hcju/isabelleps huggingface dataset:
assms_2: fixes s :: "'s"   and cfg :: "'s cfg" assumes "P s cfg" shows "state cfg = s"
lookup_pair_eqI: fixes xs :: "('a \<times> 'c) list"   and ys :: "('a \<times> 'c) list" assumes "oalist_inv_raw ko xs"   and "oalist_inv_raw ko' ys"   and "set xs = set ys" shows "lookup_pair ko xs = lookup_pair ko' ys"
interfree_Collector_Mutator: shows "interfree_aux (Some Collector, {}, Some Mutator)"
peek_and_Not: fixes P :: "'b assn"   and f :: "state \<Rightarrow> bool" shows "(P \<and>. (\<lambda>s. \<not> f s)) = (P \<and>. Not \<circ> f)"
right_neutral: fixes a :: "'g multiset" shows "a \<union># {#} = a"
abs_split: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a" shows "P \<bar>a\<bar> = (((0::'a) \<le> a \<longrightarrow> P a) \<and> (a < (0::'a) \<longrightarrow> P (- a)))"
assms_3: shows "{} \<notin> set (wall_crossings Cs)"
simps_1: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'c trm"   and x :: "Preliminaries.var" shows "subst \<pi> (Var x) = \<pi> x"
leq_trans: fixes x :: "'c llist"   and y :: "'c llist"   and z :: "'c llist" assumes "lprefix x y"   and "lprefix y z" shows "lprefix x z"
assms_4: shows "d \<in> {real_of_float dl..real_of_float du}"
rel_spmfI: fixes pq :: "('c \<times> 'd) spmf"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c spmf"   and q :: "'d spmf" assumes "\<And>x y. (x, y) \<in> set_spmf pq \<Longrightarrow> P x y"   and "map_spmf fst pq = p"   and "map_spmf snd pq = q" shows "rel_spmf P p q"
inject_3: fixes x31 :: "linear_poly"   and x32 :: "rat"   and y31 :: "linear_poly"   and y32 :: "rat" shows "(LEQ x31 x32 = LEQ y31 y32) = (x31 = y31 \<and> x32 = y32)"
norm_blinfun_compose: fixes f :: "'f \<Rightarrow>\<^sub>L 'e"   and g :: "'d \<Rightarrow>\<^sub>L 'f" shows "norm (f o\<^sub>L g) \<le> norm f * norm g"
nprv_exuI_exi: fixes F :: "'fmla set"   and x :: "'var"   and \<phi> :: "'fmla"   and y :: "'var" assumes "nprv F (exi x \<phi>)"   and "nprv (insert (subst \<phi> (Var y) x) (insert \<phi> F)) (eql (Var y) (Var x))"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "x \<in> var"   and "y \<in> var"   and "y \<noteq> x"   and "y \<notin> Fvars \<phi>"   and "x \<notin> \<Union> (Fvars ` F)"   and "y \<notin> \<Union> (Fvars ` F)" shows "nprv F (exu x \<phi>)"
equivp_gen_cong: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "equivp (step.gen_cong R)"
sum_pdevs_infinite: fixes X :: "'a set"   and f :: "'a \<Rightarrow> 'b pdevs" assumes "infinite X" shows "sum_pdevs f X = zero_pdevs"
Poincare_recurrence_thm_5: fixes A :: "'a set" assumes "A \<in> sets M" shows "emeasure M (recurrent_subset A) = emeasure M A"
res_function_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "res_function f = (\<forall>c. (\<forall>a b. cong a b c = (f c a = f c b)) \<and> pairwise (\<lambda>a b. \<not> cong a b c) (range (f c)) \<and> (\<forall>a. \<exists>k. f c a = a + k * c))"
Says_Kas_message_form: fixes A :: "agent"   and K :: "nat"   and authK :: "nat"   and Peer :: "agent"   and Ta :: "msg"   and authTicket :: "msg"   and evs :: "event list" assumes "Says Kas A \<lbrace>Crypt K \<lbrace>Key authK, Agent Peer, Ta\<rbrace>, authTicket\<rbrace> \<in> set evs"   and "evs \<in> kerbV" shows "authK \<notin> range shrK \<and> authK \<in> authKeys evs \<and> authK \<in> symKeys \<and> authTicket = Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Ta\<rbrace> \<and> K = shrK A \<and> Peer = Tgs"
