hcju/isabelleps huggingface dataset:
lexordp_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" shows "List.lexordp r xs ys = ((xs, ys) \<in> lexord {(x, y). r x y})"
leibniz_rule: fixes U :: "'b set"   and a :: "'c"   and b :: "'c"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and fx :: "'b \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow>\<^sub>L 'd"   and x0 :: "'b" assumes "\<And>x t. \<lbrakk>x \<in> U; t \<in> cbox a b\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x t) has_derivative blinfun_apply (fx x t)) (at x within U)"   and "\<And>x. x \<in> U \<Longrightarrow> f x integrable_on cbox a b"   and "continuous_on (U \<times> cbox a b) (\<lambda>(x, t). fx x t)"   and "x0 \<in> U"   and "convex U" shows "((\<lambda>x. integral (cbox a b) (f x)) has_derivative blinfun_apply (integral (cbox a b) (fx x0))) (at x0 within U)"
T2: fixes w :: "'a"   and w' :: "'a"   and \<theta> :: "'a"   and u :: "'a"   and \<theta>' :: "'a"   and \<beta> :: "'a" assumes "C.ide w"   and "C.ide w'"   and "\<guillemotleft>\<theta> : f \<star>\<^sub>C w \<Rightarrow>\<^sub>C u\<guillemotright>"   and "\<guillemotleft>\<theta>' : f \<star>\<^sub>C w' \<Rightarrow>\<^sub>C u\<guillemotright>"   and "\<guillemotleft>\<beta> : g \<star>\<^sub>C w \<Rightarrow>\<^sub>C g \<star>\<^sub>C w'\<guillemotright>"   and "(r \<star>\<^sub>C \<theta>) \<cdot>\<^sub>C \<a>\<^sub>C[r, f, w] \<cdot>\<^sub>C (\<rho> \<star>\<^sub>C w) = ((r \<star>\<^sub>C \<theta>') \<cdot>\<^sub>C \<a>\<^sub>C[r, f, w'] \<cdot>\<^sub>C (\<rho> \<star>\<^sub>C w')) \<cdot>\<^sub>C \<beta>" shows "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w \<Rightarrow>\<^sub>C w'\<guillemotright> \<and> \<beta> = g \<star>\<^sub>C \<gamma> \<and> \<theta> = \<theta>' \<cdot>\<^sub>C (f \<star>\<^sub>C \<gamma>)"
attracting_fixed_point_def: fixes f :: "'b \<Rightarrow> 'b" shows "attracting_fixed_point f = (SOME xi. xi \<in> Gromov_boundary \<and> Gromov_extension f xi = xi \<and> additive_strength f xi < 0)"
eqvt_bound: fixes p :: "perm"   and x :: "'b" shows "p \<bullet> unpermute p x \<equiv> x"
exhaust: fixes y :: "n" obtains "y = Z"   | x2 :: "n" where "y = S x2"
jpTraces_step_inv: fixes t :: "('a, 'es, 'as) BEState Trace"   and s :: "('a, 'es, 'as) BEState"   and jp :: "'a \<Rightarrow> ('cobs \<times> 'as option, 'aAct, 'b) Protocol" assumes "t \<leadsto> s \<in> SPR.jpTraces jp" shows "t \<in> SPR.jpTraces jp \<and> (\<exists>eact\<in>set (envAction (tLast t)). \<exists>aact. (\<forall>a. aact a \<in> set (SPR.actJP jp t a)) \<and> s = envTrans eact aact (tLast t))"
g_def: shows "g = (\<lambda>x. ((- 1) ^ Suc n * bernoulli (Suc n) / fact (Suc n)) *\<^sub>R (f x - f (real_of_int a)))"
map_iterator_dom_foldli_conv: fixes kvs :: "('b \<times> 'c) list" shows "map_iterator_dom (foldli kvs) = foldli (map fst kvs)"
split_asm: fixes P :: "'f \<Rightarrow> bool"   and f1 :: "'g option \<Rightarrow> 'f"   and f2 :: "'h \<Rightarrow> ('i \<Rightarrow> ('g, 'h, 'i) Resumption.resumption) \<Rightarrow> 'f"   and resumption :: "('g, 'h, 'i) Resumption.resumption" shows "P (case resumption of Resumption.resumption.Done x \<Rightarrow> f1 x | Resumption.resumption.Pause x xa \<Rightarrow> f2 x xa) = (\<not> ((\<exists>x1. resumption = Resumption.resumption.Done x1 \<and> \<not> P (f1 x1)) \<or> (\<exists>x21 x22. resumption = Resumption.resumption.Pause x21 x22 \<and> \<not> P (f2 x21 x22))))"
