hcju/isabelleps huggingface dataset:
hor_vert_finite_2: shows "finite (two_chain_horizontal_boundary two_chain)"
reachable_nodes_steps'_1: fixes u :: "'b"   and g :: "'b ugraph"   and r :: "'b"   and v :: "'b" assumes "u \<in> reachable_nodes g r"   and "(u, v) \<in> (edges g)\<^sup>*" shows "v \<in> reachable_nodes g r"
map_disc_iff: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d list" shows "(map f a = []) = (a = [])"
bind_return_val'': fixes f :: "val \<Rightarrow> 'a measure"   and x :: "val"   and M :: "'a measure" assumes "f \<in> stock_measure (val_type x) \<rightarrow>\<^sub>M subprob_algebra M" shows "return_val x \<bind> f = f x"
reachable_weakenE: fixes s :: "'e"   and A :: "('e, 'f) automaton"   and P :: "'f \<Rightarrow> bool"   and Q :: "'f \<Rightarrow> bool" assumes "s \<in> reachable A P"   and "\<And>a. P a \<Longrightarrow> Q a" shows "s \<in> reachable A Q"
order_trans: fixes x :: "'a list"   and y :: "'a list"   and z :: "'a list" assumes "suffix x y"   and "suffix y z" shows "suffix x z"
euler_mascheroni_sum: shows "(\<lambda>n. inverse (of_nat (n + 1)) + of_real (ln (real (n + 1))) - of_real (ln (real (n + 2)))) sums euler_mascheroni"
order_iff: fixes b :: "'c"   and a :: "'c" shows "(b \<le> a) = (a = a \<squnion> b)"
restrict_def: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "restrict f A = (\<lambda>x. if x \<in> A then f x else undefined)"
sums_cong: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and c :: "'a" assumes "\<And>n. f n = g n" shows "f sums c = g sums c"
sigma_finite_countable: fixes M :: "'b measure" assumes "sigma_finite_measure M" shows "\<exists>A. countable A \<and> A \<subseteq> sets M \<and> \<Union> A = space M \<and> (\<forall>a\<in>A. emeasure M a \<noteq> \<infinity>)"
is_sig_redD1: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> nat"   and term_of_pair :: "'d \<times> nat \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and fs :: "('d \<Rightarrow>\<^sub>0 'e) list"   and sing_reg :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and top_tail :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and F :: "('c \<Rightarrow>\<^sub>0 'e) set"   and p :: "'c \<Rightarrow>\<^sub>0 'e" assumes "qpm_inf_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "qpm_inf_term.is_sig_red pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict fs sing_reg top_tail F p" shows "ord.is_le_rel ord_term ord_term_strict sing_reg"
deg_le_sect_pp_def: fixes xs :: "'x list"   and d :: "nat" shows "deg_le_sect_pp xs d = punit.keys_to_list (deg_le_sect_pp_aux xs d)"
inj: shows "inj_on (\<lambda>i. i + k1) {1..k2}"
assms: shows "\<turnstile>\<^sub>! (p \<^bold>\<and>\<^sub>! q)"
et_below_j: fixes j :: "'a"   and h :: "'a"   and g :: "'a" assumes "vector j"   and "regular j"   and "j \<noteq> bot" shows "selected_edge h j g * top \<le> j"
circ_reflexive: fixes x :: "'a" shows "reflexive (x\<^sup>\<circ>)"
nth_least_list_card: fixes n :: "nat"   and A :: "nat set" assumes "enat n \<le> esize A" shows "card {k \<in> A. k < nth_least (lift A) n} = n"
