hcju/isabelleps huggingface dataset:
mrw_values_eq_SomeD: fixes P :: "'c prog"   and vs0 :: "'a \<times> addr_loc \<Rightarrow> ('a val \<times> bool) option"   and obs :: "('a, 'b) obs_event action list"   and ad :: "'a"   and al :: "addr_loc"   and v :: "'a val"   and b :: "bool" assumes "mrw_values P vs0 obs (ad, al) = \<lfloor>(v, b)\<rfloor>"   and "vs0 (ad, al) = \<lfloor>(v, b)\<rfloor> \<Longrightarrow> \<exists>wa. wa \<in> set obs \<and> is_write_action wa \<and> (ad, al) \<in> action_loc_aux P wa \<and> (b \<longrightarrow> \<not> is_new_action wa)" shows "\<exists>obs' wa obs''. obs = obs' @ wa # obs'' \<and> is_write_action wa \<and> (ad, al) \<in> action_loc_aux P wa \<and> value_written_aux P wa al = v \<and> is_new_action wa = (\<not> b) \<and> (\<forall>ob\<in>set obs''. is_write_action ob \<longrightarrow> (ad, al) \<in> action_loc_aux P ob \<longrightarrow> is_new_action ob \<and> b)"
4: shows "(\<Union>s. trees_cyl (map_tree (\<lambda>_. space M) s)) = space (tree_sigma M)"
C\<^sub>3s: shows "\<^bold>\<forall>x. ((\<^bold>\<forall>xa. (dom x) \<cdot> xa \<^bold>= xa \<^bold>\<leftarrow> E ((dom x) \<cdot> xa)) \<^bold>\<and> (\<^bold>\<forall>xa. xa \<cdot> (dom x) \<^bold>= xa \<^bold>\<leftarrow> E (xa \<cdot> (dom x)))) \<^bold>\<and> E (x \<cdot> (dom x))"
elims_2: fixes x :: "v sem_env" assumes "is_cupcake_all_env x" obtains v0 :: "(char list \<times> v) list" and c0 :: "(char list, char list, nat \<times> tid_or_exn) namespace" where "x = make_sem_env (Bind v0 []) c0" and "c0 = static_cenv \<and> list_all (is_cupcake_value \<circ> snd) v0"
simps_2: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm" shows "\<Gamma> \<turnstile> s is t : TBase = \<Gamma> \<turnstile> s \<Leftrightarrow> t : TBase"
mk_ps_chain_ccons: fixes \<sigma> :: "node \<rightharpoonup>\<^sub>f topbot"   and x :: "node + node fset \<times> command_gadget \<times> node fset"   and \<Pi> :: "(node \<rightharpoonup>\<^sub>f topbot, node + node fset \<times> command_gadget \<times> node fset) chain"   and \<pi> :: "(node + node fset \<times> command_gadget \<times> node fset) list" shows "mk_ps_chain (\<lbrace> \<sigma> \<rbrace> \<cdot> x \<cdot> \<Pi>) \<pi> = \<lbrace> \<sigma> \<rbrace> \<cdot> x \<cdot> mk_ps_chain \<Pi> \<pi>"
filter_partition_by_median_1: fixes l :: "(real, 'a) vec list"   and m :: "real"   and r :: "(real, 'a) vec list"   and k :: "'a"   and ps :: "(real, 'a) vec list" assumes "(l, m, r) = partition_by_median k ps" shows "\<forall>p\<in>set l. p $ k \<le> m"
is_Pointwise_Output_Shrink_list_eq_is_Pointwise_Output_Shrink_all: fixes pointwise_shrink :: "'d list \<Rightarrow> 'd"   and fs :: "('d \<Rightarrow> 'e message_af) list" shows "is_Pointwise_Output_Shrink_list pointwise_shrink fs = (\<forall>output_fun\<in>set fs. is_Pointwise_Output_Shrink pointwise_shrink output_fun)"
degree_oneE: fixes p :: "'a list list"   and K :: "'a list set" assumes "p \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>)"   and "degree p = 1" obtains a :: "'a list" and b :: "'a list" where "a \<in> K" and "a \<noteq> \<zero>\<^bsub>poly_ring R\<^esub>" and "b \<in> K" and "p = [a, b]"
