hcju/isabelleps huggingface dataset:
path_2: fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow> w"   and "v \<noteq> w" shows "path [v, w]"
INF_less_iff: 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 f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and a :: "'c" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "less (Inf (f ` A)) a = (\<exists>x\<in>A. less (f x) a)"
fresh_distinct_fv: fixes S :: "'e set"   and l :: "'d" assumes "atom ` S \<sharp>* l" shows "S \<inter> fv l = {}"
2: fixes x :: "real"   and n :: "int" assumes "x \<noteq> 0"   and "x * (t - s) = 2 * (real_of_int n * pi)" shows "\<bar>s - t\<bar> = \<bar>2 * (real_of_int n * pi) / x\<bar>"
prefix_to_carriers: fixes history :: "nat \<Rightarrow> 'c list"   and xs :: "'c list"   and i :: "nat" assumes "node_histories history"   and "node_histories.prefix_of_node_history history xs i" shows "set xs \<subseteq> set (history i)"
I': fixes s :: "char list \<Rightarrow> int"   and s' :: "char list \<Rightarrow> int" assumes "(c', s) \<Rightarrow> s'"   and "P s"   and "bval b' s" shows "P s'"
n2m_dtor_unfold_coG_def: fixes s1 :: "'e \<Rightarrow> ('f, 'g) coG0"   and s2 :: "'g \<Rightarrow> ('e, 'g) coF0" shows "n2m_dtor_unfold_coG s1 s2 = dtor_unfold_coG (BNF_Composition.id_bnf \<circ> BNF_Composition.id_bnf \<circ> map_pre_coG\<^sub>M id id (dtor_unfold_coF (BNF_Composition.id_bnf \<circ> BNF_Composition.id_bnf \<circ> s2)) \<circ> s1)"
intra_edge_obs_slice: fixes ms :: "'node list"   and ms'' :: "'node list"   and ms' :: "'node list"   and S :: "'node SDG_node set"   and a :: "'edge" assumes "ms \<noteq> []"   and "ms'' \<in> obs ms' \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "valid_edge a"   and "intra_kind (kind a)"   and "(\<exists>m\<in>set (tl ms). \<exists>m'. call_of_return_node m m' \<and> m' \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>) \<or> hd ms \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "hd ms = sourcenode a"   and "ms' = targetnode a # tl ms"   and "Ball (set (tl ms')) return_node" shows "ms'' \<in> obs ms \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"
insert_push: fixes xs :: "('d, 'c) bintree option list"   and v :: "'c"   and a :: "'d" assumes "distinct (vals xs)"   and "v \<notin> set (vals xs)" shows "set (elements (Binomial_Queue.insert a v xs)) = set (pq.alist_of (PQ.push v a (pqueue xs)))"
compose_first_Some: fixes xs :: "('e \<times> 'd) list"   and k :: "'e"   and v :: "'d"   and ys :: "('d \<times> 'f) list" assumes "map_of xs k = Some v" shows "map_of (compose xs ys) k = map_of ys v"
cprank_maxI: fixes BS :: "'b tensor list"   and ds :: "nat list" assumes "\<And>B. B \<in> set BS \<Longrightarrow> cprank_max1 B"   and "\<And>B. B \<in> set BS \<Longrightarrow> dims B = ds" shows "cprank_max (length BS) (listsum ds BS)"
finite_INT: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set"   and M :: "'b measure" assumes "finite I"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M" shows "\<Inter> (A ` I) \<in> sets M"
continuous_infdist: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and A :: "'c set" assumes "continuous F f" shows "continuous F (\<lambda>x. infdist (f x) A)"
inf_idem: fixes x :: "'b \<times> 'c" shows "inf_prod x x = x"
trace_callee_eqD: fixes callee1 :: "'h \<Rightarrow> 'i \<Rightarrow> ('j \<times> 'h) spmf"   and callee2 :: "'k \<Rightarrow> 'i \<Rightarrow> ('j \<times> 'k) spmf"   and A :: "'i set"   and p :: "'h spmf"   and q :: "'k spmf"   and xs :: "('i \<times> 'j) list"   and x :: "'i" assumes "trace_callee_eq callee1 callee2 A p q"   and "set xs \<subseteq> A \<times> UNIV"   and "x \<in> A" shows "trace_callee callee1 p xs x = trace_callee callee2 q xs x"
