hcju/isabelleps huggingface dataset:
REC_mono: fixes B :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> 'd"   and B' :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> 'd"   and x :: "'c" assumes "trimono B"   and "\<And>F x. B F x \<le> B' F x" shows "REC B x \<le> REC B' x"
outstanding_refs_non_volatile_Read\<^sub>s\<^sub>b_all_acquired: fixes pending_write :: "bool"   and \<O> :: "nat set"   and m :: "nat \<Rightarrow> nat"   and sb :: "'b memref list"   and \<S> :: "nat \<Rightarrow> bool option"   and a :: "nat" assumes "reads_consistent pending_write \<O> m sb"   and "non_volatile_owned_or_read_only pending_write \<S> \<O> sb"   and "a \<in> outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b sb" shows "a \<in> \<O> \<or> a \<in> all_acquired sb \<or> a \<in> read_only_reads \<O> sb"
less_irrefl: fixes x :: "'a \<times> ('a, 'b) operation" shows "\<not> hb x x"
continuous_on_rabs: fixes s :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. \<bar>f x\<bar>)"
f_props_1: shows "\<Union> (f ` one_chain1) = subdiv"
simps_4: fixes hosts :: "'b set"   and flows_fix :: "('b \<times> 'b) set"   and flows_state :: "('b \<times> 'b) set"   and more :: "'a" shows "stateful_policy.more \<lparr>hosts = hosts, flows_fix = flows_fix, flows_state = flows_state, \<dots> = more\<rparr> = more"
proj_successors_of_valid_are_valid: fixes s :: "('a, 'b) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and vs :: "'a set" assumes "s \<in> valid_states PROB" shows "state_successors (prob_proj PROB vs) s \<subseteq> valid_states PROB"
assms_1: fixes a1 :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and a2 :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "a1 \<in> A"   and "a2 \<in> A" shows "a1 - a2 \<in> A"
galois_property: fixes x :: "'a"   and y :: "'b" assumes "x \<in> carrier \<X>\<^bsub>G\<^esub>"   and "y \<in> carrier \<Y>\<^bsub>G\<^esub>" shows "(\<pi>\<^sup>*\<^bsub>G\<^esub> x \<sqsubseteq>\<^bsub>\<Y>\<^bsub>G\<^esub>\<^esub> y) = (x \<sqsubseteq>\<^bsub>\<X>\<^bsub>G\<^esub>\<^esub> \<pi>\<^sub>*\<^bsub>G\<^esub> y)"
ws: shows "set (us[i := us ! i + c \<cdot>\<^sub>v us ! j]) \<subseteq> carrier_vec n"
simps_2: fixes carrier :: "'b set"   and mult :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and add :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and more :: "'c" shows "(\<oplus>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub>) = add"
Fun_mkPoint: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and a :: "'b"   and t :: "'b" assumes "set_category S \<AA>"   and "partial_magma.ide S a"   and "t \<in> set_category_data.set S (set_category.some_img S \<AA>) a" shows "set_category_given_img.Fun S (set_category.some_img S \<AA>) (set_category.mkPoint S \<AA> a t) = (\<lambda>_\<in>{set_category_data.unity S}. t)"
intros_2: fixes m1 :: "'a \<times> ('a, 'b) operation"   and i :: "nat"   and m2 :: "'a \<times> ('a, 'b) operation" assumes "Deliver m1 \<sqsubset>\<^sup>i Broadcast m2" shows "hb m1 m2"
rtranclpD: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "R\<^sup>*\<^sup>* a b" shows "a = b \<or> a \<noteq> b \<and> R\<^sup>+\<^sup>+ a b"
assms_1: fixes a :: "'a" assumes "a \<in> set al" shows "0 < f a"
EXHcrypt_def: fixes K :: "nat"   and EK :: "nat"   and M :: "msg"   and m :: "msg" shows "EXHcrypt K EK M m = \<lbrace>Crypt K \<lbrace>M, Hash m\<rbrace>, Crypt EK \<lbrace>Key K, m, Hash M\<rbrace>\<rbrace>"
arctan_approx_def: fixes n :: "nat"   and x :: "real" shows "arctan_approx n x = x * (\<Sum>k<n. (- x\<^sup>2) ^ k / real (2 * k + 1))"
nv_OfFax_2: fixes x :: "Preliminaries.var"   and f :: "'fsym" shows "x \<notin> nvT (rOfFax f)"
