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Diffstat (limited to 'backends/hol4/testDivDefTheory.sig')
-rw-r--r-- | backends/hol4/testDivDefTheory.sig | 380 |
1 files changed, 380 insertions, 0 deletions
diff --git a/backends/hol4/testDivDefTheory.sig b/backends/hol4/testDivDefTheory.sig new file mode 100644 index 00000000..a3ce2255 --- /dev/null +++ b/backends/hol4/testDivDefTheory.sig @@ -0,0 +1,380 @@ +signature testDivDefTheory = +sig + type thm = Thm.thm + + (* Definitions *) + val even___E_def : thm + val even___P_def : thm + val even___fuel0_def_UNION_extract0 : thm + val even___fuel0_def_UNION_extract1 : thm + val even___fuel0_def_UNION_primitive : thm + val even___fuel_def_UNION_extract0 : thm + val even___fuel_def_UNION_extract1 : thm + val even___fuel_def_UNION_primitive : thm + val even_def : thm + val list_t_TY_DEF : thm + val list_t_case_def : thm + val list_t_size_def : thm + val nth_mut_fwd___E_def : thm + val nth_mut_fwd___P_def : thm + val nth_mut_fwd_def : thm + val odd___E_def : thm + val odd___P_def : thm + val odd_def : thm + + (* Theorems *) + val datatype_list_t : thm + val even___fuel0_def : thm + val even___fuel0_ind : thm + val even___fuel_def : thm + val even___fuel_ind : 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_mut_fwd___fuel0_def : thm + val nth_mut_fwd___fuel0_ind : thm + val nth_mut_fwd___fuel_def : thm + val nth_mut_fwd___fuel_ind : thm + + val testDivDef_grammars : type_grammar.grammar * term_grammar.grammar +(* + [primitives] Parent theory of "testDivDef" + + [even___E_def] Definition + + ⊢ ∀even odd i. + even___E even odd i = + if i = 0 then do b <- Return T; Return b od + else do b <- odd (i − 1); Return b od + + [even___P_def] Definition + + ⊢ ∀i $var$($n). + even___P i $var$($n) ⇔ ¬is_diverge (even___fuel0 $var$($n) i) + + [even___fuel0_def_UNION_extract0] Definition + + ⊢ ∀x x0. even___fuel0 x x0 = even___fuel0_def_UNION (INL (x,x0)) + + [even___fuel0_def_UNION_extract1] Definition + + ⊢ ∀x x0. odd___fuel0 x x0 = even___fuel0_def_UNION (INR (x,x0)) + + [even___fuel0_def_UNION_primitive] Definition + + ⊢ even___fuel0_def_UNION = + WFREC + (@R. WF R ∧ + (∀i $var$($n) $var$($m). + $var$($n) = SUC $var$($m) ∧ i ≠ 0 ⇒ + R (INR ($var$($m),i − 1)) (INL ($var$($n),i))) ∧ + ∀i $var$($n) $var$($m). + $var$($n) = SUC $var$($m) ∧ i ≠ 0 ⇒ + R (INL ($var$($m),i − 1)) (INR ($var$($n),i))) + (λeven___fuel0_def_UNION a. + case a of + INL ($var$($n),i) => + I + (case $var$($n) of + 0 => Diverge + | SUC $var$($m) => + if i = 0 then do b <- Return T; Return b od + else + do + b <- + even___fuel0_def_UNION + (INR ($var$($m),i − 1)); + Return b + od) + | INR ($var$($n'),i') => + I + (case $var$($n') of + 0 => Diverge + | SUC $var$($m) => + if i' = 0 then do b <- Return F; Return b od + else + do + b <- + even___fuel0_def_UNION (INL ($var$($m),i' − 1)); + Return b + od)) + + [even___fuel_def_UNION_extract0] Definition + + ⊢ ∀x x0. even___fuel x x0 = even___fuel_def_UNION (INL (x,x0)) + + [even___fuel_def_UNION_extract1] Definition + + ⊢ ∀x x0. odd___fuel x x0 = even___fuel_def_UNION (INR (x,x0)) + + [even___fuel_def_UNION_primitive] Definition + + ⊢ even___fuel_def_UNION = + WFREC + (@R. WF R ∧ + (∀i $var$($n) $var$($m). + $var$($n) = SUC $var$($m) ∧ i ≠ 0 ⇒ + R (INR ($var$($m),i − 1)) (INL ($var$($n),i))) ∧ + ∀i $var$($n) $var$($m). + $var$($n) = SUC $var$($m) ∧ i ≠ 0 ⇒ + R (INL ($var$($m),i − 1)) (INR ($var$($n),i))) + (λeven___fuel_def_UNION a. + case a of + INL ($var$($n),i) => + I + (case $var$($n) of + 0 => Diverge + | SUC $var$($m) => + if i = 0 then Return T + else even___fuel_def_UNION (INR ($var$($m),i − 1))) + | INR ($var$($n'),i') => + I + (case $var$($n') of + 0 => Diverge + | SUC $var$($m) => + if i' = 0 then Return F + else even___fuel_def_UNION (INL ($var$($m),i' − 1)))) + + [even_def] Definition + + ⊢ ∀i. even i = + if ∃ $var$($n). even___P i $var$($n) then + even___fuel0 ($LEAST (even___P i)) i + else Diverge + + [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_mut_fwd___E_def] Definition + + ⊢ ∀nth_mut_fwd ls i. + nth_mut_fwd___E nth_mut_fwd 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); + x <- nth_mut_fwd tl i0; + Return x + od + | ListNil => Fail Failure + + [nth_mut_fwd___P_def] Definition + + ⊢ ∀ls i $var$($n). + nth_mut_fwd___P ls i $var$($n) ⇔ + ¬is_diverge (nth_mut_fwd___fuel0 $var$($n) ls i) + + [nth_mut_fwd_def] Definition + + ⊢ ∀ls i. + nth_mut_fwd ls i = + if ∃ $var$($n). nth_mut_fwd___P ls i $var$($n) then + nth_mut_fwd___fuel0 ($LEAST (nth_mut_fwd___P ls i)) ls i + else Diverge + + [odd___E_def] Definition + + ⊢ ∀even odd i. + odd___E even odd i = + if i = 0 then do b <- Return F; Return b od + else do b <- even (i − 1); Return b od + + [odd___P_def] Definition + + ⊢ ∀i $var$($n). + odd___P i $var$($n) ⇔ ¬is_diverge (odd___fuel0 $var$($n) i) + + [odd_def] Definition + + ⊢ ∀i. odd i = + if ∃ $var$($n). odd___P i $var$($n) then + odd___fuel0 ($LEAST (odd___P i)) i + else Diverge + + [datatype_list_t] Theorem + + ⊢ DATATYPE (list_t ListCons ListNil) + + [even___fuel0_def] Theorem + + ⊢ (∀i $var$($n). + even___fuel0 $var$($n) i = + case $var$($n) of + 0 => Diverge + | SUC $var$($m) => + if i = 0 then do b <- Return T; Return b od + else do b <- odd___fuel0 $var$($m) (i − 1); Return b od) ∧ + ∀i $var$($n). + odd___fuel0 $var$($n) i = + case $var$($n) of + 0 => Diverge + | SUC $var$($m) => + if i = 0 then do b <- Return F; Return b od + else do b <- even___fuel0 $var$($m) (i − 1); Return b od + + [even___fuel0_ind] Theorem + + ⊢ ∀P0 P1. + (∀ $var$($n) i. + (∀ $var$($m). + $var$($n) = SUC $var$($m) ∧ i ≠ 0 ⇒ P1 $var$($m) (i − 1)) ⇒ + P0 $var$($n) i) ∧ + (∀ $var$($n) i. + (∀ $var$($m). + $var$($n) = SUC $var$($m) ∧ i ≠ 0 ⇒ P0 $var$($m) (i − 1)) ⇒ + P1 $var$($n) i) ⇒ + (∀v0 v1. P0 v0 v1) ∧ ∀v0 v1. P1 v0 v1 + + [even___fuel_def] Theorem + + ⊢ (∀i $var$($n). + even___fuel $var$($n) i = + case $var$($n) of + 0 => Diverge + | SUC $var$($m) => + if i = 0 then Return T else odd___fuel $var$($m) (i − 1)) ∧ + ∀i $var$($n). + odd___fuel $var$($n) i = + case $var$($n) of + 0 => Diverge + | SUC $var$($m) => + if i = 0 then Return F else even___fuel $var$($m) (i − 1) + + [even___fuel_ind] Theorem + + ⊢ ∀P0 P1. + (∀ $var$($n) i. + (∀ $var$($m). + $var$($n) = SUC $var$($m) ∧ i ≠ 0 ⇒ P1 $var$($m) (i − 1)) ⇒ + P0 $var$($n) i) ∧ + (∀ $var$($n) i. + (∀ $var$($m). + $var$($n) = SUC $var$($m) ∧ i ≠ 0 ⇒ P0 $var$($m) (i − 1)) ⇒ + P1 $var$($n) i) ⇒ + (∀v0 v1. P0 v0 v1) ∧ ∀v0 v1. P1 v0 v1 + + [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_mut_fwd___fuel0_def] Theorem + + ⊢ ∀ls i $var$($n). + nth_mut_fwd___fuel0 $var$($n) ls i = + case $var$($n) of + 0 => Diverge + | SUC $var$($m) => + 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); + x <- nth_mut_fwd___fuel0 $var$($m) tl i0; + Return x + od + | ListNil => Fail Failure + + [nth_mut_fwd___fuel0_ind] Theorem + + ⊢ ∀P. (∀ $var$($n) ls i. + (∀ $var$($m) x tl i0. + $var$($n) = SUC $var$($m) ∧ ls = ListCons x tl ∧ + u32_to_int i ≠ 0 ⇒ + P $var$($m) tl i0) ⇒ + P $var$($n) ls i) ⇒ + ∀v v1 v2. P v v1 v2 + + [nth_mut_fwd___fuel_def] Theorem + + ⊢ ∀ls i $var$($n). + nth_mut_fwd___fuel $var$($n) ls i = + case $var$($n) of + 0 => Diverge + | SUC $var$($m) => + 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_mut_fwd___fuel $var$($m) tl i0 + od + | ListNil => Fail Failure + + [nth_mut_fwd___fuel_ind] Theorem + + ⊢ ∀P. (∀ $var$($n) ls i. + (∀ $var$($m) x tl i0. + $var$($n) = SUC $var$($m) ∧ ls = ListCons x tl ∧ + u32_to_int i ≠ 0 ⇒ + P $var$($m) tl i0) ⇒ + P $var$($n) ls i) ⇒ + ∀v v1 v2. P v v1 v2 + + +*) +end |