pairs_stone: fixes phi :: "'c \<Rightarrow> 'd Filters.filter"   and x :: "'c"   and y :: "'d Filters.filter" assumes "triple phi"   and "(x, y) \<in> triple.pairs phi" shows "triple.pairs_sup (triple.pairs_uminus phi (x, y)) (triple.pairs_uminus phi (triple.pairs_uminus phi (x, y))) = (top, Filters.filter.Abs_filter {top})"
is_renaming_def: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "IsaFoR_Term.is_renaming \<sigma> = (\<exists>\<tau>. comp_subst_abbrev \<sigma> \<tau> = Var)"
P_def: shows "P = (\<lambda>y. 0 < y \<and> is_square (y\<^sup>2 * D + 1))"
pwd_spy_enc_pubk_def: fixes n :: "nat" shows "pwd_spy_enc_pubk n \<equiv> let (s, S, A, B, C, D) = pwd_spy_mult n; SK = (Some S, {A, B}, {C, D}) in (insert (Spy, Crypt (SesK SK) (PubKey C)) s, S, A, B, C, D)"
matrix_vector_mult_def: fixes m :: "(('d, 'f) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec"   and x :: "('d, 'f) Finite_Cartesian_Product.vec" shows "m *v x \<equiv> \<chi>i. \<Sum>j\<in>UNIV. m $h i $h j * x $h j"
funof_in: fixes r :: "('c \<times> 'b) set"   and x :: "'c"   and A :: "'b set" assumes "r `` {x} \<subseteq> A"   and "single_valued r"   and "x \<in> Domain r" shows "funof r x \<in> A"
dickson_less_trans: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and s :: "'a"   and t :: "'a"   and u :: "'a" assumes "dickson_less d m s t"   and "dickson_less d m t u" shows "dickson_less d m s u"
hyps_2: fixes j :: "nat"   and x :: "'a"   and y :: "'b" assumes "j \<le> length xs" shows "mset (zip (take j xs @ x # drop j xs) (take j ys @ y # drop j ys)) = add_mset (x, y) (mset (zip xs ys))"
size_annos_same2: fixes C1 :: "'a acom"   and C2 :: "'b acom" assumes "strip C1 = strip C2" shows "(length (annos C1) = length (annos C2)) = True"
interfree_def: fixes Ts :: "('b ann_com option \<times> 'b set) list" shows "interfree Ts \<equiv> \<forall>i j. i < length Ts \<and> j < length Ts \<and> i \<noteq> j \<longrightarrow> interfree_aux (OG_Hoare.com (Ts ! i), OG_Hoare.post (Ts ! i), OG_Hoare.com (Ts ! j))"
cproper_interval_unit_def: shows "cproper_interval = proper_interval"
maddux2': fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "residuated f" shows "conjugate f x \<sqinter> y \<le> conjugate f (x \<sqinter> f y)"
mult_exp_mod_exp_eq: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and m :: "nat"   and n :: "nat"   and a :: "'a" assumes "class.semiring_bits plus minus zero times one divide modulo bit"   and "m \<le> n" shows "modulo (times a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m)) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) = times (modulo a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) (n - m))) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m)"
strategyproofness_1: shows "(pmf (sds R39) d < pmf (sds R29) a \<or> pmf (sds R39) d + pmf (sds R39) c < pmf (sds R29) a + pmf (sds R29) b) \<or> pmf (sds R39) d = pmf (sds R29) a \<and> pmf (sds R39) d + pmf (sds R39) c = pmf (sds R29) a + pmf (sds R29) b"
simps_1: fixes x :: "udom"   and k :: "udom \<rightarrow> udom\<cdot>btree" shows "bindU\<cdot>(Leaf\<cdot>x)\<cdot>k = k\<cdot>x"
\<eta>': shows "\<guillemotleft>\<Phi> (g, f) \<cdot>\<^sub>D (F g \<star>\<^sub>D F f) \<cdot>\<^sub>D (\<mu> \<star>\<^sub>D F f) \<cdot>\<^sub>D \<phi> \<cdot>\<^sub>D D.inv (unit (src\<^sub>C f)) : F (src\<^sub>C f) \<Rightarrow>\<^sub>D F (g \<star>\<^sub>C f)\<guillemotright>"
Grp_def: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'f" shows "BNF_Def.Grp A f = (\<lambda>a b. b = f a \<and> a \<in> A)"
