hcju/isabelleps huggingface dataset:
assms_2: shows "P (\<sigma> |\<^sub>s i)"
Seq: fixes p :: "'a set"   and c1 :: "'a com"   and r :: "'a set"   and c2 :: "'a com"   and q :: "'a set" assumes "\<parallel>- p c1 r"   and "\<parallel>- r c2 q" shows "\<parallel>- p Seq c1 c2 q"
connectedin_closure_of: fixes X :: "'b topology"   and S :: "'b set" assumes "connectedin X S" shows "connectedin X (X closure_of S)"
multiplicity_gt_zero_iff: fixes x :: "'a"   and p :: "'a" assumes "x \<noteq> (0::'a)"   and "\<not> is_unit p" shows "(0 < multiplicity p x) = (p dvd x)"
pr1: shows "\<guillemotleft>pr1 (cod f) (cod g) : local.prod (cod f) (cod g) \<rightarrow> cod f\<guillemotright>"
max_Sup_absorb_left: fixes X :: "'b set"   and f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real" assumes "X \<noteq> {}"   and "bdd_above (f ` X)"   and "bdd_above (g ` X)"   and "Sup (g ` X) \<le> Sup (f ` X)" shows "Sup (pointwise_max f g ` X) = Sup (f ` X)"
is_rbt_fold_rbt_insert: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('b, 'c) rbt"   and f :: "'b \<Rightarrow> 'c"   and xs :: "'b list" assumes "class.linorder less_eq less"   and "ord.is_rbt less t" shows "ord.is_rbt less (List.fold (\<lambda>k. ord.rbt_insert less k (f k)) xs t)"
doesnt_read_or_modify_doesnt_modify: fixes c :: "'Com\<^sub>C"   and x :: "'Var\<^sub>C" assumes "conc.doesnt_read_or_modify c x" shows "conc.doesnt_modify c x"
assms_7: shows "w' \<equiv> rsd_winners (R(i := Ri')) alts"
ide_in_hom: fixes a :: "'c" shows "C.ide a = \<guillemotleft>a : a \<rightarrow> a\<guillemotright>"
rep_eq: fixes x :: "'b \<Rightarrow> 'a pdevs"   and xa :: "'b set" shows "pdevs_apply (sum_pdevs x xa) = (if finite xa then \<lambda>i. \<Sum>xa\<in>xa. pdevs_apply (x xa) i else (\<lambda>_. 0::'a))"
sinks_aux_less_cons: fixes I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and U :: "'b set"   and x :: "'c"   and xs :: "'c list" shows "sinks_aux_less I D U (x # xs) = (if \<exists>u\<in>U. (u, D x) \<in> I then insert (D x) (sinks_aux_less I D (insert (D x) U) xs) else sinks_aux_less I D U xs)"
neq_if_length_neq: fixes xs :: "'e list"   and ys :: "'e list" assumes "length xs \<noteq> length ys" shows "xs = ys \<equiv> False"
mbisim_thrD1: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x1"   and ln :: "'l \<Rightarrow>f nat" assumes "s1 \<approx>m s2"   and "thr s1 t = \<lfloor>(x, ln)\<rfloor>" shows "\<exists>x'. thr s2 t = \<lfloor>(x', ln)\<rfloor> \<and> t \<turnstile> (x, shr s1) \<approx> (x', shr s2) \<and> (wset s1 t = None \<or> True)"
euclid_ext2_def: fixes a :: "'a"   and b :: "'a" shows "euclid_ext2 a b = (fst (bezout_coefficients a b), snd (bezout_coefficients a b), - b div gcd a b, a div gcd a b, gcd a b)"
between_triv2: fixes a :: "'a"   and b :: "'a" shows "between (a, b) b"
simplex_retraction1: fixes x :: "'a set" assumes "x \<in> f \<turnstile> X" shows "fixespointwise f x"
mono_p: fixes n :: "nat"   and n' :: "nat"   and h :: "nat" assumes "n \<le> n'"   and "p n h" shows "p n' h"
