hcju/isabelleps huggingface dataset:
Segment_triv: fixes C :: "Class"   and m :: "Method"   and l :: "int"   and l1 :: "int"   and code :: "(int \<times> Instr) list"   and Mspec :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and Minv :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and Anno :: "(int \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and ll :: "int" assumes "Segment C m l l1 code"   and "MST\<down>(C, m) = Some (Mspec, Minv, Anno)"   and "l \<le> ll"   and "ll < l1" shows "Anno\<down>ll = None \<and> (\<exists>ins. ins_is C m ll ins \<and> code\<down>ll = Some ins)"
eqfset_imp_iff: fixes A :: "'a fset"   and B :: "'a fset"   and x :: "'a" assumes "A = B" shows "(x |\<in>| A) = (x |\<in>| B)"
divide_strict_right_mono_neg: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less b a"   and "less c zero" shows "less (divide a c) (divide b c)"
span_zero: fixes S :: "'c set" shows "(0::'c) \<in> ideal S"
ne: shows "space (Pi\<^sub>M I M) \<noteq> {}"
j_def: shows "j \<equiv> GREATEST j. k + n cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<and> j < k"
le: shows "card F \<le> card (f -` {a} \<inter> F) * card {X. X \<subseteq> E \<and> card X = s}"
ring_irreducibleE_3: fixes r :: "'a" assumes "r \<in> carrier R"   and "ring_irreducible r" shows "irreducible (Ring_Divisibility.mult_of R) r"
frameImpChainComm: fixes xvec :: "name list"   and yvec :: "name list"   and F :: "'b frame" shows "\<lparr>\<nu>*xvec\<rparr>\<lparr>\<nu>*yvec\<rparr>F \<hookrightarrow>\<^sub>F \<lparr>\<nu>*yvec\<rparr>\<lparr>\<nu>*xvec\<rparr>F"
convex_hull_Times: fixes s :: "'c set"   and t :: "'d set" shows "convex hull s \<times> t = (convex hull s) \<times> (convex hull t)"
continuous_on_eq_continuous_at: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "open s" shows "continuous_on s f = (\<forall>x\<in>s. isCont f x)"
prems_4: shows "m -a # as\<rightarrow>* (_Low_)"
finite_limit_rule: fixes A :: "nat set"   and B :: "(('b \<Rightarrow> nat option) \<times> 'a \<times> ('b \<Rightarrow> nat option)) set" assumes "\<And>x. (x \<in> A) = (r x \<in> B)" shows "finite A = (limit r \<inter> B = {})"
IHG_3: fixes HS :: "'a literal list list"   and \<Gamma> :: "'a formula multiset" assumes "disj_of_clause GR, image_mset disj_of_clause (mset (GS @ HS)) + \<Gamma> \<Rightarrow>\<^sub>n" shows "image_mset disj_of_clause (mset (GS @ HS)) + \<Gamma> \<Rightarrow>\<^sub>n"
ub_arctan: fixes k :: "int"   and prec :: "nat"   and na :: "nat" assumes "1 < k" shows "arctan (1 / real_of_int k) \<le> real_of_float (rapprox_rat prec 1 k * ub_arctan_horner prec (get_odd na) 1 (float_round_down (Suc prec) (rapprox_rat prec 1 k * rapprox_rat prec 1 k)))"
quadratic_eq_zeroes_iff: fixes D :: "real"   and b :: "real"   and a :: "real"   and c :: "real"   and x\<^sub>1 :: "real"   and x\<^sub>2 :: "real"   and x :: "real" assumes "D = b\<^sup>2 - 4 * a * c"   and "x\<^sub>1 = (- b + sqrt D) / (2 * a)"   and "x\<^sub>2 = (- b - sqrt D) / (2 * a)"   and "a \<noteq> 0" shows "(a * x\<^sup>2 + b * x + c = 0) = (0 \<le> D \<and> (x = x\<^sub>1 \<or> x = x\<^sub>2))"
