hcju/isabelleps huggingface dataset:
boundedE: fixes A :: "'c DBMEntry set"   and x :: "'c DBMEntry" assumes "finite A"   and "A \<noteq> {}"   and "x \<preceq> linordered_monoid.Min A" obtains "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
sin_conv_sinh: fixes z :: "complex" shows "sin z = - \<i> * sinh (\<i> * z)"
step_elim_cases_3: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and f :: "'b \<Rightarrow> 'b"   and s :: "('b, 'c) xstate"   and u :: "('b, 'a, 'c) com \<times> ('b, 'c) xstate" assumes "\<Gamma>\<turnstile> (Basic f, s) \<rightarrow> u" obtains sa :: "'b" where "s = Normal sa" and "u = (Skip, Normal (f__ sa))"   | f :: "'c" where "s = Fault fa__" and "u = (Skip, Fault fa__)"   | "s = Stuck" and "u = (Skip, Stuck)"   | f :: "'b" where "s = Abrupt f" and "u = (Skip, Abrupt f)"
simps_2: fixes uu :: "'a extended" shows "(uu \<le> \<infinity>) = True"
bounded_coeffs_newman_minus_deriv_zeta: shows "Bseq (fds_nth (fds_newman + fds_deriv fds_zeta))"
3: fixes l1 :: "'lbl"   and l2 :: "'lbl" assumes "l1 \<in> set (remdups (map (the_LabelN \<circ> fst) (filter (Not \<circ> is_LabelS) A)))"   and "l2 \<in> set (remdups (map (the_LabelN \<circ> fst) (filter (Not \<circ> is_LabelS) A)))"   and "l1 \<noteq> l2" shows "GSMP_disjoint (trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (proj l1 A) \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t (proj_unl l1 A)) (trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (proj l2 A) \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t (proj_unl l2 A)) (f (set C) - {m. {} \<turnstile>\<^sub>c m})"
mpt_factorE_1: fixes proj :: "(int \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow> 'b"   and M2 :: "'b measure"   and T2 :: "'b \<Rightarrow> 'b" assumes "I.mpt_factor proj M2 T2" shows "proj \<in> measure_preserving MI M2"
vote_set_def: fixes vs :: "'c \<Rightarrow> 'e \<Rightarrow> 'd option"   and Q :: "'e set" shows "vote_set vs Q \<equiv> {uu_. \<exists>a r v. uu_ = (r, v) \<and> ((r, a), v) \<in> map_graph (\<lambda>(x, y). vs x y) \<and> a \<in> Q}"
of_bool_Bex_eq_nn_integral: fixes X :: "'d set"   and P :: "'d \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> X; y \<in> X; P x; P y\<rbrakk> \<Longrightarrow> x = y" shows "of_bool (\<exists>y\<in>X. P y) = \<integral>\<^sup>+ y. of_bool (P y) \<partial>count_space X"
stop: fixes g :: "('a \<Rightarrow> bool) \<times> ('a \<Rightarrow> 'b \<times> 'a)"   and s :: "'a" assumes "\<not> fst g s" shows "terminates_onp g s"
assms_4: shows "incid B (line A' C') \<or> incid A' (line B C)"
op_norm0: shows "\<parallel>0\<parallel>\<^sub>o\<^sub>p = 0"
lessThan_Suc: fixes g :: "nat \<Rightarrow> 'e"   and n :: "nat" shows "prod g {..<Suc n} = prod g {..<n} * g n"
nn_integral_fst_count_space: fixes f :: "'c \<times> 'd \<Rightarrow> ennreal" shows "(\<Sum>\<^sup>+ x. \<Sum>\<^sup>+ y. f (x, y)) = integral\<^sup>N (count_space UNIV) f"
cp_array_def: fixes xs :: "'a list" shows "cp_array xs \<equiv> let ys = op_array_replicate (length xs) (0::'a) in nfoldli [0..<length xs] (\<lambda>_. True) (\<lambda>i ys. ASSERT (i < length xs \<and> i < length ys) \<bind> (\<lambda>_. RETURN (ys[i := xs ! i]))) ys \<bind> RETURN"
Sup_bot_conv_1: fixes A :: "'a set" shows "(Sup A = \<bottom>) = (\<forall>x\<in>A. x = \<bottom>)"
D_3: shows "U \<subseteq> (\<Union>x\<in>D. ball x 1)"
