theory_file stringlengths 5 95 | lemma_name stringlengths 5 255 | lemma_command stringlengths 17 12k ⌀ | lemma_object stringlengths 5 38.6k | template stringlengths 7 34.2k | symbols listlengths 0 85 | types listlengths 0 85 | defs listlengths 0 40 | output_key stringclasses 1
value | input stringlengths 20 32.4k | output stringlengths 22 38.7k |
|---|---|---|---|---|---|---|---|---|---|---|
Pi_Calculus/Strong_Late_Sim_Pres | Strong_Late_Sim_Pres.resPres | lemma resPres:
fixes P :: pi
and Q :: pi
and Rel :: "(pi \<times> pi) set"
and x :: name
and Rel' :: "(pi \<times> pi) set"
assumes PSimQ: "P \<leadsto>[Rel] Q"
and ResRel: "\<And>(P::pi) (Q::pi) (x::name). (P, Q) \<in> Rel \<Longrightarrow> (<\<nu>x>P, <\<nu>x>Q) \<in> Rel'"
and ... | ?P \<leadsto>[ ?Rel] ?Q \<Longrightarrow> (\<And>P Q x. (P, Q) \<in> ?Rel \<Longrightarrow> (<\<nu>x>P, <\<nu>x>Q) \<in> ?Rel') \<Longrightarrow> ?Rel \<subseteq> ?Rel' \<Longrightarrow> eqvt ?Rel \<Longrightarrow> eqvt ?Rel' \<Longrightarrow> <\<nu> ?x> ?P \<leadsto>[ ?Rel'] <\<nu> ?x> ?Q | \<lbrakk> ?H1 x_1 x_2 x_3; \<And>y_0 y_1 y_2. (y_0, y_1) \<in> x_2 \<Longrightarrow> (?H2 y_2 y_0, ?H2 y_2 y_1) \<in> x_4; ?H3 x_2 x_4; ?H4 x_2; ?H4 x_4\<rbrakk> \<Longrightarrow> ?H1 (?H2 x_5 x_1) x_4 (?H2 x_5 x_3) | [
"Rel.eqvt",
"Set.subset_eq",
"Agent.pi.Res",
"Strong_Late_Sim.simulation"
] | [
"('a \\<times> 'a) set \\<Rightarrow> bool",
"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool",
"name \\<Rightarrow> pi \\<Rightarrow> pi",
"pi \\<Rightarrow> (pi \\<times> pi) set \\<Rightarrow> pi \\<Rightarrow> bool"
] | [
"definition eqvt :: \"(('a::pt_name) \\<times> ('a::pt_name)) set \\<Rightarrow> bool\"\n where \"eqvt Rel \\<equiv> (\\<forall>x (perm::name prm). x \\<in> Rel \\<longrightarrow> perm \\<bullet> x \\<in> Rel)\"",
"abbreviation subset_eq :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool\"\n where \"subset_e... | lemma_object | ###symbols
Rel.eqvt :::: ('a \<times> 'a) set \<Rightarrow> bool
Set.subset_eq :::: 'a set \<Rightarrow> 'a set \<Rightarrow> bool
Agent.pi.Res :::: name \<Rightarrow> pi \<Rightarrow> pi
Strong_Late_Sim.simulation :::: pi \<Rightarrow> (pi \<times> pi) set \<Rightarrow> pi \<Rightarrow> bool
###defs
definition eqvt... |
###output
?P \<leadsto>[ ?Rel] ?Q \<Longrightarrow> (\<And>P Q x. (P, Q) \<in> ?Rel \<Longrightarrow> (<\<nu>x>P, <\<nu>x>Q) \<in> ?Rel') \<Longrightarrow> ?Rel \<subseteq> ?Rel' \<Longrightarrow> eqvt ?Rel \<Longrightarrow> eqvt ?Rel' \<Longrightarrow> <\<nu> ?x> ?P \<leadsto>[ ?Rel'] <\<nu> ?x> ?Q###end |
Partial_Order_Reduction/Basics/Stuttering | Stuttering.stutter_selection_stutter_sampler | lemma stutter_selection_stutter_sampler[intro]:
assumes "linfinite w" "stutter_selection s w"
shows "stutter_sampler (nth_least_ext s) (lnth w)" | linfinite ?w \<Longrightarrow> stutter_selection ?s ?w \<Longrightarrow> stutter_sampler (nth_least_ext ?s) ((?!) ?w) | \<lbrakk> ?H1 x_1; ?H2 x_2 x_1\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_2) (?H5 x_1) | [
"Coinductive_List.lnth",
"Stuttering.nth_least_ext",
"Samplers.stutter_sampler",
"Stuttering.stutter_selection",
"Coinductive_List_Extensions.linfinite"
] | [
"'a llist \\<Rightarrow> nat \\<Rightarrow> 'a",
"nat set \\<Rightarrow> nat \\<Rightarrow> nat",
"(nat \\<Rightarrow> nat) \\<Rightarrow> (nat \\<Rightarrow> 'a) \\<Rightarrow> bool",
"nat set \\<Rightarrow> 'a llist \\<Rightarrow> bool",
"'a llist \\<Rightarrow> bool"
] | [
"primrec lnth :: \"'a llist \\<Rightarrow> nat \\<Rightarrow> 'a\"\nwhere\n \"lnth xs 0 = (case xs of LNil \\<Rightarrow> undefined (0 :: nat) | LCons x xs' \\<Rightarrow> x)\"\n| \"lnth xs (Suc n) = (case xs of LNil \\<Rightarrow> undefined (Suc n) | LCons x xs' \\<Rightarrow> lnth xs' n)\"",
"definition stutte... | lemma_object | ###symbols
Coinductive_List.lnth :::: 'a llist \<Rightarrow> nat \<Rightarrow> 'a
Stuttering.nth_least_ext :::: nat set \<Rightarrow> nat \<Rightarrow> nat
Samplers.stutter_sampler :::: (nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> bool
Stuttering.stutter_selection :::: nat set \<Rightar... |
###output
linfinite ?w \<Longrightarrow> stutter_selection ?s ?w \<Longrightarrow> stutter_sampler (nth_least_ext ?s) ((?!) ?w)###end |
Iptables_Semantics/Semantics_Ternary/Primitive_Normalization | Primitive_Normalization.compress_normalize_primitive_monad(1) | lemma compress_normalize_primitive_monad:
assumes "\<And>m m' f. f \<in> set fs \<Longrightarrow> normalized_nnf_match m \<Longrightarrow> f m = Some m' \<Longrightarrow> matches \<gamma> m' a p \<longleftrightarrow> matches \<gamma> m a p"
and "\<And>m m' f. f \<in> set fs \<Longrightarrow> normalized... | (\<And>m m' f. f \<in> set ?fs \<Longrightarrow> normalized_nnf_match m \<Longrightarrow> f m = Some m' \<Longrightarrow> matches ?\<gamma> m' ?a ?p = matches ?\<gamma> m ?a ?p) \<Longrightarrow> (\<And>m m' f. f \<in> set ?fs \<Longrightarrow> normalized_nnf_match m \<Longrightarrow> f m = Some m' \<Longrightarrow> no... | \<lbrakk>\<And>y_0 y_1 y_2. \<lbrakk>y_2 \<in> ?H1 x_1; ?H2 y_0; y_2 y_0 = ?H3 y_1\<rbrakk> \<Longrightarrow> ?H4 x_2 y_1 x_3 x_4 = ?H4 x_2 y_0 x_3 x_4; \<And>y_3 y_4 y_5. \<lbrakk>y_5 \<in> ?H1 x_1; ?H2 y_3; y_5 y_3 = ?H3 y_4\<rbrakk> \<Longrightarrow> ?H2 y_4; ?H2 x_5; ?H5 x_1 x_5 = ?H3 x_6\<rbrakk> \<Longrightarrow>... | [
"Primitive_Normalization.compress_normalize_primitive_monad",
"Matching_Ternary.matches",
"Option.option.Some",
"Normalized_Matches.normalized_nnf_match",
"List.list.set"
] | [
"('a match_expr \\<Rightarrow> 'a match_expr option) list \\<Rightarrow> 'a match_expr \\<Rightarrow> 'a match_expr option",
"('a \\<Rightarrow> 'b \\<Rightarrow> ternaryvalue) \\<times> (action \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> 'a match_expr \\<Rightarrow> action \\<Rightarrow> ... | [
"datatype 'a option =\n None\n | Some (the: '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 [] = []\""
] | lemma_object | ###symbols
Primitive_Normalization.compress_normalize_primitive_monad :::: ('a match_expr \<Rightarrow> 'a match_expr option) list \<Rightarrow> 'a match_expr \<Rightarrow> 'a match_expr option
Matching_Ternary.matches :::: ('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rig... |
###output
(\<And>m m' f. f \<in> set ?fs \<Longrightarrow> normalized_nnf_match m \<Longrightarrow> f m = Some m' \<Longrightarrow> matches ?\<gamma> m' ?a ?p = matches ?\<gamma> m ?a ?p) \<Longrightarrow> (\<And>m m' f. f \<in> set ?fs \<Longrightarrow> normalized_nnf_match m \<Longrightarrow> f m = Some m' \<Longrig... |
Design_Theory/Multisets_Extras | Multisets_Extras.partition_on_msetD2 | lemma partition_on_msetD2: "partition_on_mset A P \<Longrightarrow> {#} \<notin># P" | partition_on_mset ?A ?P \<Longrightarrow> {#} \<notin># ?P | ?H1 x_1 x_2 \<Longrightarrow> ?H2 ?H3 x_2 | [
"Multiset.empty_mset",
"Multiset.not_member_mset",
"Multisets_Extras.partition_on_mset"
] | [
"'a multiset",
"'a \\<Rightarrow> 'a multiset \\<Rightarrow> bool",
"'a multiset \\<Rightarrow> 'a multiset multiset \\<Rightarrow> bool"
] | [
"abbreviation empty_mset :: \\<open>'a multiset\\<close> (\\<open>{#}\\<close>)\n where \\<open>empty_mset \\<equiv> 0\\<close>",
"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>",
... | lemma_object | ###symbols
Multiset.empty_mset :::: 'a multiset
Multiset.not_member_mset :::: 'a \<Rightarrow> 'a multiset \<Rightarrow> bool
Multisets_Extras.partition_on_mset :::: 'a multiset \<Rightarrow> 'a multiset multiset \<Rightarrow> bool
###defs
abbreviation empty_mset :: \<open>'a multiset\<close> (\<open>{#}\<close>)
w... |
###output
partition_on_mset ?A ?P \<Longrightarrow> {#} \<notin># ?P###end |
Matrices_for_ODEs/MTX_Preliminaries | MTX_Preliminaries.finite_image_of_finite2 | lemma finite_image_of_finite2:
fixes f :: "'a::finite \<Rightarrow> 'b::finite \<Rightarrow> 'c"
shows "finite {f x y |x y. P x y}" | finite { ?f x y |x y. ?P x y} | ?H1 (?H2 (\<lambda>y_0. \<exists>y_1 y_2. y_0 = x_1 y_1 y_2 \<and> x_2 y_1 y_2)) | [
"Set.Collect",
"Finite_Set.finite"
] | [
"('a \\<Rightarrow> bool) \\<Rightarrow> 'a set",
"'a set \\<Rightarrow> bool"
] | [
"class finite =\n assumes finite_UNIV: \"finite (UNIV :: 'a set)\"\nbegin"
] | lemma_object | ###symbols
Set.Collect :::: ('a \<Rightarrow> bool) \<Rightarrow> 'a set
Finite_Set.finite :::: 'a set \<Rightarrow> bool
###defs
class finite =
assumes finite_UNIV: "finite (UNIV :: 'a set)"
begin |
###output
finite { ?f x y |x y. ?P x y}###end |
CZH_Foundations/czh_digraphs/CZH_DG_Small_TDGHM | CZH_DG_Small_TDGHM.small_these_tiny_tdghms | lemma small_these_tiny_tdghms[simp]:
"small {\<NN>. \<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>}" | small {\<NN>. \<NN> : ?\<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y ?\<GG> : ?\<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub> ?\<alpha>\<^esub> ?\<BB>} | ?H1 (?H2 (?H3 x_1 x_2 x_3 x_4 x_5)) | [
"CZH_DG_Small_TDGHM.is_tiny_tdghm",
"Set.Collect",
"ZFC_in_HOL.small"
] | [
"V \\<Rightarrow> V \\<Rightarrow> V \\<Rightarrow> V \\<Rightarrow> V \\<Rightarrow> V \\<Rightarrow> bool",
"('a \\<Rightarrow> bool) \\<Rightarrow> 'a set",
"'a set \\<Rightarrow> bool"
] | [
"definition small :: \"'a set \\<Rightarrow> bool\" \n where \"small X \\<equiv> \\<exists>V_of :: 'a \\<Rightarrow> V. inj_on V_of X \\<and> V_of ` X \\<in> range elts\""
] | lemma_object | ###symbols
CZH_DG_Small_TDGHM.is_tiny_tdghm :::: V \<Rightarrow> V \<Rightarrow> V \<Rightarrow> V \<Rightarrow> V \<Rightarrow> V \<Rightarrow> bool
Set.Collect :::: ('a \<Rightarrow> bool) \<Rightarrow> 'a set
ZFC_in_HOL.small :::: 'a set \<Rightarrow> bool
###defs
definition small :: "'a set \<Rightarrow> bool"
... |
###output
small {\<NN>. \<NN> : ?\<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y ?\<GG> : ?\<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub> ?\<alpha>\<^esub> ?\<BB>}###end |
LTL/LTL | LTL.Alm_all_GF_F | lemma Alm_all_GF_F:
"\<forall>\<^sub>\<infinity>i. suffix i w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>) \<longleftrightarrow> suffix i w \<Turnstile>\<^sub>n F\<^sub>n \<psi>" | \<forall>\<^sub>\<infinity>i. suffix i ?w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n ?\<psi>) = suffix i ?w \<Turnstile>\<^sub>n F\<^sub>n ?\<psi> | ?H1 (\<lambda>y_0. ?H2 (?H3 y_0 x_1) (?H4 (?H5 x_2)) = ?H2 (?H3 y_0 x_1) (?H5 x_2)) | [
"LTL.finally\\<^sub>n",
"LTL.globally\\<^sub>n",
"Omega_Words_Fun.suffix",
"LTL.semantics_ltln",
"Filter.Alm_all"
] | [
"'a ltln \\<Rightarrow> 'a ltln",
"'a ltln \\<Rightarrow> 'a ltln",
"nat \\<Rightarrow> (nat \\<Rightarrow> 'a) \\<Rightarrow> nat \\<Rightarrow> 'a",
"(nat \\<Rightarrow> 'a set) \\<Rightarrow> 'a ltln \\<Rightarrow> bool",
"('a \\<Rightarrow> bool) \\<Rightarrow> bool"
] | [
"abbreviation finally\\<^sub>n :: \"'a ltln \\<Rightarrow> 'a ltln\" (\"F\\<^sub>n _\" [88] 87)\nwhere\n \"F\\<^sub>n \\<phi> \\<equiv> True\\<^sub>n U\\<^sub>n \\<phi>\"",
"abbreviation globally\\<^sub>n :: \"'a ltln \\<Rightarrow> 'a ltln\" (\"G\\<^sub>n _\" [88] 87)\nwhere\n \"G\\<^sub>n \\<phi> \\<equiv> Fa... | lemma_object | ###symbols
LTL.finally\<^sub>n :::: 'a ltln \<Rightarrow> 'a ltln
LTL.globally\<^sub>n :::: 'a ltln \<Rightarrow> 'a ltln
Omega_Words_Fun.suffix :::: nat \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a