A: shows "dist x (geodesic_segment_param G x t) = t"
filterlim_atI: fixes "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and c :: "'a"   and F :: "'b filter" assumes "class.topological_space open"   and "filterlim f (topological_space.nhds open c) F"   and "\<forall>\<^sub>F x in F. f x \<noteq> c" shows "filterlim f (topological_space.at_within open c UNIV) F"
monoid_axioms: shows "monoid (AND) (- (1::'a))"
eventually_at_topological: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a"   and s :: "'a set" shows "eventually P (at a within s) = (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x))"
cases: fixes v :: "letter list"   and a :: "letter list list" assumes "L v a" obtains w :: "letter list" and ws :: "letter list list" where "a = w__ # ws__" and "emb w__ v"   | ws :: "letter list list" and w :: "letter list" where "a = w # ws" and "L v ws"
semiring_normalization_rules_26: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and x :: "'b"   and p :: "nat"   and q :: "nat" assumes "class.comm_semiring_1 times one plus zero" shows "times (power.power one times x p) (power.power one times x q) = power.power one times x (p + q)"
dom_null: shows "VoV.dom VoV.null = VoV.null"
is_dghm_axioms: shows "\<HH> : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<DD>"
pts_symbolic_c_preserves_tfr\<^sub>s\<^sub>e\<^sub>t: fixes \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) extstrand_step list"   and \<S>' :: "('fun, 'var) strand_step list set"   and \<A>' :: "('fun, 'var) extstrand_step list" assumes "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet>\<^sub>c\<^sup>* (\<S>', \<A>')"   and "tfr\<^sub>s\<^sub>e\<^sub>t (\<Union> (trms\<^sub>s\<^sub>t ` \<S>) \<union> \<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set \<A>))"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (\<Union> (trms\<^sub>s\<^sub>t ` \<S>) \<union> \<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set \<A>))" shows "tfr\<^sub>s\<^sub>e\<^sub>t (\<Union> (trms\<^sub>s\<^sub>t ` \<S>') \<union> \<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set \<A>')) \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (\<Union> (trms\<^sub>s\<^sub>t ` \<S>') \<union> \<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set \<A>'))"
casts': shows "P \<turnstile> Class (last Ds) # Ts Casts Ref (a, Ds) # vs to Ref (a, Ds) # vs' "
if_SE_split_asm': fixes \<sigma> :: "'d"   and P :: "'d \<Rightarrow> bool"   and B\<^sub>1 :: "'d \<Rightarrow> ('e \<times> 'd) option"   and B\<^sub>2 :: "'d \<Rightarrow> ('e \<times> 'd) option"   and M :: "'d \<Rightarrow> (bool \<times> 'd) option" shows "(\<sigma> \<Turnstile> _if_SE P B\<^sub>1 B\<^sub>2 ;- M) = (P \<sigma> \<and> (\<sigma> \<Turnstile> B\<^sub>1 ;- M) \<or> \<not> P \<sigma> \<and> (\<sigma> \<Turnstile> B\<^sub>2 ;- M))"
31: fixes l :: "nat" assumes "l \<in> Suc ` A" shows "least (Suc ` A) \<le> l"
is_UP_monomE_3: fixes f :: "nat \<Rightarrow> 'a" assumes "is_UP_monom f" shows "f = monom (UP R) (f (deg R f)) (deg R f)"
