hcju/isabelleps huggingface dataset:
src_simps_2: fixes V\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i>\<^sub>B :: "'b \<Rightarrow> 'b"   and src\<^sub>B :: "'b \<Rightarrow> 'b"   and trg\<^sub>B :: "'b \<Rightarrow> 'b"   and F :: "('b bicategorical_language.term, 'b) concrete_category.arr" assumes "strictified_bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B"   and "partial_magma.arr (concrete_category.COMP {A. bicategorical_language.Nml V\<^sub>B src\<^sub>B trg\<^sub>B A \<and> bicategorical_language.Ide V\<^sub>B src\<^sub>B trg\<^sub>B A} (\<lambda>A B. {\<mu>. bicategorical_language.Src src\<^sub>B A = bicategorical_language.Src src\<^sub>B B \<and> bicategorical_language.Trg trg\<^sub>B A = bicategorical_language.Trg trg\<^sub>B B \<and> partial_magma.in_hom V\<^sub>B \<mu> (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) A) (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) B)}) (\<lambda>_ _ _. V\<^sub>B)) F" shows "concrete_category.Cod (strictified_bicategory.src V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B F) = bicategorical_language.Src src\<^sub>B (concrete_category.Dom F)"
ass_dvd: shows "ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))) dvd A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))"
groupWIs_code: fixes c :: "parts_connection"   and rs :: "'a simple_rule list" shows "groupWIs c rs = groupWIs2 c rs"
inv\<^sub>2_step: fixes T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat" assumes "sorted n"   and "inv\<^sub>2 T A j"   and "j < n" shows "inv\<^sub>2 (T(min\<^sub>k T m := T (min\<^sub>k T m) + t (Suc j))) (A(min\<^sub>k T m := A (min\<^sub>k T m) \<union> {Suc j})) (Suc j)"
analz_insert_pan: fixes evs :: "event list"   and K :: "nat"   and P :: "nat" assumes "evs \<in> set_pur"   and "K \<notin> range priEK" shows "(Pan P \<in> analz (insert (Key K) (knows Spy evs))) = (Pan P \<in> analz (knows Spy evs))"
component_of_map_component: fixes f :: "nat \<Rightarrow> nat"   and v :: "('a, 'b) pp \<times> nat" shows "snd (map_component_pprod f v) = f (snd v)"
simps_36: fixes vd :: "'a acom"   and ve :: "'a acom"   and v :: "'a"   and va :: "bexp"   and vb :: "'a acom"   and vc :: "'a" shows "(vd;; ve \<le> {v} WHILE va DO vb {vc}) = False"
image_ex1_eq: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and b :: "'b" assumes "inj_on f A" shows "(b \<in> f ` A) = (\<exists>!x. x \<in> A \<and> b = f x)"
a: shows "comp_par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t public arity Ana \<Gamma> Pair A M C"
polyhedron_imp_convex: fixes S :: "'a set" assumes "polyhedron S" shows "convex S"
w_simps_6: shows "cod \<pi>.chine = \<pi>.chine"
inf_r_def: fixes a :: "'a"   and b :: "'a" shows "a \<sqinter> b = (a l\<rightarrow> b) * a"
arr_cod: fixes f :: "'a" assumes "S.arr f" shows "S.arr (S.cod f)"
abc_dec_1_LE_def: shows "abc_dec_1_LE \<equiv> inv_image lex_triple abc_dec_1_measure"
walk_comp: fixes xs :: "'a list"   and ys :: "'a list" assumes "walk xs"   and "walk ys"   and "xs = [] \<or> ys = [] \<or> last xs \<rightarrow> hd ys" shows "walk (xs @ ys)"