obs_singleton: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and n :: "'b"   and S :: "'b set" assumes "CFG_wf sourcenode targetnode kind valid_edge Entry Def Use state_val" shows "(\<exists>m. CFG.obs sourcenode targetnode valid_edge n (CFG_wf.wod_backward_slice sourcenode targetnode valid_edge Def Use S) = {m}) \<or> CFG.obs sourcenode targetnode valid_edge n (CFG_wf.wod_backward_slice sourcenode targetnode valid_edge Def Use S) = {}"
simps: fixes a1 :: "(name, value) fmap"   and a2 :: "sterm"   and a3 :: "value" shows "(a1 \<turnstile>\<^sub>v a2 \<down> a3) = ((\<exists>name \<Gamma> val. a1 = \<Gamma> \<and> a2 = Sconst name \<and> a3 = val \<and> name |\<notin>| C \<and> fmlookup \<Gamma> name = Some val) \<or> (\<exists>\<Gamma> name val. a1 = \<Gamma> \<and> a2 = Svar name \<and> a3 = val \<and> fmlookup \<Gamma> name = Some val) \<or> (\<exists>\<Gamma> cs. a1 = \<Gamma> \<and> a2 = Sabs cs \<and> a3 = Vabs cs \<Gamma>) \<or> (\<exists>\<Gamma> t cs \<Gamma>' u u' env uu_ rhs val. a1 = \<Gamma> \<and> a2 = t $\<^sub>s u \<and> a3 = val \<and> \<Gamma> \<turnstile>\<^sub>v t \<down> Vabs cs \<Gamma>' \<and> \<Gamma> \<turnstile>\<^sub>v u \<down> u' \<and> vfind_match cs u' = Some (env, uu_, rhs) \<and> \<Gamma>' ++\<^sub>f env \<turnstile>\<^sub>v rhs \<down> val) \<or> (\<exists>\<Gamma> t css name \<Gamma>' cs u u' env uv_ rhs val. a1 = \<Gamma> \<and> a2 = t $\<^sub>s u \<and> a3 = val \<and> \<Gamma> \<turnstile>\<^sub>v t \<down> Vrecabs css name \<Gamma>' \<and> fmlookup css name = Some cs \<and> \<Gamma> \<turnstile>\<^sub>v u \<down> u' \<and> vfind_match cs u' = Some (env, uv_, rhs) \<and> \<Gamma>' ++\<^sub>f mk_rec_env css \<Gamma>' ++\<^sub>f env \<turnstile>\<^sub>v rhs \<down> val) \<or> (\<exists>name \<Gamma> ts us. a1 = \<Gamma> \<and> a2 = name $$ ts \<and> a3 = Vconstr name us \<and> name |\<in>| C \<and> list_all2 (veval' \<Gamma>) ts us))"
unit_def: fixes A :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" shows "unit A m n = (\<forall>i j. m \<le> j \<longrightarrow> j < n \<longrightarrow> A i j = (if i = j then 1::'b else (0::'b)))"
exec_cons: fixes \<delta> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'e"   and s :: "'e"   and r :: "'f \<times> 'g"   and rs :: "('f \<times> 'g) list" shows "pre_RDR.exec \<delta> s (Cons r rs) = \<delta> (pre_RDR.exec \<delta> s rs) r"
le_Ord_diff: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "\<alpha> \<le> \<beta>"   and "Ord \<alpha>"   and "Ord \<beta>" obtains \<gamma> :: "V" where "\<alpha> + \<gamma> = \<beta>" and "\<gamma> \<le> \<beta>" and "Ord \<gamma>"
Pair_eqvt: fixes p :: "perm"   and x :: "'b"   and y :: "'c" shows "p \<bullet> (x, y) = (p \<bullet> x, p \<bullet> y)"
myUNIV_alle: fixes x :: "'a"   and y :: "'a" shows "lang (myUNIV x y) = {xs. set xs \<subseteq> {x, y}}"
module_hom_scale_self: fixes c :: "'a" shows "linear (*s) (*s) ((*s) c)"
