hcju/isabelleps huggingface dataset:
add_le_cancel_right: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less_eq (plus a c) (plus b c) = less_eq a b"
3: fixes y :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" assumes "(y, z2) \<in> measure (card \<circ> set)" shows "y \<notin> snd ` {q \<in> Q. fst ` set (fst q) = z1}"
in_Fvars_rawpsubst_imp: fixes y :: "'var"   and \<phi> :: "'fmla"   and txs :: "('trm \<times> 'var) list" assumes "y \<in> Fvars (rawpsubst \<phi> txs)"   and "\<phi> \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm" shows "y \<in> Fvars \<phi> - snd ` set txs \<or> y \<in> \<Union> {uu_. \<exists>t x. uu_ = FvarsT t \<and> (t, x) \<in> set txs}"
pre_buf: shows "wf_mbufn' \<sigma> P V j n R r buf"
Root: fixes k :: "nat" assumes "k \<in> {1..3}" shows "(\<Sum>i<3. Root i ^ k) = f k"
pseudo_complement_pp: fixes x :: "'a"   and y :: "'a" shows "(x \<sqinter> y = bot) = (- - x \<le> - y)"
isSepaux: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "p \<noteq> []"   and "noDenyAll (a # p)"   and "separated (a # p)"   and "x \<in> dom (Cp (DenyAllFromTo (first_srcNet a) (first_destNet a) \<oplus> DenyAllFromTo (first_destNet a) (first_srcNet a) \<oplus> a))" shows "x \<notin> dom (Cp (list2FWpolicy p))"
zero_closed: shows "\<zero>\<^bsub>poly_ring L\<^esub> \<in> carrier (poly_ring L)"
rel_spmf_bindI: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'a spmf"   and q :: "'b spmf"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c spmf"   and g :: "'b \<Rightarrow> 'd spmf" assumes "rel_spmf R p q"   and "\<And>x y. R x y \<Longrightarrow> rel_spmf P (f x) (g y)" shows "rel_spmf P (p \<bind> f) (q \<bind> g)"
dbl_simps_1: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and k :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl plus (uminus (numeral.numeral one plus k)) = uminus (neg_numeral.dbl plus (numeral.numeral one plus k))"
almost_full_on_Sigma: fixes P1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A1 :: "'a set"   and P2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A2 :: "'b set" assumes "almost_full_on P1 A1"   and "almost_full_on P2 A2" shows "almost_full_on (prod_le P1 P2) (A1 \<times> A2)"
nat_pow_pow: fixes x :: "'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier (mult_of R)" shows "(x [^] n) [^] m = x [^] (n * m)"
proper_interval_set_aux_fusion_def: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and proper_interval :: "'b option \<Rightarrow> 'b option \<Rightarrow> bool"   and g1 :: "('b, 'c) generator"   and g2 :: "('b, 'd) generator"   and s1 :: "'c"   and s2 :: "'d" shows "proper_intrvl.proper_interval_set_aux_fusion less proper_interval g1 g2 s1 s2 = proper_intrvl.proper_interval_set_aux less proper_interval (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
nsigfin: shows "\<forall>n. sigma_finite_subalgebra N (F n)"
clinear_continuous_at: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "bounded_clinear f" shows "isCont f x"
f'_inj: fixes a :: "'a"   and b :: "'a" assumes "f' a = f' b" shows "a = b"
