hcju/isabelleps huggingface dataset:
convex_segment_2: fixes a :: "'b"   and b :: "'b" shows "convex {a<--<b}"
measurable_component_singleton: fixes i :: "'c"   and I :: "'c set"   and M :: "'c \<Rightarrow> 'd measure" assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> Pi\<^sub>M I M \<rightarrow>\<^sub>M M i"
isnpoly_def: fixes p :: "'b poly" shows "isnpoly p = isnpolyh p 0"
vector_sadd_rdistrib: fixes a :: "'a"   and b :: "'a"   and x :: "('a, 'b) vec" shows "(a + b) *s x = a *s x + b *s x"
THE_most_recent_writeI: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and r :: "nat"   and w :: "nat" assumes "P,E \<turnstile> r \<leadsto>mrw w" shows "(THE w. P,E \<turnstile> r \<leadsto>mrw w) = w"
mask_shift: fixes x :: "'a word"   and y :: "nat" shows "x AND NOT (mask y) >> y = x >> y"
agree1: shows "Vagree (sol 0, b) (mk_v I (OVar vid1) (sol 0, b) (sol 0)) (FVT (f1 fid2 vid1))"
good_imp_qGood_pick: fixes X :: "('c, 'd, 'varSort, 'var, 'e) qTerm set" assumes "good X" shows "qGood (pick X)"
nres_monad1: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'a nres" shows "return x \<bind> f = f x"
ecl_ord_coincide: fixes \<sigma> :: "('a \<times> 'a trm) list"   and \<sigma>' :: "('a \<times> 'a trm) list"   and C :: "'a eclause" assumes "coincide_on \<sigma> \<sigma>' (vars_of_cl (cl_ecl C))" shows "mset_ecl (C, \<sigma>) = mset_ecl (C, \<sigma>')"
unique_timestamp_authticket: fixes A :: "agent"   and X :: "msg"   and AK :: "nat"   and T :: "msg"   and evs :: "event list"   and A' :: "agent"   and X' :: "msg"   and Tgs' :: "agent"   and AK' :: "nat" assumes "Says Kas A \<lbrace>X, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key AK, T\<rbrace>\<rbrace> \<in> set evs"   and "Says Kas A' \<lbrace>X', Crypt (shrK Tgs') \<lbrace>Agent A', Agent Tgs', Key AK', T\<rbrace>\<rbrace> \<in> set evs"   and "evs \<in> kerbV" shows "A = A' \<and> X = X' \<and> Tgs = Tgs' \<and> AK = AK'"
inorder_skew: fixes t :: "('c \<times> nat) tree" shows "Tree2.inorder (skew t) = Tree2.inorder t"
sum_rel_simp_2: fixes a :: "'d"   and a' :: "'f"   and Rl :: "('c \<times> 'e) set"   and Rr :: "('d \<times> 'f) set" shows "((Inr a, Inr a') \<in> \<langle>Rl, Rr\<rangle>sum_rel) = ((a, a') \<in> Rr)"
Y'_cnv: shows "\<forall>\<omega>\<in>{0<..<1}. (\<lambda>n. Y_seq' n \<omega>) \<longlonglongrightarrow> Y' \<omega>"
PRECOND_OPT_tag_def: fixes P :: "'c" shows "PRECOND_OPT_tag P \<equiv> P"
simps: fixes Enc :: "'a \<Rightarrow> 'b"   and a1 :: "('a, 'b) Proc"   and a2 :: "('a, 'b) Proc" shows "encoding.indRelLPOp Enc a1 a2 = ((\<exists>S. a1 = TargetTerm (Enc S) \<and> a2 = SourceTerm S) \<or> (\<exists>S. a1 = SourceTerm S \<and> a2 = SourceTerm S) \<or> (\<exists>T. a1 = TargetTerm T \<and> a2 = TargetTerm T) \<or> (\<exists>P Q R. a1 = P \<and> a2 = R \<and> encoding.indRelLPOp Enc P Q \<and> encoding.indRelLPOp Enc Q R))"
nat_mod_eq_lemma: fixes x :: "nat"   and n :: "nat"   and y :: "nat" assumes "x mod n = y mod n"   and "y \<le> x" shows "\<exists>q. x = y + n * q"
polyfun_rootbound: fixes n :: "nat"   and c :: "nat \<Rightarrow> 'a" assumes "\<exists>k\<le>n. c k \<noteq> (0::'a)" shows "finite {z. (\<Sum>i\<le>n. c i * z ^ i) = (0::'a)} \<and> card {z. (\<Sum>i\<le>n. c i * z ^ i) = (0::'a)} \<le> n"
M: fixes y :: "'a set" assumes "y \<in> M" shows "f ` y \<in> M"