has_as_binary_product_def: fixes a0 :: "'a"   and a1 :: "'a"   and p0 :: "'a"   and p1 :: "'a" shows "has_as_binary_product a0 a1 p0 p1 \<equiv> ide a0 \<and> ide a1 \<and> limit_cone (concrete_category.COMP UNIV (\<lambda>a b. if a = b then {()} else {}) (\<lambda>_ _ _ _ _. ())) (\<cdot>) (binary_product_diagram.map (\<cdot>) a0 a1) (local.dom p0) (binary_product_diagram.mkCone (\<cdot>) p0 p1)"
pinduct: fixes a0 :: "nat \<times> nat \<times> 'd"   and a1 :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set"   and a2 :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and a3 :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and P :: "nat \<times> nat \<times> 'd \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list \<Rightarrow> bool" assumes "gb_schema_dummy_dom (a0, a1, a2, a3)"   and "\<And>data D bs ps. \<lbrakk>gb_schema_dummy_dom (data, D, bs, ps); \<And>x xa xb xc xd xba y. \<lbrakk>ps \<noteq> []; x = sel gs bs ps (snd data); xa = ps -- x; xb = compl gs bs xa x (snd data); xc = fst data - count_const_lt_components (fst xb); xc \<noteq> 0; xd = add_indices xb (snd data); (xba, y) = xd\<rbrakk> \<Longrightarrow> P (xc, y) (D \<union> (set xba \<times> (set gs \<union> set bs \<union> set xba) \<union> set (ps -- x) -\<^sub>p set (ap gs bs xa xba y))) (ab gs bs xba y) (ap gs bs xa xba y)\<rbrakk> \<Longrightarrow> P data D bs ps" shows "P a0 a1 a2 a3"
continuous_closedin_preimage: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and T :: "'d set" assumes "continuous_on S f"   and "closed T" shows "closedin (top_of_set S) (S \<inter> f -` T)"
asubg_costOp_commute: fixes H :: "'a set"   and x :: "'a set"   and y :: "'a set" assumes "A +> H"   and "x \<in> set_rcs (b_ag A) H"   and "y \<in> set_rcs (b_ag A) H" shows "c_top (b_ag A) H x y = c_top (b_ag A) H y x"
abs_eq: fixes x :: "('e, 'f, 'g) Tree\<^sub>\<alpha>"   and xa :: "perm" assumes "eq_onp hereditarily_fs x x" shows "xa \<bullet> Abs_formula x = Abs_formula (xa \<bullet> x)"
set_diff_eq: fixes A :: "'c set"   and B :: "'c set" shows "A - B = {x \<in> A. x \<notin> B}"
k_steps: shows "(deterministic_RP_step ^^ k) St0 = (N', P', Q', n')"
simple_function_eq_measurable: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" shows "simple_function M f = (finite (f ` space M) \<and> f \<in> M \<rightarrow>\<^sub>M count_space UNIV)"
bij: fixes x :: "'a" shows "bij_betw (Pair x) Y (S x)"
subset_fst_imageI: fixes A :: "'d set"   and B :: "'e set"   and S :: "('d \<times> 'e) set"   and y :: "'e" assumes "A \<times> B \<subseteq> S"   and "y \<in> B" shows "A \<subseteq> fst ` S"
ideal_Inter: fixes S :: "'a set set" assumes "\<forall>I\<in>S. ideal I" shows "ideal (\<Inter> S)"
measure_space_def: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)" shows "measure_space M = (sigma_algebra (measurable_sets M) \<and> positive M \<and> countably_additive M)"
unit_mult_div_div: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one" shows "times b (divide one a) = divide b a"
interleaves_equal_all_nil: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "xs \<simeq> {ys, [], P}" shows "xs = ys"
isomorphic_transitive: fixes f :: "'d"   and g :: "'d"   and h :: "'d" assumes "D.isomorphic f g"   and "D.isomorphic g h" shows "D.isomorphic f h"
