hcju/isabelleps huggingface dataset:
T_chop_reduce: fixes n :: "nat"   and xs :: "'a list" assumes "0 < n"   and "xs \<noteq> []" shows "T_chop n xs = T_take n xs + T_drop n xs + T_chop n (drop n xs)"
t_combine_bound: fixes ps :: "(int \<times> int) list"   and ps\<^sub>L :: "(int \<times> int) set"   and ps\<^sub>R :: "(int \<times> int) set"   and l :: "int"   and p\<^sub>0\<^sub>L :: "int \<times> int"   and p\<^sub>1\<^sub>L :: "int \<times> int"   and p\<^sub>0\<^sub>R :: "int \<times> int"   and p\<^sub>1\<^sub>R :: "int \<times> int" assumes "distinct ps"   and "sorted_snd ps"   and "set ps = ps\<^sub>L \<union> ps\<^sub>R"   and "\<forall>p\<in>ps\<^sub>L. fst p \<le> l"   and "\<forall>p\<in>ps\<^sub>R. l \<le> fst p"   and "sparse (dist (case p\<^sub>0\<^sub>L of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1\<^sub>L of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) ps\<^sub>L"   and "sparse (dist (case p\<^sub>0\<^sub>R of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1\<^sub>R of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) ps\<^sub>R" shows "t_combine (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps \<le> 10 * length ps + 3"
matrix_compose_gen: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and g :: "('a, 'c) vec \<Rightarrow> ('a, 'd) vec" assumes "Vector_Spaces.linear (*s) (*s) f"   and "Vector_Spaces.linear (*s) (*s) g" shows "matrix (g \<circ> f) = matrix g ** matrix f"
WatchInvariantsAfterApplyBackjump: fixes state :: "State"   and F0 :: "Literal list list" assumes "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "getConflictFlag state"   and "InvariantUniqC (getC state)"   and "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "InvariantCEntailed (getConflictFlag state) F0 (getC state)"   and "InvariantClCharacterization (getCl state) (getC state) (getM state)"   and "InvariantCllCharacterization (getCl state) (getCll state) (getC state) (getM state)"   and "InvariantClCurrentLevel (getCl state) (getM state)"   and "isUIP (opposite (getCl state)) (getC state) (getM state)"   and "0 < currentLevel (getM state)" shows "let state' = applyBackjump state in InvariantWatchesEl (getF state') (getWatch1 state') (getWatch2 state') \<and> InvariantWatchesDiffer (getF state') (getWatch1 state') (getWatch2 state') \<and> InvariantWatchCharacterization (getF state') (getWatch1 state') (getWatch2 state') (getM state') \<and> InvariantWatchListsContainOnlyClausesFromF (getWatchList state') (getF state') \<and> InvariantWatchListsUniq (getWatchList state') \<and> InvariantWatchListsCharacterization (getWatchList state') (getWatch1 state') (getWatch2 state')"
TensorDiag_in_Hom: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Diag t"   and "Diag u" shows "t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u \<in> Hom (Dom t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> Dom u) (Cod t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> Cod u)"
revis_superexpansion_ext: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a"   and \<psi> :: "'a" assumes "K = Cn A" shows "(K \<^bold>* \<phi>) \<inter> (K \<^bold>* \<psi>) \<subseteq> K \<^bold>* (\<phi> .\<or>. \<psi>)"
L2_set_triangle_ineq: fixes f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real"   and A :: "'c set" shows "L2_set (\<lambda>i. f i + g i) A \<le> L2_set f A + L2_set g A"
cut_greater_Int_conv: fixes I :: "'a set"   and t :: "'a" shows "I \<down>> t = I \<inter> {t<..}"