LTL.semantics_ltln :::: (nat \<Rightarrow> 'a set) \<Rightarrow> 'a ltln \<Rightarrow> bool... |
###output
\<forall>\<^sub>\<infinity>i. suffix i ?w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n ?\<psi>) = suffix i ?w \<Turnstile>\<^sub>n F\<^sub>n ?\<psi>###end |
Knights_Tour/KnightsTour | KnightsTour.knights_circuit_rev | null | knights_circuit ?b ?ps \<Longrightarrow> knights_circuit ?b (rev ?ps) | ?H1 x_1 x_2 \<Longrightarrow> ?H1 x_1 (?H2 x_2) | [
"List.rev",
"KnightsTour.knights_circuit"
] | [
"'a list \\<Rightarrow> 'a list",
"(int \\<times> int) set \\<Rightarrow> (int \\<times> int) list \\<Rightarrow> bool"
] | [
"primrec rev :: \"'a list \\<Rightarrow> 'a list\" where\n\"rev [] = []\" |\n\"rev (x # xs) = rev xs @ [x]\"",
"definition \"knights_circuit b ps \\<equiv> (knights_path b ps \\<and> valid_step (last ps) (hd ps))\""
] | lemma_object | ###symbols
List.rev :::: 'a list \<Rightarrow> 'a list
KnightsTour.knights_circuit :::: (int \<times> int) set \<Rightarrow> (int \<times> int) list \<Rightarrow> bool
###defs
primrec rev :: "'a list \<Rightarrow> 'a list" where
"rev [] = []" |
"rev (x # xs) = rev xs @ [x]"
definition "knights_circuit b ps \<equiv> (... |
###output
knights_circuit ?b ?ps \<Longrightarrow> knights_circuit ?b (rev ?ps)###end |
JinjaThreads/Common/TypeRel | TypeRel.sees_method_idemp | lemma sees_method_idemp:
"P \<turnstile> C sees M:Ts\<rightarrow>T=m in D \<Longrightarrow> P \<turnstile> D sees M:Ts\<rightarrow>T=m in D" | ?P \<turnstile> ?C sees ?M: ?Ts\<rightarrow> ?T = ?m in ?D \<Longrightarrow> ?P \<turnstile> ?D sees ?M: ?Ts\<rightarrow> ?T = ?m in ?D | ?H1 x_1 x_2 x_3 x_4 x_5 x_6 x_7 \<Longrightarrow> ?H1 x_1 x_7 x_3 x_4 x_5 x_6 x_7 | [
"TypeRel.Method"
] | [
"'a prog \\<Rightarrow> String.literal \\<Rightarrow> String.literal \\<Rightarrow> ty list \\<Rightarrow> ty \\<Rightarrow> 'a option \\<Rightarrow> String.literal \\<Rightarrow> bool"
] | [] | lemma_object | ###symbols
TypeRel.Method :::: 'a prog \<Rightarrow> String.literal \<Rightarrow> String.literal \<Rightarrow> ty list \<Rightarrow> ty \<Rightarrow> 'a option \<Rightarrow> String.literal \<Rightarrow> bool
###defs
|
###output
?P \<turnstile> ?C sees ?M: ?Ts\<rightarrow> ?T = ?m in ?D \<Longrightarrow> ?P \<turnstile> ?D sees ?M: ?Ts\<rightarrow> ?T = ?m in ?D###end |
Probabilistic_Noninterference/Trace_Based | Trace_Based.field_abs_le_zero_epsilon | lemma field_abs_le_zero_epsilon:
fixes x :: "'a::{linordered_field}"
assumes epsilon: "\<And>e. 0 < e \<Longrightarrow> \<bar>x\<bar> \<le> e"
shows "\<bar>x\<bar> = 0" | (\<And>e. (0:: ?'a) < e \<Longrightarrow> \<bar> ?x\<bar> \<le> e) \<Longrightarrow> \<bar> ?x\<bar> = (0:: ?'a) | (\<And>y_0. ?H1 < y_0 \<Longrightarrow> ?H2 x_1 \<le> y_0) \<Longrightarrow> ?H2 x_1 = ?H1 | [
"Groups.abs_class.abs",
"Groups.zero_class.zero"
] | [
"'a \\<Rightarrow> 'a",
"'a"
] | [
"class abs =\n fixes abs :: \"'a \\<Rightarrow> 'a\" (\"\\<bar>_\\<bar>\")",
"class zero =\n fixes zero :: 'a (\"0\")"
] | lemma_object | ###symbols
Groups.abs_class.abs :::: 'a \<Rightarrow> 'a
Groups.zero_class.zero :::: 'a
###defs
class abs =
fixes abs :: "'a \<Rightarrow> 'a" ("\<bar>_\<bar>")
class zero =
fixes zero :: 'a ("0") |
###output
(\<And>e. (0:: ?'a) < e \<Longrightarrow> \<bar> ?x\<bar> \<le> e) \<Longrightarrow> \<bar> ?x\<bar> = (0:: ?'a)###end |
Affine_Arithmetic/Straight_Line_Program | Straight_Line_Program.comparator_floatarith_simps(95) | null | comparator_floatarith (floatarith.Arctan ?x) (floatarith.Cos ?yf) = Gt | ?H1 (?H2 x_1) (?H3 x_2) = ?H4 | [
"Comparator.order.Gt",
"Approximation.floatarith.Cos",
"Approximation.floatarith.Arctan",
"Straight_Line_Program.comparator_floatarith"
] | [
"order",
"floatarith \\<Rightarrow> floatarith",
"floatarith \\<Rightarrow> floatarith",
"floatarith \\<Rightarrow> floatarith \\<Rightarrow> order"
] | [
"datatype floatarith\n = Add floatarith floatarith\n | Minus floatarith\n | Mult floatarith floatarith\n | Inverse floatarith\n | Cos floatarith\n | Arctan floatarith\n | Abs floatarith\n | Max floatarith floatarith\n | Min floatarith floatarith\n | Pi\n | Sqrt floatarith\n | Exp floatarith\n | Powr fl... | lemma_object | ###symbols
Comparator.order.Gt :::: order
Approximation.floatarith.Cos :::: floatarith \<Rightarrow> floatarith
Approximation.floatarith.Arctan :::: floatarith \<Rightarrow> floatarith
Straight_Line_Program.comparator_floatarith :::: floatarith \<Rightarrow> floatarith \<Rightarrow> order
###defs
datatype floatarith... |
###output
comparator_floatarith (floatarith.Arctan ?x) (floatarith.Cos ?yf) = Gt###end |
PCF/OpSem | OpSem.ca_lrI(1) | lemma ca_lrI [intro, simp]:
"closed P \<Longrightarrow> \<bottom> \<triangleleft> P"
"\<lbrakk> P \<Down> DBtt; closed P \<rbrakk> \<Longrightarrow> ValTT \<triangleleft> P"
"\<lbrakk> P \<Down> DBff; closed P \<rbrakk> \<Longrightarrow> ValFF \<triangleleft> P"
"\<lbrakk> P \<Down> DBNum n; closed P \<rbrakk> ... | closed ?P \<Longrightarrow> \<bottom> \<triangleleft> ?P | ?H1 x_1 \<Longrightarrow> ?H2 ?H3 x_1 | [
"Pcpo.pcpo_class.bottom",
"OpSem.ca_lr_syn",
"OpSem.closed"
] | [
"'a",
"ValD \\<Rightarrow> db \\<Rightarrow> bool",
"db \\<Rightarrow> bool"
] | [
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin",
"definition ca_lr_syn :: \"ValD \\<Rightarrow> db \\<Rightarrow> bool\" (\"_ \\<triangleleft> _\" [80,80] 80) where\n \"d \\<triangleleft> P \\<equiv> (d, P) \\<in> { (x, unProg Y) |x Y. (x, Y) \\<in> unsynlr ca.delta... | lemma_object | ###symbols
Pcpo.pcpo_class.bottom :::: 'a
OpSem.ca_lr_syn :::: ValD \<Rightarrow> db \<Rightarrow> bool
OpSem.closed :::: db \<Rightarrow> bool
###defs
class pcpo = cpo +
assumes least: "\<exists>x. \<forall>y. x \<sqsubseteq> y"
begin
definition ca_lr_syn :: "ValD \<Rightarrow> db \<Rightarrow> bool" ("_ \<triang... |
###output
closed ?P \<Longrightarrow> \<bottom> \<triangleleft> ?P###end |
Separation_Logic_Imperative_HOL/Examples/Array_Map_Impl | Array_Map_Impl.iam_lookup_abs1 | lemma iam_lookup_abs1: "k<length l \<Longrightarrow> iam_of_list l k = l!k" | ?k < length ?l \<Longrightarrow> iam_of_list ?l ?k = ?l ! ?k | x_1 < ?H1 x_2 \<Longrightarrow> ?H2 x_2 x_1 = ?H3 x_2 x_1 | [
"List.nth",
"Array_Map_Impl.iam_of_list",
"List.length"
] | [
"'a list \\<Rightarrow> nat \\<Rightarrow> 'a",
"'a option list \\<Rightarrow> nat \\<Rightarrow> 'a option",
"'a list \\<Rightarrow> nat"
] | [
"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>... | lemma_object | ###symbols
List.nth :::: 'a list \<Rightarrow> nat \<Rightarrow> 'a
Array_Map_Impl.iam_of_list :::: 'a option list \<Rightarrow> nat \<Rightarrow> 'a option
List.length :::: 'a list \<Rightarrow> nat
###defs
primrec (nonexhaustive) nth :: "'a list => nat => 'a" (infixl "!" 100) where
nth_Cons: "(x # xs) ! n = (case n... |
###output
?k < length ?l \<Longrightarrow> iam_of_list ?l ?k = ?l ! ?k###end |
Completeness/Tree | Tree.branch0 | lemma branch0: "branch subs Gamma f ==> f 0 = Gamma" | branch ?subs ?Gamma ?f \<Longrightarrow> ?f 0 = ?Gamma | ?H1 x_1 x_2 x_3 \<Longrightarrow> x_3 ?H2 = x_2 | [
"Groups.zero_class.zero",
"Tree.branch"
] | [
"'a",
"('a \\<Rightarrow> 'a set) \\<Rightarrow> 'a \\<Rightarrow> (nat \\<Rightarrow> 'a) \\<Rightarrow> bool"
] | [
"class zero =\n fixes zero :: 'a (\"0\")",
"definition\n branch :: \"['a => 'a set,'a,nat => 'a] => bool\" where\n \"branch subs Gamma f \\<longleftrightarrow> f 0 = Gamma\n & (!n . terminal subs (f n) --> f (Suc n) = f n)\n & (!n . ~ terminal subs (f n) --> f (Suc n) \\<in> subs (f n))\""
] | lemma_object | ###symbols
Groups.zero_class.zero :::: 'a
Tree.branch :::: ('a \<Rightarrow> 'a set) \<Rightarrow> 'a \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> bool
###defs
class zero =
fixes zero :: 'a ("0")
definition
branch :: "['a => 'a set,'a,nat => 'a] => bool" where
"branch subs Gamma f \<longleftrightarrow> f... |
###output
branch ?subs ?Gamma ?f \<Longrightarrow> ?f 0 = ?Gamma###end |
Integration/Integral | Integral.nnfis_integral(2) | lemma nnfis_integral:
assumes nn: "a \<in> nnfis f M" and ms: "measure_space M"
shows "integrable f M" and "\<integral> f \<partial> M = a" | ?a \<in> nnfis ?f ?M \<Longrightarrow> measure_space ?M \<Longrightarrow> \<integral> ?f \<partial> ?M = ?a | \<lbrakk>x_1 \<in> ?H1 x_2 x_3; ?H2 x_3\<rbrakk> \<Longrightarrow> ?H3 x_2 x_3 = x_1 | [
"Integral.integral",
"Measure.measure_space",
"Integral.nnfis"
] | [
"('a \\<Rightarrow> real) \\<Rightarrow> 'a set set \\<times> ('a set \\<Rightarrow> real) \\<Rightarrow> real",
"'a set set \\<times> ('a set \\<Rightarrow> real) \\<Rightarrow> bool",
"('a \\<Rightarrow> real) \\<Rightarrow> 'a set set \\<times> ('a set \\<Rightarrow> real) \\<Rightarrow> real set"
] | [
"definition\n integral:: \"('a \\<Rightarrow> real) \\<Rightarrow> ('a set set * ('a set \\<Rightarrow> real)) \\<Rightarrow> real\" (\"\\<integral> _ \\<partial>_\"(*<*)[60,61] 110(*>*)) where\n \"integrable f M \\<Longrightarrow> \\<integral> f \\<partial>M = (THE i. i \\<in> nnfis (pp f) M) -\n (THE j. j \\<i... | lemma_object | ###symbols
Integral.integral :::: ('a \<Rightarrow> real) \<Rightarrow> 'a set set \<times> ('a set \<Rightarrow> real) \<Rightarrow> real
Measure.measure_space :::: 'a set set \<times> ('a set \<Rightarrow> real) \<Rightarrow> bool
Integral.nnfis :::: ('a \<Rightarrow> real) \<Rightarrow> 'a set set \<times> ('a set... |
###output
?a \<in> nnfis ?f ?M \<Longrightarrow> measure_space ?M \<Longrightarrow> \<integral> ?f \<partial> ?M = ?a###end |
JinjaThreads/Common/TypeRel | TypeRel.sees_field_hoaux_PPiiii_PiooiI | null | sees_field_hoaux ?x (?xa, ?xb, ?xc, ?xd) \<Longrightarrow> pred.eval (sees_field_hoaux_PPiiii_Piooi ?x (?xa, ?xd)) (?xb, ?xc) | ?H1 x_1 (x_2, x_3, x_4, x_5) \<Longrightarrow> ?H2 (?H3 x_1 (x_2, x_5)) (x_3, x_4) | [
"TypeRel.sees_field_hoaux_PPiiii_Piooi",
"Predicate.pred.eval",
"TypeRel.sees_field_hoaux"
] | [
"(String.literal \\<times> String.literal) \\<times> ty \\<times> fmod \\<Rightarrow> String.literal \\<times> fmod \\<Rightarrow> (String.literal \\<times> ty) Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(String.literal \\<times> String.literal) \\<times> ty \\<times> fm... | [
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] | lemma_object | ###symbols
TypeRel.sees_field_hoaux_PPiiii_Piooi :::: (String.literal \<times> String.literal) \<times> ty \<times> fmod \<Rightarrow> String.literal \<times> fmod \<Rightarrow> (String.literal \<times> ty) Predicate.pred
Predicate.pred.eval :::: 'a Predicate.pred \<Rightarrow> 'a \<Rightarrow> bool
TypeRel.se... |
###output
sees_field_hoaux ?x (?xa, ?xb, ?xc, ?xd) \<Longrightarrow> pred.eval (sees_field_hoaux_PPiiii_Piooi ?x (?xa, ?xd)) (?xb, ?xc)###end |
Prime_Number_Theorem/Prime_Number_Theorem | Prime_Number_Theorem_Library.abs_conv_abscissa_mult_const_right | null | abs_conv_abscissa (?f * fds_const ?c) \<le> abs_conv_abscissa ?f | ?H1 (?H2 x_1 (?H3 x_2)) \<le> ?H1 x_1 | [
"Dirichlet_Series.fds_const",
"Groups.times_class.times",
"Dirichlet_Series_Analysis.abs_conv_abscissa"
] | [
"'a \\<Rightarrow> 'a fds",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a fds \\<Rightarrow> ereal"
] | [
"definition fds_const :: \"'a :: zero \\<Rightarrow> 'a fds\" where\n \"fds_const c = fds (\\<lambda>n. if n = 1 then c else 0)\"",