commute_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "commute R S = square R S S R"
is_ub_thelub_ex: fixes S :: "'b set"   and x :: "'b" assumes "\<exists>u. S <<| u"   and "x \<in> S" shows "x \<sqsubseteq> lub S"
proj_set_mono_1: fixes A :: "('d strand_label \<times> 'e) list"   and B :: "('d strand_label \<times> 'e) list"   and n :: "'d" assumes "set A \<subseteq> set B" shows "set (proj n A) \<subseteq> set (proj n B)"
IH: fixes t :: "('a, 'b) tm"   and thesis :: "bool" assumes "\<And>i. \<lbrakk>i < num_args t; args t ! i \<rightarrow>\<^sub>e\<^sub>m\<^sub>b args (emb_step_at p d t) ! i; \<And>j. \<lbrakk>j < num_args t; i \<noteq> j\<rbrakk> \<Longrightarrow> args t ! j = args (emb_step_at p d t) ! j\<rbrakk> \<Longrightarrow> thesis"   and "\<not> list_all (\<lambda>x. x = dir.Left) p"   and "position_of t (p @ [d])" shows "thesis"
assms_4: shows "a \<otimes> b \<sim> a \<otimes> b'"
u_def: shows "u = (\<lambda>(i, a). (f i, a))"
distinct_sortQ: fixes xs :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "singleCombinators xs"   and "all_in_list xs l"   and "distinct xs" shows "distinct (qsort xs l)"
stable': fixes y :: "'a" shows "filter (\<lambda>ya. compare cmp y ya = Equiv) ys = filter (\<lambda>ya. compare cmp y ya = Equiv) xs"
add_pred_subset: fixes P :: "'BExp set"   and \<S> :: "'Var set \<times> 'Var set"   and p :: "'BExp" shows "P \<subseteq> P +\<^sub>\<S> p"
a_rcos_zero: fixes I :: "'a set set"   and i :: "'a set" assumes "ideal I (R Quot I)"   and "i \<in> I" shows "I +>\<^bsub>R Quot I\<^esub> i = I"
self_append_conv2: fixes ys :: "'c list"   and xs :: "'c list" shows "(ys = xs @ ys) = (xs = [])"
less_1: fixes s :: "(accent \<times> 'a) list"   and t :: "(accent \<times> 'a) list" assumes "length s + length t < length sa__ + length ta__"   and "(s, t) \<in> greek_less r" shows "(x # s, x # t) \<in> greek_less r"
case_prodD: fixes f :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and a :: "'x"   and b :: "'y" assumes "case (a, b) of (c, d) \<Rightarrow> f c d" shows "f a b"
CIMor: fixes T :: "(ZF, ZF) Axioms"   and A :: "ZF"   and e :: "(ZF, ZF) Expression"   and B :: "ZF" assumes "ZFAxioms T" shows "([A,e,B]\<^bsub>T\<^esub> \<in> mor\<^bsub>CanonicalCat T\<^esub>) = Sig aS\<^bsub>T\<^esub> \<triangleright> Vx : A \<turnstile> e : B"
oprod_wf: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" assumes "wf r"   and "wf r'" shows "wf (r *o r')"
rel_fun_def: fixes A :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'h \<Rightarrow> bool" shows "A ===> B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
sets_restrict_space_iff: fixes \<Omega> :: "'d set"   and M :: "'d measure"   and A :: "'d set" assumes "\<Omega> \<inter> space M \<in> sets M" shows "(A \<in> sets (restrict_space M \<Omega>)) = (A \<subseteq> \<Omega> \<and> A \<in> sets M)"
alphaAbs_ex_distinct_qAFresh_def: fixes xs :: "'f"   and x :: "'g"   and X :: "('h, 'i, 'f, 'g, 'j) qTerm"   and xs' :: "'f"   and x' :: "'g"   and X' :: "('h, 'i, 'f, 'g, 'j) qTerm" shows "alphaAbs_ex_distinct_qAFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<exists>y. y \<notin> {x, x'} \<and> qAFresh xs y X \<and> qAFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