matrix'_def: fixes X :: "(('a, 'b) vec, 'b) vec"   and Y :: "(('a, 'c) vec, 'c) vec"   and f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" shows "matrix' X Y f = (\<chi>i j. coord Y (f (X $ j)) $ i)"
ker_to_rimg: fixes A :: "('c, 'd) Ring_scheme"   and R :: "('e, 'f) Ring_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "Ring A"   and "Ring R"   and "f \<in> rHom A R" shows "ker\<^bsub>A,R\<^esub> f = ker\<^bsub>A,rimg A R f\<^esub> f"
ch2ch_fst: fixes Y :: "nat \<Rightarrow> 'c \<times> 'd" assumes "chain Y" shows "chain (\<lambda>i. fst (Y i))"
V: shows "B \<subseteq> carrier (vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u})"
update_convs_1: fixes ev_counter' :: "nat \<Rightarrow> nat"   and ev_counter :: "nat"   and more :: "'a" shows "ev_counter_update ev_counter' \<lparr>ev_counter = ev_counter, \<dots> = more\<rparr> = \<lparr>ev_counter = ev_counter' ev_counter, \<dots> = more\<rparr>"
inf_prefix_correctL: fixes i :: "nat"   and xs :: "'b list"   and ys :: "'b list" assumes "i < length (xs \<sqinter> ys)" shows "(xs \<sqinter> ys) ! i = xs ! i"
DA0'_DA: shows "DA0' \<cdot> \<eta> = DA"
map_val2_conv: fixes xs :: "expr list"   and ys :: "val list"   and e :: "expr"   and zs :: "expr list" shows "(xs = map Val ys @ throw e # zs) = map_val2 xs ys (throw e # zs)"
Zero_not_eq_j1: shows "J.Zero \<noteq> J.j1"
inc_le: fixes u :: "nat list"   and y :: "nat list"   and i :: "nat" assumes "length u = length y"   and "i < length y"   and "u \<le>\<^sub>v y" shows "inc y i u \<le>\<^sub>v y"
cycle_free_loop_dest: fixes i :: "nat"   and n :: "nat"   and xs :: "nat list"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a" assumes "i \<le> n"   and "set xs \<subseteq> {0..n}"   and "cycle_free m n" shows "(0::'a) \<le> len m i i xs"
par_not_col: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and X :: "'p" assumes "A B ParStrict C D"   and "Col X A B" shows "\<not> Col X C D"
SPEC_rule_weak: fixes f :: "'d \<Rightarrow> ('b \<times> 'c) nres"   and x :: "'d"   and Q :: "'d \<Rightarrow> 'b \<times> 'c \<Rightarrow> bool"   and P :: "'d \<Rightarrow> 'b \<times> 'c \<Rightarrow> bool"   and x' :: "'d" assumes "f x \<le> SPEC (Q x)"   and "f x \<le> SPEC (P x)"   and "\<And>r1 r2. \<lbrakk>Q x (r1, r2); P x (r1, r2)\<rbrakk> \<Longrightarrow> P x' (r1, r2)" shows "f x \<le> SPEC (P x')"
linord_of_list_nth: fixes xs :: "'a list"   and i :: "nat"   and j :: "nat" assumes "(xs ! i, xs ! j) \<in> linord_of_list xs"   and "i < length xs"   and "j < length xs"   and "distinct xs" shows "j \<le> i"
lift_and_com: fixes A :: "'b \<Rightarrow> bool"   and B :: "'b \<Rightarrow> bool" shows "\<turnstile> (A \<and> B) = (B \<and> A)"
cut_same_def: fixes x :: "'b"   and s :: "'b stream" shows "cut_same x s = stake (LEAST n. sdrop n s = sconst x) s"
gen_dfs_def: shows "a.c.gen_dfs \<equiv> gds_init a.c.gds \<bind> WHILE a.c.gen_cond a.c.gen_step"
h4: shows "d p \<cdot> ad r \<le> wp y q"
language_state_for_inputs_map_fst_contained: fixes vs :: "('d \<times> 'e) list"   and M :: "('d, 'e, 'f) FSM"   and q :: "'f"   and V :: "'d list set" assumes "vs \<in> LS\<^sub>i\<^sub>n M q V" shows "map fst vs \<in> V"
