hcju/isabelleps huggingface dataset:
s_Auth: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>" shows "\<lless>\<pi>, Auth e\<ggreater> m\<rightarrow> \<lless>\<pi>', Auth e'\<ggreater>"
closed_in_D_2: fixes A :: "'a set" assumes "closed A" shows "emeasure M A = Inf (emeasure M ` {U. A \<subseteq> U \<and> open U})"
class_add_sup_state: fixes P :: "jvm_prog"   and \<tau> :: "ty list \<times> ty err list"   and \<tau>' :: "ty list \<times> ty err list"   and C :: "char list"   and cdec :: "jvm_method class" assumes "P \<turnstile> \<tau> \<le>\<^sub>i \<tau>'"   and "\<not> is_class P C" shows "class_add P (C, cdec) \<turnstile> \<tau> \<le>\<^sub>i \<tau>'"
trace_decomp_tail: fixes c :: "('p, 's, 'm) configuration"   and t :: "('p, 's, 'm) event list"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration" assumes "trace c (t @ [ev]) c'" shows "\<exists>c''. trace c t c'' \<and> c'' \<turnstile> ev \<mapsto> c'"
from_vec_to_vec: fixes M :: "('b, 'c) sq_matrix" shows "from_vec (to_vec M) = M"
BinOpRed1: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and bop :: "bop"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow> \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>e \<guillemotleft>bop\<guillemotright> e\<^sub>2,s,b\<rangle> \<rightarrow> \<langle>e' \<guillemotleft>bop\<guillemotright> e\<^sub>2,s',b'\<rangle>"
WCFG_edge_det: fixes prog :: "cmd"   and n :: "w_node"   and et :: "(char list \<Rightarrow> val option) edge_kind"   and n' :: "w_node"   and et' :: "(char list \<Rightarrow> val option) edge_kind" assumes "prog \<turnstile> n -et\<rightarrow> n'"   and "prog \<turnstile> n -et'\<rightarrow> n'" shows "et = et'"
cnorm_eq: fixes x :: "'a"   and y :: "'b" shows "(norm x = norm y) = (x \<bullet>\<^sub>C x = y \<bullet>\<^sub>C y)"
lv_evt_defs_2: fixes Quorum :: "process set set"   and mru_votes :: "process \<Rightarrow> (nat \<times> val) option"   and Q :: "process set"   and v :: "val" assumes "quorum_process Quorum" shows "quorum_process.opt_mru_guard Quorum mru_votes Q v \<equiv> Q \<in> Quorum \<and> (let mru = opt_mru_vote mru_votes Q in mru = None \<or> (\<exists>r. mru = Some (r, v)))"
measure_completion: fixes S :: "'b set"   and M :: "'b measure" assumes "S \<in> sets M" shows "Sigma_Algebra.measure (completion M) S = Sigma_Algebra.measure M S"
Pd_refl: fixes x :: "'a"   and d :: "'b" assumes "x \<in> Field (Pd'_tax d)" shows "(x, x) \<in> Pd'_tax d"
skl1_inv2D: fixes Rb :: "fid_t"   and B :: "agent"   and A :: "agent"   and x :: "skl1_state"   and gnx :: "msg"   and Ni :: "msg" assumes "guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>"   and "in_progressS (progress x Rb) {xny, xni, xnr, xgnx, xgny, xsk}"   and "sklvl1.guessed_frame Rb xgnx = Some gnx"   and "sklvl1.guessed_frame Rb xni = Some Ni"   and "can_signal x A B"   and "x \<in> skl1_inv2" shows "0 < signalsInit x (Running A B \<langle>Ni, NonceF (Rb $ nr), Exp gnx (NonceF (Rb $ ny))\<rangle>)"
matrix_def: fixes M :: "'a list list"   and rs :: "nat"   and cs :: "nat" shows "matrix M rs cs = ((\<forall>row\<in>set M. length row = cs) \<and> length M = rs)"
vsv_ex1_app2: fixes r :: "V"   and a :: "V"   and b :: "V" assumes "vsv r"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "(r\<lparr>a\<rparr> = b) = (\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r)"