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"*\" 70)",
"definition abs_conv_abscissa\n :: \"'a :: {nat_power,banach,real_normed_field,... | lemma_object | ###symbols
Dirichlet_Series.fds_const :::: 'a \<Rightarrow> 'a fds
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Dirichlet_Series_Analysis.abs_conv_abscissa :::: 'a fds \<Rightarrow> ereal
###defs
definition fds_const :: "'a :: zero \<Rightarrow> 'a fds" where
"fds_const c = fds (\<lambda>n. if... |
###output
abs_conv_abscissa (?f * fds_const ?c) \<le> abs_conv_abscissa ?f###end |
Winding_Number_Eval/Cauchy_Index_Theorem | Cauchy_Index_Theorem.poly_has_sgnx_values(1) | lemma poly_has_sgnx_values:
assumes "p\<noteq>0"
shows
"(poly p has_sgnx 1) (at_left a) \<or> (poly p has_sgnx - 1) (at_left a)"
"(poly p has_sgnx 1) (at_right a) \<or> (poly p has_sgnx - 1) (at_right a)"
"(poly p has_sgnx 1) at_top \<or> (poly p has_sgnx - 1) at_top"
"(poly p has_sgnx 1) at_bot \<... | ?p \<noteq> 0 \<Longrightarrow> (poly ?p has_sgnx 1) (at_left ?a) \<or> (poly ?p has_sgnx - 1) (at_left ?a) | x_1 \<noteq> ?H1 \<Longrightarrow> ?H2 (?H3 x_1) ?H4 (?H5 x_2) \<or> ?H2 (?H3 x_1) (?H6 ?H4) (?H5 x_2) | [
"Groups.uminus_class.uminus",
"Topological_Spaces.order_topology_class.at_left",
"Groups.one_class.one",
"Polynomial.poly",
"Cauchy_Index_Theorem.has_sgnx",
"Groups.zero_class.zero"
] | [
"'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a filter",
"'a",
"'a poly \\<Rightarrow> 'a \\<Rightarrow> 'a",
"(real \\<Rightarrow> real) \\<Rightarrow> real \\<Rightarrow> real filter \\<Rightarrow> bool",
"'a"
] | [
"class uminus =\n fixes uminus :: \"'a \\<Rightarrow> 'a\" (\"- _\" [81] 80)",
"class order_topology = order + \"open\" +\n assumes open_generated_order: \"open = generate_topology (range (\\<lambda>a. {..< a}) \\<union> range (\\<lambda>a. {a <..}))\"\nbegin",
"class one =\n fixes one :: 'a (\"1\")",
"... | lemma_object | ###symbols
Groups.uminus_class.uminus :::: 'a \<Rightarrow> 'a
Topological_Spaces.order_topology_class.at_left :::: 'a \<Rightarrow> 'a filter
Groups.one_class.one :::: 'a
Polynomial.poly :::: 'a poly \<Rightarrow> 'a \<Rightarrow> 'a
Cauchy_Index_Theorem.has_sgnx :::: (real \<Rightarrow> real) \<Rightarrow> real \... |
###output
?p \<noteq> 0 \<Longrightarrow> (poly ?p has_sgnx 1) (at_left ?a) \<or> (poly ?p has_sgnx - 1) (at_left ?a)###end |
CHERI-C_Memory_Model/Preliminary_Library | Preliminary_Library.word_integer_eq | lemma word_integer_eq:
"word_of_integer (integer_of_word w) = w" | word_of_integer (integer_of_word ?w) = ?w | ?H1 (?H2 x_1) = x_1 | [
"Preliminary_Library.integer_of_word",
"Preliminary_Library.word_of_integer"
] | [
"'a word \\<Rightarrow> integer",
"integer \\<Rightarrow> 'a word"
] | [
"definition integer_of_word :: \"'a::len word \\<Rightarrow> integer\"\n where\n \"integer_of_word x \\<equiv> integer_of_int (uint x)\"",
"definition word_of_integer :: \"integer \\<Rightarrow> 'a::len word\"\n where\n \"word_of_integer x \\<equiv> word_of_int (int_of_integer x)\""
] | lemma_object | ###symbols
Preliminary_Library.integer_of_word :::: 'a word \<Rightarrow> integer
Preliminary_Library.word_of_integer :::: integer \<Rightarrow> 'a word
###defs
definition integer_of_word :: "'a::len word \<Rightarrow> integer"
where
"integer_of_word x \<equiv> integer_of_int (uint x)"
definition word_of_integer ... |
###output
word_of_integer (integer_of_word ?w) = ?w###end |
Prim_Dijkstra_Simple/Undirected_Graph | Undirected_Graph.join_cycle_free | lemma join_cycle_free:
assumes CYCF: "cycle_free g\<^sub>1" "cycle_free g\<^sub>2"
assumes DJ: "nodes g\<^sub>1 \<inter> nodes g\<^sub>2 = {}"
assumes IN_NODES: "u\<in>nodes g\<^sub>1" "v\<in>nodes g\<^sub>2"
shows "cycle_free (ins_edge (u,v) (graph_join g\<^sub>1 g\<^sub>2))" (is "cycle_free ?g'") | cycle_free ?g\<^sub>1 \<Longrightarrow> cycle_free ?g\<^sub>2 \<Longrightarrow> nodes ?g\<^sub>1 \<inter> nodes ?g\<^sub>2 = {} \<Longrightarrow> ?u \<in> nodes ?g\<^sub>1 \<Longrightarrow> ?v \<in> nodes ?g\<^sub>2 \<Longrightarrow> cycle_free (ins_edge (?u, ?v) (graph_join ?g\<^sub>1 ?g\<^sub>2)) | \<lbrakk> ?H1 x_1; ?H1 x_2; ?H2 (?H3 x_1) (?H3 x_2) = ?H4; x_3 \<in> ?H3 x_1; x_4 \<in> ?H3 x_2\<rbrakk> \<Longrightarrow> ?H1 (?H5 (x_3, x_4) (?H6 x_1 x_2)) | [
"Undirected_Graph.graph_join",
"Undirected_Graph.ins_edge",
"Set.empty",
"Undirected_Graph.nodes",
"Set.inter",
"Undirected_Graph.cycle_free"
] | [
"'a ugraph \\<Rightarrow> 'a ugraph \\<Rightarrow> 'a ugraph",
"'a \\<times> 'a \\<Rightarrow> 'a ugraph \\<Rightarrow> 'a ugraph",
"'a set",
"'a ugraph \\<Rightarrow> 'a set",
"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a set",
"'a ugraph \\<Rightarrow> bool"
] | [
"definition \"graph_join g\\<^sub>1 g\\<^sub>2 \\<equiv> graph (nodes g\\<^sub>1 \\<union> nodes g\\<^sub>2) (edges g\\<^sub>1 \\<union> edges g\\<^sub>2)\"",
"definition \"ins_edge e g \\<equiv> graph (nodes g) (insert e (edges g))\"",
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\"",
... | lemma_object | ###symbols
Undirected_Graph.graph_join :::: 'a ugraph \<Rightarrow> 'a ugraph \<Rightarrow> 'a ugraph
Undirected_Graph.ins_edge :::: 'a \<times> 'a \<Rightarrow> 'a ugraph \<Rightarrow> 'a ugraph
Set.empty :::: 'a set
Undirected_Graph.nodes :::: 'a ugraph \<Rightarrow> 'a set
Set.inter :::: 'a set \<Rightarrow> 'a ... |
###output
cycle_free ?g\<^sub>1 \<Longrightarrow> cycle_free ?g\<^sub>2 \<Longrightarrow> nodes ?g\<^sub>1 \<inter> nodes ?g\<^sub>2 = {} \<Longrightarrow> ?u \<in> nodes ?g\<^sub>1 \<Longrightarrow> ?v \<in> nodes ?g\<^sub>2 \<Longrightarrow> cycle_free (ins_edge (?u, ?v) (graph_join ?g\<^sub>1 ?g\<^sub>2))###end |
Simple_Clause_Learning/SCL_FOL | SCL_FOL.comp_substs_nth | null | length ?\<tau>s = length ?\<sigma>s \<Longrightarrow> ?i < length ?\<tau>s \<Longrightarrow> comp_substs ?\<tau>s ?\<sigma>s ! ?i = ?\<tau>s ! ?i \<odot> ?\<sigma>s ! ?i | \<lbrakk> ?H1 x_1 = ?H1 x_2; x_3 < ?H1 x_1\<rbrakk> \<Longrightarrow> ?H2 (?H3 x_1 x_2) x_3 = ?H4 (?H2 x_1 x_3) (?H2 x_2 x_3) | [
"SCL_FOL.comp_subst_abbrev",
"SCL_FOL.comp_substs",
"List.nth",
"List.length"
] | [
"('a \\<Rightarrow> ('b, 'a) Term.term) \\<Rightarrow> ('a \\<Rightarrow> ('b, 'a) Term.term) \\<Rightarrow> 'a \\<Rightarrow> ('b, 'a) Term.term",
"('a \\<Rightarrow> ('b, 'a) Term.term) list \\<Rightarrow> ('a \\<Rightarrow> ('b, 'a) Term.term) list \\<Rightarrow> ('a \\<Rightarrow> ('b, 'a) Term.... | [
"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>... | lemma_object | ###symbols
SCL_FOL.comp_subst_abbrev :::: ('a \<Rightarrow> ('b, 'a) Term.term) \<Rightarrow> ('a \<Rightarrow> ('b, 'a) Term.term) \<Rightarrow> 'a \<Rightarrow> ('b, 'a) Term.term
SCL_FOL.comp_substs :::: ('a \<Rightarrow> ('b, 'a) Term.term) list \<Rightarrow> ('a \<Rightarrow> ('b, 'a) Term.term) lis... |
###output
length ?\<tau>s = length ?\<sigma>s \<Longrightarrow> ?i < length ?\<tau>s \<Longrightarrow> comp_substs ?\<tau>s ?\<sigma>s ! ?i = ?\<tau>s ! ?i \<odot> ?\<sigma>s ! ?i###end |
Differential_Dynamic_Logic/Syntax | Syntax.hpsafe_Test_simps | null | hpsafe (? ?p) = fsafe ?p | ?H1 (?H2 x_1) = ?H3 x_1 | [
"Syntax.fsafe",
"Syntax.hp.Test",
"Syntax.hpsafe"
] | [
"('a, 'b, 'c) formula \\<Rightarrow> bool",
"('a, 'b, 'c) formula \\<Rightarrow> ('a, 'b, 'c) hp",
"('a, 'b, 'c) hp \\<Rightarrow> bool"
] | [
"inductive hpsafe:: \"('a, 'b, 'c) hp \\<Rightarrow> bool\"\n and fsafe:: \"('a, 'b, 'c) formula \\<Rightarrow> bool\"\nwhere\n hpsafe_Pvar:\"hpsafe (Pvar x)\"\n | hpsafe_Assign:\"dsafe e \\<Longrightarrow> hpsafe (Assign x e)\"\n | hpsafe_DiffAssign:\"dsafe e \\<Longrightarrow> hpsafe (DiffAssign x e)\"\n ... | lemma_object | ###symbols
Syntax.fsafe :::: ('a, 'b, 'c) formula \<Rightarrow> bool
Syntax.hp.Test :::: ('a, 'b, 'c) formula \<Rightarrow> ('a, 'b, 'c) hp
Syntax.hpsafe :::: ('a, 'b, 'c) hp \<Rightarrow> bool
###defs
inductive hpsafe:: "('a, 'b, 'c) hp \<Rightarrow> bool"
and fsafe:: "('a, 'b, 'c) formula \<Rightarrow> bool"... |
###output
hpsafe (? ?p) = fsafe ?p###end |
Transformer_Semantics/Kleisli_Quantaloid | Kleisli_Quantaloid.kstar_unfoldl | lemma kstar_unfoldl: "\<eta> \<squnion> f \<circ>\<^sub>K kstar f \<le> kstar f" | \<eta> \<squnion> ?f \<circ>\<^sub>K kstar ?f \<le> kstar ?f | ?H1 ?H2 (?H3 x_1 (?H4 x_1)) \<le> ?H4 x_1 | [
"Kleisli_Quantaloid.kstar",
"Powerset_Monad.kcomp",
"Powerset_Monad.eta",
"Lattices.sup_class.sup"
] | [
"('a \\<Rightarrow> 'a set) \\<Rightarrow> 'a \\<Rightarrow> 'a set",
"('a \\<Rightarrow> 'b set) \\<Rightarrow> ('b \\<Rightarrow> 'c set) \\<Rightarrow> 'a \\<Rightarrow> 'c set",
"'a \\<Rightarrow> 'a set",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a"
] | [
"definition \"kstar f = (\\<Squnion>i. kpower f i)\"",
"definition kcomp :: \"('a \\<Rightarrow> 'b set) \\<Rightarrow> ('b \\<Rightarrow> 'c set) \\<Rightarrow> ('a \\<Rightarrow> 'c set)\" (infixl \"\\<circ>\\<^sub>K\" 75) where\n \"f \\<circ>\\<^sub>K g = \\<mu> \\<circ> \\<P> g \\<circ> f\"",
"abbreviatio... | lemma_object | ###symbols
Kleisli_Quantaloid.kstar :::: ('a \<Rightarrow> 'a set) \<Rightarrow> 'a \<Rightarrow> 'a set
Powerset_Monad.kcomp :::: ('a \<Rightarrow> 'b set) \<Rightarrow> ('b \<Rightarrow> 'c set) \<Rightarrow> 'a \<Rightarrow> 'c set
Powerset_Monad.eta :::: 'a \<Rightarrow> 'a set
Lattices.sup_class.sup :::: 'a \<R... |
###output
\<eta> \<squnion> ?f \<circ>\<^sub>K kstar ?f \<le> kstar ?f###end |
BNF_CC/Subtypes | Subtypes.rel_S_neg_distr_cond'_stronger | null | rel_S_neg_distr_cond' ?Co1.0 ?Co1' ?Co2.0 ?Co2' ?Contra1.0 ?Contra1' ?Contra2.0 ?Contra2' ?tytok \<Longrightarrow> rel_G_neg_distr_cond ?Co1.0 ?Co1' ?Co2.0 ?Co2' ?Contra1.0 ?Contra1' ?Contra2.0 ?Contra2' ?tytok | ?H1 x_1 x_2 x_3 x_4 x_5 x_6 x_7 x_8 x_9 \<Longrightarrow> ?H2 x_1 x_2 x_3 x_4 x_5 x_6 x_7 x_8 x_9 | [
"Axiomatised_BNF_CC.rel_G_neg_distr_cond",
"Subtypes.rel_S_neg_distr_cond'"
] | [
"('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> ('b \\<Rightarrow> 'c \\<Rightarrow> bool) \\<Rightarrow> ('d \\<Rightarrow> 'e \\<Rightarrow> bool) \\<Rightarrow> ('e \\<Rightarrow> 'f \\<Rightarrow> bool) \\<Rightarrow> ('g \\<Rightarrow> 'h \\<Rightarrow> bool) ... | [
"definition rel_G_neg_distr_cond :: \"('co1 \\<Rightarrow> 'co1' \\<Rightarrow> bool) \\<Rightarrow> ('co1' \\<Rightarrow> 'co1'' \\<Rightarrow> bool) \\<Rightarrow>\n ('co2 \\<Rightarrow> 'co2' \\<Rightarrow> bool) \\<Rightarrow> ('co2' \\<Rightarrow> 'co2'' \\<Rightarrow> bool) \\<Rightarrow>\n ('contra1 \\... | lemma_object | ###symbols
Axiomatised_BNF_CC.rel_G_neg_distr_cond :::: ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'e \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'f \<Rightarrow> bool) \<Rightarrow> ('g ... |
###output
rel_S_neg_distr_cond' ?Co1.0 ?Co1' ?Co2.0 ?Co2' ?Contra1.0 ?Contra1' ?Contra2.0 ?Contra2' ?tytok \<Longrightarrow> rel_G_neg_distr_cond ?Co1.0 ?Co1' ?Co2.0 ?Co2' ?Contra1.0 ?Contra1' ?Contra2.0 ?Contra2' ?tytok###end |
Hilbert_Space_Tensor_Product/Von_Neumann_Algebras | Von_Neumann_Algebras.commutant_weak_star_closed | lemma commutant_weak_star_closed[simp]: \<open>closedin weak_star_topology (commutant X)\<close> | closedin weak_star_topology (commutant ?X) | ?H1 ?H2 (?H3 x_1) | [
"Von_Neumann_Algebras.commutant",
"Weak_Star_Topology.weak_star_topology",
"Abstract_Topology.closedin"
