theory_file
stringlengths 5
95
| lemma_name
stringlengths 5
250
| lemma_command
stringlengths 17
21.3k
⌀ | lemma_object
stringlengths 5
62.8k
| template
stringlengths 7
63.6k
| symbols
listlengths 0
89
| defs
listlengths 0
63
|
|---|---|---|---|---|---|---|
Combinatorial_Enumeration_Algorithms/n_Subsets
|
n_Subsets.n_subset_enum_distinct
|
theorem n_subset_enum_distinct: "distinct xs \<Longrightarrow> distinct (n_subset_enum xs n)"
|
distinct ?xs \<Longrightarrow> distinct (n_subset_enum ?xs ?n)
|
?H1 x_1 \<Longrightarrow> ?H2 (?H3 x_1 x_2)
|
[
"Filter_Bool_List.filter_bool_list",
"n_Subsets.n_subset_enum_dom",
"List.distinct",
"n_Subsets.n_subset_enum",
"List.append"
] |
[
"fun filter_bool_list :: \"bool list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n \"filter_bool_list [] _ = []\"\n| \"filter_bool_list _ [] = []\"\n| \"filter_bool_list (b#bs) (x#xs) =\n (if b then x#(filter_bool_list bs xs) else (filter_bool_list bs xs))\"",
"primrec distinct :: \"'a list \\<Rightarrow> bool\" where\n\"distinct [] \\<longleftrightarrow> True\" |\n\"distinct (x # xs) \\<longleftrightarrow> x \\<notin> set xs \\<and> distinct xs\"",
"fun n_subset_enum :: \"'a list \\<Rightarrow> nat \\<Rightarrow> 'a list list\" where\n \"n_subset_enum xs n = [(filter_bool_list bs xs) . bs \\<leftarrow> (n_bool_lists n (length xs))]\"",
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\""
] |
Combinatorial_Enumeration_Algorithms/n_Subsets
|
n_Subsets.n_bool_lists2_correct
|
lemma n_bool_lists2_correct: "set (n_bool_lists n x) = n_bool_lists2 n x"
|
set (n_bool_lists ?n ?x) = n_bool_lists2 ?n ?x
|
?H1 (?H2 x_1 x_2) = ?H3 x_1 x_2
|
[
"n_Subsets.n_bool_lists2",
"n_Subsets.n_bool_lists",
"List.list.set",
"List.append",
"Nat.Suc",
"List.distinct"
] |
[
"fun n_bool_lists2 :: \"nat \\<Rightarrow> nat \\<Rightarrow> bool list set\" where\n \"n_bool_lists2 n x = (if n > x then {}\n else permutations_of_multiset (mset (replicate n True @ replicate (x-n) False)))\"",
"fun n_bool_lists :: \"nat \\<Rightarrow> nat \\<Rightarrow> bool list list\" where\n \"n_bool_lists n 0 = (if n > 0 then [] else [[]])\"\n| \"n_bool_lists n (Suc x) = (if n = 0 then [replicate (Suc x) False]\n else if n = Suc x then [replicate (Suc x) True]\n else if n > x then []\n else [False#xs . xs \\<leftarrow> n_bool_lists n x] @ [True#xs . xs \\<leftarrow> n_bool_lists (n-1) x])\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\"",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"primrec distinct :: \"'a list \\<Rightarrow> bool\" where\n\"distinct [] \\<longleftrightarrow> True\" |\n\"distinct (x # xs) \\<longleftrightarrow> x \\<notin> set xs \\<and> distinct xs\""
] |
Combinatorial_Enumeration_Algorithms/n_Subsets
|
n_Subsets.n_bool_lists_distinct
|
lemma n_bool_lists_distinct: "distinct (n_bool_lists n x)"
|
distinct (n_bool_lists ?n ?x)
|
?H1 (?H2 x_1 x_2)
|
[
"Filter_Bool_List.filter_bool_list",
"List.distinct",
"n_Subsets.n_bool_lists"
] |
[
"fun filter_bool_list :: \"bool list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n \"filter_bool_list [] _ = []\"\n| \"filter_bool_list _ [] = []\"\n| \"filter_bool_list (b#bs) (x#xs) =\n (if b then x#(filter_bool_list bs xs) else (filter_bool_list bs xs))\"",
"primrec distinct :: \"'a list \\<Rightarrow> bool\" where\n\"distinct [] \\<longleftrightarrow> True\" |\n\"distinct (x # xs) \\<longleftrightarrow> x \\<notin> set xs \\<and> distinct xs\"",
"fun n_bool_lists :: \"nat \\<Rightarrow> nat \\<Rightarrow> bool list list\" where\n \"n_bool_lists n 0 = (if n > 0 then [] else [[]])\"\n| \"n_bool_lists n (Suc x) = (if n = 0 then [replicate (Suc x) False]\n else if n = Suc x then [replicate (Suc x) True]\n else if n > x then []\n else [False#xs . xs \\<leftarrow> n_bool_lists n x] @ [True#xs . xs \\<leftarrow> n_bool_lists (n-1) x])\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Partitions
|
Integer_Partitions.integer_partitions_enum_aux_head_minus
|
lemma integer_partitions_enum_aux_head_minus:
"h \<le> m \<Longrightarrow> h > 0 \<Longrightarrow> n \<ge> h \<Longrightarrow>
ys \<in> set (integer_partitions_enum_aux (n-h) h)\<Longrightarrow> h#ys \<in> set (integer_partitions_enum_aux n m)"
|
?h \<le> ?m \<Longrightarrow> 0 < ?h \<Longrightarrow> ?h \<le> ?n \<Longrightarrow> ?ys \<in> set (integer_partitions_enum_aux (?n - ?h) ?h) \<Longrightarrow> ?h # ?ys \<in> set (integer_partitions_enum_aux ?n ?m)
|
\<lbrakk>x_1 \<le> x_2; ?H1 < x_1; x_1 \<le> x_3; x_4 \<in> ?H2 (?H3 (?H4 x_3 x_1) x_1)\<rbrakk> \<Longrightarrow> ?H5 x_1 x_4 \<in> ?H2 (?H3 x_3 x_2)
|
[
"Orderings.ord_class.min",
"Groups.zero_class.zero",
"List.list.Cons",
"Groups.minus_class.minus",
"List.list.set",
"Number_Partition.partitions",
"List.concat",
"Integer_Partitions.integer_partitions_enum_aux",
"Set_Interval.ord_class.atMost"
] |
[
"class ord =\n fixes less_eq :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\"\n and less :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\"\nbegin",
"class zero =\n fixes zero :: 'a (\"0\")",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"definition partitions :: \"(nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> bool\" (infix \"partitions\" 50)\nwhere\n \"p partitions n = ((\\<forall>i. p i \\<noteq> 0 \\<longrightarrow> 1 \\<le> i \\<and> i \\<le> n) \\<and> (\\<Sum>i\\<le>n. p i * i) = n)\"",
"primrec concat:: \"'a list list \\<Rightarrow> 'a list\" where\n\"concat [] = []\" |\n\"concat (x # xs) = x @ concat xs\"",
"fun integer_partitions_enum_aux :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat list list\" where\n \"integer_partitions_enum_aux 0 m = [[]]\"\n| \"integer_partitions_enum_aux n m =\n [h#r . h \\<leftarrow> [1..< Suc (min n m)], r \\<leftarrow> integer_partitions_enum_aux (n-h) h]\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Partitions
|
Integer_Partitions.integer_partitions_enum_aux_head_plus
|
lemma integer_partitions_enum_aux_head_plus:
"h \<le> m \<Longrightarrow> h > 0 \<Longrightarrow> ys \<in> set (integer_partitions_enum_aux n h)
\<Longrightarrow> h#ys \<in> set (integer_partitions_enum_aux (h + n) m)"
|
?h \<le> ?m \<Longrightarrow> 0 < ?h \<Longrightarrow> ?ys \<in> set (integer_partitions_enum_aux ?n ?h) \<Longrightarrow> ?h # ?ys \<in> set (integer_partitions_enum_aux (?h + ?n) ?m)
|
\<lbrakk>x_1 \<le> x_2; ?H1 < x_1; x_3 \<in> ?H2 (?H3 x_4 x_1)\<rbrakk> \<Longrightarrow> ?H4 x_1 x_3 \<in> ?H2 (?H3 (?H5 x_1 x_4) x_2)
|
[
"Integer_Partitions.integer_partitions_enum",
"List.list.set",
"List.list.Cons",
"Groups.plus_class.plus",
"Groups.zero_class.zero",
"Integer_Partitions.integer_partitions_enum_aux"
] |
[
"fun integer_partitions_enum :: \"nat \\<Rightarrow> nat list list\" where\n \"integer_partitions_enum n = integer_partitions_enum_aux n n\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"class zero =\n fixes zero :: 'a (\"0\")",
"fun integer_partitions_enum_aux :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat list list\" where\n \"integer_partitions_enum_aux 0 m = [[]]\"\n| \"integer_partitions_enum_aux n m =\n [h#r . h \\<leftarrow> [1..< Suc (min n m)], r \\<leftarrow> integer_partitions_enum_aux (n-h) h]\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Partitions
|
Integer_Partitions.integer_partitions_enum_correct_aux1
|
lemma integer_partitions_enum_correct_aux1:
assumes "0 \<notin># A "
and "\<forall>x \<in># A. x \<le> m"
shows" \<exists>xs\<in>set (integer_partitions_enum_aux (\<Sum>\<^sub># A) m). A = mset xs"
|
0 \<notin># ?A \<Longrightarrow> \<forall>x\<in>#?A. x \<le> ?m \<Longrightarrow> \<exists>xs\<in>set (integer_partitions_enum_aux (\<Sum>\<^sub># ?A) ?m). ?A = mset xs
|
\<lbrakk>?H1 ?H2 x_1; ?H3 x_1 (\<lambda>y_0. y_0 \<le> x_2)\<rbrakk> \<Longrightarrow> \<exists>y_1\<in>?H4 (?H5 (?H6 x_1) x_2). x_1 = ?H7 y_1
|
[
"Multiset.mset",
"Groups.zero_class.zero",
"List.distinct",
"List.list.set",
"Multiset.not_member_mset",
"Multiset.Ball",
"Integer_Partitions.integer_partitions_enum_aux",
"Multiset.comm_monoid_add_class.sum_mset"
] |
[
"primrec mset :: \"'a list \\<Rightarrow> 'a multiset\" where\n \"mset [] = {#}\" |\n \"mset (a # x) = add_mset a (mset x)\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"primrec distinct :: \"'a list \\<Rightarrow> bool\" where\n\"distinct [] \\<longleftrightarrow> True\" |\n\"distinct (x # xs) \\<longleftrightarrow> x \\<notin> set xs \\<and> distinct xs\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation not_member_mset :: \\<open>'a \\<Rightarrow> 'a multiset \\<Rightarrow> bool\\<close>\n where \\<open>not_member_mset a M \\<equiv> a \\<notin> set_mset M\\<close>",
"abbreviation Ball :: \"'a multiset \\<Rightarrow> ('a \\<Rightarrow> bool) \\<Rightarrow> bool\"\n where \"Ball M \\<equiv> Set.Ball (set_mset M)\"",
"fun integer_partitions_enum_aux :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat list list\" where\n \"integer_partitions_enum_aux 0 m = [[]]\"\n| \"integer_partitions_enum_aux n m =\n [h#r . h \\<leftarrow> [1..< Suc (min n m)], r \\<leftarrow> integer_partitions_enum_aux (n-h) h]\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Partitions
|
Integer_Partitions.integer_partitions_enum_aux_not_null
|
lemma integer_partitions_enum_aux_not_null:
"xs \<in> set (integer_partitions_enum_aux n m) \<Longrightarrow> x \<in> set xs \<Longrightarrow> x \<noteq> 0"
|
?xs \<in> set (integer_partitions_enum_aux ?n ?m) \<Longrightarrow> ?x \<in> set ?xs \<Longrightarrow> ?x \<noteq> 0
|
\<lbrakk>x_1 \<in> ?H1 (?H2 x_2 x_3); x_4 \<in> ?H3 x_1\<rbrakk> \<Longrightarrow> x_4 \<noteq> ?H4
|
[
"Num.num.Bit0",
"List.list.set",
"Integer_Partitions.integer_partitions_enum_aux",
"Groups.zero_class.zero",
"List.distinct"
] |
[
"datatype num = One | Bit0 num | Bit1 num",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"fun integer_partitions_enum_aux :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat list list\" where\n \"integer_partitions_enum_aux 0 m = [[]]\"\n| \"integer_partitions_enum_aux n m =\n [h#r . h \\<leftarrow> [1..< Suc (min n m)], r \\<leftarrow> integer_partitions_enum_aux (n-h) h]\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"primrec distinct :: \"'a list \\<Rightarrow> bool\" where\n\"distinct [] \\<longleftrightarrow> True\" |\n\"distinct (x # xs) \\<longleftrightarrow> x \\<notin> set xs \\<and> distinct xs\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Partitions
|
Integer_Partitions.integer_partitions_enum_aux_first
|
lemma integer_partitions_enum_aux_first:
"x # xs \<in> set (integer_partitions_enum_aux n m)
\<Longrightarrow> xs \<in> set (integer_partitions_enum_aux (n-x) x)"
|
?x # ?xs \<in> set (integer_partitions_enum_aux ?n ?m) \<Longrightarrow> ?xs \<in> set (integer_partitions_enum_aux (?n - ?x) ?x)
|
?H1 x_1 x_2 \<in> ?H2 (?H3 x_3 x_4) \<Longrightarrow> x_2 \<in> ?H2 (?H3 (?H4 x_3 x_1) x_1)
|
[
"List.list.set",
"List.list.Cons",
"Groups_List.monoid_add_class.sum_list",
"Integer_Partitions.integer_partitions",
"Integer_Partitions.integer_partitions_enum_aux",
"Groups.minus_class.minus",
"Multiset.comm_monoid_add_class.sum_mset"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"definition integer_partitions :: \"nat \\<Rightarrow> nat multiset set\" where\n \"integer_partitions i = {A. sum_mset A = i \\<and> 0 \\<notin># A}\"",
"fun integer_partitions_enum_aux :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat list list\" where\n \"integer_partitions_enum_aux 0 m = [[]]\"\n| \"integer_partitions_enum_aux n m =\n [h#r . h \\<leftarrow> [1..< Suc (min n m)], r \\<leftarrow> integer_partitions_enum_aux (n-h) h]\"",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)"
] |
Combinatorial_Enumeration_Algorithms/Integer_Partitions
|
Integer_Partitions.integer_partitions_enum_aux_max
|
lemma integer_partitions_enum_aux_max:
"xs \<in> set (integer_partitions_enum_aux n m) \<Longrightarrow> x \<in> set xs \<Longrightarrow> x \<le> m"
|
?xs \<in> set (integer_partitions_enum_aux ?n ?m) \<Longrightarrow> ?x \<in> set ?xs \<Longrightarrow> ?x \<le> ?m
|
\<lbrakk>x_1 \<in> ?H1 (?H2 x_2 x_3); x_4 \<in> ?H3 x_1\<rbrakk> \<Longrightarrow> x_4 \<le> x_3
|
[
"Fun.bij_betw",
"List.list.set",
"Integer_Partitions.integer_partitions_enum_aux",
"Number_Partition.number_partition"
] |
[
"definition bij_betw :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b set \\<Rightarrow> bool\" \\<comment> \\<open>bijective\\<close>\n where \"bij_betw f A B \\<longleftrightarrow> inj_on f A \\<and> f ` A = B\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"fun integer_partitions_enum_aux :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat list list\" where\n \"integer_partitions_enum_aux 0 m = [[]]\"\n| \"integer_partitions_enum_aux n m =\n [h#r . h \\<leftarrow> [1..< Suc (min n m)], r \\<leftarrow> integer_partitions_enum_aux (n-h) h]\"",
"definition number_partition :: \"nat \\<Rightarrow> nat multiset \\<Rightarrow> bool\"\nwhere\n \"number_partition n N = (sum_mset N = n \\<and> 0 \\<notin># N)\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Partitions
|
Integer_Partitions.integer_partitions_enum_aux_not_null_aux
|
lemma integer_partitions_enum_aux_not_null_aux:
"x#xs \<in> set (integer_partitions_enum_aux n m) \<Longrightarrow> x \<noteq> 0"
|
?x # ?xs \<in> set (integer_partitions_enum_aux ?n ?m) \<Longrightarrow> ?x \<noteq> 0
|
?H1 x_1 x_2 \<in> ?H2 (?H3 x_3 x_4) \<Longrightarrow> x_1 \<noteq> ?H4
|
[
"List.list.set",
"List.list.Cons",
"Integer_Partitions.integer_partitions_enum",
"Integer_Partitions.integer_partitions_enum_aux",
"Groups.zero_class.zero"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"fun integer_partitions_enum :: \"nat \\<Rightarrow> nat list list\" where\n \"integer_partitions_enum n = integer_partitions_enum_aux n n\"",
"fun integer_partitions_enum_aux :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat list list\" where\n \"integer_partitions_enum_aux 0 m = [[]]\"\n| \"integer_partitions_enum_aux n m =\n [h#r . h \\<leftarrow> [1..< Suc (min n m)], r \\<leftarrow> integer_partitions_enum_aux (n-h) h]\"",
"class zero =\n fixes zero :: 'a (\"0\")"
] |
Combinatorial_Enumeration_Algorithms/Integer_Partitions
|
Integer_Partitions.card_partitions_count_partitions
|
lemma card_partitions_count_partitions:
"card {p. p partitions n} = card {N. count N partitions n}"
|
card {p. p partitions ?n} = card {N. count N partitions ?n}
|
?H1 (?H2 (\<lambda>y_0. ?H3 y_0 x_1)) = ?H4 (?H5 (\<lambda>y_1. ?H3 (?H6 y_1) x_1))
|
[
"Number_Partition.partitions",
"Set.Collect",
"List.list.Cons",
"Finite_Set.card",
"Multiset.multiset.count",
"Num.num.Bit0",
"Integer_Partitions.integer_partitions_enum_aux"
] |
[
"definition partitions :: \"(nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> bool\" (infix \"partitions\" 50)\nwhere\n \"p partitions n = ((\\<forall>i. p i \\<noteq> 0 \\<longrightarrow> 1 \\<le> i \\<and> i \\<le> n) \\<and> (\\<Sum>i\\<le>n. p i * i) = n)\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype num = One | Bit0 num | Bit1 num",
"fun integer_partitions_enum_aux :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat list list\" where\n \"integer_partitions_enum_aux 0 m = [[]]\"\n| \"integer_partitions_enum_aux n m =\n [h#r . h \\<leftarrow> [1..< Suc (min n m)], r \\<leftarrow> integer_partitions_enum_aux (n-h) h]\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Partitions
