diff options
Diffstat (limited to '')
| -rw-r--r-- | backends/hol4/divDefLibTestScript.sml (renamed from backends/hol4/divDefLibExampleScript.sml) | 0 | ||||
| -rw-r--r-- | backends/hol4/divDefProto2TestScript.sml | 1222 |
2 files changed, 0 insertions, 1222 deletions
diff --git a/backends/hol4/divDefLibExampleScript.sml b/backends/hol4/divDefLibTestScript.sml index c4a57783..c4a57783 100644 --- a/backends/hol4/divDefLibExampleScript.sml +++ b/backends/hol4/divDefLibTestScript.sml diff --git a/backends/hol4/divDefProto2TestScript.sml b/backends/hol4/divDefProto2TestScript.sml deleted file mode 100644 index bc9ea9a7..00000000 --- a/backends/hol4/divDefProto2TestScript.sml +++ /dev/null @@ -1,1222 +0,0 @@ -open HolKernel boolLib bossLib Parse -open boolTheory arithmeticTheory integerTheory intLib listTheory stringTheory - -open primitivesArithTheory primitivesBaseTacLib ilistTheory primitivesTheory -open primitivesLib -open divDefProto2Theory - -val _ = new_theory "divDefProto2TestScript" - -(*====================== - * Example 1: nth - *======================*) -Datatype: - list_t = - ListCons 't list_t - | ListNil -End - -(* We use this version of the body to prove that the body is valid *) -val nth_body_def = Define ‘ - nth_body (f : (('t list_t # u32) + 't) -> (('t list_t # u32) + 't) result) - (x : (('t list_t # u32) + 't)) : - (('t list_t # u32) + 't) result = - (* Destruct the input. We need this to call the proper function in case - of mutually recursive definitions, but also to eliminate arguments - which correspond to the output value (the input type is the same - as the output type). *) - case x of - | INL x => ( - let (ls, i) = x in - case ls of - | ListCons x tl => - if u32_to_int i = (0:int) - then Return (INR x) - else - do - i0 <- u32_sub i (int_to_u32 1); - r <- f (INL (tl, i0)); - (* Eliminate the invalid outputs. This is not necessary here, - but it is in the case of non tail call recursive calls. *) - case r of - | INL _ => Fail Failure - | INR i1 => Return (INR i1) - od - | ListNil => Fail Failure) - | INR _ => Fail Failure -’ - -val dbg = ref false -fun print_dbg s = if (!dbg) then print s else () - -(* Tactic which makes progress in a proof of validity by making a case - disjunction (we use this to explore all the paths in a function body). *) -fun prove_valid_case_progress - -(* -val (asms, g) = top_goal () -*) - - -(* Tactic to prove that a body is valid: perform one step. *) -fun prove_body_is_valid_tac_step (asms, g) = - let - (* The goal has the shape: - {[ - (∀g h. ... g x = ... h x) ∨ - ∃h y. is_valid_fp_body n h ∧ ∀g. ... g x = ... od - ]} - *) - (* Retrieve the scrutinee in the goal (‘x’). - There are two cases: - - either the function has the shape: - {[ - (λ(y,z). ...) x - ]} - in which case we need to destruct ‘x’ - - or we have a normal ‘case ... of’ - *) - val body = (lhs o snd o strip_forall o fst o dest_disj) g - val scrut = - let - val (app, x) = dest_comb body - val (app, _) = dest_comb app - val {Name=name, Thy=thy, Ty = _ } = dest_thy_const app - in - if thy = "pair" andalso name = "UNCURRY" then x else failwith "not a curried argument" - end - handle HOL_ERR _ => strip_all_cases_get_scrutinee body - (* Retrieve the first quantified continuations from the goal (‘g’) *) - val qc = (hd o fst o strip_forall o fst o dest_disj) g - (* Check if the scrutinee is a recursive call *) - val (scrut_app, _) = strip_comb scrut - val _ = print_dbg ("prove_body_is_valid_step: Scrutinee: " ^ term_to_string scrut ^ "\n") - (* For the recursive calls: *) - fun step_rec () = - let - val _ = print_dbg ("prove_body_is_valid_step: rec call\n") - (* We need to instantiate the ‘h’ existantially quantified function *) - (* First, retrieve the body of the function: it is given by the ‘Return’ branch *) - val (_, _, branches) = TypeBase.dest_case body - (* Find the branch corresponding to the return *) - val ret_branch = List.find (fn (pat, _) => - let - val {Name=name, Thy=thy, Ty = _ } = (dest_thy_const o fst o strip_comb) pat - in - thy = "primitives" andalso name = "Return" - end) branches - val var = (hd o snd o strip_comb o fst o valOf) ret_branch - val br = (snd o valOf) ret_branch - (* Abstract away the input variable introduced by the pattern and the continuation ‘g’ *) - val h = list_mk_abs ([qc, var], br) - val _ = print_dbg ("prove_body_is_valid_step: h: " ^ term_to_string h ^ "\n") - (* Retrieve the input parameter ‘x’ *) - val input = (snd o dest_comb) scrut - val _ = print_dbg ("prove_body_is_valid_step: y: " ^ term_to_string input ^ "\n") - in - ((* Choose the right possibility (this is a recursive call) *) - disj2_tac >> - (* Instantiate the quantifiers *) - qexists ‘^h’ >> - qexists ‘^input’ >> - (* Unfold the predicate once *) - pure_once_rewrite_tac [is_valid_fp_body_def] >> - (* We have two subgoals: - - we have to prove that ‘h’ is valid - - we have to finish the proof of validity for the current body - *) - conj_tac >> fs [case_result_switch_eq]) - end - in - (* If recursive call: special treatment. Otherwise, we do a simple disjunction *) - (if term_eq scrut_app qc then step_rec () - else (Cases_on ‘^scrut’ >> fs [case_result_switch_eq])) (asms, g) - end - -(* Tactic to prove that a body is valid *) -fun prove_body_is_valid_tac (body_def : thm option) : tactic = - let val body_def_thm = case body_def of SOME th => [th] | NONE => [] - in - pure_once_rewrite_tac [is_valid_fp_body_def] >> gen_tac >> - (* Expand *) - fs body_def_thm >> - fs [bind_def, case_result_switch_eq] >> - (* Explore the body *) - rpt prove_body_is_valid_tac_step - end - -(* TODO: move *) -val is_valid_fp_body_tm = “is_valid_fp_body” - -(* Prove that a body satisfies the validity condition of the fixed point *) -fun prove_body_is_valid (body : term) : thm = - let - (* Explore the body and count the number of occurrences of nested recursive - calls so that we can properly instantiate the ‘N’ argument of ‘is_valid_fp_body’. - - We first retrieve the name of the continuation parameter. - Rem.: we generated fresh names so that, for instance, the continuation name - doesn't collide with other names. Because of this, we don't need to look for - collisions when exploring the body (and in the worst case, we would cound - an overapproximation of the number of recursive calls, which is perfectly - valid). - *) - val fcont = (hd o fst o strip_abs) body - val fcont_name = (fst o dest_var) fcont - fun max x y = if x > y then x else y - fun count_body_rec_calls (body : term) : int = - case dest_term body of - VAR (name, _) => if name = fcont_name then 1 else 0 - | CONST _ => 0 - | COMB (x, y) => max (count_body_rec_calls x) (count_body_rec_calls y) - | LAMB (_, x) => count_body_rec_calls x - val num_rec_calls = count_body_rec_calls body - - (* Generate the term ‘SUC (SUC ... (SUC n))’ where ‘n’ is a fresh variable. - - Remark: we first prove ‘is_valid_fp_body (SUC ... n) body’ then substitue - ‘n’ with ‘0’ to prevent the quantity from being rewritten to a bit - representation, which would prevent unfolding of the ‘is_valid_fp_body’. - *) - val nvar = genvar num_ty - (* Rem.: we stack num_rec_calls + 1 occurrences of ‘SUC’ (and the + 1 is important) *) - fun mk_n i = if i = 0 then mk_suc nvar else mk_suc (mk_n (i-1)) - val n_tm = mk_n num_rec_calls - - (* Generate the lemma statement *) - val is_valid_tm = list_mk_icomb (is_valid_fp_body_tm, [n_tm, body]) - val is_valid_thm = prove (is_valid_tm, prove_body_is_valid_tac NONE) - - (* Replace ‘nvar’ with ‘0’ *) - val is_valid_thm = INST [nvar |-> zero_num_tm] is_valid_thm - in - is_valid_thm - end - -(* -val (asms, g) = top_goal () -*) - -(* We first prove the theorem with ‘SUC (SUC n)’ where ‘n’ is a variable - to prevent this quantity from being rewritten to 2 *) -Theorem nth_body_is_valid_aux: - is_valid_fp_body (SUC (SUC n)) nth_body -Proof - prove_body_is_valid_tac (SOME nth_body_def) -QED - -Theorem nth_body_is_valid: - is_valid_fp_body (SUC (SUC 0)) nth_body -Proof - irule nth_body_is_valid_aux -QED - -val nth_raw_def = Define ‘ - nth (ls : 't list_t) (i : u32) = - case fix nth_body (INL (ls, i)) of - | Fail e => Fail e - | Diverge => Diverge - | Return r => - case r of - | INL _ => Fail Failure - | INR x => Return x -’ - -val fix_tm = “fix” - -(* Generate the raw definitions by using the grouped definition body and the - fixed point operator *) -fun mk_raw_defs (in_out_tys : (hol_type * hol_type) list) - (def_tms : term list) (body_is_valid : thm) : thm list = - let - (* Retrieve the body *) - val body = (List.last o snd o strip_comb o concl) body_is_valid - - (* Create the term ‘fix body’ *) - val fixed_body = mk_icomb (fix_tm, body) - - (* For every function in the group, generate the equation that we will - use as definition. In particular: - - add the properly injected input ‘x’ to ‘fix body’ (ex.: for ‘nth ls i’ - we add the input ‘INL (ls, i)’) - - wrap ‘fix body x’ into a case disjunction to extract the relevant output - - For instance, in the case of ‘nth ls i’: - {[ - nth (ls : 't list_t) (i : u32) = - case fix nth_body (INL (ls, i)) of - | Fail e => Fail e - | Diverge => Diverge - | Return r => - case r of - | INL _ => Fail Failure - | INR x => Return x - ]} - *) - fun mk_def_eq (i : int, def_tm : term) : term = - let - (* Retrieve the lhs of the original definition equation, and in - particular the inputs *) - val def_lhs = lhs def_tm - val args = (snd o strip_comb) def_lhs - - (* Inject the inputs into the param type *) - val input = pairSyntax.list_mk_pair args - val input = inject_in_param_sum in_out_tys i true input - - (* Compose*) - val def_rhs = mk_comb (fixed_body, input) - - (* Wrap in the case disjunction *) - val def_rhs = mk_case_select_result_sum def_rhs in_out_tys i false - - (* Create the equation *) - val def_eq_tm = mk_eq (def_lhs, def_rhs) - in - def_eq_tm - end - val raw_def_tms = map mk_def_eq (enumerate def_tms) - - (* Generate the definitions *) - val