boruvka_inner_invariant_def: fixes j :: "'a"   and f :: "'a"   and h :: "'a"   and g :: "'a"   and d :: "'a" shows "boruvka_inner_invariant j f h g d \<equiv> boruvka_outer_invariant f g \<and> g \<noteq> bot \<and> vector j \<and> regular j \<and> boruvka_outer_invariant h g \<and> forest h \<and> forest_components h \<le> forest_components f \<and> big_forest (forest_components h) d \<and> d * top \<le> - j \<and> forest_components h * j = j \<and> forest_components f = (forest_components h * (d \<squnion> d\<^sup>T))\<^sup>\<star> * forest_components h \<and> f \<squnion> f\<^sup>T = h \<squnion> h\<^sup>T \<squnion> d \<squnion> d\<^sup>T \<and> (\<forall>a b. bf_between_arcs a b (forest_components h) d \<and> a \<le> - forest_components h \<sqinter> - - g \<and> b \<le> d \<longrightarrow> sum_class.sum (b \<sqinter> g) \<le> sum_class.sum (a \<sqinter> g)) \<and> regular d"
inject_2: fixes x2 :: "Early_Semantics.freeRes"   and x1 :: "pi"   and y2 :: "Early_Semantics.freeRes"   and y1 :: "pi" shows "(x2 \<prec> x1 = y2 \<prec> y1) = (x2 = y2 \<and> x1 = y1)"
g_def: fixes xy :: "'a \<times> 'b" shows "g xy = (if P xy then Some (fst xy) else None)"
elims: fixes x :: "perm"   and xa :: "('a, 'b) L_state"   and y :: "('a, 'b) L_state" assumes "x \<bullet> xa = y" obtains p :: "perm" and x :: "'b \<times> 'b fs_set \<times> 'a" where "xa__ = pa__" and "xa = AC xb__" and "y = AC (pa__ \<bullet> xb__)"   | p :: "perm" and x :: "'b fs_set \<times> 'a" where "xa__ = p" and "xa = EF x" and "y = EF (p \<bullet> x)"
dp: fixes i :: "nat"   and n :: "nat" assumes "i \<le> n" shows "\<gamma> i fine p n"
monom_vars_list_1: shows "monom_vars_list 1 = []"
square_integrable_lmult: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and c :: "real" assumes "f square_integrable S" shows "(\<lambda>x. c * f x) square_integrable S"
ord_adds_term: fixes u :: "'t"   and v :: "'t" assumes "u adds\<^sub>t v" shows "u \<preceq>\<^sub>t v"
simps_5: fixes v :: "'a formula"   and va :: "'a formula" shows "is_cnf (v \<^bold>\<or> va) = is_disj (v \<^bold>\<or> va)"
assms_1: shows "f \<in> A \<rightarrow> B"
hs_size_def: shows "hs_size \<equiv> hm_size"
distributive_1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> R"   and "b \<in> R"   and "c \<in> R" shows "a \<cdot> (b + c) = a \<cdot> b + a \<cdot> c"
rtranclE: fixes a :: "'c"   and b :: "'c"   and r :: "('c \<times> 'c) set" assumes "(a, b) \<in> r\<^sup>*" obtains "a = b"   | y :: "'c" where "(a, y) \<in> r\<^sup>*" and "(y, b) \<in> r"
smult_mat_mult_mat_vec_assoc: fixes A :: "'a mat"   and n :: "nat"   and m :: "nat"   and w :: "'a vec"   and a :: "'a" assumes "A \<in> carrier_mat n m"   and "w \<in> carrier_vec m" shows "a \<cdot>\<^sub>m A *\<^sub>v w = a \<cdot>\<^sub>v (A *\<^sub>v w)"
meet_leftcomm: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
simps: fixes a1 :: "cmd \<times> (char list \<Rightarrow> val option)"   and a2 :: "cmd \<times> (char list \<Rightarrow> val option)" shows "red a1 a2 = ((\<exists>V e s. a1 = (V:=e, s) \<and> a2 = (Skip, s(V := interpret e s))) \<or> (\<exists>c\<^sub>1 s c\<^sub>1' s' c\<^sub>2. a1 = (c\<^sub>1;; c\<^sub>2, s) \<and> a2 = (c\<^sub>1';; c\<^sub>2, s') \<and> \<langle>c\<^sub>1,s\<rangle> \<rightarrow> \<langle>c\<^sub>1',s'\<rangle>) \<or> (\<exists>c\<^sub>2 s. a1 = (Skip;; c\<^sub>2, s) \<and> a2 = (c\<^sub>2, s)) \<or> (\<exists>b s c\<^sub>1 c\<^sub>2. a1 = (if (b) c\<^sub>1 else c\<^sub>2, s) \<and> a2 = (c\<^sub>1, s) \<and> interpret b s = Some true) \<or> (\<exists>b s c\<^sub>1 c\<^sub>2. a1 = (if (b) c\<^sub>1 else c\<^sub>2, s) \<and> a2 = (c\<^sub>2, s) \<and> interpret b s = Some false) \<or> (\<exists>b s c. a1 = (while (b) c, s) \<and> a2 = (c;; while (b) c, s) \<and> interpret b s = Some true) \<or> (\<exists>b s c. a1 = (while (b) c, s) \<and> a2 = (Skip, s) \<and> interpret b s = Some false))"
invariantI: fixes F :: "'c program"   and A :: "'c set" assumes "Init F \<subseteq> A"   and "F \<in> stable A" shows "F \<in> invariant A"
borel_measurable_nn_integral_fst: fixes f :: "'c \<times> 'a \<Rightarrow> ennreal"   and M1 :: "'c measure" assumes "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)" shows "(\<lambda>x. \<integral>\<^sup>+ y. f (x, y) \<partial>M) \<in> borel_measurable M1"
l_set_disconnected_nodes_axioms: shows "l_set_disconnected_nodes ShadowRootClass.type_wf set_disconnected_nodes set_disconnected_nodes_locs"
get_disconnected_nodes_ok: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr" assumes "type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "document_ptr |\<in>| document_ptr_kinds h" shows "h \<turnstile> ok get_disconnected_nodes document_ptr"
Gcd_fin_dvd: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A" shows "Gcd\<^sub>f\<^sub>i\<^sub>n A dvd a"
has_white_path_to_def: fixes x :: "'h"   and y :: "'h" shows "x has_white_path_to y = (\<lambda>s. (\<lambda>x y. (x points_to y) s \<and> white y s)\<^sup>*\<^sup>* x y)"
iso_in_hom_def: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" shows "VV.iso_in_hom f a b \<equiv> VV.in_hom f a b \<and> VV.iso f"
interval_lebesgue_integral_lborel_01_cong: fixes f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a" assumes "\<And>x. x \<in> {0<..<1} \<Longrightarrow> f x = g x" shows "interval_lebesgue_integral lborel 0 1 f = interval_lebesgue_integral lborel 0 1 g"
nat_not_less1: fixes n :: "nat"   and m :: "nat" assumes "n \<noteq> 0" shows "(\<not> m < n) = (\<not> m \<le> n - Suc 0)"
Inf_from_mono: fixes N :: "'f set"   and N' :: "'f set" assumes "N \<subseteq> N'" shows "Inf_from N \<subseteq> Inf_from N'"
mult_right_mono_neg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.ordered_ring plus minus zero less_eq less times uminus"   and "less_eq b a"   and "less_eq c zero" shows "less_eq (times a c) (times b c)"
rec_zero: fixes e :: "'b"   and r :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" shows "rec e r zero = e"
vimage_eq_image: fixes n :: "nat" shows "(*\<^sub>R) (root DIM('a) (real (Suc n))) -` S = (*\<^sub>R) (1 / root DIM('a) (real (Suc n))) ` S"
set_tag_name_locs_impl: shows "set_tag_name_locs = CD.a_set_tag_name_locs"
simps_1: fixes P :: "'a \<Rightarrow> bool" shows "find_indices P [] = []"
assms_2: shows "Ball (set ms\<^sub>2) valid_node"
ksup_assoc: fixes f :: "'b \<Rightarrow> 'c set"   and g :: "'b \<Rightarrow> 'c set"   and h :: "'b \<Rightarrow> 'c set" shows "f \<squnion> g \<squnion> h = f \<squnion> (g \<squnion> h)"
distrib_left: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<sqinter> (b \<squnion> c) = a \<sqinter> b \<squnion> a \<sqinter> c"