cos_aform_err_def: fixes p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real" shows "cos_aform_err p X = range_reduce_aform_err p X \<bind> (\<lambda>X. let ivl = ivl_of_aform_err p X; l = lower ivl; u = upper ivl; L = floatarith.Num l; U = floatarith.Num u in if 0 \<le> l \<and> u \<le> lb_pi p then min_range_antimono p floatarith.Cos (\<lambda>x. floatarith.Minus (Sin x)) l u X else if ub_pi p \<le> l \<and> u \<le> 2 * lb_pi p then min_range_mono p floatarith.Cos (\<lambda>x. floatarith.Minus (Sin x)) l u X else Some (ivl_err (real_interval (cos_float_interval p ivl))))"
obs_upred_def: fixes P :: "'a upred" shows "\<lbrakk>P\<rbrakk>\<^sub>o = {b. \<lbrakk>P\<rbrakk>\<^sub>e b}"
set_restriction_def: fixes f :: "'b set \<Rightarrow> 'b set" shows "set_restriction f \<equiv> \<exists>P. \<forall>A. f A = {x \<in> A. P x}"
sgn_real_def: fixes a :: "real" shows "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
simple_extension_subfield_imp_algebraic: fixes K :: "'a set"   and x :: "'a" assumes "subring K R"   and "x \<in> carrier R"   and "subfield (simple_extension K x) R" shows "(algebraic over K) x"
globI: fixes n' :: "nat"   and \<gamma> :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool"   and t :: "nat \<Rightarrow> cnf" assumes "\<forall>n\<ge>n'. \<gamma> t n" shows "(\<box>\<^sub>c\<gamma>) t n'"
insertion_zero: fixes f :: "nat \<Rightarrow> 'b" shows "insertion f 0 = (0::'b)"
inv_closed: fixes x :: "'a" assumes "x \<in> carrier G" shows "\<ominus> x \<in> carrier G"
exhaust: fixes y :: "'b llist" obtains "y = \<bottom>"   | "y = lnil"   | a :: "'b" and llist :: "'b llist" where "y = a :@ llist"
ide_char: fixes a :: "'b \<times> 'b" shows "B.VV.ide a = (B.VV.arr a \<and> B.VxV.ide a)"
h2: shows "((\<Sum>i = 0..<deg + 1. f i / D ^ i) = 0) = ((\<Sum>i = 0..<deg + 1. f i * D ^ (deg - i)) = 0)"
hyps: shows "X = A ^^ Suc n @@ X \<union> (\<Union>m\<le>n. A ^^ m @@ B)"
nth_replicate: fixes i :: "nat"   and n :: "nat"   and x :: "'a" assumes "i < n" shows "replicate n x ! i = x"
comparables_subset: fixes S :: "nat set"   and M :: "nat set" shows "\<Union> (comparables S M) \<subseteq> S \<union> M"
IH: shows "[g ^ totient (p ^ (k - 1)) \<noteq> 1] (mod p ^ k)"
lens_indep_override_def: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e" assumes "vwb_lens X"   and "vwb_lens Y" shows "(X \<bowtie> Y) = (\<forall>s\<^sub>1 s\<^sub>2 s\<^sub>3. s\<^sub>1 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>2 \<triangleleft>\<^bsub>Y\<^esub> s\<^sub>3 = s\<^sub>1 \<triangleleft>\<^bsub>Y\<^esub> s\<^sub>3 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>2)"
meet_idem: fixes x :: "'a" shows "x \<sqinter> x = x"
getRel_dom_2: fixes G :: "('b, 'c) labeled_graph"   and a :: "'c"   and b :: "'c"   and l :: "'b" assumes "graph G"   and "(a, b) \<in> getRel l G" shows "b \<in> vertices G"
natural_transformation_axioms: shows "natural_transformation A B G F \<sigma>'"
S2: shows "log2 0 = (LEAST z. 0 < 2 ^ (z + 1))"
g_tendsto_2: shows "(g \<longlongrightarrow> 0) (at_right x)"
atm_iff_pos_or_neg_lit: fixes A :: "'b"   and L :: "'b literal multiset" shows "(A \<in> atms_of L) = (Pos A \<in># L \<or> Neg A \<in># L)"
monom_is_UP_monom_2: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "is_UP_monom (monom (UP R) a n)"
cosh_minus_sinh: fixes x :: "'a" shows "cosh x - sinh x = exp (- x)"