term_variants_pred_subst: fixes P :: "'a \<Rightarrow> 'a list"   and t :: "('a, 'b) Term.term"   and s :: "('a, 'b) Term.term"   and \<delta> :: "'b \<Rightarrow> ('a, 'c) Term.term" assumes "term_variants_pred P t s" shows "term_variants_pred P (t \<cdot> \<delta>) (s \<cdot> \<delta>)"
assms: fixes i :: "nat" assumes "i \<in> S" shows "a i \<in> sigma A"
BB_pure: fixes x :: "'d" shows "pure x = B (B (pure x))"
adjuster_wit_carrier_vec: fixes f :: "'a vec"   and gs :: "'a vec list"   and mus :: "'a list" assumes "f \<in> carrier_vec n"   and "set gs \<subseteq> carrier_vec n" shows "snd (adjuster_wit mus f gs) \<in> carrier_vec n"
simps_2: fixes x :: "'d"   and \<theta> :: "('e, 'd) trm" shows "FVP (x := \<theta>) = FVT \<theta>"
dim_map: fixes C :: "'a set" assumes "domain.chamber C" shows "card (f ` C) = card C"
negchecks_model_def: fixes \<I> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and D :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) set"   and X :: "'a list"   and F :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and G :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list" shows "negchecks_model \<I> D X F G \<equiv> \<forall>\<delta>. subst_domain \<delta> = set X \<and> ground (subst_range \<delta>) \<longrightarrow> list_ex (\<lambda>f. fst f \<cdot> \<delta> \<circ>\<^sub>s \<I> \<noteq> snd f \<cdot> \<delta> \<circ>\<^sub>s \<I>) F \<or> list_ex (\<lambda>f. f \<cdot>\<^sub>p \<delta> \<circ>\<^sub>s \<I> \<notin> D) G"
CARAT_DERIV: fixes f :: "'a \<Rightarrow> 'a"   and l :: "'a"   and x :: "'a" shows "(f has_field_derivative l) (at x) = (\<exists>g. (\<forall>z. f z - f x = g z * (z - x)) \<and> isCont g x \<and> g x = l)"
Group_Img: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g" assumes "Group F"   and "Group G"   and "f \<in> gHom F G" shows "Group (Img\<^bsub>F,G\<^esub> f)"
snd_eq_Range: fixes R :: "('f \<times> 'e) set" shows "snd ` R = Range R"
path_connectedin_imp_connectedin: fixes X :: "'b topology"   and S :: "'b set" assumes "path_connectedin X S" shows "connectedin X S"
DiffE: fixes c :: "'e"   and A :: "'e set"   and B :: "'e set" assumes "c \<in> A - B" obtains "c \<in> A" and "c \<notin> B"
x_def: shows "x = [x\<lparr>0\<^sub>\<nat>\<rparr>, b]\<^sub>\<circ>"
mono_F_syn: fixes S :: "'c process"   and A :: "'c set"   and Q :: "'c process" shows "F (S \<lbrakk>A\<rbrakk> Q) = F (Q \<lbrakk>A\<rbrakk> S)"
R_G_trace_space_distr_eq: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and s :: "'s \<times> ('c \<Rightarrow> real)" assumes "cfg \<in> R_G.valid_cfg"   and "abss s = state cfg" shows "MDP.MC.T cfg = distr (MDP.MC.T (repcs s cfg)) (stream_space (count_space UNIV)) (smap absc)"
2: fixes g :: "'g"   and p :: "'val"   and q :: "'val"   and r :: "'val" assumes "notriv.phiArg g p q"   and "notriv.phiArg g p r"   and "distinct [p, q, r]"   and "p \<in> notriv.allVars g" shows "\<not> (uninst.def_dominates (uninst.phis'_all uses phis) g q p \<and> uninst.def_dominates (uninst.phis'_all uses phis) g r p)"
intros_4: fixes s :: "'b set"   and S :: "'b set set" assumes "s \<in> S" shows "generate_topology S s"
