hcju/isabelleps huggingface dataset:
member_commute_fold_once: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'b"   and xs :: "'b list" assumes "rsp_fold f"   and "x \<in> set xs" shows "fold_once f xs = fold_once f (removeAll x xs) \<circ> f x"
Fun_mapsto: fixes f :: "'s" assumes "arr f" shows "Fun f \<in> extensional (local.set (local.dom f)) \<inter> (local.set (local.dom f) \<rightarrow> local.set (cod f))"
T3_complete: fixes v :: "Vertices"   and w :: "Vertices" assumes "v \<in> V\<^bsub>T3\<^esub>"   and "w \<in> V\<^bsub>T3\<^esub>"   and "v \<noteq> w" shows "v \<rightarrow>\<^bsub>T3\<^esub> w"
that: fixes b :: "V" assumes "r\<lparr>a\<rparr> = b"   and "s\<lparr>b\<rparr> = c" shows "thesis"
surj_Sup_pres_bij_on: fixes f :: "'c \<Rightarrow> 'b" assumes "top_pres ((`) f)"   and "Sup_pres f" shows "bij_betw f (range (radj f \<circ> f)) UNIV"
content_prod_eq_1_iff: fixes p :: "'b poly"   and q :: "'b poly" shows "(content (p * q) = (1::'b)) = (content p = (1::'b) \<and> content q = (1::'b))"
simps_3: fixes \<delta> :: "'c"   and x :: "'c"   and y :: "'c" shows "gt_arctic_delta \<delta> (Num_arc_delta x) (Num_arc_delta y) = delta_gt \<delta> x y"
wf_subset: fixes r :: "('b \<times> 'b) set"   and p :: "('b \<times> 'b) set" assumes "wf r"   and "p \<subseteq> r" shows "wf p"
dghm_ArrMap_is_arr'': fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and a :: "V"   and b :: "V"   and f :: "V"   and \<FF>f :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "f : a \<mapsto>\<^bsub>\<AA>\<^esub> b"   and "\<FF>f = \<FF>\<lparr>ArrMap\<rparr>\<lparr>f\<rparr>" shows "\<FF>f : \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> \<mapsto>\<^bsub>\<BB>\<^esub> \<FF>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>"
snd_rep_of_\<psi>\<^sub>2_2: fixes i :: "nat" assumes "i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))" shows "real_of_int ((- int (1 - f (i div 2)) + int (f (i div 2))) * 1) / sqrt 2 ^ (n + 1) = (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)"
distinct_2: fixes x21 :: "'a"   and x22 :: "('a, 'b) Term.term list"   and x1 :: "'b" shows "Fun x21 x22 \<noteq> Var x1"
eq: shows "(\<lambda>y. (if 0 < y \<and> y < 1 then 1 else 0) * ln y * y powr e) = (\<lambda>y. if 0 < y \<and> y < 1 then ln y * y powr e else 0)"
genPrefix_Cons: fixes xs :: "'a list"   and y :: "'a"   and ys :: "'a list"   and r :: "('a \<times> 'a) set" shows "((xs, y # ys) \<in> genPrefix r) = (xs = [] \<or> (\<exists>z zs. xs = z # zs \<and> (z, y) \<in> r \<and> (zs, ys) \<in> genPrefix r))"
Assign_tE: fixes x :: "char list"   and a :: "aexp"   and s :: "char list \<Rightarrow> int"   and p :: "nat"   and t :: "char list \<Rightarrow> int" assumes "(x ::= a, s) \<Rightarrow> p \<Down> t" obtains "p = Suc 0" and "t = s[a/x]"
Arr_implies_Ide_Cod: fixes t :: "'a local.term" assumes "Arr t" shows "Ide (Cod t)"
range_dom_1: fixes f :: "('f \<times> 'e) set" shows "f `` Domain f = Range f"
simps_4: fixes x21 :: "'a"   and x22 :: "'a form list"   and x1 :: "nat" shows "Compound x21 x22 \<noteq> At x1"
alpha_sym: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "X #= X'" shows "X' #= X"
le_zero_eq: fixes n :: "'b multiset" shows "(n \<subseteq># {#}) = (n = {#})"
