hcju/isabelleps huggingface dataset:
tyenv_wellformed_def: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and mds :: "Mode \<Rightarrow> 'a set"   and \<Gamma> :: "'a \<Rightarrow> 'd set option"   and \<S> :: "'a set \<times> 'a set"   and P :: "'d set" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False" shows "sifum_types.tyenv_wellformed bexp_vars \<C>_vars mds \<Gamma> \<S> P \<equiv> sifum_types.mds_consistent bexp_vars \<C>_vars mds \<Gamma> \<S> P \<and> sifum_types.types_wellformed bexp_vars \<C>_vars \<Gamma> \<and> sifum_types.types_stable bexp_vars \<Gamma> \<S>"
oracle\<^sub>1_def: fixes b :: "bool"   and \<eta> :: "nat" shows "oracle\<^sub>1 b \<eta> = ekey_oracle \<eta> \<oplus>\<^sub>O (decrypt_oracle \<eta> \<oplus>\<^sub>O encrypt_oracle b \<eta>)"
privateKey_image_eq: fixes A :: "agent"   and AS :: "agent set" shows "(priK A \<in> priK ` AS) = (A \<in> AS)"
normf_def: fixes f :: "'a \<Rightarrow> real" shows "normf f \<equiv> SUP x\<in>S. \<bar>f x\<bar>"
source_target_relation_with_trans_impl_full_abstraction: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set" assumes "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel"   and "trans (Rel \<union> _Collect (P, Q) (\<exists>S. \<lbrakk>S\<rbrakk> \<in>T P \<and> S \<in>S Q))" shows "\<forall>S1 S2. ((S1, S2) \<in> _Collect (S1, S2) ((SourceTerm S1, SourceTerm S2) \<in> Rel)) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> _Collect (T1, T2) ((TargetTerm T1, TargetTerm T2) \<in> Rel))"
divisor_count_primorial_gt: fixes \<epsilon> :: "real" assumes "0 < \<epsilon>" shows "\<forall>\<^sub>F x in at_top. 2 powr ((1 - \<epsilon>) * ln (real (primorial x)) / ln (ln (real (primorial x)))) < real (divisor_count (primorial x))"
intro: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and sizes :: "int set" assumes "proper_design point_set block_collection"   and "K_block_design_axioms block_collection sizes" shows "K_block_design point_set block_collection sizes"
p_pow_rep0: fixes n :: "nat" shows "[p] \<cdot> \<one> [^] n = [(p ^ n)] \<cdot> \<one>"
rev_finite_subset: fixes B :: "'b set"   and A :: "'b set" assumes "finite B"   and "A \<subseteq> B" shows "finite A"
lappend_eq_LNil_iff: fixes xs :: "'c llist"   and ys :: "'c llist" shows "(lappend xs ys = LNil) = (xs = LNil \<and> ys = LNil)"
leg_simps_1: shows "C.arr \<mu>.cod.leg0"
first_safe_dist_def: fixes v\<^sub>e :: "real"   and a\<^sub>e :: "real" shows "first_safe_dist v\<^sub>e a\<^sub>e \<equiv> rel_dist_to_stop v\<^sub>e a\<^sub>e"
U2: shows "U2 = q.Mp (sdiv_poly (S * D' + T * H' - 1) p)"
oZero_def: shows "oZero = Abs_ordinal (ord0rel `` {ord0_Zero})"
connected_conv: fixes x :: "'a" shows "connected x = connected (x\<^sup>T)"
e1_typ: shows "CT;Map.empty \<turnstile> e1 : C1"
simps: fixes x :: "nat" shows "lg x = (if x \<le> 1 then 0 else 1 + lg (x div 2))"
assms_1: shows "{x \<in> space M. \<not> P x} \<subseteq> N"
rel_of_pred_def: fixes s :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_of_pred s \<equiv> {(a, b). s a b}"
assms_1: fixes v :: "'a vec" assumes "v \<in> set ls" shows "dim_vec v = n"
ex_un_null: shows "\<exists>!n. \<forall>f. n \<cdot>\<acute> f = n \<and> f \<cdot>\<acute> n = n"
exhaust: fixes y :: "dir" obtains "y = L"   | "y = R"
prems_3: shows "(monomial (1::'a) s, V) \<in> set (fst (ps0 @ ps1, qs0 @ qs1))"
r_amalgamate_def: shows "r_amalgamate \<equiv> Cn 2 (r_smn 1 2) [r_dummy 1 (r_const (encode r_amalgamation)), recf.Id 2 0, recf.Id 2 1]"
a_ne: fixes j :: "nat" shows "a j \<noteq> {}"
not_ccw'_eq: fixes t :: "real \<times> real"   and p :: "real \<times> real"   and s :: "real \<times> real" shows "(\<not> ccw' t p s) = (ccw' t s p \<or> coll t s p)"
