diff options
Diffstat (limited to 'backends')
| -rw-r--r-- | backends/hol4/primitivesScript.sml | 181 | ||||
| -rw-r--r-- | backends/hol4/primitivesTheory.sig | 159 |
2 files changed, 298 insertions, 42 deletions
diff --git a/backends/hol4/primitivesScript.sml b/backends/hol4/primitivesScript.sml index 4a1f5fdd..8cd54f33 100644 --- a/backends/hol4/primitivesScript.sml +++ b/backends/hol4/primitivesScript.sml @@ -1661,6 +1661,9 @@ val vec_to_list_num_bounds = new_axiom ("vec_to_list_num_bounds", “!v. (0:num) <= LENGTH (vec_to_list v) /\ LENGTH (vec_to_list v) <= Num usize_max”) +val mk_vec_axiom = new_axiom ("mk_vec_axiom", + “∀l. len l ≤ usize_max ⇒ vec_to_list (mk_vec l) = l”) + Theorem update_len: ∀ls i y. len (update ls i y) = len ls Proof @@ -1732,41 +1735,61 @@ Proof fs [] QED -val vec_index_def = Define ‘ - vec_index i v = - if usize_to_int i ≤ usize_to_int (vec_len v) - then Return (index (usize_to_int i) (vec_to_list v)) - else Fail Failure’ +Definition vec_new_def: + vec_new = mk_vec [] +End -val mk_vec_axiom = new_axiom ("mk_vec_axiom", - “∀l. len l ≤ usize_max ⇒ vec_to_list (mk_vec l) = l”) +Theorem vec_to_list_vec_new: + vec_to_list vec_new = [] +Proof + fs [vec_new_def] >> + sg_dep_rewrite_all_tac mk_vec_axiom >> fs [len_def] >> + assume_tac usize_bounds >> fs [u16_max_def] >> int_tac +QED +val _ = BasicProvers.export_rewrites ["vec_to_list_vec_new"] -(* A custom unfolding theorem for evaluation *) +Theorem vec_len_vec_new: + vec_len vec_new = int_to_usize 0 +Proof + fs [vec_len_def, vec_new_def] >> + sg_dep_rewrite_all_tac usize_to_int_int_to_usize >> fs [len_def, u16_max_def] >> + assume_tac usize_bounds >> fs [u16_max_def] >> int_tac +QED +val _ = BasicProvers.export_rewrites ["vec_len_vec_new"] + +(* A custom unfolding theorem for evaluation - we compare to “u16_max” rather + than “usize_max” on purpose. *) Theorem mk_vec_unfold: - ∀l. vec_to_list (mk_vec l) = if len l ≤ usize_max then l else vec_to_list (mk_vec l) + ∀l. vec_to_list (mk_vec l) = if len l ≤ u16_max then l else vec_to_list (mk_vec l) Proof + rw [] >> Cases_on ‘len l ≤ u16_max’ >> fs [] >> + assume_tac usize_bounds >> + sg ‘len l ≤ usize_max’ >- int_tac >> metis_tac [mk_vec_axiom] QED val _ = evalLib.add_unfold_thm "mk_vec_unfold" -(* Defining ‘vec_insert_back’ *) -val vec_insert_back_def = Define ‘ - vec_insert_back (v : 'a vec) (i : usize) (x : 'a) = - if usize_to_int i < usize_to_int (vec_len v) then - Return (mk_vec (update (vec_to_list v) (usize_to_int i) x)) - else Fail Failure’ +(* Helper *) +Definition vec_index_def: + vec_index v i = index (usize_to_int i) (vec_to_list v) +End -Theorem vec_insert_back_spec: - ∀v i x. - usize_to_int i < usize_to_int (vec_len v) ⇒ - ∃ nv. vec_insert_back v i x = Return nv ∧ - vec_len v = vec_len nv ∧ - vec_index i nv = Return x ∧ - (∀j. usize_to_int j < usize_to_int (vec_len nv) ⇒ - usize_to_int j ≠ usize_to_int i ⇒ - vec_index j nv = vec_index j v) -Proof - rpt strip_tac >> fs [vec_insert_back_def] >> +(* Helper *) +Definition vec_update_def: + vec_update v i x = mk_vec (update (vec_to_list v) (usize_to_int i) x) +End + +Theorem vec_update_eq: + ∀ v i x. + usize_to_int i < usize_to_int (vec_len v) ⇒ + let nv = vec_update v i x in + vec_len v = vec_len nv ∧ + vec_index nv i = x ∧ + (∀j. usize_to_int j < usize_to_int (vec_len nv) ⇒ + usize_to_int j ≠ usize_to_int i ⇒ + vec_index nv j = vec_index v j) +Proof + rpt strip_tac >> fs [vec_update_def] >> qspec_assume ‘update (vec_to_list v) (usize_to_int i) x’ mk_vec_axiom >> sg_dep_rewrite_all_tac update_len >> fs [] >> qspec_assume ‘v’ vec_len_spec >> gvs [] >> @@ -1778,4 +1801,110 @@ Proof sg_dep_rewrite_all_tac index_update_same >- cooper_tac >> fs [] QED +Definition vec_index_fwd_def: + vec_index_fwd v i = + if usize_to_int i ≤ usize_to_int (vec_len v) + then Return (vec_index v i) + else Fail Failure +End + +Definition vec_index_back_def: + vec_index_back v i = + if usize_to_int i < usize_to_int (vec_len v) then Return () else Fail Failure +End + +Definition vec_index_mut_fwd_def: + vec_index_mut_fwd v i = + if usize_to_int i ≤ usize_to_int (vec_len v) + then Return (vec_index v i) + else Fail Failure +End + +Definition vec_index_mut_back_def: + vec_index_mut_back v i x = + if usize_to_int i ≤ usize_to_int (vec_len v) + then Return (vec_update v i x) + else Fail Failure +End + +Theorem vec_index_fwd_spec: + ∀v i. + usize_to_int i < usize_to_int (vec_len v) ⇒ + vec_index_fwd v i = Return (vec_index v i) +Proof + rw [vec_index_fwd_def] +QED + +Theorem vec_index_back_spec: + ∀v i. + usize_to_int i < usize_to_int (vec_len v) ⇒ + vec_index_back v i = Return () +Proof + rw [vec_index_back_def] +QED + +Theorem vec_index_mut_fwd_spec: + ∀v i. + usize_to_int i < usize_to_int (vec_len v) ⇒ + vec_index_mut_fwd v i = Return (vec_index v i) +Proof + rw [vec_index_mut_fwd_def] +QED + +Theorem vec_index_mut_back_spec: + ∀v i x. + usize_to_int i < usize_to_int (vec_len v) ⇒ + vec_index_mut_back v i x = Return (vec_update v i x) +Proof + rw [vec_index_mut_back_def] +QED + +Definition vec_insert_back_def: + vec_insert_back (v : 'a vec) (i : usize) (x : 'a) = + if usize_to_int i < usize_to_int (vec_len v) then + Return (vec_update v i x) + else Fail Failure +End + +Theorem vec_insert_back_spec: + ∀v i x. + usize_to_int i < usize_to_int (vec_len v) ⇒ + vec_insert_back v i x = Return (vec_update v i x) +Proof + rw [vec_insert_back_def] +QED + +Definition vec_push_back_def: + vec_push_back (v : 'a vec) (x : 'a) : ('a vec) result = + if usize_to_int (vec_len v) < usize_max then + Return (mk_vec ((vec_to_list v) ++ [x])) + else Fail Failure +End + +Theorem vec_push_back_unfold: + ∀ v x. vec_push_back (v : 'a vec) (x : 'a) : ('a vec) result = + if usize_to_int (vec_len v) < u16_max ∨ usize_to_int (vec_len v) < usize_max then + Return (mk_vec ((vec_to_list v) ++ [x])) + else Fail Failure +Proof + assume_tac usize_bounds >> + rw [vec_push_back_def] >> fs [] >> + int_tac +QED +val _ = evalLib.add_unfold_thm "vec_push_back_unfold" + +Theorem vec_push_back_spec: + ∀ v x. + usize_to_int (vec_len v) < usize_max ⇒ + ∃ nv. vec_push_back v x = Return nv ∧ + vec_to_list nv = vec_to_list v ++ [x] +Proof + rw [vec_push_back_def, vec_len_def] >> + qspec_assume ‘v’ vec_len_spec >> + sg_dep_rewrite_all_tac