hcju/isabelleps huggingface dataset:
map_project_Int_ENC: fixes X :: "nat set"   and n :: "nat"   and Z :: "('a \<times> bool list) list set" assumes "0 \<notin> X"   and "X \<subseteq> {0..<n + 1}"   and "Z \<subseteq> lists ((set \<circ> \<sigma> \<Sigma>) (n + 1))" shows "SAMEQUOT (any, replicate n False) (map \<pi> ` (Z \<inter> lang (n + 1) (ENC (n + 1) X))) = SAMEQUOT (any, replicate n False) (map \<pi> ` Z) \<inter> lang n (ENC n ((\<lambda>x. x - 1) ` X))"
mangoldt_def: fixes n :: "nat" shows "mangoldt n = (if primepow n then of_real (ln (real (aprimedivisor n))) else (0::'a))"
axioms_2: fixes x :: "'d \<Longrightarrow> 'e" assumes "vwb_lens x" shows "mwb_lens x"
denote_while_n_dim: fixes \<rho> :: "complex mat"   and dims :: "nat list"   and M0 :: "complex mat"   and M1 :: "complex mat"   and DS :: "complex mat \<Rightarrow> complex mat"   and n :: "nat" assumes "\<rho> \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"   and "M0 \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"   and "M1 \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"   and "partial_density_operator \<rho>"   and "\<And>\<rho>'. \<lbrakk>\<rho>' \<in> carrier_mat (state_sig.d dims) (state_sig.d dims); partial_density_operator \<rho>'\<rbrakk> \<Longrightarrow> positive (DS \<rho>') \<and> trace (DS \<rho>') \<le> trace \<rho>' \<and> DS \<rho>' \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)" shows "denote_while_n M0 M1 DS n \<rho> \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"
closed_translation: fixes S :: "'b set"   and a :: "'b" assumes "closed S" shows "closed ((+) a ` S)"
map_dual_def_var: fixes f :: "'c \<Rightarrow> 'd" shows "map_dual f = \<partial> \<circ> f \<circ> \<partial>\<^sup>-"
thetaAtm_Sretr: fixes atm :: "'atom" assumes "compatAtm atm" shows "thetaAtm atm \<subseteq> Sretr (thetaAtm atm)"
TS_pairwise': fixes qs :: "nat list"   and init :: "nat list"   and x :: "nat"   and y :: "nat" assumes "qs \<in> {xs. set xs \<subseteq> set init}"   and "(x, y) \<in> {(x, y). x \<in> set init \<and> y \<in> set init \<and> x \<noteq> y}"   and "x \<noteq> y"   and "distinct init" shows "Pbefore_in x y (Partial_Cost_Model.embed (rTS [])) qs init = Pbefore_in x y (Partial_Cost_Model.embed (rTS [])) (Lxy qs {x, y}) (Lxy init {x, y})"
has_bochner_integral_inner_right: fixes c :: "'c"   and M :: "'d measure"   and f :: "'d \<Rightarrow> 'c"   and x :: "'c" assumes "c \<noteq> (0::'c) \<Longrightarrow> has_bochner_integral M f x" shows "has_bochner_integral M (\<lambda>x. c \<bullet> f x) (c \<bullet> x)"
c_top_welldefTr2: fixes N :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<triangleright> N"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "c_top G N (N \<bullet> a) (N \<bullet> b) \<subseteq> N \<bullet> (a \<cdot> b)"
zlin_qfree: fixes p :: "fm" assumes "iszlfm p" shows "qfree p"
const_g: shows "const (\<lambda>\<tau>. g (X \<tau>))"
distinct_sort: fixes xs :: "('a, 'b) Combinators list"   and l :: "'a set list" shows "distinct (FWNormalisationCore.sort xs l) = distinct xs"
part_partJoin: fixes I :: "'a set"   and P :: "'a set set"   and Q :: "'a set set" assumes "part I P"   and "part I Q" shows "part I (partJoin P Q)"
R: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and star :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.B1_algebra plus times one zero less_eq less star"   and "times x x = x" shows "star x = plus one x"