security_and_P1: fixes a1 :: "bool"   and a2 :: "bool"   and b1 :: "bool"   and b2 :: "bool"   and D :: "((bool \<times> bool) \<times> bool \<times> 'v_14_OT1) \<times> bool \<times> bool \<Rightarrow> bool spmf" shows "\<bar>spmf (and_funct (a1, a2) (b1, b2) \<bind> (\<lambda>(s1, s2). S1_and' (a1, a2) (b1, b2) s1 \<bind> D)) True - spmf (and_R1 (a1, a2) (b1, b2) \<bind> D) True\<bar> \<le> adv_14_OT"
hindrance_by_into_hindrance: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and \<epsilon> :: "ennreal" assumes "hindrance_by \<Gamma> f \<epsilon>" shows "hindrance \<Gamma> f"
le_fin_range: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list" assumes "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2" shows "set w\<^sub>1 \<subseteq> set w\<^sub>2"
statImpEnt: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b"   and \<Phi> :: "'c" assumes "AssertionStatImp \<Psi> \<Psi>'"   and "\<Psi> \<turnstile> \<Phi>" shows "\<Psi>' \<turnstile> \<Phi>"
scf_nonempty: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "is_pref_profile R" shows "scf R \<noteq> {}"
inf_top_right: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and top :: "'e"   and x :: "'e" assumes "class.bounded_semilattice_inf_top inf less_eq less top" shows "inf x top = x"
fl_dghm_surj_on_Hom: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"   and "b \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "\<FF>\<lparr>ArrMap\<rparr> `\<^sub>\<circ> Hom \<AA> a b = Hom \<BB> (\<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>) (\<FF>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>)"
is_shadow_root_ptr_kind_obtains: fixes ptr :: "(_) object_ptr" assumes "is_shadow_root_ptr_kind ptr" obtains shadow_root_ptr :: "(_) shadow_root_ptr" where "ptr = cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr"
continuous_vnm_def: fixes C :: "'b pmf set"   and P :: "('b pmf \<times> 'b pmf) set" shows "continuous_vnm C P = (\<forall>p\<in>C. \<forall>q\<in>C. \<forall>r\<in>C. p \<succeq>[P] q \<and> q \<succeq>[P] r \<longrightarrow> (\<exists>\<alpha>\<in>{0..1}. mix_pmf \<alpha> p r \<approx>[P] q))"
univ_for_pr_lm: fixes n :: "nat"   and x :: "nat" shows "univ_for_pr (c_pair n x) = nat_to_pr n x"
strict_trans2: fixes b :: "'c list"   and a :: "'c list"   and c :: "'c list" assumes "strict_suffix b a"   and "suffix c b" shows "strict_suffix c a"
exec: shows "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> Abrupt t"
thesis: fixes cs :: "'b set" assumes "finite cs"   and "(fst (f a), cs) \<in> group_by f as"   and "snd (f a) \<in> cs" shows "thesis"
p_def: shows "p \<equiv> fst \<circ> pq"
WF: fixes r :: "('c \<times> 'c) set" assumes "wo_rel r" shows "Wellfounded.wf (r - Id)"
simps_2: fixes x :: "'a"   and y :: "'a"   and p :: "'b" shows "srcNet (AllowPortFromTo x y p) = x"
map\<^sub>0_simps_2: fixes a :: "'a" assumes "B.obj a" shows "src (UP.map\<^sub>0 a) = UP.map\<^sub>0 a"
assms_2: shows "B \<equiv> fst (echelon_form_of_column_k bezout (A, i) k)"
subst_lsst_cons: fixes a :: "'a strand_label \<times> ('b, 'c) stateful_strand_step"   and A :: "('a strand_label \<times> ('b, 'c) stateful_strand_step) list"   and \<delta> :: "'c \<Rightarrow> ('b, 'c) Term.term" shows "a # A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta> = (a \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>) # (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>)"