dens': fixes M :: "val measure"   and \<rho> :: "nat \<Rightarrow> val" assumes "has_subprob_density M (stock_measure t) (f \<rho>)" shows "has_density (distr M (stock_measure t') (op_sem oper)) (stock_measure t') (g \<rho>)"
id_oracle_def: fixes s :: "'c"   and x :: "'d" shows "id_oracle s x = Pause x (\<lambda>x. Done (x, s))"
hyps: shows "ffact n (x + y) = (\<Sum>k = 0..n. of_nat (n choose k) * ffact k x * ffact (n - k) y)"
satPB_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and \<Phi> :: "('a, 'b) lit list set" shows "F.satPB \<xi> \<Phi> \<equiv> \<forall>c\<in>\<Phi>. F.satC \<xi> c"
pCons_of_rp_contains_ints: fixes a :: "int poly"   and b :: "rat"   and c :: "rat poly" assumes "of_int_poly a = pCons b c" shows "is_int_rat b"
allocationDisjointAuxiliary: fixes a :: "('c \<times> 'd set) set"   and n1 :: "'c"   and n2 :: "'c" assumes "a \<in> allocationsUniverse"   and "n1 \<in> Domain a"   and "n2 \<in> Domain a"   and "n1 \<noteq> n2" shows "a ,, n1 \<inter> a ,, n2 = {}"
B: shows "\<forall>x. a < x \<and> x < b \<longrightarrow> poly p x \<noteq> 0"
dim_unique: fixes B :: "'b set"   and V :: "'b set"   and n :: "nat" assumes "B \<subseteq> V"   and "V \<subseteq> span B"   and "independent B"   and "card B = n" shows "dim V = n"
weakPsiCongClosed: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list" assumes "\<Psi> \<rhd> P \<doteq> Q" shows "p \<bullet> \<Psi> \<rhd> p \<bullet> P \<doteq> p \<bullet> Q"
Rep_downset_inverse: fixes x :: "'a downset" shows "Abs_downset (Rep_downset x) = x"
enqueue_new_threads_def: shows "local.enqueue_new_threads = foldl local.enqueue_new_thread"
weakening_2: fixes cs :: "e list" assumes "\<S>\<bullet>\<C> \<turnstile> cs : [] _> t1s"   and "\<And>lholed. \<not> Lfilled 0 lholed [$Return] (cs @ ($* es))"   and "\<And>lholed i. \<not> Lfilled 0 lholed [$Br i] (cs @ ($* es))"   and "const_list cs"   and "\<not> const_list ($* es)"   and "i < length (s_inst \<S>)"   and "length (local \<C>) = length vs"   and "Option.is_none (memory \<C>) = Option.is_none (inst.mem (inst s ! i))" shows "\<exists>a s' vs'. Ex (reduce s vs (cs @ ($* es)) i s' vs')"
subpath_trivial: fixes g :: "real \<Rightarrow> 'b" shows "subpath 0 1 g = g"
assms_4: shows "set (labels_conv ss2) \<subseteq> T"
cscc_invar_part_def: shows "cscc_invar_part \<equiv> \<lambda>(l, p, D, pE). cscc_invar_ext G l D"
B: shows "Crypt (sesK (c * f)) \<lbrace>pubAK (c * (s + a * b')), Auth_Data g b', pubAK g, Crypt (priSK CA) (Hash (pubAK g))\<rbrace> \<in> synth (analz (A \<union> spies evsFR4))"
N_minus_commute: fixes q :: "real"   and x :: "int"   and y :: "int" assumes "fbnorm q" shows "fbnorm.N q (x - y) = fbnorm.N q (y - x)"
fps_inverse_def: fixes f :: "'a fps" shows "inverse f = fps_right_inverse f (inverse (fps_nth f 0))"
set_admissible: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "set.admissible (\<lambda>f. \<forall>x y. y \<in> f x \<longrightarrow> P x y)"
simps_201: fixes x131 :: "expr"   and x132 :: "expr"   and x133 :: "expr"   and x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list" shows "if (x131) x132 else x133 \<noteq> Call x101 x102 x103 x104"