sq_mtx_ith_def: shows "($$) \<equiv> map_fun to_vec id ($)"
restrict_modes_def: fixes mdss :: "(Mode \<Rightarrow> 'Var set) list"   and X :: "'Var set" shows "restrict_modes mdss X \<equiv> map (\<lambda>mds m. mds m \<inter> X) mdss"
finite_rhs_def: fixes ES :: "('c \<times> 'd set) set" shows "finite_rhs ES \<equiv> \<forall>(X, rhs)\<in>ES. finite rhs"
sorted_list_of_B: fixes t :: "'b tree" assumes "braun t"   and "heap t" shows "sorted (list_of_B t)"
arr_iff_in_hom: fixes f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" shows "J_C.arr f = \<guillemotleft>f : J_C.dom f \<rightarrow>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] J_C.cod f\<guillemotright>"
leq_refl: fixes x :: "'d spmf" shows "ord_spmf (=) x x"
distinguishing_formula_distinguishes: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and P :: "'b"   and Q :: "'b" assumes "indexed_nominal_ts TYPE('a) satisfies transition"   and "\<not> indexed_nominal_ts.logically_equivalent TYPE('a) satisfies transition P Q" shows "indexed_nominal_ts.is_distinguishing_formula satisfies transition (indexed_nominal_ts.distinguishing_formula satisfies transition P Q) P Q"
h: shows "prod_list (map ((!) (map_sgas sign)) (a # xa)) = (1 - (sign ! a)\<^sup>2) * prod_list (map ((!) (map_sgas sign)) xa)"
mono_neutral_cong_left: fixes T :: "'c set"   and S :: "'c set"   and h :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. h i = \<^bold>1"   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "F g S = F h T"
unit_is_iso: shows "local.iso \<eta>\<^sub>1"
subgame_\<omega>_subset: fixes V' :: "'a set" shows "\<omega>\<^bsub>G'.subgame V'\<^esub> ` V\<^bsub>G'.subgame V'\<^esub> \<subseteq> \<omega>\<^bsub>G'\<^esub> ` V\<^bsub>G'\<^esub>"
frame_thms_2: fixes P :: "assn"   and P' :: "assn"   and F :: "assn"   and F' :: "assn" assumes "P \<Longrightarrow>\<^sub>t P'"   and "F \<Longrightarrow>\<^sub>t F'" shows "F * P \<Longrightarrow>\<^sub>t F' * P'"
adj_iso3: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "f \<stileturn> g" shows "mono (f \<circ> g)"
abstFn_def: fixes cfile :: "CFile" shows "abstFn cfile \<equiv> (fileSize cfile, \<lambda>index. case cfRead cfile index of None \<Rightarrow> fillByte | Some value \<Rightarrow> value)"
real_cond_exp_cmult: fixes f :: "'a \<Rightarrow> real"   and c :: "real" assumes "integrable M f" shows "AE x in M. real_cond_exp M F (\<lambda>x. c * f x) x = c * real_cond_exp M F f x"
lve_Entry_edge: fixes e :: "'a LDCFG_node \<times> 'b edge_kind \<times> 'a LDCFG_node"   and E :: "'a"   and valid_edge :: "'c \<Rightarrow> bool"   and src :: "'c \<Rightarrow> 'a"   and trg :: "'c \<Rightarrow> 'a"   and knd :: "'c \<Rightarrow> 'b edge_kind"   and X :: "'a" assumes "e = (NewEntry, (\<lambda>s. True)\<^sub>\<surd>, LDCFG_node.Node E)" shows "lift_valid_edge valid_edge src trg knd E X e"
LeftDerivation_ge_take: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and k :: "nat"   and a :: "symbol list"   and b :: "symbol list" assumes "derivation_ge D k"   and "LeftDerivation a D b"   and "D \<noteq> []" shows "take k a = take k b \<and> is_word (take k a)"
norm_triangle_ineq: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (plus x y) \<le> norm x + norm y"
obj_def': fixes a :: "'c" shows "C.obj a = (C.arr a \<and> trg\<^sub>C a = a)"
term_equiv_def: shows "(\<leftrightarrow>) = joinablep (\<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>*)"
mono_inv_plus': fixes a1 :: "'a"   and b1 :: "'a"   and a2 :: "'a"   and b2 :: "'a"   and r :: "'a"   and r' :: "'a" assumes "a1 \<le> b1"   and "a2 \<le> b2"   and "r \<le> r'" shows "inv_plus' r a1 a2 \<le> inv_plus' r' b1 b2"