|
Integer_Partitions.integer_partitions_enum_aux_max_n
|
lemma integer_partitions_enum_aux_max_n:
"x#xs \<in> set (integer_partitions_enum_aux n m) \<Longrightarrow> x \<le> n"
|
?x # ?xs \<in> set (integer_partitions_enum_aux ?n ?m) \<Longrightarrow> ?x \<le> ?n
|
?H1 x_1 x_2 \<in> ?H2 (?H3 x_3 x_4) \<Longrightarrow> x_1 \<le> x_3
|
[
"Integer_Partitions.integer_partitions_enum_aux",
"List.list.set",
"Set.Collect",
"Groups.times_class.times",
"Multiset.comm_monoid_add_class.sum_mset",
"List.distinct",
"List.list.Cons"
] |
[
"fun integer_partitions_enum_aux :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat list list\" where\n \"integer_partitions_enum_aux 0 m = [[]]\"\n| \"integer_partitions_enum_aux n m =\n [h#r . h \\<leftarrow> [1..< Suc (min n m)], r \\<leftarrow> integer_partitions_enum_aux (n-h) h]\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"*\" 70)",
"primrec distinct :: \"'a list \\<Rightarrow> bool\" where\n\"distinct [] \\<longleftrightarrow> True\" |\n\"distinct (x # xs) \\<longleftrightarrow> x \\<notin> set xs \\<and> distinct xs\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Partitions
|
Integer_Partitions.integer_partitions_enum_aux_max_head
|
lemma integer_partitions_enum_aux_max_head:
"x#xs \<in> set (integer_partitions_enum_aux n m) \<Longrightarrow> x \<le> m"
|
?x # ?xs \<in> set (integer_partitions_enum_aux ?n ?m) \<Longrightarrow> ?x \<le> ?m
|
?H1 x_1 x_2 \<in> ?H2 (?H3 x_3 x_4) \<Longrightarrow> x_1 \<le> x_4
|
[
"Nat.Suc",
"Multiset.not_member_mset",
"Fun.bij_betw",
"List.list.Cons",
"List.list.set",
"Num.numeral_class.numeral",
"Integer_Partitions.integer_partitions_enum_aux"
] |
[
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"abbreviation not_member_mset :: \\<open>'a \\<Rightarrow> 'a multiset \\<Rightarrow> bool\\<close>\n where \\<open>not_member_mset a M \\<equiv> a \\<notin> set_mset M\\<close>",
"definition bij_betw :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b set \\<Rightarrow> bool\" \\<comment> \\<open>bijective\\<close>\n where \"bij_betw f A B \\<longleftrightarrow> inj_on f A \\<and> f ` A = B\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"fun integer_partitions_enum_aux :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat list list\" where\n \"integer_partitions_enum_aux 0 m = [[]]\"\n| \"integer_partitions_enum_aux n m =\n [h#r . h \\<leftarrow> [1..< Suc (min n m)], r \\<leftarrow> integer_partitions_enum_aux (n-h) h]\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Partitions
|
Integer_Partitions.card_partitions_number_partition
|
lemma card_partitions_number_partition:
"card {p. p partitions n} = card {N. number_partition n N}"
|
card {p. p partitions ?n} = card {N. number_partition ?n N}
|
?H1 (?H2 (\<lambda>y_0. ?H3 y_0 x_1)) = ?H4 (?H5 (?H6 x_1))
|
[
"Finite_Set.card",
"Number_Partition.partitions",
"Num.numeral_class.numeral",
"Set.Collect",
"Number_Partition.number_partition"
] |
[
"definition partitions :: \"(nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> bool\" (infix \"partitions\" 50)\nwhere\n \"p partitions n = ((\\<forall>i. p i \\<noteq> 0 \\<longrightarrow> 1 \\<le> i \\<and> i \\<le> n) \\<and> (\\<Sum>i\\<le>n. p i * i) = n)\"",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"definition number_partition :: \"nat \\<Rightarrow> nat multiset \\<Rightarrow> bool\"\nwhere\n \"number_partition n N = (sum_mset N = n \\<and> 0 \\<notin># N)\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Partitions
|
Integer_Partitions.partitions_bij_betw_count
|
lemma partitions_bij_betw_count:
"bij_betw count {N. count N partitions n} {p. p partitions n}"
|
bij_betw count {N. count N partitions ?n} {p. p partitions ?n}
|
?H1 ?H2 (?H3 (\<lambda>y_0. ?H4 (?H2 y_0) x_1)) (?H5 (\<lambda>y_1. ?H4 y_1 x_1))
|
[
"Number_Partition.partitions",
"List.list.map",
"Multiset.multiset.count",
"Set.Collect",
"List.list.Cons",
"Multiset.comm_monoid_add_class.sum_mset",
"Fun.bij_betw"
] |
[
"definition partitions :: \"(nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> bool\" (infix \"partitions\" 50)\nwhere\n \"p partitions n = ((\\<forall>i. p i \\<noteq> 0 \\<longrightarrow> 1 \\<le> i \\<and> i \\<le> n) \\<and> (\\<Sum>i\\<le>n. p i * i) = n)\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"definition bij_betw :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b set \\<Rightarrow> bool\" \\<comment> \\<open>bijective\\<close>\n where \"bij_betw f A B \\<longleftrightarrow> inj_on f A \\<and> f ` A = B\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Partitions
|
Integer_Partitions.integer_partitions_enum_aux_sum
|
lemma integer_partitions_enum_aux_sum:
"xs \<in> set (integer_partitions_enum_aux n m) \<Longrightarrow> sum_list xs = n"
|
?xs \<in> set (integer_partitions_enum_aux ?n ?m) \<Longrightarrow> sum_list ?xs = ?n
|
x_1 \<in> ?H1 (?H2 x_2 x_3) \<Longrightarrow> ?H3 x_1 = x_2
|
[
"Groups.zero_class.zero",
"Num.numeral_class.numeral",
"Groups_List.monoid_add_class.sum_list",
"List.upt",
"Integer_Partitions.integer_partitions",
"Integer_Partitions.integer_partitions_enum_aux",
"List.list.set"
] |
[
"class zero =\n fixes zero :: 'a (\"0\")",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"primrec upt :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat list\" (\"(1[_..</_'])\") where\nupt_0: \"[i..<0] = []\" |\nupt_Suc: \"[i..<(Suc j)] = (if i \\<le> j then [i..<j] @ [j] else [])\"",
"definition integer_partitions :: \"nat \\<Rightarrow> nat multiset set\" where\n \"integer_partitions i = {A. sum_mset A = i \\<and> 0 \\<notin># A}\"",
"fun integer_partitions_enum_aux :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat list list\" where\n \"integer_partitions_enum_aux 0 m = [[]]\"\n| \"integer_partitions_enum_aux n m =\n [h#r . h \\<leftarrow> [1..< Suc (min n m)], r \\<leftarrow> integer_partitions_enum_aux (n-h) h]\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Partitions
|
Integer_Partitions.integer_partitions_empty
|
lemma integer_partitions_empty: "[] \<in> set (integer_partitions_enum_aux n m) \<Longrightarrow> n = 0"
|
[] \<in> set (integer_partitions_enum_aux ?n ?m) \<Longrightarrow> ?n = 0
|
?H1 \<in> ?H2 (?H3 x_1 x_2) \<Longrightarrow> x_1 = ?H4
|
[
"Integer_Partitions.integer_partitions_enum_aux",
"List.list.Nil",
"Set_Interval.ord_class.atMost",
"List.list.set",
"Multiset.mset",
"Groups.zero_class.zero"
] |
[
"fun integer_partitions_enum_aux :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat list list\" where\n \"integer_partitions_enum_aux 0 m = [[]]\"\n| \"integer_partitions_enum_aux n m =\n [h#r . h \\<leftarrow> [1..< Suc (min n m)], r \\<leftarrow> integer_partitions_enum_aux (n-h) h]\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec mset :: \"'a list \\<Rightarrow> 'a multiset\" where\n \"mset [] = {#}\" |\n \"mset (a # x) = add_mset a (mset x)\"",
"class zero =\n fixes zero :: 'a (\"0\")"
] |
Combinatorial_Enumeration_Algorithms/Integer_Partitions
|
Integer_Partitions.integer_partitions_cardinality_aux
|
lemma integer_partitions_cardinality_aux:
"card (integer_partitions n) = (\<Sum>k\<le>n. Partition n k)"
|
card (integer_partitions ?n) = sum (Partition ?n) {..?n}
|
?H1 (?H2 x_1) = ?H3 (?H4 x_1) (?H5 x_1)
|
[
"Groups_Big.comm_monoid_add_class.sum",
"Card_Number_Partitions.Partition",
"Integer_Partitions.integer_partitions_enum",
"List.list.Cons",
"Set_Interval.ord_class.atMost",
"Integer_Partitions.integer_partitions",
"Finite_Set.card"
] |
[
"fun Partition :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat\"\nwhere\n \"Partition 0 0 = 1\"\n| \"Partition 0 (Suc k) = 0\"\n| \"Partition (Suc m) 0 = 0\"\n| \"Partition (Suc m) (Suc k) = Partition m k + Partition (m - k) (Suc k)\"",
"fun integer_partitions_enum :: \"nat \\<Rightarrow> nat list list\" where\n \"integer_partitions_enum n = integer_partitions_enum_aux n n\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"definition integer_partitions :: \"nat \\<Rightarrow> nat multiset set\" where\n \"integer_partitions i = {A. sum_mset A = i \\<and> 0 \\<notin># A}\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Partitions
|
Integer_Partitions.integer_partitions_cardinality
|
theorem integer_partitions_cardinality:
"card (integer_partitions n) = Partition (2*n) n"
|
card (integer_partitions ?n) = Partition (2 * ?n) ?n
|
?H1 (?H2 x_1) = ?H3 (?H4 (?H5 (?H6 ?H7)) x_1) x_1
|
[
"Finite_Set.card",
"Num.numeral_class.numeral",
"Integer_Partitions.integer_partitions",
"List.distinct",
"Num.num.Bit0",
"Num.num.One",
"Card_Number_Partitions.Partition",
"Groups.times_class.times"
] |
[
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"definition integer_partitions :: \"nat \\<Rightarrow> nat multiset set\" where\n \"integer_partitions i = {A. sum_mset A = i \\<and> 0 \\<notin># A}\"",
"primrec distinct :: \"'a list \\<Rightarrow> bool\" where\n\"distinct [] \\<longleftrightarrow> True\" |\n\"distinct (x # xs) \\<longleftrightarrow> x \\<notin> set xs \\<and> distinct xs\"",
"datatype num = One | Bit0 num | Bit1 num",
"datatype num = One | Bit0 num | Bit1 num",
"fun Partition :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat\"\nwhere\n \"Partition 0 0 = 1\"\n| \"Partition 0 (Suc k) = 0\"\n| \"Partition (Suc m) 0 = 0\"\n| \"Partition (Suc m) (Suc k) = Partition m k + Partition (m - k) (Suc k)\"",
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"*\" 70)"
] |
Combinatorial_Enumeration_Algorithms/Integer_Partitions
|
Integer_Partitions.integer_partitions_enum_correct
|
theorem integer_partitions_enum_correct:
"set (map mset (integer_partitions_enum n)) = integer_partitions n"
|
set (map mset (integer_partitions_enum ?n)) = integer_partitions ?n
|
?H1 (?H2 ?H3 (?H4 x_1)) = ?H5 x_1
|
[
"Num.numeral_class.numeral",
"Multiset.mset",
"Groups.times_class.times",
"Integer_Partitions.integer_partitions",
"List.list.map",
"Orderings.ord_class.min",
"Integer_Partitions.integer_partitions_enum",
"List.list.set"
] |
[
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"primrec mset :: \"'a list \\<Rightarrow> 'a multiset\" where\n \"mset [] = {#}\" |\n \"mset (a # x) = add_mset a (mset x)\"",
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"*\" 70)",
"definition integer_partitions :: \"nat \\<Rightarrow> nat multiset set\" where\n \"integer_partitions i = {A. sum_mset A = i \\<and> 0 \\<notin># A}\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"class ord =\n fixes less_eq :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\"\n and less :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\"\nbegin",
"fun integer_partitions_enum :: \"nat \\<Rightarrow> nat list list\" where\n \"integer_partitions_enum n = integer_partitions_enum_aux n n\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Partitions
|
Integer_Partitions.integer_partitions_number_partition_eq
|
lemma integer_partitions_number_partition_eq:
"integer_partitions n = {N. number_partition n N}"
|
integer_partitions ?n = {N. number_partition ?n N}
|
?H1 x_1 = ?H2 (?H3 x_1)
|
[
"List.list.set",
"Set.Collect",
"Groups.times_class.times",
"Set_Interval.ord_class.atMost",
"List.list.Cons",
"Number_Partition.number_partition",
"Integer_Partitions.integer_partitions"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"*\" 70)",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"definition number_partition :: \"nat \\<Rightarrow> nat multiset \\<Rightarrow> bool\"\nwhere\n \"number_partition n N = (sum_mset N = n \\<and> 0 \\<notin># N)\"",
"definition integer_partitions :: \"nat \\<Rightarrow> nat multiset set\" where\n \"integer_partitions i = {A. sum_mset A = i \\<and> 0 \\<notin># A}\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Partitions
|
Integer_Partitions.integer_partitions_enum_aux_distinct
|
lemma integer_partitions_enum_aux_distinct:
"distinct (integer_partitions_enum_aux n m)"
|
distinct (integer_partitions_enum_aux ?n ?m)
|
?H1 (?H2 x_1 x_2)
|
[
"Groups.plus_class.plus",
"Multiset.Ball",
"List.distinct",
"Integer_Partitions.integer_partitions_enum_aux"
] |
[
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"abbreviation Ball :: \"'a multiset \\<Rightarrow> ('a \\<Rightarrow> bool) \\<Rightarrow> bool\"\n where \"Ball M \\<equiv> Set.Ball (set_mset M)\"",
"primrec distinct :: \"'a list \\<Rightarrow> bool\" where\n\"distinct [] \\<longleftrightarrow> True\" |\n\"distinct (x # xs) \\<longleftrightarrow> x \\<notin> set xs \\<and> distinct xs\"",
"fun integer_partitions_enum_aux :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat list list\" where\n \"integer_partitions_enum_aux 0 m = [[]]\"\n| \"integer_partitions_enum_aux n m =\n [h#r . h \\<leftarrow> [1..< Suc (min n m)], r \\<leftarrow> integer_partitions_enum_aux (n-h) h]\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Partitions
|
Integer_Partitions.integer_partitions_enum_distinct
|
theorem integer_partitions_enum_distinct:
"distinct (integer_partitions_enum n)"
|
distinct (integer_partitions_enum ?n)
|
?H1 (?H2 x_1)
|
[
"List.distinct",
"Integer_Partitions.integer_partitions_enum",
"Set_Interval.ord_class.atMost",
"Groups.times_class.times",
"Integer_Partitions.integer_partitions_enum_dom"
] |
[
"primrec distinct :: \"'a list \\<Rightarrow> bool\" where\n\"distinct [] \\<longleftrightarrow> True\" |\n\"distinct (x # xs) \\<longleftrightarrow> x \\<notin> set xs \\<and> distinct xs\"",
"fun integer_partitions_enum :: \"nat \\<Rightarrow> nat list list\" where\n \"integer_partitions_enum n = integer_partitions_enum_aux n n\"",
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"*\" 70)"
] |
Combinatorial_Enumeration_Algorithms/Integer_Compositions
|
Integer_Compositions.sum_list_two_pow_aux
|
lemma sum_list_two_pow_aux:
"(\<Sum>x\<leftarrow>[0..< n]. (2::nat) ^ (n - x)) + 2 ^ (0 - 1) + 2 ^ 0 = 2 ^ (Suc n)"
|
(\<Sum>x\<leftarrow>[0..<?n]. 2 ^ (?n - x)) + 2 ^ (0 - 1) + 2 ^ 0 = 2 ^ Suc ?n
|
?H1 (?H1 (?H2 (?H3 (\<lambda>y_0. ?H4 (?H5 (?H6 ?H7)) (?H8 x_1 y_0)) (?H9 ?H10 x_1))) (?H4 (?H5 (?H6 ?H7)) (?H8 ?H10 ?H11))) (?H4 (?H5 (?H6 ?H7)) ?H10) = ?H4 (?H5 (?H6 ?H7)) (?H12 x_1)
|
[
"List.list.map",
"Nat.Suc",
"Groups.one_class.one",
"Groups.plus_class.plus",
"Num.num.One",
"Groups.zero_class.zero",
"List.upt",
"Groups_List.monoid_add_class.sum_list",
"Num.num.Bit0",
"Groups.minus_class.minus",
"List.list.Nil",
"Num.numeral_class.numeral",
"Power.power_class.power",
"Integer_Compositions.integer_compositions"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"class one =\n fixes one :: 'a (\"1\")",
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"datatype num = One | Bit0 num | Bit1 num",
"class zero =\n fixes zero :: 'a (\"0\")",
"primrec upt :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat list\" (\"(1[_..</_'])\") where\nupt_0: \"[i..<0] = []\" |\nupt_Suc: \"[i..<(Suc j)] = (if i \\<le> j then [i..<j] @ [j] else [])\"",
"datatype num = One | Bit0 num | Bit1 num",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"primrec power :: \"'a \\<Rightarrow> nat \\<Rightarrow> 'a\" (infixr \"^\" 80)\n where\n power_0: \"a ^ 0 = 1\"\n | power_Suc: \"a ^ Suc n = a * a ^ n\"",