fv_asToHeap: fixes \<Gamma> :: "assn" shows "fv (asToHeap \<Gamma>) = fv \<Gamma>"
order_consistent_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "order_consistent r s = (\<forall>a a'. r a a' \<and> s a' a \<longrightarrow> a = a')"
impr_sup_bl: fixes a :: "'a"   and b :: "'a" shows "a r\<rightarrow> b \<squnion> (b r\<rightarrow> a) = (1::'a)"
Quotient_Gromov_completion: shows "Quotient Gromov_completion_rel abs_Gromov_completion rep_Gromov_completion cr_Gromov_completion"
mono_neutral_cong_right: fixes T :: "'c set"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = (0::'d)"   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "sum g T = sum h S"
SUP_identity_eq: fixes Sup :: "'a set \<Rightarrow> 'a"   and A :: "'a set" shows "Sup ((\<lambda>x. x) ` A) = Sup A"
special_soundness_def: shows "special_soundness \<equiv> \<forall>h e e' a z z'. h \<in> valid_pub \<longrightarrow> e \<in> challenge_space \<longrightarrow> e' \<in> challenge_space \<longrightarrow> e \<noteq> e' \<longrightarrow> check h a e z \<longrightarrow> check h a e' z' \<longrightarrow> lossless_spmf (\<A>ss h (a, e, z) (a, e', z')) \<and> (\<forall>w'\<in>set_spmf (\<A>ss h (a, e, z) (a, e', z')). (h, w') \<in> Rel)"
assms_1: shows "x \<in>\<^sub>\<circ> fid_on A"
composite_simps_2: shows "trg (\<pi> \<star> \<rho>) = trg \<pi>"
extf_wf_bounded: fixes gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and n :: "nat"   and f :: "'s" assumes "\<And>z. \<not> gt z z"   and "wfP (\<lambda>x y. gt y x)" shows "wfP (\<lambda>xs ys. length ys \<le> n \<and> length xs \<le> n \<and> extf f gt ys xs)"
exhaust: fixes y :: "ivl" obtains x1 :: "int option" and x2 :: "int option" where "y = I x1 x2"
limf: shows "(sum f \<longlongrightarrow> infsetsum' f A) (finite_subsets_at_top A)"
tauLaw2SimLeft: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" assumes "\<And>\<Psi> P. (\<Psi>, P, P) \<in> Rel"   and "\<And>\<Psi> P Q R S. \<lbrakk>\<Psi> \<rhd> P \<sim> Q; (\<Psi>, Q, R) \<in> Rel; \<Psi> \<rhd> R \<sim> S\<rbrakk> \<Longrightarrow> (\<Psi>, P, S) \<in> Rel" shows "\<Psi> \<rhd> P \<oplus>\<^sub>\<top> \<tau>.P \<leadsto><Rel> \<tau>.P"
map_option_cong: fixes x :: "'c option"   and y :: "'c option"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "x = y"   and "\<And>a. y = Some a \<Longrightarrow> f a = g a" shows "map_option f x = map_option g y"
rev_swap_def: fixes M :: "'c pre_map"   and a :: "'c"   and b :: "'c" shows "pre_digraph_map.rev_swap M a b = \<lparr>edge_rev = perm_swap a b (edge_rev M), edge_succ = perm_swap a b (edge_succ M)\<rparr>"
Re_exp: fixes z :: "complex" shows "Re (exp z) = exp (Re z) * cos (Im z)"
PPs_Int: fixes X :: "'a set"   and Y :: "'a set" shows ".[X \<inter> Y] = .[X] \<inter> .[Y]"
simps_1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat" assumes "class.linordered_ab_monoid_add plus less_eq less neutral" shows "linordered_ab_monoid_add.fw plus less_eq m n 0 0 0 = linordered_ab_monoid_add.fw_upd plus less_eq m 0 0 0"
filterlim_at_top_mono: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "filterlim f at_top F"   and "\<forall>\<^sub>F x in F. f x \<le> g x" shows "filterlim g at_top F"
