hcju/isabelleps huggingface dataset:
codomain_complex: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" assumes "ChamberComplexMorphism X Y f" shows "ChamberComplex Y"
xs: shows "those (map (\<lambda>b. bound_intersect_2d_ud p (inner2_aform Z n b) g) Basis_list) = Some xs"
mod_mult_self1_is_0: 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 b :: "'a"   and a :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (times b a) b = zero"
subst_ty_def: shows "subst_ty \<equiv> \<lambda>x Y T. ty_rec (\<lambda>X. if X = Y then T else Tvar X) Top (\<lambda>T\<^sub>1 T\<^sub>2. (\<rightarrow>)) (\<lambda>X T\<^sub>2 T\<^sub>1 T\<^sub>2' T\<^sub>1'. \<forall>X<:T\<^sub>1'. T\<^sub>2') x"
analytic_fds_eval: fixes A :: "complex set"   and f :: "complex fds" assumes "A \<subseteq> {z. conv_abscissa f < ereal (Re z)}" shows "eval_fds f analytic_on A"
AllI: fixes w :: "vbl"   and A :: "formula"   and Gamma :: "formula list"   and R :: "(formula list \<times> formula list set) set" assumes "instanceF w A # Gamma \<in> deductions R"   and "w \<notin> freeVarsFL (FAll Pos A # Gamma)"   and "Alls \<subseteq> R" shows "FAll Pos A # Gamma \<in> deductions R"
insert_not_empty: fixes a :: "'c"   and A :: "'c set" shows "Set.insert a A \<noteq> {}"
assms: fixes x :: "'a"   and y :: "'b"   and z :: "'c" shows "mf x y z \<equiv> RETURN (f x y z)"
inf_sup_aci_3: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.lattice inf less_eq less sup" shows "inf x (inf y z) = inf y (inf x z)"
name_fresh_fresh: fixes a :: "name"   and x :: "'b"   and b :: "name" assumes "a \<sharp> x"   and "b \<sharp> x" shows "[(a, b)] \<bullet> x = x"
corr_q_tq1: shows "correct q (te q (i + 1))"
sup_union_left1: fixes x :: "'k"   and N :: "'k multiset"   and M :: "'k multiset" assumes "x \<notin># N" shows "add_mset x M \<union># N = add_mset x (M \<union># N)"
is_final_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" shows "relation.is_final r a \<equiv> \<nexists>b. r a b"
staticSecretA_notin_initStateB: fixes m :: "msg"   and A :: "agent"   and B :: "agent" assumes "m \<in> staticSecret A" shows "(m \<in> initState B) = (A = B)"
dist_midpoint_1: fixes a :: "'a"   and b :: "'a" shows "dist a (midpoint a b) = dist a b / 2"
Domainp_rel_gpv''_le: fixes A :: "'a \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and R :: "'c \<Rightarrow> 'f \<Rightarrow> bool" shows "Domainp (rel_gpv'' A C R) \<le> pred_gpv' (Domainp A) (Domainp C) {x. Domainp R x}"
lens_override_overshadow: fixes Y :: "'d \<Longrightarrow> 'e"   and X :: "'f \<Longrightarrow> 'e"   and S\<^sub>1 :: "'e"   and S\<^sub>2 :: "'e"   and S\<^sub>3 :: "'e" assumes "mwb_lens Y"   and "X \<subseteq>\<^sub>L Y" shows "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"
walk_array'_Cons: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and a :: "'c"   and xs :: "'c list"   and x :: "'b"   and l :: "nat"   and p :: "nat" shows "walk_array' f (Array (a # xs)) x l (Suc p) = walk_array' f (Array xs) x l p"
uniform_group_design_axioms: shows "uniform_group_design \<V> \<B> \<G> \<m>"
nrename_AndL2: fixes R :: "trm"   and x :: "name"   and y :: "name"   and z :: "name"   and N :: "trm"   and u :: "name" assumes "R[x\<turnstile>n>y] = AndL2 z.N u"   and "z \<sharp> (R, x, y)"   and "u \<sharp> (x, y)" shows "\<exists>N'. R = AndL2 z.N' u \<and> N'[x\<turnstile>n>y] = N"
adm_not_conj: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "adm (\<lambda>x. \<not> P x)"   and "adm (\<lambda>x. \<not> Q x)" shows "adm (\<lambda>x. \<not> (P x \<and> Q x))"
mersenne_mod_less: fixes k :: "int"   and m :: "nat"   and n :: "nat" assumes "k \<le> 2 ^ m"   and "n \<le> m" shows "mersenne_mod k n < 2 ^ n + 2 ^ (m - n)"
