hcju/isabelleps huggingface dataset:
IH: fixes m :: "('c, 'a) mapping" assumes "{a. case a of (t, X) \<Rightarrow> (t, X) \<in> set xs \<and> as \<in> Z X t} = {}" shows "Mapping.lookup (fold (\<lambda>a. case a of (t, X) \<Rightarrow> \<lambda>m. upd_set m (\<lambda>_. t) (Z X t)) xs m) as = Mapping.lookup m as"
continuous_map_into_fulltopology: fixes X :: "'c topology"   and X' :: "'d topology"   and T :: "'d set"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_map X (subtopology X' T) f" shows "continuous_map X X' f"
splitAt_fst: fixes xs :: "'a list"   and a :: "'a list"   and v :: "'a"   and b :: "'a list" assumes "distinct xs"   and "xs = a @ v # b" shows "fst (splitAt v xs) = a"
trace_diag: fixes c :: "'c" shows "Square_Matrix.trace (diag c) = of_nat CARD('d) * c"
rel1: shows "P \<turnstile> D \<prec>\<^sup>1 C''"
add_diff_cancel: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.group_add minus plus zero uminus" shows "minus (plus a b) b = a"
arr_Rel_components_1: fixes f :: "V"   and A :: "V"   and B :: "V" shows "[f, A, B]\<^sub>\<circ>\<lparr>ArrVal\<rparr> = f"
find_consistent_signs_def: fixes ps :: "rat poly list" shows "find_consistent_signs ps = (let (fs, data) = factorize_polys ps; sgas = find_sgas (map real_of_rat_poly fs) in Let (map (undo_factorize_polys data) sgas) (If (fs = []) [map (\<lambda>x. if poly x 0 < 0 then - 1 else if poly x 0 = 0 then 0 else 1) ps]))"
induct: fixes x :: "event list"   and P :: "event list \<Rightarrow> bool" assumes "x \<in> ns_public"   and "P []"   and "\<And>evsf X B. \<lbrakk>evsf \<in> ns_public; P evsf; X \<in> synth (analz (knows Spy evsf))\<rbrakk> \<Longrightarrow> P (Says Spy B X # evsf)"   and "\<And>evsr A B X. \<lbrakk>evsr \<in> ns_public; P evsr; Says A B X \<in> set evsr\<rbrakk> \<Longrightarrow> P (Gets B X # evsr)"   and "\<And>evs1 NA A B. \<lbrakk>evs1 \<in> ns_public; P evs1; Nonce NA \<notin> used evs1\<rbrakk> \<Longrightarrow> P (Says A B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) # evs1)"   and "\<And>evs2 NB B NA A. \<lbrakk>evs2 \<in> ns_public; P evs2; Nonce NB \<notin> used evs2; Gets B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs2\<rbrakk> \<Longrightarrow> P (Says B A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) # evs2)"   and "\<And>evs3 A B NA NB. \<lbrakk>evs3 \<in> ns_public; P evs3; Says A B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs3; Gets A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs3\<rbrakk> \<Longrightarrow> P (Says A B (Crypt (pubK B) (Nonce NB)) # evs3)" shows "P x"
trancl_subset_Sigma: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "r \<subseteq> A \<times> A" shows "r\<^sup>+ \<subseteq> A \<times> A"
assms: shows "v \<Turnstile>\<^sub>n\<^sub>s ns"
intro: fixes M :: "'b measure" assumes "\<And>A B. \<lbrakk>B \<subseteq> A; A \<in> null_sets M\<rbrakk> \<Longrightarrow> B \<in> sets M" shows "complete_measure M"
hcomp_assoc_isomorphic: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "B.ide f"   and "B.ide g"   and "B.ide h"   and "src f = trg g"   and "src g = trg h" shows "(f \<star> g) \<star> h \<cong>\<^sub>B f \<star> g \<star> h"
as2: fixes x :: "'a"   and y :: "'a" shows "ad (x \<cdot> y) + ad (x \<cdot> ad (ad y)) = ad (x \<cdot> ad (ad y))"
pmaxsimpchainD_maxsimp: fixes xs :: "'a set list"   and x :: "'a set" assumes "pgallery xs"   and "x \<in> set xs" shows "chamber x"