simple_function_finite: fixes f :: "'key \<times> 'message list \<Rightarrow> 'a" shows "simple_function \<mu> f"
new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def: fixes h :: "(_) heap" shows "new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (let new_element_ptr = element_ptr.Ref (Suc (fMax (finsert 0 (element_ptr.the_ref |`| element_ptrs h)))) in (new_element_ptr, put new_element_ptr (create_element_obj [] [] fmempty None) h))"
prv_tms_suc_var: shows "prv (eql (tms (Var xx) (suc (Var yy))) (pls (tms (Var xx) (Var yy)) (Var xx)))"
nprv_clear3_2: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and \<phi>1 :: "'c"   and \<phi>3 :: "'c"   and \<psi> :: "'c"   and \<phi>2 :: "'c" assumes "Deduct_with_False_Disj var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv"   and "Deduct_with_False_Disj.nprv cnj imp fls prv {\<phi>1, \<phi>3} \<psi>"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<phi>3 \<in> fmla"   and "\<psi> \<in> fmla" shows "Deduct_with_False_Disj.nprv cnj imp fls prv {\<phi>1, \<phi>2, \<phi>3} \<psi>"
HAStates_CompFun_SAs_mem: fixes S :: "'d"   and A :: "('d, 'e, 'f) hierauto"   and T :: "('d, 'e, 'f) seqauto" assumes "S \<in> HAStates A"   and "T \<in> the (CompFun A S)" shows "T \<in> SAs A"
t_ins_invariance: fixes Y :: "'b t_type"   and X :: "'b t_type"   and x :: "'b"   and xt :: "'b bintree" assumes "Y \<in> t_ins_set X"   and "t_ins_inv x xt X" shows "t_ins_inv x xt Y"
simps_2: fixes X :: "'a event set"   and y :: "'a event"   and t :: "'a event list" shows "setinterleaving ([], X, y # t) = (if y \<in> X then {} else {y # u |u. u setinterleaves (([], t), X)})"
lookup_def: fixes return :: "int \<Rightarrow> 'b"   and fail :: "'b"   and get :: "(('c \<Rightarrow> int option) \<Rightarrow> 'b) \<Rightarrow> 'b"   and x :: "'c" shows "memo_exp_base.lookup return fail get x = get (\<lambda>s. case s x of None \<Rightarrow> fail | Some y \<Rightarrow> return y)"
msg_nGet2: fixes n :: "'a"   and nReturn :: "'a \<Rightarrow> nat \<Rightarrow> 'b Frame list"   and recv :: "nat \<Rightarrow> 'b Frame list"   and nC :: "'a \<Rightarrow> Config"   and nStore :: "'a \<Rightarrow> nat \<Rightarrow> 'b Frame list"   and nSend :: "'a \<Rightarrow> nat \<Rightarrow> 'b Frame list"   and nGet :: "'a \<Rightarrow> nat \<Rightarrow> nat list"   and k :: "'a" assumes "\<forall>i<n. FlexRayController (nReturn i) recv (nC i) (nStore i) (nSend i) (nGet i)"   and "k < n" shows "msg (Suc 0) (nGet k)"
less_add_same_cancel2: fixes a :: "'a multiset"   and b :: "'a multiset" shows "(a \<subset># b + a) = (\<Empt> \<subset># b)"
select_first_unique: fixes x :: "'a"   and p :: "'a \<Rightarrow> real"   and y1 :: "'a"   and y2 :: "'a" assumes "select_first x p y1"   and "select_first x p y2" shows "y1 = y2"
lm068: fixes x :: "'c"   and f :: "('c \<times> 'd) set"   and X :: "'c set" assumes "x \<in> Domain (f || X)" shows "(f || X) `` {x} = f `` {x}"
add_block_wf_cond: fixes b :: "'a set" assumes "b \<subseteq> bl" shows "incidence_system bl (derived_incidence_sys.add_block b)"