top_add: fixes x :: "'c" assumes "(0::'c) \<le> x" shows "top + x = top"
swap_id: fixes i :: "'b"   and j :: "'b" shows "id(i := j, j := i) \<circ> id(i := j, j := i) = id"
lagrange: fixes G :: "('a, 'b) monoid_scheme"   and H :: "'a set" assumes "Group.group G"   and "subgroup H G" shows "card (rcosets\<^bsub>G\<^esub> H) * card H = order G"
mk_linear_orders_range: fixes r :: "'a list"   and C :: "'a set"   and B :: "'a set" assumes "r \<in> local.mk_linear_orders C B"   and "B \<subseteq> A" shows "set r \<subseteq> B"
update_convs_3: fixes subs' :: "(('d \<times> nat) \<times> 'd \<times> nat) set \<Rightarrow> (('d \<times> nat) \<times> 'd \<times> nat) set"   and red :: "('d \<times> nat) rgraph"   and black :: "('d, 'e, 'f) lts"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and init_conf :: "('e, 'f) conf"   and confs :: "'d \<times> nat \<Rightarrow> ('e, 'f) conf"   and marked :: "'d \<times> nat \<Rightarrow> bool"   and strengthenings :: "'d \<times> nat \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> bool"   and more :: "'g" shows "subs_update subs' \<lparr>red = red, black = black, subs = subs, init_conf = init_conf, confs = confs, marked = marked, strengthenings = strengthenings, \<dots> = more\<rparr> = \<lparr>red = red, black = black, subs = subs' subs, init_conf = init_conf, confs = confs, marked = marked, strengthenings = strengthenings, \<dots> = more\<rparr>"
count_Inf_multiset_nonempty: fixes A :: "'k multiset set"   and x :: "'k" assumes "A \<noteq> {}" shows "count (Inf A) x = (INF X\<in>A. count X x)"
starP_star_of: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b" shows "(*p* P) (star_of x) = P x"
set_transfer: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "(list_all2 R ===> rel_set R) set set"
taylor_deg_1_eval'': fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "f \<in> carrier P"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "c = to_fun (shift 2 (T\<^bsub>a\<^esub> f)) (\<ominus> b)" shows "to_fun f (a \<ominus> b) = to_fun f a \<ominus> deriv f a \<otimes> b \<oplus> c \<otimes> b [^] 2"
not_ord_pI: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "lookup p v \<noteq> lookup q v"   and "lookup p v \<noteq> (0::'c)"   and "\<And>u. lt_of_nat_term_order cmp_term v u \<Longrightarrow> lookup p u = lookup q u" shows "\<not> ord_p p q"
Ints_real_of_nat_divide: fixes b :: "nat"   and a :: "nat" assumes "b dvd a" shows "real a / real b \<in> \<int>"
rAlpha: fixes A\<^sub>P :: "name list"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list"   and Rs :: "('a, 'b, 'c) residual"   and C :: "'d"   and \<Psi>\<^sub>P :: "'b" assumes "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* (p \<bullet> A\<^sub>P)"   and "A\<^sub>P \<sharp>* Rs"   and "A\<^sub>P \<sharp>* C"   and "set p \<subseteq> set A\<^sub>P \<times> set (p \<bullet> A\<^sub>P)"   and "distinctPerm p"   and "Prop C \<Psi> P Rs A\<^sub>P \<Psi>\<^sub>P" shows "Prop C \<Psi> P Rs (p \<bullet> A\<^sub>P) (p \<bullet> \<Psi>\<^sub>P)"
correctCompositionDiffLevelsS10: shows "correctCompositionDiffLevels sS10"
solve_equation: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and x :: "'c"   and y :: "'c" assumes "Group.group G"   and "subgroup H G"   and "x \<in> H"   and "y \<in> H" shows "\<exists>h\<in>H. y = h \<otimes>\<^bsub>G\<^esub> x"
in_FvarsT_substTD: fixes y :: "'var"   and r :: "'trm"   and t :: "'trm"   and x :: "'var" assumes "y \<in> FvarsT (substT r t x)"   and "r \<in> atrm"   and "t \<in> trm"   and "x \<in> var" shows "y \<in> FvarsT r - {x} \<union> (if x \<in> FvarsT r then FvarsT t else {})"