HRP_fresh_iff: fixes a :: "atom"   and x :: "tm"   and x' :: "tm" shows "a \<sharp> HRP x x' = (a \<sharp> x \<and> a \<sharp> x')"
root_2: shows "0 < degree ([:- a, 1:] * p)"
seqSubstSimps_6: fixes y :: "name"   and \<sigma> :: "(name list \<times> 'a list) list"   and P :: "('a, 'b, 'c) psi" assumes "y \<sharp> \<sigma>" shows "\<lparr>\<nu>y\<rparr>P[<\<sigma>>] = \<lparr>\<nu>y\<rparr>P[<\<sigma>>]"
ih_T\<^sub>1: fixes \<Gamma> :: "binding list" assumes "\<turnstile> \<Gamma> ok"   and "T\<^sub>1 closed_in \<Gamma>" shows "\<Gamma>\<turnstile>T\<^sub>1<:T\<^sub>1"
msubstneg_I: fixes p :: "fm"   and vs :: "'b list"   and c :: "poly"   and x :: "'b"   and bs :: "'b list"   and t :: "tm" assumes "islin p"   and "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> < (0::'b)" shows "Ifm vs (x # bs) (msubstneg p c t) = Ifm vs (Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> # bs) p"
not_le_imp_less: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and y :: "'g"   and x :: "'g" assumes "class.linorder less_eq less"   and "\<not> less_eq y x" shows "less x y"
r_2: shows "S, \<Gamma> \<Rightarrow>\<^sub>n"
ev: shows "\<forall>\<^sub>F z in at 0. 1 - exp z \<noteq> 0"
Comm2: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and Q :: "pi"   and Q'' :: "pi"   and x :: "name"   and Q' :: "pi" assumes "P \<Longrightarrow>\<^sub>l\<^sup>^a[b] \<prec> P'"   and "Q \<Longrightarrow>\<^sub>lb in Q''\<rightarrow>a<x> \<prec> Q'" shows "P \<parallel> Q \<Longrightarrow>\<^sub>l\<^sup>^\<tau> \<prec> P' \<parallel> Q'"
intros_2: fixes l :: "'c list"   and u' :: "'c \<Rightarrow> 'a"   and u :: "'c \<Rightarrow> 'a" assumes "set l \<subseteq> \<X>"   and "u' = [l\<rightarrow>(0::'a)]u" shows "local.trans u u'"
not_in_keys_iff_lookup_eq_zero: fixes k :: "'c"   and f :: "'c \<Rightarrow>\<^sub>0 'd" shows "(k \<notin> keys f) = (lookup f k = (0::'d))"
order_refl: fixes a :: "'a" assumes "a \<in> S" shows "le a a"
bisimWeakCoinduct: fixes P :: "pi"   and Q :: "pi"   and X :: "(pi \<times> pi) set" assumes "(P, Q) \<in> X"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> P \<leadsto>[X] Q"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> (Q, P) \<in> X" shows "P \<sim> Q"
valid_add_new_mmuaux: fixes args :: "Monitor.args"   and cur :: "nat"   and aux :: "nat \<times> nat queue \<times> nat \<times> bool list \<times> bool list \<times> (event_data option list, nat) mapping \<times> (nat, (event_data option list, nat + nat) mapping) mapping \<times> event_data option list set list \<times> nat"   and auxlist :: "(nat \<times> event_data option list set \<times> event_data option list set) list"   and rel1 :: "event_data option list set"   and rel2 :: "event_data option list set"   and nt :: "nat" assumes "valid_mmuaux args cur aux auxlist"   and "table (args_n args) (args_L args) rel1"   and "table (args_n args) (args_R args) rel2"   and "cur \<le> nt" shows "valid_mmuaux args nt (add_new_mmuaux args rel1 rel2 nt aux) (update_until args rel1 rel2 nt auxlist)"
two_comparisons_into_case_order_5: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a"   and Q :: "'b"   and P :: "'b"   and R :: "'b" assumes "comparator comp" shows "(if lt_of_comp comp x y then Q else if le_of_comp comp x y then P else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