set_pmf_of_list: fixes xs :: "('b \<times> real) list" assumes "pmf_of_list_wf xs" shows "set_pmf (pmf_of_list xs) \<subseteq> set (map fst xs)"
assms: shows "a \<notin>\<^sub>\<circ> \<D>\<^sub>\<circ> r"
supi: fixes x__ :: "'a"   and x'__ :: "'b"   and x :: "'a"   and x' :: "'b" assumes "(x__, x'__) \<in> A"   and "(x, x') \<in> A" shows "(supi x__ x, sup x'__ x') \<in> A"
homh: shows "h \<in> hom (free_Abelian_group S) (free_Abelian_group T)"
res_diff_zero_fact': fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and k :: "nat" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "a k = b k" shows "a k \<ominus>\<^bsub>residue_ring (p ^ k)\<^esub> b k = 0"
partition_def: fixes A :: "'b set"   and B :: "'b set set" shows "Congruence.partition A B \<equiv> (\<forall>a. a \<in> A \<longrightarrow> (\<exists>!b. b \<in> B \<and> a \<in> b)) \<and> (\<forall>b. b \<in> B \<longrightarrow> b \<subseteq> A)"
assms_2: shows "functor (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>[\<^sub>A\<^sub>2,\<^sub>B\<^sub>]) G"
set_vwalk_arcs_append1: fixes p :: "'a list"   and q :: "'a list" shows "set (vwalk_arcs p) \<subseteq> set (vwalk_arcs (p @ q))"
exhaust: fixes y :: "'f option" obtains "y = None"   | x2 :: "'f" where "y = Some x2"
simps_1: fixes M :: "('fun, 'var) Term.term set" shows "\<lbrakk>M; []\<rbrakk>\<^sub>d' = (\<lambda>\<I>. True)"
OP_def: fixes x :: "'a" shows "Autoref_Tagging.OP x \<equiv> x"
rSim: fixes R :: "pi"   and S :: "pi" assumes "(R, S) \<in> X" shows "R \<leadsto>[(X \<union> bisim)] S"
midpoint_sym: fixes a :: "'a"   and b :: "'a" shows "midpoint a b = midpoint b a"
nth_default_append_trailing: fixes dflt :: "'b"   and xs :: "'b list"   and n :: "nat" shows "nth_default dflt (xs @ replicate n dflt) = nth_default dflt xs"
\<section>: shows "list.set (concat bs1) \<lless> list.set (concat bs2)"
that: fixes \<F>' :: "'a set set" assumes "\<F>' \<subseteq> \<F>"   and "countable \<F>'"   and "\<Union> \<F>' = \<Union> \<F>" shows "thesis"
process_prover_5: fixes P :: "('b event list \<times> 'b event set) set \<times> 'b event list set" shows "DIVERGENCES P = snd P"
class_cons: fixes C :: "char list"   and x :: "'b cdecl"   and P :: "'b prog" assumes "C \<noteq> fst x" shows "class (x # P) C = class P C"
retractionE: fixes g :: "'a" assumes "C.retraction g" obtains f :: "'a" where "C.ide (g \<cdot>\<^sub>C f)"
composition_closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> R"   and "b \<in> R" shows "a \<cdot> b \<in> R"
scf: fixes w :: "'a \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and pr_strict :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and pr_weak :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and least :: "'a \<Rightarrow> bool"   and scf :: "'a \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and i :: "nat"   and n :: "nat"   and f :: "'a" assumes "admissible_kbo w w0 pr_strict pr_weak least scf"   and "i < n" shows "0 < scf (f, n) i"
hom_simps_4: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b formula" shows "hom f (- x) = - hom f x"
inf_fun_def: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)"