rep_eq: fixes x :: "'a mpoly"   and xa :: "nat" shows "poly.coeff (mpoly_to_nested_poly x xa) = isolate_variable_sparse x xa"
Maclaurin_all_lt: fixes diff :: "nat \<Rightarrow> real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and n :: "nat"   and x :: "real" assumes "diff 0 = f"   and "0 < n"   and "x \<noteq> 0"   and "\<forall>m x. (diff m has_real_derivative diff (Suc m) x) (at x)" shows "\<exists>t. 0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> f x = (\<Sum>m<n. diff m 0 / fact m * x ^ m) + diff n t / fact n * x ^ n"
f_bound_True_arbitrary: fixes m :: "nat"   and fs_init :: "int vec list"   and ii :: "nat"   and fs :: "int vec list"   and outside :: "bool"   and j :: "nat" assumes "LLL.f_bound m fs_init True ii fs" shows "LLL.f_bound m fs_init outside j fs"
uminus_fds_def: fixes f :: "'a fds" shows "- f = fds (\<lambda>n. - fds_nth f n)"
Semi: fixes P :: "state \<Rightarrow> bool"   and c1 :: "com"   and Q :: "state \<Rightarrow> bool"   and c2 :: "com"   and R :: "state \<Rightarrow> bool" assumes "\<turnstile> {P} c1 {Q}"   and "\<turnstile> {Q} c2 {R}" shows "\<turnstile> {P} c1; c2 {R}"
Ifun_star_of: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "star_of f \<star> star_of x = star_of (f x)"
ldropn_lappend: fixes n :: "nat"   and xs :: "'c llist"   and ys :: "'c llist" shows "ldropn n (lappend xs ys) = (if enat n < llength xs then lappend (ldropn n xs) ys else ldropn (n - the_enat (llength xs)) ys)"
eout_knows_nonKS_k: fixes m :: "Keys"   and A :: "specID" assumes "m \<notin> specKeys A"   and "eout A (kE m)"   and "eoutKnowsECorrect A (kE m)" shows "knows A [kE m]"
e_altdef: fixes k :: "nat" shows "e k = insertion x (sym_mpoly {..<n} k)"
unit_factor_Lcm_fin: fixes A :: "'a set" shows "unit_factor (Lcm\<^sub>f\<^sub>i\<^sub>n A) = of_bool (Lcm\<^sub>f\<^sub>i\<^sub>n A \<noteq> (0::'a))"
bound: fixes b :: "nat list"   and x :: "nat"   and ys :: "nat list"   and xs :: "nat list"   and s :: "nat" assumes "Max (set b) < x" shows "cond1 b ys (x # xs) s = False"
rebalance_middle_tree_height: fixes t :: "'a btree"   and sub :: "'a btree"   and rs :: "('a btree \<times> 'a) list"   and k :: "nat"   and ls :: "('a btree \<times> 'a) list"   and sep :: "'a" assumes "height t = height sub"   and "case rs of [] \<Rightarrow> True | (rsub, rsep) # list \<Rightarrow> height rsub = height t" shows "height (rebalance_middle_tree k ls sub sep rs t) = height (Node (ls @ (sub, sep) # rs) t)"
mat_append_col_vec_index: fixes i :: "nat"   and A :: "'a mat"   and b :: "'a vec" assumes "i < dim_row A"   and "dim_row A = dim_vec b" shows "row (mat_append_col A b) i $ dim_col A = b $ i"
induced_automorphism_unique: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and f' :: "'b \<Rightarrow> 'b"   and g' :: "'b \<Rightarrow> 'b" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "OpposedThinChamberComplexFoldings X f' g' C0"   and "g' ` C0 = g ` C0" shows "OpposedThinChamberComplexFoldings.induced_automorphism X f' g' = OpposedThinChamberComplexFoldings.induced_automorphism X f g"
devExt_bc_geq: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid"   and n' :: "nat"   and n\<^sub>s :: "nat" assumes "\<not> Option.is_none (devBC t n nid n')"   and "n\<^sub>s \<le> n'" shows "devExt t n nid n\<^sub>s (n' - n\<^sub>s) = bc (\<sigma>\<^bsub>the (devBC t n nid n')\<^esub>t n')"
