hcju/isabelleps huggingface dataset:
simulation_silent1: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s1' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s1 \<approx>m s2"   and "\<tau>mred1 s1 s1'" shows "\<exists>s2'. \<tau>mRed2 s2 s2' \<and> s1' \<approx>m s2'"
exhaust: fixes y :: "('a, 'b) frame" obtains x1 :: "'a" and x2 :: "nat" and x3 :: "'b list" where "y = Frame x1 x2 x3"
distinct_29: fixes x2 :: "nat"   and x91 :: "int"   and x92 :: "MIR.num"   and x93 :: "MIR.num" shows "Bound x2 \<noteq> CF x91 x92 x93"
subring_is_ring: fixes H :: "'a set" assumes "subring H (R\<lparr>carrier := K\<rparr>)" shows "ring (R\<lparr>carrier := K, carrier := H\<rparr>)"
ex_countable_subbasis: fixes "open" :: "'d set \<Rightarrow> bool" assumes "class.second_countable_topology open" shows "\<exists>B. countable B \<and> open = generate_topology B"
lookup_update_other: fixes y :: "'a"   and z :: "'a"   and xs :: "'a list"   and zs :: "'a list"   and opt :: "('b, 'c, 'a) env option"   and env :: "('b, 'c, 'a) env"   and ys :: "'a list" assumes "y \<noteq> z" shows "lookup (update (xs @ z # zs) opt env) (xs @ y # ys) = lookup env (xs @ y # ys)"
and_int_rec: fixes k :: "int"   and l :: "int" shows "k AND l = of_bool (odd k \<and> odd l) + 2 * (k div 2 AND l div 2)"
simps_14: fixes v :: "nat" shows "[Suc v]\<^sub>1 +\<^sub>1 \<infinity>\<^sub>1 = \<infinity>\<^sub>1"
IH1: fixes m :: "nat"   and xa :: "taylor_model"   and x :: "real" assumes "m < n"   and "num_params (tm_poly xa) \<le> length I"   and "develops_at_within e a I"   and "x \<in>\<^sub>i range_tm e xa" shows "x ^ m \<in>\<^sub>i range_tm e (tm_pow prec ord I a xa m)"
invC_prop: shows "G\<turnstile>invmode sm e\<rightarrow>invC\<preceq>statT"
le_iff_inf: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.semilattice_inf inf less_eq less" shows "less_eq x y = (inf x y = x)"
1: shows "Fvars (rawpsubst \<chi> txs) \<subseteq> Fvars \<phi> - insert x (snd ` set txs) \<union> \<Union> {FvarsT ta |ta. \<exists>xa. ta = t \<and> xa = x \<or> (ta, xa) \<in> set txs}"
vc_foreach_completed_imp_post: fixes S :: "'v set"   and V :: "'v set"   and v :: "'v"   and V' :: "'v set" assumes "fe_inv S V v {} (V', False)" shows "rpost S (V, v) (V', False)"
split_list_first: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
simps_9: fixes l :: "Formula.formula list" shows "Formula.future_bounded (formula.Ands l) = list_all Formula.future_bounded l"
allDefs_disjoint': fixes n :: "'node"   and g :: "'g"   and m :: "'node"   and v :: "'var \<times> 'node \<times> Def" assumes "n \<in> set (\<alpha>n g)"   and "m \<in> set (\<alpha>n g)"   and "v \<in> braun_ssa.allDefs g n"   and "v \<in> braun_ssa.allDefs g m" shows "n = m"
svar_deriv: fixes x :: "'a"   and \<nu> :: "(real, 'a) vec" shows "((\<lambda>v. v $ x) has_derivative (\<lambda>v'. v' \<bullet> (\<chi>i. if i = x then 1 else 0))) (at \<nu>)"
trms\<^sub>s\<^sub>s\<^sub>t_proj_subset_1: fixes n :: "'c"   and A :: "('c strand_label \<times> ('a, 'b) stateful_strand_step) list" shows "trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (proj n A) \<subseteq> trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
sum_of_prideals0: fixes n :: "nat" shows "\<forall>f. (\<forall>l\<le>n. f l \<in> carrier R) \<longrightarrow> ideal R (sum_pr_ideals R f n)"
awalk_verts_conv: fixes u :: "'a"   and p :: "'b list" shows "awalk_verts u p = (if p = [] then [u] else map (tail G) p @ [head G (last p)])"
del_type_1: fixes t :: "('c \<times> 'd) bro"   and h :: "nat"   and x :: "'c" assumes "t \<in> B h" shows "del x t \<in> T h"