Sup_assertion: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and post :: "'c \<Rightarrow> 'c"   and X :: "'c set" assumes "class.complete_mbt_algebra Inf Sup inf less_eq less sup bot top one times dual dual_star omega star post"   and "X \<subseteq> mbt_algebra.assertion one times inf less_eq dual top" shows "Sup X \<in> mbt_algebra.assertion one times inf less_eq dual top"
inv_unique: fixes y :: "'a"   and x :: "'a"   and y' :: "'a" assumes "y \<otimes> x = \<one>"   and "x \<otimes> y' = \<one>"   and "x \<in> carrier \<G>"   and "y \<in> carrier \<G>"   and "y' \<in> carrier \<G>" shows "y = y'"
weakly_justified_write_seen_hb_read_committed: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat"   and J :: "nat \<Rightarrow> ('c, 'b) justifying_execution"   and r :: "nat"   and n :: "nat" assumes "P \<turnstile> (E, ws) weakly_justified_by J"   and "r \<in> read_actions (justifying_exec (J n))"   and "r \<in> committed (J n)" shows "ws (action_translation (J n) r) \<in> action_translation (J n) ` committed (J n)"
simps_1: fixes y :: "'a" shows "flat_1 (allow (allow y)) = allow y"
box_Int_box: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "box a b \<inter> box c d = box (\<Sum>i\<in>Basis. max (a \<bullet> i) (c \<bullet> i) *\<^sub>R i) (\<Sum>i\<in>Basis. min (b \<bullet> i) (d \<bullet> i) *\<^sub>R i)"
isSimplePath_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" shows "Graph.isSimplePath c u p v \<equiv> Graph.isPath c u p v \<and> distinct (cf'.pathVertices u p)"
assms_2: fixes x :: "'s"   and y :: "'s" assumes "x \<in> C"   and "(x, y) \<in> acc" shows "y \<in> C"
not_strongly_preferred_iff: fixes a :: "'alt"   and b :: "'alt" assumes "a \<in> alts"   and "b \<in> alts" shows "(\<not> a \<prec>[R i] b) = R i b a"
a_ge_0: fixes k :: "nat"   and as :: "real list"   and bs :: "real list"   and a :: "real" assumes "akra_bazzi_params k as bs"   and "a \<in> set as" shows "0 \<le> a"
arr_Rel_eqI: fixes \<alpha> :: "V"   and S :: "V"   and T :: "V" assumes "arr_Rel \<alpha> S"   and "arr_Rel \<alpha> T"   and "S\<lparr>ArrVal\<rparr> = T\<lparr>ArrVal\<rparr>"   and "S\<lparr>ArrDom\<rparr> = T\<lparr>ArrDom\<rparr>"   and "S\<lparr>ArrCod\<rparr> = T\<lparr>ArrCod\<rparr>" shows "S = T"
set_image_mset: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'b multiset" shows "set_mset (f `# M) = f ` set_mset M"
select_convs_1: fixes carrier :: "'c set"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and more :: "'d" shows "(\<otimes>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr>\<^esub>) = mult"
inject_3: fixes x41 :: "('d, 'e, 'f) com"   and x42 :: "('d, 'e, 'f) com"   and y41 :: "('d, 'e, 'f) com"   and y42 :: "('d, 'e, 'f) com" shows "(Seq x41 x42 = Seq y41 y42) = (x41 = y41 \<and> x42 = y42)"
recognises_must_be_possible_step: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and h :: "String.literal \<times> value list"   and t :: "(String.literal \<times> value list) list" assumes "recognises_execution e s r (h # t)" shows "\<exists>aa ba. (aa, ba) |\<in>| possible_steps e s r (fst h) (snd h)"
degree_add_le: fixes p :: "'b poly"   and n :: "nat"   and q :: "'b poly" assumes "degree p \<le> n"   and "degree q \<le> n" shows "degree (p + q) \<le> n"