f0v: fixes x :: "real" assumes "x \<in> {0..<v}" shows "f x \<noteq> f 1"
when_simps_1: fixes zero :: "'c"   and a :: "'c" shows "zero.when zero a True = a"
4: shows "enat i < llength (lselect (s - {least s}) w)"
induct: fixes x :: "'a \<Rightarrow> ('b, 'a) Term.term"   and P :: "('a \<Rightarrow> ('b, 'a) Term.term) \<Rightarrow> bool" assumes "x \<in> wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t_set"   and "P Var"   and "\<And>\<theta> v t. \<lbrakk>\<theta> \<in> wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t_set; P \<theta>; v \<notin> subst_domain \<theta>; v \<notin> range_vars \<theta>; fv t \<inter> insert v (subst_domain \<theta>) = {}\<rbrakk> \<Longrightarrow> P (\<theta>(v := t))" shows "P x"
M5exist_var: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a"   and w :: "'a" assumes "class.interval meets \<I>"   and "meets x y"   and "meets y z"   and "meets z w" shows "\<exists>t. meets x t \<and> meets t w"
absorb2: fixes b :: "'d"   and a :: "'d" assumes "b \<le> a" shows "min a b = b"
normalize_summation_less: fixes B :: "real"   and f :: "nat \<Rightarrow> real"   and n :: "nat" assumes "B \<noteq> 0" shows "((\<Sum>i = 0..<n + 1. f i * B ^ (n - i)) * B ^ (n mod 2) < 0) = ((\<Sum>i = 0..<n + 1. f i / B ^ i) < 0)"
intros_1: fixes f :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "f a \<noteq> b" shows "f a \<in> segment f a b"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "semiring_isom hom" shows "comm_semiring_isom hom"
new_last_neq_v1: shows "new_last \<noteq> v1"
int_def_5: shows "\<S>\<bullet>\<C>_int \<turnstile> lfilledk : [] _> ts_c"
chine_assoc_props_3: shows "\<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>1\<^sub>0 \<cdot> \<mu>_\<nu>\<pi>_\<rho>.chine_assoc = \<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>0\<^sub>1"
simps_105: fixes x51 :: "'d set"   and x52 :: "('d, 'e, 'f) com"   and x53 :: "('d, 'e, 'f) com" shows "Throw \<noteq> Cond x51 x52 x53"
subset_insertI2: fixes A :: "'b set"   and B :: "'b set"   and b :: "'b" assumes "order.greater_eq B A" shows "order.greater_eq (insert b B) A"
size_4: fixes x21 :: "dag"   and x22 :: "ref"   and x23 :: "dag" shows "size (Node x21 x22 x23) = size x21 + size x23 + Suc 0"
has_laplace_def: fixes f :: "real \<Rightarrow> complex"   and L :: "complex"   and s :: "complex" shows "(f has_laplace L) s = ((\<lambda>t. exp (t *\<^sub>R - s) * f t) has_integral L) {0..}"
reds: shows "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>"
of_hypnat_eq_iff: fixes m :: "nat star"   and n :: "nat star" shows "(of_hypnat m = of_hypnat n) = (m = n)"
set_less_aux_singleton_iff: fixes A :: "'a set"   and x :: "'a" shows "(A \<sqsubset>' {x}) = (finite A \<and> (\<forall>a\<in>A. x < a))"
inner_prod_def: fixes u :: "complex vec"   and v :: "complex vec" shows "\<langle>u|v\<rangle> \<equiv> \<Sum>i = 0..<dim_vec v. cnj (u $ i) * v $ i"
pCons_cases: fixes p :: "'b poly" obtains a :: "'b" and q :: "'b poly" where "p = pCons a q"
delta_gt_def: fixes \<delta> :: "'c" shows "delta_gt \<delta> \<equiv> \<lambda>x y. \<delta> \<le> x - y"
elims_3: fixes x :: "'a tree" assumes "\<not> wbalanced x" obtains l :: "'a tree" and xa :: "'a" and r :: "'a tree" where "x = \<langle>l, xa, r\<rangle>" and "\<not> (\<bar>int (size l) - int (size r)\<bar> \<le> 1 \<and> wbalanced l \<and> wbalanced r)"