assms: shows "a \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>"
secrecy_lemma: fixes A :: "agent"   and B :: "agent"   and evs :: "event list"   and K :: "nat"   and na :: "msg"   and nb :: "msg" assumes "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> yahalom" shows "Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace> \<in> set evs \<longrightarrow> Key K \<notin> analz (knows Spy evs)"
select_convs_1: fixes initial :: "char list \<Rightarrow> nat option"   and final :: "char list \<Rightarrow> nat option"   and more :: "'a" shows "initial \<lparr>initial = initial, final = final, \<dots> = more\<rparr> = initial"
ID_cEnum: fixes cEnum :: "('b list \<times> (('b \<Rightarrow> bool) \<Rightarrow> bool) \<times> (('b \<Rightarrow> bool) \<Rightarrow> bool)) option"   and enum :: "'b list"   and enum_all :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('b \<Rightarrow> bool) \<Rightarrow> bool" assumes "class.cenum cEnum"   and "ID cEnum = Some (enum, enum_all, enum_ex)" shows "UNIV = set enum \<and> enum_all = Ball UNIV \<and> enum_ex = Bex UNIV"
is_decoding_def: fixes s :: "'s"   and os :: "'t list"   and as :: "'s list" shows "is_decoding s os as \<equiv> emeasure (T' (I s)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>o \<omega>'. \<omega> = (s, o) ## zip as os @- \<omega>'} = max_prob s os \<and> length as = length os \<and> set as \<subseteq> \<S>"
cnf_of_encode_all_operator_preconditions_contains_clause_if: fixes \<Pi> :: "'b strips_problem"   and k :: "nat"   and t :: "nat"   and op :: "'b strips_operator"   and v :: "'b" assumes "is_valid_problem_strips \<Pi>"   and "k < t"   and "op \<in> set (\<Pi>\<^sub>\<O>)"   and "v \<in> set (precondition_of op)" shows "{(Operator k (index (\<Pi>\<^sub>\<O>) op))\<inverse>, (State k (index (\<Pi>\<^sub>\<V>) v))\<^sup>+} \<in> cnf (encode_all_operator_preconditions \<Pi> (\<Pi>\<^sub>\<O>) t)"
f: shows "\<forall>i\<le>n. i \<noteq> 0 \<longrightarrow> Le (f i) = FW M' n i 0 \<and> Le (- f i) = FW M' n 0 i"
mk_chain: fixes sel :: "nat \<times> ('d \<times> nat \<times> nat) set \<Rightarrow> ((('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) \<times> ('e \<times> nat) set) option"   and Rs :: "'f"   and init :: "nat \<times> ('d \<times> nat \<times> nat) set" shows "\<forall>i. mk_chain sel Rs init (Suc i) = make_step sel (mk_chain sel Rs init i)"
span_span: fixes W :: "rat vec set" assumes "W \<subseteq> Rn" shows "gs.span (gs.span W) = gs.span W"
intro: fixes \<alpha> :: "V"   and \<FF> :: "V"   and \<AA> :: "V"   and \<BB> :: "V" assumes "\<Z> \<alpha>"   and "vfsequence \<FF>"   and "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>"   and "is_dghm_axioms \<AA> \<BB> \<FF>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
simps_3: fixes n :: "nat"   and x :: "'b"   and xs :: "'b list" shows "delete_index (Suc n) (x # xs) = x # delete_index n xs"
l2_inv6'_init: shows "init l2 \<subseteq> l2_inv6'"
inj_on_nsets: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and n :: "nat" assumes "inj_on f A" shows "inj_on ((`) f) ([A]\<^bsup>n\<^esup>)"
distinct_30: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "Stop \<noteq> x31@[x32]"
res: shows "local.tighten_poly_bounds_epsilon l r sr = (l', r', sr')"
weight_spmf_def: fixes p :: "'a spmf" shows "weight_spmf p = Sigma_Algebra.measure (measure_spmf p) UNIV"