fpxs_of_fls_eq_iff: fixes f :: "'b fls"   and g :: "'b fls" shows "(fpxs_of_fls f = fpxs_of_fls g) = (f = g)"
Nats_mult: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_1 one times plus zero"   and "a \<in> semiring_1.Nats one plus zero"   and "b \<in> semiring_1.Nats one plus zero" shows "times a b \<in> semiring_1.Nats one plus zero"
A: shows "\<forall>xs\<in>traces P. \<forall>X. (X \<in> refusals P xs) = (X \<inter> next_events P xs = {})"
purgeIdle_def: shows "purgeIdle \<equiv> filter isState"
in_set_conv_nth: fixes x :: "'d"   and xs :: "'d list" shows "(x \<in> set xs) = (\<exists>i<length xs. xs ! i = x)"
Dexp: fixes t :: "'b tree" shows "2 ^ rlh t \<le> size t + 1"
comp\<^sub>2_correct: fixes P\<^sub>1 :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and body :: "nat exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and ls :: "val list"   and e' :: "nat exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and ls' :: "val list" assumes "P\<^sub>1 \<turnstile> C sees M: Ts\<rightarrow>T = body in C"   and "P\<^sub>1 \<turnstile>\<^sub>1 \<langle>body,(h, ls)\<rangle> \<Rightarrow> \<langle>e',(h', ls')\<rangle>" shows "compP\<^sub>2 P\<^sub>1 \<turnstile> (None, h, [([], ls, C, M, 0)]) -jvm\<rightarrow> (exception e', h', [])"
key_sets_range: fixes U :: "msg set"   and X :: "msg"   and H :: "msg set" assumes "U \<in> key_sets X H" shows "U \<subseteq> range Key"
extended_monoidal_category_def: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha> :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota> :: "'d" shows "extended_monoidal_category C T \<alpha> \<iota> \<equiv> (monoidal_category C T \<alpha> \<iota> \<and> inverse_transformation (product_category.comp C (product_category.comp C C)) C (binary_endofunctor.ToTC C T) (binary_endofunctor.ToCT C T) \<alpha> \<and> natural_isomorphism C C (\<lambda>f. T (monoidal_category.unity C \<iota>, f)) (identity_functor.map C) (\<lambda>f. if partial_magma.arr C f then C f (monoidal_category.lunit C T \<alpha> \<iota> (partial_magma.dom C f)) else partial_magma.null C)) \<and> inverse_transformation C C (\<lambda>f. T (monoidal_category.unity C \<iota>, f)) (identity_functor.map C) (\<lambda>f. if partial_magma.arr C f then C f (monoidal_category.lunit C T \<alpha> \<iota> (partial_magma.dom C f)) else partial_magma.null C) \<and> natural_isomorphism C C (\<lambda>f. T (f, monoidal_category.unity C \<iota>)) (identity_functor.map C) (\<lambda>f. if partial_magma.arr C f then C f (monoidal_category.runit C T \<alpha> \<iota> (partial_magma.dom C f)) else partial_magma.null C) \<and> inverse_transformation C C (\<lambda>f. T (f, monoidal_category.unity C \<iota>)) (identity_functor.map C) (\<lambda>f. if partial_magma.arr C f then C f (monoidal_category.runit C T \<alpha> \<iota> (partial_magma.dom C f)) else partial_magma.null C)"
eq\<^sub>h: shows "(h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h"
klift_eta_pres: shows "\<eta>\<^sup>\<dagger> = id"