row_rank_def: fixes A :: "((real, 'a) vec, 'b) vec" shows "rank A = dim (rows A)"
4: fixes n :: "nat" assumes "n \<noteq> 0"   and "x [^] n = \<one>" shows "\<exists>i. i \<noteq> 0 \<and> x [^] i = \<one>"
CollectI: fixes P :: "'d \<Rightarrow> bool"   and a :: "'d" assumes "P a" shows "a \<in> {x. P x}"
sem_eq2': shows "sterm_sem I (f1 fid1 vid1) (sol 0) = sterm_sem I (f1 fid1 vid1) (fst (mk_v I (OVar vid1) (sol 0, b) (sol 0)))"
reachable_mems_def: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem :: "'Var \<Rightarrow> 'Val" shows "reachable_mems cms mem \<equiv> {mem'. \<exists>sched cms'. (cms, mem) \<rightarrow>\<^bsub>sched\<^esub> (cms', mem')}"
bound_child_parent_neq_mems_state_set_neq_len: fixes s :: "nat set"   and k :: "nat" assumes "\<forall>x. x \<in> s \<longrightarrow> x < k" shows "finite s"
simps: fixes a1 :: "'d list"   and a2 :: "'d list" shows "pcancel1 a1 a2 = (\<exists>xss yss. a1 = concat xss \<and> a2 = concat yss \<and> list_all2 (\<lambda>xs ys. xs = ys \<or> xs = ys @ ys) xss yss)"
select_convs_1: fixes ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and pre :: "nat set"   and rreqid :: "nat"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and more :: "'a" shows "ip \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, pre = pre, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, \<dots> = more\<rparr> = ip"
c_cons_aux2: fixes u :: "nat"   and x :: "nat" assumes "0 < u" shows "c_cons x u = c_pair (c_len u) (c_pair x (c_snd (u - 1))) + 1"
integral_FTC_atLeastAtMost: fixes a :: "real"   and b :: "real"   and F :: "real \<Rightarrow> 'a"   and f :: "real \<Rightarrow> 'a" assumes "a \<le> b"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> (F has_vector_derivative f x) (at x within {a..b})"   and "continuous_on {a..b} f" shows "LBINT x. indicat_real {a..b} x *\<^sub>R f x = F b - F a"
A: shows "\<forall>xs\<in>T. \<forall>u\<in>range D \<inter> (- I) `` range D. \<forall>x\<in>D -` {u}. (ipurge_tr_rev I D u xs @ [x] \<in> T) = (xs @ [x] \<in> T)"
AgentY_Y_PPA: shows "\<lfloor>PPA (Agent Y)\<rfloor>\<^sub>Y"
split_beta': fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'g" shows "(\<lambda>(x, y). f x y) = (\<lambda>x. f (fst x) (snd x))"
cmp_simps'_3: fixes \<mu> :: "'c"   and \<nu> :: "'c" assumes "C.arr \<mu>"   and "C.arr \<nu>"   and "src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "trg\<^sub>D (\<Phi>\<^sub>F (\<mu>, \<nu>)) = F.map\<^sub>0 (trg\<^sub>C \<mu>)"
LeftDerivationFix_from_singleton_helper: fixes A :: "symbol"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and u :: "symbol list"   and B :: "symbol"   and v :: "symbol list" assumes "LeftDerivationFix [A] 0 D (length u) (u @ [B] @ v)" shows "D = []"
span_eq_iff: fixes s :: "'b set" shows "(local.span s = s) = local.subspace s"
lower_zero: fixes v :: "'t" shows "lower 0 v = 0"
a1: shows "<a>:M \<in> \<parallel><B>\<parallel>"
nb_less: fixes b' :: "contour"   and b :: "contour"   and c :: "nat" shows "(b' < nb b c) = (b' \<le> b)"
simps: fixes p :: "linear_poly"   and c :: "'a" shows "lec_of_nsc (LEQ_ns p c) = Le_Constraint Leq_Rel p c"