node_tauTE: fixes i :: "nat"   and s :: "'c"   and R :: "nat set"   and s' :: "'c"   and S :: "('c \<times> 'd seq_action \<times> 'c) set" assumes "(NodeS i s R, \<tau>, NodeS i s' R) \<in> node_sos S" obtains d :: "nat" where "(s, \<not>unicast d, s') \<in> S" and "d \<notin> R"   | "(s, \<tau>\<^sub>s, s') \<in> S"
adjoint_funcs: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and I :: "interp"   and \<omega> :: "variable \<Rightarrow> real"   and f :: "char" shows "Funcs (USubst.adjoint \<sigma> I \<omega>) f = (case SFuncs \<sigma> f of None \<Rightarrow> Funcs I f | Some r \<Rightarrow> \<lambda>d. term_sem (repc I dotid d) r \<omega>)"
ring_hom_memE_1: fixes h :: "'e \<Rightarrow> 'f"   and R :: "('e, 'g) ring_scheme"   and S :: "('f, 'h) ring_scheme"   and x :: "'e" assumes "h \<in> ring_hom R S"   and "x \<in> carrier R" shows "h x \<in> carrier S"
uint_lt2p: fixes x :: "'b word" shows "uint x < 2 ^ LENGTH('b)"
filter_append: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" shows "filter P (append xs ys) = append (filter P xs) (filter P ys)"
process_rule_2_failures: fixes xs :: "'e list"   and xs' :: "'e list"   and X :: "'e set"   and P :: "'e process" assumes "(xs @ xs', X) \<in> failures P" shows "(xs, {}) \<in> failures P"
toSink_def: fixes L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Sink :: "'a set"   and s :: "'a" assumes "Shallow_Idle L AP' S s0 \<delta> AP f Sink" shows "Shallow_Idle.toSink \<delta> Sink s \<equiv> SOME s'. s' \<in> \<delta> s \<inter> Sink"
Suc_inject: fixes x :: "nat"   and y :: "nat" assumes "Suc x = Suc y" shows "x = y"
branch_instr_low_equal_sub2: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (set_annul True (snd (fst (write_cpu (cpu_reg_val nPC s2 + 4) nPC (snd (fst (write_cpu (cpu_reg_val nPC s2) PC s1)))))))) \<and> t2 = snd (fst (set_annul True (snd (fst (write_cpu (cpu_reg_val nPC s2 + 4) nPC (snd (fst (write_cpu (cpu_reg_val nPC s2) PC s2))))))))" shows "low_equal t1 t2"
lex_fun_antisym: fixes s :: "'a \<Rightarrow> 'c"   and t :: "'a \<Rightarrow> 'c" assumes "lex_fun s t"   and "lex_fun t s" shows "s = t"
homologous_rel_add: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and a' :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and b' :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "homologous_rel p X S a a'"   and "homologous_rel p X S b b'" shows "homologous_rel p X S (a + b) (a' + b')"
transfer: shows "rel_fun pcr_word (=) (take_bit LENGTH('k)) the_int"
apply_power_self_in_orbit: fixes f :: "'a perm"   and n :: "nat"   and a :: "'a" shows "(f ^ n) \<langle>$\<rangle> a \<in> orbit f a"
is_interp_def: fixes I :: "('d, 'e, 'f) interp" shows "is_interp I \<equiv> \<forall>x i. (Functions I i has_derivative FunctionFrechet I i x) (at x) \<and> continuous_on UNIV (\<lambda>x. Blinfun (FunctionFrechet I i x))"
step_invariantE: fixes A :: "('c, 'd) automaton"   and I :: "'d \<Rightarrow> bool"   and P :: "'c \<times> 'd \<times> 'c \<Rightarrow> bool"   and s :: "'c"   and a :: "'d"   and s' :: "'c" assumes "A \<TTurnstile>\<^sub>A (I \<rightarrow>) P"   and "s \<in> reachable A I"   and "(s, a, s') \<in> automaton.trans A"   and "I a" obtains "P (s, a, s')"
1: shows "\<forall>a\<in>PROB. (case a of (s1, s2) \<Rightarrow> fmdom' s1 \<union> fmdom' s2) = {}"