"definition integer_compositions :: \"nat \\<Rightarrow> nat list set\" where\n \"integer_compositions i = {xs. sum_list xs = i \\<and> 0 \\<notin> set xs}\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Compositions
|
Integer_Compositions.integer_composition_enum_head_set
|
lemma integer_composition_enum_head_set:
assumes"x \<noteq> 0" and "x \<le> n"
shows" xs \<in> set (integer_composition_enum (n-x)) \<Longrightarrow> x#xs \<in> set (integer_composition_enum n)"
|
?x \<noteq> 0 \<Longrightarrow> ?x \<le> ?n \<Longrightarrow> ?xs \<in> set (integer_composition_enum (?n - ?x)) \<Longrightarrow> ?x # ?xs \<in> set (integer_composition_enum ?n)
|
\<lbrakk>x_1 \<noteq> ?H1; x_1 \<le> x_2; x_3 \<in> ?H2 (?H3 (?H4 x_2 x_1))\<rbrakk> \<Longrightarrow> ?H5 x_1 x_3 \<in> ?H2 (?H3 x_2)
|
[
"List.list.set",
"Integer_Compositions.integer_composition_enum",
"Groups.minus_class.minus",
"Integer_Compositions.integer_compositions",
"List.list.Cons",
"Groups.zero_class.zero",
"Integer_Compositions.integer_composition_enum_dom"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"fun integer_composition_enum :: \"nat \\<Rightarrow> nat list list\" where\n \"integer_composition_enum 0 = [[]]\"\n| \"integer_composition_enum (Suc n) =\n [Suc m #xs. m \\<leftarrow> [0..< Suc n], xs \\<leftarrow> integer_composition_enum (n-m)]\"",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"definition integer_compositions :: \"nat \\<Rightarrow> nat list set\" where\n \"integer_compositions i = {xs. sum_list xs = i \\<and> 0 \\<notin> set xs}\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"class zero =\n fixes zero :: 'a (\"0\")"
] |
Combinatorial_Enumeration_Algorithms/Integer_Compositions
|
Integer_Compositions.sum_list_two_pow
|
lemma sum_list_two_pow:
"Suc (\<Sum>x\<leftarrow>[0..<n]. 2 ^ (n - Suc x)) = 2 ^ n"
|
Suc (\<Sum>x\<leftarrow>[0..<?n]. 2 ^ (?n - Suc x)) = 2 ^ ?n
|
?H1 (?H2 (?H3 (\<lambda>y_0. ?H4 (?H5 (?H6 ?H7)) (?H8 x_1 (?H1 y_0))) (?H9 ?H10 x_1))) = ?H4 (?H5 (?H6 ?H7)) x_1
|
[
"Power.power_class.power",
"Num.num.One",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"Finite_Set.card",
"List.list.map",
"Groups.minus_class.minus",
"Set.not_member",
"List.upt",
"Groups.zero_class.zero",
"Groups_List.monoid_add_class.sum_list",
"Nat.Suc",
"List.list.set"
] |
[
"primrec power :: \"'a \\<Rightarrow> nat \\<Rightarrow> 'a\" (infixr \"^\" 80)\n where\n power_0: \"a ^ 0 = 1\"\n | power_Suc: \"a ^ Suc n = a * a ^ n\"",
"datatype num = One | Bit0 num | Bit1 num",
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"abbreviation not_member\n where \"not_member x A \\<equiv> \\<not> (x \\<in> A)\" \\<comment> \\<open>non-membership\\<close>",
"primrec upt :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat list\" (\"(1[_..</_'])\") where\nupt_0: \"[i..<0] = []\" |\nupt_Suc: \"[i..<(Suc j)] = (if i \\<le> j then [i..<j] @ [j] else [])\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Compositions
|
Integer_Compositions.integer_composition_enum_tail_elem
|
lemma integer_composition_enum_tail_elem:
"x#xs \<in> set (integer_composition_enum n) \<Longrightarrow> xs \<in> set (integer_composition_enum (n - x))"
|
?x # ?xs \<in> set (integer_composition_enum ?n) \<Longrightarrow> ?xs \<in> set (integer_composition_enum (?n - ?x))
|
?H1 x_1 x_2 \<in> ?H2 (?H3 x_3) \<Longrightarrow> x_2 \<in> ?H2 (?H3 (?H4 x_3 x_1))
|
[
"Integer_Compositions.integer_composition_enum",
"List.upt",
"List.list.set",
"Groups.minus_class.minus",
"List.list.Cons"
] |
[
"fun integer_composition_enum :: \"nat \\<Rightarrow> nat list list\" where\n \"integer_composition_enum 0 = [[]]\"\n| \"integer_composition_enum (Suc n) =\n [Suc m #xs. m \\<leftarrow> [0..< Suc n], xs \\<leftarrow> integer_composition_enum (n-m)]\"",
"primrec upt :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat list\" (\"(1[_..</_'])\") where\nupt_0: \"[i..<0] = []\" |\nupt_Suc: \"[i..<(Suc j)] = (if i \\<le> j then [i..<j] @ [j] else [])\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Compositions
|
Integer_Compositions.integer_composition_enum_not_null_aux
|
lemma integer_composition_enum_not_null_aux:
"x#xs \<in> set (integer_composition_enum n) \<Longrightarrow> x \<noteq> 0"
|
?x # ?xs \<in> set (integer_composition_enum ?n) \<Longrightarrow> ?x \<noteq> 0
|
?H1 x_1 x_2 \<in> ?H2 (?H3 x_3) \<Longrightarrow> x_1 \<noteq> ?H4
|
[
"Integer_Compositions.integer_composition_enum",
"Num.num.One",
"List.list.Cons",
"Num.numeral_class.numeral",
"List.list.set",
"Groups.zero_class.zero",
"Integer_Compositions.integer_compositions"
] |
[
"fun integer_composition_enum :: \"nat \\<Rightarrow> nat list list\" where\n \"integer_composition_enum 0 = [[]]\"\n| \"integer_composition_enum (Suc n) =\n [Suc m #xs. m \\<leftarrow> [0..< Suc n], xs \\<leftarrow> integer_composition_enum (n-m)]\"",
"datatype num = One | Bit0 num | Bit1 num",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"definition integer_compositions :: \"nat \\<Rightarrow> nat list set\" where\n \"integer_compositions i = {xs. sum_list xs = i \\<and> 0 \\<notin> set xs}\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Compositions
|
Integer_Compositions.integer_composition_enum_correct_aux
|
lemma integer_composition_enum_correct_aux:
"0 \<notin> set xs \<Longrightarrow> xs \<in> set (integer_composition_enum (sum_list xs))"
|
0 \<notin> set ?xs \<Longrightarrow> ?xs \<in> set (integer_composition_enum (sum_list ?xs))
|
?H1 ?H2 (?H3 x_1) \<Longrightarrow> x_1 \<in> ?H4 (?H5 (?H6 x_1))
|
[
"List.list.set",
"Groups_List.monoid_add_class.sum_list",
"Set.not_member",
"Groups.zero_class.zero",
"Integer_Compositions.integer_composition_enum"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation not_member\n where \"not_member x A \\<equiv> \\<not> (x \\<in> A)\" \\<comment> \\<open>non-membership\\<close>",
"class zero =\n fixes zero :: 'a (\"0\")",
"fun integer_composition_enum :: \"nat \\<Rightarrow> nat list list\" where\n \"integer_composition_enum 0 = [[]]\"\n| \"integer_composition_enum (Suc n) =\n [Suc m #xs. m \\<leftarrow> [0..< Suc n], xs \\<leftarrow> integer_composition_enum (n-m)]\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Compositions
|
Integer_Compositions.integer_composition_enum_not_null
|
lemma integer_composition_enum_not_null: "xs \<in> set (integer_composition_enum n) \<Longrightarrow> 0 \<notin> set xs"
|
?xs \<in> set (integer_composition_enum ?n) \<Longrightarrow> 0 \<notin> set ?xs
|
x_1 \<in> ?H1 (?H2 x_2) \<Longrightarrow> ?H3 ?H4 (?H5 x_1)
|
[
"List.list.set",
"Num.numeral_class.numeral",
"List.list.map",
"List.list.Cons",
"Integer_Compositions.integer_composition_enum",
"Set.not_member",
"Nat.Suc",
"Groups.zero_class.zero"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"fun integer_composition_enum :: \"nat \\<Rightarrow> nat list list\" where\n \"integer_composition_enum 0 = [[]]\"\n| \"integer_composition_enum (Suc n) =\n [Suc m #xs. m \\<leftarrow> [0..< Suc n], xs \\<leftarrow> integer_composition_enum (n-m)]\"",
"abbreviation not_member\n where \"not_member x A \\<equiv> \\<not> (x \\<in> A)\" \\<comment> \\<open>non-membership\\<close>",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"class zero =\n fixes zero :: 'a (\"0\")"
] |
Combinatorial_Enumeration_Algorithms/Integer_Compositions
|
Integer_Compositions.integer_composition_enum_sum
|
lemma integer_composition_enum_sum: "xs \<in> set (integer_composition_enum n) \<Longrightarrow> sum_list xs = n"
|
?xs \<in> set (integer_composition_enum ?n) \<Longrightarrow> sum_list ?xs = ?n
|
x_1 \<in> ?H1 (?H2 x_2) \<Longrightarrow> ?H3 x_1 = x_2
|
[
"List.list.map",
"List.list.set",
"Groups_List.monoid_add_class.sum_list",
"Integer_Compositions.integer_composition_enum"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"fun integer_composition_enum :: \"nat \\<Rightarrow> nat list list\" where\n \"integer_composition_enum 0 = [[]]\"\n| \"integer_composition_enum (Suc n) =\n [Suc m #xs. m \\<leftarrow> [0..< Suc n], xs \\<leftarrow> integer_composition_enum (n-m)]\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Compositions
|
Integer_Compositions.integer_composition_enum_length
|
lemma integer_composition_enum_length:
"length (integer_composition_enum n) = 2^(n-1)"
|
length (integer_composition_enum ?n) = 2 ^ (?n - 1)
|
?H1 (?H2 x_1) = ?H3 (?H4 (?H5 ?H6)) (?H7 x_1 ?H8)
|
[
"Integer_Compositions.integer_composition_enum",
"Groups.minus_class.minus",
"List.length",
"Groups.one_class.one",
"Num.numeral_class.numeral",
"Power.power_class.power",
"Num.num.One",
"List.list.Cons",
"List.list.Nil",
"Num.num.Bit0",
"List.list.set"
] |
[
"fun integer_composition_enum :: \"nat \\<Rightarrow> nat list list\" where\n \"integer_composition_enum 0 = [[]]\"\n| \"integer_composition_enum (Suc n) =\n [Suc m #xs. m \\<leftarrow> [0..< Suc n], xs \\<leftarrow> integer_composition_enum (n-m)]\"",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"class one =\n fixes one :: 'a (\"1\")",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"primrec power :: \"'a \\<Rightarrow> nat \\<Rightarrow> 'a\" (infixr \"^\" 80)\n where\n power_0: \"a ^ 0 = 1\"\n | power_Suc: \"a ^ Suc n = a * a ^ n\"",
"datatype num = One | Bit0 num | Bit1 num",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype num = One | Bit0 num | Bit1 num",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Compositions
|
Integer_Compositions.integer_compositions_card
|
theorem integer_compositions_card:
"card (integer_compositions n) = 2^(n-1)"
|
card (integer_compositions ?n) = 2 ^ (?n - 1)
|
?H1 (?H2 x_1) = ?H3 (?H4 (?H5 ?H6)) (?H7 x_1 ?H8)
|
[
"Finite_Set.card",
"List.list.map",
"Num.numeral_class.numeral",
"Power.power_class.power",
"Num.num.Bit0",
"Integer_Compositions.integer_composition_enum",
"Groups.minus_class.minus",
"Integer_Compositions.integer_compositions",
"List.distinct",
"Num.num.One",
"Groups.one_class.one"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"primrec power :: \"'a \\<Rightarrow> nat \\<Rightarrow> 'a\" (infixr \"^\" 80)\n where\n power_0: \"a ^ 0 = 1\"\n | power_Suc: \"a ^ Suc n = a * a ^ n\"",
"datatype num = One | Bit0 num | Bit1 num",
"fun integer_composition_enum :: \"nat \\<Rightarrow> nat list list\" where\n \"integer_composition_enum 0 = [[]]\"\n| \"integer_composition_enum (Suc n) =\n [Suc m #xs. m \\<leftarrow> [0..< Suc n], xs \\<leftarrow> integer_composition_enum (n-m)]\"",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"definition integer_compositions :: \"nat \\<Rightarrow> nat list set\" where\n \"integer_compositions i = {xs. sum_list xs = i \\<and> 0 \\<notin> set xs}\"",
"primrec distinct :: \"'a list \\<Rightarrow> bool\" where\n\"distinct [] \\<longleftrightarrow> True\" |\n\"distinct (x # xs) \\<longleftrightarrow> x \\<notin> set xs \\<and> distinct xs\"",
"datatype num = One | Bit0 num | Bit1 num",
"class one =\n fixes one :: 'a (\"1\")"
] |
Combinatorial_Enumeration_Algorithms/Integer_Compositions
|
Integer_Compositions.integer_composition_enum_empty
|
lemma integer_composition_enum_empty: "[] \<in> set (integer_composition_enum n) \<Longrightarrow> n = 0"
|
[] \<in> set (integer_composition_enum ?n) \<Longrightarrow> ?n = 0
|
?H1 \<in> ?H2 (?H3 x_1) \<Longrightarrow> x_1 = ?H4
|
[
"Integer_Compositions.integer_composition_enum_dom",
"Groups.zero_class.zero",
"List.list.set",
"List.upt",
"List.list.Nil",
"Integer_Compositions.integer_composition_enum"
] |
[
"class zero =\n fixes zero :: 'a (\"0\")",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec upt :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat list\" (\"(1[_..</_'])\") where\nupt_0: \"[i..<0] = []\" |\nupt_Suc: \"[i..<(Suc j)] = (if i \\<le> j then [i..<j] @ [j] else [])\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"fun integer_composition_enum :: \"nat \\<Rightarrow> nat list list\" where\n \"integer_composition_enum 0 = [[]]\"\n| \"integer_composition_enum (Suc n) =\n [Suc m #xs. m \\<leftarrow> [0..< Suc n], xs \\<leftarrow> integer_composition_enum (n-m)]\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Compositions
|
Integer_Compositions.integer_composition_enum_correct
|
theorem integer_composition_enum_correct:
"set (integer_composition_enum n) = integer_compositions n"
|
set (integer_composition_enum ?n) = integer_compositions ?n
|
?H1 (?H2 x_1) = ?H3 x_1
|
[
"List.list.set",
"Integer_Compositions.integer_compositions",
"Num.num.One",
"Integer_Compositions.integer_composition_enum",
"Nat.Suc"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"definition integer_compositions :: \"nat \\<Rightarrow> nat list set\" where\n \"integer_compositions i = {xs. sum_list xs = i \\<and> 0 \\<notin> set xs}\"",
"datatype num = One | Bit0 num | Bit1 num",
"fun integer_composition_enum :: \"nat \\<Rightarrow> nat list list\" where\n \"integer_composition_enum 0 = [[]]\"\n| \"integer_composition_enum (Suc n) =\n [Suc m #xs. m \\<leftarrow> [0..< Suc n], xs \\<leftarrow> integer_composition_enum (n-m)]\"",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\""
] |
Combinatorial_Enumeration_Algorithms/Integer_Compositions
|
Integer_Compositions.integer_composition_enum_distinct
|
theorem integer_composition_enum_distinct:
"distinct (integer_composition_enum n)"
|
distinct (integer_composition_enum ?n)
|
?H1 (?H2 x_1)
|
[
"List.distinct",
"Integer_Compositions.integer_composition_enum",
"List.length",
"Power.power_class.power",
"Groups.one_class.one",
"Integer_Compositions.integer_compositions"
] |
[
"primrec distinct :: \"'a list \\<Rightarrow> bool\" where\n\"distinct [] \\<longleftrightarrow> True\" |\n\"distinct (x # xs) \\<longleftrightarrow> x \\<notin> set xs \\<and> distinct xs\"",
"fun integer_composition_enum :: \"nat \\<Rightarrow> nat list list\" where\n \"integer_composition_enum 0 = [[]]\"\n| \"integer_composition_enum (Suc n) =\n [Suc m #xs. m \\<leftarrow> [0..< Suc n], xs \\<leftarrow> integer_composition_enum (n-m)]\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"primrec power :: \"'a \\<Rightarrow> nat \\<Rightarrow> 'a\" (infixr \"^\" 80)\n where\n power_0: \"a ^ 0 = 1\"\n | power_Suc: \"a ^ Suc n = a * a ^ n\"",
"class one =\n fixes one :: 'a (\"1\")",
"definition integer_compositions :: \"nat \\<Rightarrow> nat list set\" where\n \"integer_compositions i = {xs. sum_list xs = i \\<and> 0 \\<notin> set xs}\""
] |
Combinatorial_Enumeration_Algorithms/Filter_Bool_List
|
Filter_Bool_List.filter_bool_list_exist_length_card_True
|
lemma filter_bool_list_exist_length_card_True: "\<lbrakk>distinct xs; A \<subseteq> set xs; n = card A\<rbrakk>
\<Longrightarrow> \<exists>bs. length bs = length xs \<and> count_list bs True = card A \<and> A = set (filter_bool_list bs xs)"
|
distinct ?xs \<Longrightarrow> ?A \<subseteq> set ?xs \<Longrightarrow> ?n = card ?A \<Longrightarrow> \<exists>bs. length bs = length ?xs \<and> count_list bs True = card ?A \<and> ?A = set (filter_bool_list bs ?xs)
|
\<lbrakk>?H1 x_1; ?H2 x_2 (?H3 x_1); x_3 = ?H4 x_2\<rbrakk> \<Longrightarrow> \<exists>y_0. ?H5 y_0 = ?H6 x_1 \<and> ?H7 y_0 True = ?H4 x_2 \<and> x_2 = ?H3 (?H8 y_0 x_1)
|
[
"List.length",
"Set.subset_eq",
"Set.not_member",
"List.nth",
"List.list.Nil",
"List.distinct",
"Finite_Set.card",
"List.count_list",