] | [
"('a \\<Rightarrow>\\<^sub>C\\<^sub>L 'a) set \\<Rightarrow> ('a \\<Rightarrow>\\<^sub>C\\<^sub>L 'a) set",
"('a \\<Rightarrow>\\<^sub>C\\<^sub>L 'b) topology",
"'a topology \\<Rightarrow> 'a set \\<Rightarrow> bool"
] | [
"definition weak_star_topology :: \\<open>('a::chilbert_space \\<Rightarrow>\\<^sub>C\\<^sub>L'b::chilbert_space) topology\\<close>\n where \\<open>weak_star_topology = pullback_topology UNIV (\\<lambda>x. \\<lambda>t\\<in>Collect trace_class. trace (t o\\<^sub>C\\<^sub>L x))\n (produc... | lemma_object | ###symbols
Von_Neumann_Algebras.commutant :::: ('a \<Rightarrow>\<^sub>C\<^sub>L 'a) set \<Rightarrow> ('a \<Rightarrow>\<^sub>C\<^sub>L 'a) set
Weak_Star_Topology.weak_star_topology :::: ('a \<Rightarrow>\<^sub>C\<^sub>L 'b) topology
Abstract_Topology.closedin :::: 'a topology \<Rightarrow> 'a set \<Rightarrow> bool... |
###output
closedin weak_star_topology (commutant ?X)###end |
Sturm_Sequences/Sturm | Sturm_Method.poly_card_roots_leq_leq | null | card {x. ?a \<le> x \<and> x \<le> ?b \<and> poly ?p x = 0} = count_roots_between ?p ?a ?b + (if ?a \<le> ?b \<and> poly ?p ?a = 0 \<and> ?p \<noteq> 0 \<or> ?a = ?b \<and> ?p = 0 then 1 else 0) | ?H1 (?H2 (\<lambda>y_0. x_1 \<le> y_0 \<and> y_0 \<le> x_2 \<and> ?H3 x_3 y_0 = ?H4)) = ?H5 (?H6 x_3 x_1 x_2) (if x_1 \<le> x_2 \<and> ?H3 x_3 x_1 = ?H4 \<and> x_3 \<noteq> ?H4 \<or> x_1 = x_2 \<and> x_3 = ?H4 then ?H7 else ?H4) | [
"Groups.one_class.one",
"Sturm_Theorem.count_roots_between",
"Groups.plus_class.plus",
"Groups.zero_class.zero",
"Polynomial.poly",
"Set.Collect",
"Finite_Set.card"
] | [
"'a",
"real poly \\<Rightarrow> real \\<Rightarrow> real \\<Rightarrow> nat",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a",
"'a poly \\<Rightarrow> 'a \\<Rightarrow> 'a",
"('a \\<Rightarrow> bool) \\<Rightarrow> 'a set",
"'a set \\<Rightarrow> nat"
] | [
"class one =\n fixes one :: 'a (\"1\")",
"definition count_roots_between where\n\"count_roots_between p a b = (if a \\<le> b \\<and> p \\<noteq> 0 then\n (let ps = sturm_squarefree p\n in sign_changes ps a - sign_changes ps b) else 0)\"",
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarro... | lemma_object | ###symbols
Groups.one_class.one :::: 'a
Sturm_Theorem.count_roots_between :::: real poly \<Rightarrow> real \<Rightarrow> real \<Rightarrow> nat
Groups.plus_class.plus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.zero_class.zero :::: 'a
Polynomial.poly :::: 'a poly \<Rightarrow> 'a \<Rightarrow> 'a
Set.Collect... |
###output
card {x. ?a \<le> x \<and> x \<le> ?b \<and> poly ?p x = 0} = count_roots_between ?p ?a ?b + (if ?a \<le> ?b \<and> poly ?p ?a = 0 \<and> ?p \<noteq> 0 \<or> ?a = ?b \<and> ?p = 0 then 1 else 0)###end |
Taylor_Models/Polynomial_Expression | Polynomial_Expression.polymul_commute | lemma polymul_commute:
fixes p :: "'a::field_char_0 poly"
assumes np: "isnpolyh p n0"
and nq: "isnpolyh q n1"
shows "p *\<^sub>p q = q *\<^sub>p p" | isnpolyh ?p ?n0.0 \<Longrightarrow> isnpolyh ?q ?n1.0 \<Longrightarrow> ?p *\<^sub>p ?q = ?q *\<^sub>p ?p | \<lbrakk> ?H1 x_1 x_2; ?H1 x_3 x_4\<rbrakk> \<Longrightarrow> ?H2 x_1 x_3 = ?H2 x_3 x_1 | [
"Polynomial_Expression.polymul",
"Polynomial_Expression.isnpolyh"
] | [
"'a poly \\<Rightarrow> 'a poly \\<Rightarrow> 'a poly",
"'a poly \\<Rightarrow> nat \\<Rightarrow> bool"
] | [
"fun polymul :: \"'a::{plus,zero,times} poly \\<Rightarrow> 'a poly \\<Rightarrow> 'a poly\" (infixl \"*\\<^sub>p\" 60)\nwhere\n \"polymul (C c) (C c') = C (c * c')\"\n| \"polymul (C c) (CN c' n' p') =\n (if c = 0 then 0\\<^sub>p else CN (polymul (C c) c') n' (polymul (C c) p'))\"\n| \"polymul (CN c n p) (C c')... | lemma_object | ###symbols
Polynomial_Expression.polymul :::: 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly
Polynomial_Expression.isnpolyh :::: 'a poly \<Rightarrow> nat \<Rightarrow> bool
###defs
fun polymul :: "'a::{plus,zero,times} poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" (infixl "*\<^sub>p" 60)
where
"polymul (C ... |
###output
isnpolyh ?p ?n0.0 \<Longrightarrow> isnpolyh ?q ?n1.0 \<Longrightarrow> ?p *\<^sub>p ?q = ?q *\<^sub>p ?p###end |
Ribbon_Proofs/Ribbons_Interfaces | Ribbons_Interfaces.hcomp_assoc | lemma hcomp_assoc:
"(P \<otimes> (Q \<otimes> R)) = ((P \<otimes> Q) \<otimes> R)" | (?P \<otimes> (?Q \<otimes> ?R)) = ((?P \<otimes> ?Q) \<otimes> ?R) | ?H1 x_1 (?H1 x_2 x_3) = ?H1 (?H1 x_1 x_2) x_3 | [
"Ribbons_Interfaces.HComp_int"
] | [
"interface \\<Rightarrow> interface \\<Rightarrow> interface"
] | [] | lemma_object | ###symbols
Ribbons_Interfaces.HComp_int :::: interface \<Rightarrow> interface \<Rightarrow> interface
###defs
|
###output
(?P \<otimes> (?Q \<otimes> ?R)) = ((?P \<otimes> ?Q) \<otimes> ?R)###end |
Partial_Order_Reduction/Extensions/Set_Extensions | Set_Extensions.least_not_less | lemma least_not_less:
fixes A :: "'a :: wellorder set"
assumes "k < least A"
shows "k \<notin> A" | ?k < least ?A \<Longrightarrow> ?k \<notin> ?A | x_1 < ?H1 x_2 \<Longrightarrow> ?H2 x_1 x_2 | [
"Set.not_member",
"Set_Extensions.least"
] | [
"'a \\<Rightarrow> 'a set \\<Rightarrow> bool",
"'a set \\<Rightarrow> 'a"
] | [
"abbreviation not_member\n where \"not_member x A \\<equiv> \\<not> (x \\<in> A)\" \\<comment> \\<open>non-membership\\<close>"
] | lemma_object | ###symbols
Set.not_member :::: 'a \<Rightarrow> 'a set \<Rightarrow> bool
Set_Extensions.least :::: 'a set \<Rightarrow> 'a
###defs
abbreviation not_member
where "not_member x A \<equiv> \<not> (x \<in> A)" \<comment> \<open>non-membership\<close> |
###output
?k < least ?A \<Longrightarrow> ?k \<notin> ?A###end |
Knights_Tour/KnightsTour | KnightsTour.knights_path_board_m_n_geq_1 | lemma knights_path_board_m_n_geq_1: "knights_path (board n m) ps \<Longrightarrow> min n m \<ge> 1" | knights_path (board ?n ?m) ?ps \<Longrightarrow> 1 \<le> min ?n ?m | ?H1 (?H2 x_1 x_2) x_3 \<Longrightarrow> ?H3 \<le> ?H4 x_1 x_2 | [
"Orderings.ord_class.min",
"Groups.one_class.one",
"KnightsTour.board",
"KnightsTour.knights_path"
] | [
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a",
"nat \\<Rightarrow> nat \\<Rightarrow> (int \\<times> int) set",
"(int \\<times> int) set \\<Rightarrow> (int \\<times> int) list \\<Rightarrow> bool"
] | [
"class ord =\n fixes less_eq :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\"\n and less :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\"\nbegin",
"class one =\n fixes one :: 'a (\"1\")",
"definition board :: \"nat \\<Rightarrow> nat \\<Rightarrow> board\" where\n \"board n m = {(i,j) |i j. 1 \\<le> i \\<... | lemma_object | ###symbols
Orderings.ord_class.min :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.one_class.one :::: 'a
KnightsTour.board :::: nat \<Rightarrow> nat \<Rightarrow> (int \<times> int) set
KnightsTour.knights_path :::: (int \<times> int) set \<Rightarrow> (int \<times> int) list \<Rightarrow> bool
###defs
class ord =... |
###output
knights_path (board ?n ?m) ?ps \<Longrightarrow> 1 \<le> min ?n ?m###end |
CommCSL/AbstractCommutativity | AbstractCommutativity.exists_aligned_sequence | lemma exists_aligned_sequence:
assumes "possible_sequence sargs uargs s"
and "possible_sequence sargs' uargs' s'"
and "PRE_shared spre sargs sargs'"
and "\<And>k. PRE_unique (upre k) (uargs k) (uargs' k)"
shows "\<exists>s''. possible_sequence sargs' uargs' s'' \<and> PRE_sequence spre upre s ... | possible_sequence ?sargs ?uargs ?s \<Longrightarrow> possible_sequence ?sargs' ?uargs' ?s' \<Longrightarrow> PRE_shared ?spre ?sargs ?sargs' \<Longrightarrow> (\<And>k. PRE_unique (?upre k) (?uargs k) (?uargs' k)) \<Longrightarrow> \<exists>s''. possible_sequence ?sargs' ?uargs' s'' \<and> PRE_sequence ?spre ?upre ?s s... | \<lbrakk> ?H1 x_1 x_2 x_3; ?H1 x_4 x_5 x_6; ?H2 x_7 x_1 x_4; \<And>y_0. ?H3 (x_8 y_0) (x_2 y_0) (x_5 y_0)\<rbrakk> \<Longrightarrow> \<exists>y_1. ?H1 x_4 x_5 y_1 \<and> ?H4 x_7 x_8 x_3 y_1 | [
"AbstractCommutativity.PRE_sequence",
"CommCSL.PRE_unique",
"AbstractCommutativity.PRE_shared",
"AbstractCommutativity.possible_sequence"
] | [
"('a \\<Rightarrow> 'a \\<Rightarrow> bool) \\<Rightarrow> ('b \\<Rightarrow> 'c \\<Rightarrow> 'c \\<Rightarrow> bool) \\<Rightarrow> ('b, 'a, 'c) action list \\<Rightarrow> ('b, 'a, 'c) action list \\<Rightarrow> bool",
"('a \\<Rightarrow> 'a \\<Rightarrow> bool) \\<Rightarrow> 'a list \\<Righta... | [
"definition PRE_sequence :: \"('a \\<Rightarrow> 'a \\<Rightarrow> bool) \\<Rightarrow> ('i => 'b \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> ('i, 'a, 'b) action list \\<Rightarrow> ('i, 'a, 'b) action list \\<Rightarrow> bool\" where\n \"PRE_sequence spre upre s s' \\<longleftrightarrow> length s = leng... | lemma_object | ###symbols
AbstractCommutativity.PRE_sequence :::: ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b, 'a, 'c) action list \<Rightarrow> ('b, 'a, 'c) action list \<Rightarrow> bool
CommCSL.PRE_unique :::: ('a \<Rightarrow> ... |
###output
possible_sequence ?sargs ?uargs ?s \<Longrightarrow> possible_sequence ?sargs' ?uargs' ?s' \<Longrightarrow> PRE_shared ?spre ?sargs ?sargs' \<Longrightarrow> (\<And>k. PRE_unique (?upre k) (?uargs k) (?uargs' k)) \<Longrightarrow> \<exists>s''. possible_sequence ?sargs' ?uargs' s'' \<and> PRE_sequence ?spre... |
FSM_Tests/EquivalenceTesting/Intermediate_Implementations | Intermediate_Implementations.traces_to_check_set | lemma traces_to_check_set :
fixes M :: "('a,'b::linorder,'c::linorder) fsm"
assumes "observable M"
and "q \<in> states M"
shows "list.set (traces_to_check M q k) = {(\<gamma> @ [(x, y)]) | \<gamma> x y . length (\<gamma> @ [(x, y)]) \<le> k \<and> \<gamma> \<in> LS M q \<and> x \<in> inputs M \<and> y \<in> ... | observable ?M \<Longrightarrow> ?q \<in> FSM.states ?M \<Longrightarrow> list.set (traces_to_check ?M ?q ?k) = {\<gamma> @ [(x, y)] |\<gamma> x y. length (\<gamma> @ [(x, y)]) \<le> ?k \<and> \<gamma> \<in> LS ?M ?q \<and> x \<in> FSM.inputs ?M \<and> y \<in> FSM.outputs ?M} | \<lbrakk> ?H1 x_1; x_2 \<in> ?H2 x_1\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_2 x_3) = ?H5 (\<lambda>y_0. \<exists>y_1 y_2 y_3. y_0 = ?H6 y_1 (?H7 (y_2, y_3) ?H8) \<and> ?H9 (?H6 y_1 (?H7 (y_2, y_3) ?H8)) \<le> x_3 \<and> y_1 \<in> ?H10 x_1 x_2 \<and> y_2 \<in> ?H11 x_1 \<and> y_3 \<in> ?H12 x_1) | [
"FSM.outputs",
"FSM.inputs",
"FSM.LS",
"List.length",
"List.list.Nil",
"List.list.Cons",
"List.append",
"Set.Collect",
"Intermediate_Implementations.traces_to_check",
"List.list.set",
"FSM.states",
"FSM.observable"
] | [
"('a, 'b, 'c) fsm \\<Rightarrow> 'c set",
"('a, 'b, 'c) fsm \\<Rightarrow> 'b set",
"('a, 'b, 'c) fsm \\<Rightarrow> 'a \\<Rightarrow> ('b \\<times> 'c) list set",
"'a list \\<Rightarrow> nat",
"'a list",
"'a \\<Rightarrow> 'a list \\<Rightarrow> 'a list",
"'a list \\<Rightarrow> 'a list \\<Rightarrow> ... | [
"fun LS :: \"('state,'input,'output) fsm \\<Rightarrow> 'state \\<Rightarrow> ('input \\<times> 'output) list set\" where\n \"LS M q = { p_io p | p . path M q p }\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Con... | lemma_object | ###symbols
FSM.outputs :::: ('a, 'b, 'c) fsm \<Rightarrow> 'c set
FSM.inputs :::: ('a, 'b, 'c) fsm \<Rightarrow> 'b set
FSM.LS :::: ('a, 'b, 'c) fsm \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'c) list set
List.length :::: 'a list \<Rightarrow> nat