simps_1: fixes x31 :: "comb"   and x32 :: "comb"   and y31 :: "comb"   and y32 :: "comb" shows "(x31 \<bullet> x32 = y31 \<bullet> y32) = (x31 = y31 \<and> x32 = y32)"
assms_3: shows "g2 ` carrier G2 = carrier G3"
relations_zero_imp_ord_zero: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A"   and "\<forall>f\<in>relations A. f a = 0"   and "a \<in> carrier G" shows "ord a = 0"
monic_Mp: fixes f :: "int poly" assumes "monic f" shows "monic (Mp f)"
eq1: shows "int (card X) - int j - 1 gchoose (card X - Suc j) = 1"
hm_iteratei_def: fixes hm :: "('f, 'g) hashmap" shows "hm_iteratei hm \<equiv> HashMap_Impl.iteratei (impl_of_RBT_HM hm)"
obj_simps_1: fixes a :: "'c" assumes "C.obj a" shows "C.arr a"
rep_eq: fixes x :: "linear_poly" shows "valuate x = fun_valuate (Rep_linear_poly x)"
left_treeI: fixes G :: "('a, 'b) Graph_scheme"   and u :: "'a"   and s :: "'a"   and t :: "'a" assumes "Tree G"   and "u \<in> V\<^bsub>G\<^esub>"   and "s \<in> set (u \<leadsto>\<^bsub>G\<^esub> t)" shows "u \<in> Tree.left_tree G s t"
assms_3: shows "\<exists>\<rho>. tabulation (\<cdot>) (\<star>) \<a> \<i> src trg (g \<star> f\<^sup>*) \<rho> f g"
R\<^sub>N_0: fixes u :: "nat"   and l :: "nat" shows "R\<^sub>N 0 u l = return_pmf 0"
Int_not_in_eq_emp: fixes P :: "'a set" shows "P \<inter> {I. I \<notin> P} = {}"
uniformity_fps_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
star_simulation_left: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x * z \<le> z * y" shows "x\<^sup>\<star> * z \<le> z * y\<^sup>\<star>"
inj_on_to_nat: fixes S :: "'b set" shows "inj_on to_nat S"
exhaust_sel: fixes tree :: "('c, 'd) tree" obtains "tree = Tip"   | "tree = Node (lsub tree) (key tree) (nval tree) (rsub tree)"
mono: shows "monotone_on I (\<preceq>) (\<sqsubseteq>) f"
double_sum_eq_sum: shows "(\<Sum>twoCube\<in>two_chain. \<Sum>(k, g)\<in>boundary twoCube. real_of_int k * line_integral F {j} g) = (\<Sum>twoCube\<in>two_chain. integral (cubeImage twoCube) (partial_vector_derivative (\<lambda>a. F a \<bullet> j) i))"
in_set_takeD: fixes x :: "'c"   and n :: "nat"   and xs :: "'c list" assumes "x \<in> set (take n xs)" shows "x \<in> set xs"
integral_exp_times: fixes t0 :: "real"   and t :: "real"   and c :: "real" assumes "t0 \<le> t" shows "integral {t0..t} (\<lambda>t. exp (c * t)) = (if c = 0 then t - t0 else exp (c * t) / c - exp (c * t0) / c)"
DBNxt_then_FullNxt: shows "\<turnstile> \<box>BInv inp q1 mid \<and> \<box>[Nxt inp q1 mid]_(inp, q1, mid) \<and> \<box>[Nxt mid q2 out]_(mid, q2, out) \<and> \<box>[\<not> (Enq inp q1 mid \<and> Deq mid q2 out)]_vars \<longrightarrow> \<box>[FullNxt]_vars"
Inter_UNIV_conv_2: fixes A :: "'a set set" shows "(UNIV = \<Inter> A) = (\<forall>x\<in>A. x = UNIV)"
3: shows "set_mset K \<subseteq> r \<down>m J"
gcd_g: shows "degree (gcd factor local.g) \<le> degree local.g"
kop_galois_var2: fixes f :: "('d \<Rightarrow> 'c set) \<Rightarrow> 'e"   and g :: "('c \<Rightarrow> 'd set) \<Rightarrow> 'e" shows "(f \<circ> op\<^sub>K = g) = (g \<circ> op\<^sub>K = f)"
1: shows "contour_integral (linepath c a) f = contour_integral (linepath c b') f + contour_integral (linepath b' a) f"