ltrm_deg_0: fixes p :: "nat \<Rightarrow> int" assumes "p \<in> carrier (UP INTEG)"   and "deg INTEG p = 0" shows "monom (UP INTEG) (p (deg INTEG p)) (deg INTEG p) = p"
ord_eq_le_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a = b"   and "b \<le> c" shows "a \<le> c"
assms_2: fixes \<omega> :: "'a stream" assumes "P \<omega>" shows "Q1 \<omega> = Q2 \<omega>"
bin_set_ge: fixes w :: "int"   and n :: "nat" shows "w \<le> bin_sc n True w"
class_of_in_stalk: fixes A :: "'a set"   and z :: "'b" assumes "A \<in> neighborhoods x"   and "z \<in> \<O>\<^sub>X A" shows "stx.class_of A z \<in> stx.carrier_stalk"
3: shows "\<forall>x y z. (case x of (e, fe) \<Rightarrow> \<lambda>(d, fd). fe + bot \<le> fd + bot) y \<and> (case y of (e, fe) \<Rightarrow> \<lambda>(d, fd). fe + bot \<le> fd + bot) z \<longrightarrow> (case x of (e, fe) \<Rightarrow> \<lambda>(d, fd). fe + bot \<le> fd + bot) z"
DERIV_cos: fixes x :: "'a" shows "(cos has_field_derivative - sin x) (at x)"
inf_assoc: fixes inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g"   and z :: "'g" assumes "class.semilattice_inf inf less_eq less" shows "inf (inf x y) z = inf x (inf y z)"
def_body: fixes x :: "'a"   and y :: "'b" assumes "x \<noteq> \<bottom>"   and "x \<noteq> null"   and "y \<noteq> \<bottom>"   and "y \<noteq> null" shows "g x y \<noteq> \<bottom> \<and> g x y \<noteq> null"
prems_6: shows "x \<in> dom (C (AllowPortFromTo c d po))"
no_ofail_ogets: fixes f :: "'a \<Rightarrow> 'b" shows "no_ofail (\<lambda>_. True) (ogets f)"
equiv_vert_uconnected: fixes E' :: "nat uprod set" shows "equiv V (Restr (uconnected E') V)"
prefixI: fixes ys :: "'c list"   and xs :: "'c list"   and zs :: "'c list" assumes "ys = xs @ zs" shows "prefix xs ys"
cod_pr1: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "cod (pr1 a b) = a"
simps_17: fixes v :: "nat"   and va :: "atom fm" shows "push_forall (AllQ (ExN v va)) = (if freeIn 0 (ExN v va) then lowerFm 0 1 (ExN v va) else AllQ (ExN v va))"
A3: shows "\<forall>a b c. a b \<equiv> c c \<longrightarrow> a = b"
Brouwer_reduction_theorem_gen: fixes S :: "'b set"   and \<phi> :: "'b set \<Rightarrow> bool" assumes "closed S"   and "\<phi> S"   and "\<And>F. \<lbrakk>\<And>n. closed (F n); \<And>n. \<phi> (F n); \<And>n. F (Suc n) \<subseteq> F n\<rbrakk> \<Longrightarrow> \<phi> (\<Inter> (range F))" obtains T :: "'b set" where "T \<subseteq> S" and "closed T" and "\<phi> T" and "\<And>U. \<lbrakk>U \<subseteq> S; closed U; \<phi> U\<rbrakk> \<Longrightarrow> \<not> U \<subset> T"
dim_col_def: shows "dim_col \<equiv> (Rep_mat ---> id) (fst \<circ> snd)"
prems_7: shows "(af_, a'f_) \<in> \<langle>R\<rangle>ltlr_rel \<rightarrow> Rv \<rightarrow> Rv"
set_spmf_rep_eq: fixes p :: "'c spmf" shows "set_spmf p = {x. Sigma_Algebra.measure (measure_spmf p) {x} \<noteq> 0}"
null_setsD2: fixes A :: "'a set"   and M :: "'a measure" assumes "A \<in> null_sets M" shows "A \<in> sets M"
that: fixes j :: "'b" assumes "dictator swf A Is j" shows "thesis"
M_consts_1: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "\<not> w \<Turnstile>\<^sub>n false\<^sub>n M\<^sub>n \<phi>"
nat_cut_ge_finite_iff: fixes I :: "nat set"   and t :: "nat" shows "finite (I \<down>\<ge> t) = finite I"
Step1: shows "(\<lambda>g. \<lambda>h\<in>carrier G. g \<otimes> h \<otimes> inv g) \<in> carrier G \<rightarrow> carrier (BijGroup (carrier G))"