Sup_insert_finite: fixes S :: "'b set"   and x :: "'b" assumes "finite S" shows "Sup (insert x S) = (if S = {} then x else max x (Sup S))"
intros_2: fixes A :: "'f set"   and a :: "'f" assumes "finite A" shows "finite (insert a A)"
isUCont: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_linear f" shows "isUCont f"
set_map: fixes f :: "'d \<Rightarrow> 'c"   and v :: "'d dual" shows "set_dual (map_dual f v) = f ` set_dual v"
word_of_int_uint: fixes w :: "'c word" shows "word_of_int (uint w) = w"
exec_sequence_flatten_to_exec: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>sequence Seq (flatten c),s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"
commitment_correct: fixes \<eta> :: "nat" shows "okamoto_\<Sigma>_commit.abstract_com.correct \<eta>"
orbit_def: fixes m :: "'c" shows "orbit m = same_orbit_rel `` {m}"
nilrad_def: fixes R :: "('c, 'd) Ring_scheme" shows "nilrad R = {x \<in> carrier R. nilpotent R x}"
LFP_closed: fixes L :: "('a, 'b) gorder_scheme"   and f :: "'a \<Rightarrow> 'a" assumes "weak_complete_lattice L" shows "LFP\<^bsub>L\<^esub> f \<in> carrier L"
gcd_impl_main_def: fixes G1 :: "'b poly"   and G2 :: "'b poly" shows "gcd_impl_main G1 G2 = (if G1 = 0 then 0 else if G2 = 0 then normalize G1 else Polynomial.smult (gcd (content G1) (content G2)) (gcd_impl_primitive (primitive_part G1) (primitive_part G2)))"
replicate_app_Cons_same: fixes n :: "nat"   and x :: "'a"   and xs :: "'a list" shows "replicate n x @ x # xs = x # replicate n x @ xs"
preserves_comp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "CC.seq g f" shows "T (CC.comp g f) = T g \<cdot> T f"
induct: fixes G :: "prog"   and accclass :: "qtname"   and x1 :: "qtname \<times> memberdecl"   and x2 :: "qtname"   and P :: "qtname \<times> memberdecl \<Rightarrow> qtname \<Rightarrow> bool" assumes "G \<turnstile> x1 of x2 accessible_from accclass"   and "\<And>membr class. \<lbrakk>G \<turnstile> membr member_of class; G \<turnstile> Class class accessible_in pid accclass; G \<turnstile> membr in class permits_acc_from accclass\<rbrakk> \<Longrightarrow> P membr class"   and "\<And>membr class C new old supr. \<lbrakk>G \<turnstile> membr member_of class; G \<turnstile> Class class accessible_in pid accclass; membr = (C, mdecl new); G \<turnstile> (C, new) overrides\<^sub>S old; G\<turnstile>class\<prec>\<^sub>C supr; G \<turnstile>Method old of supr accessible_from accclass; P (methdMembr old) supr\<rbrakk> \<Longrightarrow> P membr class" shows "P x1 x2"
extends_to_prebicategory_with_units: shows "prebicategory_with_units (\<cdot>) (\<star>) \<a> some_unit"
scale_zero_left: fixes x :: "'a" shows "0 *\<^sub>C x = (0::'a)"
ik\<^sub>s\<^sub>s\<^sub>t_subst: fixes A :: "('e, 'f) stateful_strand_step list"   and \<delta> :: "'f \<Rightarrow> ('e, 'f) Term.term" shows "ik\<^sub>s\<^sub>s\<^sub>t (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>) = ik\<^sub>s\<^sub>s\<^sub>t A \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta>"
ResCong: fixes P :: "pi"   and Q :: "pi"   and x :: "name" assumes "(P, Q) \<in> Strong_Early_Bisim.bisim O bangRel weakBisim O Strong_Early_Bisim.bisim" shows "(<\<nu>x>P, <\<nu>x>Q) \<in> Strong_Early_Bisim.bisim O bangRel weakBisim O Strong_Early_Bisim.bisim"
rotater1_map: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "rotater1 (map f xs) = map f (rotater1 xs)"
graph_eq_iff: fixes g :: "'b ugraph"   and g' :: "'b ugraph" shows "(g = g') = (nodes g = nodes g' \<and> edges g = edges g')"