sep_dom2_aux: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "((int \<times> int) set set, int) Combinators"   and y :: "((int \<times> int) set set, int) Combinators"   and z :: "((int \<times> int) set set, int) Combinators list" assumes "x \<in> dom (C (list2FWpolicy (a \<oplus> y # z)))" shows "x \<in> dom (C (a \<oplus> list2FWpolicy (y # z)))"
hyps_1: shows "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"
composition: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set"   and g :: "'e \<Rightarrow> 'f"   and C :: "'f set" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> C" shows "compose A g f \<in> A \<rightarrow> C"
flowpipe_nonneg_2: fixes X0 :: "('a \<times> 'a \<Rightarrow>\<^sub>L 'a) set"   and hl :: "real"   and hu :: "real"   and CX :: "('a \<times> 'a \<Rightarrow>\<^sub>L 'a) set"   and X1 :: "('a \<times> 'a \<Rightarrow>\<^sub>L 'a) set" assumes "flowpipe X0 hl hu CX X1" shows "hl \<le> hu"
keys_monom_of_set_finite: fixes X :: "nat set" assumes "finite X" shows "keys (monom_of_set X) = X"
unrestricte_edges: fixes g :: "'b ugraph"   and E :: "('b \<times> 'b) set" shows "edges (restrict_edges g E) \<subseteq> edges g"
X01: fixes h :: "complex \<Rightarrow> complex"   and z :: "complex" assumes "h \<in> X"   and "z \<in> S" shows "h z \<noteq> 0 \<and> h z \<noteq> 1"
IH: fixes t :: "pratt"   and p :: "nat"   and a :: "nat"   and x :: "nat" assumes "valid_cert ys"   and "t \<in> set ys" shows "(t = Prime p \<longrightarrow> prime p) \<and> (t = Triple p a x \<longrightarrow> (\<forall>q\<in>#prime_factorization x. [a ^ ((p - 1) div q) \<noteq> 1] (mod p)) \<and> 0 < x)"
assms_2: shows "b_e_type_checker \<C> [e] (t2s _> t3s)"
closed_segment_subset_interval: fixes T :: "'a set"   and a :: "'a"   and b :: "'a" assumes "is_interval T"   and "a \<in> T"   and "b \<in> T" shows "{a--b} \<subseteq> T"
weight_space_dim_def: shows "weight_space_dim = count_weights shared_weights (deep_model_l rs)"
discovered_reachable: shows "dom (discovered s) \<subseteq> reachable"
simps: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term" shows "wpo s t = (case s of Var x \<Rightarrow> (False, case t of Var xa \<Rightarrow> x = xa | Fun g ts \<Rightarrow> (s, t) \<in> UNIV \<and> \<sigma> (g, length ts) = [] \<and> prl (g, length ts)) | Fun f ss \<Rightarrow> if (s, t) \<in> {} then (True, True) else if (s, t) \<in> UNIV then if \<exists>i\<in>set (\<sigma> (f, length ss)). wpo_ns (ss ! i) t then (True, True) else case t of Var x \<Rightarrow> (False, False \<and> False) | Fun g ts \<Rightarrow> case prc (f, length ss) (g, length ts) of (prs, prns) \<Rightarrow> if prns \<and> (\<forall>j\<in>set (\<sigma> (g, length ts)). wpo_s s (ts ! j)) then if prs then (True, True) else let ss' = map ((!) ss) (\<sigma> (f, length ss)); ts' = map ((!) ts) (\<sigma> (g, length ts)); cf = c (f, length ss); cg = c (g, length ts) in if cf = Lex \<and> cg = Lex then lex_ext wpo n ss' ts' else if cf = Mul \<and> cg = Mul then mul_ext wpo ss' ts' else (length ss' \<noteq> 0 \<and> length ts' = 0, length ts' = 0) else (False, False) else (False, False))"
intros_1: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and t :: "state" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[], t\<rangle> \<Rightarrow> t"
bind_UNION: fixes A :: "'f set"   and f :: "'f \<Rightarrow> 'e set" shows "A \<bind> f = \<Union> (f ` A)"
