hcju/isabelleps huggingface dataset:
distinct_23: fixes x3 :: "('a, 'b, 'c) formula"   and x41 :: "('a, 'b, 'c) formula"   and x42 :: "('a, 'b, 'c) formula" shows "! x3 \<noteq> (x41 && x42)"
sum_eq: shows "(\<Sum>i\<in>UNIV. g i *s X $ i) = (\<Sum>i\<in>UNIV. f i *s X $ i)"
assms_1: shows "\<forall>w. f (g w) = f w"
rules: shows "R1 \<subseteq> upRules \<and> R2 \<subseteq> nprovRules \<and> R3 \<subseteq> provRules \<and> R = Ax \<union> R1 \<union> R2 \<union> R3"
Par_def: fixes x2 :: "pi"   and x1 :: "pi" shows "x2 \<parallel> x1 \<equiv> Abs_pi (Par_Rep (Rep_pi x2) (Rep_pi x1))"
inf_closed: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - y = - - (- x \<sqinter> - y)"
int_pow_rsp: fixes G :: "('c, 'd) monoid_scheme"   and b :: "nat"   and e :: "nat"   and d :: "nat"   and c :: "nat"   and a :: "'c" assumes "Group.monoid G"   and "b + e = d + c"   and "a \<in> Units G" shows "a [^]\<^bsub>G\<^esub> b \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> (a [^]\<^bsub>G\<^esub> c) = a [^]\<^bsub>G\<^esub> d \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> (a [^]\<^bsub>G\<^esub> e)"
RPC_action_defs_3: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rst :: "state \<Rightarrow> PrIds \<Rightarrow> rpcState"   and p :: "PrIds" shows "RPCReject send rcv rst p \<equiv> $(rst!p) = #rpcA \<and> \<not> IsLegalRcvArg<arg<$(send!p)>> \<and> Return send p #BadCall \<and> unchanged (rst!p, caller rcv!p)"
wellformed_deltas: fixes s :: "'s2"   and w :: "'a2 list" assumes "wellformed2 s"   and "w \<in> lists (set alphabet2)" shows "wellformed2 (fold delta2 w s)"
countable_separating_set_linorder1: shows "\<exists>B. countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b\<in>B. x < b \<and> b \<le> y))"
absorb_iff1: fixes b :: "'a::semilattice_sup"   and a :: "'a::semilattice_sup" shows "((b::'a::semilattice_sup) \<le> (a::'a::semilattice_sup)) = (a \<squnion> b = a)"
abs_eq: fixes x :: "('e, 'f, 'g) Tree set['e]" shows "Conj\<^sub>\<alpha> (map_bset abs_Tree\<^sub>\<alpha> x) = abs_Tree\<^sub>\<alpha> (tConj x)"
IS_ID_def: fixes R :: "('a \<times> 'a) set" shows "IS_ID R \<equiv> R = Id"
weakenU: fixes s :: "'a"   and s' :: "'a" assumes "U s s'" shows "U' s s'"
characterize_root_list_p_def: fixes p :: "real poly" shows "characterize_root_list_p p \<equiv> sorted_list_of_set {x. poly p x = 0}"
plusU_zeroU_left: fixes fmapU :: "(udom \<rightarrow> udom) \<rightarrow> udom\<cdot>'b \<rightarrow> udom\<cdot>'b"   and plusU :: "udom\<cdot>'b \<rightarrow> udom\<cdot>'b \<rightarrow> udom\<cdot>'b"   and zeroU :: "udom\<cdot>'b"   and m :: "udom\<cdot>'b" assumes "class.functor_zero_plus fmapU plusU zeroU" shows "plusU\<cdot>zeroU\<cdot>m = m"
filtration_def: fixes M :: "'a measure"   and F :: "'b \<Rightarrow> 'a measure" shows "Filtration.filtration M F = ((\<forall>t. subalgebra M (F t)) \<and> (\<forall>s t. s \<le> t \<longrightarrow> subalgebra (F t) (F s)))"
that: fixes g :: "V"   and f :: "V" assumes "gf = [g, f]\<^sub>\<circ>"   and "g \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"   and "f \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>" shows "thesis"