List.list.Nil :::: 'a list
List.list.Cons :::: 'a \<Rightarrow> 'a... |
###output
observable ?M \<Longrightarrow> ?q \<in> FSM.states ?M \<Longrightarrow> list.set (traces_to_check ?M ?q ?k) = {\<gamma> @ [(x, y)] |\<gamma> x y. length (\<gamma> @ [(x, y)]) \<le> ?k \<and> \<gamma> \<in> LS ?M ?q \<and> x \<in> FSM.inputs ?M \<and> y \<in> FSM.outputs ?M}###end |
Complex_Bounded_Operators/Cblinfun_Matrix | Cblinfun_Matrix.mk_projector_orthog_correct | lemma mk_projector_orthog_correct:
fixes S :: "'a::onb_enum list"
defines "d \<equiv> length (canonical_basis :: 'a list)"
assumes ortho: "is_ortho_set (set S)" and distinct: "distinct S"
shows "mk_projector_orthog d (map vec_of_basis_enum S)
= mat_of_cblinfun (Proj (ccspan (set S)))" | is_ortho_set (set ?S) \<Longrightarrow> distinct ?S \<Longrightarrow> mk_projector_orthog (length canonical_basis) (map vec_of_basis_enum ?S) = mat_of_cblinfun (Proj (ccspan (set ?S))) | \<lbrakk> ?H1 (?H2 x_1); ?H3 x_1\<rbrakk> \<Longrightarrow> ?H4 (?H5 ?H6) (?H7 ?H8 x_1) = ?H9 (?H10 (?H11 (?H2 x_1))) | [
"Complex_Vector_Spaces.ccspan",
"Complex_Bounded_Linear_Function.Proj",
"Cblinfun_Matrix.mat_of_cblinfun",
"Cblinfun_Matrix.vec_of_basis_enum",
"List.list.map",
"Complex_Vector_Spaces.basis_enum_class.canonical_basis",
"List.length",
"Cblinfun_Matrix.mk_projector_orthog",
"List.distinct",
"List.li... | [
"'a set \\<Rightarrow> 'a ccsubspace",
"'a ccsubspace \\<Rightarrow> ('a, 'a) cblinfun",
"('a, 'b) cblinfun \\<Rightarrow> complex mat",
"'a \\<Rightarrow> complex vec",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a list \\<Rightarrow> 'b list",
"'a list",
"'a list \\<Rightarrow> nat",
"nat \\<Rightarrow>... | [
"definition mat_of_cblinfun :: \\<open>'a::{basis_enum,complex_normed_vector} \\<Rightarrow>\\<^sub>C\\<^sub>L'b::{basis_enum,complex_normed_vector} \\<Rightarrow> complex mat\\<close> where\n \\<open>mat_of_cblinfun f =\n mat (length (canonical_basis :: 'b list)) (length (canonical_basis :: 'a list)) (\n \\... | lemma_object | ###symbols
Complex_Vector_Spaces.ccspan :::: 'a set \<Rightarrow> 'a ccsubspace
Complex_Bounded_Linear_Function.Proj :::: 'a ccsubspace \<Rightarrow> ('a, 'a) cblinfun
Cblinfun_Matrix.mat_of_cblinfun :::: ('a, 'b) cblinfun \<Rightarrow> complex mat
Cblinfun_Matrix.vec_of_basis_enum :::: 'a \<Rightarrow> complex vec
... |
###output
is_ortho_set (set ?S) \<Longrightarrow> distinct ?S \<Longrightarrow> mk_projector_orthog (length canonical_basis) (map vec_of_basis_enum ?S) = mat_of_cblinfun (Proj (ccspan (set ?S)))###end |
Polynomials/Term_Order | Term_Order.LEX_eq(3) | lemma LEX_eq [code]:
"nat_term_order_eq LEX (LEX::'a nat_term_order) dg ps = True" (is ?thesis1)
"nat_term_order_eq LEX (DRLEX::'a nat_term_order) dg ps = False" (is ?thesis2)
"nat_term_order_eq LEX (DEG (to::'a nat_term_order)) dg ps =
(dg \<and> nat_term_order_eq LEX to dg ps)" (is ?thesis3)
"nat_term_ord... | nat_term_order_eq Term_Order.LEX (DEG ?to) ?dg ?ps = (?dg \<and> nat_term_order_eq Term_Order.LEX ?to ?dg ?ps) | ?H1 ?H2 (?H3 x_1) x_2 x_3 = (x_2 \<and> ?H1 ?H2 x_1 x_2 x_3) | [
"Term_Order.DEG",
"Term_Order.LEX",
"Term_Order.nat_term_order_eq"
] | [
"'a nat_term_order \\<Rightarrow> 'a nat_term_order",
"'a nat_term_order",
"'a nat_term_order \\<Rightarrow> 'a nat_term_order \\<Rightarrow> bool \\<Rightarrow> bool \\<Rightarrow> bool"
] | [
"definition DEG :: \"'a::nat_term_compare nat_term_order \\<Rightarrow> 'a nat_term_order\"\n where \"DEG to = Abs_nat_term_order (deg_comp (nat_term_compare to))\"",
"definition LEX :: \"'a::nat_term_compare nat_term_order\" where \"LEX = Abs_nat_term_order lex_comp\"",
"definition nat_term_order_eq :: \"'a n... | lemma_object | ###symbols
Term_Order.DEG :::: 'a nat_term_order \<Rightarrow> 'a nat_term_order
Term_Order.LEX :::: 'a nat_term_order
Term_Order.nat_term_order_eq :::: 'a nat_term_order \<Rightarrow> 'a nat_term_order \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool
###defs
definition DEG :: "'a::nat_term_compare nat_term_o... |
###output
nat_term_order_eq Term_Order.LEX (DEG ?to) ?dg ?ps = (?dg \<and> nat_term_order_eq Term_Order.LEX ?to ?dg ?ps)###end |
FSM_Tests/Minimisation | Minimisation.minimise_states_finite | lemma minimise_states_finite :
assumes "observable M"
and "q \<in> states (minimise M)"
shows "finite q" | observable ?M \<Longrightarrow> ?q \<in> FSM.states (minimise ?M) \<Longrightarrow> finite ?q | \<lbrakk> ?H1 x_1; x_2 \<in> ?H2 (?H3 x_1)\<rbrakk> \<Longrightarrow> ?H4 x_2 | [
"Finite_Set.finite",
"Minimisation.minimise",
"FSM.states",
"FSM.observable"
] | [
"'a set \\<Rightarrow> bool",
"('a, 'b, 'c) fsm \\<Rightarrow> ('a set, 'b, 'c) fsm",
"('a, 'b, 'c) fsm \\<Rightarrow> 'a set",
"('a, 'b, 'c) fsm \\<Rightarrow> bool"
] | [
"class finite =\n assumes finite_UNIV: \"finite (UNIV :: 'a set)\"\nbegin",
"fun minimise :: \"('a :: linorder,'b :: linorder,'c :: linorder) fsm \\<Rightarrow> ('a set,'b,'c) fsm\" where\n \"minimise M = (let\n eq_class = ofsm_table_fix M (\\<lambda>q . states M) 0;\n ts = (\\<lambda> t . (eq_class (... | lemma_object | ###symbols
Finite_Set.finite :::: 'a set \<Rightarrow> bool
Minimisation.minimise :::: ('a, 'b, 'c) fsm \<Rightarrow> ('a set, 'b, 'c) fsm
FSM.states :::: ('a, 'b, 'c) fsm \<Rightarrow> 'a set
FSM.observable :::: ('a, 'b, 'c) fsm \<Rightarrow> bool
###defs
class finite =
assumes finite_UNIV: "finite (UNIV :: 'a se... |
###output
observable ?M \<Longrightarrow> ?q \<in> FSM.states (minimise ?M) \<Longrightarrow> finite ?q###end |
Min_Max_Least_Greatest/Min_Max_Least_Greatest_Multiset | Min_Max_Least_Greatest_Multiset.is_maximal_in_mset_wrt_iff | lemma is_maximal_in_mset_wrt_iff:
"is_maximal_in_mset_wrt R X x \<longleftrightarrow> x \<in># X \<and> (\<forall>y \<in># X. y \<noteq> x \<longrightarrow> \<not> R x y)" | transp_on (set_mset ?X) ?R \<Longrightarrow> asymp_on (set_mset ?X) ?R \<Longrightarrow> is_maximal_in_mset_wrt ?R ?X ?x = (?x \<in># ?X \<and> (\<forall>y\<in># ?X. y \<noteq> ?x \<longrightarrow> \<not> ?R ?x y)) | \<lbrakk> ?H1 (?H2 x_1) x_2; ?H3 (?H2 x_1) x_2\<rbrakk> \<Longrightarrow> ?H4 x_2 x_1 x_3 = (?H5 x_3 x_1 \<and> ?H6 x_1 (\<lambda>y_0. y_0 \<noteq> x_3 \<longrightarrow> \<not> x_2 x_3 y_0)) | [
"Multiset.Ball",
"Multiset.member_mset",
"Min_Max_Least_Greatest_Multiset.is_maximal_in_mset_wrt",
"Relation.asymp_on",
"Multiset.set_mset",
"Relation.transp_on"
] | [
"'a multiset \\<Rightarrow> ('a \\<Rightarrow> bool) \\<Rightarrow> bool",
"'a \\<Rightarrow> 'a multiset \\<Rightarrow> bool",
"('a \\<Rightarrow> 'a \\<Rightarrow> bool) \\<Rightarrow> 'a multiset \\<Rightarrow> 'a \\<Rightarrow> bool",
"'a set \\<Rightarrow> ('a \\<Rightarrow> 'a \\<Rightarrow> bool) \\<Ri... | [
"abbreviation Ball :: \"'a multiset \\<Rightarrow> ('a \\<Rightarrow> bool) \\<Rightarrow> bool\"\n where \"Ball M \\<equiv> Set.Ball (set_mset M)\"",
"abbreviation member_mset :: \\<open>'a \\<Rightarrow> 'a multiset \\<Rightarrow> bool\\<close>\n where \\<open>member_mset a M \\<equiv> a \\<in> set_mset M\\<c... | lemma_object | ###symbols
Multiset.Ball :::: 'a multiset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool
Multiset.member_mset :::: 'a \<Rightarrow> 'a multiset \<Rightarrow> bool
Min_Max_Least_Greatest_Multiset.is_maximal_in_mset_wrt :::: ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a \... |
###output
transp_on (set_mset ?X) ?R \<Longrightarrow> asymp_on (set_mset ?X) ?R \<Longrightarrow> is_maximal_in_mset_wrt ?R ?X ?x = (?x \<in># ?X \<and> (\<forall>y\<in># ?X. y \<noteq> ?x \<longrightarrow> \<not> ?R ?x y))###end |
BDD/General | General.Nodes_in_pret | lemma Nodes_in_pret: "\<lbrakk>wf_ll t levellista var; nb <= length levellista\<rbrakk> \<Longrightarrow> Nodes nb levellista \<subseteq> set_of t" | wf_ll ?t ?levellista ?var \<Longrightarrow> ?nb \<le> length ?levellista \<Longrightarrow> Nodes ?nb ?levellista \<subseteq> set_of ?t | \<lbrakk> ?H1 x_1 x_2 x_3; x_4 \<le> ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_4 x_2) (?H5 x_1) | [
"BinDag.set_of",
"General.Nodes",
"Set.subset_eq",
"List.length",
"General.wf_ll"
] | [
"dag \\<Rightarrow> ref set",
"nat \\<Rightarrow> ref list list \\<Rightarrow> ref set",
"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool",
"'a list \\<Rightarrow> nat",
"dag \\<Rightarrow> ref list list \\<Rightarrow> (ref \\<Rightarrow> nat) \\<Rightarrow> bool"
] | [
"primrec set_of:: \"dag \\<Rightarrow> ref set\" where\n set_of_Tip: \"set_of Tip = {}\"\n | set_of_Node: \"set_of (Node lt a rt) = {a} \\<union> set_of lt \\<union> set_of rt\"",
"definition Nodes :: \"nat \\<Rightarrow> ref list list \\<Rightarrow> ref set\"\n where \"Nodes i levellist = (\\<Union>k\\<in>{k.... | lemma_object | ###symbols
BinDag.set_of :::: dag \<Rightarrow> ref set
General.Nodes :::: nat \<Rightarrow> ref list list \<Rightarrow> ref set
Set.subset_eq :::: 'a set \<Rightarrow> 'a set \<Rightarrow> bool
List.length :::: 'a list \<Rightarrow> nat
General.wf_ll :::: dag \<Rightarrow> ref list list \<Rightarrow> (ref \<Righta... |
###output
wf_ll ?t ?levellista ?var \<Longrightarrow> ?nb \<le> length ?levellista \<Longrightarrow> Nodes ?nb ?levellista \<subseteq> set_of ?t###end |
Riesz_Representation/Urysohn_Locally_Compact_Hausdorff | Urysohn_Locally_Compact_Hausdorff.upper_semicontinuous_map_INF | lemma upper_semicontinuous_map_INF:
fixes f :: "_ \<Rightarrow> _ \<Rightarrow> 'a :: {linorder_topology, complete_linorder}"
assumes "\<And>i. i \<in> I \<Longrightarrow> upper_semicontinuous_map X (f i)"
shows "upper_semicontinuous_map X (\<lambda>x. \<Sqinter>i\<in>I. f i x)" | (\<And>i. i \<in> ?I \<Longrightarrow> upper_semicontinuous_map ?X (?f i)) \<Longrightarrow> upper_semicontinuous_map ?X (\<lambda>x. \<Sqinter>i\<in> ?I. ?f i x) | (\<And>y_0. y_0 \<in> x_1 \<Longrightarrow> ?H1 x_2 (x_3 y_0)) \<Longrightarrow> ?H1 x_2 (\<lambda>y_1. ?H2 (?H3 (\<lambda>y_2. x_3 y_2 y_1) x_1)) | [
"Set.image",
"Complete_Lattices.Inf_class.Inf",
"Lemmas_StandardBorel.upper_semicontinuous_map"
] | [
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b set",
"'a set \\<Rightarrow> 'a",
"'a topology \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> bool"
] | [
"definition image :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b set\" (infixr \"`\" 90)\n where \"f ` A = {y. \\<exists>x\\<in>A. y = f x}\"",
"class Inf =\n fixes Inf :: \"'a set \\<Rightarrow> 'a\" (\"\\<Sqinter> _\" [900] 900)",
"definition upper_semicontinuous_map :: \"['a topolo... | lemma_object | ###symbols
Set.image :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set
Complete_Lattices.Inf_class.Inf :::: 'a set \<Rightarrow> 'a
Lemmas_StandardBorel.upper_semicontinuous_map :::: 'a topology \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool
###defs
definition image :: "('a \<Rightarrow> 'b... |
###output
(\<And>i. i \<in> ?I \<Longrightarrow> upper_semicontinuous_map ?X (?f i)) \<Longrightarrow> upper_semicontinuous_map ?X (\<lambda>x. \<Sqinter>i\<in> ?I. ?f i x)###end |
Projective_Measurements/Linear_Algebra_Complements | Linear_Algebra_Complements.tensor_mat_carrier | lemma tensor_mat_carrier:
shows "tensor_mat U V \<in> carrier_mat (dim_row U * dim_row V) (dim_col U * dim_col V)" | ?U \<Otimes> ?V \<in> carrier_mat (dim_row ?U * dim_row ?V) (dim_col ?U * dim_col ?V) | ?H1 x_1 x_2 \<in> ?H2 (?H3 (?H4 x_1) (?H4 x_2)) (?H3 (?H5 x_1) (?H5 x_2)) | [