refl: fixes a :: "'a::type" shows "(a::'a::type) \<sqsubseteq> a"
list_all2_rev1: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list" shows "list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
contD': fixes f :: "'g \<Rightarrow> 'h"   and C :: "'g set" assumes "cont f"   and "C \<noteq> {}" shows "f (Sup C) = Sup (f ` C)"
subset: shows "set (finals g) \<subseteq> set (finals g')"
fund_sol_minimal': fixes D :: "nat"   and z :: "nat \<times> nat" assumes "pell D"   and "pell.nontriv_solution D z" shows "pell.pell_valuation D (case pell.fund_sol D of (x, y) \<Rightarrow> (int x, int y)) \<le> pell.pell_valuation D (case z of (x, y) \<Rightarrow> (int x, int y))"
pd_nonz: shows "poly (pderiv (real_of_int_poly p)) (real_of_1 (p, l, r)) \<noteq> 0"
Rep_matrix_mult: fixes a :: "'a matrix"   and b :: "'a matrix"   and j :: "nat"   and i :: "nat" shows "Rep_matrix (a * b) j i = foldseq (+) (\<lambda>k. Rep_matrix a j k * Rep_matrix b k i) (max (ncols a) (nrows b))"
step_def: shows "li.step \<equiv> stutter_extend_edges UNIV li.gstep"
eq: shows "\<tau> \<Turnstile> x \<triangleq> y"
A2: shows "SRel = _Collect (S1, S2) ((SourceTerm S1, SourceTerm S2) \<in> Rel)"
cases: shows "steps0 (1, tp) A n = (s, l, r)"
assms_3: shows "\<exists>\<^sub>F k in sequentially. p k \<noteq> 0"
k1_2: shows "set_mset J \<subseteq> r \<down>s S"
prod_def: shows "Product_Type.prod = {f. \<exists>a b. f = Pair_Rep a b}"
right_inverse: fixes a :: "'b" assumes "a \<noteq> (0::'b)" shows "a * inverse a = (1::'b)"
distinct_25: fixes x2 :: "nat"   and x4 :: "ltkey" shows "cNumber x2 \<noteq> cLtK x4"
step_invariant_def: fixes A :: "('e, 'f) automaton"   and I :: "'f \<Rightarrow> bool"   and P :: "'e \<times> 'f \<times> 'e \<Rightarrow> bool" shows "(A \<TTurnstile>\<^sub>A (I \<rightarrow>) P) = (\<forall>a. I a \<longrightarrow> (\<forall>s\<in>reachable A I. \<forall>s'. (s, a, s') \<in> automaton.trans A \<longrightarrow> P (s, a, s')))"
atLeastLessThan_shift_0: fixes g :: "nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" shows "sum g {m..<n} = sum (g \<circ> (+) m) {0..<n - m}"
dom_simp: fixes f :: "'a" assumes "S.arr f" shows "S.dom f = B.dom f"
exhaust: fixes y :: "(_) node_ptr" obtains x :: "'d" where "y = node_ptr.Ext x"
exhaust_sel: fixes literal :: "'a literal" obtains "literal = Pos (get_pred literal) (get_terms literal)"   | "literal = Neg (get_pred literal) (get_terms literal)"
foundation20: fixes \<tau> :: "'d state \<times> 'd state"   and X :: "'d state \<times> 'd state \<Rightarrow> 'e" assumes "\<tau> \<Turnstile> \<delta> X" shows "\<tau> \<Turnstile> \<upsilon> X"
orbit_list_succ: fixes xs :: "'b list"   and x :: "'b" assumes "distinct xs"   and "x \<in> set xs" shows "orbit (list_succ xs) x = set xs"
rtos_hom1: fixes r :: "nat"   and s :: "nat"   and l :: "nat" assumes "0 < r"   and "0 < s"   and "l \<le> r * s - Suc 0" shows "rtos r s l \<le> s * r - Suc 0"
total_1: fixes x :: "'a" assumes "is_total x" shows "\<forall>y. y ; x = 0 \<longrightarrow> y = 0"
redex_subst_redex: fixes c :: "('a, 'b, 'c) com"   and r :: "('a, 'b, 'c) com" shows "redex (subst_redex c r) = redex r"