"Set.Collect",
"Filter_Bool_List.filter_bool_list",
"List.list.set"
] |
[
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"abbreviation subset_eq :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool\"\n where \"subset_eq \\<equiv> less_eq\"",
"abbreviation not_member\n where \"not_member x A \\<equiv> \\<not> (x \\<in> A)\" \\<comment> \\<open>non-membership\\<close>",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec distinct :: \"'a list \\<Rightarrow> bool\" where\n\"distinct [] \\<longleftrightarrow> True\" |\n\"distinct (x # xs) \\<longleftrightarrow> x \\<notin> set xs \\<and> distinct xs\"",
"primrec count_list :: \"'a list \\<Rightarrow> 'a \\<Rightarrow> nat\" where\n\"count_list [] y = 0\" |\n\"count_list (x#xs) y = (if x=y then count_list xs y + 1 else count_list xs y)\"",
"fun filter_bool_list :: \"bool list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n \"filter_bool_list [] _ = []\"\n| \"filter_bool_list _ [] = []\"\n| \"filter_bool_list (b#bs) (x#xs) =\n (if b then x#(filter_bool_list bs xs) else (filter_bool_list bs xs))\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\""
] |
Combinatorial_Enumeration_Algorithms/Filter_Bool_List
|
Filter_Bool_List.filter_bool_list_not_in
|
lemma filter_bool_list_not_in:
"distinct xs \<Longrightarrow> n < length xs\<Longrightarrow> n < length bs \<Longrightarrow> bs!n = False
\<Longrightarrow> xs!n \<notin> set (filter_bool_list bs xs)"
|
distinct ?xs \<Longrightarrow> ?n < length ?xs \<Longrightarrow> ?n < length ?bs \<Longrightarrow> ?bs ! ?n = False \<Longrightarrow> ?xs ! ?n \<notin> set (filter_bool_list ?bs ?xs)
|
\<lbrakk>?H1 x_1; x_2 < ?H2 x_1; x_2 < ?H3 x_3; ?H4 x_3 x_2 = False\<rbrakk> \<Longrightarrow> ?H5 (?H6 x_1 x_2) (?H7 (?H8 x_3 x_1))
|
[
"List.distinct",
"List.length",
"Set.subset_eq",
"Filter_Bool_List.filter_bool_list",
"Set.not_member",
"List.list.Cons",
"Finite_Set.card",
"List.count_list",
"List.list.set",
"List.nth"
] |
[
"primrec distinct :: \"'a list \\<Rightarrow> bool\" where\n\"distinct [] \\<longleftrightarrow> True\" |\n\"distinct (x # xs) \\<longleftrightarrow> x \\<notin> set xs \\<and> distinct xs\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"abbreviation subset_eq :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool\"\n where \"subset_eq \\<equiv> less_eq\"",
"fun filter_bool_list :: \"bool list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n \"filter_bool_list [] _ = []\"\n| \"filter_bool_list _ [] = []\"\n| \"filter_bool_list (b#bs) (x#xs) =\n (if b then x#(filter_bool_list bs xs) else (filter_bool_list bs xs))\"",
"abbreviation not_member\n where \"not_member x A \\<equiv> \\<not> (x \\<in> A)\" \\<comment> \\<open>non-membership\\<close>",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec count_list :: \"'a list \\<Rightarrow> 'a \\<Rightarrow> nat\" where\n\"count_list [] y = 0\" |\n\"count_list (x#xs) y = (if x=y then count_list xs y + 1 else count_list xs y)\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>"
] |
Combinatorial_Enumeration_Algorithms/Filter_Bool_List
|
Filter_Bool_List.filter_bool_list_inj_aux
|
lemma filter_bool_list_inj_aux:
assumes "length bs1 = length xs"
and "length xs = length bs2"
and "distinct xs"
shows "filter_bool_list bs1 xs = filter_bool_list bs2 xs \<Longrightarrow> bs1 = bs2"
|
length ?bs1.0 = length ?xs \<Longrightarrow> length ?xs = length ?bs2.0 \<Longrightarrow> distinct ?xs \<Longrightarrow> filter_bool_list ?bs1.0 ?xs = filter_bool_list ?bs2.0 ?xs \<Longrightarrow> ?bs1.0 = ?bs2.0
|
\<lbrakk>?H1 x_1 = ?H2 x_2; ?H2 x_2 = ?H1 x_3; ?H3 x_2; ?H4 x_1 x_2 = ?H4 x_3 x_2\<rbrakk> \<Longrightarrow> x_1 = x_3
|
[
"List.distinct",
"Filter_Bool_List.filter_bool_list",
"List.length"
] |
[
"primrec distinct :: \"'a list \\<Rightarrow> bool\" where\n\"distinct [] \\<longleftrightarrow> True\" |\n\"distinct (x # xs) \\<longleftrightarrow> x \\<notin> set xs \\<and> distinct xs\"",
"fun filter_bool_list :: \"bool list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n \"filter_bool_list [] _ = []\"\n| \"filter_bool_list _ [] = []\"\n| \"filter_bool_list (b#bs) (x#xs) =\n (if b then x#(filter_bool_list bs xs) else (filter_bool_list bs xs))\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\""
] |
Combinatorial_Enumeration_Algorithms/Filter_Bool_List
|
Filter_Bool_List.filter_bool_list_elem_nth
|
lemma filter_bool_list_elem_nth: "ys \<in> set (filter_bool_list bs xs)
\<Longrightarrow> \<exists>n. ys = xs ! n \<and> bs ! n \<and> n < length bs \<and> n < length xs"
|
?ys \<in> set (filter_bool_list ?bs ?xs) \<Longrightarrow> \<exists>n. ?ys = ?xs ! n \<and> ?bs ! n \<and> n < length ?bs \<and> n < length ?xs
|
x_1 \<in> ?H1 (?H2 x_2 x_3) \<Longrightarrow> \<exists>y_0. x_1 = ?H3 x_3 y_0 \<and> ?H4 x_2 y_0 \<and> y_0 < ?H5 x_2 \<and> y_0 < ?H6 x_3
|
[
"List.nth",
"List.count_list",
"List.length",
"List.list.set",
"Filter_Bool_List.filter_bool_list"
] |
[
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"primrec count_list :: \"'a list \\<Rightarrow> 'a \\<Rightarrow> nat\" where\n\"count_list [] y = 0\" |\n\"count_list (x#xs) y = (if x=y then count_list xs y + 1 else count_list xs y)\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"fun filter_bool_list :: \"bool list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n \"filter_bool_list [] _ = []\"\n| \"filter_bool_list _ [] = []\"\n| \"filter_bool_list (b#bs) (x#xs) =\n (if b then x#(filter_bool_list bs xs) else (filter_bool_list bs xs))\""
] |
Combinatorial_Enumeration_Algorithms/Filter_Bool_List
|
Filter_Bool_List.filter_bool_list_set_nth
|
lemma filter_bool_list_set_nth:
"set (filter_bool_list bs xs) = {xs ! n |n. bs ! n \<and> n < length bs \<and> n < length xs}"
|
set (filter_bool_list ?bs ?xs) = {?xs ! n |n. ?bs ! n \<and> n < length ?bs \<and> n < length ?xs}
|
?H1 (?H2 x_1 x_2) = ?H3 (\<lambda>y_0. \<exists>y_1. y_0 = ?H4 x_2 y_1 \<and> ?H5 x_1 y_1 \<and> y_1 < ?H6 x_1 \<and> y_1 < ?H7 x_2)
|
[
"List.length",
"Set.Collect",
"Filter_Bool_List.filter_bool_list",
"List.nth",
"List.list.set",
"List.count_list"
] |
[
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"fun filter_bool_list :: \"bool list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n \"filter_bool_list [] _ = []\"\n| \"filter_bool_list _ [] = []\"\n| \"filter_bool_list (b#bs) (x#xs) =\n (if b then x#(filter_bool_list bs xs) else (filter_bool_list bs xs))\"",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec count_list :: \"'a list \\<Rightarrow> 'a \\<Rightarrow> nat\" where\n\"count_list [] y = 0\" |\n\"count_list (x#xs) y = (if x=y then count_list xs y + 1 else count_list xs y)\""
] |
Combinatorial_Enumeration_Algorithms/Filter_Bool_List
|
Filter_Bool_List.filter_bool_list_in
|
lemma filter_bool_list_in:
"n < length xs \<Longrightarrow> n < length bs \<Longrightarrow> bs!n \<Longrightarrow> xs!n \<in> set (filter_bool_list bs xs)"
|
?n < length ?xs \<Longrightarrow> ?n < length ?bs \<Longrightarrow> ?bs ! ?n \<Longrightarrow> ?xs ! ?n \<in> set (filter_bool_list ?bs ?xs)
|
\<lbrakk>x_1 < ?H1 x_2; x_1 < ?H2 x_3; ?H3 x_3 x_1\<rbrakk> \<Longrightarrow> ?H4 x_2 x_1 \<in> ?H5 (?H6 x_3 x_2)
|
[
"List.list.Nil",
"Filter_Bool_List.filter_bool_list",
"List.list.set",
"List.length",
"List.nth"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"fun filter_bool_list :: \"bool list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n \"filter_bool_list [] _ = []\"\n| \"filter_bool_list _ [] = []\"\n| \"filter_bool_list (b#bs) (x#xs) =\n (if b then x#(filter_bool_list bs xs) else (filter_bool_list bs xs))\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>"
] |
Combinatorial_Enumeration_Algorithms/Filter_Bool_List
|
Filter_Bool_List.filter_bool_list_card
|
lemma filter_bool_list_card:
"\<lbrakk>distinct xs; length xs = length bs\<rbrakk> \<Longrightarrow> card (set (filter_bool_list bs xs)) = count_list bs True"
|
distinct ?xs \<Longrightarrow> length ?xs = length ?bs \<Longrightarrow> card (set (filter_bool_list ?bs ?xs)) = count_list ?bs True
|
\<lbrakk>?H1 x_1; ?H2 x_1 = ?H3 x_2\<rbrakk> \<Longrightarrow> ?H4 (?H5 (?H6 x_2 x_1)) = ?H7 x_2 True
|
[
"List.length",
"Set.not_member",
"List.list.set",
"Filter_Bool_List.filter_bool_list",
"List.list.Cons",
"Finite_Set.card",
"List.count_list",
"Filter_Bool_List.filter_bool_list_dom",
"List.distinct"
] |
[
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"abbreviation not_member\n where \"not_member x A \\<equiv> \\<not> (x \\<in> A)\" \\<comment> \\<open>non-membership\\<close>",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"fun filter_bool_list :: \"bool list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n \"filter_bool_list [] _ = []\"\n| \"filter_bool_list _ [] = []\"\n| \"filter_bool_list (b#bs) (x#xs) =\n (if b then x#(filter_bool_list bs xs) else (filter_bool_list bs xs))\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec count_list :: \"'a list \\<Rightarrow> 'a \\<Rightarrow> nat\" where\n\"count_list [] y = 0\" |\n\"count_list (x#xs) y = (if x=y then count_list xs y + 1 else count_list xs y)\"",
"primrec distinct :: \"'a list \\<Rightarrow> bool\" where\n\"distinct [] \\<longleftrightarrow> True\" |\n\"distinct (x # xs) \\<longleftrightarrow> x \\<notin> set xs \\<and> distinct xs\""
] |
Combinatorial_Enumeration_Algorithms/Filter_Bool_List
|
Filter_Bool_List.filter_bool_list_exist_length
|
lemma filter_bool_list_exist_length: "A \<subseteq> set xs
\<Longrightarrow> \<exists>bs. length bs = length xs \<and> A = set (filter_bool_list bs xs)"
|
?A \<subseteq> set ?xs \<Longrightarrow> \<exists>bs. length bs = length ?xs \<and> ?A = set (filter_bool_list bs ?xs)
|
?H1 x_1 (?H2 x_2) \<Longrightarrow> \<exists>y_0. ?H3 y_0 = ?H4 x_2 \<and> x_1 = ?H2 (?H5 y_0 x_2)
|
[
"List.length",
"List.list.set",
"Set.subset_eq",
"Filter_Bool_List.filter_bool_list"
] |
[
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation subset_eq :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool\"\n where \"subset_eq \\<equiv> less_eq\"",
"fun filter_bool_list :: \"bool list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n \"filter_bool_list [] _ = []\"\n| \"filter_bool_list _ [] = []\"\n| \"filter_bool_list (b#bs) (x#xs) =\n (if b then x#(filter_bool_list bs xs) else (filter_bool_list bs xs))\""
] |
Combinatorial_Enumeration_Algorithms/Filter_Bool_List
|
Filter_Bool_List.filter_bool_list_inj
|
lemma filter_bool_list_inj:
"distinct xs \<Longrightarrow> inj_on (\<lambda>bs. filter_bool_list bs xs) {bs. length bs = length xs}"
|
distinct ?xs \<Longrightarrow> inj_on (\<lambda>bs. filter_bool_list bs ?xs) {bs. length bs = length ?xs}
|
?H1 x_1 \<Longrightarrow> ?H2 (\<lambda>y_0. ?H3 y_0 x_1) (?H4 (\<lambda>y_1. ?H5 y_1 = ?H6 x_1))
|
[
"Set.Collect",
"Filter_Bool_List.filter_bool_list",
"Fun.inj_on",
"List.length",
"List.distinct"
] |
[
"fun filter_bool_list :: \"bool list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n \"filter_bool_list [] _ = []\"\n| \"filter_bool_list _ [] = []\"\n| \"filter_bool_list (b#bs) (x#xs) =\n (if b then x#(filter_bool_list bs xs) else (filter_bool_list bs xs))\"",
"definition inj_on :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> bool\" \\<comment> \\<open>injective\\<close>\n where \"inj_on f A \\<longleftrightarrow> (\\<forall>x\\<in>A. \\<forall>y\\<in>A. f x = f y \\<longrightarrow> x = y)\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"primrec distinct :: \"'a list \\<Rightarrow> bool\" where\n\"distinct [] \\<longleftrightarrow> True\" |\n\"distinct (x # xs) \\<longleftrightarrow> x \\<notin> set xs \\<and> distinct xs\""
] |
Combinatorial_Enumeration_Algorithms/Filter_Bool_List
|
Filter_Bool_List.filter_bool_list_not_elem
|
lemma filter_bool_list_not_elem: "x \<notin> set xs \<Longrightarrow> x \<notin> set (filter_bool_list bs xs)"
|
?x \<notin> set ?xs \<Longrightarrow> ?x \<notin> set (filter_bool_list ?bs ?xs)
|
?H1 x_1 (?H2 x_2) \<Longrightarrow> ?H1 x_1 (?H2 (?H3 x_3 x_2))
|
[
"List.list.set",
"Set.not_member",
"List.nth",
"Finite_Set.card",
"Filter_Bool_List.filter_bool_list",
"List.distinct"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation not_member\n where \"not_member x A \\<equiv> \\<not> (x \\<in> A)\" \\<comment> \\<open>non-membership\\<close>",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"fun filter_bool_list :: \"bool list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n \"filter_bool_list [] _ = []\"\n| \"filter_bool_list _ [] = []\"\n| \"filter_bool_list (b#bs) (x#xs) =\n (if b then x#(filter_bool_list bs xs) else (filter_bool_list bs xs))\"",
"primrec distinct :: \"'a list \\<Rightarrow> bool\" where\n\"distinct [] \\<longleftrightarrow> True\" |\n\"distinct (x # xs) \\<longleftrightarrow> x \\<notin> set xs \\<and> distinct xs\""
] |
Combinatorial_Enumeration_Algorithms/Filter_Bool_List
|
Filter_Bool_List.filter_bool_list_elem
|
lemma filter_bool_list_elem: "x \<in> set (filter_bool_list bs xs) \<Longrightarrow> x \<in> set xs"
|
?x \<in> set (filter_bool_list ?bs ?xs) \<Longrightarrow> ?x \<in> set ?xs
|
x_1 \<in> ?H1 (?H2 x_2 x_3) \<Longrightarrow> x_1 \<in> ?H1 x_3
|
[
"List.list.set",
"List.nth",
"Filter_Bool_List.filter_bool_list_dom",
"List.distinct",
"Filter_Bool_List.filter_bool_list",
"Set.not_member"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"primrec distinct :: \"'a list \\<Rightarrow> bool\" where\n\"distinct [] \\<longleftrightarrow> True\" |\n\"distinct (x # xs) \\<longleftrightarrow> x \\<notin> set xs \\<and> distinct xs\"",
"fun filter_bool_list :: \"bool list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n \"filter_bool_list [] _ = []\"\n| \"filter_bool_list _ [] = []\"\n| \"filter_bool_list (b#bs) (x#xs) =\n (if b then x#(filter_bool_list bs xs) else (filter_bool_list bs xs))\"",
"abbreviation not_member\n where \"not_member x A \\<equiv> \\<not> (x \\<in> A)\" \\<comment> \\<open>non-membership\\<close>"
] |
Combinatorial_Enumeration_Algorithms/Filter_Bool_List
|
Filter_Bool_List.filter_bool_list_distinct
|
lemma filter_bool_list_distinct: "distinct xs \<Longrightarrow> distinct (filter_bool_list bs xs)"
|
distinct ?xs \<Longrightarrow> distinct (filter_bool_list ?bs ?xs)
|
?H1 x_1 \<Longrightarrow> ?H1 (?H2 x_2 x_1)
|
[
"List.distinct",
"List.count_list",
"List.length",
"Filter_Bool_List.filter_bool_list"
] |
[
"primrec distinct :: \"'a list \\<Rightarrow> bool\" where\n\"distinct [] \\<longleftrightarrow> True\" |\n\"distinct (x # xs) \\<longleftrightarrow> x \\<notin> set xs \\<and> distinct xs\"",
"primrec count_list :: \"'a list \\<Rightarrow> 'a \\<Rightarrow> nat\" where\n\"count_list [] y = 0\" |\n\"count_list (x#xs) y = (if x=y then count_list xs y + 1 else count_list xs y)\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"fun filter_bool_list :: \"bool list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n \"filter_bool_list [] _ = []\"\n| \"filter_bool_list _ [] = []\"\n| \"filter_bool_list (b#bs) (x#xs) =\n (if b then x#(filter_bool_list bs xs) else (filter_bool_list bs xs))\""
] |
Isabelle_hoops/Totally_Ordered_Hoops
|
Totally_Ordered_Hoops.irr_test
|
lemma irr_test:
assumes "totally_ordered_hoop A PA RA a"
"\<not>totally_ordered_irreducible_hoop A PA RA a"
shows "\<exists> B C.