j_xs: shows "\<forall>j\<in>set [i + 1..<min (dim_row A) (dim_col A)]. j < min CARD('nr) CARD('nc)"
cltn2_image_def: fixes P :: "proj2 set"   and A :: "cltn2" shows "cltn2_image P A \<equiv> {apply_cltn2 p A |p. p \<in> P}"
cases: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "('a, 'b, 'c) configuration"   and a2 :: "('a, 'b, 'c) event list"   and a3 :: "('a, 'b, 'c) configuration" assumes "distributed_system channel"   and "distributed_system.trace channel trans send recv a1 a2 a3" obtains c :: "('a, 'b, 'c) configuration" where "a1 = c__" and "a2 = []" and "a3 = c__"   | c :: "('a, 'b, 'c) configuration" and ev :: "('a, 'b, 'c) event" and c' :: "('a, 'b, 'c) configuration" and t :: "('a, 'b, 'c) event list" and c'' :: "('a, 'b, 'c) configuration" where "a1 = c" and "a2 = ev # t" and "a3 = c''" and "distributed_system.next channel trans send recv c ev c'" and "distributed_system.trace channel trans send recv c' t c''"
partition_Image_element: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and X :: "'a set"   and x :: "'a" assumes "equiv A r"   and "X \<in> A // r"   and "x \<in> X" shows "r `` {x} = X"
flip: shows "bounded_bilinear (\<lambda>x y. blinfun_apply y x)"
upper_def: shows "upper \<equiv> \<lambda>n. {x. 0 \<le> x n}"
simps_97: fixes x101 :: "stmt"   and x102 :: "stmt"   and x51 :: "expr"   and x52 :: "stmt"   and x53 :: "stmt" shows "x101 Finally x102 \<noteq> If(x51) x52 Else x53"
l_coset_subset_G: fixes H :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "x <#\<^bsub>add_monoid G\<^esub> H \<subseteq> carrier G"
add_less_cancel_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less (plus c a) (plus c b) = less a b"
intros_2: fixes xs :: "'c llist"   and ys :: "'c llist"   and x :: "'c" assumes "xs \<sqsubseteq> ys" shows "LCons x xs \<sqsubseteq> LCons x ys"
emeasure_compl: fixes s :: "'c set"   and M :: "'c measure" assumes "s \<in> sets M"   and "emeasure M s \<noteq> \<infinity>" shows "emeasure M (space M - s) = emeasure M (space M) - emeasure M s"
is_Pure_def: fixes generat :: "('g, 'h, 'i) generat" shows "is_Pure generat = (\<exists>x1. generat = Pure x1)"
ball_cong: fixes A :: "'b set"   and B :: "'b set"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> P x = Q x" shows "(\<forall>x\<in>A. P x) = (\<forall>x\<in>B. Q x)"
extension_valid: fixes v :: "'a" assumes "v \<rightarrow> v0" shows "valid_path (LCons v P)"
mbuf2t_take_add'_1: fixes f :: "'b option list set \<Rightarrow> 'b option list set \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and xs :: "'b option list set list"   and ys :: "'b option list set list"   and buf :: "'b option list set list \<times> 'b option list set list"   and nts :: "nat list"   and z' :: "'a"   and buf' :: "'b option list set list \<times> 'b option list set list"   and nts' :: "nat list"   and \<sigma> :: "(char list \<times> 'b list) trace"   and j :: "nat"   and n :: "nat"   and R :: "'b list set"   and \<phi> :: "'b MFOTL.formula"   and \<psi> :: "'b MFOTL.formula"   and j' :: "nat" assumes "mbuf2t_take f z (mbuf2_add xs ys buf) nts = (z', buf', nts')"   and "wf_mbuf2' \<sigma> j n R \<phi> \<psi> buf"   and "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [min (Monitor.progress \<sigma> \<phi> j) (Monitor.progress \<sigma> \<psi> j)..