From 3adbe18d36df3767e98f30b760ccd9c6ace640ad Mon Sep 17 00:00:00 2001
From: Nadrieril
Date: Fri, 24 May 2024 17:01:16 +0200
Subject: Rename some subdirectories for consistency

---
 tests/hol4/constants/Holmakefile         |   5 +
 tests/hol4/constants/constantsScript.sml | 217 +++++++++++++
 tests/hol4/constants/constantsTheory.sig | 538 +++++++++++++++++++++++++++++++
 3 files changed, 760 insertions(+)
 create mode 100644 tests/hol4/constants/Holmakefile
 create mode 100644 tests/hol4/constants/constantsScript.sml
 create mode 100644 tests/hol4/constants/constantsTheory.sig

(limited to 'tests/hol4/constants')

diff --git a/tests/hol4/constants/Holmakefile b/tests/hol4/constants/Holmakefile
new file mode 100644
index 00000000..3c4b8973
--- /dev/null
+++ b/tests/hol4/constants/Holmakefile
@@ -0,0 +1,5 @@
+# This file was automatically generated - modify ../Holmakefile.template instead
+INCLUDES =  ../../../backends/hol4
+
+all: $(DEFAULT_TARGETS)
+.PHONY: all
diff --git a/tests/hol4/constants/constantsScript.sml b/tests/hol4/constants/constantsScript.sml
new file mode 100644
index 00000000..40a319c6
--- /dev/null
+++ b/tests/hol4/constants/constantsScript.sml
@@ -0,0 +1,217 @@
+(** THIS FILE WAS AUTOMATICALLY GENERATED BY AENEAS *)
+(** [constants] *)
+open primitivesLib divDefLib
+
+val _ = new_theory "constants"
+
+
+(** [constants::X0] *)
+Definition x0_body_def:
+  x0_body : u32 result = Return (int_to_u32 0)
+End
+Definition x0_c_def:
+  x0_c : u32 = get_return_value x0_body
+End
+
+(** [constants::X1] *)
+Definition x1_body_def:
+  x1_body : u32 result = Return core_u32_max
+End
+Definition x1_c_def:
+  x1_c : u32 = get_return_value x1_body
+End
+
+(** [constants::X2] *)
+Definition x2_body_def:
+  x2_body : u32 result = Return (int_to_u32 3)
+End
+Definition x2_c_def:
+  x2_c : u32 = get_return_value x2_body
+End
+
+val incr_fwd_def = Define ‘
+  (** [constants::incr]: forward function *)
+  incr_fwd (n : u32) : u32 result =
+    u32_add n (int_to_u32 1)
+’
+
+(** [constants::X3] *)
+Definition x3_body_def:
+  x3_body : u32 result = incr_fwd (int_to_u32 32)
+End
+Definition x3_c_def:
+  x3_c : u32 = get_return_value x3_body
+End
+
+val mk_pair0_fwd_def = Define ‘
+  (** [constants::mk_pair0]: forward function *)
+  mk_pair0_fwd (x : u32) (y : u32) : (u32 # u32) result =
+    Return (x, y)
+’
+
+Datatype:
+  (** [constants::Pair] *)
+  pair_t = <| pair_x : 't1; pair_y : 't2; |>
+End
+
+val mk_pair1_fwd_def = Define ‘
+  (** [constants::mk_pair1]: forward function *)
+  mk_pair1_fwd (x : u32) (y : u32) : (u32, u32) pair_t result =
+    Return (<| pair_x := x; pair_y := y |>)
+’
+
+(** [constants::P0] *)
+Definition p0_body_def:
+  p0_body : (u32 # u32) result = mk_pair0_fwd (int_to_u32 0) (int_to_u32 1)
+End
+Definition p0_c_def:
+  p0_c : (u32 # u32) = get_return_value p0_body
+End
+
+(** [constants::P1] *)
+Definition p1_body_def:
+  p1_body : (u32, u32) pair_t result =
+    mk_pair1_fwd (int_to_u32 0) (int_to_u32 1)
+End
+Definition p1_c_def:
+  p1_c : (u32, u32) pair_t = get_return_value p1_body
+End
+
+(** [constants::P2] *)
+Definition p2_body_def:
+  p2_body : (u32 # u32) result = Return (int_to_u32 0, int_to_u32 1)
+End
+Definition p2_c_def:
+  p2_c : (u32 # u32) = get_return_value p2_body
+End
+
+(** [constants::P3] *)
+Definition p3_body_def:
+  p3_body : (u32, u32) pair_t result =
+    Return (<| pair_x := (int_to_u32 0); pair_y := (int_to_u32 1) |>)
+End
+Definition p3_c_def:
+  p3_c : (u32, u32) pair_t = get_return_value p3_body
+End
+
+Datatype:
+  (** [constants::Wrap] *)
+  wrap_t = <| wrap_val : 't; |>
+End
+
+val wrap_new_fwd_def = Define ‘
+  (** [constants::Wrap::{0}::new]: forward function *)
+  wrap_new_fwd (val : 't) : 't wrap_t result =
+    Return (<| wrap_val := val |>)
+’
+
+(** [constants::Y] *)
+Definition y_body_def:
+  y_body : i32 wrap_t result = wrap_new_fwd (int_to_i32 2)
+End
+Definition y_c_def:
+  y_c : i32 wrap_t = get_return_value y_body
+End
+
+val unwrap_y_fwd_def = Define ‘
+  (** [constants::unwrap_y]: forward function *)
+  unwrap_y_fwd : i32 result =
+    Return y_c.wrap_val
+’
+
+(** [constants::YVAL] *)
+Definition yval_body_def:
+  yval_body : i32 result = unwrap_y_fwd
+End
+Definition yval_c_def:
+  yval_c : i32 = get_return_value yval_body
+End
+
+(** [constants::get_z1::Z1] *)
+Definition get_z1_z1_body_def:
+  get_z1_z1_body : i32 result = Return (int_to_i32 3)
+End
+Definition get_z1_z1_c_def:
+  get_z1_z1_c : i32 = get_return_value get_z1_z1_body
+End
+
+val get_z1_fwd_def = Define ‘
+  (** [constants::get_z1]: forward function *)
+  get_z1_fwd : i32 result =
+    Return get_z1_z1_c
+’
+
+val add_fwd_def = Define ‘
+  (** [constants::add]: forward function *)
+  add_fwd (a : i32) (b : i32) : i32 result =
+    i32_add a b
+’
+
+(** [constants::Q1] *)
+Definition q1_body_def:
+  q1_body : i32 result = Return (int_to_i32 5)
+End
+Definition q1_c_def:
+  q1_c : i32 = get_return_value q1_body
+End
+
+(** [constants::Q2] *)
+Definition q2_body_def:
+  q2_body : i32 result = Return q1_c
+End
+Definition q2_c_def:
+  q2_c : i32 = get_return_value q2_body
+End
+
+(** [constants::Q3] *)
+Definition q3_body_def:
+  q3_body : i32 result = add_fwd q2_c (int_to_i32 3)
+End
+Definition q3_c_def:
+  q3_c : i32 = get_return_value q3_body
+End
+
+val get_z2_fwd_def = Define ‘
+  (** [constants::get_z2]: forward function *)
+  get_z2_fwd : i32 result =
+    do
+    i <- get_z1_fwd;
+    i0 <- add_fwd i q3_c;
+    add_fwd q1_c i0
+    od
+’
+
+(** [constants::S1] *)
+Definition s1_body_def:
+  s1_body : u32 result = Return (int_to_u32 6)
+End
+Definition s1_c_def:
+  s1_c : u32 = get_return_value s1_body
+End
+
+(** [constants::S2] *)
+Definition s2_body_def:
+  s2_body : u32 result = incr_fwd s1_c
+End
+Definition s2_c_def:
+  s2_c : u32 = get_return_value s2_body
+End
+
+(** [constants::S3] *)
+Definition s3_body_def:
+  s3_body : (u32, u32) pair_t result = Return p3_c
+End
+Definition s3_c_def:
+  s3_c : (u32, u32) pair_t = get_return_value s3_body
+End
+