pr_weak_trans: fixes fn :: "'f \<times> nat"   and gm :: "'f \<times> nat"   and hk :: "'f \<times> nat" assumes "pr_weak fn gm"   and "pr_weak gm hk" shows "pr_weak fn hk"
less_tail: fixes s :: "'a list"   and t :: "'a list" assumes "s \<noteq> []"   and "s < t" shows "tl s < tl t"
eout_know_nonKS_k: fixes m :: "Keys"   and A :: "specID" assumes "m \<notin> specKeys A"   and "eout A (kE m)"   and "eoutKnowCorrect A (kKS m)" shows "know A (kKS m)"
upper: fixes x :: "'a measure"   and A :: "'a measure set" assumes "x \<in> A" shows "x \<le> Sup A"
hcomp_in_vhomE: fixes \<nu> :: "'a"   and \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<nu> \<star> \<mu> : f \<Rightarrow> g\<guillemotright>" obtains "arr \<mu>" and "arr \<nu>" and "src \<nu> = trg \<mu>" and "src \<mu> = src f" and "src \<mu> = src g" and "trg \<nu> = trg f" and "trg \<nu> = trg g"
ft_dghm_v11_on_Hom: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"   and "b \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "v11 (\<GG>\<lparr>ArrMap\<rparr> \<restriction>\<^sup>l\<^sub>\<circ> Hom \<BB> a b)"
domain_eq: shows "Domainp (pcr_poly (=)) = (\<lambda>f. \<forall>\<^sub>\<infinity>n. f n = (0::'b))"
is_ubI: fixes x :: "'b"   and u :: "'b"   and r :: "('b \<times> 'b) set"   and y :: "'b" assumes "(x, u) \<in> r"   and "(y, u) \<in> r" shows "is_ub r x y u"
span_Gram_Schmidt_upt_k: fixes k :: "nat"   and A :: "(('a, 'b) vec, 'c) vec" assumes "k < ncols A" shows "span (columns A) = span (columns (Gram_Schmidt_upt_k A k))"
that: fixes t :: "nat \<Rightarrow> 'a"   and l :: "'a" assumes "\<And>n. t n \<in> T"   and "\<And>n. t n \<noteq> l"   and "t \<longlonglongrightarrow> l"   and "l \<in> S" shows "thesis"
borel_measurable_RN_deriv: fixes M :: "'b measure"   and N :: "'b measure" shows "RN_deriv M N \<in> borel_measurable M"
lconcat_def: shows "lconcat \<equiv> llist.fixp_fun (\<lambda>lconcat xss. case xss of LNil \<Rightarrow> LNil | LCons xs xss' \<Rightarrow> lappend xs (lconcat xss'))"
A_def: shows "A = {x \<in> space M. 0 < f x}"
MN_imp_regular: fixes L :: "'b list set"   and R :: "('b list \<times> 'b list) set" assumes "MyhillNerode L R" shows "regular L"
assms_2: shows "atom x \<sharp> (e\<^sub>1, \<pi>)"
type_wf_preserved: fixes SW :: "((_) heap, 'e, 'f) prog set"   and setter :: "((_) heap, 'e, 'g) prog"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "writes SW setter h h'"   and "h \<turnstile> setter \<rightarrow>\<^sub>h h'"   and "\<And>h h' w. \<lbrakk>w \<in> SW; h \<turnstile> w \<rightarrow>\<^sub>h h'\<rbrakk> \<Longrightarrow> \<forall>object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'"   and "\<And>h h' w. \<lbrakk>w \<in> SW; h \<turnstile> w \<rightarrow>\<^sub>h h'\<rbrakk> \<Longrightarrow> \<forall>node_ptr. preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr RNode.nothing) h h'" shows "NodeClass.type_wf h = NodeClass.type_wf h'"
ws_prog_def: shows "ws_prog \<equiv> \<forall>(C, c)\<in>set Prog. C \<noteq> Object \<longrightarrow> is_class (super c) \<and> (super c, C) \<notin> subcls1\<^sup>+"
T2: fixes x :: "'c"   and y :: "'c" assumes "R2 x y" shows "T2 x (Abs2 y)"