get_scdom_component_ptrs_same_owner_document: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and owner_document :: "(_) document_ptr" assumes "heap_is_wellformed h"   and "DocumentClass.type_wf h"   and "DocumentClass.known_ptrs h"   and "h \<turnstile> get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "ptr' \<in> set sc"   and "h \<turnstile> get_owner_document ptr \<rightarrow>\<^sub>r owner_document" shows "h \<turnstile> get_owner_document ptr' \<rightarrow>\<^sub>r owner_document"
0: shows "dims Aw = replicate (2 ^ (length rs - 2)) (last rs)"
unit_ball_vol_pos: fixes n :: "real" assumes "0 \<le> n" shows "0 < unit_ball_vol n"
wt_fresh_pgwt_term_exists: fixes T :: "('fun, 'var) Term.term set"   and s :: "('fun, 'var) Term.term" assumes "finite T"   and "wf\<^sub>t\<^sub>r\<^sub>m s"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s T" obtains t :: "('fun, 'var) Term.term" where "\<Gamma> t = \<Gamma> s" and "{} \<turnstile>\<^sub>c t" and "\<forall>s\<in>T. \<forall>u\<in>subterms s. u \<notin> subterms t"
mkcan_def: fixes a :: "'c local.term set"   and b :: "'c local.term set" shows "mkcan a b = mkarr (Inv (COD b\<^bold>\<down>) \<^bold>\<cdot> DOM a\<^bold>\<down>)"
apply_guards_foldr: fixes G :: "vname gexp list"   and s :: "vname \<Rightarrow> value option" shows "apply_guards G s = (gval (foldr gAnd G (Bc True)) s = true)"
assms_1: shows "\<turnstile> imply (A # G) C"
P7: shows "B Midpoint H H'' \<and> Cong A H A H'"
assoc_naturality: fixes \<mu> :: "'a"   and \<nu> :: "'a"   and \<tau> :: "'a" assumes "B.arr \<mu>"   and "B.arr \<nu>"   and "B.arr \<tau>"   and "src\<^sub>B \<mu> = trg\<^sub>B \<nu>"   and "src\<^sub>B \<nu> = trg\<^sub>B \<tau>" shows "\<a>\<^sub>B[B.cod \<mu>, B.cod \<nu>, B.cod \<tau>] \<cdot>\<^sub>B ((\<mu> \<star>\<^sub>B \<nu>) \<star>\<^sub>B \<tau>) = (\<mu> \<star>\<^sub>B \<nu> \<star>\<^sub>B \<tau>) \<cdot>\<^sub>B \<a>\<^sub>B[B.dom \<mu>, B.dom \<nu>, B.dom \<tau>]"
absolutely_integrable_spike_set_eq: fixes S :: "'c set"   and T :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "negligible {x \<in> S - T. f x \<noteq> (0::'d)}"   and "negligible {x \<in> T - S. f x \<noteq> (0::'d)}" shows "(f absolutely_integrable_on S) = (f absolutely_integrable_on T)"
opt_bst2_opt_bst: fixes w :: "int \<Rightarrow> int \<Rightarrow> nat"   and i :: "int"   and j :: "int" assumes "Optimal_BST2 w" shows "Optimal_BST2.opt_bst2 w i j = Optimal_BST.opt_bst w i j"
pvalid: shows "wellformed_tokens p \<and> wellformed_item (finished_item \<alpha>) \<and> r \<le> length p \<and> length (chars p) = length Doc \<and> chars (take r p) = [] \<and> is_derivation (take r (terminals p) @ \<SS> # \<gamma>) \<and> derives \<alpha> (drop r (terminals p))"
intv_CL_lub: fixes a :: "'a"   and b :: "'a"   and S :: "'a set" assumes "a \<in> A"   and "b \<in> A"   and "interval r a b \<noteq> {}"   and "S \<subseteq> interval r a b" shows "\<exists>L. isLub S \<lparr>pset = interval r a b, order = induced (interval r a b) r\<rparr> L"
le_inf_word: fixes ex :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and en :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>2 :: "'a stream"   and w\<^sub>1 :: "'a stream"   and p :: "'b" assumes "transition_system_traces ex en ind"   and "traces.le_inf ind w\<^sub>2 w\<^sub>1"   and "transition_system.run ex en w\<^sub>1 p" shows "transition_system.run ex en w\<^sub>2 p"
ag_l_zero: fixes A :: "('e, 'f) aGroup_scheme"   and x :: "'e" assumes "aGroup A"   and "x \<in> carrier A" shows "\<zero>\<^bsub>A\<^esub> \<plusminus>\<^bsub>A\<^esub> x = x"
E\<^sub>i\<^sub>i: fixes x :: "i"   and y :: "i" shows "E (x \<cdot> y) \<^bold>\<leftarrow> E x \<^bold>\<and> E y \<^bold>\<and> \<^bold>\<not> (\<^bold>\<forall>x. \<^bold>\<not> (x \<cdot> x \<cong> x \<^bold>\<and> x \<cdot> x \<cong> x \<^bold>\<and> x \<cdot> y \<cong> y))"