(A = B \<union> C) \<and>
({a} = B \<inter> C) \<and>
(\<exists> y \<in> B. y \<noteq> a) \<and>
(\<exists> y \<in> C. y \<noteq> a) \<and>
(hoop B PA RA a) \<and>
(hoop C PA RA a) \<and>
(\<forall> x \<in> B-{a}. \<forall> y \<in> C. PA x y = x) \<and>
(\<forall> x \<in> B-{a}. \<forall> y \<in> C. RA x y = a) \<and>
(\<forall> x \<in> C. \<forall> y \<in> B. RA x y = y)"
|
totally_ordered_hoop ?A ?PA ?RA ?a \<Longrightarrow> \<not> totally_ordered_irreducible_hoop ?A ?PA ?RA ?a \<Longrightarrow> \<exists>B C. ?A = B \<union> C \<and> {?a} = B \<inter> C \<and> (\<exists>y\<in>B. y \<noteq> ?a) \<and> (\<exists>y\<in>C. y \<noteq> ?a) \<and> hoop B ?PA ?RA ?a \<and> hoop C ?PA ?RA ?a \<and> (\<forall>x\<in>B - {?a}. \<forall>y\<in>C. ?PA x y = x) \<and> (\<forall>x\<in>B - {?a}. \<forall>y\<in>C. ?RA x y = ?a) \<and> (\<forall>x\<in>C. \<forall>y\<in>B. ?RA x y = y)
|
\<lbrakk>?H1 x_1 x_2 x_3 x_4; \<not> ?H2 x_1 x_2 x_3 x_4\<rbrakk> \<Longrightarrow> \<exists>y_0 y_1. x_1 = ?H3 y_0 y_1 \<and> ?H4 x_4 ?H5 = ?H6 y_0 y_1 \<and> (\<exists>y_2\<in>y_0. y_2 \<noteq> x_4) \<and> (\<exists>y_3\<in>y_1. y_3 \<noteq> x_4) \<and> ?H7 y_0 x_2 x_3 x_4 \<and> ?H7 y_1 x_2 x_3 x_4 \<and> (\<forall>y_4\<in>?H8 y_0 (?H4 x_4 ?H5). \<forall>y_5\<in>y_1. x_2 y_4 y_5 = y_4) \<and> (\<forall>y_6\<in>?H8 y_0 (?H4 x_4 ?H5). \<forall>y_7\<in>y_1. x_3 y_6 y_7 = x_4) \<and> (\<forall>y_8\<in>y_1. \<forall>y_9\<in>y_0. x_3 y_8 y_9 = y_9)
|
[
"Groups.minus_class.minus",
"Set.empty",
"Totally_Ordered_Hoops.totally_ordered_irreducible_hoop",
"Set.inter",
"Set.union",
"Totally_Ordered_Hoops.totally_ordered_hoop",
"Set.insert",
"Hoops.hoop"
] |
[
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\"",
"abbreviation inter :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a set\" (infixl \"\\<inter>\" 70)\n where \"(\\<inter>) \\<equiv> inf\"",
"abbreviation union :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a set\" (infixl \"\\<union>\" 65)\n where \"union \\<equiv> sup\"",
"definition insert :: \"'a \\<Rightarrow> 'a set \\<Rightarrow> 'a set\"\n where insert_compr: \"insert a B = {x. x = a \\<or> x \\<in> B}\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_suffix_fst_1
|
lemma Interleaves_suffix_fst_1 [rule_format]:
assumes A: "\<forall>n < length ws. P (ws ! n) (drop (Suc n) ws)"
shows "xs \<cong> {ys, zs, \<lambda>v vs. P v (vs @ ws)} \<longrightarrow> xs @ ws \<cong> {ys @ ws, zs, P}"
(is "_ \<cong> {_, _, ?P'} \<longrightarrow> _")
|
(\<And>n. n < length ?ws \<Longrightarrow> ?P (?ws ! n) (drop (Suc n) ?ws)) \<Longrightarrow> ?xs \<cong> {?ys, ?zs, \<lambda>v vs. ?P v (vs @ ?ws)} \<Longrightarrow> ?xs @ ?ws \<cong> {?ys @ ?ws, ?zs, ?P}
|
\<lbrakk>\<And>y_0. y_0 < ?H1 x_1 \<Longrightarrow> x_2 (?H2 x_1 y_0) (?H3 (?H4 y_0) x_1); ?H5 x_3 x_4 x_5 (\<lambda>y_1 y_2. x_2 y_1 (?H6 y_2 x_1))\<rbrakk> \<Longrightarrow> ?H5 (?H6 x_3 x_1) (?H6 x_4 x_1) x_5 x_2
|
[
"ListInterleaving.Interleaves_syntax",
"Nat.Suc",
"List.drop",
"ListInterleaving.Interleaves_dom",
"List.nth",
"List.append",
"List.length"
] |
[
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_suffix_fst_2
|
lemma Interleaves_suffix_fst_2 [rule_format]:
assumes A: "\<forall>n < length ws. P (ws ! n) (drop (Suc n) ws)"
shows "xs @ ws \<cong> {ys @ ws, zs, P} \<longrightarrow> xs \<cong> {ys, zs, \<lambda>v vs. P v (vs @ ws)}"
(is "_ \<longrightarrow> _ \<cong> {_, _, ?P'}")
|
(\<And>n. n < length ?ws \<Longrightarrow> ?P (?ws ! n) (drop (Suc n) ?ws)) \<Longrightarrow> ?xs @ ?ws \<cong> {?ys @ ?ws, ?zs, ?P} \<Longrightarrow> ?xs \<cong> {?ys, ?zs, \<lambda>v vs. ?P v (vs @ ?ws)}
|
\<lbrakk>\<And>y_0. y_0 < ?H1 x_1 \<Longrightarrow> x_2 (?H2 x_1 y_0) (?H3 (?H4 y_0) x_1); ?H5 (?H6 x_3 x_1) (?H6 x_4 x_1) x_5 x_2\<rbrakk> \<Longrightarrow> ?H5 x_3 x_4 x_5 (\<lambda>y_1 y_2. x_2 y_1 (?H6 y_2 x_1))
|
[
"ListInterleaving.interleaves_dom",
"ListInterleaving.interleaves_syntax",
"ListInterleaving.Interleaves_syntax",
"List.append",
"List.drop",
"Nat.Suc",
"List.nth",
"List.length"
] |
[
"abbreviation interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\" \n (\"(_ \\<simeq> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<simeq> {ys, zs, P} \\<equiv> interleaves P xs ys zs\"",
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\"",
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_suffix_snd
|
lemma Interleaves_suffix_snd [rule_format]:
"\<forall>n < length ws. \<not> P (ws ! n) (drop (Suc n) ws) \<Longrightarrow>
xs \<cong> {ys, zs, \<lambda>v vs. P v (vs @ ws)} = xs @ ws \<cong> {ys, zs @ ws, P}"
|
(\<And>n. n < length ?ws \<Longrightarrow> \<not> ?P (?ws ! n) (drop (Suc n) ?ws)) \<Longrightarrow> ?xs \<cong> {?ys, ?zs, \<lambda>v vs. ?P v (vs @ ?ws)} = ?xs @ ?ws \<cong> {?ys, ?zs @ ?ws, ?P}
|
(\<And>y_0. y_0 < ?H1 x_1 \<Longrightarrow> \<not> x_2 (?H2 x_1 y_0) (?H3 (?H4 y_0) x_1)) \<Longrightarrow> ?H5 x_3 x_4 x_5 (\<lambda>y_1 y_2. x_2 y_1 (?H6 y_2 x_1)) = ?H5 (?H6 x_3 x_1) x_4 (?H6 x_5 x_1) x_2
|
[
"List.append",
"List.list.Nil",
"List.drop",
"ListInterleaving.interleaves_dom",
"Groups.plus_class.plus",
"List.nth",
"Nat.Suc",
"List.length",
"ListInterleaving.Interleaves_syntax",
"ListInterleaving.Interleaves_dom"
] |
[
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_suffix_fst
|
lemma Interleaves_suffix_fst [rule_format]:
"\<forall>n < length ws. P (ws ! n) (drop (Suc n) ws) \<Longrightarrow>
xs \<cong> {ys, zs, \<lambda>v vs. P v (vs @ ws)} = xs @ ws \<cong> {ys @ ws, zs, P}"
|
(\<And>n. n < length ?ws \<Longrightarrow> ?P (?ws ! n) (drop (Suc n) ?ws)) \<Longrightarrow> ?xs \<cong> {?ys, ?zs, \<lambda>v vs. ?P v (vs @ ?ws)} = ?xs @ ?ws \<cong> {?ys @ ?ws, ?zs, ?P}
|
(\<And>y_0. y_0 < ?H1 x_1 \<Longrightarrow> x_2 (?H2 x_1 y_0) (?H3 (?H4 y_0) x_1)) \<Longrightarrow> ?H5 x_3 x_4 x_5 (\<lambda>y_1 y_2. x_2 y_1 (?H6 y_2 x_1)) = ?H5 (?H6 x_3 x_1) (?H6 x_4 x_1) x_5 x_2
|
[
"ListInterleaving.Interleaves_syntax",
"List.nth",
"Nat.Suc",
"List.filter",
"List.list.Nil",
"List.append",
"ListInterleaving.Interleaves_dom",
"List.length",
"List.drop"
] |
[
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"primrec filter:: \"('a \\<Rightarrow> bool) \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n\"filter P [] = []\" |\n\"filter P (x # xs) = (if P x then x # filter P xs else filter P xs)\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>"
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.interleaves_prefix_fst_2
|
lemma interleaves_prefix_fst_2 [rule_format]:
"ws @ xs \<simeq> {ws @ ys, zs, P} \<longrightarrow>
(\<forall>n < length ws. P (ws ! n) (drop (Suc n) ws @ xs)) \<longrightarrow>
xs \<simeq> {ys, zs, P}"
|
?ws @ ?xs \<simeq> {?ws @ ?ys, ?zs, ?P} \<Longrightarrow> (\<And>n. n < length ?ws \<Longrightarrow> ?P (?ws ! n) (drop (Suc n) ?ws @ ?xs)) \<Longrightarrow> ?xs \<simeq> {?ys, ?zs, ?P}
|
\<lbrakk>?H1 (?H2 x_1 x_2) (?H2 x_1 x_3) x_4 x_5; \<And>y_0. y_0 < ?H3 x_1 \<Longrightarrow> x_5 (?H4 x_1 y_0) (?H2 (?H5 (?H6 y_0) x_1) x_2)\<rbrakk> \<Longrightarrow> ?H1 x_2 x_3 x_4 x_5
|
[
"Groups.plus_class.plus",
"List.length",
"Nat.Suc",
"List.nth",
"ListInterleaving.interleaves_syntax",
"List.append",
"Fun.inj",
"List.list.map",
"List.drop",
"List.list.Nil"
] |
[
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"abbreviation interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\" \n (\"(_ \\<simeq> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<simeq> {ys, zs, P} \\<equiv> interleaves P xs ys zs\"",
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\"",
"abbreviation inj :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> bool\"\n where \"inj f \\<equiv> inj_on f UNIV\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_prefix_fst_2
|
lemma Interleaves_prefix_fst_2 [rule_format]:
"ws @ xs \<cong> {ws @ ys, zs, P} \<longrightarrow>
(\<forall>n < length ws. P (ws ! n) (drop (Suc n) ws @ xs)) \<longrightarrow>
xs \<cong> {ys, zs, P}"
|
?ws @ ?xs \<cong> {?ws @ ?ys, ?zs, ?P} \<Longrightarrow> (\<And>n. n < length ?ws \<Longrightarrow> ?P (?ws ! n) (drop (Suc n) ?ws @ ?xs)) \<Longrightarrow> ?xs \<cong> {?ys, ?zs, ?P}
|
\<lbrakk>?H1 (?H2 x_1 x_2) (?H2 x_1 x_3) x_4 x_5; \<And>y_0. y_0 < ?H3 x_1 \<Longrightarrow> x_5 (?H4 x_1 y_0) (?H2 (?H5 (?H6 y_0) x_1) x_2)\<rbrakk> \<Longrightarrow> ?H1 x_2 x_3 x_4 x_5
|
[
"ListInterleaving.Interleaves_syntax",
"List.length",
"List.nth",
"List.append",
"List.drop",
"Nat.Suc"
] |
[
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\"",
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.interleaves_prefix_fst_1
|
lemma interleaves_prefix_fst_1 [rule_format]:
assumes A: "xs \<simeq> {ys, zs, P}"
shows "(\<forall>n < length ws. P (ws ! n) (drop (Suc n) ws @ xs)) \<longrightarrow>
ws @ xs \<simeq> {ws @ ys, zs, P}"
|
?xs \<simeq> {?ys, ?zs, ?P} \<Longrightarrow> (\<And>n. n < length ?ws \<Longrightarrow> ?P (?ws ! n) (drop (Suc n) ?ws @ ?xs)) \<Longrightarrow> ?ws @ ?xs \<simeq> {?ws @ ?ys, ?zs, ?P}
|
\<lbrakk>?H1 x_1 x_2 x_3 x_4; \<And>y_0. y_0 < ?H2 x_5 \<Longrightarrow> x_4 (?H3 x_5 y_0) (?H4 (?H5 (?H6 y_0) x_5) x_1)\<rbrakk> \<Longrightarrow> ?H1 (?H4 x_5 x_1) (?H4 x_5 x_2) x_3 x_4
|
[
"List.drop",
"List.filter",
"Groups.plus_class.plus",
"Nat.Suc",
"ListInterleaving.interleaves_syntax",
"List.length",
"List.append",
"List.list.map",
"ListInterleaving.interleaves_dom",
"List.nth"
] |
[
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"primrec filter:: \"('a \\<Rightarrow> bool) \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n\"filter P [] = []\" |\n\"filter P (x # xs) = (if P x then x # filter P xs else filter P xs)\"",
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"abbreviation interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\" \n (\"(_ \\<simeq> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<simeq> {ys, zs, P} \\<equiv> interleaves P xs ys zs\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>"
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_prefix_fst_1
|
lemma Interleaves_prefix_fst_1 [rule_format]:
assumes A: "xs \<cong> {ys, zs, P}"
shows "(\<forall>n < length ws. P (ws ! n) (drop (Suc n) ws @ xs)) \<longrightarrow>
ws @ xs \<cong> {ws @ ys, zs, P}"
|
?xs \<cong> {?ys, ?zs, ?P} \<Longrightarrow> (\<And>n. n < length ?ws \<Longrightarrow> ?P (?ws ! n) (drop (Suc n) ?ws @ ?xs)) \<Longrightarrow> ?ws @ ?xs \<cong> {?ws @ ?ys, ?zs, ?P}
|
\<lbrakk>?H1 x_1 x_2 x_3 x_4; \<And>y_0. y_0 < ?H2 x_5 \<Longrightarrow> x_4 (?H3 x_5 y_0) (?H4 (?H5 (?H6 y_0) x_5) x_1)\<rbrakk> \<Longrightarrow> ?H1 (?H4 x_5 x_1) (?H4 x_5 x_2) x_3 x_4
|
[
"ListInterleaving.Interleaves_syntax",
"ListInterleaving.interleaves_dom",
"List.append",
"List.drop",
"List.list.Cons",
"List.length",
"List.nth",
"Nat.Suc",
"List.filter"
] |
[
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\"",
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"primrec filter:: \"('a \\<Rightarrow> bool) \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n\"filter P [] = []\" |\n\"filter P (x # xs) = (if P x then x # filter P xs else filter P xs)\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.interleaves_prefix_snd
|
lemma interleaves_prefix_snd [rule_format]:
"\<forall>n < length ws. \<not> P (ws ! n) (drop (Suc n) ws @ xs) \<Longrightarrow>
xs \<simeq> {ys, zs, P} = ws @ xs \<simeq> {ys, ws @ zs, P}"
|
(\<And>n. n < length ?ws \<Longrightarrow> \<not> ?P (?ws ! n) (drop (Suc n) ?ws @ ?xs)) \<Longrightarrow> ?xs \<simeq> {?ys, ?zs, ?P} = ?ws @ ?xs \<simeq> {?ys, ?ws @ ?zs, ?P}
|
(\<And>y_0. y_0 < ?H1 x_1 \<Longrightarrow> \<not> x_2 (?H2 x_1 y_0) (?H3 (?H4 (?H5 y_0) x_1) x_3)) \<Longrightarrow> ?H6 x_3 x_4 x_5 x_2 = ?H6 (?H3 x_1 x_3) x_4 (?H3 x_1 x_5) x_2
|
[
"ListInterleaving.interleaves_syntax",
"List.append",
"List.nth",
"Nat.Suc",
"List.drop",
"List.length"
] |
[
"abbreviation interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\" \n (\"(_ \\<simeq> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<simeq> {ys, zs, P} \\<equiv> interleaves P xs ys zs\"",
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\"",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_prefix_snd
|
lemma Interleaves_prefix_snd [rule_format]:
"\<forall>n < length ws. \<not> P (ws ! n) (drop (Suc n) ws @ xs) \<Longrightarrow>
xs \<cong> {ys, zs, P} = ws @ xs \<cong> {ys, ws @ zs, P}"
|
(\<And>n. n < length ?ws \<Longrightarrow> \<not> ?P (?ws ! n) (drop (Suc n) ?ws @ ?xs)) \<Longrightarrow> ?xs \<cong> {?ys, ?zs, ?P} = ?ws @ ?xs \<cong> {?ys, ?ws @ ?zs, ?P}
|
(\<And>y_0. y_0 < ?H1 x_1 \<Longrightarrow> \<not> x_2 (?H2 x_1 y_0) (?H3 (?H4 (?H5 y_0) x_1) x_3)) \<Longrightarrow> ?H6 x_3 x_4 x_5 x_2 = ?H6 (?H3 x_1 x_3) x_4 (?H3 x_1 x_5) x_2
|
[
"List.nth",
"ListInterleaving.interleaves_dom",
"List.filter",
"List.length",
"List.append",
"ListInterleaving.interleaves_syntax",
"ListInterleaving.Interleaves_syntax",
"List.list.Nil",
"Nat.Suc",
"List.drop"
] |
[