get_env_bigger: fixes x :: "char list"   and e :: "'b environment"   and a :: "char list"   and X :: "'b" assumes "x \<in> env_dom (e\<lparr>a:X\<rparr>)"   and "x \<noteq> a" shows "e!x = e\<lparr>a:X\<rparr>!x"
fps_X_power_mult_nth: fixes k :: "nat"   and f :: "'a fps"   and n :: "nat" shows "fps_nth (fps_X ^ k * f) n = (if n < k then 0::'a else fps_nth f (n - k))"
parallel_converter2_map2_out: fixes conv1 :: "('h, 'j, 'l, 'n) converter"   and f :: "'i \<Rightarrow> 'p"   and g :: "'q \<Rightarrow> 'k"   and h :: "'r \<Rightarrow> 'm"   and k :: "'o \<Rightarrow> 's"   and conv2 :: "('p, 'q, 'r, 's) converter" shows "conv1 |\<^sub>= map_converter f g h k conv2 = map_converter (map_sum id f) (map_sum id g) (map_sum id h) (map_sum id k) (conv1 |\<^sub>= conv2)"
Card_order_wo_rel: fixes r :: "('a \<times> 'a) set" assumes "Card_order r" shows "wo_rel r"
Fourier_series_l2_summable: fixes S :: "real set"   and w :: "nat \<Rightarrow> real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and I :: "nat set" assumes "orthonormal_system S w"   and "\<And>i. w i square_integrable S"   and "f square_integrable S" obtains g :: "real \<Rightarrow> real" where "g square_integrable S" and "(\<lambda>n. l2norm S (\<lambda>x. (\<Sum>i\<in>I \<inter> {..n}. orthonormal_coeff S w f i * w i x) - g x)) \<longlonglongrightarrow> 0"
pr_comp3_1': fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat"   and k :: "nat \<Rightarrow> nat" assumes "pr_conv_3_to_1 f \<in> PrimRec1'"   and "g \<in> PrimRec1'"   and "h \<in> PrimRec1'"   and "k \<in> PrimRec1'" shows "(\<lambda>x. f (g x) (h x) (k x)) \<in> PrimRec1'"
matrix_invertible: fixes f :: "(real, 'a) vec \<Rightarrow> (real, 'b) vec" assumes "linear f" shows "invertible (matrix f) = (\<exists>g. linear g \<and> f \<circ> g = id \<and> g \<circ> f = id)"
dvd_div_mult2_eq: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "b * c dvd a" shows "a div (b * c) = a div b div c"
suppa_distincts: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C SuppA D E F " shows "A \<noteq> B \<and> B \<noteq> C \<and> D \<noteq> E \<and> E \<noteq> F"
zero_smallest: fixes j :: "'a" assumes "j \<in> Field r" shows "(zero, j) \<in> r"
bind_measure_pmf_cong: fixes A :: "'a \<Rightarrow> 'b measure"   and N :: "'b measure"   and B :: "'a \<Rightarrow> 'b measure"   and x :: "'a pmf" assumes "\<And>x. A x \<in> space (subprob_algebra N)"   and "\<And>x. B x \<in> space (subprob_algebra N)"   and "\<And>i. i \<in> set_pmf x \<Longrightarrow> A i = B i" shows "measure_pmf x \<bind> A = measure_pmf x \<bind> B"
IH_9: assumes "\<langle>t not in s\<rangle> \<in> set A" shows "\<langle>t \<cdot> \<theta> not in s \<cdot> \<theta>\<rangle> \<in> set (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)"
inv_inj_on_Units: shows "inj_on (m_inv G) (Units G)"
adh_lemma: fixes x :: "'a llist"   and P :: "'a llist set" assumes "x \<in> A\<^sup>\<omega>"   and "P \<subseteq> A\<^sup>\<omega>" shows "(x adh P) = (\<forall>r\<in>finpref A x. \<exists>s\<in>A\<^sup>\<omega>. r @@ s \<in> P)"
lsimulation_axioms_def: fixes R :: "('a \<times> 'b) set"   and A :: "('a, 'c, 'd) sa_rec_scheme"   and B :: "('b, 'c, 'e) sa_rec_scheme" shows "lsimulation_axioms R A B \<equiv> \<forall>a b. (a, b) \<in> R \<longrightarrow> sa_L A a = sa_L B b"
mon_def: fixes G :: "('a, 'c) monoid_scheme"   and H :: "('b, 'd) monoid_scheme" shows "mon G H = {f \<in> hom G H. inj_on f (carrier G)}"