r_leap_def: shows "r_leap \<equiv> Pr 2 (Cn 2 r_prod_encode [Cn 2 r_singleton_encode [Cn 2 r_prod_encode [recf.Id 2 0, Cn 2 r_prod_encode [recf.Id 2 1, r_constn 1 0]]], r_constn 1 0]) (Cn 4 r_step [recf.Id 4 1])"
vpairs_ex_vfst: fixes x :: "V"   and r :: "V" assumes "x \<in>\<^sub>\<circ> vpairs r" shows "\<exists>b. \<langle>vfst x, b\<rangle> \<in>\<^sub>\<circ> r"
size_multiset_def: fixes f :: "'k \<Rightarrow> nat"   and M :: "'k multiset" shows "size_multiset f M = sum (wcount f M) (set_mset M)"
set_induct: fixes x :: "'a"   and a :: "'a llist"   and P :: "'a \<Rightarrow> 'a llist \<Rightarrow> bool" assumes "x \<in> lset a"   and "\<And>z1 z2. P z1 (LCons z1 z2)"   and "\<And>z1 z2 xa. \<lbrakk>xa \<in> lset z2; P xa z2\<rbrakk> \<Longrightarrow> P xa (LCons z1 z2)" shows "P x a"
share_all_until_volatile_write_Fence_commute: fixes i :: "nat"   and ls :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and \<D> :: "'c"   and \<O> :: "'d"   and \<R> :: "'e"   and \<D>' :: "'c"   and \<R>' :: "'e"   and S :: "nat \<Rightarrow> bool option" assumes "i < length ls"   and "ls ! i = (p, Fence # is, \<theta>, [], \<D>, \<O>, \<R>)" shows "share_all_until_volatile_write (ls[i := (p, is, \<theta>, [], \<D>', \<O>, \<R>')]) S = share_all_until_volatile_write ls S"
supset_pseudominimal_inI: fixes x :: "'b set"   and P :: "'b set set" assumes "x \<in> P"   and "x \<noteq> dual_order.bottom P"   and "\<And>z. \<lbrakk>z \<in> P; order.greater z x\<rbrakk> \<Longrightarrow> z = dual_order.bottom P" shows "dual_order.pseudominimal_in P x"
exec: shows "compP2 P,t \<turnstile> Normal (xcp, h, frs) -ta-jvmd\<rightarrow> Normal (xcp', h', frs')"
is_axiom_finite_deriv: fixes s :: "(nat \<times> form) list" assumes "is_axiom (s_of_ns s)" shows "finite (deriv s)"
v_v': shows "\<forall>c\<in>X. beta_interp.v' (v c) = c"
connected_eq_components_subset_sing: fixes s :: "'c set" shows "connected s = (components s \<subseteq> {s})"
low_deg_terms_zeroTr: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and j :: "nat" shows "pol_coeff S (n, f) \<longrightarrow> polyn_expr R X (n + j) (ext_cf S j (n, f)) = X^\<^bsup>R j\<^esup> \<cdot>\<^sub>r polyn_expr R X n (n, f)"
mono_on_imp_deriv_nonneg: fixes f :: "real \<Rightarrow> real"   and A :: "real set"   and D :: "real"   and x :: "real" assumes "mono_on f A"   and "(f has_real_derivative D) (at x)"   and "x \<in> interior A" shows "0 \<le> D"
conj_subset_def: fixes A :: "'d set"   and P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" shows "(A \<subseteq> {x. P x \<and> Q x}) = (A \<subseteq> {x. P x} \<and> A \<subseteq> {x. Q x})"
lt_le_iff: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and v :: "'a" shows "(punit.lt p \<preceq> v) = (\<forall>u. v \<prec> u \<longrightarrow> lookup p u = (0::'c))"
IH: fixes k :: "'b"   and v :: "'a" assumes "m $$ k = None" shows "m ++\<^sub>f {k $$:= v} = {k $$:= v} ++\<^sub>f m"
INF_cong: fixes A :: "'g set"   and B :: "'g set"   and C :: "'g \<Rightarrow> 'h"   and D :: "'g \<Rightarrow> 'h"   and Inf :: "'h set \<Rightarrow> 'h" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Inf (C ` A) = Inf (D ` B)"
assms: assumes "nofail (m \<bind> WHILE c f)" shows "mi \<le> SPEC (\<lambda>s. rwof m c f s \<and> \<not> c s)"