select_convs_1: fixes nodes :: "'a set"   and edges :: "('a \<times> 'a) set"   and more :: "'b" shows "nodes \<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = nodes"
simps_11: fixes x3 :: "nat"   and x2 :: "nat" shows "HOLD x3 \<noteq> WAIT x2"
m\<mu>_finite_thrD_2: fixes m :: "'m"   and ts :: "'t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option"   and ts' :: "'t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option" assumes "m\<mu>t m ts ts'" shows "finite (dom ts')"
bv_to_nat_rew_msb: fixes w :: "bit list" assumes "bv_msb w = \<one>" shows "bv_to_nat w = 2 ^ (length w - 1) + bv_to_nat (tl w)"
cut_greater_image: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and I :: "'a set"   and n :: "'a" assumes "SetInterval2.strict_mono_on f A"   and "I \<subseteq> A"   and "n \<in> A" shows "(f ` I) \<down>> f n = f ` I \<down>> n"
interchange_rows_j: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and j :: "'c" shows "interchange_rows A i j $ j = A $ i"
simps_10: fixes x41 :: "('a \<Rightarrow> 'b option) \<times> 'c \<Rightarrow> bool"   and x42 :: "'d"   and x43 :: "('a \<Rightarrow> 'b option) \<Rightarrow> ('a \<Rightarrow> 'b option) \<Rightarrow> 'a \<Rightarrow> 'b option"   and x1 :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a \<Rightarrow> 'b option" shows "x41\<hookleftarrow>\<^bsub>x42\<^esub>x43 \<noteq> \<Up>x1"
nat_of_uint64_uint64_of_nat_id: fixes n :: "nat" assumes "n < 2 ^ 64" shows "nat_of_uint64 (uint64_of_nat n) = n"
prod_fun_def: fixes f1 :: "'r \<Rightarrow> 'q"   and f2 :: "'r \<Rightarrow> 'q"   and k :: "'r" shows "prod_fun f1 f2 k = (\<Sum>l. f1 l * (\<Sum>q. f2 q when k = l + q))"
finalRefE: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty" assumes "P,E,h \<turnstile> e : T"   and "is_refT T"   and "final e" obtains "e = null"   | a :: "nat" and C :: "char list" where "e = addr a__" and "T = Class C"   | a :: "nat" where "e = Throw a"
finite_set_in: fixes x :: "'a"   and FS :: "'a fset" shows "(x \<in> fset FS) = (x |\<in>| FS)"
sum_card: fixes U :: "'a set set"   and n :: "nat" assumes "\<forall>S\<in>U. card S \<le> n" shows "sum card U \<le> card U * n"
\<omega>_oexp_iff: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "Ord \<alpha>"   and "Ord \<beta>" shows "(\<omega> \<up> \<alpha> = \<omega> \<up> \<beta>) = (\<alpha> = \<beta>)"
left_def: shows "left = (\<lambda>f lb ub. {x. jumpF f (at_left x) \<noteq> 0 \<and> lb \<le> x \<and> x \<le> ub})"
prj_emb_prj: fixes x :: "udom" assumes "DEFL('a) \<sqsubseteq> DEFL('b)" shows "PRJ('a)\<cdot>(EMB('b)\<cdot>(PRJ('b)\<cdot>x)) = PRJ('a)\<cdot>x"
greatestPD2: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "greatestP P x" shows "Collect P *<= x"
h2: shows "well_def_signs (length (qs1 @ qs2)) (get_signs (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2))))"
IdE: fixes p :: "'a \<times> 'a" assumes "p \<in> Id" obtains x :: "'a" where "p = (x, x)"
dirichlet_prod_0: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" shows "dirichlet_prod f g 0 = (0::'b)"
cpx_mat_cnj_def: fixes M :: "complex mat" shows "M\<^sup>\<star> \<equiv> mat (dim_row M) (dim_col M) (\<lambda>(i, j). cnj (M $$ (i, j)))"
sfis_sum: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and S :: "'c set"   and a :: "'c \<Rightarrow> real"   and f :: "'c \<Rightarrow> 'b \<Rightarrow> real" assumes "measure_space M"   and "\<forall>i\<in>S. a i \<in> sfis (f i) M"   and "finite S" shows "sum a S \<in> sfis (\<lambda>t. \<Sum>i\<in>S. f i t) M"
cod_inv: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "category.iso C f" shows "partial_magma.cod C (category.inv C f) = partial_magma.dom C f"
1: shows "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E (C \<inter> D)"