prems_4: fixes x :: "nat option"   and k :: "nat" assumes "x \<in> set (y # xs)"   and "x = Some k" shows "k < l"
is_redI: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and a :: "'t \<Rightarrow>\<^sub>0 'c"   and b :: "'t \<Rightarrow>\<^sub>0 'c" assumes "red F a b" shows "is_red F a"
integral_scaleR_left: fixes c :: "'c"   and M :: "'d measure"   and f :: "'d \<Rightarrow> real" assumes "c \<noteq> (0::'c) \<Longrightarrow> integrable M f" shows "LINT x|M. f x *\<^sub>R c = integral\<^sup>L M f *\<^sub>R c"
FAccRedsThrow: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and a :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>throw a,s',b'\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>F{D},s,b\<rangle> \<rightarrow>* \<langle>throw a,s',b'\<rangle>"
merge_monoms_def: shows "merge_monoms = PAC_Checker_Init.merge (\<le>)"
basis_finite: fixes B :: "('a, 'b) vec set" assumes "is_basis B" shows "finite B"
ToCT_simp: fixes f :: "'c"   and g :: "'c"   and h :: "'c" assumes "C.arr f"   and "C.arr g"   and "C.arr h" shows "C.T.ToCT (f, g, h) = f \<otimes>\<^sub>C g \<otimes>\<^sub>C h"
cos_cmod_scalprod: fixes z1 :: "complex"   and z2 :: "complex" shows "cmod z1 * cmod z2 * cos (\<angle> z1 z2) = Re (scalprod z1 z2)"
SourcesA21_L1: shows "Sources level1 sA21 = {sA11}"
some_gcd_ff_list_greatest: fixes xs :: "'b fract list"   and d :: "'b fract" assumes "\<forall>x\<in>set xs. divides_ff d x" shows "divides_ff d (some_gcd_ff_list xs)"
left_neutral: fixes a :: "'b \<Rightarrow> 'b" shows "id \<circ> a = a"
order_refl: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "order r" shows "x \<sqsubseteq>\<^bsub>r\<^esub> x"
sorted_list_of_set_insert: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and x :: "'c" assumes "class.linorder less_eq less"   and "finite A" shows "linorder.sorted_list_of_set less_eq (insert x A) = linorder.insort_key less_eq (\<lambda>x. x) x (linorder.sorted_list_of_set less_eq (A - {x}))"
2: assumes "arr \<mu> \<and> B.iso (Map \<mu>)" shows "local.iso \<mu> \<and> local.inv \<mu> = MkArr (Cod \<mu>) (Dom \<mu>) (B.inv (Map \<mu>))"
simps_2: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat" shows "res_seq s (Suc k) = const_res_subseq (Suc k) (res_seq s k)"
attractive_order_axioms: shows "class.attractive_order (\<le>) (<)"
power_int_def: fixes x :: "'a"   and n :: "int" shows "x powi n = (if 0 \<le> n then x ^ nat n else inverse x ^ nat (- n))"
Max_Un: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "A \<noteq> {}"   and "finite B"   and "B \<noteq> {}" shows "Max (A \<union> B) = max (Max A) (Max B)"
comp_zimage_eq: fixes g :: "'b \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c zet" shows "zimage g (zimage f A) = zimage (g \<circ> f) A"
tlsim: shows "tllist_all2 (\<lambda>(tl1, s1'') (tl2, s2''). tl1 \<sim> tl2 \<and> s1'' \<approx> s2'') (rel_option bisim) stlsss1 stlsss2"
FvarsT_tJ: fixes \<psi> :: "'fmla" assumes "\<psi> \<in> fmla"   and "Fvars \<psi> = {xx}" shows "FvarsT (tJ \<psi>) = {}"
measure_uniform_measure: fixes M :: "'c measure"   and A :: "'c set"   and B :: "'c set" assumes "emeasure M A \<noteq> 0"   and "emeasure M A \<noteq> \<infinity>"   and "B \<in> sets M" shows "Sigma_Algebra.measure (uniform_measure M A) B = Sigma_Algebra.measure M (A \<inter> B) / Sigma_Algebra.measure M A"
AnalBinds_delete_lookup: fixes exp :: "exp \<Rightarrow> 'b \<rightarrow> 'a"   and x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> 'b\<^sub>\<bottom>" shows "(ExpAnalysis.AnalBinds exp (delete x \<Gamma>)\<cdot>ae) x = \<bottom>"
contains_carrier: assumes "F \<noteq> {}" shows "carrier \<in> F"