wf_trms_subst_compose: fixes \<theta> :: "'b \<Rightarrow> ('fun, 'b) Term.term"   and \<delta> :: "'b \<Rightarrow> ('fun, 'b) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range (\<theta> \<circ>\<^sub>s \<delta>))"
add_block_index_not_in: fixes ps :: "'a set"   and b :: "'a set" assumes "\<not> ps \<subseteq> b" shows "add_block b index ps = \<B> index ps"
map_of_SomeD: fixes xs :: "('b \<times> 'a) list"   and k :: "'b"   and y :: "'a" assumes "Map.map_of xs k = Some y" shows "(k, y) \<in> set xs"
Trans: fixes A :: "'d set"   and B :: "'d set"   and CC :: "'d set set"   and F :: "'d program"   and C :: "'d set" assumes "(A, B) \<in> elt CC F"   and "(B, C) \<in> elt CC F" shows "(A, C) \<in> elt CC F"
isSimplePath_def: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" shows "isSimplePath u p v \<equiv> isPath u p v \<and> distinct (pathVertices u p)"
simple_distributed_nonneg: fixes M :: "'c measure"   and X :: "'c \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> real"   and x :: "'d" assumes "simple_distributed M X f" shows "0 \<le> f x"
less_power_add_imp_div_less: fixes i :: "nat"   and m :: "nat"   and n :: "nat" assumes "i < 2 ^ (m + n)" shows "i div 2 ^ n < 2 ^ m"
mult_mat_of_row_vec_of_scal: fixes v :: "'b vec"   and x :: "'b" shows "mat_of_col v *\<^sub>v vec_of_scal x = x \<cdot>\<^sub>v v"
step: fixes lc :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and c' :: "'Com"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and c'' :: "'Com"   and mds'' :: "Mode \<Rightarrow> 'Var set"   and mem'' :: "'Var \<Rightarrow> 'Val" assumes "loc_reachp lc c' mds' mem'"   and "\<langle>c', mds', mem'\<rangle> \<leadsto> \<langle>c'', mds'', mem''\<rangle>" shows "loc_reachp lc c'' mds'' mem''"
vcomp_char: shows "(\<bullet>) = (\<lambda>\<nu> \<mu>. if seq \<nu> \<mu> then \<lparr>Chn = Chn \<nu> \<cdot> Chn \<mu>, Dom = Dom \<mu>, Cod = Cod \<nu>\<rparr> else null)"
col_space_eq: fixes A :: "(('b, 'd) vec, 'c) vec" shows "col_space A = {y. \<exists>x. A *v x = y}"
select_convs_1: fixes c_records :: "'c zmultiset"   and c_temp :: "'d \<Rightarrow> 'c zmultiset"   and c_msg :: "'d \<Rightarrow> 'd \<Rightarrow> 'c zmultiset list"   and c_glob :: "'d \<Rightarrow> 'c zmultiset"   and more :: "'e" shows "c_records \<lparr>c_records = c_records, c_temp = c_temp, c_msg = c_msg, c_glob = c_glob, \<dots> = more\<rparr> = c_records"
connected_closedin: fixes S :: "'c set" shows "connected S = (\<nexists>E1 E2. closedin (top_of_set S) E1 \<and> closedin (top_of_set S) E2 \<and> S \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
remove_nth_id: fixes xs :: "'b list"   and n :: "nat" assumes "length xs \<le> n" shows "remove_nth n xs = xs"
prod_cases3: fixes y :: "'d \<times> 'e \<times> 'f" obtains a :: "'d" and b :: "'e" and c :: "'f" where "y = (a, b, c)"
nip_eq: fixes x :: "i"   and y :: "i" assumes "M2" shows "PP x y = (P x y \<and> x \<noteq> y)"
LI_preserves_welltypedness_single: fixes S :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S' :: "('fun, 'var) strand_step list"   and \<theta>' :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "(S, \<theta>) \<leadsto> (S', \<theta>')"   and "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S \<theta>"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)"   and "tfr\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>s\<^sub>t S)"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>t S)"   and "list_all tfr\<^sub>s\<^sub>t\<^sub>p S" shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>' \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>')"