ns_mul_ext_local_mono: fixes xs :: "'a multiset"   and ys :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and ns' :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and s' :: "('a \<times> 'a) set" assumes "set_mset xs \<times> set_mset ys \<inter> ns \<subseteq> ns'"   and "set_mset xs \<times> set_mset ys \<inter> s \<subseteq> s'"   and "(xs, ys) \<in> ns_mul_ext ns s" shows "(xs, ys) \<in> ns_mul_ext ns' s'"
take_take: fixes m :: "nat"   and n :: "nat"   and x :: "'r tree" shows "tree_take m\<cdot>(tree_take n\<cdot>x) = tree_take (min m n)\<cdot>x"
int_finsum_eq: fixes f :: "'a \<Rightarrow> int"   and A :: "'a set" shows "finsum \<Z> f A = sum f A"
f3: fixes a :: "'a"   and aa :: "'a" shows "ad (a + aa) = ad (aa + a)"
simps_18: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a" shows "rec_finite_4 f1 f2 f3 f4 a\<^sub>2 = f2"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) G G G"
approx_hRe: fixes x :: "complex star"   and y :: "complex star" assumes "x \<approx> y" shows "hRe x \<approx> hRe y"
trg_dom: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b" assumes "horizontal_homs V src trg"   and "partial_magma.arr V \<mu>" shows "trg (partial_magma.dom V \<mu>) = trg \<mu>"
geq: fixes z :: "complex" assumes "z \<in> S" shows "f z = exp (g z)"
non_empty_cycle_root_rtc_tc: fixes r :: "'a"   and x :: "'a" assumes "non_empty_cycle_root r x" shows "x\<^sup>\<star> ; r = x\<^sup>+ ; r"
SUP_cong_simp: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b" assumes "A = B"   and "\<And>x. x \<in> B =simp=> C x = D x" shows "Sup (C ` A) = Sup (D ` B)"
det_def: fixes A :: "'a mat" shows "det A = (if dim_row A = dim_col A then \<Sum>p | p permutes {0..<dim_row A}. signof p * (\<Prod>i = 0..<dim_row A. A $$ (i, p i)) else (0::'a))"
Impl_elim_cases: fixes s :: "state"   and Cm :: "cname \<times> mname"   and n :: "nat"   and t :: "state" assumes "s -Impl Cm-n\<rightarrow> t" obtains na :: "nat" where "n = Suc na" and "s -body Cm-na\<rightarrow> t"
HF_def: fixes A :: "hf set" shows "HF A = Abs_hf (set_encode (Rep_hf ` A))"
sum_3: fixes f :: "3 \<Rightarrow> 'a" shows "sum f UNIV = f 1 + f 2 + f 3"
perron_frobenius_nonneg: fixes A :: "real mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "nonneg_mat A"   and "n \<noteq> 0" shows "\<exists>sr ks f. 0 \<le> sr \<and> 0 \<notin> set ks \<and> ks \<noteq> [] \<and> char_poly A = (\<Prod>k\<leftarrow>ks. monom 1 k - [:sr ^ k:]) * f \<and> (\<forall>x. poly (map_poly complex_of_real f) x = 0 \<longrightarrow> cmod x < sr)"
pred_intros_conj1': fixes K :: "bool"   and P :: "'d \<Rightarrow> bool"   and M :: "'d measure" assumes "K \<Longrightarrow> Measurable.pred M P" shows "Measurable.pred M (\<lambda>x. K \<and> P x)"
assoc_naturality: fixes \<mu> :: "'b"   and \<nu> :: "'b"   and \<tau> :: "'b" assumes "B.arr \<mu>"   and "B.arr \<nu>"   and "B.arr \<tau>"   and "src\<^sub>B \<mu> = trg\<^sub>B \<nu>"   and "src\<^sub>B \<nu> = trg\<^sub>B \<tau>" shows "\<a>\<^sub>B[B.cod \<mu>, B.cod \<nu>, B.cod \<tau>] \<cdot>\<^sub>B ((\<mu> \<star>\<^sub>B \<nu>) \<star>\<^sub>B \<tau>) = (\<mu> \<star>\<^sub>B \<nu> \<star>\<^sub>B \<tau>) \<cdot>\<^sub>B \<a>\<^sub>B[B.dom \<mu>, B.dom \<nu>, B.dom \<tau>]"
a2: shows "x ; 1 \<cdot> y ; 1 \<cdot> y\<^sup>T ; 1 = 0"
