hcju/isabelleps huggingface dataset:
Times_Un_distrib1: fixes A :: "'e set"   and B :: "'e set"   and C :: "'f set" shows "(A \<union> B) \<times> C = A \<times> C \<union> B \<times> C"
wca_on_V_axiom_on_preorder_on: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "f_range_on A f"   and "decisive_on A f" shows "wca_on A f = (V_axiom_on A f \<and> preorder_on A (rwp_on A f) \<and> total_on A (rwp_on A f))"
indicator_vimage: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'d set"   and x :: "'c" shows "indicator (f -` A) x = indicator A (f x)"
atLeastLessThan_reindex: fixes h :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat"   and g :: "'a \<Rightarrow> 'b" assumes "bij_betw h {m..<n} {h m..<h n}" shows "sum g {h m..<h n} = sum (g \<circ> h) {m..<n}"
G1: shows "\<turnstile>\<^sub>1 {\<lambda>l s. I l s \<and> preList upds ({(I, S, E)} WHILE b DO C) l s} WHILE b DO strip C { E \<Down> \<lambda>l s. Q l s \<and> postList upds l s}"
fmdomI: fixes m :: "'e \<rightharpoonup>\<^sub>f 'd"   and x :: "'e"   and y :: "'d" assumes "fmlookup m x = Some y" shows "x |\<in>| fmdom m"
lin_dep_cols_imp_det_0: fixes A :: "'a mat" assumes "A \<in> carrier_mat n n"   and "lin_dep (set (cols A))" shows "det A = (0::'a)"
HC_intros_7: fixes F :: "'b formula"   and H :: "'b formula"   and G :: "'b formula" shows "AX10 \<turnstile>\<^sub>H (F \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> (G \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> (F \<^bold>\<or> G) \<^bold>\<rightarrow> H"
append_assoc: fixes xs :: "'d list"   and ys :: "'d list"   and zs :: "'d list" shows "(xs @ ys) @ zs = xs @ ys @ zs"
limitNonU: fixes t :: "'a"   and n :: "nat" assumes "t \<in> S - U" shows "q n t \<le> (1 / (real k * \<delta>)) ^ n"
exhaust: fixes y :: "LambdaType.type" obtains x1 :: "nat" where "y = Atom x1"   | x21 :: "LambdaType.type" and x22 :: "LambdaType.type" where "y = x21 \<Rightarrow> x22"
Extend_def: fixes S :: "(nat, 'b) form set"   and C :: "(nat, 'b) form set set"   and f :: "nat \<Rightarrow> (nat, 'b) form" shows "Extend S C f = \<Union> (range (extend S C f))"
eq_mrexp3_eq: fixes r :: "'b mrexp3"   and s :: "'b mrexp3" assumes "ok3 r"   and "ok3 s" shows "eq_mrexp3 r s = (r = s)"
Substable_intro_box: fixes cond :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<psi> :: "'a \<Rightarrow> \<o>" assumes "Substable cond \<psi>" shows "Substable cond (\<lambda>\<phi>. \<^bold>\<box>\<psi> \<phi>)"
UN_extend_simps_6: fixes A :: "'d \<Rightarrow> 'c set"   and C :: "'d set"   and B :: "'c set" shows "\<Union> (A ` C) - B = (\<Union>x\<in>C. A x - B)"
multiset_inter_count: fixes A :: "'b multiset"   and B :: "'b multiset"   and x :: "'b" shows "count (A \<inter># B) x = min (count A x) (count B x)"
dot_def: fixes a :: "(nat \<Rightarrow> real) \<times> nat"   and b :: "(nat \<Rightarrow> real) \<times> nat" shows "a \<cdot> b = (\<Sum>j = 1..vlen a. a\<^bsub>j\<^esub> * b\<^bsub>j\<^esub>)"
abel_semigroup_axioms: shows "abel_semigroup (\<^bold>*)"
vdiff_def: fixes A :: "V"   and B :: "V" shows "A -\<^sub>\<circ> B = set {x. x \<in>\<^sub>\<circ> A \<and> x \<notin>\<^sub>\<circ> B}"
simps_4: fixes l :: "('b \<times> nat) option"   and t :: "'b" shows "lock_action_ok l t ReleaseAcquire = True"
WhileTrue: fixes b :: "bexp"   and s1 :: "char list \<Rightarrow> int option"   and c :: "com"   and x :: "nat"   and s2 :: "char list \<Rightarrow> int option"   and y :: "nat"   and s3 :: "char list \<Rightarrow> int option"   and z :: "nat" assumes "pbval b s1"   and "vars b \<subseteq> dom s1"   and "(c, s1) \<Rightarrow>\<^sub>A x \<Down> s2"   and "(WHILE b DO c, s2) \<Rightarrow>\<^sub>A y \<Down> s3"   and "1 + x + y = z" shows "(WHILE b DO c, s1) \<Rightarrow>\<^sub>A z \<Down> s3"
