Start proving that Mod2 is a semiring

int/Mod2.v

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+Require Export HoTT.
+Require Import HitTactics.
+
+Module Export modulo.
+
+Private Inductive Mod2 : Type0 :=
+  | Z : Mod2
+  | succ : Mod2 -> Mod2.
+
+Axiom mod : Z = succ(succ Z).
+
+Fixpoint Mod2_ind
+  (P : Mod2 -> Type)
+  (a : P Z)
+  (s : forall (n : Mod2), P n -> P (succ n))
+  (mod' : mod # a = s (succ Z) (s Z a))
+  (x : Mod2)
+  {struct x}
+  : P x
+  := 
+    (match x return _ -> P x with
+      | Z => fun _ => a
+      | succ n => fun _ => s n ((Mod2_ind P a s mod') n)
+    end) mod'.
+
+Axiom Mod2_ind_beta_mod : forall
+  (P : Mod2 -> Type)
+  (a : P Z)
+  (s : forall (n : Mod2), P n -> P (succ n))
+  (mod' : mod # a = s (succ Z) (s Z a))
+  , apD (Mod2_ind P a s mod') mod = mod'.
+
+Fixpoint Mod2_rec
+  (P : Type)
+  (a : P)
+  (s : P -> P)
+  (mod' : a = s (s a))
+  (x : Mod2)
+  {struct x}
+  : P
+  := 
+    (match x return _ -> P with
+      | Z => fun _ => a
+      | succ n => fun _ => s ((Mod2_rec P a s mod') n)
+    end) mod'.
+
+Axiom Mod2_rec_beta_mod : forall
+  (P : Type)
+  (a : P)
+  (s : P -> P)
+  (mod' : a = s (s a))
+  , ap (Mod2_rec P a s mod') mod = mod'.
+
+Instance: HitRecursion Mod2 := {
+  indTy := _; recTy := _; 
+  H_inductor := Mod2_ind;
+  H_recursor := Mod2_rec }.
+
+End modulo.
+
+Definition negate : Mod2 -> Mod2.
+Proof.
+hrecursion.
+- apply Z. 
+- intros. apply (succ H).
+- simpl. apply mod.
+Defined.
+
+
+Definition Bool_to_Mod2 : Bool -> Mod2.
+Proof.
+intro b.
+destruct b.
++ apply (succ Z).
++ apply Z.
+Defined.
+
+Definition Mod2_to_Bool : Mod2 -> Bool.
+Proof.
+intro x.
+hrecursion x.
+- exact false.
+- exact negb.
+- simpl. reflexivity.
+Defined.
+
+Theorem eq1 : forall n : Bool, Mod2_to_Bool (Bool_to_Mod2 n) = n.
+Proof.
+intro b.
+destruct b; compute; reflexivity.
+Qed.
+
+Theorem Bool_to_Mod2_negb : forall x : Bool, 
+  succ (Bool_to_Mod2 x) = Bool_to_Mod2 (negb x).
+Proof.
+intros.
+destruct x; compute.
++ apply mod^.
++ apply reflexivity.
+Defined.
+
+Theorem eq2 : forall n : Mod2, Bool_to_Mod2 (Mod2_to_Bool n) = n.
+Proof.
+intro n.
+hinduction n.
+- reflexivity.
+- intros n IHn.
+  symmetry. etransitivity. apply (ap succ IHn^). 
+  etransitivity. apply Bool_to_Mod2_negb.
+  hott_simpl.
+- rewrite @HoTT.Types.Paths.transport_paths_FlFr.
+  hott_simpl.
+  rewrite ap_compose. 
+  enough (ap Mod2_to_Bool mod = idpath).
+  + rewrite X. hott_simpl.
+  + apply (Mod2_rec_beta_mod Bool false negb 1).
+Defined.
+
+Theorem adj : 
+  forall x : Mod2, eq1 (Mod2_to_Bool x) = ap Mod2_to_Bool (eq2 x).
+Proof.
+intro x.
+apply hset_bool.
+Defined.
+
+Instance: IsEquiv Mod2_to_Bool.
+Proof.
+apply (BuildIsEquiv Mod2 Bool Mod2_to_Bool Bool_to_Mod2 eq1 eq2 adj).
+Qed.
+
+Definition Mod2_value : Mod2 <~> Bool := BuildEquiv _ _ Mod2_to_Bool _.
+
+Instance mod2_hset : IsHSet Mod2. 
+Proof. 
+apply (trunc_equiv Bool Mod2_value^-1). 
+Defined.
+
+Theorem modulo2 : forall n : Mod2, n = succ(succ n).
+Proof.
+hinduction.
+- apply mod.
+- intros n p.
+  apply (ap succ p).
+- apply set_path2.
+Defined.
+
+Definition plus : Mod2 -> Mod2 -> Mod2.
+Proof.
+intros n.
+hrecursion.
+- exact n.
+- apply succ.
+- apply modulo2.
+Defined.
+
+Lemma plus_x_Z_x : forall x, plus x Z = x.
+Proof.
+hinduction; cbn.
+- reflexivity.
+- intros n. refine (ap succ).
+- apply set_path2.
+Defined.
+
+Lemma plus_S_x_S : forall x y,
+ plus (succ x) y = succ (plus x y).
+Proof.
+intros x.
+hinduction; cbn.
+- reflexivity.
+- intros n Hn. refine (ap succ Hn).
+- apply set_path2.
+Defined.
+
+Lemma plus_x_x_Z : forall x, plus x x = Z.
+Proof.
+hinduction.
+- reflexivity.
+- intros n Hn. cbn. 
+  refine (ap succ (plus_S_x_S n n) @ _).
+  refine (ap (succ o succ) Hn @ _).
+  apply mod^.
+- apply set_path2.
+Defined.
+
+Definition mult : Mod2 -> Mod2 -> Mod2.
+intros x. hrecursion.
+- exact Z.
+- intros y. exact (plus x y).
+- simpl.
+  refine (_ @ ap (plus x) (plus_x_Z_x _)^).
+  apply (plus_x_x_Z x)^.
+Defined.

int/Mod2_ring.v

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+Require Import HoTT HitTactics.
+Require Export Mod2.
+
+From HoTTClasses Require Import interfaces.abstract_algebra tactics.ring_tac.
+
+Section Mod2_ring.
+
+Instance Mod2_plus : Plus Mod2 := Mod2.plus.
+Instance Mod2_mult : Mult Mod2 := Mod2.mult.
+Instance Mod2_zero : Zero Mod2 := Z.
+Instance Mod2_one  : One  Mod2 := succ Z.
+
+End Mod2_ring.

int/_CoqProject

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+-R . "" COQC = hoqc COQDEP = hoqdep
+-R ../prelude ""
+-R ../../HoTTClasses/theories HoTTClasses
+Mod2.v
+Mod2_ring.v