degree_map_poly_hom: fixes p :: "rat poly" shows "degree (map_poly of_rat p) = degree p"
exhaust: fixes y :: "htype" obtains x1 :: "String.literal" where "y = Class_type x1"   | x21 :: "ty" and x22 :: "nat" where "y = Array_type x21 x22"
lexord_same_pref_iff: fixes xs :: "'g list"   and ys :: "'g list"   and zs :: "'g list"   and r :: "('g \<times> 'g) set" shows "((xs @ ys, xs @ zs) \<in> lexord r) = ((\<exists>x\<in>set xs. (x, x) \<in> r) \<or> (ys, zs) \<in> lexord r)"
CompDefinedE: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and g :: "'e" assumes "f \<approx>>\<^bsub>C\<^esub> g" obtains "f \<in> mor\<^bsub>C\<^esub>" and "g \<in> mor\<^bsub>C\<^esub>" and "cod\<^bsub>C\<^esub> f = dom\<^bsub>C\<^esub> g"
intros_1: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "\<turnstile>\<^sub>t {P} SKIP {P}"
prems_1: fixes W :: "'a set" assumes "W \<in> I"   and "\<forall>a\<in>insert U F. W \<subseteq> a" shows "thesis"
object_ptr_kinds_eq_h3: shows "object_ptr_kinds h' = object_ptr_kinds h3"
distinct_39: fixes x5 :: "msg"   and x71 :: "nat"   and x72 :: "msg" shows "Hash x5 \<noteq> Crypt x71 x72"
map_graph_def: fixes f :: "('d \<times> 'c) set"   and G :: "('b, 'd) labeled_graph" shows "map_graph f G = LG (on_triple f `` edges G) (f `` vertices G)"
lemScaleOverAdd: fixes k :: "'a"   and u :: "'a Vector"   and v :: "'a Vector" shows "k ** (u \<oplus> v) = k ** u \<oplus> k ** v"
of_int_poly_smult: fixes c :: "int"   and p :: "int poly" shows "of_int_poly (Polynomial.smult c p) = Polynomial.smult (of_int c) (of_int_poly p)"
prod_emb_def: fixes I :: "'b set"   and M :: "'b \<Rightarrow> 'c measure"   and K :: "'b set"   and X :: "('b \<Rightarrow> 'c) set" shows "prod_emb I M K X = (\<lambda>x. restrict x K) -` X \<inter> (\<Pi>\<^sub>E i\<in>I. space (M i))"
ts_def: fixes s :: "nat \<Rightarrow> 'd list" shows "ts s \<equiv> \<forall>i. length (s i) = 1"
test_assoc2: fixes P :: "('a \<times> 'a set) set"   and R :: "('a \<times> 'a set) set"   and S :: "('a \<times> 'a set) set" assumes "P \<subseteq> 1\<^sub>\<sigma>" shows "P \<cdot> R \<cdot> S = P \<cdot> (R \<cdot> S)"
c_d: shows "(c, d) \<in> thetaSSeqD"
word_arith_nat_defs_6: fixes a :: "'b word"   and b :: "'b word" shows "a div b = word_of_nat (unat a div unat b)"
PoW_def: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "PoW t n \<equiv> LEAST x. pow_cond t n x"
distinct_89: fixes x31 :: "floatarith"   and x32 :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Mult x31 x32 \<noteq> Power x151 x152"
eval_monom_mult: fixes \<alpha> :: "'c \<Rightarrow> 'b"   and m :: "'c monom"   and n :: "'c monom" shows "eval_monom \<alpha> (m * n) = eval_monom \<alpha> m * eval_monom \<alpha> n"
restrict_rel_prod: fixes R :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and I1 :: "'c \<Rightarrow> bool"   and I2 :: "'e \<Rightarrow> bool"   and S :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and I1' :: "'d \<Rightarrow> bool"   and I2' :: "'f \<Rightarrow> bool" shows "rel_prod (R \<upharpoonleft> I1 \<otimes> I2) (S \<upharpoonleft> I1' \<otimes> I2') = rel_prod R S \<upharpoonleft> pred_prod I1 I1' \<otimes> pred_prod I2 I2'"
Cd_single: fixes x :: "'a"   and X :: "'a set"   and d :: "'b" assumes "x \<in> X"   and "inj_on Xd X"   and "x \<in> Field (TruePref.Pd_shuffle_to_top ds' (ManiPref.cop ds) d)" shows "x \<in> ManiPref'.Cd d X"