raw_defs = map (fn tm => Define ‘^tm’) raw_def_tms - in - raw_defs - end - -(* Tactic which makes progress in a proof by making a case disjunction (we use - this to explore all the paths in a function body). *) -fun case_progress (asms, g) = - let - val scrut = (strip_all_cases_get_scrutinee o lhs) g - in Cases_on ‘^scrut’ (asms, g) end - -(* Prove the final equation, that we will use as definition. *) -fun prove_def_eq_tac - (current_raw_def : thm) (all_raw_defs : thm list) (is_valid : thm) - (body_def : thm option) : tactic = - let - val body_def_thm = case body_def of SOME th => [th] | NONE => [] - in - rpt gen_tac >> - (* Expand the definition *) - pure_once_rewrite_tac [current_raw_def] >> - (* Use the fixed-point equality *) - pure_once_rewrite_left_tac [HO_MATCH_MP fix_fixed_eq is_valid] >> - (* Expand the body definition *) - pure_rewrite_tac body_def_thm >> - (* Expand all the definitions from the group *) - pure_rewrite_tac all_raw_defs >> - (* Explore all the paths - maybe we can be smarter, but this is fast and really easy *) - fs [bind_def] >> - rpt (case_progress >> fs []) - end - -(* Prove the final equations that we will give to the user as definitions *) -fun prove_def_eqs (body_is_valid : thm) (def_tms : term list) (raw_defs : thm list) : thm list= - let - val defs_tgt_raw = zip def_tms raw_defs - (* Substitute the function variables with the constants introduced in the raw - definitions *) - fun compute_fsubst (def_tm, raw_def) : {redex: term, residue: term} = - let - val (fvar, _) = (strip_comb o lhs) def_tm - val fconst = (fst o strip_comb o lhs o snd o strip_forall o concl) raw_def - in - (fvar |-> fconst) - end - val fsubst = map compute_fsubst defs_tgt_raw - val defs_tgt_raw = map (fn (x, y) => (subst fsubst x, y)) defs_tgt_raw - - fun prove_def_eq (def_tm, raw_def) : thm = - let - (* Quantify the parameters *) - val (_, params) = (strip_comb o lhs) def_tm - val def_eq_tm = list_mk_forall (params, def_tm) - (* Prove *) - val def_eq = prove (def_eq_tm, prove_def_eq_tac raw_def raw_defs body_is_valid NONE) - in - def_eq - end - val def_eqs = map prove_def_eq defs_tgt_raw - in - def_eqs - end - -Theorem nth_def: - ∀ls i. nth (ls : 't list_t) (i : u32) : 't result = - case ls of - | ListCons x tl => - if u32_to_int i = (0:int) - then (Return x) - else - do - i0 <- u32_sub i (int_to_u32 1); - nth tl i0 - od - | ListNil => Fail Failure -Proof - prove_def_eq_tac nth_raw_def [nth_raw_def] nth_body_is_valid nth_body_def -QED - -(*====================== - * Example 2: even, odd - *======================*) - -val def_qt = ‘ - (even (i : int) : bool result = - if i = 0 then Return T else odd (i - 1)) /\ - (odd (i : int) : bool result = - if i = 0 then Return F else even (i - 1)) -’ - -val result_ty = “:'a result” -val error_ty = “:error” -val alpha_ty = “:'a” -val num_ty = “:num” - -val return_tm = “Return : 'a -> 'a result” -val fail_tm = “Fail : error -> 'a result” -val fail_failure_tm = “Fail Failure : 'a result” -val diverge_tm = “Diverge : 'a result” - -val zero_num_tm = “0:num” -val suc_tm = “SUC” - -fun mk_result (ty : hol_type) : hol_type = Type.type_subst [ alpha_ty |-> ty ] result_ty -fun dest_result (ty : hol_type) : hol_type = - let - val {Args=out_ty, Thy=thy, Tyop=tyop} = dest_thy_type ty - in - if thy = "primitives" andalso tyop = "result" then hd out_ty - else failwith "dest_result: not a result" - end - -fun mk_return (x : term) : term = mk_icomb (return_tm, x) -fun mk_fail (ty : hol_type) (e : term) : term = mk_comb (inst [ alpha_ty |-> ty ] fail_tm, e) -fun mk_fail_failure (ty : hol_type) : term = inst [ alpha_ty |-> ty ] fail_failure_tm -fun mk_diverge (ty : hol_type) : term = inst [ alpha_ty |-> ty ] diverge_tm - -fun mk_suc (n : term) = mk_comb (suc_tm, n) - -(* - *) - -(* **BODY GENERATION**: - ==================== - - When generating a recursive definition, we apply a fixed-point operator to - a function body. In case we define a group of mutually recursive definitions, - we generate *one* single body for the whole group of definitions. It works - as follows. - - The input of the body is an enumeration: we start by branching over this input, and - every branch corresponds to one function in the mutually recursive group. Also, the - inputs must be grouped into tuples. Whenever we make a recursive call, we wrap the - input parameters into the proper variant, so as to call the proper function. - - Moreover, the input of the body must have the same type as its output: we also - store the outputs of the functions in some variants of the enumeration. - - In order to make this work, we need to shape the body so that: - - input values/output values are properly injected into the enumeration - - whenever we get an output value (which is an enumeration), we extract - the value from the proper variant of the enumeration - - We encode the enumeration with a nested sum type, whose constructors - are ‘INL’ and ‘INR’. - - Example: - ======== - We