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authorSon Ho2023-05-12 20:19:35 +0200
committerSon HO2023-06-04 21:54:38 +0200
commiteb7c257b30840ee947e35db0dd90f3d48894e3dc (patch)
treef8f1a1b5f1e90856181c6ee985a8601aab2e7347
parent8a5c5e4ae0cab0ab627c25ece59453a8e4bd4b64 (diff)
Do more cleanup
Diffstat (limited to '')
-rw-r--r--backends/hol4/divDefLibTestScript.sml (renamed from backends/hol4/divDefLibExampleScript.sml)0
-rw-r--r--backends/hol4/divDefProto2TestScript.sml1222
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 ()