finite_PiE_iff: fixes I :: "'a set"   and S :: "'a \<Rightarrow> 'b set" shows "finite (Pi\<^sub>E I S) = (Pi\<^sub>E I S = {} \<or> finite {i \<in> I. \<nexists>a. S i \<subseteq> {a}} \<and> (\<forall>i\<in>I. finite (S i)))"
leq_trans: fixes y :: "'b"   and x :: "'b"   and z :: "'b" assumes "y \<le> x"   and "z \<le> y" shows "z \<le> x"
inj_on_image_eq_iff: fixes f :: "'e \<Rightarrow> 'f"   and C :: "'e set"   and A :: "'e set"   and B :: "'e set" assumes "inj_on f C"   and "A \<subseteq> C"   and "B \<subseteq> C" shows "(f ` A = f ` B) = (A = B)"
length_postorder: fixes t :: "'a tree" shows "length (postorder t) = size t"
fct_gen_subalgebra_is_subalgebra: fixes X :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure" assumes "X \<in> M \<rightarrow>\<^sub>M N" shows "subalgebra M (fct_gen_subalgebra M N X)"
dsfn: shows "matrix_sum d f n \<in> carrier_mat d d"
rep_eq: fixes x :: "'c pmf" shows "rcset (cset_pmf x) = set_pmf x"
deriv_bodies: shows "\<forall>p\<in>set Procs. \<forall>Z. \<Gamma>,\<Theta> \<union> (\<Union>p\<in>set Procs. \<Union>Z. {(P p Z, p, Q p Z, A p Z)})\<turnstile>\<^bsub>/F \<^esub>(P p Z) the (\<Gamma> p) (Q p Z),(A p Z)"
Cont: fixes b :: "'a"   and conv' :: "('b, 'a, 'c, 'd) converter"   and \<I>' :: "('c, 'd) \<I>"   and conv :: "('b, 'a, 'c, 'd) converter"   and a :: "'b"   and out :: "'c"   and \<I> :: "('b, 'a) \<I>" assumes "(b, conv') \<in> results_gpv \<I>' (run_converter conv a)"   and "out \<in> outs_converter \<I> \<I>' conv'"   and "a \<in> outs_\<I> \<I>" shows "out \<in> outs_converter \<I> \<I>' conv"
assms_3: fixes w :: "'a" assumes "v \<noteq> w" shows "\<sigma> w = Var w"
abs_eq: shows "(\<le>) = (\<lambda>f g. Rep_nd_fun f \<le> Rep_nd_fun g)"
subset_eq_imp_le_multiset: fixes M :: "'a multiset"   and N :: "'a multiset" assumes "M \<subseteq># N" shows "M \<le> N"
fd: fixes x :: "complex" assumes "x \<in> interior S - K" shows "f field_differentiable at x"
seE_fringe4: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and re :: "('d \<times> nat) edge"   and prb' :: "('d, 'e, 'f) pre_RedBlack"   and c' :: "('e, 'f) conf" assumes "sub_rel_of (red prb) (subs prb)"   and "ui_edge re \<in> edges (black prb) \<and> (src re \<in> red_vertices prb \<and> tgt re \<notin> red_vertices prb \<and> red prb' = add_edge (red prb) re) \<and> src re \<notin> subsumees (subs prb) \<and> se (confs prb (src re)) (labelling (black prb) (ui_edge re)) c' \<and> prb' = \<lparr>red = red prb', black = black prb, subs = subs prb, init_conf = init_conf prb, confs = (confs prb)(tgt re := c'), marked = (marked prb)(tgt re := marked prb (src re)), strengthenings = strengthenings prb\<rparr>"   and "\<not> marked prb (src re)"   and "\<not> ui_edge ` out_edges (red prb') (src re) \<subset> out_edges (black prb) (fst (src re))"   and "out_edges (black prb) (fst (tgt re)) \<noteq> {}" shows "fringe prb' = fringe prb - {src re} \<union> {tgt re}"
pderiv_power_Suc: fixes p :: "'b poly"   and n :: "nat" shows "pderiv (p ^ Suc n) = Polynomial.smult (of_nat (Suc n)) (p ^ n) * pderiv p"