inverse_mult_eq_1': fixes f :: "'b fps" assumes "f $ 0 \<noteq> (0::'b)" shows "f * inverse f = 1"
BC_def: shows "BC \<equiv> {U. \<exists>\<psi> s. learn_bc \<psi> U s}"
WHILET_def: shows "dWHILET \<equiv> dWHILEIT (\<lambda>_. True)"
diameter_bounded_bound: fixes S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "bounded S"   and "x \<in> S"   and "y \<in> S" shows "dist x y \<le> diameter S"
renaming_distr_global_2: fixes \<alpha> :: "'a \<Rightarrow> 'a"   and \<beta> :: "'b \<Rightarrow> 'b"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a" assumes "bij \<alpha>" shows "\<R>\<^sub>G \<alpha> \<beta> (s(r := None)) = (\<R>\<^sub>G \<alpha> \<beta> s)(\<alpha> r := None)"
abs_eq: fixes x :: "('a \<times> 'b) set" shows "graph_ffun x = ffun.Abs_pfun (if finite (Domain x) then graph_pfun x else {}\<^sub>p)"
closedin_compact: fixes S :: "'a set"   and T :: "'a set" assumes "compact S"   and "closedin (top_of_set S) T" shows "compact T"
law_restrict_add: fixes R :: "('c, 'd) ring_scheme" shows "(\<oplus>\<^bsub>law_restrict R\<^esub>) = (\<lambda>a\<in>carrier R. restrict ((\<oplus>\<^bsub>R\<^esub>) a) (carrier R))"
in_charts_in_atlas: fixes x :: "('a, 'b) chart" assumes "x \<in> charts" shows "x \<in> atlas"
cring_lcf_scalar_mult: fixes p :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "p \<in> carrier P"   and "a \<in> carrier R"   and "a \<otimes> p (deg R p) \<noteq> \<zero>" shows "(a \<odot>\<^bsub>P\<^esub> p) (deg R (a \<odot>\<^bsub>P\<^esub> p)) = a \<otimes> p (deg R p)"
label_less_num_inner_nodes: fixes c :: "cmd"   and l :: "nat"   and c' :: "cmd" assumes "labels c l c'" shows "l < #:c"
blinfun_of_cblinfun_inv: fixes f :: "'a \<Rightarrow>\<^sub>L 'b" assumes "\<And>c x. f *\<^sub>v c *\<^sub>C x = c *\<^sub>C (f *\<^sub>v x)" shows "\<exists>g. blinfun_of_cblinfun g = f"
IH2: fixes p :: "net_tree" assumes "s2 \<in> init (pnet np p)"   and "i \<in> net_ips s2"   and "wf_net_tree p" shows "the (fst (netgmap sr s2) i) \<in> (fst \<circ> sr) ` init (np i)"
assms_3: shows "eval c t t' n (\<box>\<^sub>b\<gamma>)"
permutations_skip_one: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and x :: "'a" assumes "p permutes S"   and "x \<in> S" shows "p(x := x, inv p x := p x) permutes S - {x}"
simps_1: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list"   and y :: "'c"   and ys :: "'c list" shows "Term_Utils.map2 f (x # xs) (y # ys) = f x y # Term_Utils.map2 f xs ys"
g2_leadsto_a2: shows "\<turnstile> \<box>[(N1 \<or> N2) \<and> \<not> beta1]_(x, y, sem, pc1, pc2) \<and> SF(N2)_(x, y, sem, pc1, pc2) \<and> \<box>#True \<longrightarrow> (pc2 = #g \<leadsto> pc2 = #a)"
simple_function_def: fixes M :: "'d measure"   and g :: "'d \<Rightarrow> 'e" shows "simple_function M g = (finite (g ` space M) \<and> (\<forall>x\<in>g ` space M. g -` {x} \<inter> space M \<in> sets M))"
semiring_norm_52: shows "- (0::'a) = (0::'a)"
Ord_Cantor_sum: fixes \<alpha>s :: "V list"   and ms :: "nat list" assumes "list.set \<alpha>s \<subseteq> ON" shows "Ord (Cantor_sum \<alpha>s ms)"