\<tau>move2_ALengthD: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and a :: "'addr expr1"   and pc :: "nat"   and xcp :: "'addr option" assumes "\<tau>move2 P h stk (a\<bullet>length) pc xcp"   and "pc < length (compE2 a)" shows "\<tau>move2 P h stk a pc xcp"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'d tree234 \<Rightarrow> 'c"   and f2 :: "'d tree234 \<Rightarrow> 'd \<Rightarrow> 'd tree234 \<Rightarrow> 'c"   and up\<^sub>i :: "'d up\<^sub>i" shows "P (case up\<^sub>i of T\<^sub>i x \<Rightarrow> f1 x | Up\<^sub>i x xa xb \<Rightarrow> f2 x xa xb) = ((\<forall>x1. up\<^sub>i = T\<^sub>i x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22 x23. up\<^sub>i = Up\<^sub>i x21 x22 x23 \<longrightarrow> P (f2 x21 x22 x23)))"
path_component_refl_eq: fixes S :: "'b set"   and x :: "'b" shows "path_component S x x = (x \<in> S)"
integrable_subinterval: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "f integrable_on cbox a b"   and "cbox c d \<subseteq> cbox a b" shows "f integrable_on cbox c d"
Chernoff_on_def: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set" shows "Chernoff_on A f = (\<forall>B\<subseteq>A. \<forall>C\<subseteq>B. f B \<inter> C \<subseteq> f C)"
simps_7: fixes h :: "'c \<Rightarrow> 'd list \<Rightarrow> bool"   and \<phi> :: "'c fm"   and xs :: "'d list" shows "interpret h (ExQ \<phi>) xs = (\<exists>x. interpret h \<phi> (x # xs))"
of_int_of_nat_eq: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and uminus :: "'k \<Rightarrow> 'k"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (int n) = semiring_1.of_nat one plus zero n"
a_Rec_inv_V: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and eP :: "Syntax.term"   and x :: "Syntax.var"   and eV :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty" assumes "\<Gamma> \<turnstile> e, eP, Rec x eV : Fun \<tau>\<^sub>1 \<tau>\<^sub>2"   and "atom x \<sharp> \<Gamma>" obtains y :: "Syntax.var" and e' :: "Syntax.term" and eP' :: "Syntax.term" and eV' :: "Syntax.term" where "e = Rec x (Lam y e')" and "eP = Rec x (Lam y eP')" and "eV = Lam y eV'" and "atom y \<sharp> (\<Gamma>, x)" and "\<Gamma>(x $$:= Fun \<tau>\<^sub>1 \<tau>\<^sub>2) \<turnstile> Lam y e', Lam y eP', Lam y eV' : Fun \<tau>\<^sub>1 \<tau>\<^sub>2"
eucl_ivals_3: fixes a :: "'a" shows "{..a} \<in> sets borel"
predE: fixes cid :: "'id"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'cmp"   and n :: "nat"   and P :: "bool" assumes "eval cid t t' n (pred P)" shows "P"
succs_Throw: fixes \<tau> :: "ty list \<times> ty err list"   and pc :: "nat" shows "succs ThrowExc \<tau> pc = []"
cases: fixes a1 :: "'c constr list \<times> nat \<times> 'c TESL_atomic list \<times> 'c TESL_atomic list"   and a2 :: "'c constr list \<times> nat \<times> 'c TESL_atomic list \<times> 'c TESL_atomic list" assumes "a1 \<hookrightarrow>\<^sub>i a2" obtains \<Gamma> :: "'c constr list" and n :: "nat" and \<Phi> :: "'c TESL_atomic list" where "a1 = \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi>" and "a2 = \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> []"
step: shows "\<Gamma>\<turnstile> ([], ass @ css, s) \<rightarrow> (cs', css', t)"
oreturnOk_def: fixes x :: "'f" shows "oreturnOk x = K (Some (Inr x))"
simps_1: fixes C :: "char list" shows "fv (new C) = {}"
case_prodE2: fixes Q :: "'k \<Rightarrow> bool"   and P :: "'l \<Rightarrow> 'm \<Rightarrow> 'k"   and z :: "'l \<times> 'm" assumes "Q (case z of (a, b) \<Rightarrow> P a b)" obtains x :: "'l" and y :: "'m" where "z = (x, y)" and "Q (P x y)"