strategyproofness_1: shows "(pmf (sds R29) a < pmf (sds R22) d \<or> pmf (sds R29) a + (pmf (sds R29) c + pmf (sds R29) d) < pmf (sds R22) d + (pmf (sds R22) b + pmf (sds R22) a)) \<or> pmf (sds R29) a = pmf (sds R22) d \<and> pmf (sds R29) a + (pmf (sds R29) c + pmf (sds R29) d) = pmf (sds R22) d + (pmf (sds R22) b + pmf (sds R22) a)"
ms_aux_cases: fixes xs :: "('a \<times> real) msllist" obtains "xs = MSLNil"   | c :: "'a" and e :: "real" and xs' :: "('a \<times> real) msllist" where "xs = MSLCons (c, e) xs'"
intros: fixes \<Gamma> :: "'form fset"   and i :: "nat"   and n :: "('form, 'rule) graph_node"   and s :: "'subst"   and c :: "'form" assumes "\<And>f. f |\<in>| \<Gamma> \<Longrightarrow> freshenLC i ` all_local_vars n \<inter> lconsts f = {}"   and "freshenLC i ` all_local_vars n \<inter> subst_lconsts s = {}" shows "local_fresh_check n i s (\<Gamma> \<turnstile> c)"
StrongEq_refl: fixes X :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "(X \<triangleq> X) = true"
enumerator_bound: fixes ls :: "nat list"   and m :: "nat"   and nmax :: "nat"   and x :: "nat" assumes "ls \<in> set (enumerator m nmax)"   and "0 < nmax"   and "x \<in> set ls" shows "x < nmax"
WS_silent_move: fixes S :: "'node SDG_node set"   and ms\<^sub>1 :: "'node list"   and s\<^sub>1 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a :: "'edge"   and ms\<^sub>1' :: "'node list"   and s\<^sub>1' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and ms\<^sub>2 :: "'node list"   and s\<^sub>2 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,kind \<turnstile> (ms\<^sub>1,s\<^sub>1) -a\<rightarrow>\<^sub>\<tau> (ms\<^sub>1',s\<^sub>1')"   and "((ms\<^sub>1, s\<^sub>1), ms\<^sub>2, s\<^sub>2) \<in> WS S" shows "((ms\<^sub>1', s\<^sub>1'), ms\<^sub>2, s\<^sub>2) \<in> WS S"
le_sup_iff: fixes x :: "'d"   and y :: "'d"   and z :: "'d" shows "(x \<sqinter> y \<le> z) = (x \<le> z \<and> y \<le> z)"
AddSA_HAStates: fixes SA :: "('d, 'e, 'f) seqauto"   and HA :: "('d, 'e, 'f) hierauto"   and S :: "'d" assumes "States SA \<inter> HAStates HA = {}"   and "S \<in> HAStates HA" shows "HAStates (HA [++] (S, SA)) = HAStates HA \<union> States SA"
le_add_same_cancel1: fixes a :: "'a"   and b :: "'a" shows "(a \<le> a + b) = ((0::'a) \<le> b)"
order_iff_strict: fixes b :: "'t \<Rightarrow>\<^sub>0 'c"   and a :: "'t \<Rightarrow>\<^sub>0 'c" shows "(b \<preceq>\<^sub>p a) = (b \<prec>\<^sub>p a \<or> a = b)"
pwd_spy_dec_msg: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "let (s, S, A, B, C, D) = pwd_spy_dec n; SK = (Some S, {A, B}, {C, D}) in pwd_spy_dec_state n (S, A, B, C, D) \<subseteq> s \<and> {Key (Auth_ShaKey n)} \<in> key_sets (PriKey S) (crypts (Log -` spied s)) \<and> {SesKey SK} \<in> key_sets (Pwd n) (crypts (Log -` spied s))"
existence_ivl_empty2: fixes x0 :: "real"   and t0 :: "real" assumes "x0 \<notin> X" shows "existence_ivl t0 x0 = {}"
assms: shows "wf_fmla (ty_term Q constT) \<phi>"
pure_returns_heap_eq: fixes h :: "'a"   and f :: "('a, 'b, 'c) prog"   and h' :: "'a" assumes "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "pure f h" shows "h = h'"
assms_1: shows "sub_set_system \<U> \<A> \<V> \<B>"
triangularizable_0n: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat 0 n" shows "\<exists>U. U \<in> carrier_mat n n \<and> invertible_mat U \<and> lower_triangular (A * U)"