invar_merge: fixes ts\<^sub>1 :: "'b tree list"   and ts\<^sub>2 :: "'b tree list" assumes "invar ts\<^sub>1"   and "invar ts\<^sub>2" shows "invar (merge ts\<^sub>1 ts\<^sub>2)"
static_ref_comp: shows "statically_complete_calculus Bot_F Inf_F (\<Turnstile>\<inter>\<G>) no_labels.Red_I_\<G> no_labels.Red_F_\<G>_empty"
t1_def: shows "tm_floatarith prec ord [compute_bound_tm prec I a tg] [mid (compute_bound_tm prec a a tg)] f = Some tf"
simps: fixes x :: "int"   and xs :: "int list" shows "collatz x xs = (if x \<le> 1 then rev (x # xs) else if x mod 2 = 0 then even_case x (x # xs) else odd_case x xs)"
valid_msg_def: fixes m :: "'challenge" shows "valid_msg m = (m \<in> challenge_space)"
preal_less_def: fixes r :: "preal"   and s :: "preal" shows "r < s \<equiv> Rep_preal r \<subset> Rep_preal s"
unbounded_def: fixes x :: "'b"   and R :: "('b \<Rightarrow> 'c) set" shows "unbounded x R \<equiv> \<forall>t. \<exists>u\<in>R. t < u x"
1: shows "D0 = insert v (C0 \<inter> D0)"
PiE_empty_domain: fixes T :: "'c \<Rightarrow> 'd set" shows "Pi\<^sub>E {} T = {\<lambda>x. undefined}"
commute: fixes a :: "'a"   and b :: "'a" shows "a AND b = b AND a"
order_refl: fixes x :: "'t \<Rightarrow>\<^sub>0 'c" shows "x \<preceq>\<^sub>p x"
pr_algo_measure_def: shows "pr_algo_measure \<equiv> \<lambda>(f, l). Max {length p |p. \<exists>aa ba. ((f, l), p, aa, ba) \<in> trcl pr_algo_lts}"
cSup_eq_maximum: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and X :: "'a set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "z \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> less_eq x z" shows "Sup X = z"
cobounded2: fixes b :: "'x \<Rightarrow>\<^sub>0 nat"   and a :: "'x \<Rightarrow>\<^sub>0 nat" shows "b \<preceq> ordered_powerprod_lin.max a b"
A_dim_col_not1: shows "dim_col A \<noteq> 1"
list_all2_nthD2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list"   and p :: "nat" assumes "list_all2 P xs ys"   and "p < length ys" shows "P (xs ! p) (ys ! p)"
elims: fixes x :: "'b gexp"   and y :: "nat" assumes "GExp_Lexorder.height x = y" obtains uu_ :: "bool" where "x = Bc uu_" and "y = 1"   | a1 :: "'b aexp" and a2 :: "'b aexp" where "x = Eq a1__ a2__" and "y = 1 + max (AExp_Lexorder.height a1__) (AExp_Lexorder.height a2__)"   | a1 :: "'b aexp" and a2 :: "'b aexp" where "x = Gt a1 a2" and "y = 1 + max (AExp_Lexorder.height a1) (AExp_Lexorder.height a2)"   | v :: "'b" and l :: "value list" where "x = In v l" and "y = 2 + length l"   | g1 :: "'b gexp" and g2 :: "'b gexp" where "x = Nor g1 g2" and "y = 1 + max (GExp_Lexorder.height g1) (GExp_Lexorder.height g2)"
emeasure_count_space_infinite: fixes X :: "'b set"   and A :: "'b set" assumes "X \<subseteq> A"   and "infinite X" shows "emeasure (count_space A) X = \<infinity>"
m_ac_2: fixes x :: "'a"   and y :: "'a" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "x * y = y * x"
impl_of: fixes x :: "('a, 'b) RBT.rbt" shows "impl_of x \<in> {t. is_rbt t}"
left_eq_app_right: fixes u :: "'a list"   and q :: "hf"   and v :: "'a list" assumes "u \<in> left_lang q"   and "v \<in> left_lang q" shows "(u, v) \<in> eq_app_right language"
Cons_4: fixes ts :: "nat" assumes "ts \<in> set (take (length (ts__ # tss)) (lin_ts_mmuaux aux))" shows "enat ts + right (args_ivl args) < enat dt"