+(** [constants::S4] *)
+Definition s4_body_def:
+  s4_body : (u32, u32) pair_t result =
+    mk_pair1_fwd (int_to_u32 7) (int_to_u32 8)
+End
+Definition s4_c_def:
+  s4_c : (u32, u32) pair_t = get_return_value s4_body
+End
+
+val _ = export_theory ()
diff --git a/tests/hol4/constants/constantsTheory.sig b/tests/hol4/constants/constantsTheory.sig
new file mode 100644
index 00000000..287ad5f5
--- /dev/null
+++ b/tests/hol4/constants/constantsTheory.sig
@@ -0,0 +1,538 @@
+signature constantsTheory =
+sig
+  type thm = Thm.thm
+  
+  (*  Definitions  *)
+    val add_fwd_def : thm
+    val get_z1_fwd_def : thm
+    val get_z1_z1_body_def : thm
+    val get_z1_z1_c_def : thm
+    val get_z2_fwd_def : thm
+    val incr_fwd_def : thm
+    val mk_pair0_fwd_def : thm
+    val mk_pair1_fwd_def : thm
+    val p0_body_def : thm
+    val p0_c_def : thm
+    val p1_body_def : thm
+    val p1_c_def : thm
+    val p2_body_def : thm
+    val p2_c_def : thm
+    val p3_body_def : thm
+    val p3_c_def : thm
+    val pair_t_TY_DEF : thm
+    val pair_t_case_def : thm
+    val pair_t_pair_x : thm
+    val pair_t_pair_x_fupd : thm
+    val pair_t_pair_y : thm
+    val pair_t_pair_y_fupd : thm
+    val pair_t_size_def : thm
+    val q1_body_def : thm
+    val q1_c_def : thm
+    val q2_body_def : thm
+    val q2_c_def : thm
+    val q3_body_def : thm
+    val q3_c_def : thm
+    val s1_body_def : thm
+    val s1_c_def : thm
+    val s2_body_def : thm
+    val s2_c_def : thm
+    val s3_body_def : thm
+    val s3_c_def : thm
+    val s4_body_def : thm
+    val s4_c_def : thm
+    val unwrap_y_fwd_def : thm
+    val wrap_new_fwd_def : thm
+    val wrap_t_TY_DEF : thm
+    val wrap_t_case_def : thm
+    val wrap_t_size_def : thm
+    val wrap_t_wrap_val : thm
+    val wrap_t_wrap_val_fupd : thm
+    val x0_body_def : thm
+    val x0_c_def : thm
+    val x1_body_def : thm
+    val x1_c_def : thm
+    val x2_body_def : thm
+    val x2_c_def : thm
+    val x3_body_def : thm
+    val x3_c_def : thm
+    val y_body_def : thm
+    val y_c_def : thm
+    val yval_body_def : thm
+    val yval_c_def : thm
+  
+  (*  Theorems  *)
+    val EXISTS_pair_t : thm
+    val EXISTS_wrap_t : thm
+    val FORALL_pair_t : thm
+    val FORALL_wrap_t : thm
+    val datatype_pair_t : thm
+    val datatype_wrap_t : thm
+    val pair_t_11 : thm
+    val pair_t_Axiom : thm
+    val pair_t_accessors : thm
+    val pair_t_accfupds : thm
+    val pair_t_case_cong : thm
+    val pair_t_case_eq : thm
+    val pair_t_component_equality : thm
+    val pair_t_fn_updates : thm
+    val pair_t_fupdcanon : thm
+    val pair_t_fupdcanon_comp : thm
+    val pair_t_fupdfupds : thm
+    val pair_t_fupdfupds_comp : thm
+    val pair_t_induction : thm
+    val pair_t_literal_11 : thm
+    val pair_t_literal_nchotomy : thm
+    val pair_t_nchotomy : thm
+    val pair_t_updates_eq_literal : thm
+    val wrap_t_11 : thm
+    val wrap_t_Axiom : thm
+    val wrap_t_accessors : thm
+    val wrap_t_accfupds : thm
+    val wrap_t_case_cong : thm