const_lam: fixes e :: "'b" shows "const (\<lambda>_. e)"
mkIF_def: fixes x :: "'d"   and t1 :: "'d ifex"   and t2 :: "'d ifex" shows "mkIF x t1 t2 = (if t1 = t2 then t1 else IF x t1 t2)"
hypnat_of_nat_le_whn: fixes n :: "nat" shows "hypnat_of_nat n \<le> whn"
uminus_Sup: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and A :: "'d set" assumes "class.complete_boolean_algebra Inf Sup inf less_eq less sup bot top minus uminus" shows "uminus (Sup A) = Inf (uminus ` A)"
append_singl_rev: fixes a :: "'a"   and as :: "'a list" shows "a # as = [a] @ as"
edom_empty_iff_bot: fixes m :: "'c \<Rightarrow> 'd" shows "(edom m = {}) = (m = \<bottom>)"
unique_games: fixes g :: "ZF set" shows "(g = fixgames g) = (g = games_lfp)"
nat_Suc_less_le_imp: fixes k :: "nat"   and n :: "nat" assumes "k < Suc n" shows "k \<le> n"
abs_homogenous: fixes x :: "'a"   and a :: "real" assumes "x \<in> E" shows "p (a \<cdot> x) = \<bar>a\<bar> * p x"
Rep_Msg_cases: fixes y :: "freemsg set" assumes "y \<in> Msg" obtains x :: "msg" where "y = Rep_Msg x"
perm_1: fixes pi :: "(NominalSequents.var \<times> NominalSequents.var) list"   and x2 :: "nat"   and x1 :: "NominalSequents.var list" shows "pi \<bullet> At x2 x1 = At (pi \<bullet> x2) (pi \<bullet> x1)"
ff_def: shows "ff = (\<lambda>i. real_of_int (b (i + 1)) / real_of_int (a (i + 1)))"
derivable_clauses_lemma: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and C :: "'b eclause"   and P :: "'b eclause set"   and S :: "'b eclause set"   and \<sigma> :: "('b \<times> 'b trm) list"   and k :: "inferences"   and C' :: "'b literal set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms"   and "basic_superposition.derivable trm_ord sel pos_ord filter_trms C P S \<sigma> k C'" shows "cl_ecl C = subst_cl C' \<sigma>"
det_sq_matrix_eq: fixes A :: "(('c, 'd) vec, 'd) vec" shows "Square_Matrix.det (from_vec A) = det A"
Value_def: shows "Value \<equiv> snd \<circ> snd \<circ> snd \<circ> Rep_status"
Rep_formula_inverse: fixes x :: "('e, 'f, 'g) formula" shows "Abs_formula (Rep_formula x) = x"
sinks_aux_append: fixes I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and U :: "'b set"   and xs :: "'c list"   and ys :: "'c list" shows "sinks_aux I D U (xs @ ys) = sinks_aux I D (sinks_aux I D U xs) ys"
hcomp_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<nu> :: "'b arrow_of_spans_data"   and \<mu> :: "'b arrow_of_spans_data" assumes "span_bicategory C prj0 prj1" shows "span_bicategory.hcomp C prj0 prj1 \<nu> \<mu> \<equiv> if partial_magma.arr (span_vertical_category.vcomp C) \<mu> \<and> partial_magma.arr (span_vertical_category.vcomp C) \<nu> \<and> span_vertical_category.src C \<nu> = span_vertical_category.trg C \<mu> then \<lparr>Chn = span_bicategory.chine_hcomp C prj0 prj1 \<nu> \<mu>, Dom = \<lparr>Leg0 = C (Leg0 (Dom \<mu>)) (prj0 (Leg0 (Dom \<nu>)) (Leg1 (Dom \<mu>))), Leg1 = C (Leg1 (Dom \<nu>)) (prj1 (Leg0 (Dom \<nu>)) (Leg1 (Dom \<mu>)))\<rparr>, Cod = \<lparr>Leg0 = C (Leg0 (Cod \<mu>)) (prj0 (Leg0 (Cod \<nu>)) (Leg1 (Cod \<mu>))), Leg1 = C (Leg1 (Cod \<nu>)) (prj1 (Leg0 (Cod \<nu>)) (Leg1 (Cod \<mu>)))\<rparr>\<rparr> else partial_magma.null (span_vertical_category.vcomp C)"
