hcju/isabelleps huggingface dataset:
seqSubstInput: fixes x :: "name"   and \<sigma> :: "(name \<times> name) list"   and a :: "name"   and P :: "pi" assumes "x \<sharp> \<sigma>" shows "a<x>.P[<\<sigma>>] = seq_subst_name a \<sigma><x>.(P[<\<sigma>>])"
tendsto_iff: fixes f :: "'c \<Rightarrow> 'a"   and l :: "'a"   and F :: "'c filter" shows "(f \<longlongrightarrow> l) F = (\<forall>e>0. \<forall>\<^sub>F x in F. dist (f x) l < e)"
inject_4: fixes x61 :: "'b ltln"   and x62 :: "'b ltln"   and y61 :: "'b ltln"   and y62 :: "'b ltln" shows "(x61 or\<^sub>n x62 = y61 or\<^sub>n y62) = (x61 = y61 \<and> x62 = y62)"
d_def: fixes fs :: "int vec list"   and k :: "nat" shows "d fs k = gs.Gramian_determinant fs k"
rel_sel: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "('a, 'c) tllist"   and b :: "('b, 'd) tllist" shows "tllist_all2 R1 R2 a b = (is_TNil a = is_TNil b \<and> (is_TNil a \<longrightarrow> is_TNil b \<longrightarrow> R2 (terminal a) (terminal b)) \<and> (\<not> is_TNil a \<longrightarrow> \<not> is_TNil b \<longrightarrow> R1 (thd a) (thd b) \<and> tllist_all2 R1 R2 (ttl a) (ttl b)))"
carrier_eq: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "Order_Predicates.preorder_on carrier le" shows "carrier = {x. le x x}"
cinner_eq_zero_iff: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and x :: "'c" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "(cinner x x = 0) = (x = zero)"
stermsl_ctermsl_choice1: fixes q :: "('a, 'b, 'c, 'd) seqp"   and p1 :: "('a, 'b, 'c, 'd) seqp"   and p2 :: "('a, 'b, 'c, 'd) seqp" assumes "q \<in> stermsl p1" shows "q \<in> ctermsl (p1 \<oplus> p2)"
HFail_HInv5_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HFail s s' p"   and "HInv5_inner s q"   and "p \<noteq> q"   and "Inv2a s'" shows "HInv5_inner s' q"
preserves_hom: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>D b\<guillemotright>" shows "\<guillemotleft>F f : F a \<rightarrow>\<^sub>C F b\<guillemotright>"
Hermite_of_upt_row_preserves_zero_rows: fixes i :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and ass :: "'b \<Rightarrow> 'b"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and k :: "nat" assumes "is_zero_row i A"   and "echelon_form A"   and "ass_function ass"   and "res_function res"   and "k \<le> nrows A" shows "is_zero_row i (Hermite_of_upt_row_i A k ass res)"
psubstAbs_preserves_wls: fixes xs :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" assumes "wlsAbs (xs, s) A"   and "wlsEnv rho" shows "wlsAbs (xs, s) A $[rho]"
test_s_prod: fixes P :: "('a \<times> 'a set) set"   and a :: "'a"   and A :: "'a set"   and R :: "('a \<times> 'a set) set" assumes "P \<subseteq> 1\<^sub>\<sigma>" shows "((a, A) \<in> P \<cdot> R) = ((a, {a}) \<in> P \<and> (a, A) \<in> R)"
Set_member_code_2: fixes x :: "'b"   and dxs :: "'b set_dlist" shows "(x \<in> DList_set dxs) = DList_Set.member dxs x"
degree_min_int_poly_nonzero: fixes x :: "'b" shows "Polynomial.degree (min_int_poly x) \<noteq> 0"
greater_than_two_values: fixes a :: "'b"   and b :: "'b"   and z :: "'b flat_complete_lattice" assumes "a \<noteq> b"   and "flat_complete_lattice.Value a \<le> z"   and "flat_complete_lattice.Value b \<le> z" shows "z = flat_complete_lattice.Top"
rv2_Inl_last: fixes s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and kl :: "(bd_type_F + bd_type_F) list"   and xb :: "bd_type_F"   and x :: "'e" shows "\<exists>b2'. rv2 s1 s2 (kl @ [Inl xb]) x = Inl b2'"