simps_6: fixes x1 :: "rat"   and x31 :: "int poly"   and x32 :: "nat" shows "Rat_Info x1 \<noteq> Real_Alg_Info x31 x32"
exe_mode_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state" assumes "low_equal s1 s2" shows "exe_mode_val s1 = exe_mode_val s2"
Gromov_product_e_x_x: fixes e :: "'c"   and x :: "'c" shows "Gromov_product_at e x x = dist e x"
outputPermFrameSubject: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list"   and yvec :: "name list"   and zvec :: "name list" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "set p \<subseteq> set yvec \<times> set zvec"   and "yvec \<sharp>* P"   and "zvec \<sharp>* P" shows "p \<bullet> \<Psi> \<rhd> P \<longmapsto> (p \<bullet> M)\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"
assms: shows "(\<LM> A  \<RM>) \<oplus> B = \<LM> C  \<RM>"
basis_ex1_supset_circuit: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and B :: "'b set"   and x :: "'b" assumes "matroid carrier indep"   and "indep_system.basis carrier indep B"   and "x \<in> carrier - B" shows "\<exists>!C. indep_system.circuit carrier indep C \<and> C \<subseteq> insert x B"
mem_UnitsD: fixes u :: "'a \<Rightarrow> 'a" assumes "u \<in> Units" shows "invertible u \<and> u \<in> S \<rightarrow>\<^sub>E S"
induced_automorphism_bij_between_half_chamber_systems_f: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "bij_betw ((`) (OpposedThinChamberComplexFoldings.induced_automorphism X f g)) (ChamberComplex.chamber_system X - f \<turnstile> ChamberComplex.chamber_system X) (f \<turnstile> ChamberComplex.chamber_system X)"
list_of_oalist_OAlist_tc_insert: fixes x :: "'a \<times> 'b"   and xs :: "('a, 'b) oalist_tc" shows "list_of_oalist_tc (OAlist_tc_insert x xs) = update_by_pair_tc x (list_of_oalist_tc xs)"
terminates_Normal_elim_cases_7: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and b :: "'b set"   and c :: "('b, 'a, 'c) com"   and s :: "'b" assumes "\<Gamma>\<turnstile>While b c \<down> Normal s" obtains "s \<in> b" and "\<Gamma>\<turnstile>c \<down> Normal s" and "\<forall>s'. \<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> s' \<longrightarrow> \<Gamma>\<turnstile>While b c \<down> s'"   | "s \<notin> b"
drop_take: fixes n :: "nat"   and m :: "nat"   and xs :: "'a list" shows "drop n (take m xs) = take (m - n) (drop n xs)"
agree_func_fvt: fixes \<nu> :: "(real, 'd) vec \<times> (real, 'd) vec"   and \<nu>' :: "(real, 'd) vec \<times> (real, 'd) vec"   and f :: "'e"   and args :: "'d \<Rightarrow> ('e, 'd) trm"   and i :: "'d" assumes "Vagree \<nu> \<nu>' (FVT ($f f args))" shows "Vagree \<nu> \<nu>' (FVT (args i))"
assms: shows "eval r_phi [i, x] \<down>= v"
that: fixes C :: "'c list" assumes "list_all_zip (\<lambda>a c. \<exists>b. P a b c) A C"   and "list_all_zip (\<lambda>b c. \<exists>a. P a b c) B C" shows "thesis"
inverse_linear_upper: fixes a :: "real"   and x :: "real"   and b :: "real"   and alpha :: "real" assumes "0 < a"   and "x \<in> {a..b}"   and "- inverse (b * b) \<le> alpha" shows "inverse x \<le> inverse a + alpha * (x - a)"
Ball_set_replicate: fixes n :: "nat"   and a :: "'a"   and P :: "'a \<Rightarrow> bool" shows "(\<forall>x\<in>set (replicate n a). P x) = (P a \<or> n = 0)"
lm148: fixes X :: "'b set" assumes "card (Pow X) = 1 \<or> card (Pow X) = 2" shows "trivial X"