awalk_conv: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" shows "G.awalk u p v = (set (G.awalk_verts u p) \<subseteq> verts G \<and> set p \<subseteq> arcs G \<and> G.awhd u p = u \<and> G.awlast u p = v \<and> G.cas u p v)"
extract_var_non_zero_coeff: fixes p :: "'b mpoly"   and v :: "nat" shows "extract_var p v = (\<Sum>m\<in>{m'. coeff p m' \<noteq> (0::'b)}. monom (remove_key v m) (monom (Poly_Mapping.single v (lookup m v)) (coeff p m)))"
ceiling_less_iff: fixes x :: "'b"   and z :: "int" shows "(\<lceil>x\<rceil> < z) = (x \<le> of_int z - (1::'b))"
hyps: fixes x2a :: "('b, 'c) Term.term" assumes "x2a \<in> set ts" shows "vars_term_ms (x2a \<cdot> \<sigma>) = (\<Sum>x\<in>#vars_term_ms x2a. vars_term_ms (\<sigma> x))"
split_image_mset: fixes M :: "'b multiset"   and M1 :: "'b multiset"   and M2 :: "'b multiset"   and f :: "'b \<Rightarrow> 'c" assumes "M = M1 + M2" shows "image_mset f M = image_mset f M1 + image_mset f M2"
while_C_def: fixes c :: "'b control_state_scheme \<Rightarrow> bool"   and B :: "'b control_state_scheme \<Rightarrow> (unit \<times> 'b control_state_scheme) option" shows "_while_C c B \<equiv> \<lambda>\<sigma>. if exec_stop \<sigma> then Some ((), \<sigma>) else (_while_SE (\<lambda>\<sigma>. \<not> exec_stop \<sigma> \<and> c \<sigma>) B ;- unset_break_status) \<sigma>"
conseq: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and P' :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and Q' :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3 {P} c { Q}"   and "\<And>s. P' s \<Longrightarrow> P s"   and "\<And>s. Q s \<Longrightarrow> Q' s" shows "\<turnstile>\<^sub>3 {P'} c { Q'}"
Progress: 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 E :: "char list \<Rightarrow> ty option"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e :: "expr"   and T :: "ty" assumes "wf_C_prog P"   and "P,E,s \<turnstile> e : T \<surd>"   and "\<D> e \<lfloor>dom (lcl s)\<rfloor>"   and "\<not> final e" shows "\<exists>e' s'. P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>"
fv_in_subst_img: fixes s :: "'b \<Rightarrow> ('a, 'b) Term.term"   and v :: "'b"   and t :: "('a, 'b) Term.term" assumes "s v = t"   and "t \<noteq> Var v" shows "fv t \<subseteq> range_vars s"
expand: fixes prod :: "'b \<times> 'c"   and prod' :: "'b \<times> 'c" assumes "fst prod = fst prod' \<and> snd prod = snd prod'" shows "prod = prod'"
simps_4: fixes m :: "nat"   and r :: "mregex"   and s :: "mregex" shows "ok m (MPlus r s) = (ok m r \<and> ok m s)"
g_3: shows "prob_space (density M (\<lambda>x. ennreal (g x)))"
length_mat_to_polys: fixes ts :: "'c list"   and A :: "'d mat" shows "length (mat_to_polys ts A) = dim_row A"
finite_supp_B: shows "finite (supp {p \<bullet> x |p. supp (F, P) \<sharp>* p})"
urD_4: shows "root_cond (p, l, r) (real_of_1 (p, l, r))"
permutation_inverse_works_1: fixes p :: "'a \<Rightarrow> 'a" assumes "permutation p" shows "inv p \<circ> p = id"
space_lborel_f: fixes n :: "nat" shows "space (lborel_f n) = {..<n} \<rightarrow>\<^sub>E UNIV"
valid_before: shows "valid_mmsaux args cur (nt, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since) auxlist"
restr_stack_noop: fixes V :: "Vars.var set"   and S :: "stack_elem list" shows "(restr_stack V S = S) = (upds S \<subseteq> V)"
C_3: shows "C = [] \<or> (\<exists>n t. suffix [(n, receive\<langle>t\<rangle>\<^sub>s\<^sub>t)] C)"
invimage_set_def: fixes r :: "V"   and A :: "V" shows "r -`\<^sub>\<circ> A = set {a. \<exists>b\<in>elts A. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r}"