inverse_arrowsE: fixes f :: "'a2"   and g :: "'a2" assumes "C2.inverse_arrows f g" obtains "C2.ide (g \<cdot>\<^sub>2 f)" and "C2.ide (f \<cdot>\<^sub>2 g)"
ordering_axioms: shows "ordering (\<preceq>) (\<prec>)"
If2_def: fixes Q :: "tr"   and x :: "'r"   and y :: "'r" shows "If2 Q x y = If Q then x else y"
list_all_encompasses_cases: fixes p :: "real \<times> real"   and x :: "(real \<times> real) \<times> real \<times> real"   and y :: "(real \<times> real) \<times> real \<times> real"   and zs :: "((real \<times> real) \<times> real \<times> real) list" assumes "list_all (\<lambda>seg. 0 \<le> det3 (fst seg) (snd seg) p) (x # y # zs)" obtains "list_all (\<lambda>seg. 0 \<le> det3 (fst seg) (snd seg) p) [x, y, (snd y, fst x)]"   | "list_all (\<lambda>seg. 0 \<le> det3 (fst seg) (snd seg) p) ((fst x, snd y) # zs)"
bounded_byI: fixes P :: "'a \<Rightarrow> real"   and b :: "real" assumes "\<And>x. P x \<le> b" shows "bounded_by b P"
bit_shiftr_word_iff: fixes w :: "'b word"   and m :: "nat"   and n :: "nat" shows "bit (w >> m) n = bit w (m + n)"
assms_1: shows "x \<in>\<^sub>\<circ> A \<inter>\<^sub>\<circ> B"
simps_1: fixes v :: "nat" shows "var (Pos v) = v"
wsst_TYs_3: fixes w :: "'a word" shows "size w = LENGTH('a)"
direct_decomp_appendD_3: fixes A :: "'b set"   and ss1 :: "'b set list"   and ss2 :: "'b set list" assumes "direct_decomp A (ss1 @ ss2)" shows "direct_decomp A [sum_list ` listset ss1, sum_list ` listset ss2]"
ct_suffix_length: fixes ts :: "ct list"   and ts' :: "ct list" assumes "ct_suffix ts ts'" shows "length ts \<le> length ts'"
BNTr4_2: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and W1 :: "'a set"   and W2 :: "'a set" assumes "f \<in> carrier D \<rightarrow> carrier D"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "\<forall>x\<in>carrier D. x \<preceq> f x"   and "W1 \<in> WWa D f a"   and "W2 \<in> WWa D f a"   and "ord_equiv (Iod D W1) (Iod D (segment (Iod D W2) b))" shows "W1 = segment (Iod D W2) b"
fv\<^sub>e\<^sub>s\<^sub>t_append: fixes A :: "('fun, 'var) extstrand_step list"   and B :: "('fun, 'var) extstrand_step list" shows "fv\<^sub>e\<^sub>s\<^sub>t (A @ B) = fv\<^sub>e\<^sub>s\<^sub>t A \<union> fv\<^sub>e\<^sub>s\<^sub>t B"
tendsto_complex_iff: fixes f :: "'b \<Rightarrow> complex"   and x :: "complex"   and F :: "'b filter" shows "(f \<longlongrightarrow> x) F = (((\<lambda>x. Re (f x)) \<longlongrightarrow> Re x) F \<and> ((\<lambda>x. Im (f x)) \<longlongrightarrow> Im x) F)"
norm_eq_sqrt_inner: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "norm x = sqrt (inner x x)"
matchId: fixes a :: "name"   and P :: "pi" shows "[a\<frown>a]P \<simeq> P"
lex_convex_self2_1: fixes p :: "real \<times> real"   and q :: "real \<times> real"   and a :: "real" assumes "Counterclockwise_2D_Arbitrary.lex p q"   and "0 \<le> a"   and "a \<le> 1" shows "Counterclockwise_2D_Arbitrary.lex p (a *\<^sub>R p + (1 - a) *\<^sub>R q)"
elimmeets: fixes p :: "'b"   and s :: "'b"   and r :: "'b"   and q :: "'b" shows "(p \<parallel> s \<and> (\<exists>t. p \<parallel> t \<and> t \<parallel> s) \<and> (\<exists>t. r \<parallel> t \<and> t \<parallel> q)) = False"