\<alpha>ah_ileq: fixes w1 :: "('a set \<times> 'a set) list"   and w2 :: "('a set \<times> 'a set) list" assumes "w1 \<preceq> w2" shows "\<alpha>ah w1 \<le> \<alpha>ah w2"
dot_norm_neg: fixes x :: "'a"   and y :: "'a" shows "x \<bullet> y = ((norm x)\<^sup>2 + (norm y)\<^sup>2 - (norm (x - y))\<^sup>2) / 2"
Idl_subset_ideal': fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c" assumes "ring R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "(Idl\<^bsub>R\<^esub> {a} \<subseteq> Idl\<^bsub>R\<^esub> {b}) = (a \<in> Idl\<^bsub>R\<^esub> {b})"
s_TS_set: fixes i :: "nat"   and qs :: "nat list"   and init :: "nat list"   and h :: "nat list" assumes "i \<le> length qs" shows "set (s_TS init h qs i) = set init"
SigmaE2: fixes a :: "'e"   and b :: "'f"   and A :: "'e set"   and B :: "'e \<Rightarrow> 'f set" assumes "(a, b) \<in> Sigma A B" obtains "a \<in> A" and "b \<in> B a"
elims: fixes x :: "complex"   and xa :: "complex \<times> complex"   and y :: "complex \<times> complex" assumes "x *\<^sub>s\<^sub>v xa = y" obtains k :: "complex" and xb :: "complex" and ya :: "complex" where "x = k" and "xa = (xb, ya)" and "y = (k * xb, k * ya)"
rr: shows "r = sqrt ((4 - Re (- 4 * a / B * A)) / (1 + Re (a * cnj a)))"
invar_4: shows "nonneg_cexpr (set vs \<union> set vs') \<Gamma> \<delta>"
poly_cutoff_def: fixes n :: "nat"   and p :: "'b poly" shows "poly_cutoff n p = Abs_poly (\<lambda>k. if k < n then coeff p k else (0::'b))"
process_call_split_fst: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and a :: "'c rule"   and rs :: "'c rule list" shows "process_call \<Gamma> (a # rs) = process_call \<Gamma> [a] @ process_call \<Gamma> rs"
g: fixes y :: "'b" assumes "y \<in> T" shows "g y \<in> S \<and> f (g y) = y"
Compl_iff: fixes c :: "'c"   and A :: "'c set" shows "(c \<in> - A) = (c \<notin> A)"
countable_Un_Int_1: fixes I :: "'d set"   and N :: "'d \<Rightarrow> 'e set"   and M :: "'e measure" assumes "\<And>i. i \<in> I \<Longrightarrow> N i \<in> sets M" shows "\<Union> (N ` I) \<in> sets M"
Atm_ZObis: fixes atm :: "'atom" assumes "compatAtm atm" shows "Atm atm \<approx>01 Atm atm"
norm_notMT_manualQ: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and l :: "(int \<times> int \<times> protocol) set set set list" assumes "DenyAll \<in> set (policy2list p)" shows "normalize_manual_orderPrQ p l \<noteq> []"
select_convs_2: fixes worlds :: "'c set"   and relations :: "'b \<Rightarrow> ('c \<times> 'c) set"   and valuation :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and more :: "'e" shows "relations \<lparr>worlds = worlds, relations = relations, valuation = valuation, \<dots> = more\<rparr> = relations"
filter_min_append_subset: fixes xs :: "'a list"   and ys :: "'a list" shows "set (local.filter_min_append xs ys) \<subseteq> set xs \<union> set ys"
unless: fixes P :: "state \<Rightarrow> bool"   and Q :: "state \<Rightarrow> bool" shows "\<turnstile> \<box>($P \<longrightarrow> P$ \<or> Q$) \<longrightarrow> stable P \<or> \<diamond>Q"
hom_0_iff: fixes x :: "'b" shows "([:x:] = 0) = (x = (0::'b))"