convert_obs_initial_def: fixes ta :: "('a, 'b, 'c, 'd, 'e, 'f) thread_action" shows "convert_obs_initial ta = (\<lbrace>ta\<rbrace>\<^bsub>l\<^esub>, \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>, \<lbrace>ta\<rbrace>\<^bsub>c\<^esub>, \<lbrace>ta\<rbrace>\<^bsub>w\<^esub>, \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>, map NormalAction \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)"
simps_1: fixes m :: "int"   and n :: "int" shows "Const m \<squnion> Const n = (if n = m then Const m else Any)"
assms: fixes x :: "'a" assumes "x \<in> A" shows "f x \<in> {0..1}"
IH: assumes "x \<in> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta>)" shows "\<exists>y\<in>fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F. x \<in> fv (\<delta> y)"
test_comp_mult1: fixes p :: "'a" shows "! (\<iota> p) \<cdot> \<iota> p = (0::'b)"
assms: shows "\<forall>x\<in>set xs. dim_vec x = n"
sup: shows "{x} \<subseteq> R\<inverse> `` R `` {x}"
Can_norm_iff_Ide_Diagonalize: fixes t :: "'c local.term" assumes "Arr t" shows "Can \<^bold>\<parallel>t\<^bold>\<parallel> = Ide \<^bold>\<lfloor>t\<^bold>\<rfloor>"
solve_depressed_cubic_complex: fixes y :: "complex"   and e :: "complex"   and f :: "complex" shows "(y \<in> set (solve_depressed_cubic_complex e f)) = (y ^ 3 + e * y + f = 0)"
eventually_finite: fixes x :: "ereal"   and f :: "'a \<Rightarrow> ereal"   and F :: "'a filter" assumes "\<bar>x\<bar> \<noteq> \<infinity>"   and "(f \<longlongrightarrow> x) F" shows "\<forall>\<^sub>F x in F. \<bar>f x\<bar> \<noteq> \<infinity>"
ExprSound: fixes e :: "Expr"   and tp :: "TP" assumes "(e, tp) \<in> VS_expr" shows "SemExpr e tp"
complement_part: fixes x :: "i"   and y :: "i" assumes "\<not> O x y" shows "P x (\<midarrow> y)"
M5exist: fixes p :: "'a"   and q :: "'a" assumes "p \<parallel> q" shows "\<exists>r s t. r \<parallel> p \<and> p \<parallel> q \<and> q \<parallel> s \<and> r \<parallel> t \<and> t \<parallel> s"
pendulum_inv: fixes r :: "real"   and G :: "(real, 2) vec \<Rightarrow> bool" shows "\<lceil>\<lambda>s. r\<^sup>2 = (s $ 1)\<^sup>2 + (s $ 2)\<^sup>2\<rceil> \<le> wp x\<acute>= \<lambda>t s. fball (- s $ 1) s & G on \<lambda>s. {t. 0 \<le> t} UNIV @ 0 \<lceil>\<lambda>s. r\<^sup>2 = (s $ 1)\<^sup>2 + (s $ 2)\<^sup>2\<rceil>"
obtain_fresh': fixes x :: "'a" assumes "finite (supp x)" obtains a :: "'b" where "atom a \<sharp> x"
assms_2: shows "f ` outs_\<I> \<I> \<subseteq> outs_\<I> \<I>''"
eqvt: fixes x1 :: "lam"   and x2 :: "lam"   and pi :: "(name \<times> name) list" assumes " x1 \<longrightarrow>\<^sub>\<beta>\<^sup>* x2" shows " pi \<bullet> x1 \<longrightarrow>\<^sub>\<beta>\<^sup>* pi \<bullet> x2"
spec_ops_remove1: fixes xs :: "('c \<times> 'd) list"   and deps :: "'d \<Rightarrow> 'c set"   and x :: "'c \<times> 'd" assumes "spec_ops xs deps" shows "spec_ops (remove1 x xs) deps"
denote_measure_def: fixes dims :: "nat list"   and n :: "nat"   and M :: "nat \<Rightarrow> complex mat"   and DS :: "(complex mat \<Rightarrow> complex mat) list"   and \<rho> :: "complex mat" shows "state_sig.denote_measure dims n M DS \<rho> = matrix_sum (state_sig.d dims) (\<lambda>k. (DS ! k) (M k * \<rho> * adjoint (M k))) n"
inf_top_left: fixes x :: "'a" shows "1 \<cdot> x = x"