pow_generator_mod: fixes k :: "nat" shows "\<^bold>g [^] (k mod order \<G>) = \<^bold>g [^] k"
Liminf_llist_lmap_image: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b set"   and xs :: "'d llist" assumes "inj_on f (Sup_llist (lmap g xs))" shows "Liminf_llist (lmap (\<lambda>x. f ` g x) xs) = f ` Liminf_llist (lmap g xs)"
p_def: shows "p \<equiv> if ty_p = ty_p then CARD('p) else undefined"
i_def_2: shows "makespan (T(x := T x + y)) = (T(x := T x + y)) i"
N_lt_imp_dvd: fixes a :: "int"   and n :: "nat" assumes "N a < 1 / q ^ n"   and "0 < n" shows "int n dvd a"
a_set_disconnected_nodes_locs_def: fixes document_ptr :: "(_) document_ptr" shows "local.a_set_disconnected_nodes_locs document_ptr \<equiv> all_args (put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr disconnected_nodes_update)"
a_gla1: fixes x :: "'a"   and y :: "'a" assumes "ad x \<cdot> y = (0::'a)" shows "ad x \<le> ad y"
eval_Runit': fixes t :: "'c local.term" assumes "Arr t" shows "\<lbrace>\<^bold>\<r>\<^sup>-\<^sup>1\<^bold>[t\<^bold>]\<rbrace> = D.runit' \<lbrace>Cod t\<rbrace> \<cdot>\<^sub>D \<lbrace>t\<rbrace>"
inv_inv: fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a" assumes "Dgrp prod one" shows "Dmonoid.inv prod one (Dmonoid.inv prod one x) = x"
non_sdominate_succ: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "(i, j) \<in> g_E G"   and "k \<noteq> i"   and "non_strict_dominate k i" shows "non_strict_dominate k j"
le_LCons: fixes x :: "'a"   and xs :: "'a llist"   and y :: "'a"   and ys :: "'a llist" shows "(x ## xs \<le> y ## ys) = (x = y \<and> xs \<le> ys)"
accepts_def: fixes A :: "'d \<times> ('c \<Rightarrow> 'd \<Rightarrow> 'd) \<times> ('d \<Rightarrow> bool)" shows "DA.accepts A = (\<lambda>w. fin A (DA.delta A w (start A)))"
assms_2: shows "h \<turnstile> create_element document_ptr tag \<rightarrow>\<^sub>h h'"
endofunctor_H\<^sub>R: fixes f :: "'a" assumes "ide f"   and "f \<star> f \<noteq> null" shows "endofunctor (local.Right f) (H\<^sub>R f)"
st_SReal_eq: fixes x :: "real star" assumes "x \<in> \<real>" shows "st x = x"
splus_sminus: fixes t :: "'a"   and v :: "'a" shows "local.punit.sminus (local.punit.splus t v) t = v"
simps_16: fixes bs :: "real list"   and p :: "fm" shows "Ifm bs (E p) = (\<exists>x. Ifm (x # bs) p)"
setdist_refl: fixes S :: "'a set" shows "setdist S S = 0"
int_mod_ge': fixes b :: "int"   and n :: "int" assumes "b < 0"   and "0 < n" shows "b + n \<le> b mod n"
type_update_select_length1_1: fixes cts :: "ct list"   and tm :: "checker_type" assumes "type_update_select (TopType cts) = tm"   and "length cts = 1"   and "tm \<noteq> Bot" shows "ct_list_eq cts [TSome T_i32]"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "nat" assumes "Partition x xa = y" obtains "x = 0" and "xa = 0" and "y = 1"   | k :: "nat" where "x = 0" and "xa = Suc k__" and "y = 0"   | m :: "nat" where "x = Suc m__" and "xa = 0" and "y = 0"   | m :: "nat" and k :: "nat" where "x = Suc m" and "xa = Suc k" and "y = Partition m k + Partition (m - k) (Suc k)"
list_emb_code_1: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list" shows "list_emb P [] ys = True"
iso_commuting_with_\<iota>_equals_\<I>: fixes f :: "'a" assumes "\<guillemotleft>f : \<I>\<^sub>1 \<rightarrow> \<I>\<^sub>1\<guillemotright>"   and "local.iso f"   and "f \<cdot> \<iota>\<^sub>1 = \<iota>\<^sub>1 \<cdot> (f \<otimes>\<^sub>1 f)" shows "f = \<I>\<^sub>1"