cod_comp: fixes g :: "'a"   and f :: "'a" assumes "A.seq g f" shows "A.cod (g \<cdot>\<^sub>A f) = A.cod g"
smult_closed: fixes H :: "'b set"   and R :: "('c, 'd) ring_scheme"   and M :: "('c, 'b, 'e) module_scheme"   and a :: "'c"   and x :: "'b" assumes "Module.submodule H R M"   and "a \<in> carrier R"   and "x \<in> H" shows "a \<odot>\<^bsub>M\<^esub> x \<in> H"
indep_sets_mono_index: fixes J :: "'e set"   and I :: "'e set"   and F :: "'e \<Rightarrow> 'a set set" assumes "J \<subseteq> I"   and "indep_sets F I" shows "indep_sets F J"
assms: fixes i :: "'a" assumes "i \<in> Basis" shows "bounded ((\<lambda>x. x \<bullet> i) ` X)"
prj_emb_prj: fixes x :: "udom" assumes "DEFL('d) \<sqsubseteq> DEFL('e)" shows "PRJ('d)\<cdot>(EMB('e)\<cdot>(PRJ('e)\<cdot>x)) = PRJ('d)\<cdot>x"
extend_def: fixes forms :: "'b sequent"   and seq :: "'b sequent" shows "extend forms seq \<equiv>  antec forms + antec seq \<Rightarrow>* succ forms + succ seq"
find_lambda_def: fixes f :: "'b \<Rightarrow> 'b"   and x0 :: "'b" shows "fx0.find_lambda f x0 \<equiv> (\<lambda>s. s\<lparr>carrots := 1, l := 1, tortoise := x0, hare := f x0\<rparr>) ;; while (\<lambda>s. hare s \<noteq> tortoise s) ((\<lambda>s. if carrots s = l s then s\<lparr>tortoise := hare s, carrots := 2 * carrots s, l := 0\<rparr> else id s) ;; (\<lambda>s. s\<lparr>hare := f (hare s), l := l s + 1\<rparr>))"
1: fixes i :: "nat" assumes "i < length ys" shows "ys' ! i = ys ! f i"
CallUndefined: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and cs :: "('d, 'e, 'f) com list"   and css :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list"   and s :: "'d" assumes "\<Gamma> p = None" shows "\<Gamma>\<turnstile> (Call p # cs, css, Normal s) \<rightarrow> (cs, css, Stuck)"
args_apps: fixes s :: "('a, 'b) tm"   and ss :: "('a, 'b) tm list" shows "args (apps s ss) = args s @ ss"
f_Exec_Stream_Cons: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'e"   and xs :: "'e list"   and c :: "'d" shows "f_Exec_Comp_Stream trans_fun (x # xs) c = trans_fun x c # f_Exec_Comp_Stream trans_fun xs (trans_fun x c)"
bin_rsplit_len_le: fixes n :: "nat"   and ws :: "int list"   and nw :: "nat"   and w :: "int"   and m :: "nat" shows "n \<noteq> 0 \<longrightarrow> ws = bin_rsplit n (nw, w) \<longrightarrow> (length ws \<le> m) = (nw \<le> m * n)"
inject_1: fixes x1 :: "'d"   and y1 :: "'d" shows "(allow x1 = allow y1) = (x1 = y1)"
one_le_log_cancel_iff: fixes a :: "real"   and x :: "real" assumes "1 < a"   and "0 < x" shows "(1 \<le> log a x) = (a \<le> x)"
kop_galois_var: fixes f :: "'c \<Rightarrow> 'e \<Rightarrow> 'd set"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'e set" shows "(op\<^sub>K \<circ> f = g) = (op\<^sub>K \<circ> g = f)"
spmf_of_gpv_def: fixes gpv :: "('d, 'e, 'f) gpv" shows "spmf_to_gpv.spmf_of_gpv gpv = (THE p. gpv = lift_spmf p)"
Lim_transform_within_set: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and a :: "'b"   and S :: "'b set"   and T :: "'b set" assumes "(f \<longlongrightarrow> l) (at a within S)"   and "\<forall>\<^sub>F x in at a. (x \<in> S) = (x \<in> T)" shows "(f \<longlongrightarrow> l) (at a within T)"