assms_1: shows "P \<doteq>\<^sub>c Q"
fact1: fixes i :: "'c" assumes "i \<in> ODE_vars I ODE" shows "(\<lambda>t. fst (mk_v I ODE (sol 0, b) (sol t)) $ i) = (\<lambda>t. sol t $ i)"
Timpl_rule: fixes s :: "'a" assumes "I s" shows "ospec (Timpl s) (\<lambda>(ni, s'). (ni, Trueif) \<in> R s' \<and> I s' \<and> les s s')"
finprod_cong': fixes A :: "'e set"   and B :: "'e set"   and g :: "'e \<Rightarrow> 'c"   and f :: "'e \<Rightarrow> 'c" assumes "A = B"   and "g \<in> B \<rightarrow> carrier A"   and "\<And>i. i \<in> B \<Longrightarrow> f i = g i" shows "finsum A f A = finsum A g B"
vec_to_iarray_solve_consistent_rref: fixes A :: "(('d, 'e) vec, 'f) vec"   and b :: "('d, 'f) vec" assumes "reduced_row_echelon_form A" shows "vec_to_iarray (solve_consistent_rref A b) = solve_consistent_rref_iarrays (matrix_to_iarray A) (vec_to_iarray b)"
multiplicity_geI: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a"   and p :: "'a"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "\<not> dvd.dvd times p one"   and "dvd.dvd times (power.power one times p n) x" shows "n \<le> factorial_semiring.multiplicity times one p x"
partList0_addSubsetSet_equi: fixes s :: "'b set"   and ts :: "'b set list" assumes "s \<subseteq> \<Union> (set ts)" shows "addSubsetSet s (set ts) - {{}} = set (partList0 s ts) - {{}}"
"case": fixes f :: "('e \<Rightarrow> 'f) \<Rightarrow> 'd"   and x :: "'e \<Rightarrow> 'f" shows "(case StateT x of StateT x \<Rightarrow> f x) = f x"
assms_6: shows "set (labels \<sigma>) \<subseteq> {\<alpha>}"
scale_ms_aux': fixes xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and b :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and C :: "'a" assumes "is_expansion_aux xs f (b # basis)"   and "is_expansion C basis"   and "basis_wf (b # basis)" shows "is_expansion_aux (scale_ms_aux' C xs) (\<lambda>x. eval C x * f x) (b # basis)"
zero_one_infty_not_equal_6: shows "\<infinity>\<^sub>h \<noteq> 1\<^sub>h"
ipresIGWls_def: fixes h :: "'c \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'g, 'h, 'i, 'j, 'c, 'k, 'l) model_scheme"   and MOD' :: "('m, 'n, 'o, 'h, 'p, 'q, 'd, 'r, 's) model_scheme" shows "ipresIGWls h MOD MOD' \<equiv> \<forall>s X. igWls MOD s X \<longrightarrow> igWls MOD' s (h X)"
LID\<^sub>SE: fixes l :: "'a"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a, 'c) val option" assumes "l \<in> LID\<^sub>S \<sigma>" obtains "l \<in> dom \<sigma>"   | l' :: "'a" and v :: "('b, 'a, 'c) val" where "\<sigma> l' = Some v" and "l \<in> LID\<^sub>V v"
disc_iff_1: fixes A :: "'c llist set" shows "lnull (lSup A) = (\<forall>x\<in>A. lnull x)"
p_def: fixes x :: "nat" shows "p x = c_pair x (f x)"
that: fixes y :: "'a" assumes "y \<in> set (local.filter_min_append xs ys)"   and "rel y x" shows "thesis"
CR2_def: fixes \<tau> :: "ty" shows "CR2 \<tau> \<equiv> \<forall>t t'. t \<in> RED \<tau> \<and> t \<mapsto> t' \<longrightarrow> t' \<in> RED \<tau>"
sum_group_subgroup_generated: fixes I :: "'b set"   and G :: "'b \<Rightarrow> ('c, 'd) monoid_scheme"   and H :: "'b \<Rightarrow> 'c set" assumes "\<And>i. i \<in> I \<Longrightarrow> Group.group (G i)"   and "\<And>i. i \<in> I \<Longrightarrow> subgroup (H i) (G i)" shows "sum_group I (\<lambda>i. subgroup_generated (G i) (H i)) = subgroup_generated (sum_group I G) (Pi\<^sub>E I H)"
