hcju/isabelleps huggingface dataset:
newton_sqrt_aux_correct: fixes n :: "nat"   and x :: "nat" assumes "Discrete.sqrt n \<le> x" shows "newton_sqrt_aux x n = Discrete.sqrt n"
sortnMT: fixes p :: "('a, 'b) Combinators list"   and l :: "'a set list" assumes "p \<noteq> []" shows "FWNormalisationCore.sort p l \<noteq> []"
p\<^sub>1_def: shows "p\<^sub>1 = find_closest_bf p\<^sub>0 (p\<^sub>2 # p\<^sub>3 # ps)"
unfinished_dnodes_def: fixes S :: "'v set" shows "unfinished_dnodes S \<equiv> all_dnodes - S"
prod_1_def: fixes p1 :: "'d \<Rightarrow> 'f decision option"   and p2 :: "'e \<Rightarrow> 'g decision option" shows "p1 \<Otimes>\<^sub>1 p2 \<equiv> \<lambda>(x, y). case p1 x of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d1\<rfloor> \<Rightarrow> case p2 y of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d2\<rfloor> \<Rightarrow> \<lfloor>allow (d1, d2)\<rfloor> | \<lfloor>deny d2\<rfloor> \<Rightarrow> \<lfloor>allow (d1, d2)\<rfloor> | \<lfloor>deny d1\<rfloor> \<Rightarrow> case p2 y of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d2\<rfloor> \<Rightarrow> \<lfloor>deny (d1, d2)\<rfloor> | \<lfloor>deny d2\<rfloor> \<Rightarrow> \<lfloor>deny (d1, d2)\<rfloor>"
arcs_set_elem1: fixes j :: "nat"   and k :: "nat"   and i :: "nat"   and xs :: "nat list" assumes "j \<noteq> k"   and "k \<in> set (i # xs)" shows "\<exists>l. (k, l) \<in> set (arcs i j xs)"
winning_path_supergame: fixes p :: "Player"   and P :: "'a llist"   and G' :: "('a, 'c) ParityGame_scheme" assumes "G'.winning_path p P"   and "ParityGame G'"   and "G'.VV p** \<subseteq> (if p** = Even then V0\<^bsub>G'\<^esub> else V\<^bsub>G'\<^esub> - V0\<^bsub>G'\<^esub>)"   and "\<omega>\<^bsub>G'__\<^esub> = \<omega>\<^bsub>G'\<^esub>" shows "ParityGame.winning_path G' p P"
is_discrete: fixes f :: "('obj, 'obj) concrete_category.arr" shows "arr f = ide f"
prems_1: shows "set A \<subseteq> set (a # B)"
m_lcomm: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "Group.comm_monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z) = y \<otimes>\<^bsub>G\<^esub> (x \<otimes>\<^bsub>G\<^esub> z)"
span_mono: fixes A :: "'d set"   and B :: "'d set" assumes "A \<subseteq> B" shows "cspan A \<subseteq> cspan B"
in_set_conv_decomp: fixes x :: "'a"   and xs :: "'a list" shows "x mem xs = (\<exists>ys zs. xs = ys @ x # zs)"
q1_sat_props: shows "1 < length qs \<longrightarrow> p \<noteq> 0 \<and> 0 < length qs \<and> (\<forall>q. List.member qs q \<longrightarrow> coprime p q) \<longrightarrow> satisfies_properties p (take (length qs div 2) qs) (get_subsets (calculate_data p (take (length qs div 2) qs))) (get_signs (calculate_data p (take (length qs div 2) qs))) (get_matrix (calculate_data p (take (length qs div 2) qs)))"
imod_infinity: fixes n :: "enat" shows "\<infinity> mod n = \<infinity>"
alw_InvTempUpright: fixes s :: "('c, 'd) configuration stream" assumes "spec s" shows "alw (holds InvTempUpright) s"
mustT_Seq_R: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state" assumes "mustT (c1 ;; c2) s"   and "(c1, s) \<rightarrow>*t s'" shows "mustT c2 s'"
zless_add1_eq: fixes w :: "int"   and z :: "int" shows "(w < z + 1) = (w < z \<or> w = z)"