REFSTEP: fixes it :: "'S set"   and it' :: "'Sa set"   and x' :: "'Sa"   and x :: "'S"   and \<sigma> :: "'\<sigma>"   and \<sigma>' :: "'\<sigma>a" assumes "it \<subseteq> S"   and "it' \<subseteq> S'"   and "x' = \<alpha> x"   and "it' = \<alpha> ` it"   and "((it, \<sigma>), it', \<sigma>') \<in> R"   and "x \<in> it"   and "x' \<in> it'"   and "c \<sigma>"   and "c' \<sigma>'" shows "f x \<sigma> \<le> \<Down> {(\<sigma>, \<sigma>'). ((it - {x}, \<sigma>), it' - {x'}, \<sigma>') \<in> R} (f' x' \<sigma>')"
opt_bst2_is_optimal: fixes i :: "int"   and j :: "int" shows "local.wpl i j (opt_bst2 i j) = local.min_wpl i j"
zeta_partial_sum_le_pos'': fixes s :: "real" assumes "0 < s"   and "s \<noteq> 1" shows "\<exists>c>0. \<forall>x\<ge>1. \<bar>sum_upto (\<lambda>n. real n powr - s) x\<bar> \<le> c * x powr max 0 (1 - s)"
simps_2: fixes uv :: "'b" shows "typeof (OpUbx1 uv) = Some Ubx1"
wfP_accp_iff: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "wfP r = (\<forall>x. Wellfounded.accp r x)"
poly_no_roots_less_leq: fixes a :: "real"   and b :: "real"   and p :: "real poly" shows "(\<forall>x. a < x \<and> x \<le> b \<longrightarrow> poly p x \<noteq> 0) = (b \<le> a \<or> p \<noteq> 0 \<and> count_roots_between p a b = 0)"
akra_bazzi_term_floor_subtract: fixes b :: "real"   and x\<^sub>0 :: "nat"   and x\<^sub>1 :: "nat"   and c :: "real" assumes "0 < b"   and "b < 1"   and "real x\<^sub>0 \<le> b * real x\<^sub>1 - c"   and "0 < c + (1 - b) * real x\<^sub>1"   and "0 < x\<^sub>1" shows "akra_bazzi_term x\<^sub>0 x\<^sub>1 b (\<lambda>x. nat \<lfloor>b * real x - c\<rfloor>)"
discr0_transC_indis: fixes c :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "discr0 c"   and "mustT c s"   and "(c, s) \<rightarrow>c (c', s')" shows "s \<approx> s'"
akra_bazzi_term_def: fixes x\<^sub>0 :: "nat"   and x\<^sub>1 :: "nat"   and b :: "real"   and t :: "nat \<Rightarrow> nat" shows "akra_bazzi_term x\<^sub>0 x\<^sub>1 b t = (\<exists>e h. 0 < e \<and> h \<in> O(\<lambda>x. real x / ln (real x) powr (1 + e)) \<and> (\<forall>x\<ge>x\<^sub>1. x\<^sub>0 \<le> t x \<and> t x < x \<and> b * real x + h x = real (t x)))"
star_sim_left: fixes p :: "'a"   and x :: "'a" assumes "test p" shows "p \<cdot> x = x \<cdot> p \<longrightarrow> p \<cdot> x\<^sup>\<star> = p \<cdot> (x \<cdot> p)\<^sup>\<star>"
divide_nonneg_nonneg: fixes x :: "'a"   and y :: "'a" assumes "(0::'a) \<le> x"   and "(0::'a) \<le> y" shows "(0::'a) \<le> x / y"
simps_1: shows "fun_rep [] = (\<lambda>x. \<top>)"
WHILEI_body_def: fixes bind :: "'e \<Rightarrow> 'b \<Rightarrow> 'd"   and return :: "'c \<Rightarrow> 'd"   and I :: "'c \<Rightarrow> bool"   and b :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e" shows "WHILEI_body bind return I b f \<equiv> \<lambda>W s. if I s then if b s then bind (f s) W else return s else top"
exhaust_sel: fixes option :: "'f option" obtains "option = None"   | "option = Some (the option)"
closed_UNIV: fixes "open" :: "'d set \<Rightarrow> bool" assumes "class.topological_space open" shows "topological_space.closed open UNIV"
nb\<^sub>2: shows "dom (map_of (effect_to_assignments op')) = set (effect_of op) \<union> (\<Union>(v, a)\<in>set (effect_of op). {(v, a') |a'. a' \<in> \<R>\<^sub>+ \<Psi> v \<and> a' \<noteq> a})"
problem_plan_bound_UBound: fixes PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and f :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set \<Rightarrow> nat" assumes "\<forall>as s. s \<in> valid_states PROB \<and> as \<in> valid_plans PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < f PROB)"   and "finite PROB" shows "problem_plan_bound PROB < f PROB"
record_simps_5: fixes f :: "int \<Rightarrow> int"   and x :: "x" shows "b (update_b f x) = f (b x)"
