hcju/isabelleps huggingface dataset:
oseqpTEs_5: fixes \<sigma> :: "nat \<Rightarrow> 'e"   and l :: "'h"   and s\<^sub>d\<^sub>a\<^sub>t\<^sub>a :: "'e \<Rightarrow> nat"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and i :: "nat" assumes "((\<sigma>, {l}deliver(s\<^sub>d\<^sub>a\<^sub>t\<^sub>a) . p), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i" obtains "a = deliver (s\<^sub>d\<^sub>a\<^sub>t\<^sub>a (\<sigma> i))" and "\<sigma>' i = \<sigma> i" and "q = p"
simps_2: fixes I :: "'f \<Rightarrow> 'g option"   and P :: "'g \<Rightarrow> 'f \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> bool"   and ta :: "('f, 'h, 'i) new_thread_action"   and tas :: "('f, 'h, 'i) new_thread_action list" shows "upd_invs I P (ta # tas) = upd_invs (upd_inv I P ta) P tas"
col_mat_mult_index: fixes nr :: "nat"   and n :: "nat"   and m1 :: "'c list list"   and nc :: "nat"   and m2 :: "'c list list"   and j :: "nat"   and ze :: "'c"   and pl :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and ti :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "mat nr n m1"   and "mat n nc m2"   and "j < nc" shows "col (mat_multI ze pl ti nr m1 m2) j = map (\<lambda>i. scalar_prodI ze pl ti (row m1 i) (col m2 j)) [0..<nr]"
hpsafe_ODE_simps: fixes ODE :: "('a, 'c) ODE"   and p :: "('a, 'b, 'c) formula" shows "hpsafe (EvolveODE ODE p) = (osafe ODE \<and> fsafe p)"
mustT_transC: fixes c :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "mustT c s"   and "(c, s) \<rightarrow>c (c', s')" shows "mustT c' s'"
assms_1: shows "lambda C -quasi_isometry_on {A..B} c"
add_pow_ldistr: fixes a :: "'a"   and b :: "'a"   and k :: "nat" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "[k] \<cdot> a \<otimes> b = [k] \<cdot> (a \<otimes> b)"
U: shows "\<forall>(t, n)\<in>U. numbound0 t \<and> 0 < n"
strips_state_to_state_inverse_is_i: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and v :: "'c"   and s :: "'c \<Rightarrow> 'd option"   and a :: "'d" assumes "is_valid_problem_sas_plus \<Psi>"   and "v \<in> set (\<Psi>\<^sub>\<V>\<^sub>+)"   and "s v \<noteq> None"   and "a \<in> \<R>\<^sub>+ \<Psi> v" shows "(\<phi>\<^sub>S \<Psi> s) (v, a) = Some (the (s v) = a)"
ub_set_def: fixes X :: "'b set" shows "ub_set X = {y. \<forall>x\<in>X. x \<le> y}"
that_2: fixes l :: "hf" assumes "Ord l"   and "succ l = k" shows "thesis"
wf_bs_def: fixes bs :: "'b list"   and p :: "poly" shows "wf_bs bs p = (maxindex p \<le> length bs)"
simps_1: fixes ik :: "msg set"   and more :: "'a" shows "ik \<lparr>ik = ik, \<dots> = more\<rparr> = ik"
prems: shows "ts \<sim>\<^sub>h ts\<^sub>h "
iexI: fixes P :: "nat \<Rightarrow> bool"   and t :: "nat"   and I :: "nat set" assumes "P t"   and "t \<in> I" shows "\<diamond> t I. P t"