contid_maximal: fixes S :: "'a set"   and I :: "'a set"   and T :: "'a set set"   and mx :: "'a set" assumes "S = {1\<^sub>r}"   and "\<zero> \<notin> S"   and "ideal R I"   and "I \<inter> S = {}"   and "T = {J. ideal R J \<and> S \<inter> J = {} \<and> I \<subseteq> J}"   and "maximal_set T mx" shows "maximal_ideal R mx"
\<alpha>n_simps_2: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and e :: "('c, 'f) label"   and w :: "('c, 'f) label list" shows "\<alpha>n fg (e # w) = (mon_e fg e, mon_w fg w)"
h_diff_OUT_le_IN: fixes x :: "'v"   and i :: "nat" assumes "x \<noteq> source \<Delta>" shows "d_OUT (h_diff i) x \<le> d_IN (h_diff i) x"
as_1: shows "bounded_linear f' \<and> ((\<lambda>y. (f y - f (netlimit (at x within S)) - f' (y - netlimit (at x within S))) /\<^sub>R norm (y - netlimit (at x within S))) \<longlongrightarrow> (0::'b)) (at x within S)"
trimmed_pos_def: fixes F :: "'a" shows "trimmed_pos F = (trimmed F \<and> 0 < fst (dominant_term F))"
rel_ad_def: fixes R :: "('b \<times> 'b) set" shows "rel_ad R = {(x, x) |x. \<nexists>y. (x, y) \<in> R}"
simps_1: fixes carrier :: "'a set"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and zero :: "'a"   and add :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and more :: "'b" shows "\<zero>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub> = zero"
eval_fps_at_0: fixes f :: "'a fps" shows "eval_fps f (0::'a) = fps_nth f 0"
collapse_3: fixes pr_operation' :: "pr_operation'" assumes "is_SAT_PUSH' pr_operation'" shows "SAT_PUSH' (un_SAT_PUSH' pr_operation') = pr_operation'"
uminus2: fixes a :: "real \<times> real"   and b :: "real \<times> real" shows "ccw' 0 a (- b) = ccw' 0 b a"
st_vec_inject: fixes x :: "'a st_vec"   and y :: "'a st_vec" shows "(st_vec x = st_vec y) = (x = y)"
fresh_4: fixes a :: "W.tvar"   and x2 :: "trm"   and x1 :: "trm" shows "a \<sharp> App x2 x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
unl_con: fixes c :: "'a"   and w :: "'b" shows "(#c) w \<equiv> c"
indets_monomial_subset: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and c :: "'c" shows "indets (monomial c t) \<subseteq> keys t"
hyps: shows "pderivs s (Times r1 r2) \<subseteq> Timess (pderivs s r1) r2 \<union> pderivs_lang (PSuf s) r2"
subseteq_mset_def: fixes A :: "'k multiset"   and B :: "'k multiset" shows "(A \<subseteq># B) = (\<forall>a. count A a \<le> count B a)"
hermitian_eigenvalue_real: fixes A :: "complex mat"   and n :: "nat"   and es :: "complex list"   and B :: "complex mat"   and P :: "complex mat"   and Q :: "complex mat" assumes "A \<in> carrier_mat n n"   and "hermitian A"   and "char_poly A = (\<Prod>e\<leftarrow>es. [:- e, 1:])"   and "unitary_schur_decomposition A es = (B, P, Q)" shows "similar_mat_wit A B P (adjoint P) \<and> diagonal_mat B \<and> diag_mat B = es \<and> unitary P \<and> (\<forall>i<n. B $$ (i, i) \<in> \<real>)"
IH: fixes r :: "'trm" assumes "r \<in> atrm"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> atrm"   and "distinct (map snd txs)"   and "\<forall>x\<in>snd ` set txs. \<forall>t\<in>fst ` set txs. x \<notin> FvarsT t" shows "FvarsT (rawpsubstT r txs) = FvarsT r - snd ` set txs \<union> \<Union> {if x \<in> FvarsT r then FvarsT t else {} |t x. (t, x) \<in> set txs}"
cos_lt_zero_pi: fixes x :: "real" assumes "pi / 2 < x"   and "x < 3 * pi / 2" shows "cos x < 0"