CEMC_axioms_def: shows "CEMC_axioms \<equiv> u = (THE x. \<forall>y. P y x)"
emeasure_finite: fixes s :: "'s"   and A :: "'s stream set" shows "emeasure (T s) A \<noteq> \<top>"
TupleLiteralTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and elems :: "(String.literal \<times> 'a OCL_Types.type option \<times> 'a expr) list"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E Literal (TupleLiteral elems) : \<tau>" obtains "elems = []" and "\<tau> = Tuple fmempty"   | elemsa :: "(String.literal \<times> 'a OCL_Types.type option \<times> 'a expr) list" and \<xi> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" and el :: "String.literal \<times> 'a OCL_Types.type option \<times> 'a expr" and \<tau>' :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" where "elems = el # elemsa" and "\<tau> = Tuple (\<xi>(tuple_element_name el \<mapsto>\<^sub>f \<sigma>))" and "\<Gamma> \<turnstile>\<^sub>E Literal (TupleLiteral elemsa) : Tuple \<xi>" and "\<Gamma> \<turnstile>\<^sub>E tuple_element_expr el : \<tau>'" and "tuple_element_type el = Some \<sigma>" and "\<tau>' \<le> \<sigma>"
assms_3: shows "s' \<in> state_successors (prob_proj PROB vs) s"
SA_Top: fixes \<Gamma> :: "binding list"   and S :: "ty" assumes "\<turnstile> \<Gamma> ok"   and "S closed_in \<Gamma>" shows "\<Gamma>\<turnstile>S<:Top"
wlsFSw_recAll_termFSwMorph: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) Recursion.model" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.wlsFSw Delta MOD" shows "FixSyn.termFSwMorph Delta (FixSyn.rec Delta MOD) (FixSyn.recAbs Delta MOD) MOD"
5: shows "COD f = Dom (Inv (DOM f\<^bold>\<down>) \<^bold>\<cdot> \<^bold>\<lfloor>rep f\<^bold>\<rfloor> \<^bold>\<cdot> COD f\<^bold>\<down>)"
fls_base_factor_mult_both_base_factor: fixes f :: "'a fls"   and g :: "'a fls" shows "fls_base_factor (fls_base_factor f * fls_base_factor g) = fls_base_factor (f * g)"
simulation_\<tau>Runs_table1: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and s1 :: "'g"   and s2 :: "'i"   and stlsss1 :: "('h \<times> 'g, 'g option) tllist" assumes "delay_bisimulation_diverge trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2"   and "bisim s1 s2"   and "\<tau>trsys.\<tau>Runs_table trsys1 \<tau>move1 s1 stlsss1" shows "\<exists>stlsss2. \<tau>trsys.\<tau>Runs_table trsys2 \<tau>move2 s2 stlsss2 \<and> tllist_all2 (\<lambda>(tl1, s1'') (tl2, s2''). tlsim tl1 tl2 \<and> bisim s1'' s2'') (rel_option bisim) stlsss1 stlsss2"
upper_power_float_interval: fixes p :: "nat"   and n :: "nat"   and x :: "float interval" shows "upper (power_float_interval p n x) = snd (float_power_bnds p n (lower x) (upper x))"
call_edges_only: fixes a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "\<forall>a'. valid_edge a' \<and> targetnode a' = targetnode a \<longrightarrow> (\<exists>Qx rx fsx. kind a' = Qx:rx\<hookrightarrow>\<^bsub>p\<^esub>fsx)"
AG_induct: fixes p :: "'b set" shows "p \<inter> \<^bold>A\<^bold>G (p \<rightarrow> \<^bold>A\<^bold>X p) = \<^bold>A\<^bold>G p"
Kbase_nonzero: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set" assumes "distinct_pds K n P" shows "\<forall>j\<le>n. (Kb\<^bsub>K n P\<^esub>) j \<noteq> \<zero>"