+    val wrap_t_case_eq : thm
+    val wrap_t_component_equality : thm
+    val wrap_t_fn_updates : thm
+    val wrap_t_fupdfupds : thm
+    val wrap_t_fupdfupds_comp : thm
+    val wrap_t_induction : thm
+    val wrap_t_literal_11 : thm
+    val wrap_t_literal_nchotomy : thm
+    val wrap_t_nchotomy : thm
+    val wrap_t_updates_eq_literal : thm
+  
+  val constants_grammars : type_grammar.grammar * term_grammar.grammar
+(*
+   [divDef] Parent theory of "constants"
+   
+   [add_fwd_def]  Definition
+      
+      ⊢ ∀a b. add_fwd a b = i32_add a b
+   
+   [get_z1_fwd_def]  Definition
+      
+      ⊢ get_z1_fwd = Return get_z1_z1_c
+   
+   [get_z1_z1_body_def]  Definition
+      
+      ⊢ get_z1_z1_body = Return (int_to_i32 3)
+   
+   [get_z1_z1_c_def]  Definition
+      
+      ⊢ get_z1_z1_c = get_return_value get_z1_z1_body
+   
+   [get_z2_fwd_def]  Definition
+      
+      ⊢ get_z2_fwd =
+        do i <- get_z1_fwd; i0 <- add_fwd i q3_c; add_fwd q1_c i0 od
+   
+   [incr_fwd_def]  Definition
+      
+      ⊢ ∀n. incr_fwd n = u32_add n (int_to_u32 1)
+   
+   [mk_pair0_fwd_def]  Definition
+      
+      ⊢ ∀x y. mk_pair0_fwd x y = Return (x,y)
+   
+   [mk_pair1_fwd_def]  Definition
+      
+      ⊢ ∀x y. mk_pair1_fwd x y = Return <|pair_x := x; pair_y := y|>
+   
+   [p0_body_def]  Definition
+      
+      ⊢ p0_body = mk_pair0_fwd (int_to_u32 0) (int_to_u32 1)
+   
+   [p0_c_def]  Definition
+      
+      ⊢ p0_c = get_return_value p0_body
+   
+   [p1_body_def]  Definition
+      
+      ⊢ p1_body = mk_pair1_fwd (int_to_u32 0) (int_to_u32 1)
+   
+   [p1_c_def]  Definition
+      
+      ⊢ p1_c = get_return_value p1_body
+   
+   [p2_body_def]  Definition
+      
+      ⊢ p2_body = Return (int_to_u32 0,int_to_u32 1)
+   
+   [p2_c_def]  Definition
+      
+      ⊢ p2_c = get_return_value p2_body
+   
+   [p3_body_def]  Definition
+      
+      ⊢ p3_body = Return <|pair_x := int_to_u32 0; pair_y := int_to_u32 1|>
+   
+   [p3_c_def]  Definition
+      
+      ⊢ p3_c = get_return_value p3_body
+   
+   [pair_t_TY_DEF]  Definition
+      
+      ⊢ ∃rep.
+          TYPE_DEFINITION
+            (λa0'.
+                 ∀ $var$('pair_t').
+                   (∀a0'.
+                      (∃a0 a1.
+                         a0' =
+                         (λa0 a1.
+                              ind_type$CONSTR 0 (a0,a1)
+                                (λn. ind_type$BOTTOM)) a0 a1) ⇒
+                      $var$('pair_t') a0') ⇒
+                   $var$('pair_t') a0') rep
+   
+   [pair_t_case_def]  Definition
+      
+      ⊢ ∀a0 a1 f. pair_t_CASE (pair_t a0 a1) f = f a0 a1
+   
+   [pair_t_pair_x]  Definition
+      
+      ⊢ ∀t t0. (pair_t t t0).pair_x = t
+   
+   [pair_t_pair_x_fupd]  Definition
+      
+      ⊢ ∀f t t0. pair_t t t0 with pair_x updated_by f = pair_t (f t) t0
+   
+   [pair_t_pair_y]  Definition
+      
+      ⊢ ∀t t0. (pair_t t t0).pair_y = t0
+   
+   [pair_t_pair_y_fupd]  Definition
+      
