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-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
generated by cgit v1.2.3 (git 2.25.1) at 2025-12-23 06:36:37 +0000