Propagate_Black_def: shows "Propagate_Black \<equiv> \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M\<rbrace> \<acute>ind := 0;; \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> \<acute>ind = 0\<rbrace>  WHILE \<acute>ind < length \<acute>E INV \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind \<le> length \<acute>E\<rbrace>  DO \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind < length \<acute>E\<rbrace>  IF \<acute>M ! fst (\<acute>E ! \<acute>ind) = Black THEN \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind < length \<acute>E \<and> \<acute>M ! fst (\<acute>E ! \<acute>ind) = Black\<rbrace> \<acute>k := snd (\<acute>E ! \<acute>ind);; \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind < length \<acute>E \<and> \<acute>M ! fst (\<acute>E ! \<acute>ind) = Black \<and> \<acute>Auxk\<rbrace> \<langle>\<acute>M := \<acute>M[\<acute>k := Black],, \<acute>ind := \<acute>ind + 1\<rangle> ELSE \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind < length \<acute>E\<rbrace> \<langle>IF \<acute>M ! fst (\<acute>E ! \<acute>ind) \<noteq> Black THEN \<acute>ind := \<acute>ind + 1 FI\<rangle> FI OD"
w_values_mono: fixes vs0 :: "'a \<times> addr_loc \<Rightarrow> 'a val set"   and adal :: "'a \<times> addr_loc"   and P :: "'c prog"   and obs :: "('a, 'b) obs_event action list" shows "vs0 adal \<subseteq> w_values P vs0 obs adal"
seq_MP: fixes \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and I :: "('sf, 'sc, 'sz) interp"   and \<Gamma> :: "('sf, 'sc, 'sz) formula list"   and \<Delta> :: "('sf, 'sc, 'sz) formula list" assumes "\<nu> \<in> seq_sem I (\<Gamma>, \<Delta>)"   and "\<nu> \<in> fml_sem I (foldr (&&) \<Gamma> TT)" shows "\<nu> \<in> fml_sem I (foldr (||) \<Delta> FF)"
qsort'_def: fixes p :: "'b set list"   and l :: "('b, 'c) Combinators list" shows "qsort' p l = qsort l p"
pbernpoly_of_int: fixes n :: "nat"   and a :: "int" shows "pbernpoly n (real_of_int a) = bernoulli n"
finite_UNIV_finite_1_def: shows "finite_UNIV = Phantom(Enum.finite_1) True"
wellformed_items_\<J>_\<I>: fixes k :: "nat"   and u :: "nat" shows "wellformed_items (\<J> k u) \<and> wellformed_items (\<I> k)"
FO_resolution_prover_def: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and mgu :: "'b set set \<Rightarrow> 'c option"   and less_atm :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "FO_resolution_prover S subst_atm id_subst comp_subst renamings_apart atm_of_atms mgu less_atm \<equiv> FO_resolution subst_atm id_subst comp_subst renamings_apart atm_of_atms mgu less_atm \<and> selection S \<and> FO_resolution_prover_axioms S subst_atm id_subst comp_subst"
iso_tuple_surjective_proof_assistI: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e"   and y :: "'d" assumes "f x = y" shows "Record.iso_tuple_surjective_proof_assist x y f"
assms: shows "\<Gamma> \<turnstile> Snd P : \<sigma>"
nths_singleton: fixes x :: "'a"   and A :: "nat set" shows "nths [x] A = (if 0 \<in> A then [x] else [])"
doms_eq: shows "subst_domain \<I>' = subst_domain \<sigma>"
cone_Cnotzero: shows "(cone, czero) \<notin> ordIso \<and> Card_order cone"
p_1: fixes i :: "nat" assumes "i < nr" shows "f i \<le> jj"
assms_5: shows "qs \<in> lang (seq [Atom x, Atom x])"
Lp_Banach_lattice_2: fixes f :: "'b \<Rightarrow> real"   and p :: "ennreal"   and M :: "'b measure" assumes "f \<in> space\<^sub>N (\<LL> p M)" shows "Norm (\<LL> p M) (\<lambda>x. \<bar>f x\<bar>) = Norm (\<LL> p M) f"