expand: fixes writerT :: "('e, 'f, 'd) writerT"   and writerT' :: "('e, 'f, 'd) writerT" assumes "run_writer writerT = run_writer writerT'" shows "writerT = writerT'"
spectrum_imp_cxt_quotient_ring: fixes \<pp> :: "('a \<times> 'a) set set" assumes "\<pp> \<in> pi.spectrum" shows "Comm_Ring.quotient_ring (pi.carrier_local_ring_at\<setminus>\<pp>) pi.carrier_local_ring_at pi.add_local_ring_at pi.mult_local_ring_at pi.zero_local_ring_at pi.one_local_ring_at"
semifin_chain_def: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a"   and Q :: "'a set" shows "[f[x ..]Q] \<equiv> infinite Q \<and> long_ch_by_ord f Q \<and> f 0 = x"
map_prod_eq: fixes xs :: "('b \<times> 'c) list"   and ys :: "('b \<times> 'c) list" assumes "map fst xs = map fst ys"   and "map snd xs = map snd ys" shows "xs = ys"
wf_mstate_def: fixes \<phi> :: "Formula.formula"   and \<pi> :: "(String.literal \<times> event_data list) prefix"   and R :: "event_data list set"   and st :: "('msaux, 'muaux) mstate" shows "wf_mstate \<phi> \<pi> R st = (mstate_n st = Formula.nfv \<phi> \<and> (\<forall>\<sigma>. prefix_of \<pi> \<sigma> \<longrightarrow> mstate_i st = Monitor.progress \<sigma> Map.empty \<phi> (plen \<pi>) \<and> wf_mformula \<sigma> (plen \<pi>) Map.empty Map.empty (mstate_n st) R (mstate_m st) \<phi>))"
longestImpWitnessPath_def: fixes c :: "('loc, 't, 'a) configuration_scheme"   and loc1 :: "'loc"   and loc2 :: "'loc"   and xs :: "('loc \<times> 'sum \<times> 'loc) list"   and t :: "'t" shows "longestImpWitnessPath c loc1 loc2 xs t = (impWitnessPath c loc1 loc2 xs t \<and> (\<forall>loc' xs'. impWitnessPath c loc' loc2 xs' t \<longrightarrow> length xs' \<le> length xs))"
ln_\<theta>_asymptotics: assumes "prime_number_theorem" shows "(\<lambda>x. ln (\<theta> x) - ln x) \<in> o(\<lambda>_. 1)"
orbit_def: fixes s :: "'a" shows "\<gamma>\<^sup>\<phi> s = x\<acute>=\<lambda>t. f & \<lambda>s. True on \<lambda>s. T S @ 0 s"
axioms_1: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "Group_Theory.group G composition unit" shows "Group_Theory.monoid G composition unit"
height_del: fixes t :: "('c \<times> 'd) tree23"   and x :: "'c" assumes "complete t" shows "hD (Tree23_Map.del x t) = height t"
poly_lift_hom_closed: fixes S :: "('e, 'f) ring_scheme"   and \<phi> :: "'a \<Rightarrow> 'e"   and p :: "nat \<Rightarrow> 'a" assumes "cring S"   and "\<phi> \<in> ring_hom R S"   and "p \<in> carrier (UP R)" shows "poly_lift_hom R S \<phi> p \<in> carrier (UP S)"
Domain_Collect_case_prod: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "Domain {(x, y). P x y} = {x. \<exists>y. P x y}"
eq_mono: fixes x :: "real"   and bs :: "real list" shows "\<forall>a n s. Inum (x # bs) a = Inum (x # bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x # bs) (eq n s) = Ifm (x # bs) (Eq a)"
neg_def: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" shows "neg \<phi> \<pi>l \<equiv> \<not> \<phi> \<pi>l"
bind_id'_def: shows "Monad_Overloading.bind = bind_id"
permutes_inj: fixes p :: "'c \<Rightarrow> 'c"   and S :: "'c set" assumes "p permutes S" shows "inj p"