quasinorm_sum_limit: fixes f :: "'c"   and N1 :: "'c quasinorm"   and N2 :: "'c quasinorm" shows "\<exists>f1 f2. (\<forall>n. f = f1 n + f2 n) \<and> (\<lambda>n. eNorm N1 (f1 n) + eNorm N2 (f2 n)) \<longlonglongrightarrow> eNorm (N1 +\<^sub>N N2) f"
succ_iff: fixes x :: "hf"   and y :: "hf" shows "(x \<^bold>\<in> succ y) = (x = y \<or> x \<^bold>\<in> y)"
group_relative_homology_group: fixes p :: "int"   and X :: "'a topology"   and S :: "'a set" shows "Group.group (relative_homology_group p X S)"
sq: shows "2 \<cdot>\<^sub>m proj_psi * (2 \<cdot>\<^sub>m proj_psi) = 4 \<cdot>\<^sub>m proj_psi"
precondition_ind_def: fixes kstep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and output_f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and s0 :: "'a"   and current :: "'a \<Rightarrow> 'd"   and cswitch :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and kprecondition :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and realistic_execution :: "'b list list \<Rightarrow> bool"   and control :: "'a \<Rightarrow> 'd \<Rightarrow> 'b list list \<Rightarrow> 'b option \<times> 'b list list \<times> 'a"   and kinvolved :: "'b \<Rightarrow> 'd set"   and ifp :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and vpeq :: "'d \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and AS_set :: "'b list set"   and invariant :: "'a \<Rightarrow> bool"   and AS_precondition :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and aborting :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and waiting :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'a"   and execs :: "'d \<Rightarrow> 'b list list" assumes "Interruptible_Separation_Kernel kstep output_f s0 current cswitch kprecondition realistic_execution control kinvolved ifp vpeq AS_set invariant AS_precondition aborting waiting" shows "Interruptible_Separation_Kernel.precondition_ind control invariant AS_precondition s execs \<equiv> invariant s \<and> (\<forall>d. fst (control s d (execs d)) \<rightharpoonup> AS_precondition s d)"
separate_point_closed: fixes s :: "'d set"   and a :: "'d" assumes "closed s"   and "a \<notin> s" shows "\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x"
arcs_H: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d" assumes "bidel_arc G M a" shows "arcs (bidel_arc.H G M a) = arcs G - {a, edge_rev M a}"
formulas_inf: fixes x :: "'a formula"   and S :: "'a set"   and y :: "'a formula" assumes "x \<in> formulas S"   and "y \<in> formulas S" shows "x \<sqinter> y \<in> formulas S"
1: shows "\<forall>l1 l2. l1 \<noteq> l2 \<longrightarrow> GSMP_disjoint (trms\<^sub>l\<^sub>s\<^sub>t (proj l1 A')) (trms\<^sub>l\<^sub>s\<^sub>t (proj l2 A')) Sec"
points_index_alt_def: fixes ps :: "'a set" shows "\<B>\<^sup>R index ps = size (filter_mset ((\<subseteq>) ps) \<B>\<^sup>R)"
Alk: shows "A $$ (l, k) \<noteq> (0::'a)"
l_def: shows "l \<equiv> rat_precision prec (int x) (int y)"
leftderivation_implies_derivation: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and u :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.is_leftderivation \<NN> \<TT> \<RR> \<SS> u" shows "CFG.is_derivation \<NN> \<TT> \<RR> \<SS> u"