inj_on_apply_range: fixes f :: "'b perm"   and a :: "'b" shows "inj_on (\<lambda>m. (f ^ m) \<langle>$\<rangle> a) {..<order f a}"
UnCI: fixes c :: "'g"   and B :: "'g set"   and A :: "'g set" assumes "c \<notin> B \<Longrightarrow> c \<in> A" shows "c \<in> A \<union> B"
game_A_game_B: shows "map_spmf snd game_A = game_B"
range_dom_1: fixes f :: "('e \<times> 'd) set" shows "f `` Domain f = Range f"
eval_embed_true: fixes P :: "'c \<Rightarrow> bool"   and s :: "'c" assumes "P s" shows "\<guillemotleft> P \<guillemotright> s = 1"
integral_eq_zero_null_sets: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "S \<in> null_sets lebesgue" shows "integral\<^sup>L (lebesgue_on S) f = (0::'b)"
Seq: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and s :: "'e"   and c\<^sub>2 :: "('e, 'd, 'f) com" assumes "\<Gamma>\<turnstile>c\<^sub>1 \<down> Normal s"   and "\<forall>s'. \<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> \<Rightarrow> s' \<longrightarrow> \<Gamma>\<turnstile>c\<^sub>2 \<down> s'" shows "\<Gamma>\<turnstile>Seq c\<^sub>1 c\<^sub>2 \<down> Normal s"
Marker_in_channel_implies_not_done: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "trace init t final"   and "Marker \<in> set (msgs (S t i) cid)"   and "channel cid = Some (p, q)"   and "i \<le> length t" shows "snd (cs (S t i) cid) \<noteq> Done"
Nadd_assoc: fixes x :: "int \<times> int"   and y :: "int \<times> int"   and z :: "int \<times> int" shows "x +\<^sub>N y +\<^sub>N z = x +\<^sub>N (y +\<^sub>N z)"
tensor_preserves_ide: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "ide (tensor a b)"
p: shows "j + 1 + (n - j - 1) \<le> n"
game_1_def: fixes \<A> :: "(bool, '\<alpha>, '\<gamma>) gpv" shows "local.prf.game_1 \<A> = exec_gpv prf.random_oracle \<A> Map.empty \<bind> (\<lambda>(b, uu_). return_spmf b)"
map_option_case: fixes f :: "'e \<Rightarrow> 'd"   and y :: "'e option" shows "map_option f y = (case y of None \<Rightarrow> None | Some x \<Rightarrow> Some (f x))"
3: fixes S :: "'b set"   and a :: "'b" assumes "card S \<le> 2" shows "card (a \<triangleright> S) \<le> 3"
X: shows "\<forall>X\<in>S. x \<in> X"
lc0_aux: shows "{l. l = [] \<and> valid l} = {[]}"
axioms_1: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" assumes "IrrFinGroupRepresentation G smult V" shows "FinGroupRepresentation G smult V"
A: fixes x :: "'k"   and \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "x \<in> A \<Gamma>" shows "x \<in> SAT \<Gamma> f"
connected_component_of_refl: fixes X :: "'b topology"   and x :: "'b" shows "connected_component_of X x x = (x \<in> topspace X)"
inv_into_ordermap: fixes \<alpha> :: "V"   and A :: "'c set"   and r :: "('c \<times> 'c) set" assumes "\<alpha> \<in> elts (ordertype A r)" shows "inv_into A (ordermap A r) \<alpha> \<in> A"
nat_pow_Suc: fixes x :: "int"   and n :: "nat" shows "x ^ Suc n = x ^ n * x"
npowf_def: fixes K :: "('c, 'd) Ring_scheme"   and x :: "'c"   and n :: "int" shows "x\<^bsub>K\<^esub>\<^bsup>n\<^esup> = (if 0 \<le> n then x^\<^bsup>K nat n\<^esup> else x\<^bsup>\<hyphen> K\<^esup>^\<^bsup>K nat (- n)\<^esup>)"