+      ⊢ ∀f t t0. pair_t t t0 with pair_y updated_by f = pair_t t (f t0)
+   
+   [pair_t_size_def]  Definition
+      
+      ⊢ ∀f f1 a0 a1. pair_t_size f f1 (pair_t a0 a1) = 1 + (f a0 + f1 a1)
+   
+   [q1_body_def]  Definition
+      
+      ⊢ q1_body = Return (int_to_i32 5)
+   
+   [q1_c_def]  Definition
+      
+      ⊢ q1_c = get_return_value q1_body
+   
+   [q2_body_def]  Definition
+      
+      ⊢ q2_body = Return q1_c
+   
+   [q2_c_def]  Definition
+      
+      ⊢ q2_c = get_return_value q2_body
+   
+   [q3_body_def]  Definition
+      
+      ⊢ q3_body = add_fwd q2_c (int_to_i32 3)
+   
+   [q3_c_def]  Definition
+      
+      ⊢ q3_c = get_return_value q3_body
+   
+   [s1_body_def]  Definition
+      
+      ⊢ s1_body = Return (int_to_u32 6)
+   
+   [s1_c_def]  Definition
+      
+      ⊢ s1_c = get_return_value s1_body
+   
+   [s2_body_def]  Definition
+      
+      ⊢ s2_body = incr_fwd s1_c
+   
+   [s2_c_def]  Definition
+      
+      ⊢ s2_c = get_return_value s2_body
+   
+   [s3_body_def]  Definition
+      
+      ⊢ s3_body = Return p3_c
+   
+   [s3_c_def]  Definition
+      
+      ⊢ s3_c = get_return_value s3_body
+   
+   [s4_body_def]  Definition
+      
+      ⊢ s4_body = mk_pair1_fwd (int_to_u32 7) (int_to_u32 8)
+   
+   [s4_c_def]  Definition
+      
+      ⊢ s4_c = get_return_value s4_body
+   
+   [unwrap_y_fwd_def]  Definition
+      
+      ⊢ unwrap_y_fwd = Return y_c.wrap_val
+   
+   [wrap_new_fwd_def]  Definition
+      
+      ⊢ ∀val. wrap_new_fwd val = Return <|wrap_val := val|>
+   
+   [wrap_t_TY_DEF]  Definition
+      
+      ⊢ ∃rep.
+          TYPE_DEFINITION
+            (λa0.
+                 ∀ $var$('wrap_t').
+                   (∀a0.
+                      (∃a. a0 =
+                           (λa. ind_type$CONSTR 0 a (λn. ind_type$BOTTOM))
+                             a) ⇒
+                      $var$('wrap_t') a0) ⇒
+                   $var$('wrap_t') a0) rep
+   
+   [wrap_t_case_def]  Definition
+      
+      ⊢ ∀a f. wrap_t_CASE (wrap_t a) f = f a
+   
+   [wrap_t_size_def]  Definition
+      
+      ⊢ ∀f a. wrap_t_size f (wrap_t a) = 1 + f a
+   
+   [wrap_t_wrap_val]  Definition
+      
+      ⊢ ∀t. (wrap_t t).wrap_val = t
+   
+   [wrap_t_wrap_val_fupd]  Definition
+      
+      ⊢ ∀f t. wrap_t t with wrap_val updated_by f = wrap_t (f t)
+   
+   [x0_body_def]  Definition
+      
+      ⊢ x0_body = Return (int_to_u32 0)
+   
+   [x0_c_def]  Definition
+      
+      ⊢ x0_c = get_return_value x0_body
+   
+   [x1_body_def]  Definition
+      
+      ⊢ x1_body = Return core_u32_max
+   
+   [x1_c_def]  Definition
+      
+      ⊢ x1_c = get_return_value x1_body
+   
+   [x2_body_def]  Definition
+      
+      ⊢ x2_body = Return (int_to_u32 3)
+   
+   [x2_c_def]  Definition
+      
+      ⊢ x2_c = get_return_value x2_body
+   
+   [x3_body_def]  Definition
+      
+      ⊢ x3_body = incr_fwd (int_to_u32 32)
+   
+   [x3_c_def]  Definition
+      
+      ⊢ x3_c = get_return_value x3_body
+   
+   [y_body_def]  Definition
+      