<j'] nts"   and "list_all2 (\<lambda>i. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) i \<phi>)) [Monitor.progress \<sigma> \<phi> j..<Monitor.progress \<sigma> \<phi> j'] xs"   and "list_all2 (\<lambda>i. qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) i \<psi>)) [Monitor.progress \<sigma> \<psi> j..<Monitor.progress \<sigma> \<psi> j'] ys"   and "j \<le> j'" shows "wf_mbuf2' \<sigma> j' n R \<phi> \<psi> buf'"
filter_min_subset: fixes rel :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" shows "set (filter_min rel xs) \<subseteq> set xs"
is_aligned_weaken: fixes w :: "'a word"   and x :: "nat"   and y :: "nat" assumes "is_aligned w x"   and "y \<le> x" shows "is_aligned w y"
cinner_scaleC_left: fixes r :: "complex"   and x :: "'a"   and y :: "'a" shows "cinner (r *\<^sub>C x) y = cnj r * cinner x y"
set2_F_map: fixes l1 :: "'p \<Rightarrow> 'aa"   and l2 :: "'q \<Rightarrow> 'z"   and l3 :: "'r \<Rightarrow> 'ab" shows "set2_F \<circ> mapl_F l1 l2 l3 = (`) l2 \<circ> set2_F"
yinfrontofz: shows "cs ! i < z in s_TS init h \<sigma> (length (as @ [x] @ bs @ [x]) + i + 1)"
adapt_pre_def: fixes P :: "'b assn"   and Q :: "'b assn"   and Q' :: "'b assn" shows "adapt_pre P Q Q' = (\<lambda>Y s Z. \<forall>Y' s'. \<exists>Z'. P Y s Z' \<and> (Q Y' s' Z' \<longrightarrow> Q' Y' s' Z))"
pmod: fixes i :: "nat" assumes "3 \<le> i"   and "i \<le> Suc k" shows "Polynomial.smult (\<beta> i) (F i) = pseudo_mod (F (i - 2)) (F (i - 1))"
ring_of_setsI: fixes M :: "'c set set"   and \<Omega> :: "'c set" assumes "M \<subseteq> Pow \<Omega>"   and "{} \<in> M"   and "\<And>a b. \<lbrakk>a \<in> M; b \<in> M\<rbrakk> \<Longrightarrow> a \<union> b \<in> M"   and "\<And>a b. \<lbrakk>a \<in> M; b \<in> M\<rbrakk> \<Longrightarrow> a - b \<in> M" shows "ring_of_sets \<Omega> M"
simps: fixes p :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b list"   and x :: "'b"   and ws :: "'b list"   and Z :: "'b set" shows "While_Combinator.rtrancl_while_invariant p f x (ws, Z) = (x \<in> Z \<and> set ws \<subseteq> Z \<and> distinct ws \<and> {(x, y). y \<in> set (f x)} `` (Z - set ws) \<subseteq> Z \<and> Z \<subseteq> {(x, y). y \<in> set (f x)}\<^sup>* `` {x} \<and> Ball (Z - set ws) p)"
ha: shows "\<forall>b\<in>A. h b = h a"
assms_6: shows "InvariantWatchListsUniq (getWatchList state)"
1: shows "M_mat (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2) $$ (i, j) = z (subsets_smash n1 subs1 subs2 ! i) (signs_smash signs1 signs2 ! j)"
thunk: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and e :: "exp"   and S :: "stack_elem list" assumes "map_of \<Gamma> x = Some e"   and "\<not> isVal e"   and "I (\<Gamma>, Var x, S)" shows "P (\<Gamma>, Var x, S) (delete x \<Gamma>, e, Upd x # S)"
1: shows "\<forall>x. are_complementary (d x) (ad x)"
FG_constituents_def: shows "FG_constituents \<equiv> SOME Us. (\<forall>U\<in>set Us. IrrFinGroupRepresentation G (*) U) \<and> 0 \<notin> set Us \<and> group_ring = (\<Oplus>U\<leftarrow>Us. U)"