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authorSon Ho2023-02-03 19:49:49 +0100
committerSon HO2023-06-04 21:54:38 +0200
commit266bb06fea0a3ed22aad8b5862b502cb621714ac (patch)
tree12f5e4d587dca83ab6b930b1069a82cd02f70973 /backends
parenta8cf54cd23b8bd8c4cb5690ebee48a4086c4ca8d (diff)
Start working on a version of divDefLib which uses fixed point combinators
Diffstat (limited to '')
-rw-r--r--backends/hol4/divDefProtoScript.sml773
-rw-r--r--backends/hol4/divDefProtoTheory.sig210
2 files changed, 983 insertions, 0 deletions
diff --git a/backends/hol4/divDefProtoScript.sml b/backends/hol4/divDefProtoScript.sml
new file mode 100644
index 00000000..64d4b9f0
--- /dev/null
+++ b/backends/hol4/divDefProtoScript.sml
@@ -0,0 +1,773 @@
+(* Prototype: divDefLib but with general combinators *)
+
+open HolKernel boolLib bossLib Parse
+open boolTheory arithmeticTheory integerTheory intLib listTheory stringTheory
+
+open primitivesArithTheory primitivesBaseTacLib ilistTheory primitivesTheory
+open primitivesLib
+
+val _ = new_theory "divDefProto"
+
+(*
+ * Test with a general validity predicate.
+ *
+ * TODO: this works! Cleanup.
+ *)
+val fix_fuel_def = Define ‘
+ (fix_fuel (0 : num) (f : ('a -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result = Diverge) ∧
+ (fix_fuel (SUC n) (f : ('a -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result = f (fix_fuel n f) x)
+’
+
+val fix_fuel_P_def = Define ‘
+ fix_fuel_P f x n = ~(is_diverge (fix_fuel n f x))
+’
+
+val fix_def = Define ‘
+ fix (f : ('a -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result =
+ if (∃ n. fix_fuel_P f x n) then fix_fuel ($LEAST (fix_fuel_P f x)) f x else Diverge
+’
+
+val is_valid_fp_body_def = Define ‘
+ is_valid_fp_body (f : ('a -> 'b result) -> 'a -> 'b result) =
+ ∀x. (∀g h. f g x = f h x) ∨ (∃ y. ∀g. f g x = g y)
+’
+
+Theorem fix_fuel_mono:
+ ∀f. is_valid_fp_body f ⇒
+ ∀n x. fix_fuel_P f x n ⇒
+ ∀ m. n ≤ m ⇒
+ fix_fuel n f x = fix_fuel m f x
+Proof
+ ntac 2 strip_tac >>
+ Induct_on ‘n’ >> rpt strip_tac
+ >- (fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def]) >>
+ Cases_on ‘m’ >- int_tac >> fs [] >>
+ fs [is_valid_fp_body_def] >>
+ fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
+
+ (* Use the validity property *)
+ last_x_assum (qspec_assume ‘x’) >>
+ (* This makes a case disjunction on the validity property *)
+ rw []
+ >-((* Case 1: the continuation doesn't matter *) fs []) >>
+ (* Case 2: the continuation *does* matter (i.e., there is a recursive call *)
+ (* Instantiate the validity property with the different continuations *)
+ first_assum (qspec_assume ‘fix_fuel n f’) >>
+ first_assum (qspec_assume ‘fix_fuel n' f’) >>
+ last_assum (qspec_assume ‘y’) >>
+ fs []
+QED
+
+(* TODO: remove? *)
+Theorem fix_fuel_mono_least:
+ ∀f. is_valid_fp_body f ⇒
+ ∀n x. fix_fuel_P f x n ⇒
+ fix_fuel n f x = fix_fuel ($LEAST (fix_fuel_P f x)) f x
+Proof
+ rw [] >>
+ pure_once_rewrite_tac [EQ_SYM_EQ] >>
+ irule fix_fuel_mono >> fs [] >>
+ (* Use the "fundamental" property about $LEAST *)
+ qspec_assume ‘fix_fuel_P f x’ whileTheory.LEAST_EXISTS_IMP >>
+ (* Prove the premise *)
+ pop_assum sg_premise_tac >- metis_tac [] >> fs [] >>
+ spose_not_then assume_tac >>
+ fs [not_le_eq_gt]
+QED
+
+Theorem fix_fixed_diverges:
+ ∀f. is_valid_fp_body f ⇒ ∀x. ~(∃ n. fix_fuel_P f x n) ⇒ fix f x = f (fix f) x
+Proof
+ rw [fix_def] >>
+ (* Case disjunction on the validity hypothesis *)
+ fs [is_valid_fp_body_def] >>
+ last_x_assum (qspec_assume ‘x’) >>
+ rw []
+ >- (
+ last_assum (qspec_assume ‘SUC n’) >>
+ fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
+ first_assum (qspecl_assume [‘fix f’, ‘fix_fuel n f’]) >>
+ Cases_on ‘f (fix_fuel n f) x’ >> fs []
+ ) >>
+ first_assum (qspec_assume ‘fix f’) >> fs [] >>
+ fs [fix_def] >>
+ (* Case disjunction on the ‘∃n. ...’ *)
+ rw [] >>
+ (* Use the hypothesis that there is no fuel s.t. ... *)
+ last_assum (qspec_assume ‘SUC n’) >>
+ fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
+ first_assum (qspec_assume ‘fix_fuel n f’) >>
+ Cases_on ‘fix_fuel n f y’ >> fs []
+QED
+
+(* TODO: I think we can merge this with the theorem below *)
+Theorem fix_fixed_termination_rec_case:
+ ∀x y n m.