UnderS_subset_Under: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" shows "UnderS r A \<subseteq> Under r A"
map_poly_pCons_hom: fixes a :: "int"   and p :: "int poly" shows "of_int_poly (pCons a p) = pCons (of_int a) (of_int_poly p)"
vle_refl: fixes x :: "V" shows "x \<unlhd> x"
IH: shows "2 ^ length xs \<le> Suc (size (queue_to_multiset xs))"
op_image_fste_def: shows "op_image_fste = (`) fst"
assms: shows "gc' \<in> pid_gstep_succ gc"
sel: shows "select_correct_factor_int_poly (tighten_poly_bounds_binary p1 p2) bnd ((l1, r1, sgn (ipoly p1 r1)), l2, r2, sgn (ipoly p2 r2)) (poly_mult p1 p2) = z'"
assms_3: shows "src\<^sub>C f = trg\<^sub>C g"
path_decomp_1: fixes xs :: "'a list"   and ys :: "'a list" assumes "H.path (xs @ ys)" shows "H.path xs"
operative_axioms: shows "operative (\<^bold>*) \<^bold>1 g"
Right_Coset_is_orbit: fixes x :: "'a" assumes "x \<in> G" shows "H |\<cdot> x = orbit.Class x"
inorder_delete: fixes t :: "('c \<times> 'd) bro"   and h :: "nat"   and x :: "'c" assumes "t \<in> T h"   and "sorted1 (inorder t)" shows "inorder (delete x t) = del_list x (inorder t)"
coboundedI: fixes A :: "'a multiset set"   and a :: "'a multiset" assumes "finite A"   and "a \<in> A" shows "subset_mset.Inf_fin A \<subseteq># a"
order_divides_totient: fixes n :: "nat"   and a :: "nat" assumes "coprime n a" shows "ord n a dvd totient n"
A2a: fixes x :: "i" shows "cod (dom x) \<cong> dom x"
logical_relationI: fixes R :: "('a \<Rightarrow> ValD) set" assumes "\<And>fs xs. \<lbrakk>fs \<in> R; xs \<in> R\<rbrakk> \<Longrightarrow> (\<lambda>j. appF\<cdot>(fs j)\<cdot>(xs j)) \<in> R"   and "\<And>fs xs. \<lbrakk>fs \<in> R; xs \<in> R\<rbrakk> \<Longrightarrow> (\<lambda>j. strictify\<cdot>(appF\<cdot>(fs j))\<cdot>(xs j)) \<in> R"   and "\<And>fs. (\<And>xs. xs \<in> R \<Longrightarrow> (\<lambda>j. fs j\<cdot>(xs j)) \<in> R) \<Longrightarrow> (\<lambda>j. ValF\<cdot>(fs j)) \<in> R"   and "\<And>fs. (\<And>xs. xs \<in> R \<Longrightarrow> (\<lambda>j. strictify\<cdot>(fs j)\<cdot>(xs j)) \<in> R) \<Longrightarrow> (\<lambda>j. ValF\<cdot>(strictify\<cdot>(fs j))) \<in> R"   and "\<And>xs. xs \<in> R \<Longrightarrow> (\<lambda>j. fixD\<cdot>(xs j)) \<in> R"   and "\<And>cs ts es. \<lbrakk>cs \<in> R; ts \<in> R; es \<in> R\<rbrakk> \<Longrightarrow> (\<lambda>j. cond\<cdot>(cs j)\<cdot>(ts j)\<cdot>(es j)) \<in> R"   and "\<And>xs. xs \<in> R \<Longrightarrow> (\<lambda>j. succ\<cdot>(xs j)) \<in> R"   and "\<And>xs. xs \<in> R \<Longrightarrow> (\<lambda>j. pred\<cdot>(xs j)) \<in> R"   and "\<And>xs. xs \<in> R \<Longrightarrow> (\<lambda>j. isZero\<cdot>(xs j)) \<in> R" shows "logical_relation R"
unwrapForall: fixes j :: "nat"   and A :: "atom fm"   and L :: "real list" shows "eval (AllN (j + 1) A) L = eval (AllQ (AllN j A)) L"
hd_in: fixes xs :: "'x" assumes "fst (list_of_oalist xs) \<noteq> []" shows "local.hd xs \<in> set (fst (list_of_oalist xs))"
span_is_monotone: fixes S :: "rat vec set"   and T :: "rat vec set" assumes "S \<subseteq> T" shows "gs.span S \<subseteq> gs.span T"