soldAllocationVariantEquivalence: fixes N :: "integer set"   and \<Omega> :: "'d set" shows "soldAllocations'' N \<Omega> = soldAllocations''' N \<Omega>"
homogeneous_set_coneI: fixes h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set" assumes "homogeneous h" shows "homogeneous_set (cone (h, U))"
closed_imp_locally_compact: fixes S :: "'c set" assumes "closed S" shows "locally compact S"
MinimalMethodDefs'_def: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and M :: "char list"   and Cs :: "char list list"   and mthd :: "ty list \<times> ty \<times> char list list \<times> expr" shows "MinimalMethodDefs' P C M Cs mthd \<equiv> (Cs, mthd) \<in> MinimalMethodDefs P C M"
length_norm_unsigned_le: fixes w :: "bit list" shows "length (norm_unsigned w) \<le> length w"
neq: fixes p :: "tm"   and vs :: "'b list"   and bs :: "'b list" assumes "allpolys isnpoly p" shows "Ifm vs bs (neq p) = Ifm vs bs (NEq p)"
hconf_start: fixes wf_mb :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog" assumes "wf_prog wf_mb P" shows "P \<turnstile> start_heap P \<surd>"
in_carrier2: shows "inv ((\<^bold>g [^] \<alpha>1) [^] r) \<in> carrier \<G>"
induct: fixes x :: "'e"   and T :: "('e, 'f) TS_scheme"   and P :: "'e \<Rightarrow> bool" assumes "x \<in> reach T"   and "\<And>s. s \<in> init T \<Longrightarrow> P s"   and "\<And>s t. \<lbrakk>(s, t) \<in> TS.trans T; s \<in> reach T; P s\<rbrakk> \<Longrightarrow> P t" shows "P x"
sum_mset_delta: fixes y :: "'c"   and c :: "'a"   and A :: "'c multiset" shows "(\<Sum>x\<in>#A. if x = y then c else (0::'a)) = c * of_nat (count A y)"
con: fixes xk :: "'a \<times> 'a set" assumes "xk \<in> p" shows "0 \<le> content (snd xk)"
pair_imageI: fixes a :: "'b"   and b :: "'c"   and A :: "('b \<times> 'c) set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "(a, b) \<in> A" shows "f a b \<in> (\<lambda>(a, b). f a b) ` A"
span_induct_alt: fixes x :: "'c"   and S :: "'c set"   and h :: "'c \<Rightarrow> bool" assumes "x \<in> ideal S"   and "h (0::'c)"   and "\<And>c x y. \<lbrakk>x \<in> S; h y\<rbrakk> \<Longrightarrow> h (c * x + y)" shows "h x"
assms_1: shows "next_execs s execs d = aseq # aseqs"
poly_shift_0: fixes n :: "nat" shows "poly_shift n 0 = 0"
dickson_less_vD3: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and v :: "'t"   and u :: "'t" assumes "dickson_less_v d m v u" shows "v \<prec>\<^sub>t u"
A_distinct_def: shows "A_distinct \<equiv> {len m i j xs |xs. set xs \<subseteq> {0..k} \<and> i \<notin> set xs \<and> j \<notin> set xs \<and> distinct xs}"
leq_def: fixes x :: "'c"   and y :: "'c" shows "(x \<le>\<^sub>\<S> y) = (x \<oplus> y = y)"
assms: assumes "Re z = 0" shows "\<bar>Im z\<bar> < 1"
assms_1: shows "x * z\<^sup>+ * y = bot"
FormP_quot: fixes A :: "fm" shows "{} \<turnstile> FormP \<guillemotleft>A\<guillemotright>"
abs_eq: fixes xa :: "'b multiset \<times> 'b multiset"   and x :: "'b multiset \<times> 'b multiset" shows "(abs_zmultiset xa < abs_zmultiset x) = (case xa of (Mp, Mn) \<Rightarrow> \<lambda>(Np, Nn). Mp + Nn < Mn + Np) x"
us: shows "us \<equiv> getFrN (map snd txs) (suc r # map fst txs) [] (length txs)"
set_bset_to_set_bset: fixes A :: "'a set" assumes "|A| <o natLeq +c |UNIV|" shows "set_bset (the_inv set_bset A) = A"
