hcju/isabelleps huggingface dataset:
sum_le_prod1: fixes a :: "real"   and b :: "real" assumes "a \<le> 1"   and "b \<le> 1" shows "a + b \<le> 1 + a * b"
mrbst_push_down_correct: fixes A :: "'a set"   and B :: "'a set"   and x :: "'a" assumes "finite A"   and "finite B"   and "\<And>y. y \<in> A \<Longrightarrow> y < x"   and "\<And>y. y \<in> B \<Longrightarrow> x < y" shows "random_bst A \<bind> (\<lambda>l. random_bst B \<bind> mrbst_push_down l x) = random_bst ({x} \<union> A \<union> B)"
saturate_def: fixes \<Gamma> :: "'b inference set" shows "inference_system.saturate \<Gamma> \<equiv> \<lambda>CC. {x. inference_system.saturatep \<Gamma> (\<lambda>x. x \<in> CC) x}"
transitive: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and R :: "pi"   and Rel'' :: "(pi \<times> pi) set" assumes "P \<leadsto><Rel> Q"   and "Q \<leadsto><Rel'> R"   and "eqvt Rel"   and "eqvt Rel''"   and "Rel O Rel' \<subseteq> Rel''"   and "\<And>P Q. (P, Q) \<in> Rel \<Longrightarrow> P \<leadsto>\<^sup>^<Rel> Q"   and "(P, Q) \<in> Rel" shows "P \<leadsto><Rel''> R"
continuous_on_open_vimage: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "open s" shows "continuous_on s f = (\<forall>B. open B \<longrightarrow> open (f -` B \<inter> s))"
assms_2: shows "\<I> \<Turnstile> \<langle>unlabel \<A>\<rangle>"
next_imp_exchange_next: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration" assumes "Next' c0 c1" shows "cri.next' (exchange_config c0) (exchange_config c1)"
indep_in_def: fixes \<E> :: "'a set"   and X :: "'a set" shows "indep_in \<E> X = (X \<subseteq> \<E> \<and> indep X)"
carrier_subgroup_generated: fixes G :: "('b, 'c) monoid_scheme"   and S :: "'b set" shows "carrier (subgroup_generated G S) = generate G (carrier G \<inter> S)"
assms_7: fixes i :: "nat" assumes "i < length fs"   and "term_of_pair (0::'a, i) \<prec>\<^sub>t u" shows "is_RB_in d rword G (term_of_pair (0::'a, i))"
try_spmf_return_spmf1: fixes x :: "'a"   and q :: "'a spmf" shows "TRY return_spmf x ELSE q = return_spmf x"
assms_1: shows "fds_nth f 1 \<noteq> (0::'a)"
A4aRedundant: fixes x :: "i"   and y :: "i" assumes "AxiomsSet6" shows "dom x \<cdot> y \<cong> dom (dom x) \<cdot> y"
dec_I1_Extend_SO: fixes i :: "nat"   and \<AA> :: "interp"   and P :: "nat fset"   and bs1 :: "WS1S_Nameful.fo list" shows "dec_I1 (Extend SO i \<AA> P) bs1 = dec_I1 \<AA> bs1"
m_closed: fixes x :: "'a \<Rightarrow> 'b"   and y :: "'a \<Rightarrow> 'b" assumes "x \<in> carrier (DirProds G I)"   and "y \<in> carrier (DirProds G I)" shows "x \<otimes>\<^bsub>DirProds G I\<^esub> y \<in> carrier (DirProds G I)"
higher_differentiable_on_const: fixes S :: "'a set"   and c :: "'b"   and n :: "nat" shows "higher_differentiable_on S (\<lambda>x. c) n"
is_iso_def: fixes G :: "('i, 'j) monoid_scheme"   and H :: "('k, 'l) monoid_scheme" shows "G \<cong> H = (Group.iso G H \<noteq> {})"
simps_4: fixes h :: "'c \<Rightarrow> 'd list \<Rightarrow> bool"   and \<phi>\<^sub>1 :: "'c fm"   and \<phi>\<^sub>2 :: "'c fm"   and xs :: "'d list" shows "interpret h (And \<phi>\<^sub>1 \<phi>\<^sub>2) xs = (interpret h \<phi>\<^sub>1 xs \<and> interpret h \<phi>\<^sub>2 xs)"