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"primrec filter:: \"('a \\<Rightarrow> bool) \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n\"filter P [] = []\" |\n\"filter P (x # xs) = (if P x then x # filter P xs else filter P xs)\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\"",
"abbreviation interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\" \n (\"(_ \\<simeq> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<simeq> {ys, zs, P} \\<equiv> interleaves P xs ys zs\"",
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>"
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_suffix_aux
|
lemma Interleaves_suffix_aux [rule_format]:
"(\<forall>n < length ws. P (ws ! n) (drop (Suc n) ws)) \<longrightarrow>
x # xs @ ws \<cong> {ws, zs, P} \<longrightarrow>
\<not> P x (xs @ ws)"
|
(\<And>n. n < length ?ws \<Longrightarrow> ?P (?ws ! n) (drop (Suc n) ?ws)) \<Longrightarrow> ?x # ?xs @ ?ws \<cong> {?ws, ?zs, ?P} \<Longrightarrow> \<not> ?P ?x (?xs @ ?ws)
|
\<lbrakk>\<And>y_0. y_0 < ?H1 x_1 \<Longrightarrow> x_2 (?H2 x_1 y_0) (?H3 (?H4 y_0) x_1); ?H5 (?H6 x_3 (?H7 x_4 x_1)) x_1 x_5 x_2\<rbrakk> \<Longrightarrow> \<not> x_2 x_3 (?H7 x_4 x_1)
|
[
"List.drop",
"List.length",
"List.list.Cons",
"ListInterleaving.interleaves_dom",
"ListInterleaving.Interleaves_syntax",
"Nat.Suc",
"List.nth",
"List.append"
] |
[
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.interleaves_prefix_fst
|
lemma interleaves_prefix_fst [rule_format]:
"\<forall>n < length ws. P (ws ! n) (drop (Suc n) ws @ xs) \<Longrightarrow>
xs \<simeq> {ys, zs, P} = ws @ xs \<simeq> {ws @ ys, zs, P}"
|
(\<And>n. n < length ?ws \<Longrightarrow> ?P (?ws ! n) (drop (Suc n) ?ws @ ?xs)) \<Longrightarrow> ?xs \<simeq> {?ys, ?zs, ?P} = ?ws @ ?xs \<simeq> {?ws @ ?ys, ?zs, ?P}
|
(\<And>y_0. y_0 < ?H1 x_1 \<Longrightarrow> x_2 (?H2 x_1 y_0) (?H3 (?H4 (?H5 y_0) x_1) x_3)) \<Longrightarrow> ?H6 x_3 x_4 x_5 x_2 = ?H6 (?H3 x_1 x_3) (?H3 x_1 x_4) x_5 x_2
|
[
"List.nth",
"List.length",
"Nat.Suc",
"List.drop",
"List.append",
"ListInterleaving.interleaves_syntax"
] |
[
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\"",
"abbreviation interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\" \n (\"(_ \\<simeq> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<simeq> {ys, zs, P} \\<equiv> interleaves P xs ys zs\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_prefix_fst
|
lemma Interleaves_prefix_fst [rule_format]:
"\<forall>n < length ws. P (ws ! n) (drop (Suc n) ws @ xs) \<Longrightarrow>
xs \<cong> {ys, zs, P} = ws @ xs \<cong> {ws @ ys, zs, P}"
|
(\<And>n. n < length ?ws \<Longrightarrow> ?P (?ws ! n) (drop (Suc n) ?ws @ ?xs)) \<Longrightarrow> ?xs \<cong> {?ys, ?zs, ?P} = ?ws @ ?xs \<cong> {?ws @ ?ys, ?zs, ?P}
|
(\<And>y_0. y_0 < ?H1 x_1 \<Longrightarrow> x_2 (?H2 x_1 y_0) (?H3 (?H4 (?H5 y_0) x_1) x_3)) \<Longrightarrow> ?H6 x_3 x_4 x_5 x_2 = ?H6 (?H3 x_1 x_3) (?H3 x_1 x_4) x_5 x_2
|
[
"List.append",
"List.drop",
"List.length",
"ListInterleaving.Interleaves_syntax",
"List.nth",
"Nat.Suc"
] |
[
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\"",
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.interleaves_map
|
lemma interleaves_map [rule_format]:
assumes A: "inj f"
shows "xs \<simeq> {ys, zs, P} \<longrightarrow>
map f xs \<simeq> {map f ys, map f zs, \<lambda>w ws. P (inv f w) (map (inv f) ws)}"
(is "_ \<longrightarrow> _ \<simeq> {_, _, ?P'}")
|
inj ?f \<Longrightarrow> ?xs \<simeq> {?ys, ?zs, ?P} \<Longrightarrow> map ?f ?xs \<simeq> {map ?f ?ys, map ?f ?zs, \<lambda>w ws. ?P (inv ?f w) (map (inv ?f) ws)}
|
\<lbrakk>?H1 x_1; ?H2 x_2 x_3 x_4 x_5\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_2) (?H4 x_1 x_3) (?H4 x_1 x_4) (\<lambda>y_0 y_1. x_5 (?H5 x_1 y_0) (?H6 (?H5 x_1) y_1))
|
[
"Fun.inj",
"Hilbert_Choice.inv",
"ListInterleaving.interleaves_syntax",
"Nat.Suc",
"List.list.Cons",
"List.list.map",
"List.list.Nil"
] |
[
"abbreviation inj :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> bool\"\n where \"inj f \\<equiv> inj_on f UNIV\"",
"abbreviation inv :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> ('b \\<Rightarrow> 'a)\" where\n\"inv \\<equiv> inv_into UNIV\"",
"abbreviation interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\" \n (\"(_ \\<simeq> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<simeq> {ys, zs, P} \\<equiv> interleaves P xs ys zs\"",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_map
|
lemma Interleaves_map [rule_format]:
assumes A: "inj f"
shows "xs \<cong> {ys, zs, P} \<longrightarrow>
map f xs \<cong> {map f ys, map f zs, \<lambda>w ws. P (inv f w) (map (inv f) ws)}"
(is "_ \<longrightarrow> _ \<cong> {_, _, ?P'}")
|
inj ?f \<Longrightarrow> ?xs \<cong> {?ys, ?zs, ?P} \<Longrightarrow> map ?f ?xs \<cong> {map ?f ?ys, map ?f ?zs, \<lambda>w ws. ?P (inv ?f w) (map (inv ?f) ws)}
|
\<lbrakk>?H1 x_1; ?H2 x_2 x_3 x_4 x_5\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_2) (?H4 x_1 x_3) (?H4 x_1 x_4) (\<lambda>y_0 y_1. x_5 (?H5 x_1 y_0) (?H6 (?H5 x_1) y_1))
|
[
"Fun.inj",
"ListInterleaving.Interleaves_syntax",
"Hilbert_Choice.inv",
"ListInterleaving.Interleaves_dom",
"List.list.map"
] |
[
"abbreviation inj :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> bool\"\n where \"inj f \\<equiv> inj_on f UNIV\"",
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"abbreviation inv :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> ('b \\<Rightarrow> 'a)\" where\n\"inv \\<equiv> inv_into UNIV\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.interleaves_filter
|
lemma interleaves_filter [rule_format]:
assumes A: "\<forall>x xs. P x (filter Q xs) = P x xs"
shows "xs \<simeq> {ys, zs, P} \<longrightarrow> filter Q xs \<simeq> {filter Q ys, filter Q zs, P}"
|
(\<And>x xs. ?P x (filter ?Q xs) = ?P x xs) \<Longrightarrow> ?xs \<simeq> {?ys, ?zs, ?P} \<Longrightarrow> filter ?Q ?xs \<simeq> {filter ?Q ?ys, filter ?Q ?zs, ?P}
|
\<lbrakk>\<And>y_0 y_1. x_1 y_0 (?H1 x_2 y_1) = x_1 y_0 y_1; ?H2 x_3 x_4 x_5 x_1\<rbrakk> \<Longrightarrow> ?H2 (?H1 x_2 x_3) (?H1 x_2 x_4) (?H1 x_2 x_5) x_1
|
[
"List.list.Cons",
"List.list.map",
"ListInterleaving.interleaves_syntax",
"Hilbert_Choice.inv",
"List.filter",
"ListInterleaving.Interleaves_dom"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\" \n (\"(_ \\<simeq> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<simeq> {ys, zs, P} \\<equiv> interleaves P xs ys zs\"",
"abbreviation inv :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> ('b \\<Rightarrow> 'a)\" where\n\"inv \\<equiv> inv_into UNIV\"",
"primrec filter:: \"('a \\<Rightarrow> bool) \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n\"filter P [] = []\" |\n\"filter P (x # xs) = (if P x then x # filter P xs else filter P xs)\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_filter
|
lemma Interleaves_filter [rule_format]:
assumes A: "\<forall>x xs. P x (filter Q xs) = P x xs"
shows "xs \<cong> {ys, zs, P} \<longrightarrow> filter Q xs \<cong> {filter Q ys, filter Q zs, P}"
|
(\<And>x xs. ?P x (filter ?Q xs) = ?P x xs) \<Longrightarrow> ?xs \<cong> {?ys, ?zs, ?P} \<Longrightarrow> filter ?Q ?xs \<cong> {filter ?Q ?ys, filter ?Q ?zs, ?P}
|
\<lbrakk>\<And>y_0 y_1. x_1 y_0 (?H1 x_2 y_1) = x_1 y_0 y_1; ?H2 x_3 x_4 x_5 x_1\<rbrakk> \<Longrightarrow> ?H2 (?H1 x_2 x_3) (?H1 x_2 x_4) (?H1 x_2 x_5) x_1
|
[
"ListInterleaving.Interleaves_syntax",
"List.list.Nil",
"Hilbert_Choice.inv",
"List.filter",
"List.append",
"Groups.plus_class.plus"
] |
[
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation inv :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> ('b \\<Rightarrow> 'a)\" where\n\"inv \\<equiv> inv_into UNIV\"",
"primrec filter:: \"('a \\<Rightarrow> bool) \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n\"filter P [] = []\" |\n\"filter P (x # xs) = (if P x then x # filter P xs else filter P xs)\"",
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\"",
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)"
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_suffix_one_fst_1
|
lemma Interleaves_suffix_one_fst_1 [rule_format]:
"P x [] \<Longrightarrow>
xs \<cong> {ys, zs, \<lambda>w ws. P w (ws @ [x])} \<Longrightarrow> xs @ [x] \<cong> {ys @ [x], zs, P}"
|
?P ?x [] \<Longrightarrow> ?xs \<cong> {?ys, ?zs, \<lambda>w ws. ?P w (ws @ [?x])} \<Longrightarrow> ?xs @ [?x] \<cong> {?ys @ [?x], ?zs, ?P}
|
\<lbrakk>x_1 x_2 ?H1; ?H2 x_3 x_4 x_5 (\<lambda>y_0 y_1. x_1 y_0 (?H3 y_1 (?H4 x_2 ?H1)))\<rbrakk> \<Longrightarrow> ?H2 (?H3 x_3 (?H4 x_2 ?H1)) (?H3 x_4 (?H4 x_2 ?H1)) x_5 x_1
|
[
"ListInterleaving.Interleaves_syntax",
"List.list.Cons",
"List.filter",
"List.list.Nil",
"List.length",
"ListInterleaving.Interleaves_dom",
"List.nth",
"List.append"
] |
[
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec filter:: \"('a \\<Rightarrow> bool) \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n\"filter P [] = []\" |\n\"filter P (x # xs) = (if P x then x # filter P xs else filter P xs)\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_suffix_one_fst_2
|
lemma Interleaves_suffix_one_fst_2 [rule_format]:
assumes A: "P x []"
shows "xs @ [x] \<cong> {ys @ [x], zs, P} \<longrightarrow> xs \<cong> {ys, zs, \<lambda>w ws. P w (ws @ [x])}"
(is "_ \<longrightarrow> _ \<cong> {_, _, ?P'}")
|
?P ?x [] \<Longrightarrow> ?xs @ [?x] \<cong> {?ys @ [?x], ?zs, ?P} \<Longrightarrow> ?xs \<cong> {?ys, ?zs, \<lambda>w ws. ?P w (ws @ [?x])}
|
\<lbrakk>x_1 x_2 ?H1; ?H2 (?H3 x_3 (?H4 x_2 ?H1)) (?H3 x_4 (?H4 x_2 ?H1)) x_5 x_1\<rbrakk> \<Longrightarrow> ?H2 x_3 x_4 x_5 (\<lambda>y_0 y_1. x_1 y_0 (?H3 y_1 (?H4 x_2 ?H1)))
|
[
"Nat.Suc",
"List.length",
"List.list.Cons",
"List.append",
"List.list.Nil",
"ListInterleaving.Interleaves_syntax"
] |
[
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_suffix_one_snd
|
lemma Interleaves_suffix_one_snd:
"\<not> P x [] \<Longrightarrow>
xs \<cong> {ys, zs, \<lambda>w ws. P w (ws @ [x])} = xs @ [x] \<cong> {ys, zs @ [x], P}"
|
\<not> ?P ?x [] \<Longrightarrow> ?xs \<cong> {?ys, ?zs, \<lambda>w ws. ?P w (ws @ [?x])} = ?xs @ [?x] \<cong> {?ys, ?zs @ [?x], ?P}
|
\<not> x_1 x_2 ?H1 \<Longrightarrow> ?H2 x_3 x_4 x_5 (\<lambda>y_0 y_1. x_1 y_0 (?H3 y_1 (?H4 x_2 ?H1))) = ?H2 (?H3 x_3 (?H4 x_2 ?H1)) x_4 (?H3 x_5 (?H4 x_2 ?H1)) x_1
|
[
"Hilbert_Choice.inv",
"List.list.Cons",
"List.append",
"ListInterleaving.Interleaves_syntax",
"List.drop",
"ListInterleaving.interleaves_syntax",
"List.list.Nil",
"List.list.map"
] |
[
"abbreviation inv :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> ('b \\<Rightarrow> 'a)\" where\n\"inv \\<equiv> inv_into UNIV\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\"",
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"abbreviation interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\" \n (\"(_ \\<simeq> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<simeq> {ys, zs, P} \\<equiv> interleaves P xs ys zs\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_suffix_one_fst
|
lemma Interleaves_suffix_one_fst:
"P x [] \<Longrightarrow>
xs \<cong> {ys, zs, \<lambda>w ws. P w (ws @ [x])} = xs @ [x] \<cong> {ys @ [x], zs, P}"
|
?P ?x [] \<Longrightarrow> ?xs \<cong> {?ys, ?zs, \<lambda>w ws. ?P w (ws @ [?x])} = ?xs @ [?x] \<cong> {?ys @ [?x], ?zs, ?P}
|
x_1 x_2 ?H1 \<Longrightarrow> ?H2 x_3 x_4 x_5 (\<lambda>y_0 y_1. x_1 y_0 (?H3 y_1 (?H4 x_2 ?H1))) = ?H2 (?H3 x_3 (?H4 x_2 ?H1)) (?H3 x_4 (?H4 x_2 ?H1)) x_5 x_1
|
[
"List.append",
"List.list.Nil",
"ListInterleaving.Interleaves_syntax",
"List.list.Cons"
] |
[
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_all_nil_2
|
lemma Interleaves_all_nil_2 [rule_format]:
"\<forall>n < length xs. P (xs ! n) (drop (Suc n) xs) \<Longrightarrow> xs \<cong> {xs, [], P}"
|
(\<And>n. n < length ?xs \<Longrightarrow> ?P (?xs ! n) (drop (Suc n) ?xs)) \<Longrightarrow> ?xs \<cong> {?xs, [], ?P}
|
(\<And>y_0. y_0 < ?H1 x_1 \<Longrightarrow> x_2 (?H2 x_1 y_0) (?H3 (?H4 y_0) x_1)) \<Longrightarrow> ?H5 x_1 x_1 ?H6 x_2
|
[
"List.list.Cons",
"List.drop",
"Nat.Suc",
"List.length",
"List.list.Nil",
"List.nth",
"ListInterleaving.interleaves_dom",
"ListInterleaving.Interleaves_syntax"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_nil_all
|
lemma Interleaves_nil_all:
"xs \<cong> {[], xs, P} = (\<forall>n < length xs. \<not> P (xs ! n) (drop (Suc n) xs))"
|
?xs \<cong> {[], ?xs, ?P} = (\<forall>n<length ?xs. \<not> ?P (?xs ! n) (drop (Suc n) ?xs))
|
?H1 x_1 ?H2 x_1 x_2 = (\<forall>y_0<?H3 x_1. \<not> x_2 (?H4 x_1 y_0) (?H5 (?H6 y_0) x_1))
|
[
"List.nth",
"Nat.Suc",
"List.list.Nil",
"ListInterleaving.Interleaves_syntax",
"List.length",
"List.drop"
] |
[
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>"
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_all_nil_1
|
lemma Interleaves_all_nil_1 [rule_format]:
"xs \<cong> {xs, [], P} \<longrightarrow> (\<forall>n < length xs. P (xs ! n) (drop (Suc n) xs))"
|
?xs \<cong> {?xs, [], ?P} \<Longrightarrow> ?n < length ?xs \<Longrightarrow> ?P (?xs ! ?n) (drop (Suc ?n) ?xs)