+      ⊢ y_body = wrap_new_fwd (int_to_i32 2)
+   
+   [y_c_def]  Definition
+      
+      ⊢ y_c = get_return_value y_body
+   
+   [yval_body_def]  Definition
+      
+      ⊢ yval_body = unwrap_y_fwd
+   
+   [yval_c_def]  Definition
+      
+      ⊢ yval_c = get_return_value yval_body
+   
+   [EXISTS_pair_t]  Theorem
+      
+      ⊢ ∀P. (∃p. P p) ⇔ ∃t0 t. P <|pair_x := t0; pair_y := t|>
+   
+   [EXISTS_wrap_t]  Theorem
+      
+      ⊢ ∀P. (∃w. P w) ⇔ ∃u. P <|wrap_val := u|>
+   
+   [FORALL_pair_t]  Theorem
+      
+      ⊢ ∀P. (∀p. P p) ⇔ ∀t0 t. P <|pair_x := t0; pair_y := t|>
+   
+   [FORALL_wrap_t]  Theorem
+      
+      ⊢ ∀P. (∀w. P w) ⇔ ∀u. P <|wrap_val := u|>
+   
+   [datatype_pair_t]  Theorem
+      
+      ⊢ DATATYPE (record pair_t pair_x pair_y)
+   
+   [datatype_wrap_t]  Theorem
+      
+      ⊢ DATATYPE (record wrap_t wrap_val)
+   
+   [pair_t_11]  Theorem
+      
+      ⊢ ∀a0 a1 a0' a1'. pair_t a0 a1 = pair_t a0' a1' ⇔ a0 = a0' ∧ a1 = a1'
+   
+   [pair_t_Axiom]  Theorem
+      
+      ⊢ ∀f. ∃fn. ∀a0 a1. fn (pair_t a0 a1) = f a0 a1
+   
+   [pair_t_accessors]  Theorem
+      
+      ⊢ (∀t t0. (pair_t t t0).pair_x = t) ∧
+        ∀t t0. (pair_t t t0).pair_y = t0
+   
+   [pair_t_accfupds]  Theorem
+      
+      ⊢ (∀p f. (p with pair_y updated_by f).pair_x = p.pair_x) ∧
+        (∀p f. (p with pair_x updated_by f).pair_y = p.pair_y) ∧
+        (∀p f. (p with pair_x updated_by f).pair_x = f p.pair_x) ∧
+        ∀p f. (p with pair_y updated_by f).pair_y = f p.pair_y
+   
+   [pair_t_case_cong]  Theorem
+      
+      ⊢ ∀M M' f.
+          M = M' ∧ (∀a0 a1. M' = pair_t a0 a1 ⇒ f a0 a1 = f' a0 a1) ⇒
+          pair_t_CASE M f = pair_t_CASE M' f'
+   
+   [pair_t_case_eq]  Theorem
+      
+      ⊢ pair_t_CASE x f = v ⇔ ∃t t0. x = pair_t t t0 ∧ f t t0 = v
+   
+   [pair_t_component_equality]  Theorem
+      
+      ⊢ ∀p1 p2. p1 = p2 ⇔ p1.pair_x = p2.pair_x ∧ p1.pair_y = p2.pair_y
+   
+   [pair_t_fn_updates]  Theorem
+      
+      ⊢ (∀f t t0. pair_t t t0 with pair_x updated_by f = pair_t (f t) t0) ∧
+        ∀f t t0. pair_t t t0 with pair_y updated_by f = pair_t t (f t0)
+   
+   [pair_t_fupdcanon]  Theorem
+      
+      ⊢ ∀p g f.
+          p with <|pair_y updated_by f; pair_x updated_by g|> =
+          p with <|pair_x updated_by g; pair_y updated_by f|>
+   
+   [pair_t_fupdcanon_comp]  Theorem
+      
+      ⊢ (∀g f.
+           pair_y_fupd f ∘ pair_x_fupd g = pair_x_fupd g ∘ pair_y_fupd f) ∧
+        ∀h g f.
+          pair_y_fupd f ∘ pair_x_fupd g ∘ h =
+          pair_x_fupd g ∘ pair_y_fupd f ∘ h
+   
+   [pair_t_fupdfupds]  Theorem
+      
+      ⊢ (∀p g f.
+           p with <|pair_x updated_by f; pair_x updated_by g|> =
+           p with pair_x updated_by f ∘ g) ∧
+        ∀p g f.
+          p with <|pair_y updated_by f; pair_y updated_by g|> =
+          p with pair_y updated_by f ∘ g
+   
+   [pair_t_fupdfupds_comp]  Theorem
+      
+      ⊢ ((∀g f. pair_x_fupd f ∘ pair_x_fupd g = pair_x_fupd (f ∘ g)) ∧
+         ∀h g f.