diff_fun_const: fixes c :: "'c" shows "diff_fun k charts (\<lambda>x. c)"
retract_of_space_clopen: fixes X :: "'b topology"   and S :: "'b set" assumes "openin X S"   and "closedin X S"   and "S = {} \<Longrightarrow> topspace X = {}" shows "S retract_of_space X"
drop_is_Cons_take: fixes n :: "nat"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" assumes "drop n xs = y # ys" shows "take (Suc n) xs = take n xs @ [y]"
zero_le_square: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.linordered_ring plus zero minus uminus less_eq less abs times" shows "less_eq zero (times a a)"
subset_imageE: fixes B :: "'c set"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "B \<subseteq> f ` A" obtains C :: "'d set" where "C \<subseteq> A" and "B = f ` C"
numeral_less_ennreal_iff: fixes w :: "num"   and n :: "real" shows "(numeral w < ennreal n) = (numeral w < n)"
M: shows "\<forall>t\<in>M. analyzed_in t M"
llist_inf_le: fixes s :: "'b llist"   and A :: "'b set"   and t :: "'b llist" assumes "s \<in> A\<^sup>\<omega>" shows "(s \<le> t) = (s = t)"
infsetsum_Sigma': fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "countable A"   and "\<And>i. countable (B i)"   and "(\<lambda>(x, y). f x y) abs_summable_on Sigma A B" shows "(\<Sum>\<^sub>ax\<in>A. infsetsum (f x) (B x)) = (\<Sum>\<^sub>a(x, y)\<in>Sigma A B. f x y)"
history_order_def: fixes history :: "nat \<Rightarrow> 'c list"   and x :: "'c"   and i :: "nat"   and z :: "'c" assumes "node_histories history" shows "node_histories.history_order history x i z \<equiv> \<exists>xs ys zs. xs @ x # ys @ z # zs = history i"
all_acquired_append: fixes xs :: "'b memref list"   and ys :: "'b memref list" shows "all_acquired (xs @ ys) = all_acquired xs \<union> all_acquired ys"
card_gt_0_iff: fixes A :: "'e set" shows "(0 < card A) = (A \<noteq> {} \<and> finite A)"
le_less_Suc_eq: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "(n < Suc m) = (n = m)"
restrict_in_funcset: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "restrict f A \<in> A \<rightarrow> B"
not_Interp_to_interp_imp_less: fixes A :: "'b"   and N :: "'b literal multiset set"   and C :: "'b literal multiset"   and D :: "'b literal multiset" assumes "A \<notin> Interp N C"   and "A \<in> interp N D" shows "C < D"
cases: fixes a :: "('test, 'atom, 'choice) cmd" assumes "discr a" obtains c :: "('test, 'atom, 'choice) cmd" where "a = c" and "\<And>s i. i < brn c \<longrightarrow> s \<approx> eff c s i \<and> discr (local.cont c s i)"
obj_simps_2: fixes a :: "'a arrow_of_spans_data" assumes "obj a" shows "src a = a"
rotate_append: fixes l :: "'a list"   and q :: "'a list" shows "rotate (length l) (l @ q) = q @ l"
finite_rtrancl_Image: fixes R :: "('a \<times> 'a) set"   and A :: "'a set" assumes "finite R"   and "finite A" shows "finite (R\<^sup>* `` A)"
pow_carrier_mod: fixes G :: "('a, 'b) cyclic_group_scheme"   and g :: "'a"   and k :: "nat" assumes "Cyclic_Group.cyclic_group G"   and "g \<in> carrier G" shows "g [^]\<^bsub>G\<^esub> (k mod order G) = g [^]\<^bsub>G\<^esub> k"