Upper_cong: fixes L :: "('c, 'd) gorder_scheme"   and A :: "'c set"   and A' :: "'c set" assumes "weak_partial_order L"   and "A \<subseteq> carrier L"   and "A' \<subseteq> carrier L"   and "A {.=}\<^bsub>L\<^esub> A'" shows "Upper L A = Upper L A'"
Units_invertible: fixes A :: "'b mat"   and n :: "nat"   and b :: "'c" assumes "A \<in> Units (ring_mat TYPE('b) n b)" shows "invertible_mat A"
Cons_2: shows "set (a # p) \<subseteq> carrier R"
Inter_iff: fixes A :: "'b"   and C :: "'b set set" shows "(A \<in> \<Inter> C) = (\<forall>X\<in>C. A \<in> X)"
fps_deriv_add: fixes f :: "'a fps"   and g :: "'a fps" shows "fps_deriv (f + g) = fps_deriv f + fps_deriv g"
BinOp_\<tau>red1r_xt1: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e1 :: "'addr expr1"   and xs :: "'addr val list"   and e1' :: "'addr expr1"   and xs' :: "'addr val list"   and bop :: "bop"   and e2 :: "'addr expr1" assumes "\<tau>red1gr uf P t h (e1, xs) (e1', xs')" shows "\<tau>red1gr uf P t h (e1 \<guillemotleft>bop\<guillemotright> e2, xs) (e1' \<guillemotleft>bop\<guillemotright> e2, xs')"
insertion_sort_def: fixes a :: "'b array" shows "insertion_sort a = Array.len a \<bind> (\<lambda>l. for [1..<l] (\<lambda>j. (a.(j)) \<bind> (\<lambda>key. ref j \<bind> (\<lambda>i. while (!i \<bind> (\<lambda>i'. if 0 < i' then (a.(i' - 1)) \<bind> (\<lambda>x. return (key < x)) else return False)) (!i \<bind> (\<lambda>i'. (a.(i' - 1)) \<bind> (\<lambda>x. (a.(i') \<leftarrow> x) \<bind> (\<lambda>_. i := i' - 1)))) \<bind> (\<lambda>_. !i \<bind> (\<lambda>i'. a.(i') \<leftarrow> key))))))"
restrict_map_def: fixes m :: "'c \<Rightarrow> 'd option"   and A :: "'c set" shows "m |` A = (\<lambda>x. if x \<in> A then m x else None)"
weakStepSimDef_2: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi" shows "P \<leadsto><Rel> Q \<equiv> \<forall>Rs. Q \<longmapsto> Rs \<longrightarrow> weakStepSimAct P Rs P Rel"
segments_of_aform_closed: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "segments_of_aform X \<noteq> []" shows "fst (hd (segments_of_aform X)) = snd (last (segments_of_aform X))"
order_iff_strict: fixes b :: "'b word"   and a :: "'b word" shows "(b \<le>s a) = (b <s a \<or> a = b)"
add_diff_cancel_right: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and a :: "'g"   and c :: "'g"   and b :: "'g" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a c) (plus b c) = minus a b"
abs_scaleR: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and abs :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "real"   and b :: "'a" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less" shows "abs (scaleR a b) = scaleR \<bar>a\<bar> (abs b)"
vdomain_vlrestriction: fixes r :: "V"   and A :: "V" shows "\<D>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> A) = \<D>\<^sub>\<circ> r \<inter>\<^sub>\<circ> A"
b'_def: shows "b' \<equiv> snd (normalize_quot (a, b))"
simps: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" shows "gt_unary a1 a2 = (\<exists>t s. a1 = t \<and> a2 = s \<and> wt t = wt s \<and> \<not> head t \<le>\<ge>\<^sub>h\<^sub>d head s \<and> num_args t = 1 \<and> (\<exists>f\<in>local.ground_heads (head t). arity_sym f = 1 \<and> wt_sym f = 0) \<and> arg t \<ge>\<^sub>t s)"