obs_some_kth_obs: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<sigma> :: "'b \<Rightarrow> 'c"   and k :: "nat" assumes "IFC prob"   and "IFC_def.obs prob \<sigma> k \<noteq> None" obtains i :: "nat" where "IFC_def.is_kth_obs prob (IFC_def.path prob \<sigma>) k i"
mcalc2: fixes c :: "('fsym, 'psym) lit list"   and \<sigma> :: "'tp" assumes "c \<in> \<Phi>" shows "\<sigma> \<turnstile>2 c"
dpd5: shows "ad (ad (0::'a)) = (0::'a)"
fixp_induct: fixes P :: "(('b, 'call, 'ret) gpv \<Rightarrow> 's \<Rightarrow> ('b \<times> 's + 'call' \<times> ('ret \<times> 's, 'call', 'ret') rpv \<times> ('b, 'call, 'ret) rpv) spmf) \<Rightarrow> bool" assumes "spmf.admissible (\<lambda>inline1. P (curry inline1))"   and "P (\<lambda>inline1 gpv. return_pmf None)"   and "\<And>s. P s \<Longrightarrow> P (\<lambda>a aa. the_gpv a \<bind> (\<lambda>a. case a of Pure x \<Rightarrow> return_spmf (Inl (x, aa)) | IO out rpv \<Rightarrow> the_gpv (callee aa out) \<bind> (\<lambda>a. case a of Pure (x, y) \<Rightarrow> s (rpv x) y | IO out rpv' \<Rightarrow> return_spmf (Inr (out, rpv', rpv)))))" shows "P local.inline1"
shrK_def: shows "shrK \<equiv> SOME shrK. inj shrK"
poly_newton_poly: fixes n :: "nat"   and y :: "'a" shows "poly (local.newton_poly n) y = (\<Sum>i\<leftarrow>[0..<Suc n]. local.c i * poly (local.N i) y)"
drop_bit_Suc: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat"   and a :: "'c" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "drop_bit (Suc n) a = drop_bit n (divide a (numeral.numeral one plus (num.Bit0 num.One)))"
iter_mono: fixes c :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice" assumes "(c::'a::refinement_lattice) \<sqsubseteq> (d::'a::refinement_lattice)" shows "(c::'a::refinement_lattice)\<^sup>\<omega> \<sqsubseteq> (d::'a::refinement_lattice)\<^sup>\<omega>"
comp_ran_split_def: fixes P :: "('f \<Rightarrow> 'h option) \<times> ('f \<Rightarrow> 'h option)"   and p :: "'e \<Rightarrow> 'g decision option" shows "P \<Otimes>\<^sub>\<nabla> p \<equiv> \<lambda>x. case p (fst x) of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow y\<rfloor> \<Rightarrow> case fst P (snd x) of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>z\<rfloor> \<Rightarrow> \<lfloor>allow (y, z)\<rfloor> | \<lfloor>deny y\<rfloor> \<Rightarrow> case snd P (snd x) of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>z\<rfloor> \<Rightarrow> \<lfloor>deny (y, z)\<rfloor>"
fds_fds_nth: fixes f :: "'a fds" shows "fds (fds_nth f) = f"
intro: fixes num' :: "int \<Rightarrow> 'a"   and plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and test_num' :: "int \<Rightarrow> 'a \<Rightarrow> bool"   and inv_plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"   and inv_less' :: "bool \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"   and \<gamma> :: "'a \<Rightarrow> int set" assumes "Abs_Int_inv_mono num' plus' test_num' inv_plus' inv_less' \<gamma>" shows "Abs_Int_wn num' plus' test_num' inv_plus' inv_less' \<gamma>"
succ_ss_V: fixes u :: "nat" shows "h.cf.E `` {u} \<subseteq> h.cf.V"