Max_wrt_among_preorder: fixes A :: "'a set" shows "Max_wrt_among le A = {x \<in> carrier \<inter> A. \<forall>y\<in>carrier \<inter> A. le x y \<longrightarrow> le y x}"
l_heap_is_wellformed_axioms: shows "l_heap_is_wellformed ShadowRootClass.type_wf ShadowRootClass.known_ptr heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.parent_child_rel Shadow_DOM.get_child_nodes get_disconnected_nodes"
sym_trancl: fixes r :: "('b \<times> 'b) set" assumes "sym r" shows "sym (r\<^sup>+)"
convex_powr: fixes p :: "real" assumes "1 \<le> p" shows "convex_on {0..} (\<lambda>x. x powr p)"
Cons_2: shows "length (v # m1) = nc \<and> (\<forall>a\<in>set (v # m1). vec nr a)"
norm_inverse_le_norm: fixes r :: "real"   and x :: "'b" assumes "r \<le> norm x"   and "0 < r" shows "norm (inverse x) \<le> inverse r"
preserves_dom: fixes f :: "'a" assumes "Adj.C.arr f" shows "Adj.C.dom (Adj.C.map f) = Adj.C.map (Adj.C.dom f)"
matrix_to_iarray_minorM: fixes A :: "(('c, 'd) vec, 'e) vec"   and i :: "'e"   and j :: "'d" shows "matrix_to_iarray (minorM A i j) = minorM_iarrays (matrix_to_iarray A) (mod_type_class.to_nat i) (mod_type_class.to_nat j)"
le_fun_def: fixes f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f" shows "(f \<le> g) = (\<forall>x. f x \<le> g x)"
length_stake: fixes n :: "nat"   and s :: "nat \<Rightarrow> 'b" shows "length (stake n s) = n"
cases: fixes A :: "'a set"   and a1 :: "('a, 'b) resource"   and a2 :: "('a, 'b) resource" assumes "A \<turnstile>\<^sub>R a1 \<sim> a2" obtains res :: "('a, 'b) resource" and res' :: "('a, 'b) resource" where "a1 = res" and "a2 = res'" and "\<And>a. a \<in> A \<Longrightarrow> rel_spmf (rel_prod (=) (eq_resource_on A)) (run_resource res a) (run_resource res' a)"
invKey_invKey_id: fixes K :: "key" shows "invKey (invKey K) = K"
binop_app_in_vrange: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>"   and "b \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>" shows "a +\<^sub>\<int> b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vint_plus"
validE: fixes x :: "name"   and T :: "ty"   and \<Gamma> :: "(name \<times> ty) list" assumes "valid ((x, T) # \<Gamma>)" obtains "valid \<Gamma>" and "x \<sharp> \<Gamma>"
closedin_subtopology: fixes U :: "'a topology"   and V :: "'a set"   and S :: "'a set" shows "closedin (subtopology U V) S = (\<exists>T. closedin U T \<and> S = T \<inter> V)"
higher_plus: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "higher (p + q) v = higher p v + higher q v"
i_last_message_hold_def: fixes f :: "nat \<Rightarrow> 'b message_af"   and k :: "nat" shows "f \<longmapsto> k \<equiv> \<lambda>n. last_message (f \<Up> (n - n mod k) \<Down> Suc (n mod k))"
norm_vec_def: fixes x :: "('c, 'd) Finite_Cartesian_Product.vec" shows "norm x = L2_set (\<lambda>i. norm (x $h i)) UNIV"
Un_Int_distrib2: fixes B :: "'a set"   and C :: "'a set"   and A :: "'a set" shows "B \<inter> C \<union> A = (B \<union> A) \<inter> (C \<union> A)"
map: shows "(y + q - x) mod q = (y + q - y') mod q"
preserves_ide: fixes a :: "'c" assumes "Cop.ide a" shows "S.ide (Cones.map a)"
g: fixes n :: "nat" assumes "primepow n" shows "g (aprimedivisor n) (multiplicity (aprimedivisor n) n) = f n"
sets_pair_measure: fixes A :: "'b measure"   and B :: "'c measure" shows "sets (A \<Otimes>\<^sub>M B) = sigma_sets (space A \<times> space B) {a \<times> b |a b. a \<in> sets A \<and> b \<in> sets B}"