+ is_valid_fp_body f ⇒
+ (∀g. f g x = g y) ⇒
+ fix_fuel_P f x n ⇒
+ fix_fuel_P f y m ⇒
+ fix_fuel n f x = fix_fuel m f y
+Proof
+ rw [] >>
+ Cases_on ‘n’ >- fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
+ pure_asm_rewrite_tac [fix_fuel_def] >>
+ sg ‘fix_fuel_P f y n'’ >- rfs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
+ imp_res_tac fix_fuel_mono_least >>
+ fs []
+QED
+
+Theorem fix_fixed_termination_rec_case:
+ ∀ x y n m.
+ is_valid_fp_body f ⇒
+ (∀g. f g x = g y) ⇒
+ fix_fuel_P f x n ⇒
+ fix_fuel_P f y m ⇒
+ fix_fuel ($LEAST (fix_fuel_P f x)) f x =
+ fix_fuel ($LEAST (fix_fuel_P f y)) f y
+Proof
+ rw [] >>
+ imp_res_tac fix_fuel_mono_least >>
+ irule fix_fixed_termination_rec_case >>
+ fs [] >>
+ (* TODO: factorize *)
+ qspec_assume ‘fix_fuel_P f x’ whileTheory.LEAST_EXISTS_IMP >>
+ pop_assum sg_premise_tac >- metis_tac [] >>
+ qspec_assume ‘fix_fuel_P f y’ whileTheory.LEAST_EXISTS_IMP >>
+ pop_assum sg_premise_tac >- metis_tac [] >>
+ rw []
+QED
+
+Theorem fix_fixed_terminates:
+ ∀f. is_valid_fp_body f ⇒ ∀x n. fix_fuel_P f x n ⇒ fix f x = f (fix f) x
+Proof
+ rpt strip_tac >>
+ (* Case disjunction on the validity hypothesis - we don't want to consume it *)
+ last_assum mp_tac >>
+ pure_rewrite_tac [is_valid_fp_body_def] >> rpt strip_tac >>
+ pop_assum (qspec_assume ‘x’) >>
+ rw []
+ >-( (* No recursive call *)
+ fs [fix_def] >> rw [] >> fs [] >>
+ imp_res_tac fix_fuel_mono_least >>
+ Cases_on ‘$LEAST (fix_fuel_P f x)’ >> fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def]
+ ) >>
+ (* There exists a recursive call *)
+ SUBGOAL_THEN “∃m. fix_fuel_P f y m” assume_tac >-(
+ Cases_on ‘n’ >>
+ fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
+ metis_tac []
+ ) >>
+ fs [fix_def] >> rw [] >> fs [] >>
+ irule fix_fixed_termination_rec_case >>
+ fs [] >>
+ metis_tac []
+QED
+
+Theorem fix_fixed_eq:
+ ∀f. is_valid_fp_body f ⇒ ∀x. fix f x = f (fix f) x
+Proof
+ rw [] >>
+ Cases_on ‘∃n. fix_fuel_P f x n’
+ >- (irule fix_fixed_terminates >> metis_tac []) >>
+ irule fix_fixed_diverges >>
+ fs []
+QED
+
+(* Testing on an example *)
+Datatype:
+ list_t =
+ ListCons 't list_t
+ | ListNil
+End
+
+val nth_body_def = Define ‘
+ nth_body (f : ('t list_t # u32) -> 't result) (x : ('t list_t # u32)) : 't result =
+ let (ls, i) = x in
+ 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);
+ f (tl, i0)
+ od
+ | ListNil => Fail Failure
+’
+
+(* TODO: move *)
+Theorem is_valid_suffice:
+ ∃y. ∀g. g x = g y
+Proof
+ metis_tac []
+QED
+
+(* The general proof of is_valid_fp_body *)
+Theorem nth_body_is_valid:
+ is_valid_fp_body nth_body
+Proof
+ pure_rewrite_tac [is_valid_fp_body_def] >>
+ gen_tac >>
+ (* TODO: automate this *)
+ Cases_on ‘x’ >> fs [] >>
+ (* Expand *)
+ fs [nth_body_def, bind_def] >>
+ (* Explore all paths *)
+ Cases_on ‘q’ >> fs [is_valid_suffice] >>
+ Cases_on ‘u32_to_int r = 0’ >> fs [is_valid_suffice] >>
+ Cases_on ‘u32_sub r (int_to_u32 1)’ >> fs [is_valid_suffice]
+QED
+
+val nth_raw_def = Define ‘
+ nth (ls : 't list_t) (i : u32) = fix nth_body (ls, i)
+’
+
+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
+ rpt strip_tac >>
+ (* Expand the raw definition *)
+ pure_rewrite_tac [nth_raw_def] >>
+ (* Use the fixed-point equality *)
+ sg ‘fix nth_body (ls,i) = nth_body (fix nth_body) (ls,i)’
+ >- simp_tac bool_ss [HO_MATCH_MP fix_fixed_eq nth_body_is_valid] >>
+ pure_asm_rewrite_tac [] >>
+ (* Expand the body definition *)
+ qspecl_assume [‘fix nth_body’, ‘(ls, i)’] nth_body_def >>