euler_step_flowpipe: fixes X0 :: "'c set"   and h :: "real" assumes "wd TYPE('c)" shows "euler_step X0 h \<le> SPEC (\<lambda>(h', uu_, RES_ivl, RES). 0 < h' \<and> h' \<le> h \<and> flowpipe0 X0 h' h' RES_ivl RES)"
continuous_map_product_projection: fixes k :: "'c"   and I :: "'c set"   and X :: "'c \<Rightarrow> 'd topology" assumes "k \<in> I" shows "continuous_map (product_topology X I) (X k) (\<lambda>x. x k)"
chainsD: fixes c :: "'b set set"   and S :: "'b set set"   and x :: "'b set"   and y :: "'b set" assumes "c \<in> chains S"   and "x \<in> c"   and "y \<in> c" shows "x \<subseteq> y \<or> y \<subseteq> x"
le_imp_less_or_eq: fixes x :: "'b"   and y :: "'b" assumes "x \<le> y" shows "x < y \<or> x = y"
Subobjs_subclsR: 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 Cs :: "char list list"   and D :: "char list"   and D' :: "char list"   and Cs' :: "char list list" assumes "Subobjs P C (Cs @ [D, D'] @ Cs')" shows "P \<turnstile> D \<prec>\<^sub>R D'"
freeword_funlift_Abs_freeletter: fixes f :: "'c \<Rightarrow> 'b"   and s :: "'c" shows "freeword_funlift f (Abs_freeletter s) = f s"
trace_parallel_plan_strips_none_if: fixes \<Pi> :: "'b strips_problem"   and \<pi> :: "'b strips_operator list list"   and k :: "nat"   and v :: "'b" assumes "is_valid_problem_strips \<Pi>"   and "is_parallel_solution_for_problem \<Pi> \<pi>"   and "k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>)" shows "((trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi> ! k) v = None) = (v \<notin> set (\<Pi>\<^sub>\<V>))"
m_comm: fixes x :: "'a list"   and y :: "'a list" assumes "x \<in> carrier (A [X])"   and "y \<in> carrier (A [X])" shows "x \<otimes>\<^bsub>A [X]\<^esub> y = y \<otimes>\<^bsub>A [X]\<^esub> x"
intro: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V" assumes "\<exists>\<FF>. \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<AA> \<approx>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
Res: fixes P :: "pi"   and Q :: "pi"   and x :: "name" assumes "(P, Q) \<in> {(resChain lst (<\<nu>x>(P \<parallel> Q)), resChain lst (P \<parallel> <\<nu>x>Q)) |lst x P Q. x \<sharp> P} \<union> {(resChain lst (P \<parallel> <\<nu>x>Q), resChain lst (<\<nu>x>(P \<parallel> Q))) |lst x P Q. x \<sharp> P}" shows "(<\<nu>x>P, <\<nu>x>Q) \<in> {(resChain lst (<\<nu>x>(P \<parallel> Q)), resChain lst (P \<parallel> <\<nu>x>Q)) |lst x P Q. x \<sharp> P} \<union> {(resChain lst (P \<parallel> <\<nu>x>Q), resChain lst (<\<nu>x>(P \<parallel> Q))) |lst x P Q. x \<sharp> P}"
simps_3: fixes v :: "nat \<Rightarrow> 'a"   and l :: "linear_poly"   and r :: "rat" shows "v \<Turnstile>\<^sub>c LEQ l r = (l \<lbrace> v \<rbrace> \<le> r *R (1::'a))"
adds_pp_triv: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and t :: "'d"   and v :: "'c" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.adds_pp pair_of_term t (term_powerprod.splus pair_of_term term_of_pair t v)"
assms: shows "(C, D) \<in> sc \<Gamma> [] \<Delta> []"
sum: fixes a :: "real"   and g :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "a *\<^sub>R sum g A = (\<Sum>x\<in>A. a *\<^sub>R g x)"
non_zero_neighbour_pole: fixes f :: "'b \<Rightarrow> 'c"   and z :: "'b" assumes "is_pole f z" shows "\<forall>\<^sub>F w in at z. f w \<noteq> (0::'c)"
crename_OrL: fixes R :: "trm"   and a :: "coname"   and b :: "coname"   and x :: "name"   and M :: "trm"   and y :: "name"   and N :: "trm"   and z :: "name" assumes "R[a\<turnstile>c>b] = OrL x.M y.N z"   and "x \<sharp> (y, z, N, R)"   and "y \<sharp> (x, z, M, R)" shows "\<exists>M' N'. R = OrL x.M' y.N' z \<and> M'[a\<turnstile>c>b] = M \<and> N'[a\<turnstile>c>b] = N \<and> x \<sharp> N' \<and> y \<sharp> M'"