subt_trm_ord: fixes t :: "'a trm"   and s :: "'a trm" shows "t \<prec> s \<longrightarrow> (t, s) \<in> trm_ord"
symK_def: shows "symK \<equiv> SOME symK. \<exists>priSK. inj agrK \<and> inj sesK \<and> (\<forall>X X'. priSK X \<noteq> invK (priSK X')) \<and> (\<forall>X n. priSK X \<noteq> symK n) \<and> (\<forall>X n. invK (priSK X) \<noteq> symK n) \<and> (\<forall>a. invK (sesK a) = sesK a) \<and> (\<forall>n. invK (symK n) = symK n) \<and> (\<forall>m n. m \<in> bad \<longrightarrow> symK n = symK m \<longrightarrow> n \<in> bad)"
rel_fun_eq_rel: fixes R :: "'r \<Rightarrow> 's \<Rightarrow> bool" shows "rel_fun (=) R = (\<lambda>f g. \<forall>x. R (f x) (g x))"
cSUP_singleton: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "\<Squnion> (f ` {x}) = f x"
snd_ff: shows "snd fc = (if p = 0 then 1 else q2 ^ (degree p + 1))"
add_assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "a \<squnion> b \<squnion> c = a \<squnion> (b \<squnion> c)"
4: assumes "a = c"   and "b \<noteq> c" shows "\<exists>x y z. (\<Sum>i\<in>{c, b, c}. f i * i) = x * c + y * b + z * c"
min_add_distrib_right: fixes x :: "'a multiset"   and y :: "'a multiset"   and z :: "'a multiset" shows "x + subset_mset.min y z = subset_mset.min (x + y) (x + z)"
simps: fixes f :: "Value \<rightarrow> Value" shows "Fn_project\<cdot>(Fn\<cdot>f) = f"
select_convs_1: fixes nodesL :: "'b list"   and edgesL :: "('b \<times> 'b) list"   and more :: "'c" shows "nodesL \<lparr>nodesL = nodesL, edgesL = edgesL, \<dots> = more\<rparr> = nodesL"
cases: fixes a :: "trm" assumes "a \<in> value" obtains T :: "POPLmarkRecord.type" and t :: "trm" where "a = (\<lambda>:Tb__. tb__)"   | T :: "POPLmarkRecord.type" and t :: "trm" where "a = (\<lambda><:T. t)"   | fs :: "(char list \<times> trm) list" where "a = Rcd fs" and "\<forall>(l, t)\<in>set fs. t \<in> value"
assms: shows "\<exists>c'. next_change_multiplicity' c c' loc t n"
assms: fixes x :: "nat" assumes "x \<in> tvars t" shows "v x = w x"
word_not_not: fixes x :: "'a word" shows "~~ ~~ x = x"
from_F_def: fixes \<iota> :: "'f inference" shows "from_F \<iota> = {\<iota>' \<in> Inf_FL. to_F \<iota>' = \<iota>}"
count_prime_factorization_nonprime: fixes p :: "'a"   and x :: "'a" assumes "\<not> prime p" shows "count (prime_factorization x) p = 0"
supp_fin_union: fixes X :: "'a set"   and Y :: "'a set" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "fs TYPE('a) TYPE('b)"   and "finite X"   and "finite Y" shows "supp (X \<union> Y) = supp X \<union> supp Y"
DEFL_eq_resT: shows "DEFL('e\<cdot>'f resT) = DEFL('e\<^sub>\<bottom> \<oplus> ('e\<cdot>'f resT\<cdot>'f)\<^sub>\<bottom>)"
abs_eq_content: fixes y :: "real"   and x :: "real" shows "\<bar>y - x\<bar> = (if x \<le> y then content {x..y} else content {y..x})"
freshOutputAction_1: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and c :: "name" assumes "P \<longmapsto> a[b] \<prec> P'"   and "c \<sharp> P" shows "c \<noteq> a"
under_ofilter: fixes r :: "('a \<times> 'a) set"   and a :: "'a" assumes "wo_rel r" shows "ofilter r (under r a)"
subdiv1_contr_def: fixes G :: "('c, 'd) pre_digraph"   and rev_G :: "'d \<Rightarrow> 'd"   and H :: "('c, 'd) pre_digraph"   and rev_H :: "'d \<Rightarrow> 'd"   and u :: "'c"   and v :: "'c"   and w :: "'c"   and uv :: "'d"   and uw :: "'d"   and vw :: "'d"   and HM :: "'d pre_map" shows "subdiv1_contr G rev_G H rev_H u v w uv uw vw HM \<equiv> subdiv_step G rev_G H rev_H u v w uv uw vw \<and> subdiv1_contr_axioms H rev_H HM"