h3: shows "insertion (nth_default 0 (L[var := x])) p = (\<Sum>i = 0..<d + 1. insertion (nth_default 0 (L[var := x])) (isolate_variable_sparse p var i * Var var ^ i))"
igvuin: shows "InvGlobVacantUptoImpliesNrec (shd s)"
wf_leadsto: fixes r :: "('a \<times> 'a) set"   and F :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and G :: "'b \<Rightarrow> bool"   and sigma :: "behavior"   and x :: "'a" assumes "wf r"   and "\<And>x. sigma \<Turnstile> F x \<leadsto> G \<or> (\<exists>y. #((y, x) \<in> r) \<and> F y)" shows "sigma \<Turnstile> F x \<leadsto> G"
Subring_zero_ring_zero: fixes R :: "('e, 'f) Ring_scheme"   and S :: "('e, 'g) Ring_scheme" assumes "Ring R"   and "Subring R S" shows "\<zero>\<^bsub>S\<^esub> = \<zero>\<^bsub>R\<^esub>"
exp_add: fixes x :: "'e"   and y :: "'e" shows "exp (x + y) = exp x * exp y"
happened_before_same: fixes i :: "nat"   and j :: "nat"   and k :: "'a" assumes "i < j"   and "j < length (events k)" shows "happened_immediately_before\<^sup>+\<^sup>+ (k, i) (k, j)"
finite_bot_fun: shows "bot = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n UNIV"
assms: shows "\<S>\<bullet>\<C> \<turnstile> [$e1, $e2] : t1s _> t2s"
inf_bot_right: fixes x :: "'b multiset" shows "x \<inter># {#} = {#}"
prod_z_x2: shows "poly (prod_list_var qs) x2 = 0"
adapt_stoch_proc_borel_measurable: fixes X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'a"   and N :: "'a measure" assumes "adapt_stoch_proc F X N" shows "\<forall>n. random_variable N (X n)"
bvars\<^sub>s\<^sub>t_singleton: fixes x :: "('d, 'c) strand_step" shows "bvars\<^sub>s\<^sub>t [x] = set (bvars\<^sub>s\<^sub>t\<^sub>p x)"
leadsTo_Un_Un: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set"   and B' :: "'c set" assumes "F \<in> A \<longmapsto> A'"   and "F \<in> B \<longmapsto> B'" shows "F \<in> A \<union> B \<longmapsto> A' \<union> B'"
assms_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<cdot> d y \<cdot> z = x \<cdot> (d y \<cdot> z)"
SUP_cong: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "\<Squnion> (C ` A) = \<Squnion> (D ` B)"
WTCond1: fixes P :: "(char list list \<times> char list exp) prog"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and e\<^sub>1 :: "char list exp"   and T\<^sub>1 :: "ty"   and e\<^sub>2 :: "char list exp"   and T\<^sub>2 :: "ty" assumes "P,E \<turnstile> e :: Boolean"   and "P,E \<turnstile> e\<^sub>1 :: T\<^sub>1"   and "P,E \<turnstile> e\<^sub>2 :: T\<^sub>2"   and "P \<turnstile> T\<^sub>1 \<le> T\<^sub>2"   and "P \<turnstile> T\<^sub>2 \<le> T\<^sub>1 \<longrightarrow> T\<^sub>2 = T\<^sub>1" shows "P,E \<turnstile> if (e) e\<^sub>1 else e\<^sub>2 :: T\<^sub>2"
assms: shows "WriteMem ad al v \<in> set start_heap_obs"
all_top: fixes P :: "indi \<Rightarrow> alt \<Rightarrow> real"   and b :: "alt" assumes "\<forall>i. P i <\<cdot> b" shows "F P <\<cdot> b"
IH: assumes "P" shows "\<Turnstile>\<^sub>3 {Q} c { R}"
nprv_eql_substE02: fixes t1 :: "'trm"   and t2 :: "'trm"   and F :: "'fmla set"   and \<phi> :: "'fmla"   and y :: "'var"   and \<psi> :: "'fmla" assumes "eql t1 t2 \<in> F"   and "nprv F (subst \<phi> t2 y)"   and "\<psi> = subst \<phi> t1 y \<or> \<psi> \<in> F"   and "F \<subseteq> fmla"   and "finite F"   and "\<psi> \<in> fmla"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "\<phi> \<in> fmla"   and "y \<in> var" shows "nprv F \<psi>"