P1E: fixes f :: "nat \<Rightarrow> nat option" assumes "f \<in> \<P>" obtains r :: "recf" where "recfn 1 r" and "\<forall>x. eval r [x] = f x"
assms_14: shows "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"
assms_1: shows "i < dim_row (Matrix.mat (2 ^ n) (2 ^ n) (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> (- 1) ^ i \<cdot>\<^bsub>n\<^esub>  j / sqrt 2 ^ n)))"
add_order_preserving_right: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and v :: "'b" assumes "class.lgroup minus plus zero uminus inf less_eq less sup"   and "less_eq a b" shows "less_eq (plus a v) (plus b v)"
cod_comp: fixes g :: "'c"   and f :: "'c" assumes "C.seq g f" shows "C.cod (g \<cdot> f) = C.cod g"
measurable_on_0: fixes S :: "'a set" shows "(\<lambda>x. 0::'b) measurable_on S"
Psi_def: shows "Psi \<equiv> Init InitPsi \<and> \<box>[N1 \<or> N2]_(x, y, sem, pc1, pc2) \<and> SF(N1)_(x, y, sem, pc1, pc2) \<and> SF(N2)_(x, y, sem, pc1, pc2)"
integrable_cong_AE_imp: fixes M :: "'b measure"   and g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" assumes "integrable M g"   and "f \<in> borel_measurable M"   and "AE x in M. g x = f x" shows "integrable M f"
vimage2p_Grp: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "BNF_Def.vimage2p f g P = BNF_Def.Grp UNIV f OO P OO (BNF_Def.Grp UNIV g)\<inverse>\<inverse>"
proots_unbounded_line_card_def: fixes p :: "complex poly"   and st :: "complex"   and tt :: "complex" shows "proots_unbounded_line_card p st tt = card (proots_within p (unbounded_line st tt))"
simps_4: fixes v :: "poly"   and va :: "poly" shows "isnpolyh (Add v va) = (\<lambda>k. False)"
elims_3: fixes x :: "('form \<times> nat list) \<times> ('form, 'var) out_port" assumes "\<not> valid_out_port x" obtains v :: "'form \<times> nat list" and p :: "('form, 'var) out_port" where "x = (v, p)" and "\<not> (v |\<in>| vertices \<and> p |\<in>| outPorts (nodeOf v))"
one_chain_i_integrals: shows "one_chain_line_integral F {i} one_chain_typeI = one_chain_line_integral F {i} one_chain_typeII \<and> (\<forall>(k, \<gamma>)\<in>one_chain_typeI. line_integral_exists F {i} \<gamma>) \<and> (\<forall>(k, \<gamma>)\<in>one_chain_typeII. line_integral_exists F {i} \<gamma>)"
sim3: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi" assumes "P \<leadsto>\<^sup>^<Rel> Q"   and "\<And>Q'. Q \<longmapsto> \<tau> \<prec> Q' \<Longrightarrow> (P, Q') \<notin> Rel" shows "P \<leadsto><Rel> Q"
crdt_ops_ref_exists: fixes pre :: "('c \<times> 'd) list"   and oid :: "'c"   and oper :: "'d"   and suf :: "('c \<times> 'd) list"   and deps :: "'d \<Rightarrow> 'c set"   and ref :: "'c" assumes "crdt_ops (pre @ (oid, oper) # suf) deps"   and "ref \<in> deps oper" shows "ref \<in> fst ` set pre"
t_loop_4: shows "\<C>' = \<C>\<lparr>label := [t1s] @ label \<C>\<rparr>"
D_def: shows "D = {x. Some (fst x) = snd x}"
\<gamma>\<^sub>h: shows "\<guillemotleft>\<gamma>\<^sub>h : w\<^sub>h \<rightarrow> w\<^sub>h'\<guillemotright> \<and> \<beta>\<^sub>h = tab\<^sub>1 h \<star> \<gamma>\<^sub>h \<and> \<theta>\<^sub>h = \<theta>\<^sub>h' \<cdot> (tab\<^sub>0 h \<star> \<gamma>\<^sub>h)"