jumpF_pathfinish_part_circlepath: fixes st :: "real"   and tt :: "real"   and r :: "real"   and z :: "complex"   and z0 :: "complex" assumes "st < tt"   and "0 < r"   and "cmod (z - z0) \<noteq> r" shows "jumpF_pathfinish (part_circlepath z r st tt) z0 = (if r * cos tt + Re z - Re z0 = 0 then let \<Delta> = r * sin tt + Im z - Im z0 in if (0 < sin tt \<or> cos tt = - 1) \<and> \<Delta> < 0 \<or> (sin tt < 0 \<or> cos tt = 1) \<and> 0 < \<Delta> then - 1 / 2 else 1 / 2 else 0)"
betw4_strong: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "[[a b c d]]" shows "[[a b d]] \<and> [[a c d]]"
elliptic_isometryI: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "bounded {(f ^^ n) x |n. True}"   and "isometry f" shows "elliptic_isometry f"
opnet_cast2: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d net_state"   and H :: "nat set"   and K :: "nat set"   and m :: "'e"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and s' :: "'d net_state"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set"   and t :: "'d net_state"   and R :: "nat set"   and t' :: "'d net_state"   and T :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set" assumes "((\<sigma>, s), H\<not>K:arrive(m), \<sigma>', s') \<in> S"   and "((\<sigma>, t), R:*cast(m), \<sigma>', t') \<in> T"   and "H \<subseteq> R"   and "K \<inter> R = {}" shows "((\<sigma>, SubnetS s t), R:*cast(m), \<sigma>', SubnetS s' t') \<in> opnet_sos S T"
intro: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "semiring_ow U plus times"   and "comm_monoid_add_ow U plus zero"   and "mult_zero_ow U times zero" shows "semiring_0_ow U plus zero times"
reach_def: fixes x :: "'node" shows "reach x \<equiv> {y. (x, y) \<in> next\<^sup>* \<and> y \<noteq> nil}"
cut_less_mem_iff: fixes x :: "'a"   and I :: "'a set"   and t :: "'a" shows "(x \<in> I \<down>< t) = (x \<in> I \<and> x < t)"
no_writes_unchanged0: fixes k :: "nat"   and v :: "'var"   and \<sigma> :: "'var \<Rightarrow> 'val" assumes "\<forall>l<k. v \<notin> local.writes (path \<sigma> l)" shows "(\<sigma>\<^bsup>k\<^esup>) v = \<sigma> v"
top_add: fixes x :: "'b" assumes "(0::'b) \<le> x" shows "top + x = top"
hom_0_iff: fixes x :: "'a" shows "(to_fract x = 0) = (x = (0::'a))"
paths_Cons_nxt: fixes t :: "'a ttree"   and x :: "'a"   and xs :: "'a list" assumes "possible t x"   and "xs \<in> paths (nxt t x)" shows "x # xs \<in> paths t"
indep_card_eq_dim_span: fixes B :: "'b set" assumes "independent B" shows "finite B \<and> card B = dim (span B)"
inmarked_def: fixes x :: "'v" shows "inmarked x = (x \<in> A \<Gamma> \<or> Inr x \<in> S)"
remove_max_max: fixes empty :: "'a"   and is_empty :: "'a \<Rightarrow> bool"   and of_list :: "'b list \<Rightarrow> 'a"   and multiset :: "'a \<Rightarrow> 'b multiset"   and remove_max :: "'a \<Rightarrow> 'b \<times> 'a"   and inv :: "'a \<Rightarrow> bool"   and l :: "'a"   and m :: "'b"   and l' :: "'a" assumes "RemoveMax empty is_empty of_list multiset remove_max inv"   and "\<not> is_empty l"   and "inv l"   and "(m, l') = remove_max l" shows "m = Max (Collection.set multiset l)"
gauss_cnj_mult: fixes a :: "gauss_int"   and b :: "gauss_int" shows "gauss_cnj (a * b) = gauss_cnj a * gauss_cnj b"