C'_p: shows "instance_of\<^sub>l\<^sub>s C' C \<and> falsifies\<^sub>g [] C'"
Limsup_eqI: fixes F :: "'b filter"   and x :: "'d"   and f :: "'b \<Rightarrow> 'd" assumes "\<And>P. eventually P F \<Longrightarrow> x \<le> Sup (f ` Collect P)"   and "\<And>y. (\<And>P. eventually P F \<Longrightarrow> y \<le> Sup (f ` Collect P)) \<Longrightarrow> y \<le> x" shows "Limsup F f = x"
adds_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and s :: "'b" assumes "class.ninv_comm_monoid_add plus zero" shows "plus.adds plus s zero = (s = zero)"
size_mset_set: fixes A :: "'c set" shows "size (mset_set A) = card A"
integrableI_nonneg: fixes f :: "'c \<Rightarrow> real"   and M :: "'c measure" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M < \<infinity>" shows "integrable M f"
someI2_ex: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<exists>a. P a"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (SOME x. P x)"
get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('c, 'a, 'b, 'd, 'g option) RElement_scheme \<Rightarrow> 'l" assumes "DocumentClass.type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter"
gbm_\<alpha>_def: fixes m1 :: "'m1" shows "gbm_\<alpha> m1 \<equiv> \<lparr>nodes = dom (m1.\<alpha> m1), edges = {(v, w, v'). \<exists>m2 s3. m1.\<alpha> m1 v = Some m2 \<and> m2.\<alpha> m2 v' = Some s3 \<and> w \<in> s3.\<alpha> s3}\<rparr>"
konigDtree: fixes inv :: "'sequent \<times> 'a \<Rightarrow> bool"   and pred :: "'sequent \<times> 'a \<Rightarrow> 'rule \<Rightarrow> 'sequent \<times> 'a \<Rightarrow> bool"   and t :: "('sequent \<times> 'rule) tree"   and a :: "'a" assumes "\<And>r s sl a. \<lbrakk>r \<in> R; eff r s sl; inv (s, a)\<rbrakk> \<Longrightarrow> \<exists>s' a'. s' |\<in>| sl \<and> inv (s', a') \<and> pred (s, a) r (s', a')"   and "local.wf t"   and "inv (fst (tree.root t), a)" shows "alw (\<lambda>stepas. let ((s, r), a) = shd stepas; ((s', uu_), a') = shd (stl stepas) in inv (s, a) \<and> pred (s, a) r (s', a')) (konigDtree inv pred t a)"
intro: fixes U :: "'a set"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and inverse :: "'a \<Rightarrow> 'a" assumes "semigroup_ow U f"   and "group_ow_axioms U f z inverse" shows "group_ow U f z inverse"
assms: shows "targets \<mu> \<inter> targets \<nu> \<noteq> {}"
worecZSL_def: fixes Z :: "'c"   and S :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and L :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" shows "worecZSL Z S L \<equiv> worecSL S (\<lambda>f a. if a = zero then Z else L f a)"
Invariants_quasi_Invariants_sets: fixes A :: "'a set" assumes "A \<in> sets M" shows "(\<exists>B\<in>sets Invariants. A \<Delta> B \<in> null_sets M) = (T --` A \<Delta> A \<in> null_sets M)"
deriv: fixes \<xi> :: "real" assumes "x \<le> \<xi>"   and "\<xi> \<le> x + a" shows "\<exists>D. ((\<lambda>xa. (xa - x)\<^sup>2 * f' / 2 + xa * inverse x - ln xa) has_real_derivative D) (at \<xi>) \<and> 0 \<le> D"
A: shows "ipurge_tr I (c_dom D) (D x) (c_tr step out (step (foldl step s\<^sub>0 xs) x) ys) = ipurge_tr I (c_dom D) (D x) (c_tr step out (foldl step s\<^sub>0 xs) ys)"
lists_drop_emp: fixes us :: "'b list list"   and C :: "'b list set" assumes "us \<in> lists C\<^sub>+" shows "us \<in> lists C"
1: assumes "s\<^sub>1 \<leadsto> s\<^sub>2"   and "s\<^sub>1 \<leadsto> s\<^sub>2'" shows "\<exists>s\<^sub>3 s\<^sub>3'. s\<^sub>3 \<approx> s\<^sub>3' \<and> (s\<^sub>2, s\<^sub>3) \<in> [\<leadsto>]\<^sup>= \<and> (s\<^sub>2', s\<^sub>3') \<in> [\<leadsto>]\<^sup>="
WT_send: fixes \<eta> :: "nat"   and s :: "'a list cstate" shows "\<I>_uniform (nlists UNIV \<eta>) UNIV \<turnstile>c S_CHAN.send_oracle s \<surd>"