+ pure_asm_rewrite_tac [LET_THM] >>
+ simp_tac arith_ss []
+QED
+
+(************************** FAILED TESTS *)
+
+(*
+Datatype:
+ fresult = FReturn 'b | FFail error | FRec 'a
+End
+
+(* TODO: monad converter from fresult to result + rewrite rules *)
+val fres_to_res_def = Define ‘
+ fres_to_res (f : 'a -> 'b result) (r : ('a, 'b) fresult) : 'b result =
+ case r of
+ | FReturn x => Return x
+ | FFail e => Fail e
+ | FRec y => f y
+’
+
+val fixa_fuel_def = Define ‘
+ (fixa_fuel (0 : num) (f : 'a -> ('a, 'b) fresult) (x : 'a) : 'b result = Diverge) ∧
+ (fixa_fuel (SUC n) (f : 'a -> ('a, 'b) fresult) (x : 'a) : 'b result =
+ fres_to_res (fixa_fuel n f) (f x))
+’
+
+val fixa_fuel_P_def = Define ‘
+ fixa_fuel_P f x n = ~(is_diverge (fixa_fuel n f x))
+’
+
+val fixa_def = Define ‘
+ fixa (f : 'a -> ('a, 'b) fresult) (x : 'a) : 'b result =
+ if (∃ n. fixa_fuel_P f x n) then fixa_fuel ($LEAST (fixa_fuel_P f x)) f x else Diverge
+’
+
+Theorem fixa_fuel_mono:
+ ∀n m. n <= m ⇒ (∀f x. fixa_fuel_P f x n ⇒ fixa_fuel n f x = fixa_fuel m f x)
+Proof
+ Induct_on ‘n’ >> rpt strip_tac
+ >- (fs [fixa_fuel_P_def, is_diverge_def, fixa_fuel_def, fres_to_res_def]) >>
+ Cases_on ‘m’ >- int_tac >> fs [] >>
+ last_x_assum imp_res_tac >>
+ fs [fixa_fuel_P_def, is_diverge_def, fixa_fuel_def, fres_to_res_def] >>
+ Cases_on ‘f x’ >> fs []
+QED
+
+(* TODO: remove ? *)
+Theorem fixa_fuel_P_least:
+ ∀f x n. fixa_fuel_P f x n ⇒ fixa_fuel_P f x ($LEAST (fixa_fuel_P f x))
+Proof
+ rw [] >>
+ (* Use the "fundamental" property about $LEAST *)
+ qspec_assume ‘fixa_fuel_P f x’ whileTheory.LEAST_EXISTS_IMP >>
+ (* Prove the premise *)
+ pop_assum sg_premise_tac >- metis_tac [] >>
+ rw []
+QED
+
+Theorem fixa_fuel_mono_least:
+ ∀n f x. fixa_fuel_P f x n ⇒ fixa_fuel n f x = fixa_fuel ($LEAST (fixa_fuel_P f x)) f x
+Proof
+ rw [] >>
+ (* Use the "fundamental" property about $LEAST *)
+ qspec_assume ‘fixa_fuel_P f x’ whileTheory.LEAST_EXISTS_IMP >>
+ (* Prove the premise *)
+ pop_assum sg_premise_tac >- metis_tac [] >>
+ (* Prove that $LEAST ... ≤ n *)
+ sg ‘n >= $LEAST (fixa_fuel_P f x)’ >-(
+ spose_not_then assume_tac >>
+ fs [not_ge_eq_lt]) >>
+ pure_once_rewrite_tac [EQ_SYM_EQ] >>
+ irule fixa_fuel_mono >>
+ fs []
+QED
+
+Theorem fixa_fixed_eq:
+ ∀f x. fixa f x = fres_to_res (fixa f) (f x)
+Proof
+ rw [fixa_def, fres_to_res_def]
+ >- (
+ (* Termination case *)
+ imp_res_tac fixa_fuel_P_least >>
+ Cases_on ‘$LEAST (fixa_fuel_P f x)’ >>
+ fs [fixa_fuel_P_def, is_diverge_def, fixa_fuel_def, fres_to_res_def] >>
+ Cases_on ‘f x’ >> fs [] >>
+ rw [] >> fs [] >>
+ irule fixa_fuel_mono_least >>
+ fs [fixa_fuel_P_def, is_diverge_def, fixa_fuel_def]) >>
+ (* Divergence case *)
+ fs [fres_to_res_def] >>
+ sg ‘∀n. ~(fixa_fuel_P f x (SUC n))’ >- fs [] >>
+ Cases_on ‘f x’ >> fs [] >>
+ fs [fixa_fuel_P_def, is_diverge_def, fixa_fuel_def, fres_to_res_def] >>
+ rw []
+QED
+
+val fix_fuel_def = Define ‘
+ (fix_fuel (0 : num) (f : ('a -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result = Diverge) ∧
+ (fix_fuel (SUC n) (f : ('a -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result = f (fix_fuel n f) x)
+’
+
+val fix_fuel_P_def = Define ‘
+ fix_fuel_P f x n = ~(is_diverge (fix_fuel n f x))
+’
+
+val fix_def = Define ‘
+ fix (f : ('a -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result =
+ if (∃ n. fix_fuel_P f x n) then fix_fuel ($LEAST (fix_fuel_P f x)) f x else Diverge
+’
+
+Theorem fixa_fuel_eq_fix_fuel:
+ ∀n (f : 'a -> ('a, 'b) fresult) (x : 'a).