simps_9: fixes sh :: "sheap"   and e\<^sub>1 :: "'a exp"   and F :: "char list"   and D :: "char list"   and e\<^sub>2 :: "'a exp" shows "iconf sh (e\<^sub>1\<bullet>F{D} := e\<^sub>2) = (case val_of e\<^sub>1 of None \<Rightarrow> iconf sh e\<^sub>1 \<and> \<not> sub_RI e\<^sub>2 | \<lfloor>v\<rfloor> \<Rightarrow> iconf sh e\<^sub>2)"
transform_irule_set_iter_def: fixes rs :: "(name \<times> (Term_Class.term list \<times> pterm) fset) fset" shows "Compiler.transform_irule_set_iter rs = ((transform_irule_set \<circ> compress_irule_set) ^^ max_arity rs) rs"
rel_fun_map_fun2: fixes h :: "'g \<Rightarrow> 'd"   and A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" assumes "rel_fun (eq_on (range h)) A f g" shows "rel_fun (BNF_Def.Grp UNIV h)\<inverse>\<inverse> A f (map_fun h id g)"
simps_3: fixes c :: "poly"   and n :: "nat"   and p :: "poly"   and c' :: "int \<times> int" shows "CN c n p *\<^sub>p C c' = (if c' = 0\<^sub>N then 0\<^sub>p else CN (c *\<^sub>p C c') n (p *\<^sub>p C c'))"
in_fimage_Suc: fixes x :: "nat"   and A :: "nat fset" shows "(x |\<in>| Suc |`| A) = (\<exists>y. y |\<in>| A \<and> x = Suc y)"
prob_compl: fixes A :: "real set"   and n :: "nat" assumes "A \<in> Mn.events n" shows "Mn.prob n (space (M n) - A) = 1 - Mn.prob n A"
relInvImage_mono: fixes R1 :: "('d \<times> 'd) set"   and R2 :: "('d \<times> 'd) set"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'd" assumes "R1 \<subseteq> R2" shows "BNF_Greatest_Fixpoint.relInvImage A R1 f \<subseteq> BNF_Greatest_Fixpoint.relInvImage A R2 f"
Beta: fixes r :: "dB"   and s :: "dB"   and ss :: "dB list" assumes "IT (r[s/0] \<degree>\<degree> ss)"   and "IT s" shows "IT (Abs r \<degree> s \<degree>\<degree> ss)"
meta_adjunction_axioms: shows "meta_adjunction C D F G (\<lambda>x. G) (\<lambda>y. F)"
eq_iff: fixes a :: "'c multiset"   and b :: "'c multiset" shows "(a = b) = (b \<subseteq># a \<and> a \<subseteq># b)"
cong_mult: 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 division_segment :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a"   and d :: "'a"   and e :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a"   and "unique_euclidean_semiring.cong modulo d e a" shows "unique_euclidean_semiring.cong modulo (times b d) (times c e) a"
assms: shows "RETURN c \<le> \<Down> R (RETURN a)"
card_of_ordIsoI: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "bij_betw f A B" shows "|A| =o |B|"
induct: fixes P :: "'c match_expr \<Rightarrow> bool"   and a0 :: "'c match_expr" assumes "P MatchAny"   and "\<And>uu_. P (Match uu_)"   and "P (MatchNot MatchAny)"   and "\<And>uv_. P (MatchNot (Match uv_))"   and "\<And>m. P m \<Longrightarrow> P (MatchNot (MatchNot m))"   and "\<And>m1 m2. \<lbrakk>P (MatchNot m1); P (MatchNot m2)\<rbrakk> \<Longrightarrow> P (MatchNot (MatchAnd m1 m2))"   and "\<And>m1 m2. \<lbrakk>P m1; P m2\<rbrakk> \<Longrightarrow> P (MatchAnd m1 m2)" shows "P a0"