image_mset_cong: fixes M :: "'a multiset"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in># M \<Longrightarrow> f x = g x" shows "f `# M = g `# M"
normalized_sort_subset: fixes s :: "String.literal set" shows "normalize_sort cs s \<subseteq> s"
some_img_def: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set" assumes "set_category S \<AA>" shows "set_category.some_img S \<AA> = (SOME img. set_category_given_img S img \<AA>)"
simps_5: fixes f :: "typ \<Rightarrow> typ"   and t :: "Core.term"   and u :: "Core.term" shows "map_types f (t $ u) = map_types f t $ map_types f u"
convex_hull_scaling: fixes c :: "real"   and S :: "'c set" shows "convex hull (*\<^sub>R) c ` S = (*\<^sub>R) c ` (convex hull S)"
rep_eq: fixes x :: "('d, 'c) sq_matrix"   and xa :: "'c \<Rightarrow> 'c" shows "to_fun (perm_rows x xa) = (\<lambda>i. to_fun x (xa i))"
copm_1: shows "comm_monoid_mult_class.coprime (lead_coeff f) m"
Max_gr_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less x (linorder.Max less_eq A) = (\<exists>a\<in>A. less x a)"
separated_separating: fixes V :: "nat list"   and g :: "graph"   and P :: "face set" assumes "set V \<subseteq> \<V> g"   and "separated g (set V)"   and "\<forall>f\<in>P. |vertices f| \<le> 4" shows "separating (set V) (\<lambda>v. set (facesAt g v) \<inter> P)"
H: assumes "emeasure M (space M) = \<infinity>" shows "0 \<in> I"
kmp_result: fixes s :: "'a list"   and t :: "'a list" shows "kmp_SPEC s t = RETURN (if sublist s t then Some (LEAST i. sublist_at s t i) else None)"
st_asi_def: fixes instr :: "sparc_operation \<times> inst_operand list"   and s_val :: "word1" shows "st_asi instr s_val \<equiv> if fst instr \<in> {load_store_type STD, load_store_type ST, load_store_type STH, load_store_type STB} then if s_val = 0 then word_of_int 10 else word_of_int 11 else get_operand_asi (snd instr ! 3)"
open_euler1: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c" assumes "fin_digraph G"   and "Digraph_Component.connected G"   and "u \<in> verts G"   and "v \<in> verts G"   and "\<And>w. \<lbrakk>w \<in> verts G; u \<noteq> w; v \<noteq> w\<rbrakk> \<Longrightarrow> in_degree G w = out_degree G w"   and "in_degree G u + 1 = out_degree G u"   and "out_degree G v + 1 = in_degree G v" shows "\<exists>p. pre_digraph.euler_trail G u p v"
pax5: fixes F :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool"   and v :: "'c \<Rightarrow> 'd" shows "|~ \<circle>\<box>F \<longrightarrow> \<box>[\<circle>F]_v"
subspace_0: fixes S :: "'b set" assumes "csubspace S" shows "(0::'b) \<in> S"
msg_distinct_119: fixes X :: "msg"   and Y :: "msg"   and X' :: "msg"   and Y' :: "msg" shows "Sign X Y \<noteq> Enc X' Y'"
is_universal: fixes a' :: "'a"   and \<chi>' :: "'b \<Rightarrow> 'a" assumes "D1.cone a' \<chi>'" shows "\<exists>!f. \<guillemotleft>f : a' \<rightarrow> a1\<guillemotright> \<and> D1.cones_map f \<pi>1 = \<chi>'"