rsquarefree_def: fixes p :: "'c poly" shows "rsquarefree p = (p \<noteq> 0 \<and> (\<forall>a. order a p = 0 \<or> order a p = 1))"
cases: fixes x :: "variable" obtains n :: "String.literal" where "x = Free na__"   | n :: "String.literal" and uu_ :: "int" where "x = Var (n, uu_)"
UnionE: fixes A :: "'b"   and C :: "'b set set" assumes "(A::'b) \<in> \<Union> (C::'b set set)" obtains X :: "'b set" where "(A::'b) \<in> (X::'b set)" and "(X::'b set) \<in> (C::'b set set)"
set_dropWhileD: fixes x :: "'d"   and P :: "'d \<Rightarrow> bool"   and xs :: "'d list" assumes "x \<in> set (dropWhile P xs)" shows "x \<in> set xs"
is_tm_dghm_axioms: shows "\<HH> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<CC>"
common_sudiv_exists_comm: fixes C1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and C2 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" shows "common_sudiv_exists C1 C2 = common_sudiv_exists C2 C1"
iso_subset: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and Q :: "'b set" assumes "OrderingSetIso less_eq less less_eq' less' P f"   and "order.greater_eq P Q" shows "OrderingSetIso less_eq less less_eq' less' Q f"
logic_actual: fixes \<phi> :: "\<o>" shows "[\<^bold>\<A>\<phi> \<^bold>\<equiv> \<phi>]"
order_trans: fixes x :: "'d list"   and y :: "'d list"   and z :: "'d list" assumes "subseq x y"   and "subseq y z" shows "subseq x z"
ex_walls: fixes C :: "'a set"   and D :: "'a set" assumes "chamber C"   and "chamber D"   and "C \<sim> D"   and "C \<noteq> D" shows "\<exists>f g. OpposedThinChamberComplexFoldings X f g C \<and> D = g ` C"
lookup_pair_update_by_pair: fixes xs :: "('a \<times> 'c) list"   and k1 :: "'a"   and v :: "'c"   and k2 :: "'a" assumes "oalist_inv_raw xs" shows "lookup_pair (update_by_pair (k1, v) xs) k2 = (if k1 = k2 then v else lookup_pair xs k2)"
semigroup_axioms: shows "Type_Semigroups.semigroup f\<^sub>b"
arr_cod: fixes f :: "'a \<times> 'a" assumes "VV.arr f" shows "VV.arr (VV.cod f)"
path_from_toE: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and xs :: "'c list"   and w :: "'c" assumes "Digraph G"   and "v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w" shows "(Digraph.walk G xs \<and> distinct xs) \<and> xs \<noteq> [] \<and> hd xs = v \<and> last xs = w"
assms: fixes P :: "'a llist" assumes "vmc_path G P v0 p \<sigma>" shows "visits_via P A W"
open_segment_eq_real_ivl: fixes a :: "real"   and b :: "real" shows "open_segment a b = (if a \<le> b then {a<..<b} else {b<..<a})"
Fvars_rawpsubst_su: fixes \<phi> :: "'fmla"   and txs :: "('trm \<times> 'var) list" assumes "\<phi> \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm" shows "Fvars (rawpsubst \<phi> txs) \<subseteq> Fvars \<phi> - snd ` set txs \<union> \<Union> {uu_. \<exists>t x. uu_ = FvarsT t \<and> (t, x) \<in> set txs}"
vdisjnt_singleton2: fixes A :: "V"   and a :: "V" shows "vdisjnt A (set {a}) = (a \<notin>\<^sub>\<circ> A)"
mono: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c llist" shows "mono_llist (\<lambda>ldropWhile. case x of LNil \<Rightarrow> LNil | LCons xa xs' \<Rightarrow> if P xa then ldropWhile xs' else x)"
is_ub_def: fixes S :: "'a set"   and x :: "'a" shows "S <| x = (\<forall>y\<in>S. y \<sqsubseteq> x)"
exu_def_var: fixes x :: "'var"   and y :: "'var"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "y \<in> var"   and "y \<noteq> x"   and "y \<notin> Fvars \<phi>"   and "\<phi> \<in> fmla" shows "exu x \<phi> = cnj (exi x \<phi>) (exi y (all x (imp \<phi> (eql (Var x) (Var y)))))"