Sup_least: fixes A :: "'a set"   and z :: "'a" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<le> z" shows "\<Squnion>A \<le> z"
that_1: shows "Negligible.negligible (\<lambda>n. local.ddh.advantage n (P1_DDH_mal_adv_\<sigma>_true n M z \<A> D))"
assms_2: fixes n :: "nat"   and x :: "'g"   and \<Sigma>1 :: "Unboxed.type option list"   and x' :: "('b, 'c, 'd, 'op, 'opinl, 'opubx, 'e, 'f) Ubx.instr"   and \<Sigma>2 :: "Unboxed.type option list" assumes "f n x \<Sigma>1 = Ok (x', \<Sigma>2)" shows "Subx.sp_instr F x' \<Sigma>1 = Ok \<Sigma>2"
rep_eq: fixes x :: "'a vector"   and xa :: "'a vector" shows "(x \<le> xa) = (Rep_vector x \<le> Rep_vector xa)"
ts_conc: fixes c :: "'form" assumes "c |\<in>| conc_forms" shows "snd (fst (tree.root (ts c))) = c"
lang_ENC_split: fixes X :: "nat set"   and Y1 :: "nat set"   and Y2 :: "nat set"   and n :: "nat" assumes "finite X"   and "X = Y1 \<union> Y2"   and "n = 0 \<or> (\<forall>p\<in>X. p < n)" shows "lang n (ENC n X) = lang n (ENC n Y1) \<inter> lang n (ENC n Y2)"
rsplit_ex: fixes bs :: "real list"   and f :: "int \<Rightarrow> MIR.num \<Rightarrow> fm"   and a :: "MIR.num" shows "Ifm bs (rsplit f a) = (\<exists>(\<phi>, n, s)\<in>set (rsplit0 a). Ifm bs (MIR.conj \<phi> (f n s)))"
SHOnextConfig_def: fixes A :: "('a, 'b, 'c, 'd) CHOAlgorithm_scheme"   and r :: "nat"   and cfg :: "'a \<Rightarrow> 'b"   and HO :: "'a \<Rightarrow> 'a set"   and SHO :: "'a \<Rightarrow> 'a set"   and cfg' :: "'a \<Rightarrow> 'b" shows "SHOnextConfig A r cfg HO SHO cfg' \<equiv> CSHOnextConfig A r cfg HO SHO (\<lambda>q. undefined) cfg'"
assms_1: shows "w \<le>p u\<^sup>\<omega>"
hRe_hcis: fixes a :: "real star" shows "hRe (hcis a) = (*f* cos) a"
stieltjes_gamma_aux3: shows "fps_expansion (\<lambda>s. (s - 1) * pre_zeta 1 s + 1) 1 = fps_X * fps_expansion (pre_zeta 1) 1 + 1"
transitionRelation_def: fixes F0 :: "Literal list list"   and decisionVars :: "nat set" shows "transitionRelation F0 decisionVars \<equiv> {(stateA, stateB). transition stateA stateB F0 decisionVars}\<^sup>*"
Rep_trm_cases: fixes y :: "trm_Rep" assumes "y \<in> Collect trm_Rep_set" obtains x :: "trm" where "y = Rep_trm x"
frech_linear: fixes I :: "('d, 'e, 'f) interp"   and \<theta> :: "('d, 'f) trm"   and x :: "real"   and \<nu> :: "(real, 'f) vec"   and \<nu>' :: "(real, 'f) vec" assumes "is_interp I"   and "dfree \<theta>" shows "x * frechet I \<theta> \<nu> \<nu>' = frechet I \<theta> \<nu> (x *\<^sub>R \<nu>')"
assoc_simps_1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "arr (assoc a b c)"
for_rec2_eq: fixes f :: "'b \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b nres"   and a :: "'b"   and n :: "nat" shows "for_rec2 f a n n n = nfoldli [0..<n + 1] (\<lambda>x. True) (\<lambda>i. nfoldli [0..<n + 1] (\<lambda>x. True) (\<lambda>j a. f a i j)) a"
reds_def: shows "reds \<equiv> \<lambda>P. {((xb, xa, x), xe, xd, xc). redsp P xb xa x xe xd xc}"
covector_complement_closed: fixes x :: "'a" assumes "top \<sqinter> x = x" shows "top \<sqinter> - x = - x"
type_bexprI: fixes t :: "bexp set"   and \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option"   and e :: "bexp" assumes "t = \<Union> (to_total \<Gamma> ` bexp_vars e)" shows "type_bexpr \<Gamma> e t"
