import macros tactic
variable Sigma {A : (Type 1)} (B : A → (Type 1)) : (Type 1)
variable Pair {A : (Type 1)} {B : A → (Type 1)} (a : A) (b : B a) : Sigma B
variable Fst {A : (Type 1)} {B : A → (Type 1)} (p : Sigma B) : A
variable Snd {A : (Type 1)} {B : A → (Type 1)} (p : Sigma B) : B (Fst p)
axiom FstAx {A : (Type 1)} {B : A → (Type 1)} (a : A) (b : B a) : @Fst A B (Pair a b) = a
axiom SndAx {A : (Type 1)} {B : A → (Type 1)} (a : A) (b : B a) : @Snd A B (Pair a b) == b

scope
-- Remark: A1 a1 A2 a2 A3 a3 A3 a4 are parameters for the definitions and theorems in this scope.
variable A1 : (Type 1)
variable a1 : A1
variable A2 : A1 → (Type 1)
variable a2 : A2 a1
variable A3 : ∀ a1 : A1, A2 a1 → (Type 1)
variable a3 : A3 a1 a2
variable A4 : ∀ (a1 : A1) (a2 : A2 a1), A3 a1 a2 → (Type 1)
variable a4 : A4 a1 a2 a3
-- Pair type parameterized by a1 and a2
definition A1A2_tuple_ty (a1 : A1) (a2 : A2 a1) : (Type 1)
:= @Sigma (A3 a1 a2) (A4 a1 a2)
-- Pair a3 a4
definition tuple_34 : A1A2_tuple_ty a1 a2
:= @Pair (A3 a1 a2) (A4 a1 a2) a3 a4
-- Triple type parameterized by a1 a2 a3
definition A1_tuple_ty (a1 : A1) : (Type 1)
:= @Sigma (A2 a1) (A1A2_tuple_ty a1)
-- Triple a2 a3 a4
definition tuple_234 : A1_tuple_ty a1
:= @Pair (A2 a1) (A1A2_tuple_ty a1) a2 tuple_34
-- Quadruple type
definition tuple_ty : (Type 1)
:= @Sigma A1 A1_tuple_ty
-- Quadruple a1 a2 a3 a4
definition tuple_1234 : tuple_ty
:= @Pair A1 A1_tuple_ty a1 tuple_234
-- First element of the quadruple
definition f_1234 : A1
:= @Fst A1 A1_tuple_ty tuple_1234
-- Rest of the quadruple (i.e., a triple)
definition s_1234 : A1_tuple_ty f_1234
:= @Snd A1 A1_tuple_ty tuple_1234
theorem H_eq_a1 : f_1234 = a1
:= @FstAx A1 A1_tuple_ty a1 tuple_234
theorem H_eq_triple : s_1234 == tuple_234
:= @SndAx A1 A1_tuple_ty a1 tuple_234
-- Second element of the quadruple
definition fs_1234 : A2 f_1234
:= @Fst (A2 f_1234) (A1A2_tuple_ty f_1234) s_1234
-- Rest of the triple (i.e., a pair)
definition ss_1234 : A1A2_tuple_ty f_1234 fs_1234
:= @Snd (A2 f_1234) (A1A2_tuple_ty f_1234) s_1234
theorem H_eq_a2 : fs_1234 == a2
:= have H1 : @Fst (A2 f_1234) (A1A2_tuple_ty f_1234) s_1234 == @Fst (A2 a1) (A1A2_tuple_ty a1) tuple_234,
     from hcongr (hcongr (hrefl (λ x, @Fst (A2 x) (A1A2_tuple_ty x))) (to_heq H_eq_a1)) H_eq_triple,
   have H2 : @Fst (A2 a1) (A1A2_tuple_ty a1) tuple_234 = a2,
     from FstAx _ _,
   htrans H1 (to_heq H2)
theorem H_eq_pair : ss_1234 == tuple_34
:= have H1 : @Snd (A2 f_1234) (A1A2_tuple_ty f_1234) s_1234 == @Snd (A2 a1) (A1A2_tuple_ty a1) tuple_234,
     from hcongr (hcongr (hrefl (λ x, @Snd (A2 x) (A1A2_tuple_ty x))) (to_heq H_eq_a1)) H_eq_triple,
   have H2 : @Snd (A2 a1) (A1A2_tuple_ty a1) tuple_234 == tuple_34,
     from SndAx _ _,
   htrans H1 H2
-- Third element of the quadruple
definition fss_1234 : A3 f_1234 fs_1234
:= @Fst (A3 f_1234 fs_1234) (A4 f_1234 fs_1234) ss_1234
-- Fourth element of the quadruple
definition sss_1234 : A4 f_1234 fs_1234 fss_1234
:= @Snd (A3 f_1234 fs_1234) (A4 f_1234 fs_1234) ss_1234
theorem H_eq_a3 : fss_1234 == a3
:= have H1 : @Fst (A3 f_1234 fs_1234) (A4 f_1234 fs_1234) ss_1234 == @Fst (A3 a1 a2) (A4 a1 a2) tuple_34,
     from hcongr (hcongr (hcongr (hrefl (λ x y z, @Fst (A3 x y) (A4 x y) z)) (to_heq H_eq_a1)) H_eq_a2) H_eq_pair,
   have H2 : @Fst (A3 a1 a2) (A4 a1 a2) tuple_34 = a3,
     from FstAx _ _,
   htrans H1 (to_heq H2)
theorem H_eq_a4 : sss_1234 == a4
:= have H1 : @Snd (A3 f_1234 fs_1234) (A4 f_1234 fs_1234) ss_1234 == @Snd (A3 a1 a2) (A4 a1 a2) tuple_34,
     from hcongr (hcongr (hcongr (hrefl (λ x y z, @Snd (A3 x y) (A4 x y) z)) (to_heq H_eq_a1)) H_eq_a2) H_eq_pair,
   have H2 : @Snd (A3 a1 a2) (A4 a1 a2) tuple_34 == a4,
     from SndAx _ _,
   htrans H1 H2
eval tuple_1234
eval f_1234
eval fs_1234
eval fss_1234
eval sss_1234
check H_eq_a1
check H_eq_a2
check H_eq_a3
check H_eq_a4
theorem f_quadruple : Fst (Pair a1 (Pair a2 (Pair a3 a4))) = a1
:= H_eq_a1
theorem fs_quadruple : Fst (Snd (Pair a1 (Pair a2 (Pair a3 a4)))) == a2
:= H_eq_a2
theorem fss_quadruple : Fst (Snd (Snd (Pair a1 (Pair a2 (Pair a3 a4))))) == a3
:= H_eq_a3
theorem sss_quadruple : Snd (Snd (Snd (Pair a1 (Pair a2 (Pair a3 a4))))) == a4
:= H_eq_a4
end
-- Show the theorems with their parameters
check f_quadruple
check fs_quadruple
check fss_quadruple
check sss_quadruple
exit