lookup_minus_fun: fixes s :: "'c \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 'd" shows "lookup (s - t) = lookup s - lookup t"
as_needed_asses_submap_exec_viii': fixes f :: "('d, 'e) fmap"   and g :: "('d, 'e) fmap" assumes "f \<subseteq>\<^sub>f g" shows "fmdom' f \<subseteq> fmdom' g"
unicity_of_evaluation: fixes e :: "trm"   and e\<^sub>1 :: "trm"   and e\<^sub>2 :: "trm" assumes "e \<Down> e\<^sub>1"   and "e \<Down> e\<^sub>2" shows "e\<^sub>1 = e\<^sub>2"
t2: shows "local.iso \<lbrace>t2\<rbrace>"
constructive_security_trivial: fixes \<I> :: "nat \<Rightarrow> ('i, 'k) \<I>"   and \<I>_common :: "nat \<Rightarrow> ('j, 'l) \<I>"   and res :: "nat \<Rightarrow> ('i + 'j, 'k + 'l) resource"   and bound :: "nat \<Rightarrow> enat"   and lossless :: "bool" assumes "\<And>\<eta>. \<I> \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta> \<turnstile>res res \<eta> \<surd>" shows "constructive_security res res (\<lambda>_. 1\<^sub>C) \<I> \<I> \<I>_common bound lossless (\<lambda>_. (id, id))"
strict_prefixE': fixes xs :: "'a list"   and ys :: "'a list" assumes "xs < ys" obtains z :: "'a" and zs :: "'a list" where "ys = xs @ z # zs"
length_cols_mat_to_cols_list: fixes j :: "nat"   and A :: "'a Matrix.mat" assumes "j < dim_col A" shows "length (map (\<lambda>i. A $$ (i, j)) [0..<dim_row A]) = dim_row A"
hyp2_cltn2_id: fixes p :: "hyp2" shows "hyp2_cltn2 p cltn2_id = p"
map_U_parametric: fixes L1 :: "'n \<Rightarrow> 'p \<Rightarrow> bool"   and L1' :: "'o \<Rightarrow> 'q \<Rightarrow> bool"   and Co1 :: "'r \<Rightarrow> 'aa \<Rightarrow> bool"   and Co1' :: "'s \<Rightarrow> 'ab \<Rightarrow> bool"   and Co2 :: "'t \<Rightarrow> 'ac \<Rightarrow> bool"   and Co2' :: "'u \<Rightarrow> 'ad \<Rightarrow> bool"   and Contra1' :: "'v \<Rightarrow> 'ae \<Rightarrow> bool"   and Contra1 :: "'w \<Rightarrow> 'af \<Rightarrow> bool"   and Contra2' :: "'x \<Rightarrow> 'ag \<Rightarrow> bool"   and Contra2 :: "'y \<Rightarrow> 'ah \<Rightarrow> bool" shows "((L1 ===> L1') ===> (Co1 ===> Co1') ===> (Co2 ===> Co2') ===> (Contra1' ===> Contra1) ===> (Contra2' ===> Contra2) ===> rel_U L1 Co1 Co2 Contra1 Contra2 ===> rel_U L1' Co1' Co2' Contra1' Contra2') map_U map_U"
ccpo_Sup_least: fixes Sup :: "'f set \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and A :: "'f set"   and z :: "'f" assumes "class.ccpo Sup less_eq less"   and "Complete_Partial_Order.chain less_eq A"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq x z" shows "less_eq (Sup A) z"
prime_dvd_power: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and 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 unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and p :: "'c"   and x :: "'c"   and n :: "nat" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "dvd.dvd times p (power.power one times x n)" shows "dvd.dvd times p x"