surjective: fixes r :: "'b gState_scheme" shows "r = \<lparr>gState.vars = gState.vars r, channels = gState.channels r, timeout = timeout r, procs = procs r, \<dots> = gState.more r\<rparr>"
not_on_closed_segmentI_complex: fixes b :: "complex"   and z :: "complex"   and a :: "complex" assumes "(Re b - Re z) * (Im z - Im a) \<noteq> (Im b - Im z) * (Re z - Re a)" shows "z \<notin> closed_segment a b"
select_convs_1: fixes Pol :: "('b \<times> 'b) set"   and Map :: "'c \<Rightarrow> 'b"   and Doms :: "'b set"   and List :: "'c list"   and ListOp :: "'c list option"   and Set :: "'c set"   and more :: "'d" shows "Pol \<lparr>Pol = Pol, Map = Map, Doms = Doms, List = List, ListOp = ListOp, Set = Set, \<dots> = more\<rparr> = Pol"
intro_elim_6_c: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> \<^bold>\<equiv> \<psi> in v]"   and "[\<^bold>\<not>\<phi> in v]" shows "[\<^bold>\<not>\<psi> in v]"
hom_one: shows "h \<one> = \<one>\<^bsub>H\<^esub>"
in_set_insert: fixes x :: "'e"   and xs :: "'e list" assumes "x \<in> set xs" shows "List.insert x xs = xs"
35: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and u :: "'a" shows "x \<sqinter> y \<squnion> z \<sqinter> (u \<sqinter> y) = (x \<squnion> z \<sqinter> u) \<sqinter> y"
le_imp_inverse_le: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b"   and "(0::'a) < a" shows "inverse b \<le> inverse a"
nat_pow_one: fixes n :: "nat" shows "[n] \<cdot>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>R\<^esub>"
awalk_verts_arc1: fixes e :: "'b"   and p :: "'b list"   and u :: "'a" assumes "e \<in> set p" shows "tail G e \<in> set (awalk_verts u p)"
assms: shows "C1.ide (fst a) \<and> C2.ide (snd a)"
prj_simps_1: shows "C.arr \<nu>\<pi>.prj\<^sub>1"
elims: fixes x :: "interval_tree"   and y :: "interval_tree" assumes "delete_elt_tree x = y" obtains "xa__ = Tip" and "y = undefined"   | lt :: "interval_tree" and x :: "nat idx_interval" and m :: "nat" and rt :: "interval_tree" where "xa__ = Node lt x m rt" and "y = (if lt = Tip then rt else if rt = Tip then lt else let x' = fst (del_min rt); rt' = snd (del_min rt); m' = max3 x' (tmax lt) (tmax rt') in Node lt (fst (del_min rt)) m' rt')"
comp_char: fixes g :: "'a"   and f :: "'a" shows "g \<cdot> f = (if S.arr f \<and> S.arr g \<and> C.seq g f then C g f else C.null)"
nres_of_def: fixes r :: "'d dres" shows "nres_of r \<equiv> case r of dSUCCEEDi \<Rightarrow> SUCCEED | dFAILi \<Rightarrow> FAIL | dRETURN x \<Rightarrow> RETURN x"
map_add_comm: fixes m1 :: "'d \<Rightarrow> 'e option"   and m2 :: "'d \<Rightarrow> 'e option" assumes "dom m1 \<inter> dom m2 = {}" shows "m1 ++ m2 = m2 ++ m1"
od_fm_fun_indPr1: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g"   and C :: "('a set \<times> ('a \<Rightarrow> 'g)) set" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N"   and "Chain (od_fm_fun R M N f H) C"   and "C \<noteq> {}" shows "(\<Union> {a. \<exists>b. (a, b) \<in> C}, \<lambda>x\<in>fgs R M (\<Union> {a. \<exists>b. (a, b) \<in> C}). THE y. y \<in> carrier N \<and> (\<exists>a b. (a, b) \<in> C \<and> x \<in> fgs R M a \<and> y = b x)) \<in> fsps R M N f H"
hom_0: fixes x :: "rat" assumes "of_rat x = (0::'a)" shows "x = 0"