+ fixa_fuel n f x = fix_fuel n (\g y. fres_to_res g (f y)) x
+Proof
+ Induct_on ‘n’ >> rw [fixa_fuel_def, fix_fuel_def, fres_to_res_def]
+QED
+
+Theorem fixa_fuel_P_eq_fix_fuel_P:
+ ∀(f : 'a -> ('a, 'b) fresult) (x : 'a).
+ fixa_fuel_P f x = fix_fuel_P (\g y. fres_to_res g (f y)) x
+Proof
+ rw [] >> irule EQ_EXT >> rw [] >>
+ qspecl_assume [‘x'’, ‘f’, ‘x’] fixa_fuel_eq_fix_fuel >>
+ fs [fixa_fuel_P_def, fix_fuel_P_def]
+QED
+
+Theorem fixa_fuel_eq_fix_fuel:
+ ∀(f : 'a -> ('a, 'b) fresult) (x : 'a).
+ fixa f x = fix (\g y. fres_to_res g (f y)) x
+Proof
+ rw [fixa_def, fix_def, fixa_fuel_P_eq_fix_fuel_P, fixa_fuel_eq_fix_fuel]
+QED
+
+(*
+ The annoying thing is that to make this work, we need to rewrite the
+ body by swapping matches, which is not easy to automate.
+*)
+
+(*
+f x =
+ case _ of
+ | _ => Fail e
+ | _ => Return x
+ | _ => f y
+
+f x (g : ('a, 'b) result => 'c)
+
+f x (fres_to_res g) = f x g
+
+*)
+
+(*
+val fix_fuel_def = Define ‘
+ (fix_fuel (0 : num) (f : ('a -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result = Diverge) ∧
+ (fix_fuel (SUC n) (f : ('a -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result = f (fix_fuel n f) x)
+’
+
+val fix_fuel_P_def = Define ‘
+ fix_fuel_P f x n = ~(is_diverge (fix_fuel n f x))
+’
+*)
+
+
+(*
+val is_valid_FP_body_def = Define ‘
+ is_valid_FP_body (f : (('a, 'b) fresult -> 'c) -> ('a, 'b) fresult -> 'c) =
+ ∃ (fa: (('a, 'b) fresult -> ('a, 'b) fresult) -> ('a, 'b) fresult -> ('a, 'b) fresult).
+ ∀ (g : ('a, 'b) fresult -> 'c) x.
+ f g x = g (fa (\x. x) x)
+
+val fix_fuel_def = Define ‘
+ (fix_fuel (0 : num) (f : (('a, 'b) fresult -> 'b result) -> ('a, 'b) fresult -> 'b result) (x : ('a, 'b) fresult) : 'b result = Diverge) ∧
+ (fix_fuel (SUC n) (f : (('a, 'b) fresult -> 'b result) -> ('a, 'b) fresult -> 'b result) (x : ('a, 'b) fresult) : 'b result = f (fix_fuel n f) x)
+’
+
+val fix_fuel_P_def = Define ‘
+ fix_fuel_P f x n = ~(is_diverge (fix_fuel n f x))
+’
+
+val fix_def = Define ‘
+ fix f x : 'b result =
+ if (∃ n. fix_fuel_P f x n) then fix_fuel ($LEAST (fix_fuel_P f x)) f x else Diverge
+
+
+Theorem fix_fuel_mono:
+ ∀f. is_valid_FP_body f ⇒
+ ∀n m. n <= m ⇒ (∀x. fix_fuel_P f x n ⇒ fix_fuel n f x = fix_fuel m f x)
+Proof
+ strip_tac >> strip_tac >>
+ Induct_on ‘n’ >> rpt strip_tac
+ >- (fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def]) >>
+ fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
+ Cases_on ‘m’ >- int_tac >>
+ fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
+ fs [is_valid_FP_body] >>
+ last_x_assum (qspec_assume ‘fix_fuel n f’) >>
+ fs []
+QED
+
+’*)
+
+(*
+ * Test with a validity predicate which gives monotonicity.
+
+ TODO: not enough to prove the fixed-point equation.