\<phi>T_prv_fls: fixes T :: "'fmla" assumes "T \<in> fmla"   and "Fvars T = {xx}"   and "\<And>\<phi>. \<lbrakk>\<phi> \<in> fmla; Fvars \<phi> = {}\<rbrakk> \<Longrightarrow> prv (eqv (subst T \<langle>\<phi>\<rangle> xx) \<phi>)" shows "prv fls"
ground_fm_aux_simps_3: fixes t :: "tm"   and u :: "tm"   and S :: "atom set" shows "ground_fm_aux (t EQ u) S = (ground_aux t S \<and> ground_aux u S)"
interior_maximal: fixes T :: "'b set"   and S :: "'b set" assumes "T \<subseteq> S"   and "open T" shows "T \<subseteq> interior S"
UEA1: fixes \<AA> :: "'b formula"   and \<Gamma> :: "'b formula list"   and \<Theta> :: "'b formula list"   and \<BB> :: "'b formula" assumes "\<AA> # \<Gamma> \<Rightarrow> \<Theta>" shows "\<AA> \<^bold>\<and> \<BB> # \<Gamma> \<Rightarrow> \<Theta>"
paste_2: fixes \<nu> :: "'a"   and \<mu> :: "'a" shows "\<nu> \<star> \<mu> = (\<nu> \<star> cod \<mu>) \<cdot> (local.dom \<nu> \<star> \<mu>)"
attach_wiring_right_simps: fixes f :: "'k \<Rightarrow> 'm"   and g :: "'n \<Rightarrow> 'l" shows "attach_wiring_right (f, g) = map_fun id (map_fun id (map_gpv' id f g))"
extend_Constrains: fixes F :: "'a program"   and A :: "'a set"   and B :: "'a set" shows "(extend h F \<in> extend_set h A Co extend_set h B) = (F \<in> A Co B)"
nn_integral_limsup: fixes u :: "nat \<Rightarrow> 'b \<Rightarrow> ennreal"   and M :: "'b measure"   and w :: "'b \<Rightarrow> ennreal" assumes "\<And>i. u i \<in> borel_measurable M"   and "w \<in> borel_measurable M"   and "\<And>i. AE x in M. u i x \<le> w x"   and "integral\<^sup>N M w < \<infinity>" shows "limsup (\<lambda>n. integral\<^sup>N M (u n)) \<le> \<integral>\<^sup>+ x. limsup (\<lambda>n. u n x) \<partial>M"
map_of_sunion_with: fixes xs :: "('a \<times> 'c) list"   and ys :: "('a \<times> 'c) list"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'a" assumes "sorted (map fst xs)"   and "sorted (map fst ys)" shows "map_of (sunion_with f xs ys) k = (case map_of xs k of None \<Rightarrow> map_of ys k | Some v \<Rightarrow> case map_of ys k of None \<Rightarrow> Some v | Some w \<Rightarrow> Some (f k v w))"
simps_1: fixes \<Gamma> :: "('k, 'l) web_scheme"   and x3 :: "'m"   and y3 :: "'m" assumes "countable_bipartite_web \<Gamma>" shows "(bi.Inner x3 = bi.Inner y3) = (x3 = y3)"
hv_props_2: shows "\<forall>(k, \<gamma>)\<in>\<H>. \<exists>(k', \<gamma>')\<in>two_chain_horizontal_boundary two_chain. \<exists>a\<in>{0..1}. \<exists>b\<in>{0..1}. a \<le> b \<and> subpath a b \<gamma>' = \<gamma>"
hyps: fixes j :: "nat" assumes "i - j < i - ja__"   and "k < j"   and "j < i" shows "\<pi> i pd\<rightarrow> \<pi> j"
conj_commute: fixes conj :: "'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice"   and chaos :: "'a::refinement_lattice"   and a :: "'a::refinement_lattice"   and b :: "'a::refinement_lattice" assumes "conjunction (conj::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) (chaos::'a::refinement_lattice)" shows "(conj::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) (a::'a::refinement_lattice) (b::'a::refinement_lattice) = conj b a"