walk_decomp_1: fixes xs :: "'a list"   and ys :: "'a list" assumes "H.walk (xs @ ys)" shows "H.walk xs"
D_base_finite': fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat"   and k :: "nat" shows "finite {len m i j xs |xs. set xs \<subseteq> {0..k} \<and> distinct (i # j # xs)}"
min_dist_minD: fixes v :: "nat"   and d :: "nat"   and v' :: "nat" assumes "local.dist v d v'" shows "min_dist v v' \<le> d"
widen_antisym_lemma: fixes G :: "prog"   and S :: "ty"   and T :: "ty" assumes "G\<turnstile>S\<preceq>T"   and "\<forall>I J. G\<turnstile>I\<preceq>I J \<and> G\<turnstile>J\<preceq>I I \<longrightarrow> I = J"   and "\<forall>C D. G\<turnstile>C\<preceq>\<^sub>C D \<and> G\<turnstile>D\<preceq>\<^sub>C C \<longrightarrow> C = D"   and "\<forall>I. G\<turnstile>Object\<leadsto>I \<longrightarrow> False"   and "G\<turnstile>T\<preceq>S" shows "S = T"
sum_code: fixes g :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "sum g (set xs) = sum_list (map g (remdups xs))"
simps: fixes f :: "event_data option list set list \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and buf :: "event_data option list set list list" shows "mbufn_take f z buf = (if buf = [] \<or> [] \<in> set buf then (z, buf) else mbufn_take f (f (map hd buf) z) (map tl buf))"
ZObis_5: fixes cs :: "('test, 'atom, 'choice) cmd list" assumes "properL cs"   and "\<And>c. c \<in> set cs \<Longrightarrow> c \<approx>s c" shows "ParT cs \<approx>01 ParT cs"
xm: shows "\<forall>x\<in>set xs. x < m"
ValK_copy_rec_ValK_copy: shows "fix\<cdot>ValK_copy_rec = (KM_map\<cdot>ValK_copy, cfun_map\<cdot>(KM_map\<cdot>ValK_copy)\<cdot>(KM_map\<cdot>ValK_copy))"
set_unsat_simps_4: fixes I :: "'b list"   and s :: "('b, 'a) state" shows "\<V> (set_unsat I s) = \<V> s"
simps_1: fixes f :: "'c \<Rightarrow> 'd" shows "isort_key f [] = []"
mult_nrows: fixes fmul :: "'d \<Rightarrow> 'b \<Rightarrow> 'c"   and fadd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'d matrix"   and B :: "'b matrix" assumes "\<forall>a. fmul (0::'d) a = (0::'c)"   and "\<forall>a. fmul a (0::'b) = (0::'c)"   and "fadd (0::'c) (0::'c) = (0::'c)" shows "nrows (mult_matrix fmul fadd A B) \<le> nrows A"
ClassD: fixes b :: "'a set \<times> 'b"   and a :: "'a set \<times> 'b" assumes "b \<in> rel.Class a"   and "a \<in> Sigma I \<FF>" shows "(b, a) \<in> {(x, y). x \<sim> y}"
finite_glitOfL: fixes l :: "('fsym, 'psym) lit" shows "finite (glitOfL l)"
preserves_src: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "src\<^sub>D (F \<mu>) = F.map\<^sub>0 (src\<^sub>C \<mu>)"
ofilter_Under: fixes A :: "'a set" assumes "A \<subseteq> Field r" shows "r.ofilter (r.Under A)"
M1: shows "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"
iso_char: fixes f :: "'c \<times> 'c \<times> 'c" shows "C.VxVxV.iso f = (C.iso (fst f) \<and> C.VxV.iso (snd f))"
Cons_path_2: shows "m'' \<in> set (predecessors g m')"
ipresCons_imp_ipresSubstAll: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and hA :: "('b, 'c, 'd, 'a, 'f) qAbs set \<Rightarrow> 'h"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h, 'i) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.ipresCons Delta h hA MOD"   and "FixSyn.igSubstCls Delta MOD"   and "FixSyn.igConsIPresIGWls Delta MOD"   and "FixSyn.igFreshCls Delta MOD" shows "FixSyn.ipresSubstAll Delta h hA MOD"