ArrayCpy: fixes \<pi> :: "char list \<Rightarrow> com option"   and x :: "char list"   and y :: "char list"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "\<pi>: (x[] ::= y, s) \<Rightarrow> s(x := s y)"
CX: fixes h' :: "real" assumes "0 < h'"   and "h' < h" shows "(flow0 x0 h', Dflow x0 h' o\<^sub>L d0) \<in> CX"
tarski_aux: fixes R :: "('a \<times> 'a set) set"   and a :: "'a"   and A :: "'a set" assumes "R - 1\<^sub>\<pi> \<noteq> {}"   and "(a, A) \<in> NC" shows "(a, A) \<in> NC \<cdot> ((R - 1\<^sub>\<pi>) \<cdot> NC)"
size_4: fixes x21 :: "'f"   and x22 :: "'f list" shows "length (x21 # x22) = length x22 + Suc 0"
fmdom'_notD: fixes x :: "'c"   and m :: "('c, 'd) fmap" assumes "x \<notin> fmdom' m" shows "m $$ x = None"
HPhase1or2ReadThen_Inv2a: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "Inv2a s"   and "HPhase1or2ReadThen s s' p d q" shows "Inv2a s'"
x: shows "h \<turnstile> get_ancestors_di x \<rightarrow>\<^sub>r a # ancestors"
ker_inc_zero: fixes F :: "('c, 'd) aGroup_scheme"   and G :: "('e, 'f) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "aGroup F"   and "aGroup G"   and "f \<in> aHom F G" shows "\<zero>\<^bsub>F\<^esub> \<in> ker\<^bsub>F,G\<^esub> f"
\<phi>_def: fixes \<alpha> :: "'i"   and x :: "'a" shows "\<phi> \<alpha> x = (\<Sum>i | i \<in> N \<and> a i = \<alpha> \<and> x \<in> W i. g i x)"
AbruptProp: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "'b"   and n :: "nat" shows "\<Gamma>\<turnstile> \<langle>c,Abrupt s\<rangle> =n\<Rightarrow> Abrupt s"
inj_comm_semiring_hom_axioms: shows "inj_comm_semiring_hom (\<lambda>a. [:a:])"
uniformly_continuous_imp_continuous: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "uniformly_continuous_on s f" shows "continuous_on s f"
coeffs_def: fixes p :: "'c poly" shows "Polynomial.coeffs p = (if p = 0 then [] else map (poly.coeff p) [0..<Suc (Polynomial.degree p)])"
cs: fixes y :: "nat" shows "c (s, y) = (if y \<le> n then f y else 0)"
equiv_out_refl: fixes a :: "'prog_out" shows "equiv_out a a"
finite_nominals_set: fixes S :: "(('b, 'c) fm list \<times> 'c) set" assumes "finite S" shows "finite (\<Union> (block_nominals ` S))"
tendsto_add_Pair: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.topological_monoid_add plus zero open" shows "LIM x topological_space.nhds open a \<times>\<^sub>F topological_space.nhds open b. plus (fst x) (snd x) :> topological_space.nhds open (plus a b)"
distinct_44: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "Stop \<noteq> x41 ;; x42"
assms: shows "(SubnetS s t, a, st') \<in> cnet_sos (pnet_sos (automaton.trans (pnet np p1)) (automaton.trans (pnet np p2)))"
fmlookup_dom'_iff: fixes x :: "'a"   and m :: "('a, 'b) fmap" shows "(x \<in> fmdom' m) = (\<exists>a. m $$ x = Some a)"
simple_bochner_integral_def: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'c" shows "Bochner_Integration.simple_bochner_integral M f = (\<Sum>y\<in>f ` space M. \<P>(x in M. f x = y) *\<^sub>R y)"
up_repr: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P" shows "(\<Oplus>\<^bsub>P\<^esub>i\<in>{..deg R p}. monom P (coeff P p i) i) = p"