table_join': fixes t :: "nat"   and ys :: "'a option list set" assumes "(t, ys) \<in> set auxlist" shows "table n R (join ys pos X)"
le_infE: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.semilattice_inf inf less_eq less"   and "less_eq x (inf a b)" obtains "less_eq x a" and "less_eq x b"
simps_1: fixes q :: "('c monom \<times> 'd) list" shows "poly_mult [] q = []"
add_0_right: fixes m :: "nat" shows "(m::nat) + 0 = m"
cases: fixes a1 :: "'a list"   and a2 :: "'a list multiset"   and a3 :: "('b, 'c) label list"   and a4 :: "'a list"   and a5 :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "((a1, a2), a3, a4, a5) \<in> ntrs fg" obtains u :: "'a" and r :: "'a list" and ce :: "'a list multiset" and p :: "'b" and u' :: "'a" and w :: "('b, 'c) label list" and v :: "'a" and ce' :: "'a list multiset" where "a1 = u # r" and "a2 = ce" and "a3 = LCall p # w" and "a4 = v # u' # r" and "a5 = ce'" and "((u # r, ce), LCall p, entry fg p # u' # r, ce) \<in> trss fg" and "(([entry fg p], ce), w, [v], ce') \<in> trcl (trss fg)"
DERIV_continuous: fixes f :: "'b \<Rightarrow> 'b"   and D :: "'b"   and x :: "'b"   and s :: "'b set" assumes "(f has_field_derivative D) (at x within s)" shows "continuous (at x within s) f"
AtomicP_Mem: fixes x :: "tm"   and y :: "tm" shows "{TermP x, TermP y} \<turnstile> AtomicP (Q_Mem x y)"
Rep_ell2_inject: fixes x :: "'c ell2"   and y :: "'c ell2" shows "(Rep_ell2 x = Rep_ell2 y) = (x = y)"
ha: shows "\<forall>x. insertion (nth_default 0 (L[var := x])) a = A"
eq_inf_le_inf_transp: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a stream"   and w\<^sub>2 :: "'a stream"   and w\<^sub>3 :: "'a stream" assumes "traces ind"   and "traces.eq_inf ind w\<^sub>1 w\<^sub>2"   and "traces.le_inf ind w\<^sub>2 w\<^sub>3" shows "traces.le_inf ind w\<^sub>1 w\<^sub>3"
2: shows "\<nexists>t p. (c\<^sub>1;; c\<^sub>2, s) \<Rightarrow> p \<Down> t \<and> Q t < \<infinity>"
bounded_nat_set_is_finite: fixes N :: "nat set"   and n :: "nat" assumes "\<forall>i\<in>N. i < n" shows "finite N"
fpxs_of_fls_mult: fixes f :: "'b fls"   and g :: "'b fls" shows "fpxs_of_fls (f * g) = fpxs_of_fls f * fpxs_of_fls g"
subst2: fixes x :: "name"   and M :: "'c"   and xvec :: "name list"   and Tvec :: "'a list" assumes "x \<sharp> M"   and "x \<sharp> xvec"   and "x \<sharp> Tvec" shows "x \<sharp> substCond M xvec Tvec"
h1: shows "\<forall>x\<in>set eq. case x of (d, e, f) \<Rightarrow> d * (c / b)\<^sup>2 + e * - (c / b) + f = 0"
recv_insert_once: fixes i :: "'a"   and j :: "nat"   and s :: "'a \<times> nat"   and m :: "('a \<times> nat, 'b) insert_message"   and k :: "nat"   and t :: "'a \<times> nat" assumes "event_at (i, j) (Receive s (Insert m))"   and "event_at (i, k) (Receive t (Insert m))" shows "j = k"
Pool_3: fixes i :: "nat" assumes "i < count_weights s (Pool m1 m2)" shows "w1 i = w2 i"
div_mult_self3: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and modulo :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and euclidean_size :: "'e \<Rightarrow> nat"   and b :: "'e"   and c :: "'e"   and a :: "'e" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "divide (plus (times c b) a) b = plus c (divide a b)"
simps_2: fixes b :: "bool"   and bs :: "bool list"   and w :: "int" shows "bl_to_bin_aux (b # bs) w = bl_to_bin_aux bs (of_bool b + 2 * w)"
elims: fixes x :: "(int \<times> int) list"   and y :: "(int \<times> int) \<times> int \<times> int" assumes "closest_pair x = y" obtains "x = []" and "y = undefined"   | uu_ :: "int \<times> int" where "x = [uu_]" and "y = undefined"   | v :: "int \<times> int" and vb :: "int \<times> int" and vc :: "(int \<times> int) list" where "x = v # vb # vc" and "y = (let (_, x, y) = closest_pair_rec (mergesort fst (v # vb # vc)) in (x, y))"