Vitali_covering_theorem_balls: fixes S :: "'b set"   and K :: "'c set"   and a :: "'c \<Rightarrow> 'b"   and r :: "'c \<Rightarrow> real" assumes "\<And>x d. \<lbrakk>x \<in> S; 0 < d\<rbrakk> \<Longrightarrow> \<exists>i. i \<in> K \<and> x \<in> ball (a i) (r i) \<and> r i < d" obtains C :: "'c set" where "countable C" and "C \<subseteq> K" and "pairwise (\<lambda>i j. disjnt (ball (a i) (r i)) (ball (a j) (r j))) C" and "negligible (S - (\<Union>i\<in>C. ball (a i) (r i)))"
intros: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and \<R> :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and lc\<^sub>1 :: "('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)"   and lc\<^sub>2 :: "('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)" assumes "sifum_security_init dma \<C>_vars eval"   and "sifum_security_init.strong_low_bisim_mm dma \<C>_vars eval \<R>"   and "(lc\<^sub>1, lc\<^sub>2) \<in> \<R>" shows "(lc\<^sub>1, lc\<^sub>2) \<in> sifum_security_init.mm_equiv dma \<C>_vars eval"
construct_fun_from_rel_def: fixes R :: "('d \<times> 'c) set"   and x :: "'d" shows "construct_fun_from_rel R x = (THE y. (x, y) \<in> R)"
domIff: fixes a :: "'b"   and m :: "'b \<Rightarrow> 'c option" shows "(a \<in> dom m) = (m a \<noteq> None)"
CoCallAritySafe_axioms: shows "CoCallAritySafe ccExp Aexp ccHeap Aheap"
if_SE_D1: fixes P :: "'b \<Rightarrow> bool"   and \<sigma> :: "'b"   and B\<^sub>1 :: "'b \<Rightarrow> (bool \<times> 'b) option"   and B\<^sub>2 :: "'b \<Rightarrow> (bool \<times> 'b) option" assumes "P \<sigma>" shows "(\<sigma> \<Turnstile> if\<^sub>S\<^sub>E P B\<^sub>1 B\<^sub>2) = (\<sigma> \<Turnstile> B\<^sub>1)"
reaches_steps: fixes x :: "'a"   and y :: "'a" assumes "x \<rightarrow>* y" shows "\<exists>xs. hd xs = x \<and> last xs = y \<and> steps xs"
fin: fixes K :: "('a, 'b) ring_scheme"   and V :: "('a, 'c, 'd) module_scheme"   and A :: "'c set" assumes "vectorspace K V"   and "vectorspace.fin_dim K V"   and "\<not> module.lin_dep K V A"   and "A \<subseteq> carrier V" shows "finite A"
generate_valid_stateful_policy_IFSACS_stateful_policy_compliance: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph"   and edgesList :: "('b \<times> 'b) list"   and \<T> :: "'b stateful_policy" assumes "valid_reqs M"   and "wf_graph G"   and "all_security_requirements_fulfilled M G"   and "set edgesList = edges G"   and "\<T> = generate_valid_stateful_policy_IFSACS G M edgesList" shows "stateful_policy_compliance \<T> G M"
halfspace_eq_empty_gt: fixes b :: "real"   and a :: "'a" shows "({x. b < a \<bullet> x} = {}) = (a = (0::'a) \<and> 0 \<le> b)"
1: shows "set_integrable lborel (einterval (ereal a) \<infinity>) f"
eq_infI: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a stream"   and w\<^sub>2 :: "'a stream" assumes "traces ind"   and "traces.le_inf ind w\<^sub>1 w\<^sub>2"   and "traces.le_inf ind w\<^sub>2 w\<^sub>1" shows "traces.eq_inf ind w\<^sub>1 w\<^sub>2"
12: shows "\<forall>d. f d \<noteq> bot \<longrightarrow> f m + bot \<le> f d + bot"
isomorphic_symmetric: fixes f :: "'a"   and g :: "'a" assumes "f \<cong>\<^sub>C g" shows "g \<cong>\<^sub>C f"
fcompose_const: fixes a :: "'a"   and q :: "'a poly"   and r :: "'a poly" shows "fcompose [:a:] q r = [:a:]"
all_not_in_conv: fixes A :: "'a set" shows "(\<forall>x. \<not> x \<in> A) = (A = {})"
less_iff_add1_le_hmset: fixes m :: "hmultiset"   and n :: "hmultiset" shows "(m < n) = (m + 1 \<le> n)"