mset_le_add_mset_decr_left1: fixes c :: "'c"   and a :: "'c multiset"   and b :: "'c multiset" assumes "add_mset c a \<subseteq># b" shows "a \<subseteq># b"
case_prodE: fixes c :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and p :: "'h \<times> 'i" assumes "case p of (a, b) \<Rightarrow> c a b" obtains x :: "'h" and y :: "'i" where "p = (x, y)" and "c x y"
DA0_is: shows "DA0 \<cdot> \<eta> = DA"
agree_trans: fixes \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and \<mu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and A :: "('a + 'a) set"   and \<omega> :: "(real, 'a) vec \<times> (real, 'a) vec"   and B :: "('a + 'a) set" assumes "Vagree \<nu> \<mu> A"   and "Vagree \<mu> \<omega> B" shows "Vagree \<nu> \<omega> (A \<inter> B)"
measurable_shift: fixes f :: "'b \<Rightarrow> 'c stream"   and N :: "'b measure"   and M :: "'c measure"   and g :: "'b \<Rightarrow> 'c stream"   and n :: "nat" assumes "f \<in> N \<rightarrow>\<^sub>M stream_space M"   and "g \<in> N \<rightarrow>\<^sub>M stream_space M" shows "(\<lambda>x. stake n (f x) @- g x) \<in> N \<rightarrow>\<^sub>M stream_space M"
servTicket_authentic_Kas: fixes B :: "agent"   and A :: "agent"   and servK :: "nat"   and Ts :: "nat"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace> \<in> parts (knows Spy evs)"   and "B \<noteq> Tgs"   and "B \<notin> bad"   and "evs \<in> kerbIV" shows "\<exists>authK Ta. Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>\<rbrace>) \<in> set evs"
fold_congs_5: fixes r :: "('b, 'c) state_scheme"   and r' :: "('b, 'c) state_scheme"   and v' :: "'b"   and f :: "'b \<Rightarrow> 'b"   and f' :: "'b \<Rightarrow> 'b" assumes "r = r'"   and "tortoise r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "tortoise_update f r = tortoise_update f' r'"
adverse_in_R1: fixes z :: "nat"   and i :: "nat"   and j :: "nat" shows "local.adverse z i j \<in> \<R>"
errorT_eqI: fixes a :: "'e\<cdot>('g, 'f) errorT"   and b :: "'e\<cdot>('g, 'f) errorT" assumes "runErrorT\<cdot>a = runErrorT\<cdot>b" shows "a = b"
fmadd_restrict_right_dom: fixes n :: "('c, 'd) fmap"   and m :: "('c, 'd) fmap" shows "fmrestrict_fset (fmdom n) (m ++\<^sub>f n) = n"
open_complex_homo_def: fixes U :: "complex_homo set" shows "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U)"
\<tau>multithreaded_wf_init_fin: shows "\<tau>multithreaded_wf init_fin_final init_fin init_fin_\<tau>move"
osc_def: fixes ur :: "('s \<times> 's) set" shows "osc ur \<equiv> \<forall>s1\<in>S\<^bsub>SES\<^esub>. \<forall>s1'\<in>S\<^bsub>SES\<^esub>. \<forall>s2'\<in>S\<^bsub>SES\<^esub>. \<forall>e\<in>E\<^bsub>SES\<^esub> - C\<^bsub>\<V>\<^esub>. reachable SES s1 \<and> reachable SES s1' \<and> s1' e\<longrightarrow>\<^bsub>SES\<^esub> s2' \<and> (s1', s1) \<in> ur \<longrightarrow> (\<exists>s2\<in>S\<^bsub>SES\<^esub>. \<exists>\<delta>. \<delta> \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [] \<and> \<delta> \<upharpoonleft> V\<^bsub>\<V>\<^esub> = [e] \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> s1 \<delta>\<Longrightarrow>\<^bsub>SES\<^esub> s2 \<and> (s2', s2) \<in> ur)"