Df: fixes t :: "real" assumes "0 < t"   and "t < a" shows "dist (f' t / g' t) x < e / 4"
monoid_add_hom_def: fixes hom :: "'c \<Rightarrow> 'd" shows "monoid_add_hom hom \<equiv> zero_hom hom \<and> semigroup_add_hom hom"
AAss_\<tau>execI3: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e'' :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and v :: "'addr val"   and v' :: "'addr val" assumes "\<tau>exec_move ci P t e'' h (stk, loc, pc, xcp) (stk', loc', pc', xcp')" shows "\<tau>exec_move ci P t (e\<lfloor>e'\<rceil> := e'') h (stk @ [v, v'], loc, length (compE2 e) + length (compE2 e') + pc, xcp) (stk' @ [v, v'], loc', length (compE2 e) + length (compE2 e') + pc', xcp')"
td_th: fixes g :: "'a \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c"   and x :: "'a" assumes "g \<circ> Abs = f" shows "f (Rep x) = g x"
evtEN: fixes n :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'cmp"   and \<gamma> :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool" assumes "\<not> (\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>)"   and "eval c t t' n (\<diamond>\<^sub>b\<gamma>)" shows "\<exists>n'\<ge>n. eval c t t' n' \<gamma>"
carrier_subgroup_generated_alt: fixes G :: "('f, 'g) monoid_scheme"   and S :: "'f set" assumes "Group.group G"   and "S \<subseteq> carrier G" shows "carrier (subgroup_generated G S) = \<Inter> {H. subgroup H G \<and> carrier G \<inter> S \<subseteq> H}"
less_eq_vec_def: fixes v :: "'a vec"   and w :: "'a vec" shows "(v \<le> w) = (dim_vec v = dim_vec w \<and> (\<forall>i<dim_vec w. v $ i \<le> w $ i))"
linear: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'f"   and y :: "'f" assumes "class.linorder less_eq less" shows "less_eq x y \<or> less_eq y x"
hyps: fixes y :: "('s, 'v) tm" assumes "size y < size s"   and "y >\<^sub>t y" shows "False"
reorder_def: fixes ko :: "'o"   and xs :: "'x" shows "reorder ko xs = oalist_of_list (sort_oalist_aux ko (list_of_oalist xs), ko)"
closed_map_inclusion: fixes X :: "'c topology"   and S :: "'c set" assumes "closedin X S" shows "closed_map (subtopology X S) X id"
ESem_subst: fixes e :: "exp"   and \<sigma> :: "Vars.var \<Rightarrow> Value"   and x :: "Vars.var"   and y :: "Vars.var" shows "\<lbrakk> e \<rbrakk>\<^bsub>\<sigma>(x := \<sigma> y)\<^esub> = \<lbrakk> e[x::=y] \<rbrakk>\<^bsub>\<sigma>\<^esub>"
gen_dfs_upper_aux: fixes x :: "'a"   and y :: "'a"   and E' :: "('a \<times> 'a) set"   and u0 :: "'a"   and E :: "('a \<times> 'a) set"   and U :: "'a set" assumes "(x, y) \<in> E'\<^sup>*"   and "(u0, x) \<in> E"   and "u0 \<in> U"   and "E' \<subseteq> E"   and "E `` U \<subseteq> U" shows "y \<in> U"
measurable_fst: fixes M1 :: "'d measure"   and M2 :: "'e measure" shows "fst \<in> M1 \<Otimes>\<^sub>M M2 \<rightarrow>\<^sub>M M1"
elims_2: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq x" obtains "x = []"   | xa :: "'c" and ys :: "'c list" where "x = xa # ys" and "Ball (set ys) (less_eq xa) \<and> linorder.sorted less_eq ys"
coprime_pq: fixes k :: "nat" shows "coprime (p k) (q k)"