evtIA: fixes n :: "nat"   and c :: "'kid"   and t :: "nat \<Rightarrow> cnf"   and n' :: "nat"   and t' :: "nat \<Rightarrow> 'KS"   and \<gamma> :: "(nat \<Rightarrow> 'KS) \<Rightarrow> nat \<Rightarrow> bool" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n'"   and "\<exists>i\<ge>n'. \<parallel>c\<parallel>\<^bsub>t i\<^esub> \<Longrightarrow> \<exists>n''\<ge>\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n'\<^esub>. n'' \<le> \<langle>c \<rightarrow> t\<rangle>\<^bsub>n'\<^esub> \<and> sb.eval c t t' n'' \<gamma>"   and "\<not> (\<exists>i\<ge>n'. \<parallel>c\<parallel>\<^bsub>t i\<^esub>) \<Longrightarrow> sb.eval c t t' n' \<gamma>" shows "sb.eval c t t' n (\<diamond>\<^sub>b\<gamma>)"
simps_2: fixes v :: "('c, 'd) Node"   and xs :: "('c, 'd) Node list" shows "app3 (Single v) xs Empty = rconsNlist (Single v) xs"
wf_RcdT: fixes fs :: "(char list \<times> POPLmarkRecord.type) list"   and \<Gamma> :: "binding list" assumes "unique fs"   and "\<forall>(l, T)\<in>set fs. \<Gamma> \<turnstile>\<^sub>w\<^sub>f T" shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f RcdT fs"
sigma_protocol: fixes \<eta> :: "nat" shows "Schnorr_\<Sigma>.\<Sigma>_protocol \<eta>"
timpl_closure_set_idem: fixes M :: "(('a, 'b, 'c) prot_fun, 'd) Term.term set"   and TI :: "('c set \<times> 'c set) set" shows "timpl_closure_set (timpl_closure_set M TI) TI = timpl_closure_set M TI"
not_neutral_contains_not_neutral: fixes g :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "sum g A \<noteq> (0::'c)" obtains a :: "'d" where "a \<in> A" and "g a \<noteq> (0::'c)"
ccpo_Sup_singleton: fixes Sup :: "'b::type set \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type" assumes "class.ccpo (Sup::'b::type set \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)" shows "(Sup::'b::type set \<Rightarrow> 'b::type) {x::'b::type} = x"
two_not_abs_cube: fixes x :: "int" assumes "\<bar>x ^ 3\<bar> = 2" shows "False"
prec_def: fixes c :: "com"   and E :: "(char list \<Rightarrow> int) \<Rightarrow> 'a" shows "prec c E \<equiv> \<lambda>s. E (\<down>\<^sub>s (c, s))"
min_arg_not_ge_2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a"   and n :: "'a" assumes "class.linorder less_eq less" shows "(\<not> less (ord.min less_eq m n) n) = (ord.min less_eq m n = n)"
qSwap_preserves_alpha: fixes X :: "('e, 'f, 'varSort, 'var, 'g) qTerm"   and X' :: "('e, 'f, 'varSort, 'var, 'g) qTerm"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "qGood X \<or> qGood X'"   and "X #= X'" shows "X #[[z1 \<and> z2]]_zs #= X' #[[z1 \<and> z2]]_zs"
unitary_ex_mat_Ph: shows "unitary (tensor_P mat_Ph (1\<^sub>m K))"
gr_implies_not_zero: fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and m :: "'f"   and n :: "'f" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "less m n" shows "n \<noteq> zero"
abs_pdevs_def: shows "abs_pdevs = unop_pdevs abs"
iso_hcomp: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.iso \<mu>"   and "Span.iso \<nu>"   and "Span.src \<nu> = Span.trg \<mu>" shows "Span.iso (\<nu> \<circ> \<mu>)"
ORLEFT_elim: fixes x :: "name"   and M :: "trm"   and B :: "ty"   and C :: "ty" assumes "x:M \<in> ORLEFT (B OR C) (\<parallel>(B)\<parallel>) (\<parallel>(C)\<parallel>)" obtains y' :: "name" and M' :: "trm" and z' :: "name" and N' :: "trm" where "M = OrL y'.M' z'.N' x" and "fin (OrL y'.M' z'.N' x) x" and "y':M' \<in> \<parallel>(B)\<parallel>" and "z':N' \<in> \<parallel>(C)\<parallel>"
gt_right_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Gt C D" shows "A B Gt D C"