add_assoc4: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.semigroup_add plus" shows "plus (plus (plus a b) c) d = plus a (plus (plus b c) d)"
split_asm: fixes P :: "'h \<Rightarrow> bool"   and f1 :: "'h"   and f2 :: "'i \<Rightarrow> 'h"   and option :: "'i option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (\<not> (option = None \<and> \<not> P f1 \<or> (\<exists>x2. option = Some x2 \<and> \<not> P (f2 x2))))"
set_sorted_list_of_set: fixes A :: "'a word set" assumes "finite A" shows "set (signed.sorted_list_of_set A) = A"
elims: fixes x :: "nat set"   and xa :: "(nat set \<times> 'a option list set) set"   and xb :: "nat set \<times> 'a option list"   and y :: "(nat set \<times> 'a option list set) set" assumes "newQuery x xa xb = y" obtains V :: "nat set" and Q :: "(nat set \<times> 'a option list set) set" and st :: "nat set" and t :: "'a option list" where "x = V" and "xa = Q" and "xb = (st, t)" and "y = (\<lambda>tab. projectTable V (semiJoin tab (st, t))) ` Q"
intros: fixes s :: "('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 t :: "'t"   and x :: "'x1"   and s' :: "('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 x' :: "'x1"   and ln' :: "'l \<Rightarrow>f nat"   and ta :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action" assumes "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "thr s' t = \<lfloor>(x', ln')\<rfloor>"   and "\<tau>move1 (x, shr s) ta (x', shr s')" shows "m\<tau>move1 s (t, ta) s'"
implies_galois: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "(z \<le> x \<squnion> y) = (z - y \<le> x)"
run_axioms: shows "run cs\<^sub>0 clstep cs"
proj_tuple_replicate: fixes bs :: "bool list"   and as :: "'b option list" assumes "\<And>i. i \<in> set bs \<Longrightarrow> \<not> i"   and "length bs = length as" shows "proj_tuple bs as = replicate (length bs) None"
prod_bm: shows "\<forall>i. (\<lambda>w. X w * indpre i w) \<in> borel_measurable M"
phi_pos: fixes n :: "nat" shows "0 \<le> \<Phi> n"
diagseq_fold_reduce: fixes n :: "nat" shows "diagseq n = fold_reduce 0 n n"
wt_getTvars: fixes \<sigma>l :: "'tp list" shows "list_all GE.wt (GE.getTvars \<sigma>l)"
homeomorphic_map_maps: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" shows "homeomorphic_map X Y f = (\<exists>g. homeomorphic_maps X Y f g)"
injective_map_def: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set" shows "injective_map \<alpha> S T \<equiv> map \<alpha> S T \<and> injective_map_axioms \<alpha> S"
inner_mult_right: fixes 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 a :: "'b"   and m :: "real"   and b :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner a (of_real m * b) = m * inner a b"
simps: fixes x :: "'t"   and xs :: "'t list" shows "ord_term_lin.max_list (x # xs) = (case xs of [] \<Rightarrow> x | a # list \<Rightarrow> ord_term_lin.max x (ord_term_lin.max_list xs))"