abs_square_le_1: fixes x :: "'a" shows "(x\<^sup>2 \<le> (1::'a)) = (\<bar>x\<bar> \<le> (1::'a))"
LLL_mu_abs_bound: fixes i :: "nat"   and j :: "nat" assumes "i < m"   and "j < i" shows "\<bar>fs.gs.\<mu> i j\<bar> \<le> rat_of_nat (N ^ (m - 1) * 2 ^ (m - 1) * m)"
preserves_reflects_arr: fixes f :: "('a S.E.term, 'a) S.arr" shows "S.arr (S.trg f) = S.arr f"
substitution: fixes E :: "char list \<Rightarrow> 'a"   and F :: "char list \<Rightarrow> 'a list \<Rightarrow> 'a"   and G :: "char list \<Rightarrow> 'a list \<Rightarrow> bool"   and l :: "fterm literal"   and \<sigma> :: "char list \<Rightarrow> fterm" shows "eval\<^sub>l E F G (l \<cdot>\<^sub>l \<sigma>) = eval\<^sub>l (evalsub E F \<sigma>) F G l"
tllist_ord_refl: fixes b :: "'f"   and xs :: "('g, 'f) tllist" shows "tllist_ord b xs xs"
simps_6: fixes n :: "nat"   and execs :: "'dom_t \<Rightarrow> 'action_t list list"   and s :: "'state_t" assumes "\<not> interrupt (Suc n)"   and "\<not> thread_empty (execs (current s))"   and "precondition (next_state s execs) (next_action s execs)" shows "run (Suc n) (Some s) execs = run n (Some (step (next_state s execs) (next_action s execs))) (next_execs s execs)"
remove_guard_add_update_def: fixes t :: "transition"   and inputX :: "nat"   and outputX :: "nat" shows "remove_guard_add_update t inputX outputX = \<lparr>Label = Label t, Arity = Arity t, Guards = filter (\<lambda>g. \<not> gexp_constrains g (V (I inputX))) (Guards t), Outputs = Outputs t, Updates = (outputX, V (I inputX)) # Updates t\<rparr>"
simps_2: fixes l :: "'b llist"   and i :: "nat" shows "l \<up> Suc i = (case l of LNil \<Rightarrow> LNil | x ## xs \<Rightarrow> xs \<up> i)"
id_func_functor: shows "Functor id_func AA : AA \<longrightarrow> AA"
dbl_simps_5: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and k :: "Num.num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl plus (numeral.numeral one plus k) = numeral.numeral one plus (num.Bit0 k)"
singlePairInInjectionsUniverse: fixes x :: "'c"   and X :: "'d" shows "{(x, X)} \<in> injectionsUniverse"
prod_in_span: fixes b :: "'a vec"   and S :: "'a vec set"   and a :: "'a" assumes "b \<in> carrier_vec n"   and "S \<subseteq> carrier_vec n"   and "a = (0::'a) \<or> b \<in> local.span S" shows "a \<cdot>\<^sub>v b \<in> local.span S"
arbitrary_union_of_Int_eq: fixes P :: "'b set \<Rightarrow> bool" shows "(\<forall>S T. (arbitrary union_of P) S \<and> (arbitrary union_of P) T \<longrightarrow> (arbitrary union_of P) (S \<inter> T)) = (\<forall>S T. P S \<and> P T \<longrightarrow> (arbitrary union_of P) (S \<inter> T))"
x_2: shows "(0::'a) < \<bar>x\<bar> + (1::'a)"
that: fixes B :: "nat \<Rightarrow> 'a set" assumes "inj B"   and "\<And>n. open (B n)"   and "\<And>S. open S \<Longrightarrow> \<exists>k. S = \<Union> {B n |n. n \<in> k}" shows "thesis"
homotopy_equivalence_relative_homology_group_isomorphisms: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and T :: "'c set"   and g :: "'c \<Rightarrow> 'b"   and p :: "int" assumes "continuous_map X Y f"   and "f ` S \<subseteq> T"   and "continuous_map Y X g"   and "g ` T \<subseteq> S"   and "homotopic_with (\<lambda>h. h ` S \<subseteq> S) X X (g \<circ> f) id"   and "homotopic_with (\<lambda>k. k ` T \<subseteq> T) Y Y (f \<circ> g) id" shows "group_isomorphisms (relative_homology_group p X S) (relative_homology_group p Y T) (hom_induced p X S Y T f) (hom_induced p Y T X S g)"