normalize_idem: fixes R :: "('b, 'c) ring_scheme"   and p :: "'b list"   and q :: "'b list" assumes "ring R" shows "ring.normalize R (ring.normalize R p @ q) = ring.normalize R (p @ q)"
Body: fixes K :: "nat"   and X :: "msg"   and H :: "msg set" assumes "Crypt K X \<in> items H" shows "X \<in> items H"
one_strongly_connected_iff_4_eq: fixes x :: "'a" shows "one_strongly_connected x = (x\<^sup>T ; 1 ; x = x\<^sup>+)"
forward: shows "comm_monoid_mult_class.coprime p q \<longrightarrow> comm_monoid_mult_class.coprime (real_of_rat_poly p) (real_of_rat_poly q)"
fsubsetI: fixes A :: "'x fset"   and B :: "'x fset" assumes "\<And>x. x |\<in>| A \<Longrightarrow> x |\<in>| B" shows "A |\<subseteq>| B"
TS: shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) qs = length qs - 2"
EM_remainder_def: fixes n :: "nat"   and f :: "real \<Rightarrow> 'a"   and a :: "int" shows "EM_remainder n f a = (if EM_remainder_converges n f a then Lim at_top (\<lambda>x. EM_remainder' n f (real_of_int a) (real_of_int x)) else (0::'a))"
weakBisimScopeExtChainSym: fixes xvec :: "name list"   and \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi" assumes "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* Q" shows "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>P \<parallel> Q \<approx> (\<lparr>\<nu>*xvec\<rparr>P) \<parallel> Q"
in_sccsI: fixes c :: "('a, 'b) pre_digraph" assumes "induced_subgraph c (G \<restriction> {v. u \<rightarrow>\<^sup>* v})"   and "strongly_connected c"   and "\<nexists>c'. induced_subgraph c' (G \<restriction> {v. u \<rightarrow>\<^sup>* v}) \<and> strongly_connected c' \<and> verts c \<subset> verts c'" shows "c \<in> c.sccs"
ennreal_prod_eq_top: fixes f :: "'d \<Rightarrow> ennreal"   and I :: "'d set" shows "(prod f I = top) = (finite I \<and> (\<forall>i\<in>I. f i \<noteq> 0) \<and> (\<exists>i\<in>I. f i = top))"
buf: shows "wf_mbuf2' \<sigma> j n R \<phi>'' \<psi>'' buf"
space_T: fixes K :: "'c \<Rightarrow> 'c pmf"   and s :: "'c" shows "space (MC_syntax.T K s) = space (stream_space (count_space UNIV))"
phis'_def: fixes g :: "'g"   and x :: "'node \<times> 'val" shows "uninst.phis' (usesOf \<circ> uses) (\<lambda>g. Mapping.lookup (phis g)) g x \<equiv> case x of (n, v) \<Rightarrow> if v = uninst_code.chooseNext uses phis g then None else map_option (map (uninst.substNext (usesOf \<circ> uses) (\<lambda>g. Mapping.lookup (phis g)) g)) (Mapping.lookup (phis g) (n, v))"
steps_append2: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and x :: "'c"   and ys :: "'c list" assumes "Graph_Defs.steps E (xs @ [x])"   and "Graph_Defs.steps E (x # ys)" shows "Graph_Defs.steps E (xs @ x # ys)"
vconn_implies_hpar_3: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b"   and f :: "'b"   and g :: "'b" assumes "horizontal_homs V src trg"   and "partial_magma.in_hom V \<mu> f g" shows "src g = src f"
map_pmf_rep_eq: fixes f :: "'c \<Rightarrow> 'b"   and M :: "'c pmf" shows "measure_pmf (map_pmf f M) = distr (measure_pmf M) (count_space UNIV) f"