assms: shows "\<AA> \<approx>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
assms_2: fixes \<sigma> :: "nat \<Rightarrow> state"   and m :: "msg" assumes "P \<sigma> m" shows "P (\<sigma>(i := \<sigma> i\<lparr>msg := m\<rparr>)) m"
simps: fixes b :: "'a" shows "the_In2 (In2 b) = b"
isomTr2: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and H :: "('i, 'j) Group_scheme" assumes "Group F"   and "Group G"   and "Group H"   and "F \<cong> G"   and "G \<cong> H" shows "F \<cong> H"
l1_inv5_init: shows "init l1 \<subseteq> l1_inv5"
ford_fulkerson_2: shows "(\<not> Ex isAugmentingPath) = (\<exists>k. NCut c s t k \<and> val = NCut.cap c k)"
simps_18: fixes h :: "hash"   and t :: "Syntax.term"   and t' :: "Syntax.term"   and x :: "Syntax.var" shows "Hashed h t[t' / x] = Hashed h (t[t' / x])"
measure_lborel_cbox: fixes l :: "'b"   and u :: "'b" assumes "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b" shows "Sigma_Algebra.measure lborel (cbox l u) = prod ((\<bullet>) (u - l)) Basis"
le_name_code_1: fixes x :: "char"   and xs :: "char list" shows "(Name (x # xs) \<le> Name []) = False"
comp_tuple_arr: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a"   and l :: "'a" assumes "C.commutative_square f g h k"   and "C.seq h l" shows "\<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> \<cdot> l = \<langle>h \<cdot> l \<lbrakk>f, g\<rbrakk> k \<cdot> l\<rangle>"
st_vec_cases: fixes y :: "(real, 'a) Finite_Cartesian_Product.vec" assumes "y \<in> {v. non_neg_vec v \<and> stoch_vec v}" obtains x :: "'a st_vec" where "y = st_vec x"
timpl_closure_Fun_inv: fixes s :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and f :: "('a, 'b, 'c) prot_fun"   and T :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and TI :: "('c set \<times> 'c set) set" assumes "s \<in> timpl_closure (Fun f T) TI" shows "\<exists>g S. s = Fun g S"
intros_8: fixes f :: "'d"   and F :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'd) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and g :: "'f set"   and P :: "'f set"   and c :: "('f, 'e, 'd) com"   and Q :: "'f set"   and A :: "'f set" assumes "f \<in> F"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (g \<inter> P) c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Guard f g c Q,A"
induct: fixes P :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and a0 :: "'a option"   and a1 :: "'a option" assumes "\<And>uu_. P None uu_"   and "\<And>v. P (Some v) None"   and "\<And>a b. P (Some a) (Some b)" shows "P a0 a1"
extended_distance_set_Cauchy_1: fixes u :: "nat \<Rightarrow> 'c Bonk_Schramm_extension_unfolded"   and n :: "nat" assumes "would_be_Cauchy u \<in> extended_distance_set" shows "u n \<in> extended_distance_set \<inter> underS Bonk_Schramm_extension_unfolded_wo (would_be_Cauchy u)"
roots1_def: fixes p :: "'a poly" shows "roots1 p = - coeff p 0 / coeff p 1"
infs_def: fixes x :: "'a"   and y :: "'a" shows "infs x y = Infs {x, y}"
X_minus_closed: fixes a :: "'a" assumes "a \<in> carrier R" shows "X_poly_minus R a \<in> carrier P"
encR: fixes S :: "'procS" shows "SourceTerm S \<R>\<lbrakk>\<cdot>\<rbrakk> TargetTerm (\<lbrakk>S\<rbrakk>)"
card_permutations: fixes S :: "'a set"   and n :: "nat" assumes "card S = n"   and "finite S" shows "card {p. p permutes S} = fact n"