|
\<lbrakk>?H1 x_1 x_1 ?H2 x_2; x_3 < ?H3 x_1\<rbrakk> \<Longrightarrow> x_2 (?H4 x_1 x_3) (?H5 (?H6 x_3) x_1)
|
[
"Nat.Suc",
"List.nth",
"ListInterleaving.Interleaves_syntax",
"List.drop",
"List.length",
"List.list.map",
"List.list.Nil"
] |
[
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_all_nil
|
lemma Interleaves_all_nil:
"xs \<cong> {xs, [], P} = (\<forall>n < length xs. P (xs ! n) (drop (Suc n) xs))"
|
?xs \<cong> {?xs, [], ?P} = (\<forall>n<length ?xs. ?P (?xs ! n) (drop (Suc n) ?xs))
|
?H1 x_1 x_1 ?H2 x_2 = (\<forall>y_0<?H3 x_1. x_2 (?H4 x_1 y_0) (?H5 (?H6 y_0) x_1))
|
[
"List.nth",
"ListInterleaving.Interleaves_syntax",
"List.list.Nil",
"List.drop",
"List.length",
"Nat.Suc"
] |
[
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.interleaves_equal_fst
|
lemma interleaves_equal_fst [rule_format]:
"xs \<simeq> {ys, zs, P} \<longrightarrow> xs \<simeq> {ys', zs, P} \<longrightarrow> ys = ys'"
|
?xs \<simeq> {?ys, ?zs, ?P} \<Longrightarrow> ?xs \<simeq> {?ys', ?zs, ?P} \<Longrightarrow> ?ys = ?ys'
|
\<lbrakk>?H1 x_1 x_2 x_3 x_4; ?H1 x_1 x_5 x_3 x_4\<rbrakk> \<Longrightarrow> x_2 = x_5
|
[
"List.list.Nil",
"List.list.map",
"Fun.inj",
"ListInterleaving.interleaves_syntax"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation inj :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> bool\"\n where \"inj f \\<equiv> inj_on f UNIV\"",
"abbreviation interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\" \n (\"(_ \\<simeq> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<simeq> {ys, zs, P} \\<equiv> interleaves P xs ys zs\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.interleaves_equal_snd
|
lemma interleaves_equal_snd:
"xs \<simeq> {ys, zs, P} \<Longrightarrow> xs \<simeq> {ys, zs', P} \<Longrightarrow> zs = zs'"
|
?xs \<simeq> {?ys, ?zs, ?P} \<Longrightarrow> ?xs \<simeq> {?ys, ?zs', ?P} \<Longrightarrow> ?zs = ?zs'
|
\<lbrakk>?H1 x_1 x_2 x_3 x_4; ?H1 x_1 x_2 x_5 x_4\<rbrakk> \<Longrightarrow> x_3 = x_5
|
[
"ListInterleaving.Interleaves_syntax",
"List.list.Cons",
"ListInterleaving.interleaves_syntax",
"List.list.map"
] |
[
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\" \n (\"(_ \\<simeq> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<simeq> {ys, zs, P} \\<equiv> interleaves P xs ys zs\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_equal_fst
|
lemma Interleaves_equal_fst:
"xs \<cong> {ys, zs, P} \<Longrightarrow> xs \<cong> {ys', zs, P} \<Longrightarrow> ys = ys'"
|
?xs \<cong> {?ys, ?zs, ?P} \<Longrightarrow> ?xs \<cong> {?ys', ?zs, ?P} \<Longrightarrow> ?ys = ?ys'
|
\<lbrakk>?H1 x_1 x_2 x_3 x_4; ?H1 x_1 x_5 x_3 x_4\<rbrakk> \<Longrightarrow> x_2 = x_5
|
[
"ListInterleaving.Interleaves_syntax",
"List.nth"
] |
[
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>"
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_equal_snd
|
lemma Interleaves_equal_snd:
"xs \<cong> {ys, zs, P} \<Longrightarrow> xs \<cong> {ys, zs', P} \<Longrightarrow> zs = zs'"
|
?xs \<cong> {?ys, ?zs, ?P} \<Longrightarrow> ?xs \<cong> {?ys, ?zs', ?P} \<Longrightarrow> ?zs = ?zs'
|
\<lbrakk>?H1 x_1 x_2 x_3 x_4; ?H1 x_1 x_2 x_5 x_4\<rbrakk> \<Longrightarrow> x_3 = x_5
|
[
"ListInterleaving.Interleaves_syntax",
"List.length",
"List.list.Nil",
"Nat.Suc",
"List.filter"
] |
[
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"primrec filter:: \"('a \\<Rightarrow> bool) \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n\"filter P [] = []\" |\n\"filter P (x # xs) = (if P x then x # filter P xs else filter P xs)\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.interleaves_swap
|
lemma interleaves_swap:
"xs \<simeq> {ys, zs, P} = xs \<simeq> {zs, ys, \<lambda>w ws. \<not> P w ws}"
|
?xs \<simeq> {?ys, ?zs, ?P} = ?xs \<simeq> {?zs, ?ys, \<lambda>w ws. \<not> ?P w ws}
|
?H1 x_1 x_2 x_3 x_4 = ?H1 x_1 x_3 x_2 (\<lambda>y_0 y_1. \<not> x_4 y_0 y_1)
|
[
"ListInterleaving.interleaves_syntax",
"List.nth",
"Groups.plus_class.plus",
"ListInterleaving.Interleaves_dom"
] |
[
"abbreviation interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\" \n (\"(_ \\<simeq> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<simeq> {ys, zs, P} \\<equiv> interleaves P xs ys zs\"",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)"
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_swap
|
lemma Interleaves_swap:
"xs \<cong> {ys, zs, P} = xs \<cong> {zs, ys, \<lambda>w ws. \<not> P w ws}"
|
?xs \<cong> {?ys, ?zs, ?P} = ?xs \<cong> {?zs, ?ys, \<lambda>w ws. \<not> ?P w ws}
|
?H1 x_1 x_2 x_3 x_4 = ?H1 x_1 x_3 x_2 (\<lambda>y_0 y_1. \<not> x_4 y_0 y_1)
|
[
"List.list.map",
"ListInterleaving.Interleaves_syntax",
"ListInterleaving.Interleaves_dom"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.interleaves_length
|
lemma interleaves_length [rule_format]:
"xs \<simeq> {ys, zs, P} \<longrightarrow> length xs = length ys + length zs"
|
?xs \<simeq> {?ys, ?zs, ?P} \<Longrightarrow> length ?xs = length ?ys + length ?zs
|
?H1 x_1 x_2 x_3 x_4 \<Longrightarrow> ?H2 x_1 = ?H3 (?H2 x_2) (?H2 x_3)
|
[
"List.length",
"ListInterleaving.Interleaves_dom",
"ListInterleaving.interleaves_syntax",
"Groups.plus_class.plus",
"Nat.Suc"
] |
[
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"abbreviation interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\" \n (\"(_ \\<simeq> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<simeq> {ys, zs, P} \\<equiv> interleaves P xs ys zs\"",
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_length
|
lemma Interleaves_length:
"xs \<cong> {ys, zs, P} \<Longrightarrow> length xs = length ys + length zs"
|
?xs \<cong> {?ys, ?zs, ?P} \<Longrightarrow> length ?xs = length ?ys + length ?zs
|
?H1 x_1 x_2 x_3 x_4 \<Longrightarrow> ?H2 x_1 = ?H3 (?H2 x_2) (?H2 x_3)
|
[
"Groups.plus_class.plus",
"List.length",
"Hilbert_Choice.inv",
"ListInterleaving.Interleaves_syntax",
"List.list.map"
] |
[
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"abbreviation inv :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> ('b \\<Rightarrow> 'a)\" where\n\"inv \\<equiv> inv_into UNIV\"",
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_suffix_one_aux
|
lemma Interleaves_suffix_one_aux:
assumes A: "P x []"
shows "\<not> xs @ [x] \<cong> {[], zs, P}"
|
?P ?x [] \<Longrightarrow> \<not> ?xs @ [?x] \<cong> {[], ?zs, ?P}
|
x_1 x_2 ?H1 \<Longrightarrow> \<not> ?H2 (?H3 x_3 (?H4 x_2 ?H1)) ?H1 x_4 x_1
|
[
"List.list.Nil",
"List.list.Cons",
"ListInterleaving.Interleaves_syntax",
"List.append"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.interleaves_nil
|
lemma interleaves_nil:
"[] \<simeq> {ys, zs, P} \<Longrightarrow> ys = [] \<and> zs = []"
|
[] \<simeq> {?ys, ?zs, ?P} \<Longrightarrow> ?ys = [] \<and> ?zs = []
|
?H1 ?H2 x_1 x_2 x_3 \<Longrightarrow> x_1 = ?H2 \<and> x_2 = ?H2
|
[
"List.list.Cons",
"List.append",
"List.list.Nil",
"ListInterleaving.interleaves_syntax",
"List.drop",
"Groups.plus_class.plus"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\" \n (\"(_ \\<simeq> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<simeq> {ys, zs, P} \\<equiv> interleaves P xs ys zs\"",
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)"
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_nil
|
lemma Interleaves_nil:
"[] \<cong> {ys, zs, P} \<Longrightarrow> ys = [] \<and> zs = []"
|
[] \<cong> {?ys, ?zs, ?P} \<Longrightarrow> ?ys = [] \<and> ?zs = []
|
?H1 ?H2 x_1 x_2 x_3 \<Longrightarrow> x_1 = ?H2 \<and> x_2 = ?H2
|
[
"ListInterleaving.interleaves_dom",
"ListInterleaving.Interleaves_syntax",
"List.drop",
"List.list.Cons",
"List.list.Nil"
] |
[
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_interleaves
|
lemma Interleaves_interleaves [rule_format]:
"xs \<cong> {ys, zs, P} \<longrightarrow> xs \<simeq> {ys, zs, P}"
|
?xs \<cong> {?ys, ?zs, ?P} \<Longrightarrow> ?xs \<simeq> {?ys, ?zs, ?P}
|
?H1 x_1 x_2 x_3 x_4 \<Longrightarrow> ?H2 x_1 x_2 x_3 x_4
|
[
"List.list.Cons",
"List.list.map",
"ListInterleaving.Interleaves_syntax",
"Fun.inj",
"ListInterleaving.interleaves_syntax"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"abbreviation inj :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> bool\"\n where \"inj f \\<equiv> inj_on f UNIV\"",
"abbreviation interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\" \n (\"(_ \\<simeq> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<simeq> {ys, zs, P} \\<equiv> interleaves P xs ys zs\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.interleaves_equal_all_nil
|
lemma interleaves_equal_all_nil:
"xs \<simeq> {ys, [], P} \<Longrightarrow> xs = ys"
|
?xs \<simeq> {?ys, [], ?P} \<Longrightarrow> ?xs = ?ys
|
?H1 x_1 x_2 ?H2 x_3 \<Longrightarrow> x_1 = x_2
|
[
"ListInterleaving.interleaves_syntax",
"List.list.Nil"
] |
[
"abbreviation interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\" \n (\"(_ \\<simeq> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<simeq> {ys, zs, P} \\<equiv> interleaves P xs ys zs\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.interleaves_equal_nil_all
|
lemma interleaves_equal_nil_all:
"xs \<simeq> {[], zs, P} \<Longrightarrow> xs = zs"
|
?xs \<simeq> {[], ?zs, ?P} \<Longrightarrow> ?xs = ?zs
|
?H1 x_1 ?H2 x_2 x_3 \<Longrightarrow> x_1 = x_2
|
[
"List.list.Nil",
"ListInterleaving.interleaves_dom",
"Fun.inj",
"List.nth",
"ListInterleaving.interleaves_syntax",
"ListInterleaving.Interleaves_syntax"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation inj :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> bool\"\n where \"inj f \\<equiv> inj_on f UNIV\"",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"abbreviation interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\" \n (\"(_ \\<simeq> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<simeq> {ys, zs, P} \\<equiv> interleaves P xs ys zs\"",
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_equal_all_nil
|
lemma Interleaves_equal_all_nil:
"xs \<cong> {ys, [], P} \<Longrightarrow> xs = ys"
|
?xs \<cong> {?ys, [], ?P} \<Longrightarrow> ?xs = ?ys
|
?H1 x_1 x_2 ?H2 x_3 \<Longrightarrow> x_1 = x_2
|
[
"List.list.map",
"ListInterleaving.Interleaves_syntax",
"List.list.Nil",
"List.list.Cons"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.Interleaves_equal_nil_all
|
lemma Interleaves_equal_nil_all:
"xs \<cong> {[], zs, P} \<Longrightarrow> xs = zs"
|
?xs \<cong> {[], ?zs, ?P} \<Longrightarrow> ?xs = ?zs
|
?H1 x_1 ?H2 x_2 x_3 \<Longrightarrow> x_1 = x_2
|
[
"List.drop",
"List.filter",
"ListInterleaving.Interleaves_dom",
"List.list.Nil",
"ListInterleaving.interleaves_dom",
"ListInterleaving.Interleaves_syntax"
] |
[
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"primrec filter:: \"('a \\<Rightarrow> bool) \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n\"filter P [] = []\" |\n\"filter P (x # xs) = (if P x then x # filter P xs else filter P xs)\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation Interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\"\n (\"(_ \\<cong> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<cong> {ys, zs, P} \\<equiv> Interleaves P xs ys zs\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.interleaves_all_nil
|
lemma interleaves_all_nil:
"xs \<simeq> {xs, [], P}"
|
?xs \<simeq> {?xs, [], ?P}
|
?H1 x_1 x_1 ?H2 x_2
|
[
"Groups.plus_class.plus",
"List.list.Nil",
"ListInterleaving.interleaves_syntax"
] |
[
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\" \n (\"(_ \\<simeq> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<simeq> {ys, zs, P} \\<equiv> interleaves P xs ys zs\""
] |
List_Interleaving/ListInterleaving
|
ListInterleaving.interleaves_nil_all
|
lemma interleaves_nil_all:
"xs \<simeq> {[], xs, P}"
|
?xs \<simeq> {[], ?xs, ?P}
|
?H1 x_1 ?H2 x_1 x_2
|
[
"ListInterleaving.interleaves_syntax",
"List.list.Nil"
] |
[
"abbreviation interleaves_syntax ::\n \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> ('a \\<Rightarrow> 'a list \\<Rightarrow> bool) \\<Rightarrow> bool\" \n (\"(_ \\<simeq> {_, _, _})\" [60, 60, 60] 51)\n where \"xs \\<simeq> {ys, zs, P} \\<equiv> interleaves P xs ys zs\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.unfold_zipWithStep
|
lemma unfold_zipWithStep:
fixes f :: "'a \<rightarrow> 'b \<rightarrow> 'c"
fixes ha :: "'s \<rightarrow> ('a, 's) Step"
fixes hb :: "'t \<rightarrow> ('b, 't) Step"
defines h_def: "h \<equiv> zipWithStep\<cdot>f\<cdot>ha\<cdot>hb"
shows
"(\<forall>sa sb. sa \<noteq> \<bottom> \<longrightarrow> sb \<noteq> \<bottom> \<longrightarrow>
unfold\<cdot>h\<cdot>(sa :!: sb :!: Nothing) =
zipWithL\<cdot>f\<cdot>(unfold\<cdot>ha\<cdot>sa)\<cdot>(unfold\<cdot>hb\<cdot>sb)) \<and>
(\<forall>sa sb a. sa \<noteq> \<bottom> \<longrightarrow> sb \<noteq> \<bottom> \<longrightarrow>
unfold\<cdot>h\<cdot>(sa :!: sb :!: Just\<cdot>(L\<cdot>a)) =
zipWithL\<cdot>f\<cdot>(LCons\<cdot>a\<cdot>(unfold\<cdot>ha\<cdot>sa))\<cdot>(unfold\<cdot>hb\<cdot>sb))"
|