hash_blind_transaction: fixes transaction :: "transaction" shows "hash_transaction (blind_transaction transaction) = hash_transaction (embed_transaction transaction)"
avl_delete_root: fixes t :: "'b tree" assumes "avl t"   and "t \<noteq> ET" shows "avl (delete_root t)"
Max_eq_iff: fixes A :: "'c DBMEntry set"   and m :: "'c DBMEntry" assumes "finite A"   and "A \<noteq> {}" shows "(linordered_monoid.Max A = m) = (m \<in> A \<and> (\<forall>a\<in>A. a \<preceq> m))"
lexordp_into_lexordp_eq: fixes less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'d list"   and ys :: "'d list" assumes "ord.lexordp less xs ys" shows "ord.lexordp_eq less xs ys"
v: shows "us ! i + c \<cdot>\<^sub>v us ! j \<in> carrier_vec n"
AGMC_SC_def: fixes Cn :: "'b set \<Rightarrow> 'b set"   and contraction :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "AGMC_SC Cn contraction true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL \<equiv> AGM_Contraction Cn contraction \<and> Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL \<and> Compact_logic Cn"
Ana_invar_subst_subset: fixes M :: "('fun, 'var) Term.term set"   and N :: "('fun, 'var) Term.term set" assumes "Ana_invar_subst M"   and "N \<subseteq> M" shows "Ana_invar_subst N"
setops\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('b, 'c) stateful_strand_step list" shows "setops\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (setops\<^sub>s\<^sub>s\<^sub>t\<^sub>p ` set S)"
right_adjoints_compose: fixes g :: "('a S.E.term, 'a) S.arr"   and g' :: "('a S.E.term, 'a) S.arr" assumes "S.is_right_adjoint g"   and "S.is_right_adjoint g'"   and "S.src g = S.trg g'" shows "S.is_right_adjoint (g \<star>\<^sub>S g')"
group_strategy_proof_def: fixes ds :: "'d set"   and \<phi> :: "('d \<Rightarrow> ('x \<times> 'x) set) \<Rightarrow> ('h \<Rightarrow> 'x set \<Rightarrow> 'x set) \<Rightarrow> 'd set \<Rightarrow> 'x set" shows "group_strategy_proof ds \<phi> = (\<forall>Pd Ch. ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh Pd Ch \<longrightarrow> \<not> (\<exists>ds'\<subseteq>ds. ds' \<noteq> {} \<and> (\<exists>Pd'. ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh (override_on Pd Pd' ds') Ch \<and> (\<forall>d\<in>ds'. \<exists>y\<in>\<phi> (override_on Pd Pd' ds') Ch ds. y \<in> AboveS (Pd d) (dX (\<phi> Pd Ch ds) d)))))"
add_minus_cancel: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.group_add minus plus zero uminus" shows "plus a (plus (uminus a) b) = b"
1: shows "|~ \<box>[P]_v \<longrightarrow> \<circle>\<box>[P]_v"
involved_ifp: fixes kstep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and output_f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and current :: "'a \<Rightarrow> 'd"   and cswitch :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and kprecondition :: "'a \<Rightarrow> 'b \<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" assumes "Separation_Kernel kstep output_f current cswitch kprecondition control kinvolved ifp vpeq" shows "\<forall>s a. \<forall>d\<in>Kernel.involved kinvolved a. kprecondition s (the a) \<longrightarrow> ifp d (current s)"