assms: shows "K\<^bsub>n\<^esub> (with_proj G)"
normal_non_zero: fixes p :: "'a poly" assumes "normal_poly p" shows "p \<noteq> 0"
le_bound_of: fixes f :: "'c \<Rightarrow> real"   and x :: "'c" assumes "Expectations.bounded f" shows "f x \<le> bound_of f"
reduced_labelling_nonzero: fixes j :: "nat"   and n :: "nat"   and x :: "nat \<Rightarrow> nat" assumes "j < n"   and "x j \<noteq> 0" shows "reduced n x \<le> j"
frontier_outside_subset: fixes S :: "'a set" assumes "closed S" shows "frontier (outside S) \<subseteq> S"
IncompatibleClassChangeC_def: shows "IncompatibleClassChangeC \<equiv> (IncompatibleClassChangeError, Object, [], [])"
domains_char: fixes f :: "'a" shows "domains f = C.codomains f"
eq: shows "\<forall>d\<in>ds'. length b \<le> d"
not_dbm_lt_le_impl: fixes a :: "'b"   and b :: "'b" assumes "\<not> Lt a \<prec> Le b" shows "b < a"
bind_singleton_conv_image: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'c" shows "Set.bind A (\<lambda>x. {f x}) = f ` A"
lists_succ_simps_pf_1: fixes xs :: "'b list"   and xss :: "'b list list" shows "lists_succ (xs # xss) = list_succ xs \<circ> lists_succ xss"
ord_le_eq_trans: fixes a :: "'d list"   and b :: "'d list"   and c :: "'d list" assumes "prefix a b"   and "b = c" shows "prefix a c"
saturated_def: fixes \<Gamma> :: "'b inference set"   and N :: "'b literal multiset set" shows "inference_system.saturated \<Gamma> N = (concls_of (inference_system.inferences_from \<Gamma> N) \<subseteq> N)"
w: shows "y = (gcd c d)\<^sup>2 * w"
assms_1: shows "list_all2 (\<approx>) x x'"
comm_monoid_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" shows "comm_monoid f z \<equiv> abel_semigroup f \<and> comm_monoid_axioms f z"
compat_append_left: fixes gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ys :: "'a list"   and xs :: "'a list"   and zs :: "'a list" assumes "ext gt ys xs" shows "ext gt (zs @ ys) (zs @ xs)"
1: shows "Fun f T \<in> subterms\<^sub>s\<^sub>e\<^sub>t S \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta> \<or> Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t \<theta> ` (fv\<^sub>s\<^sub>e\<^sub>t S \<inter> subst_domain \<theta>)"
finalNonConflictState: fixes state :: "State"   and decisionVars :: "nat set" assumes "\<not> applicableDecide state decisionVars" shows "decisionVars \<subseteq> vars (elements (getM state))"
extremum_uniqueI: fixes a :: "'a" assumes "\<bottom> \<le> a" shows "a = \<bottom>"
can_syncI: fixes r :: "('d, 'a, 'b, 'c, 'e, 'f) semantics"   and t :: "'a"   and x :: "'b"   and m :: "'c"   and ta :: "('d, 'a, 'b, 'c, 'e, 'f) thread_action"   and x' :: "'b"   and m' :: "'c"   and LT :: "('d + 'a + 'a) set" assumes "r t (x, m) ta (x', m')"   and "LT = collect_waits ta" shows "multithreaded_base.can_sync r t x m LT"
ok_extend_iff: fixes F :: "'a program"   and G :: "'a program" shows "extend h F ok extend h G = F ok G"
Spy_dont_know_RPwd: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> certified_mail"   and "Key (RPwd A) \<in> parts (knows Spy evs)" shows "A \<in> bad"
full\<^sub>i_ins: fixes n :: "nat"   and t :: "'b tree234"   and a :: "'b" assumes "full n t" shows "full\<^sub>i n (ins a t)"
neg_numeral_less_floor: fixes v :: "num"   and x :: "'b" shows "(- numeral v < \<lfloor>x\<rfloor>) = (- numeral v + (1::'b) \<le> x)"
If_le: fixes b :: "bexp"   and p1 :: "'a"   and C1 :: "'a acom"   and p2 :: "'a"   and C2 :: "'a acom"   and S :: "'a"   and C :: "'a acom" shows "(IF b THEN {p1} C1 ELSE {p2} C2 {S} \<le> C) = (\<exists>p1' p2' C1' C2' S'. C = IF b THEN {p1'} C1' ELSE {p2'} C2' {S'} \<and> p1 \<le> p1' \<and> p2 \<le> p2' \<and> C1 \<le> C1' \<and> C2 \<le> C2' \<and> S \<le> S')"