A_inst: shows "has_all_wt_instances_of \<Gamma> (set A') (set A)"
gtrp2gtr: fixes s :: "'c"   and c :: "'c multiset"   and w :: "'d el_step list"   and s' :: "'c"   and c' :: "'c multiset"   and T :: "(('c \<times> 'c multiset) \<times> 'd \<times> 'c \<times> 'c multiset) set" assumes "((s, c), w, s', c') \<in> trcl (gtrp T)" shows "(add_mset s c, map le_rem_s w, add_mset s' c') \<in> trcl (gtr T)"
assms_3: shows "\<forall>p\<in>set (smods p q). \<forall>x. a < x \<and> x \<le> a' \<or> a' \<le> x \<and> x < a \<longrightarrow> poly p x \<noteq> 0"
sec2: assumes "exception_\<I> (\<I>_real' \<oplus>\<^sub>\<I> \<I>_common') \<turnstile>g \<A> \<surd>" shows "constructive_security_sim_obsf real_resource ideal_resource sim \<I>_real \<I>_common (absorb \<A> (obsf_converter (w_adv_real |\<^sub>= w_usr))) adv"
assms_1: shows "\<Gamma>\<turnstile> \<langle>Guard f g c,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F)"
sup_red_f_in_red_liminf: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set"   and Ns :: "'a set llist" assumes "calculus Bot Inf entails Red_I Red_F"   and "chain (calculus.derive Red_F) Ns" shows "Sup_llist (lmap Red_F Ns) \<subseteq> Red_F (Liminf_llist Ns)"
left_neutral: fixes a :: "'a" shows "\<zero> \<oplus> a = a"
Runs_lappendE: fixes \<sigma> :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ((status \<times> 'x) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set"   and tls :: "('thread_id \<times> ('addr, 'thread_id, status \<times> 'x, 'heap, 'addr, ('addr, 'thread_id) obs_event action) thread_action) llist"   and tls' :: "('thread_id \<times> ('addr, 'thread_id, status \<times> 'x, 'heap, 'addr, ('addr, 'thread_id) obs_event action) thread_action) llist" assumes "h.mthr.if.mthr.Runs \<sigma> (lappend tls tls')"   and "lfinite tls" obtains \<sigma>' :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ((status \<times> 'x) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set" where "h.mthr.if.mthr.Trsys \<sigma> (list_of tls) \<sigma>'" and "h.mthr.if.mthr.Runs \<sigma>' tls'"
x_dif_y: shows "xs \<noteq> ys \<or> x \<noteq> y"
elims: fixes x :: "real"   and xa :: "nat"   and y :: "real" assumes "det_discount x xa = y" obtains r :: "real" where "x = ra__" and "xa = 0" and "y = 1"   | r :: "real" and n :: "nat" where "x = r" and "xa = Suc n" and "y = inverse (1 + r) * det_discount r n"
spaceN_cmult: fixes x :: "'c"   and N :: "'c quasinorm"   and c :: "real" assumes "x \<in> space\<^sub>N N" shows "c *\<^sub>R x \<in> space\<^sub>N N"
assms_3: fixes x :: "'a" assumes "x \<in> A" shows "B x \<noteq> {}"
mkAutoSim_simps_1: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'f list"   and envTrans :: "'f \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'g"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'h"   and jviewInit :: "'b \<Rightarrow> 'h \<Rightarrow> 'g"   and jviewIncr :: "'b \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> 'g"   and simf :: "'e Trace \<Rightarrow> 'i"   and simRels :: "'b \<Rightarrow> ('i \<times> 'i) set"   and simVal :: "'i \<Rightarrow> 'c \<Rightarrow> bool"   and simAbs :: "'j \<Rightarrow> 'i set"   and simObs :: "'b \<Rightarrow> 'j \<Rightarrow> 'h"   and simInit :: "'b \<Rightarrow> 'h \<Rightarrow> 'j"   and simTrans :: "'b \<Rightarrow> 'j \<Rightarrow> 'j list"   and simAction :: "'b \<Rightarrow> 'j \<Rightarrow> 'd list"   and a :: "'b" assumes "AlgSimIncrEnvironment jkbp envInit envAction envTrans envVal jview envObs jviewInit jviewIncr simf simRels simVal simAbs simObs simInit simTrans simAction" shows "pInit (AlgSimIncrEnvironment.mkAutoSim simObs simInit simTrans simAction a) = simInit a"