measure_pmf_zero_iff: fixes p :: "'c pmf"   and s :: "'c set" shows "(measure_pmf.prob p s = 0) = (set_pmf p \<inter> s = {})"
u_2: fixes x :: "'a" shows "0 \<le> u x"
lasso_run_rel_def: fixes R :: "('e \<times> 'f) set" shows "\<langle>R\<rangle>lasso_run_rel = br run_of_lasso (\<lambda>_. True) O (nat_rel \<rightarrow> R)"
Smith_normal_form_upt_k_intro: fixes k :: "nat"   and A :: "(('c, 'b) vec, 'a) vec" assumes "\<And>a b. to_nat a = to_nat b \<and> to_nat a + 1 < k \<and> to_nat b + 1 < k \<Longrightarrow> A $ a $ b dvd A $ (a + (1::'a)) $ (b + (1::'b))"   and "\<And>a b. to_nat a \<noteq> to_nat b \<and> (to_nat a < k \<or> to_nat b < k) \<Longrightarrow> A $ a $ b = (0::'c)" shows "Smith_normal_form_upt_k A k"
prepend_in_P1: fixes f :: "nat \<Rightarrow> nat option"   and vs :: "nat list" assumes "f \<in> \<P>" shows "vs \<odot> f \<in> \<P>"
not_parts_not_analz: fixes c :: "msg"   and H :: "msg set" assumes "c \<notin> parts H" shows "c \<notin> analz H"
C: fixes cfs :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list" assumes "ccomp c\<^sub>2 \<Turnstile> cfs\<box>" shows "cpred c\<^sub>2 (cfs ! 0) (cfs ! (length cfs - Suc 0))"
retraction: fixes v :: "'a" assumes "v \<in> \<Union> X" shows "f (f v) = f v"
orthogonal_iff_angle: fixes A :: "'b"   and B :: "'b"   and C :: "'b" shows "orthogonal (A - B) (C - B) = (angle A B C = pi / 2)"
sat_get_and: fixes \<sigma> :: "(String.literal \<times> event_data list) Trace.trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and \<phi> :: "formula" shows "sat \<sigma> V v i \<phi> = list_all (sat \<sigma> V v i) (get_and_list \<phi>)"
regular_weak_composition_axioms: shows "regular_weak_composition (\<cdot>) (\<star>)"
evpm2: shows "\<forall>\<^sub>F x in at x0 within S. \<forall>s>0. s \<le> return_time p1 x \<longrightarrow> flow0 x s \<notin> p2"
hd_eq_first: fixes xs :: "'d list" assumes "xs \<noteq> []" shows "xs ! 0 = hd xs"
select_trap_result1: fixes s :: "'b sparc_state" assumes "reset_trap_val s = True" shows "snd (select_trap () s) = False"
snd_edge_at: fixes c :: "'form"   and "is" :: "nat list" shows "snd (edge_at c is) = edge_to c is"
assms_2: shows "sign_changes' \<equiv> \<lambda>ps x. \<Sum>ps'\<leftarrow>split_sign_changes ps x. sign_changes ps' x"
mod2_eq_if: fixes a :: "'a" shows "a mod (2::'a) = (if even a then 0::'a else (1::'a))"
simps_1: fixes a :: "'a"   and w :: "nat \<Rightarrow> 'a" shows "(a ## w) 0 = a"
split_list_prop: fixes xs :: "'c list"   and P :: "'c \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. P x" shows "\<exists>ys x zs. xs = ys @ x # zs \<and> P x"
a1: shows "((\<forall>i t. i \<le> n \<and> SEQ i s1 = Some t \<longrightarrow> good_context t \<and> get_delayed_pool t = [] \<and> get_trap_set t = {}) \<and> (\<forall>i t. i \<le> n \<and> SEQ i s2 = Some t \<longrightarrow> good_context t \<and> get_delayed_pool t = [] \<and> get_trap_set t = {}) \<longrightarrow> (\<exists>t1. Some t1 = SEQ n s1 \<and> (\<exists>t2. Some t2 = SEQ n s2 \<and> get_S (cpu_reg_val PSR t1) = 0 \<and> get_S (cpu_reg_val PSR t2) = 0 \<and> low_equal t1 t2))) \<and> (\<forall>i t. i \<le> Suc n \<and> SEQ i s1 = Some t \<longrightarrow> good_context t \<and> get_delayed_pool t = [] \<and> get_trap_set t = {}) \<and> (\<forall>i t. i \<le> Suc n \<and> SEQ i s2 = Some t \<longrightarrow> good_context t \<and> get_delayed_pool t = [] \<and> get_trap_set t = {})"