consider the following group of mutually recursive definitions: *) - -val even_odd_qt = Defn.parse_quote ‘ - (even (i : int) : bool result = if i = 0 then Return T else odd (i - 1)) /\ - (odd (i : int) : bool result = if i = 0 then Return F else even (i - 1)) -’ - -(* From those equations, we generate the following body: *) - -val even_odd_body_def = Define ‘ - even_odd_body - (* The body takes a continuation - required by the fixed-point operator *) - (f : (int + bool + int + bool) -> (int + bool + int + bool) result) - - (* The type of the input is: - input of even + output of even + input of odd + output of odd *) - (x : int + bool + int + bool) : - - (* The output type is the same as the input type - this constraint - comes from limitations in the way we can define the fixed-point - operator inside the HOL logic *) - (int + bool + int + bool) result = - - (* Case disjunction over the input, in order to figure out which - function from the group is actually called (even, or odd). *) - case x of - | INL i => (* Input of even *) - (* Body of even *) - if i = 0 then Return (INR (INL T)) - else - (* Recursive calls are calls to the continuation f, wrapped - in the proper variant: here we call odd *) - (case f (INR (INR (INL (i - 1)))) of - | Fail e => Fail e - | Diverge => Diverge - | Return r => - (* Extract the proper value from the enumeration: here, the - call is tail-call so this is not really necessary, but we - might need to retrieve the output of the call to odd, which - is a boolean, and do something more complex with it. *) - case r of - | INL _ => Fail Failure - | INR (INL _) => Fail Failure - | INR (INR (INL _)) => Fail Failure - | INR (INR (INR b)) => (* Extract the output of odd *) - (* Return: inject into the variant for the output of even *) - Return (INR (INL b)) - ) - | INR (INL _) => (* Output of even *) - (* We must ignore this one *) - Fail Failure - | INR (INR (INL i)) => - (* Body of odd *) - if i = 0 then Return (INR (INR (INR F))) - else - (* Call to even *) - (case f (INL (i - 1)) of - | Fail e => Fail e - | Diverge => Diverge - | Return r => - (* Extract the proper value from the enumeration *) - case r of - | INL _ => Fail Failure - | INR (INL b) => (* Extract the output of even *) - (* Return: inject into the variant for the output of odd *) - Return (INR (INR (INR b))) - | INR (INR (INL _)) => Fail Failure - | INR (INR (INR _)) => Fail Failure - ) - | INR (INR (INR _)) => (* Output of odd *) - (* We must ignore this one *) - Fail Failure -’ - -(* Small helper to generate wrappers of the shape: ‘INL x’, ‘INR (INL x)’, etc. - Note that we should have: ‘length before_tys + 1 + length after tys >= 2’ - - Ex.: - ==== - The enumeration has type: “: 'a + 'b + 'c + 'd”. - We want to generate the variant which injects “x:'c” into this enumeration. - - We need to split the list of types into: - {[ - before_tys = [“:'a”, “'b”] - tm = “x: 'c” - after_tys = [“:'d”] - ]} - - The function will generate: - {[ - INR (INR (INL x) : 'a + 'b + 'c + 'd - ]} - - (* Debug *) - val before_tys = [“:'a”, “:'b”, “:'c”] - val tm = “x:'d” - val after_tys = [“:'e”, “:'f”] - - val before_tys = [“:'a”, “:'b”, “:'c”] - val tm = “x:'d” - val after_tys = [] - - mk_inl_inr_wrapper before_tys tm after_tys - *) -fun list_mk_inl_inr (before_tys : hol_type list) (tm : term) (after_tys : hol_type list) : - term = - let - val (before_tys, pat) = - if after_tys = [] - then - let - val just_before_ty = List.last before_tys - val before_tys = List.take (before_tys, List.length before_tys - 1) - val pat = sumSyntax.mk_inr (tm, just_before_ty) - in - (before_tys, pat) - end - else (before_tys, sumSyntax.mk_inl (tm, sumSyntax.list_mk_sum after_tys)) - val pat = foldr (fn (ty, pat) => sumSyntax.mk_inr (pat, ty)) pat before_tys - in - pat - end - - -(* This function wraps a term into the proper variant of the input/output - sum. - - Ex.: - ==== - For the input of the first function, we generate: ‘INL x’ - For the output of the first function, we generate: ‘INR (INL x)’ - For the input of the 2nd function, we generate: ‘INR (INR (INL x))’ - etc. - - If ‘is_input’ is true: we are wrapping an input. Otherwise we are wrapping - an output. - - (* Debug *) - val tys = [(“:'a”, “:'b”), (“:'c”, “:'d”), (“:'e”, “:'f”)] - val j = 1 - val tm = “x:'c” - val tm = “y:'d” - val is_input = true - *) -fun inject_in_param_sum (tys : (hol_type * hol_type) list) (j : int) (is_input : bool) - (tm : term) : term = - let - fun flatten ls = List.concat (map (fn (x, y) => [x, y]) ls) - val before_tys = flatten (List.take (tys, j)) - val (input_ty, output_ty) = List.nth (tys, j) - val after_tys = flatten (List.drop (tys, j + 1)) - val (before_tys, after_tys) = - if is_input then (before_tys, output_ty :: after_tys) - else (before_tys @ [input_ty], after_tys) - in - list_mk_inl_inr before_tys tm after_tys - end - -(* Remark: the order of the branches when creating matches is important. - For instance, in the case of ‘result’ it must be: ‘Return’, ‘Fail’, ‘Diverge’. - - For the purpose of stability and maintainability, we introduce this small helper - which reorders the cases in a pattern before actually creating the case - expression. - *) -fun unordered_mk_case (scrut: term, pats: (term * term) list) : term = - let - (* Retrieve the constructors *) - val cl = TypeBase.constructors_of (type_of scrut) - (* Retrieve the names of the constructors *) - val names = map (fst o dest_const) cl - (* Use those to reorder the patterns *) - fun is_pat (name : string) (pat, _) = - let - val app = (fst o strip_comb) pat - val app_name = (fst o dest_const) app - in - app_name = name - end - val pats = map (fn name => valOf (List.find (is_pat name) pats)) names - in - (* Create the case *) - TypeBase.mk_case (scrut, pats) - end - -(* Wrap a term of type “:'a result” into a ‘case of’ which matches over - the result. - - Ex.: - ==== - {[ - f x - - ~~> - - case f x of - | Fail e => Fail e - | Diverge => Diverge - | Return y => ... (* The branch content is generated by the continuation *) - ]} - - ‘gen_ret_branch’ is a *continuation* which generates the content of the - ‘Return’ branch (i.e., the content of the ‘...’ in the example above). - It receives as input the value contained by the ‘Return’ (i.e., the variable - ‘y’ in the example above). - - Remark.: the type of the term generated by ‘gen_ret_branch’ must have - the type ‘result’, but it can change the content of the result (i.e., - if ‘scrut’ has type ‘:'a result’, we can change the type of the wrapped - expression to ‘:'b result’). - - (* Debug *) - val scrut = “x: int result” - fun gen_ret_branch x = mk_return x - - val scrut = “x: int result” - fun gen_ret_branch _ = “Return T” - - mk_result_case scrut gen_ret_branch - *) -fun mk_result_case (scrut : term) (gen_ret_branch : term -> term) : term = - let - val scrut_ty = dest_result (type_of scrut) - (* Return branch *) - val ret_var = genvar scrut_ty - val ret_pat = mk_return ret_var - val ret_br = gen_ret_branch ret_var - val ret_ty = dest_result (type_of ret_br) - (* Failure branch *) - val fail_var = genvar error_ty - val fail_pat = mk_fail scrut_ty fail_var - val fail_br = mk_fail ret_ty fail_var - (* Diverge branch *) - val div_pat = mk_diverge scrut_ty - val div_br = mk_diverge ret_ty - in - unordered_mk_case (scrut, [(ret_pat, ret_br), (fail_pat, fail_br), (div_pat, div_br)]) - end - -(* Generate a ‘case ... of’ over a sum type. - - Ex.: - ==== - If the scrutinee is: “x : 'a + 'b + 'c” (i.e., the tys list is: [“:'a”, “:b”, “:c”]), - we generate: - - {[ - case x of - | INL y0 => ... (* Branch of index 0 *) - | INR (INL y1) => ... (* Branch of index 1 *) - | INR (INR (INL y2)) => ... (* Branch of index 2 *) - | INR (INR (INR y3)) => ... (* Branch of index 3 *) - ]} - - The content of the branches is generated by the ‘gen_branch’ continuation, - which receives as input the index of the branch as well as the variable - introduced by the pattern (in the example above: ‘y0’ for the branch 0, - ‘y1’ for the branch 1, etc.) - - (* Debug *) - val tys = [“:'a”, “:'b”] - val scrut = mk_var ("x", sumSyntax.list_mk_sum tys) - fun gen_branch i (x : term) = “F” - - val tys = [“:'a”, “:'b”, “:'c”, “:'d”] - val scrut = mk_var ("x", sumSyntax.list_mk_sum tys) - fun gen_branch i (x : term) = if type_of x = “:'c” then mk_return x else mk_fail_failure “:'c” - - list_mk_sum_case scrut tys gen_branch - *) -(* For debugging *) -val list_mk_sum_case_case = ref (“T”, [] : (term * term) list) -(* -val (scrut, [(pat1, br1), (pat2, br2)]) = !list_mk_sum_case_case -*) -fun list_mk_sum_case (scrut : term) (tys : hol_type list) - (gen_branch : int -> term -> term) : term = - let - (* Create the cases. Note that without sugar, the match actually looks like this: - {[ - case x of - | INL y0 => ... (* Branch of index 0 *) - | INR x1 - case x1 of - | INL y1 => ... (* Branch of index 1 *) - | INR x2 => - case x2 of - | INL y2 => ... (* Branch of index 2 *) - | INR y3 => ... (* Branch of index 3 *) - ]} - *) - fun create_case (j : int) (scrut : term) (tys : hol_type list) : term = - let - val _ = print_dbg ("list_mk_sum_case: " ^ - String.concatWith ", " (map type_to_string tys) ^ "\n") - in - case tys of - [] => failwith "tys is too short" - | [ ty ] => - (* Last element: no match to perform *) - gen_branch j scrut - | ty1 :: tys => - (* Not last: we create a pattern: - {[ - case scrut of - | INL pat_var1 => ... (* Branch of index i *) - | INR pat_var2 => - ... (* Generate this term recursively *) - ]} - *) - let - (* INL branch *) - val after_ty = sumSyntax.list_mk_sum tys - val pat_var1 = genvar ty1 - val pat1 = sumSyntax.mk_inl (pat_var1, after_ty) - val br1 = gen_branch j pat_var1 - (* INR branch *) - val pat_var2 = genvar after_ty - val pat2 = sumSyntax.mk_inr (pat_var2, ty1) - val br2 = create_case (j+1) pat_var2 tys - val _ = print_dbg ("list_mk_sum_case: assembling:\n" ^ - term_to_string scrut ^ ",\n" ^ - "[(" ^ term_to_string pat1 ^ ",\n " ^ term_to_string br1 ^ "),\n\n" ^ - " (" ^ term_to_string pat2 ^ ",\n " ^ term_to_string br2 ^ ")]\n\n") - val case_elems = (scrut, [(pat1, br1), (pat2, br2)]) - val _ = list_mk_sum_case_case := case_elems - in - (* Put everything together *) - TypeBase.mk_case case_elems - end - end - in - create_case 0 scrut tys - end - -(* Generate a ‘case ... of’ to select the input/output of the ith variant of - the param enumeration. - - Ex.: - ==== - There are two functions in the group, and we select the input of the function of index 1: - {[ - case x of - | INL _ => Fail Failure (* Input of function of index 0 *) - | INR (INL _) => Fail Failure (* Output of function of index 0 *) - | INR (INR (INL y)) => Return y (* Input of the function of index 1: select this one *) - | INR (INR (INR _)) => Fail Failure (* Output of the function of index 1 *) - ]} - - (* Debug *) - val tys = [(“:'a”, “:'b”)] - val scrut = “x : 'a + 'b” - val fi = 0 - val is_input = true - - val tys = [(“:'a”, “:'b”), (“:'c”, “:'d”)] - val scrut = “x : 'a + 'b + 'c + 'd” - val fi = 1 - val is_input = false - - val scrut = mk_var ("x", sumSyntax.list_mk_sum (flatten tys)) - - list_mk_case_select scrut tys fi is_input - *) -fun list_mk_case_sum_select (scrut : term) (tys : (hol_type * hol_type) list) - (fi : int) (is_input : bool) : term = - let - (* The index of the element in the enumeration that we will select *) - val i = 2 * fi + (if is_input then 0 else 1) - (* Flatten the types and numerotate them *) - fun flatten ls = List.concat (map (fn (x, y) => [x, y]) ls) - val tys = flatten tys - (* Get the return type *) - val ret_ty = List.nth (tys, i) - (* The continuation which will generate the content of the branches *) - fun gen_branch j var = if j = i then mk_return var else mk_fail_failure ret_ty - in - (* Generate the ‘case ... of’ *) - list_mk_sum_case scrut tys gen_branch - end - -(* Generate a ‘case ... of’ to select the input/output of the ith variant of - the param enumeration. - - Ex.: - ==== - There are two functions in the group, and we select the input of the function of index 1: - {[ - case x of - | Fail e => Fail e - | Diverge => Diverge - | Return r => - case r of - | INL _ => Fail Failure (* Input of function of index 0 *) - | INR (INL _) => Fail Failure (* Output of function of index 0 *) - | INR (INR (INL y)) => Return y (* Input of the function of index 1: select this one *) - | INR (INR (INR _)) => Fail Failure (* Output of the function of index 1 *) - ]} - *) -fun mk_case_select_result_sum (scrut : term) (tys : (hol_type * hol_type) list) - (fi : int) (is_input : bool) : term = - (* We match over the result, then over the enumeration *) - mk_result_case scrut (fn x => list_mk_case_sum_select x tys fi is_input) - -(* -val scrut = call -val tys = in_out_tys -val is_input = false -val call = mk_case_select_result_sum call in_out_tys fi false -*) - -(* TODO: move *) -fun enumerate (ls : 'a list) : (int * 'a) list = - zip (List.tabulate (List.length ls, fn i => i)) ls - -(* Generate a body for the fixed-point operator from a quoted group of mutually - recursive definitions. - - See TODO for detailed explanations: from the quoted equations for ‘nth’ - (or for [‘even’, ‘odd’]) we generate the body ‘nth_body’ (or ‘even_odd_body’, - respectively). - *) -fun mk_body (fnames : string list) (in_out_tys : (hol_type * hol_type) list) - (def_tms : term list) : term = - let - val fnames_set = Redblackset.fromList String.compare fnames - - (* Compute a map from function name to function index *) - val fnames_map = Redblackmap.fromList String.compare - (map (fn (x, y) => (y, x)) (enumerate fnames)) - - (* Compute the input/output type, that we dub the "parameter type" *) - fun flatten ls = List.concat (map (fn (x, y) => [x, y]) ls) - val param_type = sumSyntax.list_mk_sum (flatten in_out_tys) - - (* Introduce a variable for the confinuation *) - val fcont = genvar (param_type --> mk_result param_type) - - (* In the function equations, replace all the recursive calls with calls to the continuation. - - When replacing a recursive call, we have to do two things: - - we need to inject the input parameters into the parameter type - Ex.: - - ‘nth tl i’ becomes ‘f (INL (tl, i))’ where ‘f’ is the continuation - - ‘even i’ becomes ‘f (INL i)’ where ‘f’ is the continuation - - we need to wrap the the call to the continuation into a ‘case ... of’ - to extract its output (we need to make sure that the transformation - preserves the type of the expression!) - Ex.: ‘nth tl i’ becomes: - {[ - case f (INL (tl, i)) of - | Fail e => Fail e - | Diverge => Diverge - | Return r => - case r of - | INL _ => Fail Failure - | INR x => Return (INR x) - ]} - *) - (* For debugging *) - val replace_rec_calls_rec_call_tm = ref “T” - fun replace_rec_calls (fnames_set : string Redblackset.set) (tm : term) : term = - let - val _ = print_dbg ("replace_rec_calls: original expression:\n" ^ - term_to_string tm ^ "\n\n") - val ntm = - case dest_term tm of - VAR (name, ty) => - (* Check that this is not one of the functions in the group - remark: - we could handle that by introducing lambdas. - *) - if Redblackset.member (fnames_set, name) - then failwith ("mk_body: not well-formed definition: found " ^ name ^ - " in an improper position") - else tm - | CONST _ => tm - | LAMB (x, tm) => - let - (* The variable might shadow one of the functions *) - val fnames_set = Redblackset.delete (fnames_set, (fst o dest_var) x) - (* Update the term in the lambda *) - val tm = replace_rec_calls fnames_set tm - in - (* Reconstruct *) - mk_abs (x, tm) - end - | COMB (_, _) => - let - (* Completely destruct the application, check if this is a recursive call *) - val (app, args) = strip_comb tm - val is_rec_call = Redblackset.member (fnames_set, (fst o dest_var) app) - handle HOL_ERR _ => false - (* Whatever the case, apply the transformation to all the inputs *) - val args = map (replace_rec_calls fnames_set) args - in - (* If this is not a recursive call: apply the transformation to all the - terms. Otherwise, replace. *) - if not is_rec_call then list_mk_comb (replace_rec_calls fnames_set app, args) - else - (* Rec call: replace *) - let - val _ = replace_rec_calls_rec_call_tm := tm - (* First, find the index of the function *) - val fname = (fst o dest_var) app - val fi = Redblackmap.find (fnames_map, fname) - (* Inject the input values into the param type *) - val input = pairSyntax.list_mk_pair args - val input = inject_in_param_sum in_out_tys fi true input - (* Create the recursive call *) - val call = mk_comb (fcont, input) - (* Wrap the call into a ‘case ... of’ to extract the output *) - val call = mk_case_select_result_sum call in_out_tys fi false - in - (* Return *) - call - end - end - val _ = print_dbg ("replace_rec_calls: new expression:\n" ^ term_to_string ntm ^ "\n\n") - in - ntm - end - handle HOL_ERR e => - let - val _ = print_dbg ("replace_rec_calls: failed on:\n" ^ term_to_string tm ^ "\n\n") - in - raise (HOL_ERR e) - end - fun replace_rec_calls_in_eq (eq : term) : term = - let - val (l, r) = dest_eq eq - in - mk_eq (l, replace_rec_calls fnames_set r) - end - val def_tms_with_fcont = map replace_rec_calls_in_eq def_tms - - (* Wrap all the function bodies to inject their result into the param type. - - We collect the function inputs at the same time, because they will be - grouped into a tuple that we will have to deconstruct. - *) - fun inject_body_to_enums (i : int, def_eq : term) : term list * term = - let - val (l, body) = dest_eq def_eq - val (_, args) = strip_comb l - (* We have the deconstruct the result, then, in the ‘Return’ branch, - properly wrap the returned value *) - val body = mk_result_case body (fn x => mk_return (inject_in_param_sum in_out_tys i false x)) - in - (args, body) - end - val def_tms_inject = map inject_body_to_enums (enumerate def_tms_with_fcont) - - (* Currify the body inputs. - - For instance, if the body has inputs: ‘x’, ‘y’; we return the following: - {[ - (‘z’, ‘case z of (x, y) => ... (* body *) ’) - ]} - where ‘z’ is fresh. - - We return: (curried input, body). - - (* Debug *) - val body = “(x:'a, y:'b, z:'c)” - val args = [“x:'a”, “y:'b”, “z:'c”] - currify_body_inputs (args, body) - *) - fun currify_body_inputs (args : term list, body : term) : term * term = - let - fun mk_curry (args : term list) (body : term) : term * term = - case args of - [] => failwith "no inputs" - | [x] => (x, body) - | x1 :: args => - let - val (x2, body) = mk_curry args body - val scrut = genvar (pairSyntax.list_mk_prod (map type_of (x1 :: args))) - val pat = pairSyntax.mk_pair (x1, x2) - val br = body - in - (scrut, TypeBase.mk_case (scrut, [(pat, br)])) - end - in - mk_curry args body - end - val def_tms_currified = map currify_body_inputs def_tms_inject - - (* Group all the functions into a single body, with an outer ‘case .. of’ - which selects the appropriate body depending on the input *) - val param_ty = sumSyntax.list_mk_sum (flatten in_out_tys) - val input = genvar param_ty - fun mk_mut_rec_body_branch (i : int) (patvar : term) : term = - (* Case disjunction on whether the branch is for an input value (in - which case we call the proper body) or an output value (in which - case we return ‘Fail ...’ *) - if i mod 2 = 0 then - let - val fi = i div 2 - val (x, def_tm) = List.nth (def_tms_currified, fi) - (* The variable in the pattern and the variable expected by the - body may not be the same: we introduce a let binding *) - val def_tm = mk_let (mk_abs (x, def_tm), patvar) - in - def_tm - end - else - (* Output value: fail *) - mk_fail_failure param_ty - val mut_rec_body = list_mk_sum_case input (flatten in_out_tys) mk_mut_rec_body_branch - - - (* Abstract away the parameters to produce the final body of the fixed point *) - val mut_rec_body = list_mk_abs ([fcont, input], mut_rec_body) - in - mut_rec_body - end - -(* For explanations about the different steps, see TODO *) -fun DefineDiv (def_qt : term quotation) = - let - (* Parse the definitions. - - Example: - {[ - (even (i : int) : bool result = if i = 0 then Return T else odd (i - 1)) /\ - (odd (i : int) : bool result = if i = 0 then Return F else even (i - 1)) - ]} - *) - val def_tms = (strip_conj o list_mk_conj o rev) (Defn.parse_quote def_qt) - - (* Compute the names and the input/output types of the functions *) - fun compute_names_in_out_tys (tm : term) : string * (hol_type * hol_type) = - let - val app = lhs tm - val name = (fst o dest_var o fst o strip_comb) app - val out_ty = dest_result (type_of app) - val input_tys = pairSyntax.list_mk_prod (map type_of ((snd o strip_comb) app)) - in - (name, (input_tys, out_ty)) - end - val (fnames, in_out_tys) = unzip (map compute_names_in_out_tys def_tms) - - (* Generate the body. - - See the comments at the beginning of the file (lookup "BODY GENERATION"). - *) - val body = mk_body fnames in_out_tys def_tms - - (* Prove that the body satisfies the validity property required by the fixed point *) - val body_is_valid = prove_body_is_valid body - - (* Generate the definitions for the various functions by using the fixed point - and the body. *) - val raw_defs = mk_raw_defs in_out_tys def_tms body_is_valid - - (* Prove the final equations *) - val def_eqs = prove_def_eqs body_is_valid def_tms raw_defs - in - def_eqs - end - -val [even_def, odd_def] = DefineDiv ‘ - (even (i : int) : bool result = - if i = 0 then Return T else odd (i - 1)) /\ - (odd (i : int) : bool result = - if i = 0 then Return F else even (i - 1)) -’ - -val [nth_def] = DefineDiv ‘ - nth (ls : 't list_t) (i : u32) : 't result = - case ls of - | ListCons x tl => - if u32_to_int i = (0:int) - then (Return x) - else - do - i0 <- u32_sub i (int_to_u32 1); - nth tl i0 - od - | ListNil => Fail Failure -’ - -val even_odd_body_def = Define ‘ - even_odd_body - (* The body takes a continuation - required by the fixed-point operator *) - (f : (int + bool + int + bool) -> (int + bool + int + bool) result) - (* The type of the input/output is: - input of even + output of even + input of odd + output of odd - *) - (x : int + bool + int + bool) : - (int + bool + int + bool) result = - (* Case disjunction over the input, in order to figure out which - function from the group is actually called (even , or odd). *) - case x of - | INL i => (* Input of even *) - (* Body of even *) - if i = 0 then Return (INR (INL T)) - else - (* Recursive calls are calls to the continuation f, wrapped - in the proper variant: here we call odd *) - (case f (INR (INR (INL (i - 1)))) of - | Fail e => Fail e - | Diverge => Diverge - | Return r => - (* Eliminate the unwanted results *) - case r of - | INL _ => Fail Failure - | INR (INL _) => Fail Failure - | INR (INR (INL _)) => Fail Failure - | INR (INR (INR b)) => (* Extract the output of odd *) - (* Inject into the variant for the output of even *) - Return (INR (INL b)) - ) - | INR (INL _) => (* Output of even *) - (* We must ignore this one *) - Fail Failure - | INR (INR (INL i)) => - (* Body of odd *) - if i = 0 then Return (INR (INR (INR F))) - else - (* Call to even *) - (case f (INL (i - 1)) of - | Fail e => Fail e - | Diverge => Diverge - | Return r => - (* Eliminate the unwanted results *) - case r of - | INL _ => Fail Failure - | INR (INL b) => (* Extract the output of even *) - (* Inject into the variant for the output of odd *) - Return (INR (INR (INR b))) - | INR (INR (INL _)) => Fail Failure - | INR (INR (INR _)) => Fail Failure - ) - | INR (INR (INR _)) => (* Output of odd *) - (* We must ignore this one *) - Fail Failure -’ - -Theorem even_odd_body_is_valid_aux: - is_valid_fp_body (SUC (SUC n)) even_odd_body -Proof - prove_body_is_valid_tac (SOME even_odd_body_def) -QED - -Theorem even_odd_body_is_valid: - is_valid_fp_body (SUC (SUC 0)) even_odd_body -Proof - irule even_odd_body_is_valid_aux -QED - -val even_raw_def = Define ‘ - even (i : int) = - case fix even_odd_body (INL i) of - | Fail e => Fail e - | Diverge => Diverge - | Return r => - case r of - | INL _ => Fail Failure - | INR (INL b) => Return b - | INR (INR (INL _)) => Fail Failure - | INR (INR (INR _)) => Fail Failure -’ - -val odd_raw_def = Define ‘ - odd (i : int) = - case fix even_odd_body (INR (INR (INL i))) of - | Fail e => Fail e - | Diverge => Diverge - | Return r => - case r of - | INL _ => Fail Failure - | INR (INL b) => Fail Failure - | INR (INR (INL _)) => Fail Failure - | INR (INR (INR b)) => Return b -’ - -Theorem even_def: - ∀i. even (i : int) : bool result = - if i = 0 then Return T else odd (i - 1) -Proof - prove_def_eq_tac even_raw_def [even_raw_def, odd_raw_def] even_odd_body_is_valid even_odd_body_def -QED - -Theorem odd_def: - ∀i. odd (i : int) : bool result = - if i = 0 then Return F else even (i - 1) -Proof - prove_def_eq_tac odd_raw_def [even_raw_def, odd_raw_def] even_odd_body_is_valid even_odd_body_def -QED - -val _ = export_theory () |