main: assumes "root_cond plr x"   and "sgn l = sgn r"   and "sgn x = real_of_rat (sgn r)"   and "\<exists>!x. root_cond plr x"   and "poly_cond p"   and "0 < degree p"   and "primitive p" shows "thesis"
y2: shows "lst (snd t', ts) = lst_conv (snd t', map (Pair True) ts)"
assms_2: shows "B \<subseteq> U\<^sub>3"
simps_3: fixes a :: "Cooper.num" shows "bound0 (Lt a) = numbound0 a"
new_vars_private_def: fixes \<R> :: "((((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) \<times> ((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)) set" shows "new_vars_private \<R> \<equiv> \<forall>c\<^sub>1\<^sub>A mds\<^sub>A mem\<^sub>1\<^sub>A c\<^sub>1\<^sub>C mds\<^sub>C mem\<^sub>1\<^sub>C. (\<langle>c\<^sub>1\<^sub>A, mds\<^sub>A, mem\<^sub>1\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<longrightarrow> (\<forall>c\<^sub>1\<^sub>C' mds\<^sub>C' mem\<^sub>1\<^sub>C'. eval_abv\<^sub>C \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C \<langle>c\<^sub>1\<^sub>C', mds\<^sub>C', mem\<^sub>1\<^sub>C'\<rangle>\<^sub>C \<longrightarrow> (\<forall>v\<^sub>C. (mem\<^sub>1\<^sub>C' v\<^sub>C \<noteq> mem\<^sub>1\<^sub>C v\<^sub>C \<or> dma\<^sub>C mem\<^sub>1\<^sub>C' v\<^sub>C < dma\<^sub>C mem\<^sub>1\<^sub>C v\<^sub>C) \<and> v\<^sub>C \<notin> range Eg2_var\<^sub>C_of_Eg1 \<longrightarrow> v\<^sub>C \<in> mds\<^sub>C' AsmNoReadOrWrite) \<and> mds\<^sub>C AsmNoReadOrWrite - range Eg2_var\<^sub>C_of_Eg1 \<subseteq> mds\<^sub>C' AsmNoReadOrWrite - range Eg2_var\<^sub>C_of_Eg1)"
facetrel_subset: fixes y :: "'c set"   and x :: "'c set" assumes "y \<lhd> x" shows "order.greater_eq x y"
not_arr_null: shows "\<not> Cop_S.arr Cop_S.null"
a_evalF_iterative: fixes x :: "'a \<aproc> \<times> 'a \<aproc> set list \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set) \<times> 'a" shows "\<aF>\<cdot>(Discr x) = \<^ps> abs_g\<cdot>(\<Union>i. (\<^ps> abs_R)\<^bsup>i\<^esup>\<cdot>{Discr (Inl x)})"
monoid_add_hom_axioms: shows "monoid_add_hom hom"
cMatch: assumes "P \<longmapsto> Rs" shows "F a a"
fset_inject: fixes x :: "'b fset"   and y :: "'b fset" shows "(fset x = fset y) = (x = y)"
is_process7: fixes P :: "('a event list \<times> 'a event set) set \<times> 'a event list set" assumes "is_process P" shows "\<forall>s t. s \<notin> DIVERGENCES P \<or> \<not> tickFree s \<or> \<not> front_tickFree t \<or> s @ t \<in> DIVERGENCES P"
assms_1: fixes a1 :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and a2 :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "a1 \<in> A"   and "a2 \<in> A" shows "a1 + a2 \<in> A"
pp_leq_nc: shows "\<forall>f. pivot_fun A f (dim_col A) \<longrightarrow> (\<forall>i<dim_row A. f i \<le> dim_col A)"
that: fixes ab :: "'a set" assumes "path ab a b"   and "prolongation a b \<subseteq> ab" shows "thesis"
nat_Suc_predicate_change_exists: fixes a :: "nat"   and b :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "a \<le> b"   and "\<not> P a"   and "P b" shows "\<exists>n\<ge>a. n < b \<and> \<not> P n \<and> P (Suc n)"
return_spmf_None_parametric: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_spmf A (return_pmf None) (return_pmf None)"