vars_monom_single_cases: fixes v :: "nat"   and k :: "nat"   and a :: "'a" shows "vars (MPoly_Type.monom (monomial k v) a) = (if k = 0 \<or> a = (0::'a) then {} else {v})"
rhs_step_bind_SPEC: fixes \<Phi> :: "'a \<Rightarrow> bool"   and x' :: "'a"   and m :: "'b nres"   and R :: "('b \<times> 'c) set"   and f' :: "'a \<Rightarrow> 'c nres" assumes "\<Phi> x'"   and "m \<le> \<Down> R (f' x')" shows "m \<le> \<Down> R (SPEC \<Phi> \<bind> f')"
euclidean_size_gcd_less1: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (0::'a)"   and "\<not> a dvd b" shows "euclidean_size (gcd a b) < euclidean_size a"
pcompose_add: fixes p :: "'b poly"   and q :: "'b poly"   and r :: "'b poly" shows "(p + q) \<circ>\<^sub>p r = p \<circ>\<^sub>p r + q \<circ>\<^sub>p r"
separatedin_def: fixes X :: "'b topology"   and S :: "'b set"   and T :: "'b set" shows "separatedin X S T \<equiv> S \<subseteq> topspace X \<and> T \<subseteq> topspace X \<and> S \<inter> X closure_of T = {} \<and> T \<inter> X closure_of S = {}"
sup_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x + y + z = x + (y + z)"
simps_5: fixes bs :: "real list"   and a :: "MIR.num" shows "Ifm bs (Le a) = (Inum bs a \<le> 0)"
wqo_on_imp_almost_full_on: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "wqo_on P A" shows "almost_full_on P A"
monotone_restrict_spmf: fixes A :: "'b set" shows "monotone (ord_spmf (=)) (ord_spmf (=)) (\<lambda>p. p \<upharpoonleft> A)"
flat_Stars: fixes vs :: "'b val list" shows "flat (Stars vs) = concat (map flat vs)"
filter_tab_all_False: fixes t :: "'a \<Rightarrow> 'b option"   and p :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>k y. t k = Some y \<longrightarrow> \<not> p k y" shows "filter_tab p t = Map.empty"
homotopy_equivalent_space_sym: fixes X :: "'a topology"   and Y :: "'b topology" shows "(X homotopy_equivalent_space Y) = (Y homotopy_equivalent_space X)"
assms_3: shows "g \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"
simps_3: fixes c\<^sub>1 :: "com"   and c\<^sub>2 :: "com"   and X :: "char list set" shows "Lb (c\<^sub>1;; c\<^sub>2) X = (Lb c\<^sub>1 \<circ> Lb c\<^sub>2) X"
simps: fixes a1 :: "'c"   and a2 :: "'c list" shows "ListMem a1 a2 = ((\<exists>x xs. a1 = x \<and> a2 = x # xs) \<or> (\<exists>x xs y. a1 = x \<and> a2 = y # xs \<and> ListMem x xs))"
Syntax_Arith_aux_axioms: shows "Syntax_Arith_aux var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num zer suc pls tms"
NAKTr1: assumes "M fgover R" shows "\<exists>H. finite_generator R M H \<and> (LEAST j. \<exists>L. finite_generator R M L \<and> j = card L) = card H"
covering_space_locally: fixes \<phi> :: "'c set \<Rightarrow> bool"   and C :: "'c set"   and p :: "'c \<Rightarrow> 'd"   and S :: "'d set"   and \<psi> :: "'d set \<Rightarrow> bool" assumes "locally \<phi> C"   and "covering_space C p S"   and "\<And>T. \<lbrakk>T \<subseteq> C; \<phi> T\<rbrakk> \<Longrightarrow> \<psi> (p ` T)" shows "locally \<psi> S"
IdD: fixes a :: "'d"   and b :: "'d" assumes "(a, b) \<in> Id" shows "a = b"
less_nat: fixes v :: "val"   and n :: "nat" assumes "v \<sqsubseteq> VNat n" obtains "v = VNat n"
select_convs_3: fixes arrays :: "typerep \<Rightarrow> nat \<Rightarrow> nat list"   and refs :: "typerep \<Rightarrow> nat \<Rightarrow> nat"   and lim :: "nat"   and more :: "'b" shows "lim \<lparr>arrays = arrays, refs = refs, lim = lim, \<dots> = more\<rparr> = lim"