integrable_const: fixes a :: "'c" shows "integrable M (\<lambda>x. a)"
isomorphic_transitive: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "f \<cong> g"   and "g \<cong> h" shows "f \<cong> h"
simps: fixes n :: "nat"   and v :: "nat"   and c :: "nat" shows "Fast_Dice_Roll.fast_dice_roll n v c = (if n \<le> v then if c < n then return_spmf c else Fast_Dice_Roll.fast_dice_roll n (v - n) (c - n) else coin_spmf \<bind> (\<lambda>b. Fast_Dice_Roll.fast_dice_roll n (2 * v) (2 * c + (if b then 1 else 0))))"
fundamental_theorem_of_linear_inequalities_C_imp_B: fixes A :: "'a vec set"   and b :: "'a vec" assumes "A \<subseteq> carrier_vec n"   and "finite A"   and "b \<in> carrier_vec n"   and "\<nexists>c A'. c \<in> carrier_vec n \<and> A' \<subseteq> A \<and> Suc (card A') = dim_span (insert b A) \<and> (\<forall>a\<in>A'. c \<bullet> a = (0::'a)) \<and> lin_indpt A' \<and> (\<forall>a\<^sub>i\<in>A. (0::'a) \<le> c \<bullet> a\<^sub>i) \<and> c \<bullet> b < (0::'a)" shows "\<exists>B\<subseteq>A. lin_indpt B \<and> card B = dim_span A \<and> b \<in> finite_cone B"
sub_trees_refl: fixes t :: "'a factor_tree" shows "t \<in> sub_trees t"
star_pnat_def: fixes x :: "pnat" shows "x\<^sup>\<star> \<equiv> 1"
OUTfromVCorrect1_data15: shows "OUTfromVCorrect1 data15"
prod_mset_delta': fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and y :: "'c"   and c :: "'b"   and A :: "'c multiset" assumes "class.comm_monoid_mult times one" shows "comm_monoid_mult.prod_mset times one {#if y = x then c else one. x \<in># A#} = power.power one times c (count A y)"
inject_1: fixes x6 :: "Arraytype"   and y6 :: "Arraytype" shows "(ArrT x6 = ArrT y6) = (x6 = y6)"
Confidentiality_lemma: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and evs :: "event list" assumes "Says Tgs A \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, Number Ts\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace>\<rbrace> \<in> set evs"   and "Key authK \<notin> analz (knows Spy evs)"   and "servK \<in> symKeys"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbV"   and "Key servK \<in> analz (knows Spy evs)" shows "expiredSK Ts evs"
fmap_of_list_insort_key_fst: fixes ps :: "('c \<times> 'd) list"   and k :: "'c"   and v :: "'d" assumes "distinct (map fst ps)"   and "k \<notin> set (map fst ps)" shows "fmap_of_list (insort_key fst (k, v) ps) = fmap_of_list ps(k $$:= v)"
test_comp_mult2: fixes p :: "'a" shows "\<iota> p \<cdot> ! (\<iota> p) = (0::'b)"
inj_on_opp_chambers: shows "inj_on ((`) f) (\<C> - f \<turnstile> \<C>)"
FAssNone\<^sub>1: fixes P :: "nat exp prog"   and e\<^sub>1 :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and a :: "nat"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and e\<^sub>2 :: "nat exp"   and v :: "val"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "val list"   and sh\<^sub>2 :: "sheap"   and C :: "char list"   and fs :: "Objects.fields"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v,(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"   and "h\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"   and "\<nexists>b t. P \<turnstile> C has F,b:t in D" shows "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NoSuchFieldError,(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"
one_fun_apply: fixes x :: "'b" shows "1 x = (1::'a)"