hfun_to_arr_arr_to_hfun: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f" shows "hfun_to_arr (ide_to_hf (local.dom f)) (ide_to_hf (cod f)) (arr_to_hfun f) = f"
simps_6: fixes ns :: "nat list" shows "bn_count (Suc (Suc (Suc 0)) # ns) = Suc (Suc (Suc 0)) + bn_count ns"
simps_3: fixes x :: "('a, 'b) aformula"   and v :: "('a, 'b) aformula"   and va :: "('a, 'b) aformula list" shows "nFAND (x # v # va) = FAnd x (nFAND (v # va))"
quotient_imp_continuous_map: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "quotient_map X X' f" shows "continuous_map X X' f"
planning_dimacs_sound: fixes \<Pi> :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and h :: "nat"   and dimacs_M :: "int list" assumes "sat_solve_sasp \<Pi>"   and "let cnf_formula = map_formula (cnf_to_dimacs.var_to_dimacs (Suc h) (Suc (length (ast_problem.ast\<delta> \<Pi>)))) (\<Phi>\<^sub>\<forall> \<phi> prob_with_noop (ast_problem.abs_prob \<Pi>)  h) in dimacs_model dimacs_M (cnf_to_dimacs cnf_formula)" shows "ast_problem.valid_plan \<Pi> (ast_problem.decode_abs_plan \<Pi> (rem_noops (map (strips_op_to_sasp (prob_with_noop (ast_problem.abs_prob \<Pi>))) (concat (\<Phi>\<inverse> \<phi> prob_with_noop (ast_problem.abs_prob \<Pi>)  dimacs_model_to_abs dimacs_M (\<lambda>_. False) \<circ> cnf_to_dimacs.var_to_dimacs (Suc h) (Suc (length (ast_problem.ast\<delta> \<Pi>))) h)))))"
elims: fixes x :: "'a ns_constraint"   and y :: "'a ns_constraint" assumes "normalize_ns_constraint x = y" obtains l :: "linear_poly" and r :: "'a" where "x = LEQ_ns l__ r__" and "y = (let v = Abstract_Linear_Poly.max_var l__; c = coeff l__ v in if c = 0 then LEQ_ns l__ r__ else let ic = inverse c in if c < 0 then GEQ_ns (ic *R l__) (ic *R r__) else LEQ_ns (ic *R l__) (ic *R r__))"   | l :: "linear_poly" and r :: "'a" where "x = GEQ_ns l r" and "y = (let v = Abstract_Linear_Poly.max_var l; c = coeff l v in if c = 0 then GEQ_ns l r else let ic = inverse c in if c < 0 then LEQ_ns (ic *R l) (ic *R r) else GEQ_ns (ic *R l) (ic *R r))"
dist_below_tr_2: shows "FF \<notsqsubseteq> \<bottom>"
igSubstIGVar1_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Iteration.model_scheme" shows "igSubstIGVar1 MOD \<equiv> \<forall>ys y Y xs x. igWls MOD (asSort ys) Y \<longrightarrow> ys \<noteq> xs \<or> y \<noteq> x \<longrightarrow> igSubst MOD ys Y y (igVar MOD xs x) = igVar MOD xs x"
lower_asymptotic_density_lim: fixes A :: "nat set"   and l :: "real" assumes "(\<lambda>n. real (card (A \<inter> {..<n})) / real n) \<longlonglongrightarrow> l" shows "lower_asymptotic_density A = l"
double_swap_qSwap: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and zs' :: "'varSort"   and x' :: "'var"   and y' :: "'var"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "good X" shows "qGood (pick X #[[x \<and> y]]_zs #[[x' \<and> y']]_zs') \<and> X #[x \<and> y]_zs #[x' \<and> y']_zs' = asTerm (pick X #[[x \<and> y]]_zs #[[x' \<and> y']]_zs')"
augment_cf_empty: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and cap :: "'a" shows "Graph.augment_cf c {} cap = c"
assert_code_def: shows "assert_code = Assert'.assert assert_bound_code check_code"
dsg3: fixes x :: "'a"   and y :: "'a" shows "d (d x \<cdot> y) = d x \<cdot> d y"
sup_ge2: fixes y :: "'c multiset"   and x :: "'c multiset" shows "y \<subseteq># x \<union># y"