simps_1: fixes bal_i :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and e :: "real"   and n :: "nat"   and d :: "nat"   and x :: "'b" assumes "RBTi2 TYPE('b) bal_i e" shows "RBTi2.ins2 bal_i n d x \<langle>\<rangle> = (if bal_i (n + 1) (d + 1) then Bal2 \<langle>\<langle>\<rangle>, x, \<langle>\<rangle>\<rangle> else Unbal2 \<langle>\<langle>\<rangle>, x, \<langle>\<rangle>\<rangle> 1 1)"
eq_add_iff2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and e :: "'c"   and c :: "'c"   and b :: "'c"   and d :: "'c" assumes "class.ring plus zero minus uminus times" shows "(plus (times a e) c = plus (times b e) d) = (c = plus (times (minus b a) e) d)"
\<sigma>_prop_Suc: fixes k :: "nat" shows "\<sigma> k \<in> qs \<and> \<sigma> (Suc k) \<in> qs \<and> name (\<sigma> k) \<in> incoming (\<sigma> (Suc k)) \<and> (\<forall>\<psi>\<in>old (\<sigma> (Suc k)). suffix (Suc k) \<xi> \<Turnstile>\<^sub>r \<psi>) \<and> (\<forall>\<psi>\<in>next (\<sigma> (Suc k)). suffix (Suc k) \<xi> \<Turnstile>\<^sub>r X\<^sub>r \<psi>) \<and> {\<eta>. \<exists>\<mu>. \<mu> U\<^sub>r \<eta> \<in> old (\<sigma> (Suc k)) \<and> suffix (Suc k) \<xi> \<Turnstile>\<^sub>r \<eta>} \<subseteq> old (\<sigma> (Suc k))"
utility_le_iff: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier"   and "y \<in> carrier" shows "(u x \<le> u y) = le x y"
Invariants_def: shows "Invariants = sigma (space M) {A \<in> sets M. T -` A \<inter> space M = A}"
dgrad_set_def: fixes d :: "'b \<Rightarrow> nat"   and m :: "nat" shows "dgrad_set d m = {t. d t \<le> m}"
comp_fun_commute_apply_cm: fixes f :: "'c \<Rightarrow> 't \<Rightarrow> int" shows "comp_fun_commute (\<lambda>(loc, t) c. apply_cm c loc t (f loc t))"
interleave_length: fixes x :: "'b list"   and a :: "'b list"   and b :: "'b list" assumes "x \<in> a \<otimes> b" shows "length x = length a + length b"
Framed_Link_Diagram_Equivalence_def: fixes T1 :: "Link_Diagram"   and T2 :: "Link_Diagram" shows "T1 ~L T2 \<equiv> Rep_Link_Diagram T1 ~ Rep_Link_Diagram T2"
map_raw_cong: fixes xs :: "('c \<times> 'd) list \<times> 'e"   and f :: "'c \<times> 'd \<Rightarrow> 'c \<times> 'f"   and g :: "'c \<times> 'd \<Rightarrow> 'c \<times> 'f" assumes "\<And>kv. kv \<in> set (fst xs) \<Longrightarrow> f kv = g kv" shows "map_raw f xs = map_raw g xs"
normalize_simps_24: fixes gs :: "('f \<times> 'd set) list"   and b :: "'d set"   and I :: "'d set"   and V :: "('d \<times> 'd) set"   and c :: "('d, 'e, 'f) com" shows "Language.normalize (whileAnnoG gs b I V c) = guards gs (While b (sequence Seq (flatten (Language.normalize c) @ flatten (guards gs Skip))))"
HaddP_imp_OrdP: fixes x :: "tm"   and y :: "tm"   and z :: "tm" shows "{HaddP x y z} \<turnstile> OrdP y"
isGuard_grdLit: fixes x :: "Preliminaries.var" shows "isGuard x (grdLit x)"
is_variable_imp_incr_bv_unchanged: fixes inc :: "nat"   and lev :: "nat"   and v :: "variable"   and T :: "typ" shows "incr_bv inc lev (Fv v T) = Fv v T"
res_c: shows "dmu_array_cost (IArray fs_init) (IArray []) 0 = (res, cost)"
f_unreachable: shows "f \<in> \<emptyset> ae b"
igFreshIGAbs2STR_imp_igFreshIGAbs2: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('g, 'h, 'd, 'e, 'i, 'j, 'k, 'l, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "igFreshIGAbs2STR MOD" shows "FixSyn.igFreshIGAbs2 Delta MOD"
compl_fam_def: fixes S :: "'c \<Rightarrow> 'a set"   and I :: "'c set" shows "compl_fam S I = (\<forall>i\<in>I. complementary (S i) (IDirProds G S (I - {i})))"