discounted_measurable: fixes X :: "'c \<Rightarrow> real"   and N :: "'c measure"   and r :: "real"   and m :: "nat" assumes "X \<in> borel_measurable N" shows "discounted_value r (\<lambda>m. X) m \<in> borel_measurable N"
assms_1: shows "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<CC>"
decr_grading_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and d :: "'c \<Rightarrow> nat" assumes "class.monoid_add plus zero" shows "monoid_add.decr_grading plus d = (SOME f. monoid_add.hom_grading_fun plus d f)"
distinct_1: fixes x1 :: "'d"   and x2 :: "'e" shows "Normal x1 \<noteq> Fault x2"
assms_2: shows "Bisimulation_Invariant B C sim2 PB PC"
Ord_Sup: fixes X :: "V set" assumes "\<And>x. x \<in> X \<Longrightarrow> Ord x" shows "Ord (\<Squnion> X)"
at_within_ball_bot_iff: fixes x :: "'b"   and y :: "'b"   and r :: "real" shows "(at x within ball y r = bot) = (r = 0 \<or> x \<notin> cball y r)"
restr_eqI: fixes A :: "'a set"   and A' :: "'a set"   and F :: "'a \<Rightarrow> 'b"   and F' :: "'a \<Rightarrow> 'b" assumes "A = A'"   and "\<And>x. x \<in> A \<Longrightarrow> F x = F' x" shows "restrict F A = restrict F' A'"
simps_2: fixes c :: "'d"   and cs :: "'d list"   and t :: "'e"   and u :: "'d \<Rightarrow> 'e" shows "[c # cs\<rightarrow>t]u = ([cs\<rightarrow>t]u)(c := t)"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'c tree \<Rightarrow> 'b"   and f3 :: "'c tree \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b"   and up2 :: "'c up2" shows "P (case up2 of Same2 \<Rightarrow> f1 | Bal2 x \<Rightarrow> f2 x | Unbal2 x xa xb \<Rightarrow> f3 x xa xb) = ((up2 = Same2 \<longrightarrow> P f1) \<and> (\<forall>x2. up2 = Bal2 x2 \<longrightarrow> P (f2 x2)) \<and> (\<forall>x31 x32 x33. up2 = Unbal2 x31 x32 x33 \<longrightarrow> P (f3 x31 x32 x33)))"
map_gpv'_map_gpv_swap: fixes f :: "'d \<Rightarrow> 'a"   and g :: "'e \<Rightarrow> 'b"   and h :: "'c \<Rightarrow> 'f"   and f' :: "'g \<Rightarrow> 'd"   and gpv :: "('g, 'e, 'f) gpv" shows "map_gpv' f g h (map_gpv f' id gpv) = map_gpv (f \<circ> f') id (map_gpv' id g h gpv)"
emeasure_lborel_cbox: fixes l :: "'b"   and u :: "'b" assumes "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b" shows "emeasure lborel (cbox l u) = ennreal (prod ((\<bullet>) (u - l)) Basis)"
not_eq_order_implies_strict: fixes a :: "'k multiset"   and b :: "'k multiset" assumes "a \<noteq> b"   and "b \<subseteq># a" shows "b \<subset># a"
of_char_of: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "comm_semiring_1.of_char times one plus zero (unique_euclidean_semiring_with_bit_shifts.char_of plus times one bit a) = modulo a (numeral.numeral one plus (num.Bit0 (num.Bit0 (num.Bit0 (num.Bit0 (num.Bit0 (num.Bit0 (num.Bit0 (num.Bit0 num.One)))))))))"
getFr_var: fixes xs :: "'var list"   and ts :: "'trm list"   and \<phi>s :: "'fmla list"   and t :: "'trm" assumes "set xs \<subseteq> var"   and "set ts \<subseteq> trm"   and "set \<phi>s \<subseteq> fmla"   and "t \<in> set ts" shows "getFr xs ts \<phi>s \<notin> set xs"
A1': fixes a :: "hyp2"   and b :: "hyp2" shows "a b \<congruent>\<^sub>K b a"
indpre_def: shows "indpre = (\<lambda>i. indicat_real (prY i \<inter> space M))"
psi_split: fixes n :: "nat" shows "psi n = theta n + psi_even n + psi_odd n"
