45 lines
1.9 KiB
Lua
45 lines
1.9 KiB
Lua
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-- Create a HoTT compatible environment.
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-- That is,
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-- Type.{0} is predicative
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-- No proof irrelevance for Type.{0}
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-- Proof irrelevance for Id types
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local env = empty_environment({prop_proof_irrel=false, impredicative=false, cls_proof_irrel={"Id"}})
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assert(not env:prop_proof_irrel())
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assert(not env:impredicative())
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assert(env:cls_proof_irrel():head() == name("Id"))
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assert(env:cls_proof_irrel():tail():is_nil())
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local l = mk_param_univ("l")
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local U_l = mk_sort(l)
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local A = Const("A")
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env = add_decl(env, mk_var_decl("Id", {l}, Pi(A, U_l, mk_arrow(A, A, U_l))))
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local Set = mk_sort(mk_level_zero())
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env = add_decl(env, mk_var_decl("N", Set))
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local N = Const("N")
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env = add_decl(env, mk_var_decl("a", N))
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env = add_decl(env, mk_var_decl("b", N))
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local a = Const("a")
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local b = Const("b")
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local Id_z = Const("Id", {mk_level_zero()})
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env = add_decl(env, mk_axiom("H1", Id_z(N, a, b)))
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env = add_decl(env, mk_axiom("H2", Id_z(N, a, b)))
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local tc = type_checker(env)
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-- H1 and H2 are definitionally equal since both have type Id.{0} N a b
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-- and Id is in env:cls_proof_irrel()
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assert(tc:is_def_eq(Const("H1"), Const("H2")))
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env = add_decl(env, mk_var_decl("Path", {l}, Pi(A, U_l, mk_arrow(A, A, U_l))))
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local Path_z = Const("Path", {mk_level_zero()})
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env = add_decl(env, mk_axiom("H3", Path_z(N, a, b)))
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env = add_decl(env, mk_axiom("H4", Path_z(N, a, b)))
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local tc = type_checker(env)
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assert(tc:is_def_eq(Const("H1"), Const("H2")))
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assert(not tc:is_def_eq(Const("H3"), Const("H4")))
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assert(tc:is_def_eq(tc:check(Const("H3")), tc:check(Const("H4"))))
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print("H1 : " .. tostring(tc:check(Const("H1"))))
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print("H2 : " .. tostring(tc:check(Const("H2"))))
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print("H3 : " .. tostring(tc:check(Const("H3"))))
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print("H4 : " .. tostring(tc:check(Const("H4"))))
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print("N : " .. tostring(get_formatter()(env, tc:check(Const("N")))))
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-- N : Type.{0}
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assert(not tc:is_def_eq(Const("a"), Const("b")))
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