gaussian_def: shows "gaussian = map_real_pair (\<lambda>m s. density (stock_measure REAL) (gaussian_density (m, s) \<circ> extract_real)) undefined"
triangle_num_le_iff: fixes m :: "nat"   and n :: "nat" shows "(triangle_num m \<le> triangle_num n) = (m \<le> n)"
InvariantConflictClauseCharacterizationAfterApplyBackjump: fixes state :: "State" assumes "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)" shows "let state' = applyBackjump state in InvariantConflictClauseCharacterization (getConflictFlag state') (getConflictClause state') (getF state') (getM state')"
nearly_healthy_wlp_Seq: fixes t :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "nearly_healthy (wlp t)"   and "nearly_healthy (wlp u)" shows "nearly_healthy (wlp (t ;; u))"
pp_mapping_linorder_matrix: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.pp_mapping f = (\<forall>i. \<exists>j. f (i, j) \<noteq> bot \<and> (\<forall>k. j \<noteq> k \<longrightarrow> f (i, k) = bot))"
relfun_S: fixes S :: "'b set"   and s :: "'b" assumes "PreCoxeterSystem S"   and "s \<in> S" shows "add_order (s + s) = 1"
process_ret_add_match_dist2: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and rs :: "'c rule list"   and s :: "state"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>add_match m (process_ret rs), s\<rangle> \<Rightarrow> t" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>process_ret (add_match m rs), s\<rangle> \<Rightarrow> t"
intros_2: fixes \<pi> :: "nat \<Rightarrow> 'n"   and m :: "nat"   and n :: "nat" assumes "(\<pi>, m) \<in> scp"   and "n cd\<^bsup>\<pi>\<^esup>\<rightarrow> m" shows "(\<pi>, n) \<in> scp"
zero_le_mult_iff: 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"   and b :: "'a" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less_eq zero (times a b) = (less_eq zero a \<and> less_eq zero b \<or> less_eq a zero \<and> less_eq b zero)"
relcompp_assoc: fixes r :: "'c \<Rightarrow> 'f \<Rightarrow> bool"   and s :: "'f \<Rightarrow> 'e \<Rightarrow> bool"   and t :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "(r OO s) OO t = r OO s OO t"
simps_15: fixes v :: "'a local.term" shows "Nml \<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[v\<^bold>] = False"
poly_deg_one: shows "poly_deg 1 = (0::'c)"
sums_unique: fixes f :: "nat \<Rightarrow> 'b"   and s :: "'b" assumes "f sums s" shows "s = suminf f"
iffalse: fixes BMap :: "'a \<Rightarrow> bool"   and E :: "'b \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'a"   and b :: "'b"   and m :: "'c \<Rightarrow> 'a"   and c1 :: "('b, 'c) MWLfCom"   and c2 :: "('b, 'c) MWLfCom" assumes "BMap (E b m) = False" shows "((if b then c1 else c2 fi, m), [c2], m) \<in> MWLf_semantics.MWLfSteps_det E BMap"
deg_pols_add2: fixes p :: "'a"   and q :: "'a" assumes "p \<in> carrier R"   and "q \<in> carrier R"   and "deg R S X p = deg R S X q" shows "deg R S X (p \<plusminus> q) \<le> deg R S X q"
range_sessionkeys_not_priK: fixes KK :: "nat set"   and B :: "agent" assumes "KK \<subseteq> range sessionK" shows "priEK B \<notin> KK"
CFGExit_wf_axioms: shows "CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit"
\<beta>_intro: shows "LeftDerivation a E \<beta> \<and> LeftDerivation \<beta> (derivation_shift F 0 (Suc j)) b"
arccos: fixes y :: "real" assumes "- 1 \<le> y"   and "y \<le> 1" shows "0 \<le> arccos y \<and> arccos y \<le> pi \<and> cos (arccos y) = y"
cl_inv_vars_update_1: fixes g :: "'a gState_scheme"   and p :: "pState"   and vs :: "(String.literal, variable) assoc_list \<Rightarrow> (String.literal, variable) assoc_list" assumes "cl_inv (g, p)" shows "cl_inv (g, pState.vars_update vs p)"