act_sup_distr: fixes x :: "'a"   and y :: "'a"   and p :: "'b" shows "\<alpha> (x \<squnion> y) p = \<alpha> x p \<squnion> \<alpha> y p"
assms_2: shows "z \<cdot> x = (0::'a)"
Unifier_sym: fixes s :: "('a, 'b) Term.term"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" assumes "Unifier \<sigma> s t" shows "Unifier \<sigma> t s"
dist_measure_has_subprob_density: fixes v :: "val"   and dst :: "pdf_dist" assumes "v \<in> type_universe (dist_param_type dst)" shows "has_subprob_density (dist_measure dst v) (stock_measure (dist_result_type dst)) (dist_dens dst v)"
Abs_cfun_inverse2: fixes f :: "'a \<Rightarrow> 'b" assumes "cont f" shows "Rep_cfun (Abs_cfun f) = f"
mset_le_single_cases: fixes M :: "'f multiset"   and a :: "'f" assumes "M \<subseteq># {#a#}" obtains "M = {#}"   | "M = {#a#}"
fmadd_def: shows "(++\<^sub>f) \<equiv> map_fun fmlookup (map_fun fmlookup Abs_fmap) (++)"
subdivide_interval: fixes a :: "real"   and x :: "real"   and y :: "real" assumes "a < \<bar>x - y\<bar>"   and "0 < a" obtains n :: "real" where "n \<in> \<int>" and "x < n * a \<and> n * a < y \<or> y < n * a \<and> n * a < x"
igSubstBinpIPresIGWlsBinp_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igSubstBinpIPresIGWlsBinp MOD \<equiv> \<forall>ys y Y delta binp. igWls MOD (asSort ys) Y \<and> igWlsBinp MOD delta binp \<longrightarrow> igWlsBinp MOD delta (igSubstBinp MOD ys Y y binp)"
insertion_fun_add: fixes f :: "nat \<Rightarrow> 'a"   and p :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and q :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "insertion_fun f (poly_mapping.lookup (p + q)) = insertion_fun f (poly_mapping.lookup p) + insertion_fun f (poly_mapping.lookup q)"
non_exec_call: fixes \<sigma> :: "'a control_state_scheme"   and M :: "'c \<Rightarrow> 'a control_state_scheme \<Rightarrow> ('b \<times> 'a control_state_scheme) option"   and A\<^sub>1 :: "'a control_state_scheme \<Rightarrow> 'c"   and M' :: "'b \<Rightarrow> 'a control_state_scheme \<Rightarrow> (bool \<times> 'a control_state_scheme) option" assumes "\<not> exec_stop \<sigma>" shows "(\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E (call\<^sub>C M A\<^sub>1) M') = (\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E (M (A\<^sub>1 \<sigma>)) M')"
ih: shows "graph_homomorphism G\<^sub>1 G\<^sub>2 f"
move3_mrexps3: fixes r :: "'b mrexp3"   and s :: "'b rexp"   and x :: "'b"   and a :: "bool" assumes "r \<in> mrexps3 s" shows "move3 x r a \<in> mrexps3 s"
good_ruleset_alt: fixes rs :: "'c rule list" shows "good_ruleset rs = (\<forall>r\<in>set rs. get_action r = Accept \<or> get_action r = Drop \<or> get_action r = Reject \<or> get_action r = Log \<or> get_action r = Empty)"
tranclD3: fixes S :: "'a"   and T :: "'a"   and R :: "('a \<times> 'a) set" assumes "(S, T) \<in> R\<^sup>+"   and "(S, T) \<notin> R" shows "\<exists>U. (S, U) \<in> R \<and> (U, T) \<in> R\<^sup>+"
\<Sigma>_protocols_to_commitments_axioms: fixes \<eta> :: "nat" shows "\<Sigma>_protocols_to_commitments (init \<eta>) (response \<eta>) (check \<eta>) (R \<eta>) (S2 \<eta>) (ss_adversary \<eta>) (challenge_space \<eta>) (valid_pub \<eta>) (G \<eta>)"
card_arcs: shows "card (arcs H) = card (arcs G) + 2"