op_eventually_within_sctn_def: fixes X :: "'c set"   and sctn :: "'c sctn"   and S :: "'c set" shows "op_eventually_within_sctn X sctn S = ivl_rep S \<bind> (\<lambda>(l, u). op_ivl_rep_of_set X \<bind> (\<lambda>(xl, xu). CHECK (\<lambda>_. print_msg (ST ''op_eventually_within_sctn: empty ivl'')) (l \<le> u) \<bind> (\<lambda>_. CHECK (\<lambda>_. print_msg (ST ''op_eventually_within_sctn: not in Basis'')) (\<bar>normal sctn\<bar> \<in> set Basis_list) \<bind> (\<lambda>_. subset_spec_plane S sctn \<bind> (\<lambda>b. CHECK (\<lambda>_. print_msg (ST ''op_eventually_within_sctn: subset_spec_plane 1'')) b \<bind> (\<lambda>_. subset_spec_plane ({xl..xu} ::: lvivl_rel) sctn \<bind> (\<lambda>b. CHECK (\<lambda>_. print_msg (ST ''op_eventually_within_sctn: subset_spec_plane 2'')) b \<bind> (\<lambda>_. RETURN (b \<and> (\<forall>i\<in>set Basis_list - {\<bar>normal sctn\<bar>}. l \<bullet> i < xl \<bullet> i \<and> xu \<bullet> i < u \<bullet> i))))))))))"
Un_Diff: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d set" shows "A \<union> B - C = A - C \<union> (B - C)"
ln_Gamma_series_complex_of_real: fixes x :: "real"   and n :: "nat" assumes "0 < x"   and "0 < n" shows "ln_Gamma_series (complex_of_real x) n = complex_of_real (ln_Gamma_series x n)"
A_Issues_B: fixes A :: "agent"   and B :: "agent"   and ST :: "msg"   and servK :: "nat"   and T3 :: "nat"   and evs :: "event list" assumes "Says A B \<lbrace>ST, Crypt servK \<lbrace>Agent A, Number T3\<rbrace>\<rbrace> \<in> set evs"   and "Key servK \<notin> analz (knows Spy evs)"   and "B \<noteq> Tgs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbV" shows "A Issues B with Crypt servK \<lbrace>Agent A, Number T3\<rbrace> on evs"
weight_le_Var_Var: fixes x :: "'b"   and y :: "'b" shows "weight_le (Var x) (Var y) = (x = y)"
approx_un_eq_Some: fixes p :: "nat"   and f :: "float interval \<Rightarrow> float interval"   and r :: "(real \<times> real pdevs) \<times> real"   and s :: "(real \<times> real pdevs) \<times> real" shows "(approx_un p (\<lambda>ivl. Some (f ivl)) (Some r) = Some s) = (s = ivl_err (real_interval (f (ivl_of_aform_err p r))))"
add_increasing2: fixes c :: "'a multiset"   and b :: "'a multiset"   and a :: "'a multiset" assumes "{#} \<subseteq># c"   and "b \<subseteq># a" shows "b \<subseteq># a + c"
ecl_def: fixes x :: "'a" shows "[x]\<^sub>\<sim> \<equiv> {y \<in> A. x \<sim> y} \<union> {x}"
qstermCf_def: fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state" shows "qstermCf cf = (\<forall>cfT. enabled cf cfT \<longrightarrow> qsend (cf ## cfT) < \<infinity>)"
bal_consE: fixes c\<^sub>1 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and c\<^sub>2 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and T :: "((Vars.var \<times> exp) list \<times> exp \<times> stack_elem list) list"   and c\<^sub>5 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and s :: "stack_elem" assumes "c\<^sub>1 \<Rightarrow>\<^sup>b\<^sup>*\<^bsub>c\<^sub>2 # T\<^esub> c\<^sub>5"   and "stack c\<^sub>2 = s # stack c\<^sub>1" obtains T\<^sub>1 :: "((Vars.var \<times> exp) list \<times> exp \<times> stack_elem list) list" and c\<^sub>3 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" and c\<^sub>4 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" and T\<^sub>2 :: "((Vars.var \<times> exp) list \<times> exp \<times> stack_elem list) list" where "T = T\<^sub>1 @ c\<^sub>4 # T\<^sub>2" and "c\<^sub>2 \<Rightarrow>\<^sup>b\<^sup>*\<^bsub>T\<^sub>1\<^esub> c\<^sub>3" and "c\<^sub>3 \<Rightarrow> c\<^sub>4" and "c\<^sub>4 \<Rightarrow>\<^sup>b\<^sup>*\<^bsub>T\<^sub>2\<^esub> c\<^sub>5"