+ *)
+val fix_fuel_def = Define ‘
+ (fix_fuel (0 : num) (f : ('a -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result = Diverge) ∧
+ (fix_fuel (SUC n) (f : ('a -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result = f (fix_fuel n f) x)
+’
+
+val fix_fuel_P_def = Define ‘
+ fix_fuel_P f x n = ~(is_diverge (fix_fuel n f x))
+’
+
+val fix_def = Define ‘
+ fix (f : ('a -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result =
+ if (∃ n. fix_fuel_P f x n) then fix_fuel ($LEAST (fix_fuel_P f x)) f x else Diverge
+’
+
+Theorem fix_eq:
+ fix f = \x. if (∃ n. fix_fuel_P f x n) then fix_fuel ($LEAST (fix_fuel_P f x)) f x else Diverge
+Proof
+ irule EQ_EXT >> fs [fix_def]
+QED
+
+val is_valid_fp_body_def = Define ‘
+ is_valid_fp_body (f : ('a -> 'b result) -> 'a -> 'b result) =
+ ∀n m. n ≤ m ⇒ ∀x. fix_fuel_P f x n ⇒ fix_fuel n f x = fix_fuel m f x
+’
+
+Theorem fix_fuel_mono_least:
+ ∀f. is_valid_fp_body f ⇒
+ ∀n x. fix_fuel_P f x n ⇒ fix_fuel n f x = fix_fuel ($LEAST (fix_fuel_P f x)) f x
+Proof
+ rw [] >>
+ (* Use the "fundamental" property about $LEAST *)
+ qspec_assume ‘fix_fuel_P f x’ whileTheory.LEAST_EXISTS_IMP >>
+ (* Prove the premise *)
+ pop_assum sg_premise_tac >- metis_tac [] >> fs [] >>
+ (* Prove that $LEAST ... ≤ n *)
+ sg ‘n >= $LEAST (fix_fuel_P f x)’ >-(
+ spose_not_then assume_tac >>
+ fs [not_ge_eq_lt]) >>
+ pure_once_rewrite_tac [EQ_SYM_EQ] >>
+ (* Finish by using the monotonicity property *)
+ fs [is_valid_fp_body_def]
+QED
+
+
+(* TODO: remove ? *)
+Theorem fix_fuel_P_least:
+ ∀f x n. fix_fuel_P f x n ⇒ fix_fuel_P f x ($LEAST (fix_fuel_P f x))
+Proof
+ rw [] >>
+ (* Use the "fundamental" property about $LEAST *)
+ qspec_assume ‘fix_fuel_P f x’ whileTheory.LEAST_EXISTS_IMP >>
+ (* Prove the premise *)
+ pop_assum sg_premise_tac >- metis_tac [] >>
+ rw []
+QED
+
+(*Theorem test:
+ fix_fuel_P f x (SUC n) ⇒
+ (∀m. n ≤ m ⇒ f (fix_fuel n f) x = f (fix_fuel m f) x) ⇒
+ f (fix_fuel n f) x = f (fix f) x
+Proof
+ rw [] >>
+ spose_not_then assume_tac >>
+ fs [fix_eq]
+*)
+Theorem fix_fixed_eq:
+ ∀f. is_valid_fp_body f ⇒ ∀x. fix f x = f (fix f) x
+Proof
+ rw [fix_def] >> CASE_TAC >> fs []
+ >- (
+ (* Termination case *)
+ sg ‘$LEAST (fix_fuel_P f x) <= n’ >- cheat >>
+ imp_res_tac fix_fuel_P_least >>
+ Cases_on ‘$LEAST (fix_fuel_P f x)’ >>
+ fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
+ Cases_on ‘n’ >>
+ fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
+ (* Use the validity assumption *)
+ fs [is_valid_fp_body_def] >>
+
+(* last_x_assum imp_res_tac >> *)
+ last_assum (qspecl_assume [‘SUC n'’, ‘SUC n''’]) >> fs [] >>
+ pop_assum imp_res_tac >>
+ pop_assum (qspec_assume ‘x’) >>
+ pop_assum sg_premise_tac >- fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
+
+ fs [fix_fuel_def] >>
+
+
+ fs [fix_fuel_P_def] >>
+ rfs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
+
+ fs [fix_eq] >>
+
+ Cases_on ‘fix_fuel n' f x’ >> fs [] >>
+ last_assum (qspecl_then [‘n'’, ‘’
+ f (fix_fuel n' f) x >>
+
+ Cases_on ‘f x’ >> fs [] >>
+ rw [] >> fs [] >>
+ irule fix_fuel_mono_least >>
+ fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def]) >>
+ (* Divergence case *)
+ fs [fres_to_res_def] >>
+ sg ‘∀n. ~(fix_fuel_P f x (SUC n))’ >- fs [] >>
+ Cases_on ‘f x’ >> fs [] >>
+ fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
+ rw []
+QED
+
+
+(* A slightly weaker and more precise validity criteria.
+
+ We simply extracted the part of the inductive proof that we can't prove without
+ knowing ‘f’.
+ *)
+val is_valid_fp_body_weak_def = Define ‘
+ is_valid_fp_body_weak f =
+ ∀n m x.