ins_mset: fixes t :: "('c, 'd) SkewBinomialTree"   and q :: "('c, 'd) SkewBinomialTree list" assumes "tree_invar t"   and "queue_invar q" shows "queue_to_multiset (ins t q) = tree_to_multiset t + queue_to_multiset q"
lspasl_iu_eq: fixes h1 :: "'a"   and h2 :: "'a" shows "(h1,h2\<triangleright>0::'a) = ((0::'a,0::'a\<triangleright>0::'a) \<and> h1 = (0::'a) \<and> h2 = (0::'a))"
split_face_f12_f21_neq_norm: fixes oldF :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and vs :: "nat list"   and f12 :: "face"   and f21 :: "face" assumes "pre_split_face oldF ram1 ram2 vs"   and "2 < |vertices oldF|"   and "2 < |vertices f12|"   and "2 < |vertices f21|"   and "(f12, f21) = split_face oldF ram1 ram2 vs" shows "normFace f12 \<noteq> normFace f21"
cos_arg: fixes z :: "complex" assumes "z \<noteq> 0" shows "cos (arg z) = Re z / cmod z"
L: shows "\<Union> (P\<^sub>1 \<union> wrap (B\<^sub>1 \<union> {u}) \<union> P\<^sub>2 \<union> wrap B\<^sub>2) = \<Union> (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2) \<union> {u}"
div_minus_right: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "divide a (uminus b) = divide (uminus a) b"
select_nonBV: fixes x :: "variable"   and \<alpha> :: "game"   and I :: "interp"   and X :: "(variable \<Rightarrow> real) set"   and \<omega> :: "variable \<Rightarrow> real" assumes "x \<notin> BVG \<alpha>" shows "selectlike (game_sem I \<alpha> (selectlike X \<omega> {x})) \<omega> {x} = selectlike (game_sem I \<alpha> X) \<omega> {x}"
apartment_min_gallery: fixes A :: "'a set set"   and Cs :: "'a set list" assumes "A \<in> \<A>"   and "SimplicialComplex.min_maxsimpchain A Cs" shows "min_gallery Cs"
X: shows "topology (arbitrary union_of (finite intersection_of (\<lambda>x. x \<in> \<B>) relative_to U)) = X"
residualFreshChainSimp2_1: fixes xvec :: "name list"   and M :: "'d"   and N :: "'d"   and P :: "('d, 'e, 'f) psi" shows "xvec \<sharp>* RIn M N P = (xvec \<sharp>* M \<and> xvec \<sharp>* N \<and> xvec \<sharp>* P)"
assms: shows "\<forall>xs\<in>set xss. distinct xs"
dX_Int_Field_Pd: fixes X :: "'a set"   and d :: "'b" assumes "Singleton_for_d.dX X d \<subseteq> Field (Pd' d)" shows "X \<inter> Field (Pd' d) = Singleton_for_d.dX X d"
map_id: fixes t :: "'d \<Rightarrow> 'e" shows "id \<circ> t = t"
inv_from_bot_def: fixes n :: "nat"   and p :: "'b mat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'b mat"   and mi :: "nat" shows "inv_from_bot n p A mi \<equiv> \<forall>i\<ge>mi. i < n \<longrightarrow> p A i"
nres_order_simps_19: shows "Sup UNIV = FAIL"
recurrent_def: fixes s :: "'s" shows "recurrent s = almost_everywhere (T s) (ev (HLD {s}))"
simps: fixes uu :: "'a extN" shows "uu\<^sup>\<star> = extN.Top"
weak_conv_m_def: fixes M_seq :: "nat \<Rightarrow> real measure"   and M :: "real measure" shows "weak_conv_m M_seq M \<equiv> weak_conv (\<lambda>n. cdf (M_seq n)) (cdf M)"
fun_open_ball_aux: fixes U :: "('a \<Rightarrow> 'b) set" shows "open U = (\<forall>x\<in>U. \<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> y \<in> U)"
to_stream_def: fixes X :: "nat \<Rightarrow> 'a" shows "to_stream X = smap X nats"
simps: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and s :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a1 :: "'c"   and a2 :: "'e" shows "(r OO s) a1 a2 = (\<exists>a b c. a1 = a \<and> a2 = c \<and> r a b \<and> s b c)"
Int_insert_right: fixes A :: "'d set"   and a :: "'d"   and B :: "'d set" shows "A \<inter> insert a B = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"
polynomial_of_list_remove_powers_polynomial_bool: fixes xs :: "(char list multiset \<times> int) multiset" shows "polynomial_of_mset xs - polynomial_of_mset (remove_powers xs) \<in> More_Modules.ideal polynomial_bool"