L2_set_mono: fixes K :: "'c set"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "\<And>i. i \<in> K \<Longrightarrow> f i \<le> g i"   and "\<And>i. i \<in> K \<Longrightarrow> 0 \<le> f i" shows "L2_set f K \<le> L2_set g K"
t: shows "t \<in> subterms\<^sub>s\<^sub>e\<^sub>t (subst_range (\<theta>1 \<circ>\<^sub>s \<theta>2)) - range Var"
split_option_all: fixes P :: "'a option \<Rightarrow> bool" shows "(\<forall>x. P x) = (P None \<and> (\<forall>x. P (Some x)))"
lemma_B: fixes B :: "agent"   and evs :: "event list"   and K :: "nat"   and A :: "agent"   and Tk :: "nat"   and X :: "msg"   and Ta :: "nat" assumes "B \<notin> bad"   and "evs \<in> bankerb_gets"   and "Key K \<notin> analz (knows Spy evs)"   and "Says Server A (Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, X\<rbrace>) \<in> set evs"   and "Crypt K (Number Ta) \<in> parts (knows Spy evs)" shows "Says B A (Crypt K (Number Ta)) \<in> set evs"
Init: fixes E :: "env"   and C :: "qtname"   and dt :: "loc \<Rightarrow> ty option" assumes "is_class (prg E) C" shows "E,dt\<Turnstile>Init C\<Colon>\<surd>"
landau_symbol_at_top_imp_at_left: fixes f :: "real \<Rightarrow> 'f"   and a :: "real"   and g :: "real \<Rightarrow> 'f" assumes "(\<lambda>x. f (a - inverse x)) \<in> L' at_top (\<lambda>x. g (a - inverse x))" shows "f \<in> L (at_left a) g"
until_imp_eventually: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" shows "(P \<U> Q) \<sigma> \<longrightarrow> (\<diamond>Q) \<sigma>"
matrix_mul_assoc: fixes A :: "(('b, 'e) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and B :: "(('b, 'f) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec"   and C :: "(('b, 'c) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" shows "A ** (B ** C) = A ** B ** C"
div_const_poly_conv_map_poly: fixes c :: "'b"   and p :: "'b poly" assumes "[:c:] dvd p" shows "p div [:c:] = map_poly (\<lambda>x. x div c) p"
get_host_def: shows "get_host \<equiv> l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_host get_shadow_root"
norm_unsigned_append1: fixes xs :: "bit list"   and ys :: "bit list" assumes "norm_unsigned xs \<noteq> []" shows "norm_unsigned (xs @ ys) = norm_unsigned xs @ ys"
s_pre: shows "s = (\<lambda>i j. if j \<in> upd ` {..<i} then Suc (base j) else base j) ` {..n}"
iso_is_arr: fixes f :: "'c \<times> 'c" assumes "C.VV.iso f" shows "C.VV.arr f"
isomorphic_implies_hpar_4: fixes f :: "'a"   and f' :: "'a" assumes "f \<cong>\<^sub>B f'" shows "trg f = trg f'"
de: shows "[A,d,B]\<^bsub>T\<^esub> = [A,e,B]\<^bsub>T\<^esub>"
A: shows "case_nat undefined \<rho> \<circ> Suc = \<rho>"
prod_unit_l: fixes a :: "'a"   and b :: "'a" assumes "a \<oplus> b \<in> Units (add_monoid R)"   and "a \<in> Units (add_monoid R)"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "b \<in> Units (add_monoid R)"
inverse_less_1_iff: fixes x :: "'a" shows "(inverse x < (1::'a)) = (x \<le> (0::'a) \<or> (1::'a) < x)"
eval_\<pi>: shows "primes_pi = (\<lambda>x. \<Sum>p\<leftarrow>map fst (filter snd (List.enumerate 2 (sieve 1 (replicate (nat \<lfloor>x\<rfloor> - 1) True)))). 1)"
sorted_wrt_nth_less: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and i :: "nat"   and j :: "nat" assumes "sorted_wrt P xs"   and "i < j"   and "j < length xs" shows "P (xs ! i) (xs ! j)"
remove_key_lookup: fixes k0 :: "'d"   and f :: "'d \<Rightarrow>\<^sub>0 'c"   and k :: "'d" shows "lookup (remove_key k0 f) k = (lookup f k when k \<noteq> k0)"