plus_nmultiset_mono: fixes X :: "unit nmultiset"   and Y :: "unit nmultiset"   and X' :: "unit nmultiset"   and Y' :: "unit nmultiset" assumes "(X, Y) < (X', Y')"   and "no_elem X"   and "no_elem Y"   and "no_elem X'"   and "no_elem Y'" shows "plus_nmultiset X Y < plus_nmultiset X' Y'"
simps: fixes uu :: "'d"   and v :: "'c"   and uv :: "('d, 'c) bintree list" shows "val (Node uu v uv) = v"
sup_comp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<squnion> y) * z = x * z \<squnion> y * z"
assms_5: fixes p :: "'s_core spmf"   and q :: "'s_core' spmf"   and ia :: "'iadv_core"   and s1 :: "'s_core"   and s2 :: "'s_core'"   and oa :: "'oadv_core"   and s1' :: "'s_core"   and s2' :: "'s_core'" assumes "S p q"   and "ia \<in> IA"   and "s1 \<in> set_spmf p"   and "s2 \<in> set_spmf q"   and "(oa, s1') \<in> set_spmf (cfunc_adv core1 s1 ia)"   and "(oa, s2') \<in> set_spmf (cfunc_adv core2 s2 ia)" shows "trace_eq_simcl S (cond_spmf_fst (p \<bind> (\<lambda>s1. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (q \<bind> (\<lambda>s2. cfunc_adv core2 s2 ia)) oa)"
class_add_norm_eff_pc: fixes i :: "instr"   and P :: "jvm_prog"   and pc :: "nat"   and \<tau> :: "ty list \<times> ty err list"   and mpc :: "nat"   and C :: "char list"   and cdec :: "jvm_method class" assumes "\<forall>(pc', \<tau>')\<in>set (norm_eff i P pc \<tau>). pc' < mpc" shows "\<forall>(pc', \<tau>')\<in>set (norm_eff i (class_add P (C, cdec)) pc \<tau>). pc' < mpc"
insert_Nil: fixes x :: "'c" shows "List.insert x [] = [x]"
maintainedD2: fixes A :: "('b, 'c) labeled_graph"   and B :: "('b, 'c) labeled_graph"   and G :: "('b, 'd) labeled_graph"   and f :: "('c \<times> 'd) set" assumes "maintained (A, B) G"   and "graph_homomorphism A G f" obtains g :: "('c \<times> 'd) set" where "graph_homomorphism B G g" and "f \<subseteq> g"
disc_1: fixes x1 :: "'b" shows "is_Prim \<^bold>\<langle>x1\<^bold>\<rangle>"
smallest_prime_beyond_smallest: fixes p :: "nat"   and n :: "nat" assumes "prime p"   and "n \<le> p" shows "smallest_prime_beyond n \<le> p"
E: shows "ipurge_tr_rev_aux I D (D ` (X \<union> set zs)) xs = ipurge_tr_rev_aux I D (D ` (X \<union> set zs)) (ys @ [y])"
galois_aux2: fixes x :: "'a"   and y :: "'a" shows "(x \<cdot> - y = 0) = (x \<le> y)"
set_iterator_def: fixes it :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and S0 :: "'b set" shows "set_iterator it S0 \<equiv> set_iterator_genord it S0 (\<lambda>_ _. True)"
amd_assn_def: fixes dflt :: "'f"   and K :: "'d \<Rightarrow> nat \<Rightarrow> assn"   and V :: "'e \<Rightarrow> 'f \<Rightarrow> assn" shows "amd_assn dflt K V \<equiv> hr_comp (hr_comp (is_nff dflt) (nat_rel \<rightarrow>\<^sub>f dflt_option_rel_aux dflt)) (\<langle>the_pure K, the_pure V\<rangle>map_rel)"
nat_pow_eone: fixes x :: "'a" assumes "x \<in> carrier G" shows "x [^] 1 = x"
cone_iff_finite_cone: fixes A :: "'b vec set"   and n :: "nat" assumes "A \<subseteq> carrier_vec n"   and "finite A" shows "gram_schmidt.cone n A = gram_schmidt.finite_cone n A"
elims_3: fixes x :: "'at Literal" assumes "\<not> negative x" obtains A :: "'at" where "x = Pos A"