closed_inv: fixes f :: "int \<Rightarrow> int" assumes "f \<in> CofGroups.Ex1" shows "inv f \<in> CofGroups.Ex1"
induct: fixes x :: "['b]"   and P :: "['b] \<Rightarrow> bool" assumes "finite_list x"   and "P []"   and "\<And>x xs. \<lbrakk>finite_list xs; P xs\<rbrakk> \<Longrightarrow> P (x : xs)" shows "P x"
NewInitDoneRed: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and l :: "sfields" assumes "sh C = \<lfloor>(sfs, Done)\<rfloor>" shows "redp P (new C) (h, l, sh) False (new C) (h, l, sh) True"
cod_inv: fixes f :: "'b" assumes "D.iso f" shows "D.cod (D.inv f) = D.dom f"
case_swap: fixes f :: "'i \<Rightarrow> 'h \<Rightarrow> 'g"   and p :: "'i \<times> 'h" shows "(case prod.swap p of (y, x) \<Rightarrow> f x y) = (case p of (x, y) \<Rightarrow> f x y)"
PiE_eq_singleton: fixes I :: "'f set"   and S :: "'f \<Rightarrow> 'g set"   and f :: "'f \<Rightarrow> 'g" shows "(Pi\<^sub>E I S = {restrict f I}) = (\<forall>i\<in>I. S i = {f i})"
RPCReject_def: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rst :: "state \<Rightarrow> PrIds \<Rightarrow> rpcState"   and p :: "PrIds" shows "RPCReject send rcv rst p \<equiv> $(rst!p) = #rpcA \<and> \<not> IsLegalRcvArg<arg<$(send!p)>> \<and> Return send p #BadCall \<and> unchanged (rst!p, caller rcv!p)"
igFreshBinp_termMOD: shows "igFreshBinp termMOD = freshBinp"
Elem_Ext_ZF: fixes R :: "(ZF \<times> ZF) set"   and x :: "ZF"   and s :: "ZF" assumes "set_like R" shows "Elem x (Ext_ZF R s) = ((x, s) \<in> R)"
rw2: shows "monom (g (Suc b)) (Suc b) mod f = monom (g (Suc b)) (Suc b)"
naturality: fixes f :: "'c \<times> 'c" assumes "C.CC.arr f" shows "\<phi> (C.CC.cod f) \<cdot>\<^sub>D T\<^sub>DoFF.map f = FoT\<^sub>C.map f \<cdot>\<^sub>D \<phi> (C.CC.dom f)"
tendsto_neg_powr: fixes s :: "real"   and f :: "'b \<Rightarrow> real"   and F :: "'b filter" assumes "s < 0"   and "filterlim f at_top F" shows "((\<lambda>x. f x powr s) \<longlongrightarrow> 0) F"
\<Psi>_sel_4: shows "sink \<Psi> = \<langle>a\<rangle>"
sots1: fixes n :: "nat"   and k :: "nat" assumes "is_sum2sq_nat n" shows "prime (4 * k + 3) \<longrightarrow> even (multiplicity (4 * k + 3) n)"
measure_of_of_measure: fixes M :: "'b measure" shows "measure_of (space M) (sets M) (emeasure M) = M"
trg_src: fixes \<mu> :: "'b" assumes "D.arr \<mu>" shows "trg\<^sub>D (src\<^sub>D \<mu>) = src\<^sub>D \<mu>"
filter_inf: fixes F :: "'a set"   and G :: "'a set" assumes "ord_class.filter F"   and "ord_class.filter G" shows "ord_class.filter (F \<inter> G)"
extend_act_inverse: fixes act :: "('a \<times> 'a) set" shows "project_act h (extend_act h act) = act"
step_preserves_length_ts: fixes ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option"   and ts' :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m' :: "nat \<Rightarrow> nat"   and \<S>' :: "nat \<Rightarrow> bool option" assumes "sbh_computation.concurrent_step (ts, m, \<S>) (ts', m', \<S>')" shows "length ts' = length ts"
us_facts_4: shows "z \<notin> FvarsT t1"