mod_ring_uminus_integer: shows "(mod_ring_rel_integer ===> mod_ring_rel_integer) (uminus_p_integer pp) uminus"
less_le: fixes x :: "'a list"   and y :: "'a list" shows "strict_border x y = (border x y \<and> x \<noteq> y)"
assms_3: shows "f sn\<^sub>0 = Suc n"
is_scc_pointwise: fixes E :: "('a \<times> 'a) set"   and U :: "'a set" shows "is_scc E U = (U \<noteq> {} \<and> (\<forall>u\<in>U. \<forall>v\<in>U. (u, v) \<in> (Restr E U)\<^sup>*) \<and> (\<forall>u\<in>U. \<forall>v. v \<notin> U \<and> (u, v) \<in> E\<^sup>* \<longrightarrow> (\<forall>u'\<in>U. (v, u') \<notin> E\<^sup>*)))"
abs_int_poly_eq_0_iff: fixes p :: "int poly" shows "(abs_int_poly p = 0) = (p = 0)"
enn2real_positive_iff: fixes x :: "ennreal" shows "(0 < enn2real x) = (0 < x \<and> x < top)"
Not_eqvt: fixes p :: "perm"   and x :: "('a, 'b, 'c, 'd) formula" shows "p \<bullet> FL_Formula.Not x = FL_Formula.Not (p \<bullet> x)"
cong_mult_lcancel: fixes k :: "'b"   and m :: "'b"   and a :: "'b"   and b :: "'b" assumes "coprime k m" shows "[k * a = k * b] (mod m) = [a = b] (mod m)"
that: fixes C :: "'a set" assumes "C \<in> components S"   and "\<not> connected (- C)" shows "thesis"
transpose_transpose: fixes A :: "(('d, 'e) vec, 'f) vec" shows "Finite_Cartesian_Product.transpose (Finite_Cartesian_Product.transpose A) = A"
open_neg_translation: fixes S :: "'b set"   and a :: "'b" assumes "open S" shows "open ((-) a ` S)"
g_bnd_swap: fixes i :: "nat"   and fs :: "int vec list"   and fs' :: "int vec list"   and B :: "rat" assumes "i < m"   and "i \<noteq> 0"   and "LLL_invariant_weak fs"   and "\<bar>gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) i (i - 1)\<bar> \<le> 1 / 2"   and "\<alpha> * \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) i\<parallel>\<^sup>2 < \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) (i - 1)\<parallel>\<^sup>2"   and "fs' = fs[i := fs ! (i - 1), i - 1 := fs ! i]"   and "g_bnd B fs" shows "g_bnd B fs'"
simps_4: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b" shows "(case NoMsg of NoMsg \<Rightarrow> f1 | Msg x \<Rightarrow> f2 x) = f1"
Rep_lifted_pair_inverse: fixes x :: "('c, 'd) lifted_pair" shows "Abs_lifted_pair (Rep_lifted_pair x) = x"
sat_precond_drest_sat_precond: fixes vs :: "'a set"   and s :: "('a, bool) fmap"   and as :: "(('a, bool) fmap \<times> ('a, bool) fmap) list" assumes "sat_precond_as (fmrestrict_set vs s) as" shows "sat_precond_as s as"
simps_4: fixes a :: "real"   and b :: "real"   and c :: "real"   and x :: "real" shows "aEvalUni (NeqUni (a, b, c)) x = (a * x\<^sup>2 + b * x + c \<noteq> 0)"
eo: shows "signs ! a = 1 \<or> signs ! a = 0 \<or> signs ! a = - 1"
exhaust_sel: fixes kripke :: "('a, 'b) kripke" obtains "kripke = Kripke (\<pi> kripke) (\<K> kripke)"
\<gamma>: shows "\<gamma> \<in> inferences_from (Liminf_llist Ns - Rf (Liminf_llist Ns))"
r_cancel_one: fixes x :: "'a list"   and a :: "'a list" assumes "x \<in> carrier (poly_ring R)"   and "a \<in> carrier (poly_ring R)" shows "(a \<oplus>\<^bsub>poly_ring R\<^esub> x = x) = (a = \<zero>\<^bsub>poly_ring R\<^esub>)"
StepStatus_def: fixes ST :: "('d, 'e, 'f) status"   and TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and U :: "'f update" shows "StepStatus ST TS U = (let (A, C, E, D) = Rep_status ST; C' = StepConf A C TS; E' = StepActEvent TS; D' = U !!! D in Abs_status (A, C', E', D'))"