linear_orderable_6_implies_3: fixes x :: "'a" assumes "linear_orderable_6 x" shows "linear_orderable_3 (x\<^sup>+)"
Diagonal_to_Smith_row_i_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and i :: "nat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b" shows "Diagonal_to_Smith_row_i A i bezout = diagonal_to_Smith_i [i + 1..<min (nrows A) (ncols A)] A i bezout"
homeomorphic_imp_perfect_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "homeomorphic_map X Y f" shows "perfect_map X Y f"
F_f_inv: fixes y :: "'index \<times> 'trap"   and x :: "'range" shows "y \<in> set_spmf I \<longrightarrow> x \<in> range (fst y) \<longrightarrow> F\<^sub>i\<^sub>n\<^sub>v (fst y) (snd y) (F (fst y) x) = x"
nat_pow_eq: fixes x :: "nat"   and k :: "nat" shows "x [^]\<^bsub>units_of R\<^esub> k = x ^ k mod p"
R_def: fixes \<V> :: "'b V_rec"   and Tr :: "'b list set" shows "R \<V> Tr \<equiv> \<forall>\<tau>\<in>Tr. \<exists>\<tau>'\<in>Tr. \<tau>' \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [] \<and> \<tau>' \<upharpoonleft> V\<^bsub>\<V>\<^esub> = \<tau> \<upharpoonleft> V\<^bsub>\<V>\<^esub>"
seqlsimp: fixes i :: "'e"   and P :: "'f \<times> 'g \<Rightarrow> bool"   and \<sigma> :: "'e \<Rightarrow> 'f"   and p :: "'g" shows "seql i P (\<sigma>, p) = P (\<sigma> i, p)"
eval_Cn2: shows "eval (Cn 2 r_result [Mn 2 (Cn 3 r_not [r_result]), recf.Id 2 0, recf.Id 2 1]) [i, x] \<down>"
subsetCE: fixes A :: "'c set"   and B :: "'c set"   and c :: "'c" assumes "A \<subseteq> B" obtains "c \<notin> A"   | "c \<in> B"
step_def: fixes s :: "'b list"   and r :: "'b"   and a :: "nat \<times> nat list" shows "step s r a = (let (k, sws) = a in mtf2 k r (swaps sws s))"
remove1_mset_eqE: fixes x1 :: "'b multiset"   and L :: "'b"   and M :: "'b multiset" assumes "remove1_mset L x1 = M" obtains "L \<in># x1" and "x1 = M + {#L#}"   | "L \<notin># x1" and "x1 = M"
Legendre_values: fixes p :: "int"   and q :: "int" shows "Legendre p q \<in> {1, - 1, 0}"
transitionWeakTransition: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs" assumes "P \<longmapsto>\<alpha> \<prec> P'" shows "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'"
orthogonal_extension: fixes S :: "'a set"   and T :: "'a set" assumes "pairwise orthogonal S" obtains U :: "'a set" where "pairwise orthogonal (S \<union> U)" and "span (S \<union> U) = span (S \<union> T)"
simps: fixes s :: "'d"   and f :: "'d \<Rightarrow> 'e" shows "put_state s (StateT f) = StateT (\<lambda>_. f s)"
converse_converse: fixes r :: "('h \<times> 'i) set" shows "(r\<inverse>)\<inverse> = r"
infsafety_closed_iff: fixes P :: "'a llist set" assumes "P \<subseteq> A\<^sup>\<omega>" shows "infsafety A P = (P closed)"
assms_2: fixes \<psi> :: "'fmla" assumes "\<psi> \<in> G" shows "nprv (insert \<psi> F) \<chi>"
quote_all_def: fixes p :: "perm"   and V :: "name set" shows "quote_all p V = {QuoteP (Var i) (Var (p \<bullet> i)) |i. i \<in> V}"
ideD_3: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "category C"   and "partial_magma.ide C a" shows "partial_magma.cod C a = a"
disj_def: fixes \<gamma> :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma>' :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool" shows "\<gamma> \<or>\<^sup>b \<gamma>' \<equiv> \<lambda>t n. \<gamma> t n \<or> \<gamma>' t n"