simp at ha
+  simp only [Set.mem_diff, Set.mem_setOf_eq] at ha
   use a; exact ha.symm
 
 private lemma for_fresh_iff₂ : EquivariantRelation 𝔸 R → (∃ a, a # x ∧ R a x) → (∀ a, a # x → R a x) := by
@@ -220,7 +220,7 @@ private lemma for_fresh_iff₃ : EquivariantRelation 𝔸 R → (∀ a, a # x �
     intro a ha
     rw [Set.mem_compl_iff] at ha
     exact h a ha
-  simp [NominalSet.ForFresh]
+  simp only [NominalSet.ForFresh]
   apply Set.infinite_of_finite_compl
   apply Set.Finite.subset (NominalSet.supp_finite 𝔸) hsub
 
diff --git a/Nominal/Helpers/Finset.lean b/Nominal/Helpers/Finset.lean
index 1287c8e..9a74ffc 100644
--- a/Nominal/Helpers/Finset.lean
+++ b/Nominal/Helpers/Finset.lean
@@ -16,7 +16,7 @@ theorem induction
     · apply insert
       · assumption
       · apply h_1; assumption
-    · simp at h
+    · simp only [not_nonempty_iff_eq_empty] at h
       rw [h]
       apply single
 
diff --git a/Nominal/Instances/Abstraction.lean b/Nominal/Instances/Abstraction.lean
index 891f00f..d7fade9 100644
--- a/Nominal/Instances/Abstraction.lean
+++ b/Nominal/Instances/Abstraction.lean
@@ -88,7 +88,7 @@ private lemma alpha_eq_pred_equivariant : EquivariantRelation 𝔸 (AlphaEqPred
   rw [AlphaEqPred, AlphaEqPred]
   intro h
   have h' : π • ⸨a, x.fst.atom⸩ • x.fst.body = π • ⸨a, x.snd.atom⸩ • x.snd.body := by congr
-  simp [smul_smul, FPerm.mul_swap_eq_swap_mul] at h'
+  simp only [smul_smul, FPerm.mul_swap_eq_swap_mul] at h'
   simpa [smul_smul]
 
 lemma alpha_eq_iff_exists {x y : NameAbstraction' 𝔸 X} : AlphaEq x y ↔ ∃ c, c # x ∧ c # y ∧ ⸨c, x.atom⸩ • x.body = ⸨c, y.atom⸩ • y.body := by
@@ -96,14 +96,14 @@ lemma alpha_eq_iff_exists {x y : NameAbstraction' 𝔸 X} : AlphaEq x y ↔ ∃
     rw [alpha_eq_iff_pred, this]
     apply Equivariance.for_fresh_iff_exists
     exact alpha_eq_pred_equivariant
-  simp only [Prod.fresh, AlphaEqPred, @and_assoc]
+  simp [Prod.fresh, AlphaEqPred, @and_assoc]
 
 lemma alpha_eq_iff_all {x y : NameAbstraction' 𝔸 X} : AlphaEq x y ↔ ∀ c, c # x ∧ c # y → ⸨c, x.atom⸩ • x.body = ⸨c, y.atom⸩ • y.body := by
   suffices (∀ c, c # x ∧ c # y → ⸨c, x.atom⸩ • x.body = ⸨c, y.atom⸩ • y.body) = (∀ c, c # (x, y) → AlphaEqPred c (x, y)) by
     rw [alpha_eq_iff_pred, this]
     apply Equivariance.for_fresh_iff_all
     exact alpha_eq_pred_equivariant
-  simp only [Prod.fresh, AlphaEqPred]
+  simp [Prod.fresh, AlphaEqPred]
 
 lemma alpha_eq_iff_cases {x y : NameAbstraction' 𝔸 X} : AlphaEq x y ↔ (x.atom = y.atom ∧ x.body = y.body) ∨ (x.atom ≠ y.atom ∧ x.atom # y ∧ x.body = ⸨x.atom, y.atom⸩ • y.body) := by sorry
 