mcont: fixes f :: "'c" assumes "mcont lubb ordb Sup (\<le>) (U f)" shows "mcont lubb ordb Sup (\<le>) (U (F f))"
horiz_vector_helper_pos_ind: fixes p :: "real poly"   and qs :: "real poly list"   and x :: "real"   and sign :: "rat list"   and I :: "nat list" assumes "p \<noteq> 0"   and "\<forall>q. List.member qs q \<longrightarrow> coprime p q"   and "x \<in> {x. poly p x = 0}"   and "sign = consistent_sign_vec_copr qs x" shows "list_constr I (length qs) \<longrightarrow> (0 < poly (prod_list (retrieve_polys qs I)) x) = (z I sign = 1)"
chain_sup_graph: fixes S :: "nat \<Rightarrow> ('e, 'f) labeled_graph" assumes "chain S" shows "graph (chain_sup S)"
vdiff_iff: fixes x :: "V"   and A :: "V"   and B :: "V" shows "(x \<in>\<^sub>\<circ> A -\<^sub>\<circ> B) = (x \<in>\<^sub>\<circ> A \<and> x \<notin>\<^sub>\<circ> B)"
step: fixes r :: "('a, 'call, 'ret) Resumption.resumption"   and s :: "'s" shows "exec_gpv' (lift_resumption r) s = execp_resumption' r s"
conf_widen: fixes P :: "'a prog"   and h :: "heap"   and v :: "val"   and T :: "ty"   and T' :: "ty" assumes "P,h \<turnstile> v :\<le> T"   and "subtype P T T'" shows "P,h \<turnstile> v :\<le> T'"
flip: shows "bounded_cbilinear (\<lambda>x y. y ** x)"
split: fixes P :: "'a \<Rightarrow> bool"   and f :: "(oref \<Rightarrow> obj option) \<Rightarrow> (lname \<Rightarrow> val option) \<Rightarrow> 'a"   and st :: "st" shows "P (case st of st x xa \<Rightarrow> f x xa) = (\<forall>x1 x2. st = st x1 x2 \<longrightarrow> P (f x1 x2))"
satisfiable_def: fixes g :: "vname gexp" shows "satisfiable g \<equiv> \<exists>i r. gval g (join_ir i r) = true"
smult_vec_zero: fixes v :: "'b vec" shows "(0::'b) \<cdot>\<^sub>v v = 0\<^sub>v (dim_vec v)"
ssubst_preserves_body: fixes t :: "sterm"   and t' :: "sterm"   and x :: "char list" assumes "body t"   and "lc t'" shows "body ([x \<rightarrow> t'] t)"
antisym: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> x" shows "x = y"
idom_hom_axioms: shows "idom_hom (inv hom)"
trans: fixes b :: "'c \<times> 'd"   and a :: "'c \<times> 'd"   and c :: "'c \<times> 'd" assumes "less_eq_prod b a"   and "less_eq_prod c b" shows "less_eq_prod c a"
nsum_add_nmTr: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and m :: "nat"   and g :: "nat \<Rightarrow> 'a" shows "(\<forall>j\<le>n. f j \<in> carrier A) \<and> (\<forall>j\<le>m. g j \<in> carrier A) \<longrightarrow> \<Sigma>\<^sub>e A jointfun n f m g Suc (n + m) = \<Sigma>\<^sub>e A f n \<plusminus> \<Sigma>\<^sub>e A g m"
fmap_return: fixes f :: "'g \<rightarrow> 'e"   and x :: "'g" shows "fmap\<cdot>f\<cdot>(return\<cdot>x) = return\<cdot>(f\<cdot>x)"
inf_sup_aci_7: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and x :: "'b::type"   and y :: "'b::type"   and z :: "'b::type" assumes "class.lattice (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type)" shows "(sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) (sup (y::'b::type) (z::'b::type)) = sup y (sup x z)"
prob_space_def: fixes M :: "'b measure" shows "prob_space M \<equiv> finite_measure M \<and> prob_space_axioms M"