weakenSimulation_def: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<leadsto>\<^sub>w<Rel> Q \<equiv> \<forall>\<alpha> Q'. \<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q' \<longrightarrow> bn \<alpha> \<sharp>* \<Psi> \<longrightarrow> bn \<alpha> \<sharp>* P \<longrightarrow> (\<exists>P'. \<Psi> \<rhd> P \<Longrightarrow>\<alpha> \<prec> P' \<and> (\<Psi>, P', Q') \<in> Rel)"
sup_state_conv: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and ST :: "ty list"   and LT :: "ty err list"   and ST' :: "ty list"   and LT' :: "ty err list" shows "P \<turnstile> (ST, LT) \<le>\<^sub>i (ST', LT') = (P \<turnstile> ST [\<le>] ST' \<and> P \<turnstile> LT [\<le>\<^sub>\<top>] LT')"
diff_mult_distrib1: fixes x :: "'a"   and y :: "'a"   and a :: "real" assumes "x \<in> V"   and "y \<in> V" shows "a \<cdot> (x - y) = a \<cdot> x - a \<cdot> y"
M_consts_4: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n \<phi> M\<^sub>n true\<^sub>n = w \<Turnstile>\<^sub>n F\<^sub>n \<phi>"
poly_smult: fixes a :: "'a"   and p :: "'a poly"   and x :: "'a" shows "poly (Polynomial.smult a p) x = a * poly p x"
inner_gt_zero_iff: fixes x :: "'a" shows "(0 < real_of (inner_field x x)) = (x \<noteq> (0::'a))"
join_right: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c" assumes "weak_upper_semilattice L"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "y \<sqsubseteq>\<^bsub>L\<^esub> x \<squnion>\<^bsub>L\<^esub> y"
satisfies_T0: shows "is_left_adjoint p\<^sub>0"
axioms_1: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and \<sigma>' :: "'c \<Rightarrow> 'c" assumes "vmc2_path G P v0 p \<sigma> \<sigma>'" shows "vmc_path G P v0 p** \<sigma>'"
nterm_to_term_subst_replace_bound: fixes u' :: "nterm"   and n :: "nat"   and \<Gamma> :: "name list"   and x :: "name"   and u :: "nterm" assumes "closed u'"   and "n \<le> length \<Gamma>"   and "x \<notin> set (take n \<Gamma>)" shows "nterm_to_term \<Gamma> (subst_single u x u') = replace_bound n (nterm_to_term (insert_nth n x \<Gamma>) u) (nterm_to_term \<Gamma> u')"
section_retraction_of_iso_1: fixes f :: "'s"   and g :: "'s" assumes "local.iso f"   and "ide (g \<cdot> f)" shows "inverse_arrows f g"
Y_def: fixes n :: "nat" shows "Y n = \<Inter> (Z' ` {1..n})"
changes_0: fixes xs :: "'a list" shows "changes ((0::'a) # xs) = changes xs"
f2: shows "\<forall>x y z. x \<in> carrier rec_rng_of_frac \<longrightarrow> y \<in> carrier rec_rng_of_frac \<longrightarrow> z \<in> carrier rec_rng_of_frac \<longrightarrow> z \<otimes>\<^bsub>rec_rng_of_frac\<^esub> (x \<oplus>\<^bsub>rec_rng_of_frac\<^esub> y) = z \<otimes>\<^bsub>rec_rng_of_frac\<^esub> x \<oplus>\<^bsub>rec_rng_of_frac\<^esub> z \<otimes>\<^bsub>rec_rng_of_frac\<^esub> y"
ss_imp_se_fm: fixes A :: "fm" assumes "ss_fm A" shows "se_fm A"
dg_Cod_vdomain: shows "\<D>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) = \<CC>\<lparr>Arr\<rparr>"
honest_never_says_current_timestamp_in_auth: fixes evs :: "event list"   and T :: "nat"   and X :: "msg"   and A :: "agent"   and B :: "agent"   and Y :: "msg" assumes "CT evs = T"   and "Number T \<in> parts {X}"   and "A \<notin> bad"   and "evs \<in> kerbV" shows "Says A B \<lbrace>Y, X\<rbrace> \<notin> set evs"
degree_less: fixes p :: "'n \<Rightarrow> 'n" assumes "p \<noteq> id" shows "degree (\<Prod>i\<in>UNIV. to_fun (\<^bold>X - \<^bold>C A) i (p i)) < degree (\<Prod>i\<in>UNIV. to_fun (\<^bold>X - \<^bold>C A) i i)"