simps_3: fixes f :: "('c \<Rightarrow> 'b) tree"   and x :: "'c tree" shows "right (f \<diamondop> x) = right f \<diamondop> right x"
elims_3: fixes x :: "protocol"   and xa :: "8 word" assumes "\<not> match_proto x xa" obtains p :: "8 word" and p_p :: "8 word" where "x = Proto p" and "xa = p_p" and "p_p \<noteq> p"
conj_UINF_dist: fixes P :: "'a upred"   and S :: "'b set"   and F :: "'b \<Rightarrow> 'a upred" shows "(P \<and> (\<Sqinter> Q \<in> S \<bullet> F Q)) = (\<Sqinter> Q \<in> S \<bullet> P \<and> F Q)"
list_assn_append_Cons_left: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> assn"   and xs :: "'b list"   and x :: "'b"   and ys :: "'b list"   and zs :: "'c list" shows "list_assn A (xs @ x # ys) zs = (\<exists>\<^sub>Azs1 z zs2. list_assn A xs zs1 * A x z * list_assn A ys zs2 * \<up> (zs1 @ z # zs2 = zs))"
Numb_inject: fixes x :: "nat"   and y :: "nat" assumes "Numb x = Numb y" shows "x = y"
right_diff_distrib_numeral: fixes v :: "num"   and b :: "'b"   and c :: "'b" shows "numeral v * (b - c) = numeral v * b - numeral v * c"
assms_4: shows "thread_oks ts \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>"
strict_not_outside_2: fixes x :: "'alt"   and y :: "'alt" assumes "x \<prec>[Pareto R] y" shows "y \<in> alts"
linorder_def: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "class.linorder less_eq less \<equiv> class.order less_eq less \<and> class.linorder_axioms less_eq"
se_preserves_finiteness1: fixes l :: "('d, 'e) label"   and c :: "('d, 'e) conf"   and c' :: "('d, 'e) conf" assumes "finite_label l"   and "se c l c'"   and "\<forall>e\<in>pred c. finite (Bexp.vars e)" shows "\<forall>e\<in>pred c'. finite (Bexp.vars e)"
in_carrier1: shows "inv \<^bold>g [^] (u1 + (order \<G> - s)) \<in> carrier \<G>"
acc_iff: fixes x :: "'a"   and y :: "'a" shows "((x, y) \<in> K1.acc) = (\<exists>n. 0 < K1.p x y n)"
trans: fixes b :: "'t \<Rightarrow>\<^sub>0 'c"   and a :: "'t \<Rightarrow>\<^sub>0 'c"   and c :: "'t \<Rightarrow>\<^sub>0 'c" assumes "b \<preceq>\<^sub>p a"   and "c \<preceq>\<^sub>p b" shows "c \<preceq>\<^sub>p a"
simps_24: fixes e1 :: "'b aexp"   and e2 :: "'b aexp"   and v :: "'b aexp"   and va :: "'b aexp" shows "AExp_Lexorder.less_aexp_aux (Times e1 e2) (Plus v va) = False"
preallocated_start: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list" shows "preallocated (start_heap P)"
isCont_minus: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "isCont f a" shows "isCont (\<lambda>x. - f x) a"
less_eq_option_Some: fixes x :: "'a"   and y :: "'a" shows "(Some x \<le> Some y) = (x \<le> y)"
ft_invar_def: fixes t :: "('b, 'c) FingerTreeStruc" shows "ft_invar t \<equiv> is_leveln_ftree 0 t \<and> is_measured_ftree t"
ft: shows "(\<lambda>x. (f (x + t) - f t) / sin (x / 2)) absolutely_integrable_on {- pi..pi}"
strict_preference_iff_strict_utility: fixes outcomes :: "'b set"   and relation :: "('b pmf \<times> 'b pmf) set"   and u :: "'b \<Rightarrow> real"   and p :: "'b pmf"   and q :: "'b pmf" assumes "vNM_utility outcomes relation u"   and "p \<in> \<P> outcomes"   and "q \<in> \<P> outcomes" shows "(p \<succeq>[relation] q \<and> (q, p) \<notin> relation) = (measure_pmf.expectation q u < measure_pmf.expectation p u)"