"Matrix.dim_col",
"Matrix.dim_row",
"Groups.times_class.times",
"Matrix.carrier_mat",
"Tensor.tensor_mat"
] | [
"'a Matrix.mat \\<Rightarrow> nat",
"'a Matrix.mat \\<Rightarrow> nat",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"nat \\<Rightarrow> nat \\<Rightarrow> 'a Matrix.mat set",
"complex Matrix.mat \\<Rightarrow> complex Matrix.mat \\<Rightarrow> complex Matrix.mat"
] | [
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"*\" 70)",
"definition carrier_mat :: \"nat \\<Rightarrow> nat \\<Rightarrow> 'a mat set\"\n where \"carrier_mat nr nc = { m . dim_row m = nr \\<and> dim_col m = nc}\"",
"definition tensor_mat:: \"[complex Matrix.mat, complex... | lemma_object | ###symbols
Matrix.dim_col :::: 'a Matrix.mat \<Rightarrow> nat
Matrix.dim_row :::: 'a Matrix.mat \<Rightarrow> nat
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Matrix.carrier_mat :::: nat \<Rightarrow> nat \<Rightarrow> 'a Matrix.mat set
Tensor.tensor_mat :::: complex Matrix.mat \<Rightarrow> ... |
###output
?U \<Otimes> ?V \<in> carrier_mat (dim_row ?U * dim_row ?V) (dim_col ?U * dim_col ?V)###end |
Earley_Parser/Earley_Recognizer | Earley_Recognizer.wf_bins_Scan\<^sub>L' | null | wf_bins ?\<G> ?\<omega> ?bs \<Longrightarrow> ?k < length ?bs \<Longrightarrow> ?x \<in> set (items (?bs ! ?k)) \<Longrightarrow> ?k < length ?\<omega> \<Longrightarrow> next_symbol ?x \<noteq> None \<Longrightarrow> ?y = inc_item ?x (?k + 1) \<Longrightarrow> wf_item ?\<G> ?\<omega> ?y \<and> end_item ?y = ?k + 1 | \<lbrakk> ?H1 x_1 x_2 x_3; x_4 < ?H2 x_3; x_5 \<in> ?H3 (?H4 (?H5 x_3 x_4)); x_4 < ?H2 x_2; ?H6 x_5 \<noteq> ?H7; x_6 = ?H8 x_5 (?H9 x_4 ?H10)\<rbrakk> \<Longrightarrow> ?H11 x_1 x_2 x_6 \<and> ?H12 x_6 = ?H9 x_4 ?H10 | [
"Earley.item.end_item",
"Earley.wf_item",
"Groups.one_class.one",
"Groups.plus_class.plus",
"Earley_Fixpoint.inc_item",
"Option.option.None",
"Earley.next_symbol",
"List.nth",
"Earley_Recognizer.items",
"List.list.set",
"List.length",
"Earley_Recognizer.wf_bins"
] | [
"'a item \\<Rightarrow> nat",
"'a cfg \\<Rightarrow> 'a list \\<Rightarrow> 'a item \\<Rightarrow> bool",
"'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a item \\<Rightarrow> nat \\<Rightarrow> 'a item",
"'a option",
"'a item \\<Rightarrow> 'a option",
"'a list \\<Rightarrow> nat \\<Rightarrow> 'a"... | [
"datatype 'a item = \n Item (rule_item: \"'a rule\") (dot_item : nat) (start_item : nat) (end_item : nat)",
"definition wf_item :: \"'a cfg \\<Rightarrow> 'a list => 'a item \\<Rightarrow> bool\" where \n \"wf_item \\<G> \\<omega> x \\<equiv>\n rule_item x \\<in> set (\\<RR> \\<G>) \\<and> \n dot_item x \... | lemma_object | ###symbols
Earley.item.end_item :::: 'a item \<Rightarrow> nat
Earley.wf_item :::: 'a cfg \<Rightarrow> 'a list \<Rightarrow> 'a item \<Rightarrow> bool
Groups.one_class.one :::: 'a
Groups.plus_class.plus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Earley_Fixpoint.inc_item :::: 'a item \<Rightarrow> nat \<Rightarrow>... |
###output
wf_bins ?\<G> ?\<omega> ?bs \<Longrightarrow> ?k < length ?bs \<Longrightarrow> ?x \<in> set (items (?bs ! ?k)) \<Longrightarrow> ?k < length ?\<omega> \<Longrightarrow> next_symbol ?x \<noteq> None \<Longrightarrow> ?y = inc_item ?x (?k + 1) \<Longrightarrow> wf_item ?\<G> ?\<omega> ?y \<and> end_item ?y = ... |
Goedel_HFSet_Semanticless/Instance | Instance.prv_cnj_imp_scnj2 | null | ?\<phi> \<in> UNIV \<Longrightarrow> ?\<psi> \<in> UNIV \<Longrightarrow> {} \<turnstile> ?\<phi> AND ?\<psi> IMP scnj { ?\<phi>, ?\<psi>} | \<lbrakk>x_1 \<in> ?H1; x_2 \<in> ?H1\<rbrakk> \<Longrightarrow> ?H2 ?H3 (?H4 (?H5 x_1 x_2) (?H6 (?H7 x_1 (?H7 x_2 ?H3)))) | [
"Set.insert",
"Instance.scnj",
"SyntaxN.Conj",
"SyntaxN.Imp",
"Set.empty",
"SyntaxN.hfthm",
"Set.UNIV"
] | [
"'a \\<Rightarrow> 'a set \\<Rightarrow> 'a set",
"fm set \\<Rightarrow> fm",
"fm \\<Rightarrow> fm \\<Rightarrow> fm",
"fm \\<Rightarrow> fm \\<Rightarrow> fm",
"'a set",
"fm set \\<Rightarrow> fm \\<Rightarrow> bool",
"'a set"
] | [
"definition insert :: \"'a \\<Rightarrow> 'a set \\<Rightarrow> 'a set\"\n where insert_compr: \"insert a B = {x. x = a \\<or> x \\<in> B}\"",
"definition Conj :: \"fm \\<Rightarrow> fm \\<Rightarrow> fm\" (infixr \"AND\" 135)\n where \"Conj A B \\<equiv> Neg (Disj (Neg A) (Neg B))\"",
"abbreviation Imp :: ... | lemma_object | ###symbols
Set.insert :::: 'a \<Rightarrow> 'a set \<Rightarrow> 'a set
Instance.scnj :::: fm set \<Rightarrow> fm
SyntaxN.Conj :::: fm \<Rightarrow> fm \<Rightarrow> fm
SyntaxN.Imp :::: fm \<Rightarrow> fm \<Rightarrow> fm
Set.empty :::: 'a set
SyntaxN.hfthm :::: fm set \<Rightarrow> fm \<Rightarrow> bool
Set.UN... |
###output
?\<phi> \<in> UNIV \<Longrightarrow> ?\<psi> \<in> UNIV \<Longrightarrow> {} \<turnstile> ?\<phi> AND ?\<psi> IMP scnj { ?\<phi>, ?\<psi>}###end |
Native_Word/Uint16 | Uint16.shiftr_uint16_code | lemma shiftr_uint16_code [code]: "drop_bit n x = (if n < 16 then uint16_shiftr x (integer_of_nat n) else 0)" | drop_bit ?n ?x = (if ?n < 16 then uint16_shiftr ?x (integer_of_nat ?n) else 0) | ?H1 x_1 x_2 = (if x_1 < ?H2 (?H3 (?H3 (?H3 (?H3 ?H4)))) then ?H5 x_2 (?H6 x_1) else ?H7) | [
"Groups.zero_class.zero",
"Code_Numeral.integer_of_nat",
"Uint16.uint16_shiftr",
"Num.num.One",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"Bit_Operations.semiring_bit_operations_class.drop_bit"
] | [
"'a",
"nat \\<Rightarrow> integer",
"uint16 \\<Rightarrow> integer \\<Rightarrow> uint16",
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"nat \\<Rightarrow> 'a \\<Rightarrow> 'a"
] | [
"class zero =\n fixes zero :: 'a (\"0\")",
"definition uint16_shiftr :: \"uint16 \\<Rightarrow> integer \\<Rightarrow> uint16\"\nwhere [code del]:\n \"uint16_shiftr x n = (if n < 0 \\<or> 16 \\<le> n then undefined (drop_bit :: nat \\<Rightarrow> uint16 \\<Rightarrow> _) x n else drop_bit (nat_of_integer n) x)... | lemma_object | ###symbols
Groups.zero_class.zero :::: 'a
Code_Numeral.integer_of_nat :::: nat \<Rightarrow> integer
Uint16.uint16_shiftr :::: uint16 \<Rightarrow> integer \<Rightarrow> uint16
Num.num.One :::: num
Num.num.Bit0 :::: num \<Rightarrow> num
Num.numeral_class.numeral :::: num \<Rightarrow> 'a
Bit_Operations.semiring_... |
###output
drop_bit ?n ?x = (if ?n < 16 then uint16_shiftr ?x (integer_of_nat ?n) else 0)###end |
FOL_Seq_Calc3/Encoding | Encoding.lt_list_encode | lemma lt_list_encode: \<open>n [\<in>] ns \<Longrightarrow> n < list_encode ns\<close> | ?n [\<in>] ?ns \<Longrightarrow> ?n < list_encode ?ns | ?H1 x_1 x_2 \<Longrightarrow> x_1 < ?H2 x_2 | [
"Nat_Bijection.list_encode",
"List_Syntax.list_member_syntax"
] | [
"nat list \\<Rightarrow> nat",
"'a \\<Rightarrow> 'a list \\<Rightarrow> bool"
] | [
"fun list_encode :: \"nat list \\<Rightarrow> nat\"\n where\n \"list_encode [] = 0\"\n | \"list_encode (x # xs) = Suc (prod_encode (x, list_encode xs))\"",
"abbreviation list_member_syntax :: \\<open>'a \\<Rightarrow> 'a list \\<Rightarrow> bool\\<close> (\\<open>_ [\\<in>] _\\<close> [51, 51] 50) where\n \... | lemma_object | ###symbols
Nat_Bijection.list_encode :::: nat list \<Rightarrow> nat
List_Syntax.list_member_syntax :::: 'a \<Rightarrow> 'a list \<Rightarrow> bool
###defs
fun list_encode :: "nat list \<Rightarrow> nat"
where
"list_encode [] = 0"
| "list_encode (x # xs) = Suc (prod_encode (x, list_encode xs))"
abbreviation ... |
###output
?n [\<in>] ?ns \<Longrightarrow> ?n < list_encode ?ns###end |
POPLmark-deBruijn/POPLmarkRecord | POPLmarkRecord.substT_liftT(1) | theorem substT_liftT [simp]:
"k \<le> k' \<Longrightarrow> k' < k + n \<Longrightarrow> (\<up>\<^sub>\<tau> n k T)[k' \<mapsto>\<^sub>\<tau> U]\<^sub>\<tau> = \<up>\<^sub>\<tau> (n - 1) k T"
"k \<le> k' \<Longrightarrow> k' < k + n \<Longrightarrow> (\<up>\<^sub>r\<^sub>\<tau> n k rT)[k' \<mapsto>\<^sub>\<tau> U]\<... | ?k \<le> ?k' \<Longrightarrow> ?k' < ?k + ?n \<Longrightarrow> \<up>\<^sub>\<tau> ?n ?k ?T[ ?k' \<mapsto>\<^sub>\<tau> ?U]\<^sub>\<tau> = \<up>\<^sub>\<tau> (?n - 1) ?k ?T | \<lbrakk>x_1 \<le> x_2; x_2 < ?H1 x_1 x_3\<rbrakk> \<Longrightarrow> ?H2 (?H3 x_3 x_1 x_4) x_2 x_5 = ?H3 (?H4 x_3 ?H5) x_1 x_4 | [
"Groups.one_class.one",
"Groups.minus_class.minus",
"POPLmarkRecord.liftT",
"POPLmarkRecord.substTT",
"Groups.plus_class.plus"
] | [
"'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"nat \\<Rightarrow> nat \\<Rightarrow> POPLmarkRecord.type \\<Rightarrow> POPLmarkRecord.type",
"POPLmarkRecord.type \\<Rightarrow> nat \\<Rightarrow> POPLmarkRecord.type \\<Rightarrow> POPLmarkRecord.type",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a"
] | [
"class one =\n fixes one :: 'a (\"1\")",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"primrec liftT :: \"nat \\<Rightarrow> nat \\<Rightarrow> type \\<Rightarrow> type\" (\"\\<up>\\<^sub>\\<tau>\")\n and liftrT :: \"nat \\<Rightarrow> nat \\<Rightarrow> r... | lemma_object | ###symbols
Groups.one_class.one :::: 'a
Groups.minus_class.minus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
POPLmarkRecord.liftT :::: nat \<Rightarrow> nat \<Rightarrow> POPLmarkRecord.type \<Rightarrow> POPLmarkRecord.type
POPLmarkRecord.substTT :::: POPLmarkRecord.type \<Rightarrow> nat \<Rightarrow> POPLmar... |
###output
?k \<le> ?k' \<Longrightarrow> ?k' < ?k + ?n \<Longrightarrow> \<up>\<^sub>\<tau> ?n ?k ?T[ ?k' \<mapsto>\<^sub>\<tau> ?U]\<^sub>\<tau> = \<up>\<^sub>\<tau> (?n - 1) ?k ?T###end |
Nested_Multisets_Ordinals/Syntactic_Ordinal | Syntactic_Ordinal.head_\<omega>_plus | null | head_\<omega> (?m + ?n) = sup (head_\<omega> ?m) (head_\<omega> ?n) | ?H1 (?H2 x_1 x_2) = ?H3 (?H1 x_1) (?H1 x_2) | [
"Lattices.sup_class.sup",
"Groups.plus_class.plus",
"Syntactic_Ordinal.head_\\<omega>"
] | [
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"hmultiset \\<Rightarrow> hmultiset"
] | [
"class sup =\n fixes sup :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"\\<squnion>\" 65)",
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"definition head_\\<omega> :: \"hmultiset \\<Rightarrow> hmultiset\" where\n \"head_\\<omega> M = (if M = 0 then 0... | lemma_object | ###symbols
Lattices.sup_class.sup :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.plus_class.plus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Syntactic_Ordinal.head_\<omega> :::: hmultiset \<Rightarrow> hmultiset
###defs
class sup =
fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)
class ... |
###output
head_\<omega> (?m + ?n) = sup (head_\<omega> ?m) (head_\<omega> ?n)###end |
CHERI-C_Memory_Model/CHERI_C_Concrete_Memory_Model | CHERI_C_Concrete_Memory_Model.store_bytes_u64(4) | lemma store_bytes_u64:
shows "off \<in> Mapping.keys (store_bytes m off (flatten_u64 v))"
and "Suc off \<in> Mapping.keys (store_bytes m off (flatten_u64 v))"
and "Suc (Suc off) \<in> Mapping.keys (store_bytes m off (flatten_u64 v))"
and "Suc (Suc (Suc off)) \<in> Mapping.keys (store_bytes m off (flatten_... | Suc (Suc (Suc ?off)) \<in> Mapping.keys (store_bytes ?m ?off (flatten_u64 ?v)) | ?H1 (?H1 (?H1 x_1)) \<in> ?H2 (?H3 x_2 x_1 (?H4 x_3)) | [
"Preliminary_Library.flatten_u64",
"CHERI_C_Concrete_Memory_Model.store_bytes",
"Mapping.keys",
"Nat.Suc"
] | [
"64 word \\<Rightarrow> 8 word list",