+ ((∀x. fix_fuel_P f x n ⇒ fix_fuel n f x = fix_fuel m f x) ⇒
+ n ≤ m ⇒
+ fix_fuel_P f x (SUC n) ⇒
+ fix_fuel (SUC n) f x = fix_fuel (SUC m) f x)
+’
+
+Theorem is_valid_fp_body_weak_imp_is_valid:
+ ∀f. is_valid_fp_body_weak f ⇒ is_valid_fp_body f
+Proof
+ rpt strip_tac >> fs [is_valid_fp_body_def] >>
+ Induct_on ‘n’ >- fs [fix_fuel_P_def, fix_fuel_def, is_diverge_def] >>
+ Cases_on ‘m’ >> fs [] >>
+ fs [is_valid_fp_body_weak_def]
+QED
+
+(* Testing on an example *)
+Datatype:
+ list_t =
+ ListCons 't list_t
+ | ListNil
+End
+
+val nth_body_def = Define ‘
+ nth_body (f : ('t list_t # u32) -> 't result) (x : ('t list_t # u32)) : 't result =
+ let (ls, i) = x in
+ 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);
+ f (tl, i0)
+ od
+ | ListNil => (Fail Failure)
+’
+
+(* The general proof of is_valid_fp_body. We isolate a more precise property below. *)
+Theorem nth_body_is_valid:
+ is_valid_fp_body nth_body
+Proof
+ pure_rewrite_tac [is_valid_fp_body_def] >>
+ Induct_on ‘n’ >- fs [fix_fuel_P_def, fix_fuel_def, is_diverge_def] >>
+ Cases_on ‘m’ >- fs [] >>
+ rpt strip_tac >>
+ (* TODO: here *)
+ last_assum imp_res_tac >>
+ (* TODO: here? *)
+ fs [fix_fuel_def, nth_body_def, fix_fuel_P_def, is_diverge_def, bind_def] >>
+ Cases_on ‘x’ >> fs [] >> (* TODO: automate this *)
+ (* Explore all paths *)
+ Cases_on ‘q’ >> fs [] >>
+ Cases_on ‘u32_to_int r = 0’ >> fs [] >>
+ Cases_on ‘u32_sub r (int_to_u32 1)’ >> fs []
+QED
+
+
+Theorem nth_body_is_valid_weak:
+ is_valid_fp_body_weak nth_body
+Proof
+ pure_rewrite_tac [is_valid_fp_body_weak_def] >>
+ rpt strip_tac >>
+ (* TODO: automate this - we may need to destruct a variable number of times *)
+ Cases_on ‘x’ >> fs [] >>
+ (* Expand all *)
+ fs [fix_fuel_def, nth_body_def, fix_fuel_P_def, is_diverge_def, bind_def] >>
+ (* Explore all paths *)
+ Cases_on ‘q’ >> fs [] >>
+ Cases_on ‘u32_to_int r = 0’ >> fs [] >>
+ Cases_on ‘u32_sub r (int_to_u32 1)’ >> fs []
+QED
+
+(*
+ * Test with a more general validity predicate.
+ *)
+(* We need to control the way calls to the continuation are performed, so
+ that we can inspect the inputs (otherwise we can't prove anything).
+ *)
+val is_valid_FP_body_def = Define ‘
+ is_valid_FP_body (f : (('a, 'b) fresult -> 'c) -> 'a -> 'c) =
+ ∃ (fa: (('a, 'b) fresult -> ('a, 'b) fresult) -> 'a -> ('a, 'b) fresult).
+ ∀ (g : ('a, 'b) fresult -> 'c) x.
+ f g x = g (fa (\x. x) x)
+’
+
+
+val fix_fuel_def = Define ‘
+ (fix_fuel (0 : num) (f : (('a, 'b) fresult -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result = Diverge) ∧
+ (fix_fuel (SUC n) (f : (('a, 'b) fresult -> 'b result) -> 'a -> 'b result) (x : 'a) : 'b result =
+ f (fres_to_res (fix_fuel n f)) x)
+’
+
+val fix_fuel_P_def = Define ‘
+ fix_fuel_P f x n = ~(is_diverge (fix_fuel n f x))
+’
+
+val fix_def = Define ‘
+ fix f x : 'b result =
+ if (∃ n. fix_fuel_P f x n) then fix_fuel ($LEAST (fix_fuel_P f x)) f x else Diverge
+’
+
+Theorem fix_fuel_mono:
+ ∀f. is_valid_FP_body f ⇒
+ ∀n m. n <= m ⇒ (∀x. fix_fuel_P f x n ⇒ fix_fuel n f x = fix_fuel m f x)
+Proof
+ strip_tac >> strip_tac >>
+ Induct_on ‘n’ >> rpt strip_tac
+ >- (fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def]) >>
+ fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def, fres_to_res_def] >>
+ Cases_on ‘m’ >- int_tac >>
+ fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
+ fs [is_valid_FP_body_def, fres_to_res_def] >>
+ CASE_TAC >>
+ last_x_assum (qspec_assume ‘fres_to_res (fix_fuel n f)’) >>
+ fs [fres_to_res_def]
+QED
+
+(* Tests *)
+
+
+
+Datatype:
+ list_t =
+ ListCons 't list_t