+           pair_x_fupd f ∘ pair_x_fupd g ∘ h = pair_x_fupd (f ∘ g) ∘ h) ∧
+        (∀g f. pair_y_fupd f ∘ pair_y_fupd g = pair_y_fupd (f ∘ g)) ∧
+        ∀h g f. pair_y_fupd f ∘ pair_y_fupd g ∘ h = pair_y_fupd (f ∘ g) ∘ h
+   
+   [pair_t_induction]  Theorem
+      
+      ⊢ ∀P. (∀t t0. P (pair_t t t0)) ⇒ ∀p. P p
+   
+   [pair_t_literal_11]  Theorem
+      
+      ⊢ ∀t01 t1 t02 t2.
+          <|pair_x := t01; pair_y := t1|> = <|pair_x := t02; pair_y := t2|> ⇔
+          t01 = t02 ∧ t1 = t2
+   
+   [pair_t_literal_nchotomy]  Theorem
+      
+      ⊢ ∀p. ∃t0 t. p = <|pair_x := t0; pair_y := t|>
+   
+   [pair_t_nchotomy]  Theorem
+      
+      ⊢ ∀pp. ∃t t0. pp = pair_t t t0
+   
+   [pair_t_updates_eq_literal]  Theorem
+      
+      ⊢ ∀p t0 t.
+          p with <|pair_x := t0; pair_y := t|> =
+          <|pair_x := t0; pair_y := t|>
+   
+   [wrap_t_11]  Theorem
+      
+      ⊢ ∀a a'. wrap_t a = wrap_t a' ⇔ a = a'
+   
+   [wrap_t_Axiom]  Theorem
+      
+      ⊢ ∀f. ∃fn. ∀a. fn (wrap_t a) = f a
+   
+   [wrap_t_accessors]  Theorem
+      
+      ⊢ ∀t. (wrap_t t).wrap_val = t
+   
+   [wrap_t_accfupds]  Theorem
+      
+      ⊢ ∀w f. (w with wrap_val updated_by f).wrap_val = f w.wrap_val
+   
+   [wrap_t_case_cong]  Theorem
+      
+      ⊢ ∀M M' f.
+          M = M' ∧ (∀a. M' = wrap_t a ⇒ f a = f' a) ⇒
+          wrap_t_CASE M f = wrap_t_CASE M' f'
+   
+   [wrap_t_case_eq]  Theorem
+      
+      ⊢ wrap_t_CASE x f = v ⇔ ∃t. x = wrap_t t ∧ f t = v
+   
+   [wrap_t_component_equality]  Theorem
+      
+      ⊢ ∀w1 w2. w1 = w2 ⇔ w1.wrap_val = w2.wrap_val
+   
+   [wrap_t_fn_updates]  Theorem
+      
+      ⊢ ∀f t. wrap_t t with wrap_val updated_by f = wrap_t (f t)
+   
+   [wrap_t_fupdfupds]  Theorem
+      
+      ⊢ ∀w g f.
+          w with <|wrap_val updated_by f; wrap_val updated_by g|> =
+          w with wrap_val updated_by f ∘ g
+   
+   [wrap_t_fupdfupds_comp]  Theorem
+      
+      ⊢ (∀g f. wrap_val_fupd f ∘ wrap_val_fupd g = wrap_val_fupd (f ∘ g)) ∧
+        ∀h g f.
+          wrap_val_fupd f ∘ wrap_val_fupd g ∘ h = wrap_val_fupd (f ∘ g) ∘ h
+   
+   [wrap_t_induction]  Theorem
+      
+      ⊢ ∀P. (∀t. P (wrap_t t)) ⇒ ∀w. P w
+   
+   [wrap_t_literal_11]  Theorem
+      
+      ⊢ ∀u1 u2. <|wrap_val := u1|> = <|wrap_val := u2|> ⇔ u1 = u2
+   
+   [wrap_t_literal_nchotomy]  Theorem
+      
+      ⊢ ∀w. ∃u. w = <|wrap_val := u|>
+   
+   [wrap_t_nchotomy]  Theorem
+      
+      ⊢ ∀ww. ∃t. ww = wrap_t t
+   
+   [wrap_t_updates_eq_literal]  Theorem
+      
+      ⊢ ∀w u. w with wrap_val := u = <|wrap_val := u|>
+   
+   
+*)
+end
-- 
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