match_concat: fixes bss :: "'b list list" shows "\<forall>as. length as = length (concat bss) \<longrightarrow> (\<exists>css. as = concat css \<and> list_all2 (\<lambda>xs ys. length xs = length ys) css bss)"
connected_closedin_eq: fixes S :: "'c set" shows "connected S = (\<nexists>E1 E2. closedin (top_of_set S) E1 \<and> closedin (top_of_set S) E2 \<and> E1 \<union> E2 = S \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
\<iota>\<^sub>0_is: shows "\<iota>\<^sub>0 = Infer (CAs @ [DA]) E"
lappend_ltl: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "\<not> lnull xs" shows "ltl xs $ ys = ltl (xs $ ys)"
permute_name_def: fixes p :: "perm"   and a :: "name" shows "p \<bullet> a = Abs_name (p \<bullet> Rep_name a)"
generate_consistent: fixes K :: "'c set"   and H :: "'c set" assumes "K \<subseteq> H"   and "subgroup H H" shows "generate (H\<lparr>carrier := H\<rparr>) K = generate H K"
Ch_range': fixes x :: "'a"   and h :: "'c"   and X :: "'a set" assumes "x \<in> Ch h X" shows "x \<in> X \<and> Xh x = h"
subst_new: fixes c :: "'a"   and s :: "'a FOL_Fitting.term"   and p :: "('a, 'b) form"   and m :: "nat" assumes "new_term c s"   and "new c p" shows "new c (p[s/m])"
word_eq_iff: fixes x :: "'b word"   and y :: "'b word" shows "(x = y) = (\<forall>n<LENGTH('b). x !! n = y !! n)"
rotate_Suc: fixes n :: "nat"   and xs :: "'a list" shows "rotate (Suc n) xs = rotate1 (rotate n xs)"
RedFAssNull: fixes F :: "char list"   and D :: "char list"   and v :: "val"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" shows "P \<turnstile> \<langle>null\<bullet>F{D} := Val v,s\<rangle> \<rightarrow> \<langle>THROW NullPointer,s\<rangle>"
comp_in_carr: fixes x :: "'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> ('d, 'e) monoid_scheme"   and I :: "'c set"   and i :: "'c" assumes "x \<in> carrier (DirProds G I)"   and "i \<in> I" shows "x i \<in> carrier (G i)"
intros: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "'d" assumes "r a b" shows "Rangep r b"
sup_bot_left: fixes sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and bot :: "'g"   and x :: "'g" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "sup bot x = x"
joint_w_cmpser: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and m :: "nat"   and g :: "nat \<Rightarrow> 'a set" assumes "w_cmpser G n f"   and "w_cmpser G m g"   and "\<natural>f n \<triangleright> g 0" shows "w_cmpser G (Suc (n + m)) (jointfun n f m g)"
N_1: shows "fb.N 1 = 1 / (2 * (2 - 1))"
init_ex_mor: fixes s1 :: "('d, 'e, 'f) F1 \<Rightarrow> 'e"   and s2 :: "('d, 'e, 'f) F2 \<Rightarrow> 'f" shows "\<exists>f g. mor UNIV UNIV ctor1 ctor2 UNIV UNIV s1 s2 f g"
chain_subdiv_chain_character: fixes one_chain1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and subdiv :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" shows "chain_subdiv_chain one_chain1 subdiv = (\<exists>f. \<Union> (f ` one_chain1) = subdiv \<and> (\<forall>(k, \<gamma>)\<in>one_chain1. if k = 1 then chain_subdiv_path \<gamma> (f (k, \<gamma>)) else chain_subdiv_path (reversepath \<gamma>) (f (k, \<gamma>))) \<and> (\<forall>p\<in>one_chain1. \<forall>p'\<in>one_chain1. p \<noteq> p' \<longrightarrow> f p \<inter> f p' = {}) \<and> (\<forall>x\<in>one_chain1. finite (f x)))"
connected_space_iff_components_subset_singleton: fixes X :: "'b topology" shows "connected_space X = (\<exists>a. connected_components_of X \<subseteq> {a})"
sort_key_simps_1: fixes f :: "'b \<Rightarrow> 'a" shows "sort_key f [] = []"