measurable_sigma_sets: fixes N :: "'c measure"   and \<Omega> :: "'c set"   and A :: "'c set set"   and f :: "'d \<Rightarrow> 'c"   and M :: "'d measure" assumes "sets N = sigma_sets \<Omega> A"   and "A \<subseteq> Pow \<Omega>"   and "f \<in> space M \<rightarrow> \<Omega>"   and "\<And>y. y \<in> A \<Longrightarrow> f -` y \<inter> space M \<in> sets M" shows "f \<in> M \<rightarrow>\<^sub>M N"
sorted_wrt_append: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" shows "List.sorted_wrt P (xs @ ys) = (List.sorted_wrt P xs \<and> List.sorted_wrt P ys \<and> (\<forall>x\<in>set xs. \<forall>y\<in>set ys. P x y))"
representation_diff: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and basis :: "'c set"   and v :: "'c"   and u :: "'c" assumes "module scale"   and "\<not> module.dependent scale basis"   and "v \<in> module.span scale basis"   and "u \<in> module.span scale basis" shows "module.representation scale basis (u - v) = (\<lambda>b. module.representation scale basis u b - module.representation scale basis v b)"
eventually_ln_const: fixes C :: "real" assumes "0 < C" shows "\<forall>\<^sub>F x in at_top. 1 / 2 < ln (C * x) / ln x"
complex_roots_complex_def: fixes p :: "complex poly" shows "complex_roots_complex p = (SOME as. smult (lead_coeff p) (\<Prod>a\<leftarrow>as. [:- a, 1:]) = p \<and> length as = degree p)"
cExt: fixes R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi"   and \<Psi>'' :: "'b" assumes "(\<Psi>, R, S) \<in> Rel'" shows "(\<Psi> \<otimes> \<Psi>'', R, S) \<in> Rel'"
semigroup_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "semigroup f \<equiv> \<forall>a b c. f (f a b) c = f a (f b c)"
subset_insertI: fixes B :: "'b set"   and a :: "'b" shows "B \<subseteq> insert a B"
cone_mkCone: fixes e :: "'a" assumes "D.is_equalized_by e" shows "D.cone (local.dom e) (D.mkCone e)"
relInvImage_Id_on: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set" assumes "\<And>a1 a2. (f a1 = f a2) = (a1 = a2)" shows "BNF_Greatest_Fixpoint.relInvImage A (Id_on B) f \<subseteq> Id"
cod_dom: fixes f :: "'a" assumes "R.arr f" shows "R.cod (R.dom f) = R.dom f"
induct: fixes P :: "'b common_primitive match_expr \<Rightarrow> bool"   and a0 :: "'b common_primitive match_expr" assumes "P MatchAny"   and "\<And>uu_. P (Match (Src_Ports (L4Ports uu_ [])))"   and "\<And>uv_ uw_. P (Match (Src_Ports (L4Ports uv_ [uw_])))"   and "\<And>v vb va vd. P (Match (Src_Ports (L4Ports v (vb # va # vd))))"   and "\<And>v. P (Match (Src v))"   and "\<And>v. P (Match (Dst v))"   and "\<And>v. P (Match (IIface v))"   and "\<And>v. P (Match (OIface v))"   and "\<And>v. P (Match (Prot v))"   and "\<And>v. P (Match (Dst_Ports v))"   and "\<And>v. P (Match (MultiportPorts v))"   and "\<And>v. P (Match (L4_Flags v))"   and "\<And>v. P (Match (CT_State v))"   and "\<And>v. P (Match (Extra v))"   and "\<And>uz_. P (MatchNot (Match (Src_Ports uz_)))"   and "\<And>v. P (MatchNot (Match (Src v)))"   and "\<And>v. P (MatchNot (Match (Dst v)))"   and "\<And>v. P (MatchNot (Match (IIface v)))"   and "\<And>v. P (MatchNot (Match (OIface v)))"   and "\<And>v. P (MatchNot (Match (Prot v)))"   and "\<And>v. P (MatchNot (Match (Dst_Ports v)))"   and "\<And>v. P (MatchNot (Match (MultiportPorts v)))"   and "\<And>v. P (MatchNot (Match (L4_Flags v)))"   and "\<And>v. P (MatchNot (Match (CT_State v)))"   and "\<And>v. P (MatchNot (Match (Extra v)))"   and "\<And>m1 m2. \<lbrakk>P m1; P m2\<rbrakk> \<Longrightarrow> P (MatchAnd m1 m2)"   and "\<And>vb_ vc_. P (MatchNot (MatchAnd vb_ vc_))"   and "\<And>vd_. P (MatchNot (MatchNot vd_))"   and "P (MatchNot MatchAny)" shows "P a0"
strict_implies_order: fixes a :: "'c list"   and b :: "'c list" assumes "strict_suffix a b" shows "suffix a b"
obligation1'': fixes h :: "nat list"   and init :: "nat list"   and qs :: "nat list" shows "Partial_Cost_Model.config'_rand (COMB h) (fst (COMB h) init \<bind> (\<lambda>is. return_pmf (init, is))) qs = map_pmf noc (Sum_pmf (8 / 10) (Partial_Cost_Model.config'_rand BIT (fst BIT init \<bind> (\<lambda>is. return_pmf (init, is))) qs) (Partial_Cost_Model.config'_rand (Partial_Cost_Model.embed (rTS h)) (fst (Partial_Cost_Model.embed (rTS h)) init \<bind> (\<lambda>is. return_pmf (init, is))) qs))"
a_get_owner_document_tups_def: shows "a_get_owner_document_tups = [(is_element_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r, a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<circ> the \<circ> cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r), (is_character_data_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r, a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<circ> the \<circ> cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r), (is_document_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r, a_get_owner_document\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<circ> the \<circ> cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r)]"
Method_i_i_i_o_o_o_oI: fixes x :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and xa :: "char list"   and xb :: "char list"   and xc :: "ty list"   and xd :: "ty"   and xe :: "'b"   and xf :: "char list" assumes "x \<turnstile> xa sees xb: xc\<rightarrow>xd = xe in xf" shows "pred.eval (Method_i_i_i_o_o_o_o x xa xb) (xc, xd, xe, xf)"
lattice_of_def: fixes fs :: "'a vec list" shows "lattice_of fs = range (\<lambda>c. M.sumlist (map (\<lambda>i. of_int (c i) \<cdot>\<^sub>v fs ! i) [0..<length fs]))"
in_X: fixes x :: "'c" assumes "v x \<le> n" shows "x \<in> X"
Inf_hd_in_Eps: fixes n :: "nat"   and m :: "nat" assumes "n < m" shows "Inf (hd (F m)) \<in> Eps (\<Phi> (F n))"
bv_slice_def: fixes w :: "bit list" shows "bv_slice w = (\<lambda>(b, e). fst (bv_chop (snd (bv_chop w (b + 1))) e))"
FSC_def: 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 C :: "'a"   and D :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a"   and D' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.FSC Bet Cong A B C D A' B' C' D' \<equiv> Tarski_neutral_dimensionless.Col Bet A B C \<and> Tarski_neutral_dimensionless.Cong3 Cong A B C A' B' C' \<and> Cong A D A' D' \<and> Cong B D B' D'"
subprob_space_distr: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and M' :: "'c measure" assumes "subprob_space M"   and "f \<in> M \<rightarrow>\<^sub>M M'"   and "space M' \<noteq> {}" shows "subprob_space (distr M M' f)"