load_word_mem3_mod_state: fixes s :: "'b sparc_state"   and addr :: "word32"   and asi :: "word8"   and new_undef :: "bool"   and new_traps :: "Trap set"   and new_state_var :: "sparc_state_var"   and new_dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and new_user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and new_cpu_reg :: "CPU_register \<Rightarrow> word32" shows "fst (case load_word_mem s addr asi of None \<Rightarrow> (None, s) | Some w \<Rightarrow> (Some w, add_instr_cache s addr w 15)) = fst (case load_word_mem (s\<lparr>cpu_reg := new_cpu_reg, user_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var, traps := new_traps, undef := new_undef\<rparr>) addr asi of None \<Rightarrow> (None, s\<lparr>cpu_reg := new_cpu_reg, user_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var, traps := new_traps, undef := new_undef\<rparr>) | Some w \<Rightarrow> (Some w, add_instr_cache (s\<lparr>cpu_reg := new_cpu_reg, user_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var, traps := new_traps, undef := new_undef\<rparr>) addr w 15))"
psubstEnv_compose: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and rho'' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" assumes "goodEnv rho"   and "goodEnv rho'"   and "goodEnv rho''" shows "rho &[rho'] &[rho''] = rho &[rho' &[rho'']]"
intros_3: fixes a :: "nat"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" shows "(Write False a (D, f) A L R W # is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is, \<theta>, sb @ [Write\<^sub>s\<^sub>b False a (D, f) (f \<theta>) A L R W], m, \<D>, \<O>, \<R>, \<S>)"
subseqs_length_simple: fixes b :: "'b list"   and xs :: "'b list" assumes "b \<in> set (subseqs xs)" shows "length b \<le> length xs"
ccBindsExtra_eq: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and G :: "CoCalls" shows "ccBindsExtra \<Gamma>\<cdot>(ae, G) = G \<squnion> ccBinds \<Gamma>\<cdot>(ae, G) \<squnion> (\<Squnion>x\<mapsto>e\<in>map_of \<Gamma>. fv e G\<times> ccNeighbors x G)"
asymmetric_irreflexive: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" assumes "matrix_stone_relation_algebra.asymmetric x" shows "matrix_stone_relation_algebra.irreflexive x"
Elem_zimplode: fixes x :: "ZF"   and z :: "ZF zet" shows "Elem x (zimplode z) = zin x z"
simps_3: fixes i :: "name"   and x :: "tm"   and t :: "tm"   and u :: "tm" shows "subst i x (Eats t u) = Eats (subst i x t) (subst i x u)"
inv_max_hi_insert: fixes t :: "('b ivl \<times> 'b) tree"   and x :: "'b ivl" assumes "inv_max_hi t" shows "inv_max_hi (Interval_Tree.insert x t)"
rtrancl_imp_UN_relpow: fixes p :: "'a \<times> 'a"   and R :: "('a \<times> 'a) set" assumes "p \<in> R\<^sup>*" shows "p \<in> \<Union> (range ((^^) R))"
int_pow_neg: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b"   and i :: "int" assumes "Group.group G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> - i = inv\<^bsub>G\<^esub> (x [^]\<^bsub>G\<^esub> i)"
of_nat_2p: shows "word_of_nat (2 ^ LENGTH('b)) = 0"
index_one_empty_rm_blv: fixes blv :: "'a set"   and p :: "'a set" assumes "\<Lambda> = 1"   and "blv \<in># \<B>"   and "p \<subseteq> blv"   and "card p = 2" shows "filter_mset ((\<subseteq>) p) (remove1_mset blv \<B>) = {#}"
efficient_power_correct: fixes m :: "nat"   and y :: "nat"   and x :: "nat"   and n :: "nat" shows "mod_exp_nat.efficient_power m y x n = ((\<lambda>y. x * y mod m) ^^ n) y"
as_needed_asses_submap_exec_ii: fixes v :: "'d"   and f :: "('d, 'e) fmap"   and g :: "('d, 'e) fmap" assumes "v \<in> fmdom' f"   and "f \<subseteq>\<^sub>f g" shows "fmlookup f v = fmlookup g v"
prems: shows "\<lbrakk>M; map (\<lambda>d. \<forall>X\<langle>\<or>\<noteq>: [(pair (t, s), pair (snd d))]\<rangle>\<^sub>s\<^sub>t) (filter (\<lambda>d. d \<notin> set Di) (di # D))\<rbrakk>\<^sub>d \<I>"
PPs_insertE: fixes t :: "'a \<Rightarrow>\<^sub>0 nat"   and x :: "'a"   and X :: "'a set" assumes "t \<in> .[insert x X]" obtains e :: "nat" and tx :: "'a \<Rightarrow>\<^sub>0 nat" where "tx \<in> .[X]" and "t = monomial e x + tx"