gt_subst: fixes \<rho> :: "'v \<Rightarrow> ('s, 'v) tm"   and t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "wary_subst \<rho>"   and "wary t"   and "wary s"   and "local.kbo_std.gt t s" shows "local.kbo_std.gt (subst \<rho> t) (subst \<rho> s)"
lam_eval_prod: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and h :: "'a" assumes "ide a"   and "ide b"   and "ide c"   and "\<guillemotleft>h : a \<rightarrow> exp b c\<guillemotright>" shows "\<Lambda> a b c (eval b c \<cdot> (h \<otimes> b)) = h"
right_unique_rel_pred: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "right_total A" shows "right_unique (rel_pred A)"
size_eq_sum_mset: fixes M :: "'a multiset" shows "size M = (\<Sum>a\<in>#M. 1)"
equivP_pick_in: fixes P :: "'c \<Rightarrow> bool"   and \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and X :: "'c set" assumes "equivP P \<phi>"   and "(P /// \<phi>) X" shows "pick X \<in> X"
converse_inject: fixes r :: "('d \<times> 'c) set"   and s :: "('d \<times> 'c) set" shows "(r\<inverse> = s\<inverse>) = (r = s)"
ty_exhaust: fixes ty :: "ty" shows "ty = TBase \<or> (\<exists>\<sigma> \<tau>. ty = \<sigma> \<rightarrow> \<tau>) \<or> (\<exists>\<sigma>. ty = T \<sigma>)"
matrix'_eq_matrix: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" shows "matrix' cart_basis' cart_basis' f = matrix f"
rpsem_Seq_simps: fixes \<alpha> :: "prog"   and \<beta> :: "prog"   and \<nu> :: "fin_string \<Rightarrow> real"   and b :: "fin_string \<Rightarrow> real" shows "([\<alpha> ;; \<beta>]\<nu> \<downharpoonleft> b) = (\<exists>\<mu>. ([\<alpha>]\<nu> \<downharpoonleft> \<mu>) \<and> ([\<beta>]\<mu> \<downharpoonleft> b))"
min_2: fixes m :: "nat" assumes "sdrop m s = sconst x" shows "min \<le> m"
norm_diff_triangle_ineq: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (minus (plus a b) (plus c d)) \<le> norm (minus a c) + norm (minus b d)"
B3: shows "TRel = _Collect (T1, T2) ((TargetTerm T1, TargetTerm T2) \<in> Rel)"
assms_4: shows "\<guillemotleft>\<psi> : g \<Rightarrow> g'\<guillemotright>"
ipset_from_netmask_base_mask_consume: fixes base :: "'a word"   and m :: "nat" shows "ipset_from_netmask (base && ~~ mask (LENGTH('a) - m)) (~~ mask (LENGTH('a) - m)) = ipset_from_netmask base (~~ mask (LENGTH('a) - m))"
B: fixes x :: "'a" assumes "x \<in> A" shows "f x \<in> B"
consistent_def: fixes valuation :: "Literal list" shows "consistent valuation \<equiv> \<not> inconsistent valuation"
assms: fixes c :: "nat"   and n :: "nat" assumes "0 < c" shows "\<exists>m>n. c * g m < f m"
\<Sigma>_protocols_to_commitments_axioms_def: fixes init :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd) spmf"   and response :: "'c \<Rightarrow> 'b \<Rightarrow> 'e \<Rightarrow> 'f spmf"   and check :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and Rel :: "('a \<times> 'b) set"   and S_raw :: "'a \<Rightarrow> 'e \<Rightarrow> ('d \<times> 'f) spmf"   and \<A>ss :: "'a \<Rightarrow> 'd \<times> 'e \<times> 'f \<Rightarrow> 'd \<times> 'e \<times> 'f \<Rightarrow> 'b spmf"   and challenge_space :: "'e set"   and valid_pub :: "'a set"   and G :: "('a \<times> 'b) spmf" shows "\<Sigma>_protocols_to_commitments_axioms init response check Rel S_raw \<A>ss challenge_space valid_pub G \<equiv> (\<Sigma>_protocols_base.\<Sigma>_protocol init response check Rel S_raw \<A>ss challenge_space valid_pub \<and> (\<forall>h w. (h, w) \<in> set_spmf G \<longrightarrow> (h, w) \<in> Rel)) \<and> lossless_spmf G \<and> (\<forall>h w. lossless_spmf (init h w)) \<and> (\<forall>r w e. lossless_spmf (response r w e))"