set_iterator_map_linord_def: fixes it :: "('c \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'd \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and S0 :: "('a \<times> 'd) set" shows "set_iterator_map_linord it S0 \<equiv> set_iterator_genord it S0 (\<lambda>(k, uu_) (k', uu_). k \<le> k')"
rec_2: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b list \<Rightarrow> 'a \<Rightarrow> 'a"   and x21 :: "'b"   and x22 :: "'b list" shows "rec_list f1 f2 (x21 # x22) = f2 x21 x22 (rec_list f1 f2 x22)"
OV_inv4_inductive_1: shows "init obsv_TS \<subseteq> OV_inv4"
keys_minus: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and q :: "'b \<Rightarrow>\<^sub>0 'c" shows "keys (p - q) \<subseteq> keys p \<union> keys q"
eint_minus_ineq: fixes c :: "eint"   and d :: "eint"   and b :: "eint" assumes "c \<noteq> \<infinity>"   and "d \<le> b" shows "d - c \<le> b - c"
RRREF: fixes x :: "'a"   and y :: "'a" assumes "x \<in> S"   and "y \<in> S"   and "RR x y" shows "RR' x y"
reciprocal_1: fixes p :: "nat" shows "reciprocal_poly p 1 = monom (1::'b) p"
raw_match'_restriction_on_tvsT: fixes T :: "typ"   and U :: "typ"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "raw_match' T U subs = Some subs'" shows "raw_match' T U (subs |` tvsT T) = Some (subs' |` tvsT T)"
rank_mul_left_invertible_mat: fixes A :: "'b mat"   and n :: "nat"   and B :: "'b mat"   and nc :: "nat" assumes "invertible_mat A"   and "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n nc" shows "vec_space.rank n (A * B) = vec_space.rank n B"
inres_tape: fixes st :: "nat"   and l :: "cell list"   and r :: "cell list"   and tp :: "(action \<times> nat) list"   and stp :: "nat"   and a :: "nat"   and b :: "cell list"   and c :: "cell list"   and l' :: "cell list"   and r' :: "cell list"   and a' :: "nat"   and b' :: "cell list"   and c' :: "cell list" assumes "steps0 (st, l, r) tp stp = (a, b, c)"   and "steps0 (st, l', r') tp stp = (a', b', c')"   and "tinres l l'"   and "tinres r r'" shows "a = a' \<and> tinres b b' \<and> tinres c c'"
countable: shows "countable (\<Union> (set_spmf ` Y))"
cut_le_Max_le: fixes I :: "'a set"   and t :: "'a" assumes "finite (I \<down>\<le> t)"   and "I \<down>\<le> t \<noteq> {}" shows "Max (I \<down>\<le> t) \<le> t"
elims: fixes x :: "nat"   and y :: "(char list, 'a) process_term" assumes "sys' x = y" obtains "x = 0" and "y = CH (PROC x) (ACT ''b'' (VAR (0::'a)))"   | "x = Suc 0" and "y = CH (ACT ''a'' (VAR (0::'a))) (PROC x)"   | va :: "nat" where "x = Suc (Suc va)" and "y = ACT ''a'' (VAR (0::'a))"
distinct_8: fixes ty1' :: "ty"   and ty2' :: "ty" shows "ty1' \<rightarrow> ty2' \<noteq> Top"
vector_to_cblinfun_scaleC: fixes a :: "complex"   and \<psi> :: "'b" shows "vector_to_cblinfun (a *\<^sub>C \<psi>) = a *\<^sub>C vector_to_cblinfun \<psi>"
sign_changes_two: fixes x :: "'a"   and y :: "'a" shows "sign_changes [x, y] = (if (0::'a) < x \<and> y < (0::'a) \<or> x < (0::'a) \<and> (0::'a) < y then 1 else 0)"
set_valued_functor_axioms: shows "set_valued_functor (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Y a')"