objE: fixes a :: "'b" assumes "B.obj a" obtains "B.ide a" and "src\<^sub>B a = a" and "trg\<^sub>B a = a"
awalk_cost_Nil: fixes f :: "'b \<Rightarrow> real" shows "awalk_cost f [] = 0"
charslength_of_butlast: fixes p :: "(symbol \<times> character list) list"   and k :: "nat"   and u :: "nat"   and q :: "(symbol \<times> character list) list"   and t :: "symbol \<times> character list" assumes "p \<in> \<P> k u"   and "p = q @ [t]" shows "charslength q \<le> k"
Max_wrt_total_preorder: shows "Max_wrt R = {x \<in> alts. \<forall>y\<in>alts. R y x}"
IH: assumes "is_cupcake_all_env env"   and "is_cupcake_exp e" shows "cupcake_evaluate_single env e (snd (s2, Rval v1))"
preserves_arr: fixes f :: "'a" assumes "arr f" shows "S.arr (S.UP f)"
q'_def: fixes t :: "real" shows "ego2.q' t = v\<^sub>e"
bin_sign_rest: fixes w :: "int" shows "bin_sign (w div 2) = bin_sign w"
le_minus_one_simps_1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (uminus one) zero"
rem_implicit_pres_ops_well_formed: fixes prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" assumes "\<And>op. op \<in> set (ast_problem.ast\<delta> prob) \<Longrightarrow> consistent_pres_op op"   and "ast_problem.well_formed prob" shows "ast_problem.well_formed (rem_implicit_pres_ops prob)"
FullInit: shows "\<turnstile> FullInit \<longrightarrow> BInit inp qc out"
ars_r_def: fixes x :: "'a" shows "r x = ar (ar x)"
det_rows_mul: fixes a :: "nat \<Rightarrow> 'b Matrix.vec"   and n :: "nat"   and c :: "nat \<Rightarrow> 'b" assumes "a \<in> {0..<n} \<rightarrow> carrier_vec n" shows "Determinant.det (mat\<^sub>r n n (\<lambda>i. c i \<cdot>\<^sub>v a i)) = prod c {0..<n} * Determinant.det (mat\<^sub>r n n a)"
div_instr_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (div_instr instr s1)) \<and> t2 = snd (fst (div_instr instr s2))" shows "low_equal t1 t2"
Par1: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs"   and Q :: "ccs" assumes "P \<longmapsto>\<alpha> \<prec> P'" shows "P \<parallel> Q \<longmapsto>\<alpha> \<prec> P' \<parallel> Q"
interpret_floatariths_nth: fixes n :: "nat"   and fas :: "floatarith list"   and xs :: "real list" assumes "n < length fas" shows "interpret_floatariths fas xs ! n = interpret_floatarith (fas ! n) xs"
finalStateCharacterization: fixes state0 :: "State"   and F0 :: "Literal list list"   and state :: "State"   and decisionVars :: "nat set" assumes "isInitialState state0 F0"   and "(state0, state) \<in> transitionRelation F0 decisionVars"   and "isFinalState state F0 decisionVars" shows "getConflictFlag state = False \<and> \<not> formulaFalse (getF state) (elements (getM state)) \<and> decisionVars \<subseteq> vars (elements (getM state)) \<or> getConflictFlag state = True \<and> getC state = []"