iTILL_cut_less: fixes n :: "nat"   and t :: "nat" shows "[\<dots>n] \<down>< t = (if n < t then [\<dots>n] else if t = 0 then {} else [\<dots>t - Suc 0])"
map_sel_1: fixes a :: "'c list"   and f :: "'c \<Rightarrow> 'd" assumes "a \<noteq> []" shows "hd (map f a) = f (hd a)"
Zleast_in_mI_pos: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set"   and j :: "nat" assumes "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "I \<noteq> {\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>}"   and "j \<le> n" shows "0 \<le> mL K P I j"
higher_differentiable_on_id: fixes S :: "'b set"   and n :: "nat" shows "higher_differentiable_on S (\<lambda>x. x) n"
cindex_pathE_reversepath: fixes g :: "real \<Rightarrow> complex"   and z :: "complex" shows "cindex_pathE (reversepath g) z = - cindex_pathE g z"
fps_XD_mult_const: fixes c :: "'a"   and a :: "'a fps" shows "fps_XD (fps_const c * a) = fps_const c * fps_XD a"
hyps: shows "\<Union> (extend A S f ` {..m}) = extend A S f m"
big_step_induct: fixes x1a :: "char list \<Rightarrow> com option"   and x2a :: "com"   and x2b :: "char list \<Rightarrow> int \<Rightarrow> int"   and x3a :: "char list \<Rightarrow> int \<Rightarrow> int"   and P :: "(char list \<Rightarrow> com option) \<Rightarrow> com \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" assumes "x1a: (x2a, x2b) \<Rightarrow> x3a"   and "\<And>\<pi> s. P \<pi> SKIP s s"   and "\<And>\<pi> x i a s. P \<pi> (x[i] ::= a) s (s(x := (s x)(aval i s := aval a s)))"   and "\<And>\<pi> x y s. P \<pi> (x[] ::= y) s (s(x := s y))"   and "\<And>\<pi> x s. P \<pi> (CLEAR x[]) s (s(x := \<lambda>_. 0))"   and "\<And>\<pi> l s. P \<pi> (Assign_Locals l) s <l|s>"   and "\<And>\<pi> c\<^sub>1 s\<^sub>1 s\<^sub>2 c\<^sub>2 s\<^sub>3. \<lbrakk>\<pi>: (c\<^sub>1, s\<^sub>1) \<Rightarrow> s\<^sub>2; P \<pi> c\<^sub>1 s\<^sub>1 s\<^sub>2; \<pi>: (c\<^sub>2, s\<^sub>2) \<Rightarrow> s\<^sub>3; P \<pi> c\<^sub>2 s\<^sub>2 s\<^sub>3\<rbrakk> \<Longrightarrow> P \<pi> (c\<^sub>1;; c\<^sub>2) s\<^sub>1 s\<^sub>3"   and "\<And>b s \<pi> c\<^sub>1 t c\<^sub>2. \<lbrakk>bval b s; \<pi>: (c\<^sub>1, s) \<Rightarrow> t; P \<pi> c\<^sub>1 s t\<rbrakk> \<Longrightarrow> P \<pi> (IF b THEN c\<^sub>1 ELSE c\<^sub>2) s t"   and "\<And>b s \<pi> c\<^sub>2 t c\<^sub>1. \<lbrakk>\<not> bval b s; \<pi>: (c\<^sub>2, s) \<Rightarrow> t; P \<pi> c\<^sub>2 s t\<rbrakk> \<Longrightarrow> P \<pi> (IF b THEN c\<^sub>1 ELSE c\<^sub>2) s t"   and "\<And>\<pi> c s s'. \<lbrakk>\<pi>: (c, <<>|s>) \<Rightarrow> s'; P \<pi> c <<>|s> s'\<rbrakk> \<Longrightarrow> P \<pi> (SCOPE c) s <s|s'>"   and "\<And>b s \<pi> c. \<not> bval b s \<Longrightarrow> P \<pi> (WHILE b DO c) s s"   and "\<And>b s\<^sub>1 \<pi> c s\<^sub>2 s\<^sub>3. \<lbrakk>bval b s\<^sub>1; \<pi>: (c, s\<^sub>1) \<Rightarrow> s\<^sub>2; P \<pi> c s\<^sub>1 s\<^sub>2; \<pi>: (WHILE b DO c, s\<^sub>2) \<Rightarrow> s\<^sub>3; P \<pi> (WHILE b DO c) s\<^sub>2 s\<^sub>3\<rbrakk> \<Longrightarrow> P \<pi> (WHILE b DO c) s\<^sub>1 s\<^sub>3"   and "\<And>\<pi> p c s t. \<lbrakk>\<pi> p = Some c; \<pi>: (c, s) \<Rightarrow> t; P \<pi> c s t\<rbrakk> \<Longrightarrow> P \<pi> (PCall p) s t"   and "\<And>\<pi>' c s t \<pi>. \<lbrakk>\<pi>': (c, s) \<Rightarrow> t; P \<pi>' c s t\<rbrakk> \<Longrightarrow> P \<pi> (PScope \<pi>' c) s t" shows "P x1a x2a x2b x3a"