step: fixes 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 r :: "nat \<Rightarrow>f value option"   and t :: "(String.literal \<times> value list) list" assumes "\<exists>(s'', T)|\<in>|possible_steps e s' r' l i. obtains s r e s'' (evaluate_updates T i r') t" shows "obtains s r e s' r' ((l, i) # t)"
finfun_Ball_except_update: fixes xs :: "'a list"   and A :: "'a \<Rightarrow>f bool"   and a :: "'a"   and b :: "bool"   and P :: "'a \<Rightarrow> bool" shows "finfun_Ball_except xs A(a $:= b) P = ((a \<in> set xs \<or> (b \<longrightarrow> P a)) \<and> finfun_Ball_except (a # xs) A P)"
spr_simRels_def: fixes jkbp :: "'a \<Rightarrow> ('a, 'b, 'c \<times> 'd) GC list"   and envObsC :: "'f \<Rightarrow> 'g" assumes "FiniteBroadcastEnvironment jkbp" shows "FiniteBroadcastEnvironment.spr_simRels envObsC \<equiv> \<lambda>a. {(s, s') |s s'. (envObsC (es (sprFst s)), ps (sprFst s) a, pubActs (sprFst s)) = (envObsC (es (sprFst s')), ps (sprFst s') a, pubActs (sprFst s')) \<and> (envObsC (es (sprLst s)), ps (sprLst s) a, pubActs (sprLst s)) = (envObsC (es (sprLst s')), ps (sprLst s') a, pubActs (sprLst s')) \<and> sprCRel s = sprCRel s'}"
H'_1: shows "Aenc \<langle>SecureTag, Messages.Agent A, M\<rangle> (pubK B) = X \<and> priK A = Y"
Hintikka_Extend: fixes H :: "('a, 'b) fm set" assumes "consistent H"   and "maximal H"   and "saturated H"   and "infinite UNIV" shows "Hintikka H"
simps_6: fixes v :: "'a formula"   and va :: "'a formula" shows "is_cnf (v \<^bold>\<rightarrow> va) = is_disj (v \<^bold>\<rightarrow> va)"
perm_swap_2: fixes a :: "name"   and b :: "name"   and x :: "'a" shows "[(a, b)] \<bullet> [(b, a)] \<bullet> x = x"
insert_ident: fixes x :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "x \<notin> A"   and "x \<notin> B" shows "(insert x A = insert x B) = (A = B)"
canI_associator_1_2: fixes f :: "'a"   and g :: "'a" assumes "ide f"   and "ide g"   and "src f = trg g" shows "\<a>[f, src f, g] = can (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>src f\<^bold>\<rangle>\<^sub>0 \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>) ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>src f\<^bold>\<rangle>\<^sub>0) \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>)"
in_lset_ldropnD: fixes x :: "'c"   and n :: "nat"   and xs :: "'c llist" assumes "x \<in> lset (ldropn n xs)" shows "x \<in> lset xs"
extremum: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and top :: "'c"   and a :: "'c" assumes "ordering_top less_eq less top" shows "less_eq a top"
mono_aval'': fixes S :: "'a Abs_State.st option"   and S' :: "'a Abs_State.st option"   and e :: "aexp" assumes "S \<sqsubseteq> S'" shows "aval'' e S \<sqsubseteq> aval'' e S'"
abs_pdevs_val_le_tdev: fixes e :: "nat \<Rightarrow> real"   and x :: "'a pdevs" assumes "e \<in> UNIV \<rightarrow> {- 1..1}" shows "\<bar>pdevs_val e x\<bar> \<le> tdev x"
Rep_Abs_mod: fixes m :: "int" shows "Rep_bit1 (Abs_bit1 (m mod int CARD('a bit1))) = m mod int CARD('a bit1)"
bs_eq_correct: fixes s1 :: "integer"   and s2 :: "integer" shows "bs_eq s1 s2 = (bs_\<alpha> s1 = bs_\<alpha> s2)"