ordermap_mono_less: fixes w :: "'c"   and x :: "'c"   and r :: "('c \<times> 'c) set"   and A :: "'c set" assumes "(w, x) \<in> r"   and "wf r"   and "trans r"   and "w \<in> A"   and "x \<in> A"   and "small A" shows "ordermap A r w < ordermap A r x"
path_total: shows "path loc1 loc1 (path1 @ path2)"
Call_Sound: fixes \<Sigma> :: "((Class \<times> Method) \<times> nat) list"   and f :: "Class \<times> Method"   and n :: "nat"   and l :: "int"   and code :: "(int \<times> Instr) list"   and args :: "Var list"   and code1 :: "(int \<times> Instr) list"   and l1 :: "int"   and C :: "Class"   and m :: "Method"   and T :: "(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 MI :: "(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 G :: "((Class \<times> Method \<times> 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) \<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> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<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> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and nn :: "nat"   and k :: "nat" assumes "Sig_good \<Sigma>"   and "\<Sigma>\<down>f = Some n"   and "(l, code, CallPrim f args, code1, l1) \<in> compilePrim"   and "MST\<down>(C, m) = Some (T, MI, Anno)"   and "Segment C m l l1 code1"   and "derivAssum G C m l1 (Cachera nn)"   and "k = n + nn" shows "deriv G C m l (Cachera k)"
hyps_2: shows "|{x. \<not> qAFresh xs x X}| <o |UNIV|"
clear_locals_def: fixes \<xi> :: "state" shows "clear_locals \<xi> = \<xi>\<lparr>msg := SOME x. True, data := SOME x. True, dests := SOME x. True, pre := SOME x. True, rreqid := SOME x. True, dip := SOME x. True, oip := SOME x. True, hops := SOME x. True, dsn := SOME x. True, dsk := SOME x. True, osn := SOME x. True, sip := SOME x. x \<noteq> ip \<xi>, handled := SOME x. True\<rparr>"
lossless1: fixes s1 :: "'b"   and x :: "'c" assumes "bad1 s1" shows "lossless_spmf (oracle1 s1 x)"
genset_uminus: fixes S :: "'b set"   and a :: "'b" assumes "PreCoxeterSystem S"   and "a \<in> S" shows "- a = a"
row_length_domain_block: fixes ls :: "brick list" shows "rat_poly.row_length (blockmat ls) = 2 ^ nat (domain_block ls)"
prod_emb_id: fixes B :: "('d \<Rightarrow> 'e) set"   and L :: "'d set"   and M :: "'d \<Rightarrow> 'e measure" assumes "B \<subseteq> (\<Pi>\<^sub>E i\<in>L. space (M i))" shows "prod_emb L M L B = B"
ptensor_mat_def: fixes m1 :: "'a mat"   and m2 :: "'a mat" shows "tensor_P m1 m2 = ps_P.tensor_mat m1 m2"
ordinal_exp_monoR: fixes x :: "ordinal"   and y :: "ordinal"   and y' :: "ordinal" assumes "0 < x"   and "y \<le> y'" shows "x ** y \<le> x ** y'"
assms_2: shows "m * n \<le> \<^bold>|w\<^bold>|"
countable_Int1: fixes A :: "'k set"   and B :: "'k set" assumes "countable A" shows "countable (A \<inter> B)"
fpxs_root_order_uminus: fixes f :: "'b fpxs" shows "fpxs_root_order (- f) = fpxs_root_order f"
AssignAux: fixes e :: "Expr"   and \<beta> :: "(Location \<times> Location) set"   and s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and x :: "Var" assumes "Expr_low e"   and "s \<equiv>\<^sub>\<beta> t" shows "(update (fst s) x (evalE e (fst s)), snd s) \<equiv>\<^sub>\<beta> (update (fst t) x (evalE e (fst t)), snd t)"
polynomial_incl: fixes K :: "'a list set"   and p :: "'a list list" assumes "polynomial\<^bsub>K [X]\<^esub> K p" shows "set p \<subseteq> K"