Mf_def: fixes cfs :: "int \<times> int poly multiset" shows "Mf cfs \<equiv> case cfs of (c, fs) \<Rightarrow> (M c, image_mset Mp fs)"
allocationInjectionsUnivervseProperty: fixes G :: "'d set"   and N :: "'c set" shows "allAllocations N G = allAllocations' N G"
IH: assumes "\<forall>y<z. eval f (y # xs) \<notin> {None, Some 0}" shows "reachable ((Mn n f, xs, []) # rest, None) ((f, z # xs, []) # (Mn n f, xs, [z]) # rest, None)"
map_ident: fixes t :: "'d rexp" shows "map_rexp (\<lambda>x. x) t = t"
transitionE_2: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "P \<Longrightarrow>a<\<nu>x> \<prec> P'" shows "\<exists>P'' P'''. P \<Longrightarrow>\<^sub>\<tau> P''' \<and> P''' \<longmapsto> a<\<nu>x> \<prec> P'' \<and> P'' \<Longrightarrow>\<^sub>\<tau> P'"
exec_cons: fixes s :: "val option"   and r :: "proc \<times> val"   and rs :: "(proc \<times> val) list" shows "s \<star> Cons r rs = (s \<star> rs) \<bullet> r"
to_rat_real_alg_main: fixes x :: "real_alg" shows "to_rat_real_alg_main x = (if real_of x \<in> \<rat> then Some (THE q. real_of x = real_of_rat q) else None)"
eq_mask_less: fixes w :: "'b word"   and n :: "nat" assumes "w = w AND mask n"   and "n < LENGTH('b)" shows "w < 2 ^ n"
consistentI: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and r :: "Vars.var list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list"   and ce :: "Vars.var \<Rightarrow> one\<^sub>\<bottom>" assumes "a_consistent (ae, a, as) (restr_conf (- set r) (\<Gamma>, e, S))"   and "edom ae = edom ce"   and "prognosis ae as a (\<Gamma>, e, S) \<sqsubseteq> ce"   and "\<And>x. \<lbrakk>x \<in> thunks \<Gamma>; many \<sqsubseteq> ce x\<rbrakk> \<Longrightarrow> ae x = up\<cdot>0"   and "set r \<subseteq> domA \<Gamma> \<union> upds S - edom ce" shows "consistent (ae, ce, a, as, r) (\<Gamma>, e, S)"
wit: shows "similar_mat_wit (char_matrix A ev ^\<^sub>m k) (char_matrix B ev ^\<^sub>m k) P Q"
bij: shows "bij_betw f {0..<card UNIV} UNIV"
below_ID: fixes d :: "'a defl" shows "cast\<cdot>d \<sqsubseteq> ID"
continuous_on_components: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "locally connected S"   and "\<And>C. C \<in> components S \<Longrightarrow> continuous_on C f" shows "continuous_on S f"
WT_plus_oracle_iff: fixes \<I>l :: "('a, 'c) \<I>"   and \<I>r :: "('b, 'd) \<I>"   and left :: "'e \<Rightarrow> 'a \<Rightarrow> ('c \<times> 'e) spmf"   and right :: "'e \<Rightarrow> 'b \<Rightarrow> ('d \<times> 'e) spmf"   and s :: "'e" shows "\<I>l \<oplus>\<^sub>\<I> \<I>r \<turnstile>c (left \<oplus>\<^sub>O right) s \<surd> = (\<I>l \<turnstile>c left s \<surd> \<and> \<I>r \<turnstile>c right s \<surd>)"
cdloop: shows "k cd\<^bsup>\<pi> @\<^bsup>k\<^esup> \<pi>l\<^esup>\<rightarrow> i \<or> (\<forall>j'>i'. j' cd\<^bsup>\<pi>'\<^esup>\<rightarrow> i')"
run_one_step_basic_grow_memory_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Grow_memory) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
simps_1: fixes P :: "pi" shows "P[<[]>] = P"
C_ss_VIS: fixes v :: "nat" assumes "v \<in> C" shows "v \<in> dom PRED"
divisor_sigma_naive: fixes x :: "'b"   and n :: "nat" shows "divisor_sigma x n = (if n = 0 then 0::'b else fold_atLeastAtMost_nat (\<lambda>d acc. if d dvd n then nat_power d x + acc else acc) 1 n (0::'b))"