step: fixes possibilities :: "(nat \<times> transition) fset"   and e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and s' :: "nat"   and t :: "transition"   and p :: "value option list"   and r' :: "nat \<Rightarrow>f value option" assumes "possibilities = possible_steps e s r l i"   and "random_member possibilities = Some (s', t)"   and "evaluate_outputs t i r = p"   and "evaluate_updates t i r = r'" shows "step e s r l i = Some (t, s', p, r')"
QTrans: shows "Q \<Longrightarrow>\<^sub>lu in Q''\<rightarrow>a<x> \<prec> Q'"
zero_less_abs_iff: fixes abs :: "'b \<Rightarrow> 'b"   and 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 a :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less zero (abs a) = (a \<noteq> zero)"
gen_one: fixes G :: "('c, 'd) monoid_scheme"   and gens :: "'c set" shows "\<one>\<^bsub>G\<^esub> \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>"
rep_eq: fixes x :: "('a, 'b) pp"   and xa :: "('a, 'b) pp" shows "pp.mapping_of (x - xa) = pp.mapping_of x - pp.mapping_of xa"
exec_gpv_invariant: fixes x :: "'a"   and s' :: "('b, 'c) resource"   and gpv :: "('a, 'b, 'c) gpv"   and s :: "('b, 'c) resource"   and \<I> :: "('b, 'c) \<I>" assumes "(x, s') \<in> set_spmf (exec_gpv run_resource gpv s)"   and "lossless_resource \<I> s \<and> \<I> \<turnstile>res s \<surd>"   and "\<I> \<turnstile>g gpv \<surd>" shows "lossless_resource \<I> s' \<and> \<I> \<turnstile>res s' \<surd>"
empty_inside: assumes "connected S"   and "\<And>C. C \<in> components (- S) \<Longrightarrow> \<not> bounded C" shows "inside S = {}"
neutral: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "\<forall>x\<in>A. g x = \<^bold>1" shows "F.F g A = \<^bold>1"
bst_del_min: fixes t :: "'b tree" assumes "bst_wrt (\<le>) t" shows "bst_wrt (\<le>) (del_min t)"
n2_type3: fixes t1 :: "'c bro"   and h :: "nat"   and t2 :: "'c bro"   and a :: "'c" assumes "t1 \<in> T h"   and "t2 \<in> T h" shows "n2 t1 a t2 \<in> T (Suc h)"
tfinite_induct: fixes xs :: "('e, 'f) tllist"   and P :: "('e, 'f) tllist \<Rightarrow> bool" assumes "tfinite xs"   and "\<And>y. P (TNil y)"   and "\<And>x xs. \<lbrakk>tfinite xs; P xs\<rbrakk> \<Longrightarrow> P (TCons x xs)" shows "P xs"
semClosed: fixes \<phi> :: "'aprop dfmla"   and \<pi>l :: "('state \<times> 'aprop set) stream list"   and env :: "pvar \<Rightarrow> nat" assumes "wff \<phi>"   and "FV \<phi> = {}"   and "pointerOf \<pi>l = s0" shows "semClosed \<phi> = sem \<phi> env \<pi>l"
document_ptr_kinds_eq_h: shows "document_ptr_kinds h' = document_ptr_kinds h |\<union>| {|new_document_ptr|}"
translation_isomorphism: shows "monoid_isomorphism translation M (\<cdot>) \<one> (translation ` M) (compose M) (identity M)"
assms_2: shows "theta\<inverse> \<subseteq> ZOretrT (theta\<inverse>)"
5: shows "integral Dx_pair (\<lambda>a. - partial_vector_derivative (\<lambda>a. F a \<bullet> j) i a) = integral (cbox a b) (\<lambda>x. F (g2 x, x) \<bullet> j - F (g1 x, x) \<bullet> j)"
complex_is_Int_iff: fixes z :: "complex" shows "(z \<in> \<int>) = (Im z = 0 \<and> (\<exists>i. Re z = real_of_int i))"
nlcons_invmeas: fixes t :: "('a, 'b) FingerTreeStruc"   and nd :: "('a, 'b) Node" assumes "FingerTreeStruc.is_measured_ftree t"   and "FingerTreeStruc.is_measured_node nd" shows "FingerTreeStruc.is_measured_ftree (FingerTreeStruc.nlcons nd t)"