that: fixes y :: "('idx, 'pred, 'act) formula" assumes "supp y \<subseteq> supp P"   and "y distinguishes P from Q" shows "thesis"
arr_def: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "arr f \<equiv> domains f \<noteq> {} \<or> codomains f \<noteq> {}"
le: fixes x :: "'a" shows "norm (f x - x) \<le> c * norm x"
strict_implies_not_eq: fixes b :: "'a list"   and a :: "'a list" assumes "strict_prefix b a" shows "a \<noteq> b"
eval': shows "(cms, subst (to_partial mem |` Y) mem) \<leadsto> (cms', subst (to_partial mem |` Y) mem')"
thr0: shows "- row i A = (\<Sum>j\<in>UNIV - {i}. (1::'a) / c i *s (c j *s row j A))"
that: fixes r :: "'b \<Rightarrow>\<^sub>0 'a" assumes "keys r \<subseteq> B"   and "x = rep r" shows "thesis"
ipd_1: shows "\<rho> m = ipd (\<rho> k)"
prod_map_def: shows "prod_map = (\<Lambda> f g p. (f\<cdot>(fst p), g\<cdot>(snd p)))"
hypnat_induct_obj: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat star" shows "(*p* P) 0 \<and> (\<forall>n. (*p* P) n \<longrightarrow> (*p* P) (n + 1)) \<longrightarrow> (*p* P) n"
simps_2: fixes cmp :: "'b comparator"   and xs :: "'b list" shows "Sorting_Algorithms.merge cmp xs [] = xs"
intro: fixes A :: "('h, 'j) graph_rec_scheme"   and R :: "('h \<times> 'i) set"   and B :: "('i, 'k) graph_rec_scheme" assumes "\<And>a b. \<lbrakk>a \<in> g_V A; (a, b) \<in> R\<rbrakk> \<Longrightarrow> b \<in> g_V B"   and "\<And>a0. a0 \<in> g_V0 A \<Longrightarrow> \<exists>b0. b0 \<in> g_V0 B \<and> (a0, b0) \<in> R"   and "\<And>a a' b. \<lbrakk>(a, a') \<in> g_E A; (a, b) \<in> R\<rbrakk> \<Longrightarrow> \<exists>b'. (b, b') \<in> g_E B \<and> (a', b') \<in> R" shows "simulation_axioms R A B"
atMost_atLeastAtMost_0_conv: fixes i :: "nat" shows "{..i} = {0..i}"
cong_exp_trans_1: fixes a :: "'c"   and b :: "nat"   and c :: "'c"   and n :: "'c"   and d :: "'c" assumes "[a ^ b = c] (mod n)"   and "[a = d] (mod n)" shows "[d ^ b = c] (mod n)"
h: fixes i :: "'a" assumes "i \<in> I" shows "h i \<in> Group.iso (Gs i) (Hs (f i))"
dfs_comp_distinct: fixes f :: "'f \<Rightarrow> 'c"   and g :: "('d, 'e) bintree \<Rightarrow> 'f"   and xs :: "('d, 'e) bintree option list" assumes "distinct (dfs (f \<circ> g) xs)" shows "distinct (dfs g xs)"
h1: assumes "\<exists>(a', b', c')\<in>set eq. (a' = 0 \<and> b' \<noteq> 0) \<and> (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0)" shows "\<exists>x. (\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)"
addition_sub_aug_pm_addition: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e" shows "addition_set f A \<subseteq> addition_set f (aug_pm_set z i A)"
A_to_prodag_def: fixes A :: "('d, 'g) aGroup_scheme"   and I :: "'e set"   and S :: "'e \<Rightarrow> 'd \<Rightarrow> 'f"   and B :: "'e \<Rightarrow> ('f, 'h) aGroup_scheme" shows "A_to_prodag A I S B = (\<lambda>a\<in>carrier A. \<lambda>k\<in>I. S k a)"
simps_12: fixes v :: "\<I>"   and va :: "formula" shows "get_and_list (Prev v va) = [Prev v va]"
Tau: fixes P :: "pi" shows "\<tau>.P \<longmapsto>\<^sub>l \<tau> \<prec>\<^sub>l P"
\<epsilon>_2: shows "ball 0 \<epsilon> \<subseteq> eval_fps h -` {..<2} \<inter> eball 0 (fps_conv_radius f)"
Immediate: fixes l :: "Location"   and s :: "Store" assumes "ref l \<noteq> nullV" shows "s\<turnstile> l reachable_from ref l"
f0_2: shows "f $ 0 * y = (1::'a)"
ereal_mult_le_0_iff: fixes a :: "ereal"   and b :: "ereal" shows "(a * b \<le> 0) = (0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b)"
sorted_distinct: fixes h :: "'b \<Rightarrow> int"   and a :: "'b"   and b :: "'b"   and t :: "'b Tree" shows "sorted_distinct_pred h a b t"
lead_coeff_cf_pos_poly: fixes f :: "int poly" shows "(0 < lead_coeff (cf_pos_poly f)) = (f \<noteq> 0)"