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
diff --git a/Nominal/Permutations.lean b/Nominal/Permutations.lean
index a583b4c..b8d510e 100644
--- a/Nominal/Permutations.lean
+++ b/Nominal/Permutations.lean
@@ -31,7 +31,7 @@ theorem mem_supp (π : Equiv.Perm X) (x : X) : x ∈ π.supp ↔ π x ≠ x := b
 
 /-- If an element does not occur in the support, the permutation doesn't move it. -/
 lemma supp_notin {π : Equiv.Perm X} {x : X} : x ∉ π.supp → π x = x := by
-  rw [show (π x = x) ↔ ¬(π x ≠ x) by simp]
+  rw [show (π x = x) ↔ ¬(π x ≠ x) by simp only [ne_eq, not_not]]
   simp [← (mem_supp π x).not]
 
 /-- The support of the identity function is the empty set. -/
@@ -46,7 +46,7 @@ lemma supp_inv (π : Equiv.Perm X) : π⁻¹.supp = π.supp := by
 
 /-- The support of two concatenated permutations is a subset of the individual supports as a union. -/
 lemma supp_mul_subset (π τ : Equiv.Perm X) : (π * τ).supp ⊆ π.supp ∪ τ.supp := by
-  simp [supp]
+  simp only [supp, coe_mul, Function.comp_apply, ne_eq]
   apply set_support_mul_subset
 
 end Equiv.Perm
diff --git a/Nominal/Support.lean b/Nominal/Support.lean
index 89021a0..430aa8c 100644
--- a/Nominal/Support.lean
+++ b/Nominal/Support.lean
@@ -62,7 +62,8 @@ lemma support_inter {x : X} {S T : Set 𝔸} : HasSupport 𝔸 x S → S.Finite
   -- First we obtain an injection m : S \ T → (π'.supp ∪ S ∪ T)ᶜ.
   have h_cod_inf : ((π'.supp ∪ S ∪ T)ᶜ).Infinite := by
     apply Set.Finite.infinite_compl
-    simp; exact ⟨⟨h_π_fin, h_S_finite⟩, h_T_finite⟩
+    simp only [Set.finite_union]
+    exact ⟨⟨h_π_fin, h_S_finite⟩, h_T_finite⟩
   have h_dom_fin : (S \ T).Finite :=
     Set.Finite.diff h_S_finite
 
@@ -308,7 +309,7 @@ lemma fresh_exists {x : X} : ∃ (a : 𝔸), a # x := by
 lemma fresh_swap_fix {a b : 𝔸} {x : X} : a # x → b # x → ⸨a, b⸩ ∈ PermSet.fix' 𝔸 x := by
   intro ha hb
   suffices ⸨a, b⸩ • x = (1 : FPerm 𝔸) • x by
-    simp only [one_smul, PermSet.fix', Set.mem_setOf_eq] at *; assumption
+    simp [one_smul, PermSet.fix'] at *; assumption
   apply PermSet.support_permutations_eq
   · exact supp_is_support 𝔸
   · intro c hc
@@ -316,13 +317,13 @@ lemma fresh_swap_fix {a b : 𝔸} {x : X} : a # x → b # x → ⸨a, b⸩ ∈ P
 
 /-- If a proposition holds for suitably fresh a, then one can always pick such an a. -/
 lemma for_fresh_exists {P : 𝔸 → Prop} : (И a, P a) → (∃ a, P a) := by
-  simp [NominalSet.ForFresh]; intro h
+  simp only [ForFresh]; intro h
   exact h.nonempty
 
 /-- Extensionality lemma for the freshness quantifier. -/
 lemma for_fresh_ext {P Q : 𝔸 → Prop} : (∀ a, P a ↔ Q a) → (И a, P a) = (И a, Q a) := by
   intro h
-  simp [NominalSet.ForFresh]
+  simp only [ForFresh, eq_iff_iff]
   suffices {a | P a} = {a | Q a} by rw [this]
   exact Set.ext h