mstep_mverdicts: fixes \<pi> :: "(String.literal \<times> event_data list) prefix"   and R :: "event_data list set"   and st :: "('a, 'b) mstate"   and tdb :: "(String.literal \<times> event_data list) set \<times> nat"   and v :: "event_data option list"   and i :: "nat" assumes "wf_mstate \<phi> \<pi> R st"   and "last_ts \<pi> \<le> snd tdb"   and "mem_restr R v" shows "((i, v) \<in> flatten_verdicts (fst (mstep (map_prod mk_db id tdb) st))) = ((i, v) \<in> M (psnoc \<pi> tdb) - M \<pi>)"
identify_block_main: fixes j :: "nat"   and i :: "nat" assumes "identify_block A j = i" shows "i \<le> j \<and> (i = 0 \<or> A $$ (i - 1, i) \<noteq> (1::'a)) \<and> (\<forall>k\<ge>i. k < j \<longrightarrow> A $$ (k, Suc k) = (1::'a))"
wlog_linorder_le: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b"   and a :: "'b" assumes "\<And>a b. a \<le> b \<Longrightarrow> P a b"   and "P b a \<Longrightarrow> P a b" shows "P a b"
bi_unique_alt_def: fixes A :: "'i \<Rightarrow> 'j \<Rightarrow> bool" shows "bi_unique A = (left_unique A \<and> right_unique A)"
simps: fixes f :: "'b \<Rightarrow> nat"   and l :: "'b list" shows "arg_max_list f l = (let m = Max (set (map f l)) in arg_min_list (\<lambda>x. m - f x) l)"
openin_topspace: fixes U :: "'c topology" shows "openin U (topspace U)"
sel_2: fixes x :: "quat" shows "Im1 (- x) = - Im1 x"
greaterThanLessThan_empty_iff2: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.dense_linorder less_eq less" shows "({} = ord.greaterThanLessThan less a b) = less_eq b a"
assms_3: fixes x :: "'a" assumes "x \<in> carrier" shows "measure_pmf.prob q (preferred_alts le x) \<le> measure_pmf.prob p (preferred_alts le x)"
simps_1: fixes fill :: "'b"   and xs :: "'b list" shows "takefill fill 0 xs = []"
lexp_tail: fixes order :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list"   and x :: "'b" assumes "lexp order xs ys" shows "lexp order (x # xs) (x # ys)"
eq_id_iff: fixes f :: "'f \<Rightarrow> 'f" shows "(\<forall>x. f x = x) = (f = id)"
spr_simRels_def: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and agent :: "'b"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'f" assumes "FiniteSingleAgentEnvironment jkbp agent" shows "FiniteSingleAgentEnvironment.spr_simRels envObs \<equiv> \<lambda>a. rel_ext (\<lambda>uu_. \<exists>U u V v. uu_ = ((U, u), V, v) \<and> U = V \<and> {u, v} \<subseteq> U \<and> envObs a u = envObs a v)"
maxsimp_connect: fixes X :: "'b set set"   and x :: "'b set"   and y :: "'b set" assumes "ChamberComplex X"   and "x \<noteq> y"   and "SimplicialComplex.maxsimp X x"   and "SimplicialComplex.maxsimp X y" shows "\<exists>xs. SimplicialComplex.maxsimpchain X (x # xs @ [y])"
distrib1: fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and G :: "'b literal list list" assumes "F, \<Gamma> \<Rightarrow>\<^sub>n"   and "image_mset disj_of_clause (mset G) + \<Gamma> \<Rightarrow>\<^sub>n" shows "mset (map (\<lambda>d. F \<^bold>\<or> disj_of_clause d) G) + \<Gamma> \<Rightarrow>\<^sub>n"
dis_rews_1: shows "is_MkI\<cdot>\<bottom> = \<bottom>"
infinity_bot_demand: fixes f :: "C \<rightarrow> 'a" shows "(f\<cdot>C\<^sup>\<infinity> = \<bottom>) = (demand f = C\<^sup>\<infinity>)"