f_Max\<^sub>0_mono_2: fixes T :: "nat \<Rightarrow> nat"   and x :: "nat"   and y :: "nat"   and m :: "nat" assumes "T x \<le> y" shows "f_Max\<^sub>0 T m \<le> f_Max\<^sub>0 (T(x := y)) m"
imp_qr: fixes \<pp> :: "'a set" assumes "\<pp> \<in> U" shows "Comm_Ring.quotient_ring (R\<setminus>\<pp>) R (+) (\<cdot>) \<zero> \<one>"
id: shows "map_poly (complex_of_real \<circ> Re) p = p"
distinct_18: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "x41 ;; x42 \<noteq> Skip"
2: shows "Dom (local.comp f (pr1 (local.dom f) (local.dom g))) = local.set (local.prod (local.dom f) (local.dom g))"
simps_3: fixes n :: "nat"   and k :: "nat"   and T :: "POPLmark.type"   and U :: "POPLmark.type" shows "\<up>\<^sub>\<tau> n k (T \<rightarrow> U) = \<up>\<^sub>\<tau> n k T \<rightarrow> \<up>\<^sub>\<tau> n k U"
INF_ereal_add_right: fixes I :: "'a set"   and c :: "ereal"   and f :: "'a \<Rightarrow> ereal" assumes "I \<noteq> {}"   and "c \<noteq> - \<infinity>"   and "\<And>x. x \<in> I \<Longrightarrow> 0 \<le> f x" shows "(INF i\<in>I. c + f i) = c + Inf (f ` I)"
set_cases_2: fixes e :: "'c"   and a :: "('d, 'c) list_op" assumes "e \<in> set2_list_op a" obtains z1 :: "'d option" where "a = Insert z1 e"
bigstepT_the_cost: fixes c :: "com"   and s :: "char list \<Rightarrow> int"   and P :: "nat"   and T :: "char list \<Rightarrow> int" assumes "(c, s) \<Rightarrow> P \<Down> T" shows "\<down>\<^sub>t (c, s) = P"
TP_def: fixes \<Sigma> :: "((Class \<times> Method) \<times> nat) list"   and F :: "((Class \<times> Method) \<times> Var list \<times> Expr) list" shows "TP \<Sigma> F = ((\<forall>f. (\<Sigma>\<down>f = None) = (F\<down>f = None)) \<and> (\<forall>f n par e. \<Sigma>\<down>f = Some n \<longrightarrow> F\<down>f = Some (par, e) \<longrightarrow> (\<Sigma>, e, n) \<in> TP_expr))"
rep_eq: fixes x :: "real_alg_3" shows "to_rat_3 x = to_rat_2 (rep_real_alg_3 x)"
assms_2: shows "VPow a \<in>\<^sub>\<circ> Vset (ZFC_in_HOL.succ \<alpha>)"
that_1: assumes "x \<in> cipher_queries ose"   and "x \<notin> set ciphers" shows "thesis"
simps: fixes Rr :: "nat list \<Rightarrow> bool"   and xs :: "nat list"   and w :: "nat" shows "Minr Rr xs w = (let setx = {y |y. y \<le> w \<and> Rr (xs @ [y])} in if setx = {} then Suc w else Min setx)"
elims_3: fixes x :: "bool list"   and xa :: "tree" assumes "\<not> path x xa" obtains d :: "bool" and ds :: "bool list" and T1 :: "tree" and T2 :: "tree" where "x = d # ds" and "xa = Branching T1 T2" and "\<not> (if d then path ds T1 else path ds T2)"   | v :: "bool" and va :: "bool list" where "x = v # va" and "xa = Leaf"
genby_diff_closed: fixes w :: "'b"   and S :: "'b set"   and w' :: "'b" assumes "w \<in> \<langle>S\<rangle>"   and "w' \<in> \<langle>S\<rangle>" shows "w - w' \<in> \<langle>S\<rangle>"
inducts: fixes P :: "'e list \<Rightarrow> bool"   and list :: "'e list" assumes "P []"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (x1 # x2)" shows "P list"
mod_add1_eq_if: fixes a :: "nat"   and b :: "nat"   and m :: "nat" shows "(a + b) mod m = (if a mod m + b mod m < m then a mod m + b mod m else a mod m + b mod m - m)"
rlowa_rlow: shows "(rep \<propto> lowa) q = (rep \<propto> low) q"
covector_sup_closed: fixes x :: "'a"   and y :: "'a" assumes "top \<sqinter> x = x"   and "top \<sqinter> y = y" shows "top \<sqinter> (x \<squnion> y) = x \<squnion> y"