rev_conj_cong: fixes Q :: "bool"   and Q' :: "bool"   and P :: "bool"   and P' :: "bool" assumes "Q = Q'"   and "Q' \<Longrightarrow> P = P'" shows "(P \<and> Q) = (P' \<and> Q')"
collapse_2: fixes tllist :: "('f, 'g) tllist" assumes "\<not> is_TNil tllist" shows "TCons (thd tllist) (ttl tllist) = tllist"
mult_neg_pos: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less a zero"   and "less zero b" shows "less (times a b) zero"
thd_tSup: fixes Y :: "('d, 'b) tllist set" assumes "\<exists>xs\<in>Y. \<not> is_TNil xs" shows "thd (local.tSup Y) = (THE x. x \<in> thd ` (Y \<inter> {xs. \<not> is_TNil xs}))"
axioms_1: fixes sinvar :: "'b graph \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> bool"   and default_node_properties :: "'c"   and receiver_violation :: "bool" assumes "SecurityInvariant sinvar default_node_properties receiver_violation" shows "SecurityInvariant_preliminaries sinvar"
returns_to_earlierI: fixes P :: "'a set"   and x :: "'a"   and t :: "real" assumes "returns_to P (flow0 x t)"   and "closed P"   and "0 \<le> t"   and "t \<in> existence_ivl0 x"   and "\<forall>\<^sub>F t in at_right 0. flow0 x t \<notin> P" shows "returns_to P x"
B: shows "strict_mono f \<and> (\<forall>i. f i \<in> T P) \<and> (\<forall>i. \<exists>t. f i = f 0 @ t \<and> set t \<subseteq> ev ` A)"
sup_aci_4: fixes x :: "'b"   and y :: "'b" shows "x + (x + y) = x + y"
intro: fixes A :: "'b set set"   and \<A> :: "'b set set set"   and B :: "'b set set"   and B' :: "'b set set"   and X :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set" assumes "A \<in> \<A>"   and "B \<in> \<A>"   and "B' \<in> \<A>"   and "SimplicialComplex.maxsimp X C"   and "SimplicialComplex.maxsimp X D"   and "SimplicialComplex.maxsimp X E"   and "z \<lhd> C"   and "z \<lhd> D"   and "z \<lhd> E"   and "C \<in> A \<inter> B"   and "D \<in> A \<inter> B'"   and "E \<in> B \<inter> B'"   and "D \<noteq> C"   and "E \<noteq> D"   and "C \<noteq> E" shows "ChamberComplexApartmentSystemTriangle_axioms X \<A> A B B' C D E z"
some_opt_false_trivial: shows "Eps_Opt (\<lambda>_. False) = None"
gap_monotone: fixes t\<^sub>1 :: "nat"   and t\<^sub>2 :: "nat"   and i :: "nat" assumes "t\<^sub>1 \<le> t\<^sub>2" shows "gap i t\<^sub>1 \<le> gap i t\<^sub>2"
homotopic_with_refl: fixes P :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" shows "homotopic_with P X Y f f = (continuous_map X Y f \<and> P f)"
classes_above_sees_field2: fixes C :: "char list"   and P :: "'b prog"   and P' :: "'b prog"   and F :: "char list"   and b :: "staticb"   and t :: "ty"   and C' :: "char list" assumes "classes_above P C \<inter> classes_changed P P' = {}"   and "P' \<turnstile> C sees F,b:t in C'" shows "P \<turnstile> C sees F,b:t in C'"
containID: fixes i :: "nat"   and \<Gamma> :: "'b form multiset"   and \<Delta> :: "'b form multiset"   and R :: "('b sequent list \<times> 'b sequent) set" assumes "At i \<in># \<Gamma> \<and> At i \<in># \<Delta>"   and "Ax \<subseteq> R" shows "( \<Gamma> \<Rightarrow>* \<Delta>, 0) \<in> derivable R*"
IntE: fixes c :: "'d"   and A :: "'d set"   and B :: "'d set" assumes "c \<in> A \<inter> B" obtains "c \<in> A" and "c \<in> B"