+ | ListNil
+End
+
+(*
+val def_qt = ‘
+ nth_mut_fwd (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_mut_fwd tl i0
+ od
+ | ListNil =>
+ Fail Failure
+’
+*)
+
+val nth_body_def = Define ‘
+ nth_body (f : ('t list_t # u32, 't) fresult -> 'c) (ls : 't list_t, i : u32) : 'c =
+ case ls of
+ | ListCons x tl =>
+ if u32_to_int i = (0:int)
+ then f (FReturn x)
+ else
+ do
+ i0 <- u32_sub i (int_to_u32 1);
+ f (FRec (tl, i0))
+ od
+ | ListNil => f (FFail Failure)
+’
+
+Theorem nth_body_is_valid_FP_body:
+ is_valid_FP_body nth_body
+Proof
+ fs [is_valid_FP_body_def] >>
+ exists_tac “nth_body” >>
+QED
+
+“nth_body”
+
+(* TODO: abbreviation for ‘(\g y. fres_to_res g (f y))’ *)
+Theorem fix_fixed_eq:
+ ∀f x. fix (\g y. fres_to_res g (f y)) x =
+ (f x)
+ fres_to_res (\g y. fres_to_res g (f y))
+
+
+
+(*
+TODO: can't prove that
+
+Theorem fix_fuel_mono:
+ ∀n m. n <= m ⇒ (∀f x. fix_fuel_P f x n ⇒ fix_fuel n f x = fix_fuel m f x)
+Proof
+ Induct_on ‘n’ >> rpt strip_tac
+ >- (fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def]) >>
+ fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
+ Cases_on ‘m’ >- int_tac >>
+ fs [fix_fuel_P_def, is_diverge_def, fix_fuel_def] >>
+
+ sg ‘fix_fuel n f = fix_fuel n' f’ >>
+
+ last_x_assum imp_res_tac >>
+ pop_assum (qspecl_assume [‘x’, ‘f’]) >>
+
+Theorem fix_def_eq:
+ ∀f x. fix f x = f (fix f) x
+Proof
+
+*)
+*)
+
+
+val _ = export_theory ()
diff --git a/backends/hol4/divDefProtoTheory.sig b/backends/hol4/divDefProtoTheory.sig
new file mode 100644
index 00000000..6e4e5950
--- /dev/null
+++ b/backends/hol4/divDefProtoTheory.sig
@@ -0,0 +1,210 @@
+signature divDefProtoTheory =
+sig
+ type thm = Thm.thm
+
+ (* Definitions *)
+ val fix_def : thm
+ val fix_fuel_P_def : thm
+ val fix_fuel_def : thm
+ val is_valid_fp_body_def : thm
+ val list_t_TY_DEF : thm
+ val list_t_case_def : thm
+ val list_t_size_def : thm
+ val nth_body_def : thm
+
+ (* Theorems *)
+ val datatype_list_t : thm
+ val fix_fixed_diverges : thm
+ val fix_fixed_eq : thm
+ val fix_fixed_terminates : thm
+ val fix_fixed_termination_rec_case : thm
+ val fix_fuel_compute : thm
+ val fix_fuel_mono : thm
+ val fix_fuel_mono_least : thm
+ val is_valid_suffice : thm
+ val list_t_11 : thm
+ val list_t_Axiom : thm
+ val list_t_case_cong : thm
+ val list_t_case_eq : thm
+ val list_t_distinct : thm
+ val list_t_induction : thm
+ val list_t_nchotomy : thm
+ val nth_body_is_valid : thm
+ val nth_def : thm
+
+ val divDefProto_grammars : type_grammar.grammar * term_grammar.grammar
+(*
+ [primitives] Parent theory of "divDefProto"
+
+ [fix_def] Definition
+
+ ⊢ ∀f x.
+ fix f x =
+ if ∃n. fix_fuel_P f x n then
+ fix_fuel ($LEAST (fix_fuel_P f x)) f x
+ else Diverge
+
+ [fix_fuel_P_def] Definition
+
+ ⊢ ∀f x n. fix_fuel_P f x n ⇔ ¬is_diverge (fix_fuel n f x)
+
+ [fix_fuel_def] Definition
+
+ ⊢ (∀f x. fix_fuel 0 f x = Diverge) ∧
+ ∀n f x. fix_fuel (SUC n) f x = f (fix_fuel n f) x
+
+ [is_valid_fp_body_def] Definition
+
+ ⊢ ∀f. is_valid_fp_body f ⇔
+ ∀x. (∀g h. f g x = f h x) ∨ ∃y. ∀g. f g x = g y
+
+ [list_t_TY_DEF] Definition
+
+ ⊢ ∃rep.
+ TYPE_DEFINITION
+ (λa0'.
+ ∀ $var$('list_t').
+ (∀a0'.
+ (∃a0 a1.
+ a0' =
+ (λa0 a1.
+ ind_type$CONSTR 0 a0
+ (ind_type$FCONS a1 (λn. ind_type$BOTTOM)))
+ a0 a1 ∧ $var$('list_t') a1) ∨
+ a0' =
+ ind_type$CONSTR (SUC 0) ARB (λn. ind_type$BOTTOM) ⇒
+ $var$('list_t') a0') ⇒
+ $var$('list_t') a0') rep
+
+ [list_t_case_def] Definition
+
+ ⊢ (∀a0 a1 f v. list_t_CASE (ListCons a0 a1) f v = f a0 a1) ∧
+ ∀f v. list_t_CASE ListNil f v = v
+
+ [list_t_size_def] Definition
+
+ ⊢ (∀f a0 a1.