finite_gpvI: fixes gpv :: "('c, 'out, 'in) gpv" assumes "\<And>out c input. \<lbrakk>IO out c \<in> set_spmf (the_gpv gpv); input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> local.finite_gpv (c input)" shows "local.finite_gpv gpv"
LCons_lprefix_conv: fixes x :: "'b"   and xs :: "'b llist"   and ys :: "'b llist" shows "lprefix (LCons x xs) ys = (\<exists>ys'. ys = LCons x ys' \<and> lprefix xs ys')"
Seq_NoFaultStuckD2': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and F :: "'f set" assumes "\<Gamma>\<turnstile> \<langle>Seq c1 c2,s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` F" shows "\<forall>t. \<Gamma>\<turnstile> \<langle>c1,s\<rangle> \<Rightarrow> t \<longrightarrow> t \<notin> {Stuck} \<union> Fault ` F \<longrightarrow> \<Gamma>\<turnstile> \<langle>c2,t\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` F"
ereal_mult_minus_right: fixes a :: "ereal"   and b :: "ereal" shows "a * - b = - (a * b)"
f1_def: shows "f1 = pr_conv_3_to_2 f"
ta_empty_def: shows "ta_empty \<equiv> \<lparr>ta_initial = {}, ta_rules = {}\<rparr>"
arrE: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'a \<times> 'b" assumes "product_category C1 C2"   and "partial_magma.arr (product_category.comp C1 C2) f" obtains "partial_magma.arr C1 (fst f) \<and> partial_magma.arr C2 (snd f)"
prems_3: fixes x :: "Vars.var"   and ea :: "exp"   and aa :: "'a" assumes "(x, ea) \<in> set as'" shows "eqvt_at transform_sumC (aa, ea)"
smap2_applicative: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and x :: "'e stream"   and y :: "'f stream" shows "smap2 f x y = pure f \<diamondop> x \<diamondop> y"
tan_half: fixes x :: "'a" shows "tan x = sin ((2::'a) * x) / (cos ((2::'a) * x) + (1::'a))"
domain_invol: fixes x :: "'a" shows "ar ar ar ar x = ar ar x"
Ext: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and \<Psi>'' :: "'b" assumes "(\<Psi>', S, T) \<in> Rel" shows "(\<Psi>' \<otimes> \<Psi>'', S, T) \<in> Rel"
snd_case: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'b"   and p :: "'d \<times> 'e" shows "snd (case p of (x, y) \<Rightarrow> (f x, g y)) = g (snd p)"
idem: fixes a :: "'a multiset" shows "a \<union># a = a"
elims_2: fixes x :: "nat list \<times> nat list"   and xa :: "nat list \<times> nat list" assumes "x <\<^sub>r\<^sub>l\<^sub>e\<^sub>x\<^sub>2 xa" obtains xs :: "nat list" and ys :: "nat list" and us :: "nat list" and vs :: "nat list" where "x = (xs, ys)" and "xa = (us, vs)" and "xs @ ys <\<^sub>r\<^sub>l\<^sub>e\<^sub>x us @ vs"
adds_deg_pm_antisym: fixes s :: "'c \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 'd" assumes "s adds t"   and "deg_pm t \<le> deg_pm s" shows "s = t"
less_oSuc: fixes x :: "ordinal" shows "x < oSuc x"
map_comp: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'c"   and v :: "'d option" shows "map_option g (map_option f v) = map_option (g \<circ> f) v"
monoid_homomorphism_axioms: shows "monoid_homomorphism (\<rho> z' w) (\<FF> z') \<cdot>\<^bsub>z'\<^esub> \<one>\<^bsub>z'\<^esub> (\<FF> w) \<cdot>\<^bsub>w\<^esub> \<one>\<^bsub>w\<^esub>"
prems: shows "\<Gamma>\<turnstile>mark_guards f (Catch c1 c2) \<down> Normal s"