simps_99: fixes x71 :: "char list"   and x72 :: "expr"   and x4 :: "val" shows "x71:=x72 \<noteq> Val x4"
nodes_sim: fixes a :: "'b"   and b :: "'f" assumes "a \<in> s1.b.V"   and "(a, b) \<in> R2" shows "b \<in> s2.b.V"
that_2: assumes "x \<^bold>\<in> y" shows "thesis"
filterlim_mono_eventually: fixes f :: "'c \<Rightarrow> 'd"   and F :: "'d filter"   and G :: "'c filter"   and F' :: "'d filter"   and G' :: "'c filter"   and f' :: "'c \<Rightarrow> 'd" assumes "filterlim f F G"   and "F \<le> F'"   and "G' \<le> G"   and "\<forall>\<^sub>F x in G'. f x = f' x" shows "filterlim f' F' G'"
rel_arg_cong: fixes x :: "'a"   and y :: "'a"   and f :: "'a \<Rightarrow> 'c" assumes "(x, y) \<in> Id" shows "(f x, f y) \<in> Id"
WT: shows "\<I> \<turnstile>g \<A> \<surd>"
abs: fixes s :: "dB"   and t :: "dB" assumes "s \<rightarrow>\<^sub>\<eta> t" shows "Abs s \<rightarrow>\<^sub>\<eta> Abs t"
circ_isolate: fixes x :: "'a" shows "x\<^sup>\<circ> = x\<^sup>\<circ> * bot \<squnion> x\<^sup>\<star>"
FL_valid_Tree\<^sub>\<alpha>_Act\<^sub>\<alpha>: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and effect_apply :: "'e \<Rightarrow> 'b \<Rightarrow> 'b"   and L :: "'d \<times> 'e fs_set \<times> 'e \<Rightarrow> 'e fs_set"   and P :: "'b"   and f :: "'e"   and \<alpha> :: "'d"   and t\<^sub>\<alpha> :: "('f, 'c, 'd, 'e) Tree\<^sub>\<alpha>" assumes "effect_nominal_ts satisfies transition effect_apply L" shows "effect_nominal_ts.FL_valid_Tree\<^sub>\<alpha> satisfies transition effect_apply P (Act\<^sub>\<alpha> f \<alpha> t\<^sub>\<alpha>) = (\<exists>\<alpha>'' t\<^sub>\<alpha>' P'. Act\<^sub>\<alpha> f \<alpha> t\<^sub>\<alpha> = Act\<^sub>\<alpha> f \<alpha>'' t\<^sub>\<alpha>' \<and> transition (effect_apply f P) \<langle>\<alpha>'',P'\<rangle> \<and> effect_nominal_ts.FL_valid_Tree\<^sub>\<alpha> satisfies transition effect_apply P' t\<^sub>\<alpha>')"
IH: fixes Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "(X, Y) \<in> swapped O rel" shows "phi Y"
zone_delay_def: fixes Z :: "('e \<Rightarrow> 'f) set" shows "Z\<^sup>\<up> = {u \<oplus> d |u d. u \<in> Z \<and> (0::'f) \<le> d}"
matrix_vector_mul_2: fixes f :: "(real, 'a) vec \<Rightarrow> (real, 'b) vec" assumes "linear f" shows "(*v) (matrix f) = f"
in_orthogonal_complement_imp_orthogonal: fixes y :: "'a"   and S :: "'a set"   and x :: "'a" assumes "y \<in> S"   and "x \<in> orthogonal_complement S" shows "orthogonal x y"
max_atm_imp_Interp: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and C :: "'b literal multiset"   and N :: "'b literal multiset set"   and A :: "'b" assumes "ground_resolution_with_selection S"   and "C \<in> N"   and "Pos A \<in># C"   and "A = Max (atms_of C)"   and "S C = {#}" shows "ground_resolution_with_selection.Interp S N C \<Turnstile> C"
advantage_def: fixes \<A> :: "('i, 'j) distinguisher"   and res1 :: "('i, 'j) resource"   and res2 :: "('i, 'j) resource" shows "advantage \<A> res1 res2 = \<bar>spmf (connect \<A> res1) True - spmf (connect \<A> res2) True\<bar>"
\<nu>_def: shows "\<nu> \<equiv> \<lambda>f. if arr f then mkarr (COD f\<^bold>\<down>) \<cdot> f else null"