"(nat, memval) mapping \\<Rightarrow> nat \\<Rightarrow> 8 word list \\<Rightarrow> (nat, memval) mapping",
"('a, 'b) mapping \\<Rightarrow> 'a set",
"nat \\<Rightarrow> nat"
] | [
"abbreviation flatten_u64 :: \"64 word \\<Rightarrow> 8 word list\"\n where\n \"flatten_u64 x \\<equiv> (word_rsplit :: 64 word \\<Rightarrow> 8 word list) x\"",
"primrec store_bytes :: \"(nat, memval) mapping \\<Rightarrow> nat \\<Rightarrow> 8 word list \\<Rightarrow> (nat, memval) mapping\"\n where\n \"sto... | lemma_object | ###symbols
Preliminary_Library.flatten_u64 :::: 64 word \<Rightarrow> 8 word list
CHERI_C_Concrete_Memory_Model.store_bytes :::: (nat, memval) mapping \<Rightarrow> nat \<Rightarrow> 8 word list \<Rightarrow> (nat, memval) mapping
Mapping.keys :::: ('a, 'b) mapping \<Rightarrow> 'a set
Nat.Suc :::: nat \<Rightarrow>... |
###output
Suc (Suc (Suc ?off)) \<in> Mapping.keys (store_bytes ?m ?off (flatten_u64 ?v))###end |
Myhill-Nerode/Myhill_1 | Myhill_1.every_eqclass_has_transition | lemma every_eqclass_has_transition:
assumes has_str: "s @ [c] \<in> X"
and in_CS: "X \<in> UNIV // \<approx>A"
obtains Y where "Y \<in> UNIV // \<approx>A" and "Y \<cdot> {[c]} \<subseteq> X" and "s \<in> Y" | ?s @ [ ?c] \<in> ?X \<Longrightarrow> ?X \<in> UNIV // \<approx> ?A \<Longrightarrow> (\<And>Y. Y \<in> UNIV // \<approx> ?A \<Longrightarrow> Y \<cdot> {[ ?c]} \<subseteq> ?X \<Longrightarrow> ?s \<in> Y \<Longrightarrow> ?thesis) \<Longrightarrow> ?thesis | \<lbrakk> ?H1 x_1 (?H2 x_2 ?H3) \<in> x_3; x_3 \<in> ?H4 ?H5 (?H6 x_4); \<And>y_0. \<lbrakk>y_0 \<in> ?H4 ?H5 (?H6 x_4); ?H7 (?H8 y_0 (?H9 (?H2 x_2 ?H3) ?H10)) x_3; x_1 \<in> y_0\<rbrakk> \<Longrightarrow> x_5\<rbrakk> \<Longrightarrow> x_5 | [
"Set.empty",
"Set.insert",
"Regular_Set.conc",
"Set.subset_eq",
"Myhill_1.str_eq",
"Set.UNIV",
"Equiv_Relations.quotient",
"List.list.Nil",
"List.list.Cons",
"List.append"
] | [
"'a set",
"'a \\<Rightarrow> 'a set \\<Rightarrow> 'a set",
"'a list set \\<Rightarrow> 'a list set \\<Rightarrow> 'a list set",
"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool",
"'a list set \\<Rightarrow> ('a list \\<times> 'a list) set",
"'a set",
"'a set \\<Rightarrow> ('a \\<times> 'a) set \\<Rig... | [
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\"",
"definition insert :: \"'a \\<Rightarrow> 'a set \\<Rightarrow> 'a set\"\n where insert_compr: \"insert a B = {x. x = a \\<or> x \\<in> B}\"",
"definition conc :: \"'a lang \\<Rightarrow> 'a lang \\<Rightarrow> 'a lang\" (infixr \"@@\" ... | lemma_object | ###symbols
Set.empty :::: 'a set
Set.insert :::: 'a \<Rightarrow> 'a set \<Rightarrow> 'a set
Regular_Set.conc :::: 'a list set \<Rightarrow> 'a list set \<Rightarrow> 'a list set
Set.subset_eq :::: 'a set \<Rightarrow> 'a set \<Rightarrow> bool
Myhill_1.str_eq :::: 'a list set \<Rightarrow> ('a list \<times> 'a li... |
###output
?s @ [ ?c] \<in> ?X \<Longrightarrow> ?X \<in> UNIV // \<approx> ?A \<Longrightarrow> (\<And>Y. Y \<in> UNIV // \<approx> ?A \<Longrightarrow> Y \<cdot> {[ ?c]} \<subseteq> ?X \<Longrightarrow> ?s \<in> Y \<Longrightarrow> ?thesis) \<Longrightarrow> ?thesis###end |
Cauchy/CauchysMeanTheorem | CauchysMeanTheorem.prod_list_split | lemma prod_list_split:
fixes xs::"real list"
shows "\<Prod>:xs = (\<Prod>:(list_neq xs m) * \<Prod>:(list_eq xs m))" | prod_list ?xs = prod_list (list_neq ?xs ?m) * prod_list (list_eq ?xs ?m) | ?H1 x_1 = ?H2 (?H1 (?H3 x_1 x_2)) (?H1 (?H4 x_1 x_2)) | [
"CauchysMeanTheorem.list_eq",
"CauchysMeanTheorem.list_neq",
"Groups.times_class.times",
"Groups_List.monoid_mult_class.prod_list"
] | [
"'a list \\<Rightarrow> 'a \\<Rightarrow> 'a list",
"'a list \\<Rightarrow> 'a \\<Rightarrow> 'a list",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a list \\<Rightarrow> 'a"
] | [
"abbreviation\n list_eq :: \"('a list) \\<Rightarrow> 'a \\<Rightarrow> ('a list)\" where\n \"list_eq xs el == filter (\\<lambda>x. x=el) xs\"",
"abbreviation\n list_neq :: \"('a list) \\<Rightarrow> 'a \\<Rightarrow> ('a list)\" where\n \"list_neq xs el == filter (\\<lambda>x. x\\<noteq>el) xs\"",
"class t... | lemma_object | ###symbols
CauchysMeanTheorem.list_eq :::: 'a list \<Rightarrow> 'a \<Rightarrow> 'a list
CauchysMeanTheorem.list_neq :::: 'a list \<Rightarrow> 'a \<Rightarrow> 'a list
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups_List.monoid_mult_class.prod_list :::: 'a list \<Rightarrow> 'a
###defs
ab... |
###output
prod_list ?xs = prod_list (list_neq ?xs ?m) * prod_list (list_eq ?xs ?m)###end |
BNF_Operations/Lift | Lifting.relator_distr_raw(15) | null | left_unique ?R2 \<Longrightarrow> right_total ?R2 \<Longrightarrow> right_unique ?R'2 \<Longrightarrow> left_total ?R'2 \<Longrightarrow> Lifting.POS (?R2 OO ?R'2) ?A1 \<Longrightarrow> Lifting.NEG (?S2 OO ?S'2) ?B1 \<Longrightarrow> Lifting.NEG (rel_fun ?R2 ?S2 OO rel_fun ?R'2 ?S'2) (rel_fun ?A1 ?B1) | \<lbrakk> ?H1 x_1; ?H2 x_1; ?H3 x_2; ?H4 x_2; ?H5 (?H6 x_1 x_2) x_3; ?H7 (?H6 x_4 x_5) x_6\<rbrakk> \<Longrightarrow> ?H7 (?H6 (?H8 x_1 x_4) (?H8 x_2 x_5)) (?H8 x_3 x_6) | [
"BNF_Def.rel_fun",
"Lifting.NEG",
"Relation.relcompp",
"Lifting.POS",
"Transfer.left_total",
"Transfer.right_unique",
"Transfer.right_total",
"Transfer.left_unique"
] | [
"('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> ('c \\<Rightarrow> 'd \\<Rightarrow> bool) \\<Rightarrow> ('a \\<Rightarrow> 'c) \\<Rightarrow> ('b \\<Rightarrow> 'd) \\<Rightarrow> bool",
"('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> ('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarro... | [
"definition\n rel_fun :: \"('a \\<Rightarrow> 'c \\<Rightarrow> bool) \\<Rightarrow> ('b \\<Rightarrow> 'd \\<Rightarrow> bool) \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> ('c \\<Rightarrow> 'd) \\<Rightarrow> bool\"\nwhere\n \"rel_fun A B = (\\<lambda>f g. \\<forall>x y. A x y \\<longrightarrow> B (f x)... | lemma_object | ###symbols
BNF_Def.rel_fun :::: ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> bool
Lifting.NEG :::: ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Right... |
###output
left_unique ?R2 \<Longrightarrow> right_total ?R2 \<Longrightarrow> right_unique ?R'2 \<Longrightarrow> left_total ?R'2 \<Longrightarrow> Lifting.POS (?R2 OO ?R'2) ?A1 \<Longrightarrow> Lifting.NEG (?S2 OO ?S'2) ?B1 \<Longrightarrow> Lifting.NEG (rel_fun ?R2 ?S2 OO rel_fun ?R'2 ?S'2) (rel_fun ?A1 ?B1)###end |
Partial_Function_MR/Partial_Function_MR | Partial_Function_MR.option_map_id | lemma option_map_id: "map_option (\<lambda> x. x) x = x" | map_option (\<lambda>x. x) ?x = ?x | ?H1 (\<lambda>y_0. y_0) x_1 = x_1 | [
"Option.option.map_option"
] | [
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a option \\<Rightarrow> 'b option"
] | [
"datatype 'a option =\n None\n | Some (the: 'a)"
] | lemma_object | ###symbols
Option.option.map_option :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a option \<Rightarrow> 'b option
###defs
datatype 'a option =
None
| Some (the: 'a) |
###output
map_option (\<lambda>x. x) ?x = ?x###end |
LTL_to_GBA/LTL_to_GBA | LTL_to_GBA.expand_new_name_expand_init | lemma expand_new_name_expand_init: "expand_init < expand_new_name nm" | expand_init < expand_new_name ?nm | ?H1 < ?H2 x_1 | [
"LTL_to_GBA.expand_new_name",
"LTL_to_GBA.expand_init"
] | [
"nat \\<Rightarrow> nat",
"'a"
] | [
"definition \"expand_new_name \\<equiv> Suc\"",
"definition \"expand_init \\<equiv> 0\""
] | lemma_object | ###symbols
LTL_to_GBA.expand_new_name :::: nat \<Rightarrow> nat
LTL_to_GBA.expand_init :::: 'a
###defs
definition "expand_new_name \<equiv> Suc"
definition "expand_init \<equiv> 0" |
###output
expand_init < expand_new_name ?nm###end |
Slicing/While/Com | Complete_Lattices.INF1_E | null | Inf (?B ` ?A) ?b \<Longrightarrow> (?B ?a ?b \<Longrightarrow> ?thesis) \<Longrightarrow> (?a \<notin> ?A \<Longrightarrow> ?thesis) \<Longrightarrow> ?thesis | \<lbrakk> ?H1 (?H2 x_1 x_2) x_3; x_1 x_4 x_3 \<Longrightarrow> x_5; ?H3 x_4 x_2 \<Longrightarrow> x_5\<rbrakk> \<Longrightarrow> x_5 | [
"Set.not_member",
"Set.image",
"Complete_Lattices.Inf_class.Inf"
] | [
"'a \\<Rightarrow> 'a set \\<Rightarrow> bool",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b set",
"'a set \\<Rightarrow> 'a"
] | [
"abbreviation not_member\n where \"not_member x A \\<equiv> \\<not> (x \\<in> A)\" \\<comment> \\<open>non-membership\\<close>",
"definition image :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b set\" (infixr \"`\" 90)\n where \"f ` A = {y. \\<exists>x\\<in>A. y = f x}\"",
"class Inf =\... | lemma_object | ###symbols
Set.not_member :::: 'a \<Rightarrow> 'a set \<Rightarrow> bool
Set.image :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set
Complete_Lattices.Inf_class.Inf :::: 'a set \<Rightarrow> 'a
###defs
abbreviation not_member
where "not_member x A \<equiv> \<not> (x \<in> A)" \<comment> \<open>n... |
###output
Inf (?B ` ?A) ?b \<Longrightarrow> (?B ?a ?b \<Longrightarrow> ?thesis) \<Longrightarrow> (?a \<notin> ?A \<Longrightarrow> ?thesis) \<Longrightarrow> ?thesis###end |
Riesz_Representation/Riesz_Representation | Riesz_Representation.pos_lin_functional_on_CX_complex_decompose | null | positive_linear_functional_on_CX ?X ?\<phi> \<Longrightarrow> continuous_map ?X euclidean ?f \<Longrightarrow> ?f has_compact_support_on ?X \<Longrightarrow> ?\<phi> (restrict ?f (topspace ?X)) = complex_of_real (Re (?\<phi> (\<lambda>x\<in>topspace ?X. complex_of_real ((\<lambda>x\<in>topspace ?X. Re (?f x)) x)))) + \... | \<lbrakk> ?H1 x_1 x_2; ?H2 x_1 ?H3 x_3; ?H4 x_3 x_1\<rbrakk> \<Longrightarrow> x_2 (?H5 x_3 (?H6 x_1)) = ?H7 (?H8 (?H9 (x_2 (?H5 (\<lambda>y_1. ?H8 (?H5 (\<lambda>y_2. ?H9 (x_3 y_2)) (?H6 x_1) y_1)) (?H6 x_1))))) (?H10 ?H11 (?H8 (?H9 (x_2 (?H5 (\<lambda>y_3. ?H8 (?H5 (\<lambda>y_4. ?H12 (x_3 y_4)) (?H6 x_1) y_3)) (?H6 ... | [
"Complex.complex.Im",
"Complex.imaginary_unit",
"Groups.times_class.times",
"Complex.complex.Re",
"Complex.complex_of_real",
"Groups.plus_class.plus",
"Abstract_Topology.topspace",
"FuncSet.restrict",
"Riesz_Representation.has_compact_support_on",
"Abstract_Topology.euclidean",
"Abstract_Topolog... | [
"complex \\<Rightarrow> real",
"complex",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"complex \\<Rightarrow> real",
"real \\<Rightarrow> complex",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a topology \\<Rightarrow> 'a set",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'a \\<Rightarrow... | [
"codatatype complex = Complex (Re: real) (Im: real)",
"primcorec imaginary_unit :: complex (\"\\<i>\")\n where\n \"Re \\<i> = 0\"\n | \"Im \\<i> = 1\"",
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"*\" 70)",
"abbreviation complex_of_real :: \"real \\<Rightarrow>... | lemma_object | ###symbols
Complex.complex.Im :::: complex \<Rightarrow> real
Complex.imaginary_unit :::: complex
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Complex.complex.Re :::: complex \<Rightarrow> real
Complex.complex_of_real :::: real \<Rightarrow> complex
Groups.plus_class.plus :::: 'a \<Rightarrow... |
###output
positive_linear_functional_on_CX ?X ?\<phi> \<Longrightarrow> continuous_map ?X euclidean ?f \<Longrightarrow> ?f has_compact_support_on ?X \<Longrightarrow> ?\<phi> (restrict ?f (topspace ?X)) = complex_of_real (Re (?\<phi> (\<lambda>x\<in>topspace ?X. complex_of_real ((\<lambda>x\<in>topspace ?X. Re (?f x)... |