inject_2: fixes x1 :: "pi"   and y1 :: "pi" shows "(\<tau>.x1 = \<tau>.y1) = (x1 = y1)"
axioms: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set" assumes "Module smult M" shows "RModule UNIV smult M"
vsingletonD: fixes y :: "V"   and x :: "V" assumes "y \<in>\<^sub>\<circ> ZFC_in_HOL.set {x}" shows "y = x"
exec_move_AAssI1: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and e' :: "'addr expr1"   and e'' :: "'addr expr1" assumes "exec_move ci P t e h s ta h' s'" shows "exec_move ci P t (e\<lfloor>e'\<rceil> := e'') h s ta h' s'"
vector_1: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> '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 bot :: "'b"   and top :: "'b"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and converse :: "'b \<Rightarrow> 'b"   and unit :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.relation_algebra minus uminus inf less_eq less sup bot top composition converse unit"   and "relation_algebra.is_vector top composition x" shows "composition (inf x y) z = inf x (composition y z)"
msubst_const: fixes P :: "('e, 'f) uexpr"   and v :: "('g, 'f) uexpr" shows "P\<lbrakk>x\<rightarrow>v\<rbrakk> = P"
res_eq: shows "Lrule_result EquivForwardL j (SG ! i) = [(close (q # \<Gamma>) (\<Gamma> ! j), \<Delta>), (close \<Gamma> (\<Gamma> ! j), p # \<Delta>)]"
stut_eventually: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "STUTINV F" shows "STUTINV \<diamond>F"
ps0_def: shows "ps0 \<equiv> ap gs [] [] bs data"
zero_order_3: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "'c" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "\<not> less n zero"
assms_3: shows "\<A>' = \<A> @ [Step \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t]"
face_of_imp_compact: fixes S :: "'b set"   and T :: "'b set" assumes "convex S"   and "compact S"   and "T face_of S" shows "compact T"
ivl_disj_un_singleton_4: fixes l :: "'b"   and u :: "'b" assumes "l < u" shows "{l<..<u} \<union> {u} = {l<..u}"
steps_Skip_impl_exec: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c :: "('a, 'b, 'c) com"   and s :: "('a, 'c) xstate"   and t :: "('a, 'c) xstate" assumes "\<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>* (Skip, t)" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"
bot_empty_eq2: shows "bot = (\<lambda>x y. (x, y) \<in> {})"
simps_1: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and x :: "Preliminaries.var" shows "GE.int \<xi> (Var x) = \<xi> x"
simps: fixes a :: "32 word" shows "dotdecimal_of_ipv4addr a = (nat_of_ipv4addr ((a >> 24) && 255), nat_of_ipv4addr ((a >> 16) && 255), nat_of_ipv4addr ((a >> 8) && 255), nat_of_ipv4addr (a && 255))"
assms: fixes j :: "nat" assumes "j \<in> set inds" shows "j < dim_col A"
stop_doesnt_access: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and x :: "'a" assumes "sifum_modes ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars" shows "sifum_security_init.doesnt_modify dma (sifum_lang_no_dma.eval\<^sub>w ev\<^sub>A ev\<^sub>B) Stop x \<and> sifum_security_init.doesnt_read_or_modify \<C>_vars (sifum_lang_no_dma.eval\<^sub>w ev\<^sub>A ev\<^sub>B) Stop x"
Lcm_gauss_int_def: shows "Lcm \<equiv> normalization_euclidean_semiring_class.Lcm"
trimmed_pos_real_iff: fixes x :: "real" shows "trimmed_pos x = (0 < x)"
mat_cons2: fixes p :: "pat"   and env :: "v sem_env"   and s :: "'b state"   and v1 :: "v"   and ck :: "bool"   and pes :: "(pat \<times> exp0) list"   and err_v :: "v"   and bv :: "'b state \<times> (v, v) result"   and e :: "exp0" assumes "allDistinct (pat_bindings p []) \<and> pmatch (c env) (refs s) p v1 [] = No_match \<and> evaluate_match ck env s v1 pes err_v bv" shows "evaluate_match ck env s v1 ((p, e) # pes) err_v bv"