+ list_t_size f (ListCons a0 a1) = 1 + (f a0 + list_t_size f a1)) ∧
+ ∀f. list_t_size f ListNil = 0
+
+ [nth_body_def] Definition
+
+ ⊢ ∀f x.
+ nth_body f x =
+ (let
+ (ls,i) = x
+ in
+ case ls of
+ ListCons x tl =>
+ if u32_to_int i = 0 then Return x
+ else do i0 <- u32_sub i (int_to_u32 1); f (tl,i0) od
+ | ListNil => Fail Failure)
+
+ [datatype_list_t] Theorem
+
+ ⊢ DATATYPE (list_t ListCons ListNil)
+
+ [fix_fixed_diverges] Theorem
+
+ ⊢ ∀f. is_valid_fp_body f ⇒
+ ∀x. ¬(∃n. fix_fuel_P f x n) ⇒ fix f x = f (fix f) x
+
+ [fix_fixed_eq] Theorem
+
+ ⊢ ∀f. is_valid_fp_body f ⇒ ∀x. fix f x = f (fix f) x
+
+ [fix_fixed_terminates] Theorem
+
+ ⊢ ∀f. is_valid_fp_body f ⇒
+ ∀x n. fix_fuel_P f x n ⇒ fix f x = f (fix f) x
+
+ [fix_fixed_termination_rec_case] Theorem
+
+ ⊢ ∀x y n m.
+ is_valid_fp_body f ⇒
+ (∀g. f g x = g y) ⇒
+ fix_fuel_P f x n ⇒
+ fix_fuel_P f y m ⇒
+ fix_fuel ($LEAST (fix_fuel_P f x)) f x =
+ fix_fuel ($LEAST (fix_fuel_P f y)) f y
+
+ [fix_fuel_compute] Theorem
+
+ ⊢ (∀f x. fix_fuel 0 f x = Diverge) ∧
+ (∀n f x.
+ fix_fuel (NUMERAL (BIT1 n)) f x =
+ f (fix_fuel (NUMERAL (BIT1 n) − 1) f) x) ∧
+ ∀n f x.
+ fix_fuel (NUMERAL (BIT2 n)) f x =
+ f (fix_fuel (NUMERAL (BIT1 n)) f) x
+
+ [fix_fuel_mono] Theorem
+
+ ⊢ ∀f. is_valid_fp_body f ⇒
+ ∀n x.
+ fix_fuel_P f x n ⇒
+ ∀m. n ≤ m ⇒ fix_fuel n f x = fix_fuel m f x
+
+ [fix_fuel_mono_least] Theorem
+
+ ⊢ ∀f. is_valid_fp_body f ⇒
+ ∀n x.
+ fix_fuel_P f x n ⇒
+ fix_fuel n f x = fix_fuel ($LEAST (fix_fuel_P f x)) f x
+
+ [is_valid_suffice] Theorem
+
+ ⊢ ∃y. ∀g. g x = g y
+
+ [list_t_11] Theorem
+
+ ⊢ ∀a0 a1 a0' a1'.
+ ListCons a0 a1 = ListCons a0' a1' ⇔ a0 = a0' ∧ a1 = a1'
+
+ [list_t_Axiom] Theorem
+
+ ⊢ ∀f0 f1. ∃fn.
+ (∀a0 a1. fn (ListCons a0 a1) = f0 a0 a1 (fn a1)) ∧
+ fn ListNil = f1
+
+ [list_t_case_cong] Theorem
+
+ ⊢ ∀M M' f v.
+ M = M' ∧ (∀a0 a1. M' = ListCons a0 a1 ⇒ f a0 a1 = f' a0 a1) ∧
+ (M' = ListNil ⇒ v = v') ⇒
+ list_t_CASE M f v = list_t_CASE M' f' v'
+
+ [list_t_case_eq] Theorem
+
+ ⊢ list_t_CASE x f v = v' ⇔
+ (∃t l. x = ListCons t l ∧ f t l = v') ∨ x = ListNil ∧ v = v'
+
+ [list_t_distinct] Theorem
+
+ ⊢ ∀a1 a0. ListCons a0 a1 ≠ ListNil
+
+ [list_t_induction] Theorem
+
+ ⊢ ∀P. (∀l. P l ⇒ ∀t. P (ListCons t l)) ∧ P ListNil ⇒ ∀l. P l
+
+ [list_t_nchotomy] Theorem
+
+ ⊢ ∀ll. (∃t l. ll = ListCons t l) ∨ ll = ListNil
+
+ [nth_body_is_valid] Theorem
+
+ ⊢ is_valid_fp_body nth_body
+
+ [nth_def] Theorem
+
+ ⊢ ∀ls i.
+ nth ls i =
+ case ls of
+ ListCons x tl =>
+ if u32_to_int i = 0 then Return x
+ else do i0 <- u32_sub i (int_to_u32 1); nth tl i0 od
+ | ListNil => Fail Failure
+
+
+*)
+end