77 lines
4.1 KiB
Lua
77 lines
4.1 KiB
Lua
local env = bare_environment()
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local l = mk_param_univ("l")
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local A = Const("A")
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local a = Const("a")
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local b = Const("b")
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local P = Const("P")
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env = add_decl(env, mk_definition("id", {l},
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Pi(A, mk_sort(l), mk_arrow(A, mk_arrow(A, Bool))),
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Fun({{A, mk_sort(l)}, {a, A}, {b, A}}, Pi(P, mk_arrow(A, Bool), mk_arrow(P(a), P(b))))))
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local id_l = Const("id", {l})
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local H = Const("H")
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env = add_decl(env, mk_theorem("refl", {l},
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Pi({{A, mk_sort(l)}, {a, A}}, id_l(A, a, a)),
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Fun({{A, mk_sort(l)}, {a, A}, {P, mk_arrow(A, Bool)}, {H, P(a)}}, H)))
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local H1 = Const("H1")
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local H2 = Const("H2")
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env = add_decl(env, mk_theorem("subst", {l},
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Pi({{A, mk_sort(l)}, {P, mk_arrow(A, Bool)}, {a, A}, {b, A}, {H1, id_l(A, a, b)}, {H2, P(a)}}, P(b)),
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Fun({{A, mk_sort(l)}, {P, mk_arrow(A, Bool)}, {a, A}, {b, A}, {H1, id_l(A, a, b)}, {H2, P(a)}},
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H1(P, H2))))
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local refl_l = Const("refl", {l})
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local subst_l = Const("subst", {l})
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local x = Const("x")
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env = add_decl(env, mk_theorem("symm", {l},
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Pi({{A, mk_sort(l)}, {a, A}, {b, A}, {H, id_l(A, a, b)}}, id_l(A, b, a)),
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Fun({{A, mk_sort(l)}, {a, A}, {b, A}, {H, id_l(A, a, b)}},
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subst_l(A, Fun(x, A, id_l(A, x, a)), a, b, H, refl_l(A, a)))))
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local c = Const("c")
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env = add_decl(env, mk_theorem("trans", {l},
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Pi({{A, mk_sort(l)}, {a, A}, {b, A}, {c, A}, {H1, id_l(A, a, b)}, {H2, id_l(A, b, c)}}, id_l(A, a, c)),
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Fun({{A, mk_sort(l)}, {a, A}, {b, A}, {c, A}, {H1, id_l(A, a, b)}, {H2, id_l(A, b, c)}},
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subst_l(A, Fun(x, A, id_l(A, a, x)), b, c, H2, H1))))
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local symm_l = Const("symm", {l})
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local trans_l = Const("trans", {l})
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print(env:get("trans"):value())
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env = env:add_universe("u")
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local u = mk_global_univ("u")
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local tc = type_checker(env)
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print(tc:check(Const("trans", {u})))
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local id_u = Const("id", {u})
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local refl_u = Const("refl", {u})
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local subst_u = Const("subst", {u})
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local symm_u = Const("symm", {u})
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local trans_u = Const("trans", {u})
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local d = Const("d")
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local H3 = Const("H3")
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print(tc:check(Fun({{A, mk_sort(u)}, {a, A}, {b, A}, {c, A}, {d, A},
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{H1, id_u(A, b, a)},
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{H2, id_u(A, b, c)},
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{H3, id_u(A, c, d)}},
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trans_u(A, a, b, d,
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symm_u(A, b, a, H1),
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trans_u(A, b, c, d, H2, H3)))))
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local g = name_generator("tst")
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local tc2 = type_checker(env, g)
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print("=================")
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local f = Const("f")
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local mf_ty = mk_metavar("f_ty", Pi(A, mk_sort(u), mk_sort(mk_meta_univ("l_f"))))
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local mA1 = mk_metavar("A1", Pi({{A, mk_sort(u)}, {f, mf_ty(A)}, {a, A}}, mk_sort(mk_meta_univ("l_A1"))))
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print(tc2:check(Fun({{A, mk_sort(u)}, {f, mf_ty(A)}, {a, A}},
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id_u(mA1(A, f, a), f(a), a))))
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local tc2 = type_checker(env, g)
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local scope = {{A, mk_sort(u)}, {a, A}, {b, A}, {c, A}, {d, A}, {H1, id_u(A, b, a)},
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{H2, id_u(A, b, c)}, {H3, id_u(A, c, d)}}
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local mP = mk_metavar("P", Pi(scope, mk_metavar("P_ty", Pi(scope, mk_sort(mk_meta_univ("l_P"))))(A, a, b, c, d, H1, H2, H3)))(A, a, b, c, d, H1, H2, H3)
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local ma = mk_metavar("a", Pi(scope, mk_metavar("a_ty", Pi(scope, mk_sort(mk_meta_univ("l_a"))))(A, a, b, c, d, H1, H2, H3)))(A, a, b, c, d, H1, H2, H3)
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local mb = mk_metavar("b", Pi(scope, mk_metavar("b_ty", Pi(scope, mk_sort(mk_meta_univ("l_b"))))(A, a, b, c, d, H1, H2, H3)))(A, a, b, c, d, H1, H2, H3)
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local mA1 = mk_metavar("A1", Pi(scope, mk_sort(mk_meta_univ("l_A1"))))(A, a, b, c, d, H1, H2, H3)
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local mA2 = mk_metavar("A2", Pi(scope, mk_sort(mk_meta_univ("l_A2"))))(A, a, b, c, d, H1, H2, H3)
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print("================")
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print(tc2:check(Fun(scope,
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subst_u(mA1, mP, mb, ma,
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H1,
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trans_u(mA2, b, c, d, H2, H3)))))
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