wtPB_def: fixes \<Phi> :: "(('fsym, 'tp) efsym, 'psym) lit list set" shows "TE.wtPB \<Phi> \<equiv> Ball \<Phi> TE.wtC"
CondT\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and e\<^sub>1 :: "nat exp"   and e' :: "nat exp"   and s\<^sub>2 :: "heap \<times> val list \<times> sheap"   and e\<^sub>2 :: "nat exp" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>true,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1,s\<^sub>1\<rangle> \<Rightarrow> \<langle>e',s\<^sub>2\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',s\<^sub>2\<rangle>"
e_type_local_shallow: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and n :: "nat"   and i :: "nat"   and vs :: "v list"   and es :: "e list"   and ts :: "t list"   and ts' :: "t list" assumes "\<S>\<bullet>\<C> \<turnstile> [Local n i vs es] : ts _> ts'" shows "\<exists>tls. length tls = n \<and> ts' = ts @ tls \<and> \<S>\<bullet>Some tls \<tturnstile>_ i vs;es : tls"
eligible_literal_def: fixes L :: "'a literal"   and C :: "'a eclause"   and \<sigma> :: "('a \<times> 'a trm) list" shows "eligible_literal L C \<sigma> = (L \<in> sel (cl_ecl C) \<or> sel (cl_ecl C) = {} \<and> maximal_literal (subst_lit L \<sigma>) (subst_cl (cl_ecl C) \<sigma>))"
0: shows "D.cone (C.dom e) (D.mkCone e)"
simps_2: fixes f :: "'a"   and ts :: "('a, 'b) Term.term list" shows "num_syms (Fun f ts) = Suc (sum_list (map num_syms ts))"
arr_char: fixes f :: "(bool, bool list) concrete_category.arr" shows "J.arr f = (f = J.Zero \<or> f = J.One \<or> f = J.j0 \<or> f = J.j1)"
resultant_impl_def: shows "resultant_impl = resultant_impl_generic dichotomous_Lazard"
fstE: fixes x :: "'a \<times> 'b"   and a :: "'a"   and b :: "'b"   and P :: "'a \<Rightarrow> bool" assumes "x = (a, b)"   and "P (fst x)" shows "P a"
m: fixes n :: "nat" assumes "n < length cl" shows "mC_C Sbis (cl ! n) (dl ! n) s t (P n) (F n)"
abs_eq: shows "1 = toQ \<lparr>mag = 1::'a, dim = QD('b), unit_sys = unit\<rparr>"
connected_space_iff_components_subset_singleton: fixes X :: "'a topology" shows "connected_space X = (\<exists>a. connected_components_of X \<subseteq> {a})"
prime_factors_gt_0_int: fixes p :: "int"   and x :: "int" assumes "p \<in># prime_factorization x" shows "0 < p"
image_UN: fixes f :: "'q \<Rightarrow> 'p"   and B :: "'r \<Rightarrow> 'q set"   and A :: "'r set" shows "f ` \<Union> (B ` A) = (\<Union>x\<in>A. f ` B x)"
assms_4: shows "p2 \<inter> (CX \<union> fst ` P1) = {}"
pterm_to_sterm_consts: fixes t :: "pterm" assumes "pre_strong_term_class.wellformed t" shows "consts (pterm_to_sterm t) = consts t"
acyclicP_wf_subcls1: fixes P :: "'a prog" assumes "acyclicP (subcls1 P)" shows "wfP (subcls1 P)\<inverse>\<inverse>"
f1_l0: fixes l :: "real" assumes "l0 \<le> l / c" shows "f1 * (l / c) \<le> f1 * l - 1"
PREC_case_aux: fixes f :: "nat list \<Rightarrow> nat"   and kf :: "nat"   and g :: "nat list \<Rightarrow> nat"   and kg :: "nat"   and l :: "nat list" assumes "\<And>l. f l + sum_list l < ack kf (sum_list l)"   and "\<And>l. g l + sum_list l < ack kg (sum_list l)" shows "PREC f g l + sum_list l < ack (Suc (kf + kg)) (sum_list l)"
l8_20_1_R1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and P :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A = B" shows "Tarski_neutral_dimensionless.Per Bet Cong B A P"