invar1: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and Q :: "'a \<Rightarrow> enat"   and \<pi> :: "'a \<Rightarrow> 'a option" assumes "Prim_Invar2_ctd_Presentation_Loc w g r Q \<pi>" shows "Prim.prim_invar1 w g r (A Q \<pi>)"
sum_mset_mono: fixes A :: "'a multiset"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in># A \<Longrightarrow> f x \<le> g x" shows "\<Sum>\<^sub># (image_mset f A) \<le> \<Sum>\<^sub># (image_mset g A)"
regular_bot_top: fixes uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and top :: "'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.linorder_stone_algebra_expansion uminus less_eq less bot top inf sup" shows "(x = uminus (uminus x)) = (x = bot \<or> x = top)"
is_none_alt_def: fixes x :: "'c option" shows "Option.is_none x = (case x of None \<Rightarrow> True | Some x \<Rightarrow> False)"
ran_def: fixes m :: "'e \<Rightarrow> \<langle>'d\<rangle>\<^sub>\<bottom>" shows "ran m = {b. \<exists>a. m a = \<lfloor>b\<rfloor>}"
subject_reduction: fixes v :: "val"   and e :: "exp"   and \<rho> :: "(nat \<times> val) list"   and e' :: "exp" assumes "v \<in> E e \<rho>"   and "e \<longrightarrow> e'" shows "v \<in> E e' \<rho>"
3: shows "gurun A w (sdrop n r) (gtarget (stake n r) v)"
compact_Times: fixes s :: "'a set"   and t :: "'b set" assumes "compact s"   and "compact t" shows "compact (s \<times> t)"
deg_pm_eq_0_iff: fixes s :: "'b \<Rightarrow>\<^sub>0 'a" shows "(deg_pm s = (0::'a)) = (s = 0)"
bind_pmf_const: fixes M :: "'b pmf"   and c :: "'a pmf" shows "M \<bind> (\<lambda>x. c) = c"
add_independentS_doubleD: fixes A :: "'a set"   and B :: "'a set"   and b :: "'a"   and a :: "'a" assumes "add_independentS [A, B]"   and "b \<in> B"   and "a \<in> A"   and "a + b = (0::'a)" shows "a = (0::'a)"
filtered_to_directed: fixes X :: "'a set" shows "filtered X = directed (\<partial> ` X)"
naturality: fixes f :: "'a" assumes "arr f" shows "\<d>[cod f] \<cdot> local.map f = (T \<circ> \<Delta>.map) f \<cdot> \<d>[local.dom f]"
trg_some_trg: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "some_trg (some_trg \<mu>) = some_trg \<mu>"
var_def1: shows "A = Fraction_Field.Fract x 1"
nat_term_comp_proj_comp: fixes cmp :: "(('c, 'd) pp \<times> nat) nat_term_order" shows "nat_term_comp (proj_comp cmp)"
second_safe_dist_def: fixes v\<^sub>e :: "real"   and a\<^sub>e :: "real"   and v\<^sub>o :: "real"   and a\<^sub>o :: "real" shows "second_safe_dist v\<^sub>e a\<^sub>e v\<^sub>o a\<^sub>o \<equiv> rel_dist_to_stop v\<^sub>e a\<^sub>e - rel_dist_to_stop v\<^sub>o a\<^sub>o"
simps: fixes a1 :: "('f, 'g, 'h) Resumption.resumption"   and a2 :: "('f, 'g, 'h) Resumption.resumption" shows "resumption_ord a1 a2 = ((\<exists>a a'. a1 = Resumption.resumption.Done a \<and> a2 = Resumption.resumption.Done a' \<and> option_ord a a') \<or> (\<exists>out c. a1 = ABORT \<and> a2 = Resumption.resumption.Pause out c) \<or> (\<exists>c c' out. a1 = Resumption.resumption.Pause out c \<and> a2 = Resumption.resumption.Pause out c' \<and> rel_fun (=) resumption_ord c c'))"
frag_extend_compose: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow>\<^sub>0 int"   and g :: "'e \<Rightarrow> 'd"   and c :: "'e \<Rightarrow>\<^sub>0 int" shows "frag_extend f (frag_extend (frag_of \<circ> g) c) = frag_extend (f \<circ> g) c"
atLeastLessThan_less: fixes m :: "nat"   and n :: "nat" shows "({0..<m} \<subset> {0..<n}) = (m < n)"
borel_measurable_add': fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "f + g \<in> borel_measurable M"
length_merges: fixes ps :: "(nat \<times> nat list) list"   and ss :: "nat list list" shows "length (merges f ps ss) = length ss"