usize_to_int_int_to_usize >- int_tac >> fs [] >> + sg_dep_rewrite_all_tac mk_vec_axiom + >- (fs [len_append, len_def, vec_len_def] >> int_tac) >> + fs [] +QED + val _ = export_theory () diff --git a/backends/hol4/primitivesTheory.sig b/backends/hol4/primitivesTheory.sig index 45caf234..2407e2f2 100644 --- a/backends/hol4/primitivesTheory.sig +++ b/backends/hol4/primitivesTheory.sig @@ -201,9 +201,16 @@ sig val usize_mul_def : thm val usize_rem_def : thm val usize_sub_def : thm + val vec_index_back_def : thm val vec_index_def : thm + val vec_index_fwd_def : thm + val vec_index_mut_back_def : thm + val vec_index_mut_fwd_def : thm val vec_insert_back_def : thm val vec_len_def : thm + val vec_new_def : thm + val vec_push_back_def : thm + val vec_update_def : thm (* Theorems *) val datatype_error : thm @@ -327,9 +334,18 @@ sig val usize_rem_eq : thm val usize_sub_eq : thm val usize_to_int_int_to_usize_unfold : thm + val vec_index_back_spec : thm + val vec_index_fwd_spec : thm + val vec_index_mut_back_spec : thm + val vec_index_mut_fwd_spec : thm val vec_insert_back_spec : thm val vec_len_spec : thm + val vec_len_vec_new : thm + val vec_push_back_spec : thm + val vec_push_back_unfold : thm val vec_to_list_int_bounds : thm + val vec_to_list_vec_new : thm + val vec_update_eq : thm val primitives_grammars : type_grammar.grammar * term_grammar.grammar (* @@ -1285,12 +1301,39 @@ sig ⊢ ∀x y. usize_sub x y = mk_usize (usize_to_int x − usize_to_int y) + [vec_index_back_def] Definition + + ⊢ ∀v i. + vec_index_back v i = + if usize_to_int i < usize_to_int (vec_len v) then Return () + else Fail Failure + [vec_index_def] Definition - ⊢ ∀i v. - vec_index i v = + ⊢ ∀v i. vec_index v i = index (usize_to_int i) (vec_to_list v) + + [vec_index_fwd_def] Definition + + ⊢ ∀v i. + vec_index_fwd v i = + if usize_to_int i ≤ usize_to_int (vec_len v) then + Return (vec_index v i) + else Fail Failure + + [vec_index_mut_back_def] Definition + + ⊢ ∀v i x. + vec_index_mut_back v i x = + if usize_to_int i ≤ usize_to_int (vec_len v) then + Return (vec_update v i x) + else Fail Failure + + [vec_index_mut_fwd_def] Definition + + ⊢ ∀v i. + vec_index_mut_fwd v i = if usize_to_int i ≤ usize_to_int (vec_len v) then - Return (index (usize_to_int i) (vec_to_list v)) + Return (vec_index v i) else Fail Failure [vec_insert_back_def] Definition @@ -1298,13 +1341,31 @@ sig ⊢ ∀v i x. vec_insert_back v i x = if usize_to_int i < usize_to_int (vec_len v) then - Return (mk_vec (update (vec_to_list v) (usize_to_int i) x)) + Return (vec_update v i x) else Fail Failure [vec_len_def] Definition ⊢ ∀v. vec_len v = int_to_usize (len (vec_to_list v)) + [vec_new_def] Definition + + ⊢ vec_new = mk_vec [] + + [vec_push_back_def] Definition + + ⊢ ∀v x. + vec_push_back v x = + if usize_to_int (vec_len v) < usize_max then + Return (mk_vec (vec_to_list v ⧺ [x])) + else Fail Failure + + [vec_update_def] Definition + + ⊢ ∀v i x. + vec_update v i x = + mk_vec (update (vec_to_list v) (usize_to_int i) x) + [datatype_error] Theorem ⊢ DATATYPE (error Failure) @@ -1851,9 +1912,9 @@ sig [mk_vec_unfold] Theorem - [oracles: DISK_THM] [axioms: mk_vec_axiom] [] + [oracles: DISK_THM] [axioms: mk_vec_axiom, usize_bounds] [] ⊢ ∀l. vec_to_list (mk_vec l) = - if len l ≤ usize_max then l else vec_to_list (mk_vec l) + if len l ≤ u16_max then l else vec_to_list (mk_vec l) [num2error_11] Theorem @@ -2295,20 +2356,35 @@ sig if 0 ≤ n ∧ n ≤ u16_max then n else usize_to_int (int_to_usize n) + [vec_index_back_spec] Theorem + + ⊢ ∀v i. + usize_to_int i < usize_to_int (vec_len v) ⇒ + vec_index_back v i = Return () + + [vec_index_fwd_spec] Theorem + + ⊢ ∀v i. + usize_to_int i < usize_to_int (vec_len v) ⇒ + vec_index_fwd v i = Return (vec_index v i) + + [vec_index_mut_back_spec] Theorem + + ⊢ ∀v i x. + usize_to_int i < usize_to_int (vec_len v) ⇒ + vec_index_mut_back v i x = Return (vec_update v i x) + + [vec_index_mut_fwd_spec] Theorem + + ⊢ ∀v i. + usize_to_int i < usize_to_int (vec_len v) ⇒ + vec_index_mut_fwd v i = Return (vec_index v i) + [vec_insert_back_spec] Theorem - [oracles: DISK_THM] - [axioms: vec_to_list_num_bounds, usize_bounds, - int_to_usize_usize_to_int, usize_to_int_bounds, - usize_to_int_int_to_usize, mk_vec_axiom] [] ⊢ ∀v i x. usize_to_int i < usize_to_int (vec_len v) ⇒ - ∃nv. - vec_insert_back v i x = Return nv ∧ vec_len v = vec_len nv ∧ - vec_index i nv = Return x ∧ - ∀j. usize_to_int j < usize_to_int (vec_len nv) ⇒ - usize_to_int j ≠ usize_to_int i ⇒ - vec_index j nv = vec_index j v + vec_insert_back v i x = Return (vec_update v i x) [vec_len_spec] Theorem @@ -2318,11 +2394,62 @@ sig ⊢ ∀v. vec_len v = int_to_usize (len (vec_to_list v)) ∧ 0 ≤ len (vec_to_list v) ∧ len (vec_to_list v) ≤ usize_max + [vec_len_vec_new] Theorem + + [oracles: DISK_THM] + [axioms: mk_vec_axiom, int_to_usize_usize_to_int, + usize_to_int_int_to_usize, usize_bounds] [] + ⊢ vec_len vec_new = int_to_usize 0 + + [vec_push_back_spec] Theorem + + [oracles: DISK_THM] + [axioms: usize_to_int_int_to_usize, mk_vec_axiom, + int_to_usize_usize_to_int, usize_bounds, vec_to_list_num_bounds] [] + ⊢ ∀v x. + usize_to_int (vec_len v) < usize_max ⇒ + ∃nv. + vec_push_back v x = Return nv ∧ + vec_to_list nv = vec_to_list v ⧺ [x] + + [vec_push_back_unfold] Theorem + + [oracles: DISK_THM] [axioms: usize_bounds] [] + ⊢ ∀v x. + vec_push_back v x = + if + usize_to_int (vec_len v) < u16_max ∨ + usize_to_int (vec_len v) < usize_max + then + Return (mk_vec (vec_to_list v ⧺ [x])) + else Fail Failure + [vec_to_list_int_bounds] Theorem [oracles: DISK_THM] [axioms: usize_bounds, vec_to_list_num_bounds] [] ⊢ ∀v. 0 ≤ len (vec_to_list v) ∧ len (vec_to_list v) ≤ usize_max + [vec_to_list_vec_new] Theorem + + [oracles: DISK_THM] [axioms: mk_vec_axiom, usize_bounds] [] + ⊢ vec_to_list vec_new = [] + + [vec_update_eq] Theorem + + [oracles: DISK_THM] + [axioms: vec_to_list_num_bounds, usize_bounds, + int_to_usize_usize_to_int, usize_to_int_bounds, + usize_to_int_int_to_usize, mk_vec_axiom] [] + ⊢ ∀v i x. + usize_to_int i < usize_to_int (vec_len v) ⇒ + (let + nv = vec_update v i x + in + vec_len v = vec_len nv ∧ vec_index nv i = x ∧ + ∀j. usize_to_int j < usize_to_int (vec_len nv) ⇒ + usize_to_int j ≠ usize_to_int i ⇒ + vec_index nv j = vec_index v j) + *) end |