CoSMed/Post_Confidentiality/Post | Post.reachNT_step_induct | null | reachNT ?s \<Longrightarrow> ?P istate \<Longrightarrow> (\<And>s a ou s'. reachNT s \<Longrightarrow> step s a = (ou, s') \<Longrightarrow> \<not> T (Trans s a ou s') \<Longrightarrow> ?P s \<Longrightarrow> ?P s') \<Longrightarrow> ?P ?s | \<lbrakk> ?H1 x_1; x_2 ?H2; \<And>y_0 y_1 y_2 y_3. \<lbrakk> ?H1 y_0; ?H3 y_0 y_1 = (y_2, y_3); \<not> ?H4 (?H5 y_0 y_1 y_2 y_3); x_2 y_0\<rbrakk> \<Longrightarrow> x_2 y_3\<rbrakk> \<Longrightarrow> x_2 x_1 | [
"IO_Automaton.trans.Trans",
"Post.T",
"System_Specification.step",
"System_Specification.istate",
"Post.reachNT"
] | [
"'a \\<Rightarrow> 'b \\<Rightarrow> 'c \\<Rightarrow> 'a \\<Rightarrow> ('a, 'b, 'c) trans",
"(state, act, out) trans \\<Rightarrow> bool",
"state \\<Rightarrow> act \\<Rightarrow> out \\<times> state",
"state",
"state \\<Rightarrow> bool"
] | [
"datatype ('state,'act,'out) trans = Trans (srcOf: 'state) (actOf: 'act) (outOf: 'out) (tgtOf: 'state)",
"fun step :: \"state \\<Rightarrow> act \\<Rightarrow> out * state\" where\n\"step s (Cact ca) = cStep s ca\"\n|\n\"step s (Uact ua) = uStep s ua\"\n|\n\"step s (UUact uua) = uuStep s uua\"\n|\n\"step s (Ract ... | lemma_object | ###symbols
IO_Automaton.trans.Trans :::: 'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> ('a, 'b, 'c) trans
Post.T :::: (state, act, out) trans \<Rightarrow> bool
System_Specification.step :::: state \<Rightarrow> act \<Rightarrow> out \<times> state
System_Specification.istate :::: state
Post.r... |
###output
reachNT ?s \<Longrightarrow> ?P istate \<Longrightarrow> (\<And>s a ou s'. reachNT s \<Longrightarrow> step s a = (ou, s') \<Longrightarrow> \<not> T (Trans s a ou s') \<Longrightarrow> ?P s \<Longrightarrow> ?P s') \<Longrightarrow> ?P ?s###end |
ConcurrentIMP/CIMP | CIMP_vcg_rules.vcg_conj | null | valid_syn ?coms ?p ?aft ?I ?c ?Q \<Longrightarrow> valid_syn ?coms ?p ?aft ?I ?c ?R \<Longrightarrow> valid_syn ?coms ?p ?aft ?I ?c (\<lambda>s. ?Q s \<and> ?R s) | \<lbrakk> ?H1 x_1 x_2 x_3 x_4 x_5 x_6; ?H1 x_1 x_2 x_3 x_4 x_5 x_7\<rbrakk> \<Longrightarrow> ?H1 x_1 x_2 x_3 x_4 x_5 (\<lambda>y_0. x_6 y_0 \<and> x_7 y_0) | [
"CIMP_vcg_rules.valid_syn"
] | [
"('a \\<Rightarrow> ('b, 'c, 'd, 'e) com) \\<Rightarrow> 'a \\<Rightarrow> ('e \\<Rightarrow> 'c set) \\<Rightarrow> (('b, 'c, 'a, 'd, 'e) system_state \\<Rightarrow> bool) \\<Rightarrow> ('b, 'c, 'd, 'e) com \\<Rightarrow> (('b, 'c, 'a, 'd, 'e) system_state \\... | [
"abbreviation\n valid_syn :: \"('answer, 'location, 'proc, 'question, 'state) state_pred\n \\<Rightarrow> ('answer, 'location, 'question, 'state) com\n \\<Rightarrow> ('answer, 'location, 'proc, 'question, 'state) state_pred \\<Rightarrow> bool\" where\n \"valid_syn P c Q \\<equiv> coms, p... | lemma_object | ###symbols
CIMP_vcg_rules.valid_syn :::: ('a \<Rightarrow> ('b, 'c, 'd, 'e) com) \<Rightarrow> 'a \<Rightarrow> ('e \<Rightarrow> 'c set) \<Rightarrow> (('b, 'c, 'a, 'd, 'e) system_state \<Rightarrow> bool) \<Rightarrow> ('b, 'c, 'd, 'e) com \<Rightarrow> (('b, 'c,... |
###output
valid_syn ?coms ?p ?aft ?I ?c ?Q \<Longrightarrow> valid_syn ?coms ?p ?aft ?I ?c ?R \<Longrightarrow> valid_syn ?coms ?p ?aft ?I ?c (\<lambda>s. ?Q s \<and> ?R s)###end |
Diophantine_Eqns_Lin_Hom/List_Vector | List_Vector.le_append_swap | lemma le_append_swap:
assumes "length y = length v"
and "x @ y \<le>\<^sub>v w @ v"
shows "y @ x \<le>\<^sub>v v @ w" | length ?y = length ?v \<Longrightarrow> ?x @ ?y \<le>\<^sub>v ?w @ ?v \<Longrightarrow> ?y @ ?x \<le>\<^sub>v ?v @ ?w | \<lbrakk> ?H1 x_1 = ?H1 x_2; ?H2 (?H3 x_3 x_1) (?H3 x_4 x_2)\<rbrakk> \<Longrightarrow> ?H2 (?H3 x_1 x_3) (?H3 x_2 x_4) | [
"List.append",
"List_Vector.less_eq",
"List.length"
] | [
"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list",
"nat list \\<Rightarrow> nat list \\<Rightarrow> bool",
"'a list \\<Rightarrow> nat"
] | [
"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 less_eq :: \"nat list \\<Rightarrow> nat list \\<Rightarrow> bool\" (\"_/ \\<le>\\<^sub>v _\" [51, 51] 50)\n where\n \... | lemma_object | ###symbols
List.append :::: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list
List_Vector.less_eq :::: nat list \<Rightarrow> nat list \<Rightarrow> bool
List.length :::: 'a list \<Rightarrow> nat
###defs
primrec append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65) where
append_Nil: "[] @... |
###output
length ?y = length ?v \<Longrightarrow> ?x @ ?y \<le>\<^sub>v ?w @ ?v \<Longrightarrow> ?y @ ?x \<le>\<^sub>v ?v @ ?w###end |
Universal_Turing_Machine/Uncomputable | Uncomputable.inv_loop2_Oc_via_1 | lemma inv_loop2_Oc_via_1[elim]: "\<lbrakk>0 < x; inv_loop1 x (b, Oc # list)\<rbrakk> \<Longrightarrow> inv_loop2 x (Oc # b, list)" | 0 < ?x \<Longrightarrow> inv_loop1 ?x (?b, Oc # ?list) \<Longrightarrow> inv_loop2 ?x (Oc # ?b, ?list) | \<lbrakk> ?H1 < x_1; ?H2 x_1 (x_2, ?H3 ?H4 x_3)\<rbrakk> \<Longrightarrow> ?H5 x_1 (?H3 ?H4 x_2, x_3) | [
"Uncomputable.inv_loop2",
"Turing.cell.Oc",
"List.list.Cons",
"Uncomputable.inv_loop1",
"Groups.zero_class.zero"
] | [
"nat \\<Rightarrow> cell list \\<times> cell list \\<Rightarrow> bool",
"cell",
"'a \\<Rightarrow> 'a list \\<Rightarrow> 'a list",
"nat \\<Rightarrow> cell list \\<times> cell list \\<Rightarrow> bool",
"'a"
] | [
"datatype cell = Bk | Oc",
"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\")"
] | lemma_object | ###symbols
Uncomputable.inv_loop2 :::: nat \<Rightarrow> cell list \<times> cell list \<Rightarrow> bool
Turing.cell.Oc :::: cell
List.list.Cons :::: 'a \<Rightarrow> 'a list \<Rightarrow> 'a list
Uncomputable.inv_loop1 :::: nat \<Rightarrow> cell list \<times> cell list \<Rightarrow> bool
Groups.zero_class.zero ::... |
###output
0 < ?x \<Longrightarrow> inv_loop1 ?x (?b, Oc # ?list) \<Longrightarrow> inv_loop2 ?x (Oc # ?b, ?list)###end |
Standard_Borel_Spaces/Set_Based_Metric_Space | Set_Based_Metric_Space.mdist_set_bounded | lemma mdist_set_bounded:
assumes "\<And>y. y \<in> A \<Longrightarrow> mdist m x y < K" "K > 0"
shows "mdist_set m A x < K" | (\<And>y. y \<in> ?A \<Longrightarrow> mdist ?m ?x y < ?K) \<Longrightarrow> 0 < ?K \<Longrightarrow> mdist_set ?m ?A ?x < ?K | \<lbrakk>\<And>y_0. y_0 \<in> x_1 \<Longrightarrow> ?H1 x_2 x_3 y_0 < x_4; ?H2 < x_4\<rbrakk> \<Longrightarrow> ?H3 x_2 x_1 x_3 < x_4 | [
"Set_Based_Metric_Space.mdist_set",
"Groups.zero_class.zero",
"Abstract_Metric_Spaces.mdist"
] | [
"'a metric \\<Rightarrow> 'a set \\<Rightarrow> 'a \\<Rightarrow> real",
"'a",
"'a metric \\<Rightarrow> 'a \\<Rightarrow> 'a \\<Rightarrow> real"
] | [
"definition mdist_set :: \"'a metric \\<Rightarrow> 'a set \\<Rightarrow> 'a \\<Rightarrow> real\" where\n\"mdist_set m \\<equiv> Metric_space.d_set (mspace m) (mdist m)\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"definition mdist where \"mdist m \\<equiv> snd (dest_metric m)\""
] | lemma_object | ###symbols
Set_Based_Metric_Space.mdist_set :::: 'a metric \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> real
Groups.zero_class.zero :::: 'a
Abstract_Metric_Spaces.mdist :::: 'a metric \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> real
###defs
definition mdist_set :: "'a metric \<Rightarrow> 'a set \<Rightarr... |
###output
(\<And>y. y \<in> ?A \<Longrightarrow> mdist ?m ?x y < ?K) \<Longrightarrow> 0 < ?K \<Longrightarrow> mdist_set ?m ?A ?x < ?K###end |
Optics/Scenes | Scenes.scene_union_unit(1) | lemma scene_union_unit [simp]: "X \<squnion>\<^sub>S \<bottom>\<^sub>S = X" "\<bottom>\<^sub>S \<squnion>\<^sub>S X = X" | ?X \<squnion>\<^sub>S \<bottom>\<^sub>S = ?X | ?H1 x_1 ?H2 = x_1 | [
"Scenes.bot_scene",
"Scenes.union_scene"
] | [
"'a scene",
"'a scene \\<Rightarrow> 'a scene \\<Rightarrow> 'a scene"
] | [
"abbreviation bot_scene :: \"'s scene\" (\"\\<bottom>\\<^sub>S\")\nwhere \"bot_scene \\<equiv> bot\"",
"abbreviation union_scene :: \"'s scene \\<Rightarrow> 's scene \\<Rightarrow> 's scene\" (infixl \"\\<squnion>\\<^sub>S\" 65)\nwhere \"union_scene \\<equiv> sup\""
] | lemma_object | ###symbols
Scenes.bot_scene :::: 'a scene
Scenes.union_scene :::: 'a scene \<Rightarrow> 'a scene \<Rightarrow> 'a scene
###defs
abbreviation bot_scene :: "'s scene" ("\<bottom>\<^sub>S")
where "bot_scene \<equiv> bot"
abbreviation union_scene :: "'s scene \<Rightarrow> 's scene \<Rightarrow> 's scene" (infixl "\<squ... |
###output
?X \<squnion>\<^sub>S \<bottom>\<^sub>S = ?X###end |
Separation_Logic_Imperative_HOL/Automation | Automation.SLN_left | lemma SLN_left: "SLN * P = P" | Automation.SLN * ?P = ?P | ?H1 ?H2 x_1 = x_1 | [
"Automation.SLN",
"Groups.times_class.times"
] | [
"assn",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a"
] | [
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"*\" 70)"
] | lemma_object | ###symbols
Automation.SLN :::: assn
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
###defs
class times =
fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70) |
###output
Automation.SLN * ?P = ?P###end |
Refine_Imperative_HOL/IICF/Impl/IICF_HOL_List | IICF_HOL_List.HOL_list_replicate_hnr | lemma HOL_list_replicate_hnr[sepref_fr_rules]: "CONSTRAINT is_pure A \<Longrightarrow> (uncurry (return \<circ>\<circ> op_list_replicate), uncurry (RETURN \<circ>\<circ> op_list_replicate)) \<in> nat_assn\<^sup>k *\<^sub>a A\<^sup>k \<rightarrow>\<^sub>a list_assn A" | CONSTRAINT is_pure ?A \<Longrightarrow> (uncurry (return \<circ>\<circ> op_list_replicate), uncurry (RETURN \<circ>\<circ> op_list_replicate)) \<in> nat_assn\<^sup>k *\<^sub>a ?A\<^sup>k \<rightarrow>\<^sub>a list_assn ?A | ?H1 ?H2 x_1 \<Longrightarrow> (?H3 (?H4 ?H5 ?H6), ?H3 (?H4 ?H7 ?H6)) \<in> ?H8 (?H9 (?H10 ?H11) (?H10 x_1)) (?H12 x_1) | [
"Sepref_HOL_Bindings.list_assn",
"Sepref_HOL_Bindings.nat_assn",
"Sepref_Rules.hfkeep",
"Sepref_Rules.hfprod",
"Sepref_Rules.hfreft",
"Refine_Basic.RETURN",
"IICF_List.op_list_replicate",
"Heap_Monad.return",
"Misc.comp2",
"Misc.uncurry",
"Sepref_Basic.is_pure",
"Sepref_Constraints.CONSTRAINT"... | [
"('a \\<Rightarrow> 'b \\<Rightarrow> assn) \\<Rightarrow> 'a list \\<Rightarrow> 'b list \\<Rightarrow> assn",
"nat \\<Rightarrow> nat \\<Rightarrow> assn",
"('a \\<Rightarrow> 'b \\<Rightarrow> assn) \\<Rightarrow> ('a \\<Rightarrow> 'b \\<Rightarrow> assn) \\<times> ('a \\<Rightarrow> 'b \\<Rightarrow> assn)... | [
"fun list_assn :: \"('a \\<Rightarrow> 'c \\<Rightarrow> assn) \\<Rightarrow> 'a list \\<Rightarrow> 'c list \\<Rightarrow> assn\" where\n \"list_assn P [] [] = emp\"\n| \"list_assn P (a#as) (c#cs) = P a c * list_assn P as cs\"\n| \"list_assn _ _ _ = False\"",
"abbreviation \"nat_assn \\<equiv> (id_assn::nat \\<... | lemma_object | ###symbols
Sepref_HOL_Bindings.list_assn :::: ('a \<Rightarrow> 'b \<Rightarrow> assn) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> assn
Sepref_HOL_Bindings.nat_assn :::: nat \<Rightarrow> nat \<Rightarrow> assn
Sepref_Rules.hfkeep :::: ('a \<Rightarrow> 'b \<Rightarrow> assn) \<Rightarrow> ('a \<Rightar... |
###output
CONSTRAINT is_pure ?A \<Longrightarrow> (uncurry (return \<circ>\<circ> op_list_replicate), uncurry (RETURN \<circ>\<circ> op_list_replicate)) \<in> nat_assn\<^sup>k *\<^sub>a ?A\<^sup>k \<rightarrow>\<^sub>a list_assn ?A###end |
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