equiv_up_to_True: fixes c :: "com"   and c' :: "com" shows "((\<lambda>_. True) \<Turnstile> c \<sim> c') = (c \<sim> c')"
th1: shows "inv ` {p. p permutes S} = {p. p permutes S}"
equivalence: shows "equivalence (R \<times> S) {(x, y). (x, y) \<in> (R \<times> S) \<times> R \<times> S \<and> x \<sim> y}"
ide_char: fixes a :: "'c" shows "C'.ide a = (C'.arr a \<and> C.ide a)"
topological_basis_imp_subbasis: fixes "open" :: "'c set \<Rightarrow> bool"   and B :: "'c set set" assumes "class.topological_space open"   and "topological_space.topological_basis open B" shows "open = generate_topology B"
assms_3: shows "fst ` edges G \<subseteq> L"
ExE: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" assumes "[\<^bold>\<exists>x. \<phi> x in v]" shows "\<exists>y. [\<phi> y in v]"
is_processT5: fixes P :: "'a process" shows "\<forall>s X Y. (s, X) \<in> F P \<and> (\<forall>c. c \<in> Y \<longrightarrow> (s @ [c], {}) \<notin> F P) \<longrightarrow> (s, X \<union> Y) \<in> F P"
cases: fixes a1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and a2 :: "'t \<times> ('l, 't, 'x, 'm, 'w, 'o) thread_action"   and a3 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "redT a1 a2 a3" obtains t :: "'t" and x :: "'x" and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action" and x' :: "'x" and m' :: "'m" and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" where "a1 = sa__" and "a2 = (tb__, ta)" and "a3 = s'a__" and "tb__ \<turnstile> \<langle>x__, shr sa__\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>" and "thr sa__ tb__ = \<lfloor>(x__, no_wait_locks)\<rfloor>" and "actions_ok sa__ tb__ ta" and "redT_upd sa__ tb__ ta x' m' s'a__"   | s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and t :: "'t" and x :: "'x" and n :: "'l" and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and ln :: "'l \<Rightarrow>f nat" where "a1 = s" and "a2 = (t, K$ [], [], [], [], [], convert_RA ln)" and "a3 = s'" and "thr s t = \<lfloor>(x, ln)\<rfloor>" and "\<not> waiting (wset s t)" and "may_acquire_all (locks s) t ln" and "0 < ln $ n" and "s' = (acquire_all (locks s) t ln, (thr s(t \<mapsto> (x, no_wait_locks)), shr s), wset s, interrupts s)"
callee_invariant_on_True: fixes callee :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and \<I> :: "('e, 'f) \<I>" shows "callee_invariant_on callee (\<lambda>_. True) \<I> = (\<forall>s. \<I> \<turnstile>c callee s \<surd>)"
least_UpperI: fixes A :: "'c set"   and L :: "('c, 'd) gorder_scheme"   and s :: "'c" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq>\<^bsub>L\<^esub> s"   and "\<And>y. y \<in> Upper L A \<Longrightarrow> s \<sqsubseteq>\<^bsub>L\<^esub> y"   and "A \<subseteq> carrier L"   and "s \<in> carrier L" shows "is_lub L s A"
length_dims0: shows "length dims0 = card ({0..<length dims} \<inter> vars0)"
intro: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" assumes "\<And>x. f x \<circ> f x = f x" shows "comp_fun_idem_axioms f"
filter_Un: fixes xs :: "'a list"   and A :: "'a set"   and B :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "set xs = A \<union> B" shows "set (filter P xs) = {x \<in> A. P x} \<union> {x \<in> B. P x}"