injection_union: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" assumes "inj_on f A"   and "inj_on f B"   and "f ` A \<inter> f ` B = {}" shows "inj_on f (A \<union> B)"
convergent_cong: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in sequentially. f x = g x" shows "convergent f = convergent g"
degree_le: fixes n :: "nat"   and p :: "'b poly" assumes "\<forall>i>n. coeff p i = (0::'b)" shows "degree p \<le> n"
K_Boole: fixes p :: "'b fm"   and G :: "'b fm list" assumes "\<turnstile> imply ((\<^bold>\<not> p) # G) \<^bold>\<bottom>" shows "\<turnstile> imply G p"
eq_inf_symp: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a stream"   and w\<^sub>2 :: "'a stream" assumes "traces ind"   and "traces.eq_inf ind w\<^sub>1 w\<^sub>2" shows "traces.eq_inf ind w\<^sub>2 w\<^sub>1"
simps: fixes a1 :: "ccs"   and a2 :: "ccs" shows "a1 \<sim> a2 = (\<exists>P Q. a1 = P \<and> a2 = Q \<and> P \<leadsto>[bisim] Q \<and> Q \<sim> P)"
blinfun_scaleR_def: fixes a :: "'e \<Rightarrow>\<^sub>L real"   and b :: "'f" shows "blinfun_scaleR a b = blinfun_scaleR_left b o\<^sub>L a"
\<theta>_upper_bound: fixes x :: "real" assumes "1 \<le> x" shows "primes_theta x < ln 4 * x"
d_OUT_restrict_current_le: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" shows "d_OUT (g \<upharpoonleft> \<Gamma> / f) x \<le> d_OUT g x"
heap_del_locs: fixes s :: "state" shows "heap (del_locs s) = heap s"
mem_equal_mod_cache: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36"   and new_cache1 :: "cpu_cache"   and new_cache2 :: "cpu_cache" assumes "mem_equal s1 s2 pa" shows "mem_equal (s1\<lparr>cache := new_cache1\<rparr>) (s2\<lparr>cache := new_cache2\<rparr>) pa"
env_axioms: shows "env Finca_empty Finca_get Finca_add Finca_to_list"
chainE: fixes Y :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "chain Y" shows "Y i \<sqsubseteq> Y (Suc i)"
covector_conv_vector: fixes v :: "'a" shows "covector v = (v[[top]] = v\<^sup>T)"
extends_cons: fixes S :: "'b list"   and x :: "'b" shows "S \<lesssim> x # S"
span_is_subset2: fixes R :: "('g, 'h) ring_scheme"   and M :: "('g, 'i, 'j) module_scheme"   and S :: "'i set" assumes "module R M"   and "S \<subseteq> carrier M" shows "LinearCombinations.module.span R M S \<subseteq> carrier M"
simps_2: fixes x :: "brick"   and y :: "brick list" shows "domain_block (x # y) = domain x + domain_block y"
elims_3: fixes x :: "'a option"   and xa :: "'a option" assumes "\<not> x \<le> xa" obtains xb :: "'a" and y :: "'a" where "x = Some xb" and "xa = Some y" and "\<not> xb \<le> y"   | uu_ :: "'a" where "x = Some uu_" and "xa = None"
assms_2: shows "is_delete m \<or> m \<notin> T"
covim: shows "ov\<inverse> O m \<subseteq> ov \<union> f\<inverse> \<union> d\<inverse>"
fmdom'_fmupd: fixes a :: "'c"   and b :: "'d"   and m :: "('c, 'd) fmap" shows "fmdom' (m(a $$:= b)) = insert a (fmdom' m)"
prop_restrict: fixes x :: "'a"   and Z :: "'a set"   and X :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "x \<in> Z"   and "Z \<subseteq> {x \<in> X. P x}" shows "P x"
completely_multiplicative_fds_inverse': fixes f :: "'b fds" assumes "completely_multiplicative_function (fds_nth f)" shows "inverse f = fds (\<lambda>n. moebius_mu n * fds_nth f n)"
inf_sup_aci_7: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.lattice inf less_eq less sup" shows "sup x (sup y z) = sup y (sup x z)"
hyps: shows "FWNormalisationCore.sorted p l \<longrightarrow> all_in_list p l \<longrightarrow> distinct p \<longrightarrow> allNetsDistinct p \<longrightarrow> singleCombinators p \<longrightarrow> wellformed_policy2 p"