length_split: fixes sensors :: "cars \<Rightarrow> traffic \<Rightarrow> cars \<Rightarrow> real" assumes "hmlsl sensors" shows "\<forall>ts v. 0 < \<parallel>ext v\<parallel> \<longrightarrow> (\<exists>va u. v=va\<parallel>u \<and> 0 < \<parallel>ext va\<parallel> \<and> 0 < \<parallel>ext u\<parallel>)"
maxi_ub: fixes x :: "'c"   and xs :: "'c list"   and key :: "'c \<Rightarrow> 'd" assumes "x \<in> set xs" shows "key x \<le> key (xs ! maxi xs key)"
runit'_closed: fixes f :: "'a" assumes "Arr f"   and "B.ide f" shows "Arr (B.inv (B.\<rr> f))"
stone: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.stone_algebra inf less_eq less sup bot top uminus" shows "sup (uminus x) (uminus (uminus x)) = top"
unifiers_def: fixes E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) set" shows "unifiers E = {\<sigma>. \<forall>p\<in>E. fst p \<cdot> \<sigma> = snd p \<cdot> \<sigma>}"
content_ff_to_fract: fixes p :: "'b fract poly" assumes "set (coeffs p) \<subseteq> range to_fract" shows "content_ff p \<in> range to_fract"
Seq_decomp: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and p\<^sub>1 :: "('e, 'd, 'f) com"   and p\<^sub>2 :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and p' :: "('e, 'd, 'f) com"   and s' :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (Seq p\<^sub>1 p\<^sub>2, s) \<rightarrow> (p', s')"   and "p\<^sub>1 \<noteq> Skip"   and "p\<^sub>1 \<noteq> Throw" shows "\<exists>p\<^sub>1'. \<Gamma>\<turnstile> (p\<^sub>1, s) \<rightarrow> (p\<^sub>1', s') \<and> p' = Seq p\<^sub>1' p\<^sub>2"
A_fdia: fixes x :: "'a"   and y :: "'a" shows "A x y =  |x\<rangle> y \<cdot> ad y"
\<i>_simps_3: fixes a :: "'a" assumes "B.obj a" shows "trg (UP \<i>\<^sub>B[a] \<cdot> cmp\<^sub>U\<^sub>P (a, a)) = UP.map\<^sub>0 a"
FIXvarFIX: fixes b :: "BExpr" shows "PhiWhile b = (\<lambda>\<Phi> (s, t, \<beta>). (b, \<Phi>, \<beta>, s, t) \<in> var)"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "group_add_hom hom" shows "ab_group_add_hom hom"
partition_on_insert_singleton: fixes A :: "'b set"   and P :: "'b set set"   and a :: "'b"   and A' :: "'b set" assumes "partition_on A P"   and "a \<notin> A"   and "insert a A = A'" shows "partition_on A' (insert {a} P)"
the_apartment_iso_apartment_chamber_map: fixes X :: "'a set set"   and Y :: "'a set set"   and Ca :: "'a set"   and C :: "'a set" assumes "X \<in> \<A>"   and "Y \<in> \<A>"   and "chamber Ca"   and "Ca \<in> X \<inter> Y"   and "SimplicialComplex.maxsimp X C" shows "SimplicialComplex.maxsimp Y (the_apartment_iso X Y ` C)"
arr_def: fixes f :: "'a \<times> 'a" shows "B.VV.arr f \<equiv> B.VV.domains f \<noteq> {} \<or> B.VV.codomains f \<noteq> {}"
rbt_less_simps_2: fixes k :: "'a"   and c :: "color"   and lt :: "('a, 'c) rbt"   and kt :: "'a"   and v :: "'c"   and rt :: "('a, 'c) rbt" shows "(Branch c lt kt v rt |\<guillemotleft> k) = (kt < k \<and> lt |\<guillemotleft> k \<and> rt |\<guillemotleft> k)"
new_document_ptr_new: fixes h :: "(_) heap" shows "document_ptr.Ref (Suc (fMax (finsert 0 (document_ptr.the_ref |`| document_ptrs h)))) |\<notin>| document_ptrs h"
uminus_Neg: fixes A :: "'a" shows "- Neg A = Pos A"
splits_at_append_suffix_prevails: fixes a :: "symbol list"   and b :: "symbol list"   and i :: "nat"   and u :: "symbol list"   and N :: "symbol"   and v :: "symbol list" assumes "splits_at (a @ b) i u N v"   and "i < length a" shows "\<exists>v'. v = v' @ b \<and> a = u @ [N] @ v'"