extremum_uniqueI: fixes a :: "'b" assumes "\<top> \<le> a" shows "a = \<top>"
induct: fixes P1 :: "expr \<Rightarrow> bool"   and P2 :: "expr list \<Rightarrow> bool"   and expr_list :: "expr list" assumes "\<And>x. P1 (new x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (Cast x1 x2)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (\<lparr>x1\<rparr>x2)"   and "\<And>x. P1 (Val x)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x3\<rbrakk> \<Longrightarrow> P1 (x1 \<guillemotleft>x2\<guillemotright> x3)"   and "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (x1:=x2)"   and "\<And>x1 x2 x3. P1 x1 \<Longrightarrow> P1 (x1\<bullet>x2{x3})"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x1; P1 x4\<rbrakk> \<Longrightarrow> P1 (x1\<bullet>x2{x3} := x4)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x1; P2 x4\<rbrakk> \<Longrightarrow> P1 (Call x1 x2 x3 x4)"   and "\<And>x1 x2 x3. P1 x3 \<Longrightarrow> P1 {x1:x2; x3}"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1;; x2)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x2; P1 x3\<rbrakk> \<Longrightarrow> P1 (if (x1) x2 else x3)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (while (x1) x2)"   and "\<And>x. P1 x \<Longrightarrow> P1 (throw x)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P2 expr_list"
mbisim_deadlocked_preserve1: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and s1 :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and s2 :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('g \<times> 'd \<Rightarrow>f nat) option) \<times> 'h) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and t :: "'b" assumes "FWbisimulation final1 r1 final2 r2 bisim"   and "FWbisimulation_base.mbisim bisim (\<lambda>x1 x2. True) s1 s2"   and "t \<in> multithreaded_base.deadlocked final1 r1 s1" shows "t \<in> multithreaded_base.deadlocked final2 r2 s2"
vec_first_last_append: fixes v :: "'b vec"   and n :: "nat"   and m :: "nat" assumes "v \<in> carrier_vec (n + m)" shows "vec_first v n @\<^sub>v vec_last v m = v"
normalize_mult_normalize_right: fixes a :: "'b poly"   and b :: "'b poly" shows "smult (inverse (lead_coeff (a * smult (inverse (lead_coeff b)) b))) (a * smult (inverse (lead_coeff b)) b) = smult (inverse (lead_coeff (a * b))) (a * b)"
positions_in_row: fixes w :: "('b \<times> bool list) list"   and i :: "nat" shows "positions_in_row w i = {p. p < length w \<and> snd (w ! p) ! i}"
charpoly_erase_mat_sr: fixes i :: "'n" shows "0 < poly (charpoly (erase_mat A i i)) sr"
del_left_height: fixes t :: "'b tree"   and v :: "'b"   and t' :: "'b tree" assumes "del_left t = (v, t')"   and "t \<noteq> \<langle>\<rangle>" shows "height t' \<le> height t"
equal_refl: fixes equal :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.equal equal" shows "equal x x = True"
exp_conj_mono_left: fixes P :: "'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real"   and R :: "'b \<Rightarrow> real" assumes "P \<tturnstile> Q" shows "P && R \<tturnstile> Q && R"
3: shows "v\<^sup>T = r\<^sup>T * t\<^sup>\<star> \<and> vector v"