IH: fixes P :: "'b set" assumes "simple_ruleset rs"   and "p \<in> P" shows "approximating_bigstep_fun \<gamma> p (rmshadow \<gamma> rs P) = approximating_bigstep_fun \<gamma> p rs"
K: shows "Key (sesK (c * (d * (s' + a' * b')))) \<in> A"
chamber_preimage: fixes C :: "'a set" assumes "C \<in> X"   and "codomain.chamber (f ` C)" shows "domain.chamber C"
ideI: fixes a :: "'c \<times> 'c" assumes "C.arr (fst a) \<and> C.arr (snd a) \<and> src\<^sub>C (fst a) = trg\<^sub>C (snd a)"   and "C.VxV.ide a" shows "C.VV.ide a"
Nset2_finiteTr: fixes n :: "nat" shows "\<forall>A. finite A \<and> card A = Suc n \<longrightarrow> (\<exists>f. f \<in> {i. i \<le> n} \<rightarrow> A \<and> surj_to f {i. i \<le> n} A)"
connected_def: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set" assumes "class.topological_space open" shows "topological_space.connected open S = (\<nexists>A B. open A \<and> open B \<and> S \<subseteq> A \<union> B \<and> A \<inter> B \<inter> S = {} \<and> A \<inter> S \<noteq> {} \<and> B \<inter> S \<noteq> {})"
alternating_sum_list_conv_nataction: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "nat"   and s :: "'b"   and t :: "'b" assumes "class.monoid_add plus zero" shows "monoid_add.sum_list plus zero (alternating_list (2 * n) s t) = power.power zero plus (plus s t) n"
Receive: fixes t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list"   and \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) strand_step list" assumes "receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>" shows "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet> (update\<^sub>s\<^sub>t \<S> (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S), \<A> @ [send\<langle>t\<rangle>\<^sub>s\<^sub>t])"
assms_1: shows "((n', ns'), n, ns) \<in> expand_ord \<phi> \<and> expand_inv \<phi> (n', ns')"
inv_closed: fixes x :: "'a" assumes "x \<in> carrier R" shows "\<ominus> x \<in> carrier R"
abs_eq: shows "sup_sls = Abs_semilattice_set sup"
analz_Key_image_insert_eq: fixes K :: "nat"   and evs :: "event list"   and KK :: "nat set" assumes "Key K \<notin> used evs"   and "K \<in> symKeys"   and "evs \<in> set_pur" shows "analz (Key ` insert K KK \<union> knows Spy evs) = insert (Key K) (analz (Key ` KK \<union> knows Spy evs))"
crel_vs_def: fixes R :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and v :: "'f"   and s :: "('b, 'g) state" shows "crel_vs R v s \<equiv> \<forall>M. cmem M \<and> P M \<longrightarrow> (case run_state s M of (v', M') \<Rightarrow> R v v' \<and> cmem M' \<and> P M')"
moebius_sum_tail_bound: fixes d :: "nat" assumes "0 < d" shows "\<bar>\<Sum>n. moebius_mu (Suc n + d) / (real (Suc n + d))\<^sup>2\<bar> \<le> 1 / real d"
well_related_set_axioms: shows "well_related_set X (\<sqsubseteq>)"
assms: fixes v :: "nat" assumes "v \<in> V"   and "(u, v) \<notin> E" shows "g (u, v) = (0::'b)"
assms: shows "set adjs = {(v, d). (u, v) \<in> edges g \<and> w {u, v} = d}"
cl_subset_carrier: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and X :: "'b set" assumes "matroid carrier indep" shows "matroid.cl carrier indep X \<subseteq> carrier"
lassocr_swap_sum_def: shows "lassocr_swap_sum = rsuml \<circ> map_sum swap_sum id \<circ> lsumr"
prems_3: fixes i :: "nat" assumes "i < Suc n" shows "exponential_order M c (nderiv i f {0<..})"
swap_eq_on: fixes arr1 :: "'a \<Rightarrow> 'b"   and arr2 :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "arr1 = arr2 on S"   and "x \<notin> S"   and "y \<notin> S" shows "arr1 = swap arr2 x y on S"
time_split_at_tm: fixes n :: "nat"   and xs :: "'a list" shows "time (split_at_tm n xs) = min n (length xs) + 1"