assms_3: shows "\<forall>p q. dfa_is_node M q \<and> p < q \<longrightarrow> tr_lookup T q p = (\<exists>n<m. dist_nodes M n v p q)"
simps_2: fixes sys :: "'b \<Rightarrow> ('a, 'b) process_term"   and p :: "'a process" shows "CH1 sys (PROC p) = PROC (ch1Of p)"
seteq_qle_def: fixes M :: "'a set"   and N :: "'a set" shows "M =\<^sub>\<preceq> N \<equiv> M \<subseteq>\<^sub>\<preceq> N \<and> N \<subseteq>\<^sub>\<preceq> M"
basis_to_basis_subspace_isomorphism: fixes S :: "'a set"   and T :: "'b set"   and B :: "'a set"   and C :: "'b set" assumes "subspace S"   and "subspace T"   and "dim S = dim T"   and "B \<subseteq> S"   and "independent B"   and "S \<subseteq> span B"   and "card B = dim S"   and "C \<subseteq> T"   and "independent C"   and "T \<subseteq> span C"   and "card C = dim T" shows "\<exists>f. linear f \<and> f ` B = C \<and> f ` S = T \<and> inj_on f S"
neg_involutive: fixes x :: "'a" assumes "test x" shows "! (! x) = x"
indspaceD_add: fixes f :: "('f, 'g) aezfun \<Rightarrow> 'v"   and G :: "'g set"   and FH :: "('f, 'g) aezfun set"   and V :: "'v set"   and x :: "('f, 'g) aezfun"   and y :: "('f, 'g) aezfun" assumes "f \<in> BaseFGMod.indspace G FH V"   and "x \<in> FH"   and "y \<in> FH" shows "f (x + y) = f x + f y"
l2_inv2I: fixes x :: "l2_state" assumes "\<And>A B KE. \<lbrakk>Auth A B \<langle>Number 0, KE\<rangle> \<in> chan x; A \<notin> bad x \<and> B \<notin> bad x\<rbrakk> \<Longrightarrow> \<exists>Ra. guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> in_progress (progress x Ra) xpkE \<and> KE = epubKF (Ra $ kE)" shows "x \<in> l2_inv2"
vmc_path_lnull_ltl_no_deadend: assumes "\<not> lnull (ltl P)" shows "vmc_path_no_deadend G P v p \<sigma>"
U_bounded_2: fixes x :: "'s"   and y :: "'s" shows "U x y \<le> 1"
path_connected_continuous_image: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_on S f"   and "path_connected S" shows "path_connected (f ` S)"
h2: shows "satisfy_equation_R p (qs1 @ qs2) (subsets_smash_R (length qs1) subsets1 subsets2) (signs_smash signs1 signs2)"
Collect_disj_eq: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" shows "{x. P x \<or> Q x} = {x. P x} \<union> {x. Q x}"
setdist_empty2: fixes t :: "'b set" shows "setdist t {} = 0"
CH_def: fixes X :: "'a set" shows "CH X = (\<Union>h. Ch h X)"
bij_phi: shows "bij_betw ((*) (1 / pi) \<circ> (\<lambda>x. if - 1 \<le> x \<and> x \<le> 1 then arccos x else if x < - 1 then - x + pi else 1 - x) \<circ> (\<lambda>t. 2 * x_coord (1 - t))) {0..1} {0..1}"
leq_pred_eq: shows "local.leq_pred = (\<le>)"
unit_char_1: fixes a :: "'a" assumes "B.obj a" shows "\<guillemotleft>UP.unit a : UP.map\<^sub>0 a \<Rightarrow> UP a\<guillemotright>"
zero_divisor_mult_left: fixes a :: "'b"   and b :: "'b" assumes "zero_divisor a" shows "zero_divisor (a * b)"
lo_rgt: fixes r :: "nat" shows "lo (trpl 0 0 r) (Pi 2) = r"
seq_compact_closed_subset: fixes s :: "'c set"   and t :: "'c set" assumes "closed s"   and "s \<subseteq> t"   and "seq_compact t" shows "seq_compact s"
lens_override_overshadow_left: fixes X :: "'a \<Longrightarrow> 'b"   and S\<^sub>1 :: "'b"   and S\<^sub>2 :: "'b"   and S\<^sub>3 :: "'b" assumes "mwb_lens X" shows "S\<^sub>1 \<triangleleft>\<^bsub>X\<^esub> S\<^sub>2 \<triangleleft>\<^bsub>X\<^esub> S\<^sub>3 = S\<^sub>1 \<triangleleft>\<^bsub>X\<^esub> S\<^sub>3"