st_zero_ge: fixes x :: "real star" assumes "x \<le> 0"   and "x \<in> HFinite" shows "st x \<le> 0"
has_fixpoint_def: fixes f :: "'a \<Rightarrow> 'a" shows "has_fixpoint f \<equiv> \<exists>x. is_fixpoint f x"
LIMSEQ_const_mult: fixes f :: "nat \<Rightarrow> real"   and a :: "real"   and c :: "real" assumes "f \<longlonglongrightarrow> a" shows "(\<lambda>x. c * f x) \<longlonglongrightarrow> c * a"
fin_subst1: fixes M :: "trm"   and x :: "name"   and y :: "name"   and P :: "trm"   and c :: "coname" assumes "fin M x"   and "x \<noteq> y"   and "x \<sharp> P" shows "fin (M{y:=<c>.P}) x"
sigma_le_iff: fixes \<A> :: "'b set set"   and \<Omega> :: "'b set"   and x :: "'b measure" assumes "\<A> \<subseteq> Pow \<Omega>" shows "(sigma \<Omega> \<A> \<le> x) = (\<Omega> \<subseteq> space x \<and> (space x = \<Omega> \<longrightarrow> \<A> \<subseteq> sets x))"
assms_5: shows "\<T> = generate_valid_stateful_policy_IFSACS G M edgesList"
order_iff_strict: fixes b :: "'c DBMEntry"   and a :: "'c DBMEntry" shows "(b \<preceq> a) = (b \<prec> a \<or> a = b)"
zero_hom_0_axioms: shows "zero_hom_0 (inv (inv (inv (inv hom))))"
ge_lbound_def: fixes c :: "'a"   and b :: "'a option" shows "c \<ge>\<^sub>l\<^sub>b b = \<unrhd>\<^sub>l\<^sub>b (<) c b"
thetaIfZOT_ZObisT: shows "thetaIfZOT \<subseteq> ZObisT"
E_sup_mono: fixes K :: "'a \<Rightarrow> 'a pmf set"   and f :: "'a stream \<Rightarrow> ennreal"   and g :: "'a stream \<Rightarrow> ennreal"   and s :: "'a" assumes "Markov_Decision_Process K"   and "\<And>\<omega>. f \<omega> \<le> g \<omega>" shows "Markov_Decision_Process.E_sup K s f \<le> Markov_Decision_Process.E_sup K s g"
prems: shows "sat_precond_as s ((a # as) @ as')"
sum_representation_eq: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and basis :: "'d set"   and v :: "'d"   and B :: "'d set" assumes "module scale"   and "\<not> module.dependent scale basis"   and "v \<in> module.span scale basis"   and "finite B"   and "basis \<subseteq> B" shows "(\<Sum>b\<in>B. scale (module.representation scale basis v b) b) = v"
decisive_on_mono: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and B :: "'a set" assumes "decisive_on A f"   and "B \<subseteq> A" shows "decisive_on B f"
tail_minus_higher: fixes v :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys p" shows "punit.tail (p - punit.higher p v) = punit.lower p v"
eq_refl: fixes x :: "'b list"   and y :: "'b list" assumes "x = y" shows "x \<le>s y"
push_forward_def: fixes X :: "('c \<Rightarrow> real) \<Rightarrow> real" shows "inv.push_forward X = restrict0 src.diff_fun_space (\<lambda>g. X (restrict0 dest.carrier (g \<circ> f')))"
list_assn_prod_map: fixes A :: "'a \<Rightarrow> 'c \<Rightarrow> assn"   and B :: "'b \<Rightarrow> 'd \<Rightarrow> assn"   and xs :: "('a \<times> 'b) list"   and ys :: "('c \<times> 'd) list" shows "list_assn (A \<times>\<^sub>a B) xs ys = list_assn B (separators xs) (separators ys) * list_assn A (subtrees xs) (subtrees ys)"
idom_hom_def: fixes hom :: "'c \<Rightarrow> 'd" shows "idom_hom hom \<equiv> comm_ring_hom hom"