Key_analz_image_Key: fixes evs :: "event list"   and SK :: "nat"   and KK :: "nat set" assumes "evs \<in> kerbV"   and "SK \<in> symKeys \<and> KK \<subseteq> - range shrK"   and "\<forall>K\<in>KK. \<not> AKcryptSK K SK evs" shows "(Key SK \<in> analz (Key ` KK \<union> knows Spy evs)) = (SK \<in> KK \<or> Key SK \<in> analz (knows Spy evs))"
Inf_subdistl_iso: fixes f :: "'b \<times> 'c \<Rightarrow> 'd" assumes "f \<circ> Inf_prod \<le> Inf \<circ> (`) f" shows "prod.mono f"
simps: fixes w :: "'a list"   and I :: "(nat + nat set) list"   and \<phi> :: "'a formula" shows "wf_interp_for_formula (w, I) \<phi> = ((0 < length w \<and> (\<forall>a\<in>set w. a \<in> set \<Sigma>) \<and> Ball (set I) (case_sum (\<lambda>p. p < length w) (\<lambda>P. \<forall>p\<in>P. p < length w))) \<and> (\<forall>n\<in>FOV \<phi>. case I ! n of Inl x \<Rightarrow> True | Inr b \<Rightarrow> False) \<and> (\<forall>n\<in>SOV \<phi>. case I ! n of Inl x \<Rightarrow> False | Inr b \<Rightarrow> True))"
ex_bij_betw_nat_finite: fixes M :: "'b set" assumes "finite M" shows "\<exists>h. bij_betw h {0..<card M} M"
p_eq_f: fixes k :: "nat" assumes "0 < k"   and "k \<le> n" shows "p k = f k"
obtain_nonzero: assumes "V \<noteq> 0" obtains g :: "'b" where "g \<in> V" and "g \<noteq> (0::'b)"
col_ABB: fixes A :: "Points"   and B :: "Points" shows "col A B B"
to_fun_inject: fixes x :: "('d, 'c) sq_matrix"   and y :: "('d, 'c) sq_matrix" shows "(to_fun x = to_fun y) = (x = y)"
joinI: fixes a :: "'a"   and c :: "'a"   and A :: "('a \<times> 'a) set"   and b :: "'a" assumes "(a, c) \<in> A\<^sup>*"   and "(b, c) \<in> A\<^sup>*" shows "(a, b) \<in> A\<^sup>\<down>"
comp_null_2: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "partial_magma C" shows "C f (partial_magma.null C) = partial_magma.null C"
iwlsFSbSwTR_termFSbImorph_iff: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) model_scheme"   and h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd" assumes "iwlsFSbSwTR MOD" shows "termFSbImorph h hA MOD = ipresCons h hA MOD"
not_dlist_member_empty: fixes x :: "'b" shows "\<not> member DList.empty x"
dghm_comp_components_1: fixes \<GG> :: "V"   and \<FF> :: "V" shows "(\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>)\<lparr>ObjMap\<rparr> = \<GG>\<lparr>ObjMap\<rparr> \<circ>\<^sub>\<circ> \<FF>\<lparr>ObjMap\<rparr>"
monoG: shows "mono (G::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice)"
append_is_Nil_conv: fixes xs :: "'c list"   and ys :: "'c list" shows "(xs @ ys = []) = (xs = [] \<and> ys = [])"
enn2ereal_eq_top_iff: fixes x :: "ennreal" shows "(enn2ereal x = \<infinity>) = (x = top)"
equivalence_functor_axioms: shows "equivalence_functor (\<cdot>\<^sub>S) (\<cdot>\<^sub>S) R"
summandsIdemp: fixes Q :: "pi"   and P :: "pi"   and Q' :: "pi" assumes "Q \<in> summands P"   and "Q \<equiv>\<^sub>e Q'" shows "P \<oplus> Q' \<equiv>\<^sub>e P"
AbGroup_inner_dirsum_pairwise_int0: fixes Gs :: "'a set list"   and G :: "'a set"   and G' :: "'a set" assumes "\<forall>G\<in>set Gs. AbGroup G"   and "add_independentS Gs"   and "G \<in> set Gs"   and "G' \<in> set Gs"   and "G \<noteq> G'" shows "G \<inter> G' = 0"
Abs_iso_inverse: fixes y :: "'b \<Rightarrow> 'b" assumes "y \<in> {f. mono f}" shows "Rep_iso (Abs_iso y) = y"