(\<forall>sa sb. sa \<noteq> \<bottom> \<longrightarrow> sb \<noteq> \<bottom> \<longrightarrow> unfold\<cdot>(zipWithStep\<cdot>?f\<cdot>?ha\<cdot>?hb)\<cdot>(sa :!: sb :!: Nothing) = zipWithL\<cdot>?f\<cdot>(unfold\<cdot>?ha\<cdot>sa)\<cdot>(unfold\<cdot>?hb\<cdot>sb)) \<and> (\<forall>sa sb a. sa \<noteq> \<bottom> \<longrightarrow> sb \<noteq> \<bottom> \<longrightarrow> unfold\<cdot>(zipWithStep\<cdot>?f\<cdot>?ha\<cdot>?hb)\<cdot>(sa :!: sb :!: Just\<cdot>(L\<cdot>a)) = zipWithL\<cdot>?f\<cdot>(LCons\<cdot>a\<cdot>(unfold\<cdot>?ha\<cdot>sa))\<cdot>(unfold\<cdot>?hb\<cdot>sb))
|
(\<forall>y_0 y_1. y_0 \<noteq> ?H1 \<longrightarrow> y_1 \<noteq> ?H2 \<longrightarrow> ?H3 (?H4 ?H5 (?H6 (?H7 (?H8 ?H9 x_1) x_2) x_3)) (?H10 (?H11 ?H12 (?H13 (?H14 ?H15 y_0) y_1)) ?H16) = ?H17 (?H18 (?H19 ?H20 x_1) (?H21 (?H22 ?H23 x_2) y_0)) (?H24 (?H25 ?H26 x_3) y_1)) \<and> (\<forall>y_2 y_3 y_4. y_2 \<noteq> ?H1 \<longrightarrow> y_3 \<noteq> ?H2 \<longrightarrow> ?H3 (?H4 ?H5 (?H6 (?H7 (?H8 ?H9 x_1) x_2) x_3)) (?H10 (?H11 ?H12 (?H13 (?H14 ?H15 y_2) y_3)) (?H27 ?H28 (?H29 ?H30 y_4))) = ?H17 (?H18 (?H19 ?H20 x_1) (?H31 (?H32 ?H33 y_4) (?H21 (?H22 ?H23 x_2) y_2))) (?H24 (?H25 ?H26 x_3) y_3))
|
[
"StreamFusion.Maybe.Nothing",
"StreamFusion.Maybe.Just",
"LazyList.LList.LCons",
"StreamFusion.L.L",
"LazyList.zipWithL",
"StreamFusion.Both.Both",
"StreamFusion.zipWithStep",
"StreamFusion.enumFromToS",
"StreamFusion.L_defl",
"Pcpo.pcpo_class.bottom",
"Stream.unfold",
"Cfun.cfun.Rep_cfun"
] |
[
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.zipWithS_fix_ind
|
lemma zipWithS_fix_ind:
assumes x: "x = fix\<cdot>f" and y: "y = fix\<cdot>g"
assumes adm_P: "adm (\<lambda>x. P (fst x) (snd x))"
assumes adm_Q: "adm (\<lambda>x. Q (fst x) (snd x))"
assumes P_0: "\<And>b. P \<bottom> b" and P_Suc: "\<And>a b. P a b \<Longrightarrow> Q a b \<Longrightarrow> P (f\<cdot>a) b"
assumes Q_0: "\<And>a. Q a \<bottom>" and Q_Suc: "\<And>a b. P a b \<Longrightarrow> Q a b \<Longrightarrow> Q a (g\<cdot>b)"
shows "P x y \<and> Q x y"
|
?x = fix\<cdot>?f \<Longrightarrow> ?y = fix\<cdot>?g \<Longrightarrow> adm (\<lambda>x. ?P (fst x) (snd x)) \<Longrightarrow> adm (\<lambda>x. ?Q (fst x) (snd x)) \<Longrightarrow> (\<And>b. ?P \<bottom> b) \<Longrightarrow> (\<And>a b. ?P a b \<Longrightarrow> ?Q a b \<Longrightarrow> ?P (?f\<cdot>a) b) \<Longrightarrow> (\<And>a. ?Q a \<bottom>) \<Longrightarrow> (\<And>a b. ?P a b \<Longrightarrow> ?Q a b \<Longrightarrow> ?Q a (?g\<cdot>b)) \<Longrightarrow> ?P ?x ?y \<and> ?Q ?x ?y
|
\<lbrakk>x_1 = ?H1 ?H2 x_2; x_3 = ?H3 ?H4 x_4; ?H5 (\<lambda>y_0. x_5 (?H6 y_0) (?H7 y_0)); ?H5 (\<lambda>y_1. x_6 (?H6 y_1) (?H7 y_1)); \<And>y_2. x_5 ?H8 y_2; \<And>y_3 y_4. \<lbrakk>x_5 y_3 y_4; x_6 y_3 y_4\<rbrakk> \<Longrightarrow> x_5 (?H9 x_2 y_3) y_4; \<And>y_5. x_6 y_5 ?H10; \<And>y_6 y_7. \<lbrakk>x_5 y_6 y_7; x_6 y_6 y_7\<rbrakk> \<Longrightarrow> x_6 y_6 (?H11 x_4 y_7)\<rbrakk> \<Longrightarrow> x_5 x_1 x_3 \<and> x_6 x_1 x_3
|
[
"Pcpo.pcpo_class.bottom",
"Adm.adm",
"StreamFusion.Maybe.match_Just",
"Product_Type.prod.fst",
"Stream.Step.Yield",
"Fix.fix",
"Cfun.cfun.Rep_cfun",
"Tr.cifte_syn",
"Product_Type.prod.snd",
"StreamFusion.Either.Left"
] |
[
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin",
"definition adm :: \"('a::cpo \\<Rightarrow> bool) \\<Rightarrow> bool\"\n where \"adm P \\<longleftrightarrow> (\\<forall>Y. chain Y \\<longrightarrow> (\\<forall>i. P (Y i)) \\<longrightarrow> P (\\<Squnion>i. Y i))\"",
"definition \"prod = {f. \\<exists>a b. f = Pair_Rep (a::'a) (b::'b)}\"",
"definition \"fix\" :: \"('a \\<rightarrow> 'a) \\<rightarrow> 'a\"\n where \"fix = (\\<Lambda> F. \\<Squnion>i. iterate i\\<cdot>F\\<cdot>\\<bottom>)\"",
"abbreviation cifte_syn :: \"[tr, 'c, 'c] \\<Rightarrow> 'c\" (\"(If (_)/ then (_)/ else (_))\" [0, 0, 60] 60)\n where \"If b then e1 else e2 \\<equiv> tr_case\\<cdot>e1\\<cdot>e2\\<cdot>b\"",
"definition \"prod = {f. \\<exists>a b. f = Pair_Rep (a::'a) (b::'b)}\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.unfold_concatMapStep
|
lemma unfold_concatMapStep:
fixes ha :: "'s \<rightarrow> ('a, 's) Step"
fixes f :: "'a \<rightarrow> ('b, 't) Stream"
defines h_def: "h \<equiv> concatMapStep\<cdot>f\<cdot>ha"
defines f'_def: "f' \<equiv> unstream oo f"
shows
"(\<forall>sa. sa \<noteq> \<bottom> \<longrightarrow>
unfold\<cdot>h\<cdot>(sa :!: Nothing) = concatMapL\<cdot>f'\<cdot>(unfold\<cdot>ha\<cdot>sa)) \<and>
(\<forall>sa hb sb. sa \<noteq> \<bottom> \<longrightarrow> sb \<noteq> \<bottom> \<longrightarrow>
unfold\<cdot>h\<cdot>(sa :!: Just\<cdot>(Stream\<cdot>hb\<cdot>sb)) =
appendL\<cdot>(unfold\<cdot>hb\<cdot>sb)\<cdot>(concatMapL\<cdot>f'\<cdot>(unfold\<cdot>ha\<cdot>sa)))"
|
(\<forall>sa. sa \<noteq> \<bottom> \<longrightarrow> unfold\<cdot>(concatMapStep\<cdot>?f\<cdot>?ha)\<cdot>(sa :!: Nothing) = concatMapL\<cdot>(unstream oo ?f)\<cdot>(unfold\<cdot>?ha\<cdot>sa)) \<and> (\<forall>sa hb sb. sa \<noteq> \<bottom> \<longrightarrow> sb \<noteq> \<bottom> \<longrightarrow> unfold\<cdot>(concatMapStep\<cdot>?f\<cdot>?ha)\<cdot>(sa :!: Just\<cdot>(Stream\<cdot>hb\<cdot>sb)) = appendL\<cdot>(unfold\<cdot>hb\<cdot>sb)\<cdot> (concatMapL\<cdot>(unstream oo ?f)\<cdot>(unfold\<cdot>?ha\<cdot>sa)))
|
(\<forall>y_0. y_0 \<noteq> ?H1 \<longrightarrow> ?H2 (?H3 ?H4 (?H5 (?H6 ?H7 x_1) x_2)) (?H8 (?H9 ?H10 y_0) ?H11) = ?H12 (?H13 ?H14 (?H15 ?H16 x_1)) (?H17 (?H18 ?H19 x_2) y_0)) \<and> (\<forall>y_1 y_2 y_3. y_1 \<noteq> ?H1 \<longrightarrow> y_3 \<noteq> ?H20 \<longrightarrow> ?H2 (?H3 ?H4 (?H5 (?H6 ?H7 x_1) x_2)) (?H8 (?H9 ?H10 y_1) (?H21 ?H22 (?H23 (?H24 ?H25 y_2) y_3))) = ?H26 (?H27 ?H28 (?H29 (?H30 ?H31 y_2) y_3)) (?H12 (?H13 ?H14 (?H15 ?H16 x_1)) (?H17 (?H18 ?H19 x_2) y_1)))
|
[
"StreamFusion.Maybe.Just",
"Cfun.cfun.Rep_cfun",
"StreamFusion.Maybe.Nothing",
"Stream.unstream",
"StreamFusion.zipWithS",
"Stream.unfold",
"StreamFusion.Both.Both",
"StreamFusion.concatMapStep",
"Pcpo.pcpo_class.bottom",
"LazyList.concatMapL",
"LazyList.appendL",
"Cfun.cfcomp_syn",
"Stream.Stream.Stream"
] |
[
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin",
"abbreviation cfcomp_syn :: \"['b \\<rightarrow> 'c, 'a \\<rightarrow> 'b] \\<Rightarrow> 'a \\<rightarrow> 'c\" (infixr \"oo\" 100)\n where \"f oo g == cfcomp\\<cdot>f\\<cdot>g\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.unfold_appendStep
|
lemma unfold_appendStep:
fixes ha :: "'s \<rightarrow> ('a, 's) Step"
fixes hb :: "'t \<rightarrow> ('a, 't) Step"
assumes sb0 [simp]: "sb0 \<noteq> \<bottom>"
shows
"(\<forall>sa. sa \<noteq> \<bottom> \<longrightarrow> unfold\<cdot>(appendStep\<cdot>ha\<cdot>hb\<cdot>sb0)\<cdot>(Left\<cdot>sa) =
appendL\<cdot>(unfold\<cdot>ha\<cdot>sa)\<cdot>(unfold\<cdot>hb\<cdot>sb0)) \<and>
(\<forall>sb. sb \<noteq> \<bottom> \<longrightarrow> unfold\<cdot>(appendStep\<cdot>ha\<cdot>hb\<cdot>sb0)\<cdot>(Right\<cdot>sb) =
unfold\<cdot>hb\<cdot>sb)"
|
?sb0.0 \<noteq> \<bottom> \<Longrightarrow> (\<forall>sa. sa \<noteq> \<bottom> \<longrightarrow> unfold\<cdot>(appendStep\<cdot>?ha\<cdot>?hb\<cdot>?sb0.0)\<cdot>(Either.Left\<cdot>sa) = appendL\<cdot>(unfold\<cdot>?ha\<cdot>sa)\<cdot>(unfold\<cdot>?hb\<cdot>?sb0.0)) \<and> (\<forall>sb. sb \<noteq> \<bottom> \<longrightarrow> unfold\<cdot>(appendStep\<cdot>?ha\<cdot>?hb\<cdot>?sb0.0)\<cdot>(Either.Right\<cdot>sb) = unfold\<cdot>?hb\<cdot>sb)
|
x_1 \<noteq> ?H1 \<Longrightarrow> (\<forall>y_0. y_0 \<noteq> ?H2 \<longrightarrow> ?H3 (?H4 ?H5 (?H6 (?H7 (?H8 ?H9 x_2) x_3) x_1)) (?H10 ?H11 y_0) = ?H12 (?H13 ?H14 (?H15 (?H16 ?H17 x_2) y_0)) (?H18 (?H19 ?H20 x_3) x_1)) \<and> (\<forall>y_1. y_1 \<noteq> ?H1 \<longrightarrow> ?H3 (?H4 ?H5 (?H6 (?H7 (?H8 ?H9 x_2) x_3) x_1)) (?H21 ?H22 y_1) = ?H18 (?H19 ?H20 x_3) y_1)
|
[
"LazyList.appendL",
"Cfun.cfun.Rep_cfun",
"Product_Type.prod.snd",
"StreamFusion.Either.Left",
"Pcpo.pcpo_class.bottom",
"StreamFusion.appendStep",
"Stream.unfold",
"StreamFusion.L.match_L",
"StreamFusion.Either.Right",
"StreamFusion.Either.is_Right",
"StreamFusion.Maybe_finite"
] |
[
"definition \"prod = {f. \\<exists>a b. f = Pair_Rep (a::'a) (b::'b)}\"",
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.zipWithS_fix_ind_lemma
|
lemma zipWithS_fix_ind_lemma:
fixes P Q :: "nat \<Rightarrow> nat \<Rightarrow> bool"
assumes P_0: "\<And>j. P 0 j" and P_Suc: "\<And>i j. P i j \<Longrightarrow> Q i j \<Longrightarrow> P (Suc i) j"
assumes Q_0: "\<And>i. Q i 0" and Q_Suc: "\<And>i j. P i j \<Longrightarrow> Q i j \<Longrightarrow> Q i (Suc j)"
shows "P i j \<and> Q i j"
|
(\<And>j. ?P 0 j) \<Longrightarrow> (\<And>i j. ?P i j \<Longrightarrow> ?Q i j \<Longrightarrow> ?P (Suc i) j) \<Longrightarrow> (\<And>i. ?Q i 0) \<Longrightarrow> (\<And>i j. ?P i j \<Longrightarrow> ?Q i j \<Longrightarrow> ?Q i (Suc j)) \<Longrightarrow> ?P ?i ?j \<and> ?Q ?i ?j
|
\<lbrakk>\<And>y_0. x_1 ?H1 y_0; \<And>y_1 y_2. \<lbrakk>x_1 y_1 y_2; x_2 y_1 y_2\<rbrakk> \<Longrightarrow> x_1 (?H2 y_1) y_2; \<And>y_3. x_2 y_3 ?H1; \<And>y_4 y_5. \<lbrakk>x_1 y_4 y_5; x_2 y_4 y_5\<rbrakk> \<Longrightarrow> x_2 y_4 (?H2 y_5)\<rbrakk> \<Longrightarrow> x_1 x_3 x_4 \<and> x_2 x_3 x_4
|
[
"Groups.zero_class.zero",
"Nat.Suc"
] |
[
"class zero =\n fixes zero :: 'a (\"0\")",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.foldrS_Stream
| null |
?s \<noteq> \<bottom> \<Longrightarrow> foldrS\<cdot>?f\<cdot>?z\<cdot>(Stream\<cdot>?h\<cdot>?s) = (case ?h\<cdot>?s of Done \<Rightarrow> ?z | Skip\<cdot>s' \<Rightarrow> foldrS\<cdot>?f\<cdot>?z\<cdot>(Stream\<cdot>?h\<cdot>s') | Yield\<cdot>x\<cdot>s' \<Rightarrow> ?f\<cdot>x\<cdot>(foldrS\<cdot>?f\<cdot>?z\<cdot>(Stream\<cdot>?h\<cdot>s')))
|
x_1 \<noteq> ?H1 \<Longrightarrow> ?H2 (?H3 (?H4 ?H5 x_2) x_3) (?H6 (?H7 ?H8 x_4) x_1) = ?H9 (?H10 (?H11 (?H12 ?H13 x_3) (?H14 (\<lambda>y_0. ?H2 (?H3 (?H4 ?H5 x_2) x_3) (?H6 (?H7 ?H8 x_4) y_0)))) (?H15 (\<lambda>y_1. ?H14 (\<lambda>y_2. ?H16 (?H17 x_2 y_1) (?H2 (?H3 (?H4 ?H5 x_2) x_3) (?H6 (?H7 ?H8 x_4) y_2)))))) (?H18 x_4 x_1)
|
[
"Cfun.cfun.Abs_cfun",
"Stream.Step.Step_case",
"Cfun.cfun.Rep_cfun",
"StreamFusion.foldrS",
"Pcpo.pcpo_class.bottom",
"Stream.Stream.Stream"
] |
[
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
Stream-Fusion/StreamFusion
|
StreamFusion.concatMapS_cong
|
lemma concatMapS_cong:
fixes f :: "'a \<Rightarrow> ('b, 's) Stream"
fixes g :: "'a \<Rightarrow> ('b, 't) Stream"
fixes a :: "('a, 'u) Stream"
fixes b :: "('a, 'v) Stream"
shows "(\<And>x. f x \<approx> g x) \<Longrightarrow> a \<approx> b \<Longrightarrow> cont f \<Longrightarrow> cont g \<Longrightarrow>
concatMapS\<cdot>(\<Lambda> x. f x)\<cdot>a \<approx> concatMapS\<cdot>(\<Lambda> x. g x)\<cdot>b"
|
(\<And>x. ?f x \<approx> ?g x) \<Longrightarrow> ?a \<approx> ?b \<Longrightarrow> cont ?f \<Longrightarrow> cont ?g \<Longrightarrow> concatMapS\<cdot>(\<Lambda> x. ?f x)\<cdot>?a \<approx> concatMapS\<cdot>(\<Lambda> x. ?g x)\<cdot>?b
|
\<lbrakk>\<And>y_0. ?H1 (x_1 y_0) (x_2 y_0); ?H2 x_3 x_4; ?H3 x_1; ?H4 x_2\<rbrakk> \<Longrightarrow> ?H5 (?H6 (?H7 ?H8 (?H9 x_1)) x_3) (?H10 (?H11 ?H12 (?H13 x_2)) x_4)
|
[
"StreamFusion.Switch.Switch_case",
"Cfun.cfun.Rep_cfun",
"LazyList.enumFromToL",
"Cont.cont",
"Cfun.cfun.Abs_cfun",
"StreamFusion.concatMapS",
"StreamFusion.filterS",
"Stream.bisimilar",
"StreamFusion.Switch_bisim"
] |
[
"definition cont :: \"('a::cpo \\<Rightarrow> 'b::cpo) \\<Rightarrow> bool\"\n where \"cont f = (\\<forall>Y. chain Y \\<longrightarrow> range (\\<lambda>i. f (Y i)) <<| f (\\<Squnion>i. Y i))\"",
"definition\n bisimilar :: \"('a, 's) Stream \\<Rightarrow> ('a, 't) Stream \\<Rightarrow> bool\" (infix \"\\<approx>\" 50)\nwhere\n \"a \\<approx> b \\<longleftrightarrow> unstream\\<cdot>a = unstream\\<cdot>b \\<and> a \\<noteq> \\<bottom> \\<and> b \\<noteq> \\<bottom>\""
] |
Stream-Fusion/StreamFusion
|
StreamFusion.unstream_zipWithS
|
lemma unstream_zipWithS:
"a \<noteq> \<bottom> \<Longrightarrow> b \<noteq> \<bottom> \<Longrightarrow>
unstream\<cdot>(zipWithS\<cdot>f\<cdot>a\<cdot>b) = zipWithL\<cdot>f\<cdot>(unstream\<cdot>a)\<cdot>(unstream\<cdot>b)"
|
?a \<noteq> \<bottom> \<Longrightarrow> ?b \<noteq> \<bottom> \<Longrightarrow> unstream\<cdot>(zipWithS\<cdot>?f\<cdot>?a\<cdot>?b) = zipWithL\<cdot>?f\<cdot>(unstream\<cdot>?a)\<cdot>(unstream\<cdot>?b)
|
\<lbrakk>x_1 \<noteq> ?H1; x_2 \<noteq> ?H2\<rbrakk> \<Longrightarrow> ?H3 ?H4 (?H5 (?H6 (?H7 ?H8 x_3) x_1) x_2) = ?H9 (?H10 (?H11 ?H12 x_3) (?H13 ?H14 x_1)) (?H15 ?H16 x_2)
|
[
"Stream.unstream",
"StreamFusion.Both_rep",
"Cfun.cfun.Rep_cfun",
"Fix.fix",
"Product_Type.prod.snd",
"LazyList.zipWithL",
"Pcpo.pcpo_class.bottom",
"StreamFusion.zipWithS"
] |
[
"definition \"fix\" :: \"('a \\<rightarrow> 'a) \\<rightarrow> 'a\"\n where \"fix = (\\<Lambda> F. \\<Squnion>i. iterate i\\<cdot>F\\<cdot>\\<bottom>)\"",
"definition \"prod = {f. \\<exists>a b. f = Pair_Rep (a::'a) (b::'b)}\"",
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin"
] |
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