ne_matchers_imp_empty: fixes \<sigma> :: "'d \<Rightarrow> ('e, 'f) Term.term option"   and P :: "(('e, 'd) Term.term \<times> ('e, 'f) Term.term) multiset" assumes "matchers_map \<sigma> \<inter> matchers (set_mset P) \<noteq> {}" shows "\<exists>\<sigma>'. ((P, \<sigma>), {#}, \<sigma>') \<in> matchrel\<^sup>*"
Always_eq_invariant_reachable: fixes A :: "'b set" shows "Always A = {F. F \<in> invariant (reachable F \<inter> A)}"
enum_idx_def: fixes x :: "'c" shows "enum_idx x = index_of x enum_class.enum"
mgf_ps: shows "G,A\<turnstile>{=:n} In3 ps\<succ> {G\<rightarrow>}"
less: fixes b :: "int"   and a :: "int" assumes "nat \<bar>b\<bar> < nat \<bar>ba__\<bar>" shows "gcd (of_int a) (of_int b) = of_int (gcd a b)"
simps_22: fixes v :: "'a rexp"   and vb :: "'a" shows "local.le_rexp (Star v) (Atom vb) = False"
zero_imp_le_mult: fixes y :: "V"   and x :: "V" assumes "0 \<in> elts y" shows "x \<le> x * y"
lucas_lehmer_distrib_left: fixes m :: "nat"   and z :: "nat \<times> nat"   and x :: "nat \<times> nat"   and y :: "nat \<times> nat" assumes "1 < m" shows "lucas_lehmer_mult m z (lucas_lehmer_add m x y) = lucas_lehmer_add m (lucas_lehmer_mult m z x) (lucas_lehmer_mult m z y)"
ennreal_top_neq_of_nat: fixes i :: "nat" shows "top \<noteq> of_nat i"
Quotient3_refl2: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c"   and r :: "'c"   and s :: "'c" assumes "Quotient3 R Abs Rep"   and "R r s" shows "R s s"
add3: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ccw (0::'a) a b \<and> ccw (0::'a) a c" shows "ccw (0::'a) a (b + c)"
m1: fixes x :: "'av"   and y :: "'av" assumes "x \<le> y" shows "m y \<le> m x"
c: shows "\<forall>col_index\<in>J. col_index < m"
thetaSeqZOD_def: shows "thetaSeqZOD \<equiv> {(c1 ;; c2, d1 ;; d2) |c1 c2 d1 d2. c1 \<approx>01 d1 \<and> discr c2 \<and> discr d2}"
simps_2: fixes v :: "nat"   and s :: "nat" shows "encode_Op (Inc v s) = c_pair 1 (c_pair v s)"
components_are_iso: fixes a :: "'c \<times> 'c" assumes "C.VV.ide a" shows "D.iso (\<Phi>\<^sub>G a)"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "P 0"   and "\<And>v. (\<And>x. P (Suc v div 2)) \<Longrightarrow> P (Suc v)" shows "P a0"
interior_eq: fixes S :: "'a set" shows "(interior S = S) = open S"
in_set_conv_nth: fixes x :: "'f"   and xs :: "'f list" shows "(x \<in> set xs) = (\<exists>i<length xs. xs ! i = x)"
countably_additive_def: fixes M :: "'c set set"   and f :: "'c set \<Rightarrow> ennreal" shows "countably_additive M f = (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> \<Union> (range A) \<in> M \<longrightarrow> (\<Sum>i. f (A i)) = f (\<Union> (range A)))"
stutter_sampler_comp: fixes f :: "nat \<Rightarrow> nat"   and \<sigma> :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> nat" assumes "stutter_sampler f \<sigma>"   and "stutter_sampler g (\<sigma> \<circ> f)" shows "stutter_sampler (f \<circ> g) \<sigma>"
assms_2: shows "\<bar>n\<bar> \<noteq> 1"
b_acc_subset_\<delta>_statesI: fixes x :: "'a"   and \<delta> :: "('a, 'b) ta_rule set" assumes "x \<in> b_accessible \<delta>" shows "x \<in> \<delta>_states \<delta>"
isCont_arsinh: fixes x :: "real" shows "isCont arsinh x"
linorder_rank_singleton: fixes R :: "('b \<times> 'b) set"   and y :: "'b"   and x :: "'b" shows "linorder_rank R {y} x = (if x \<noteq> y \<and> (y, x) \<in> R then 1 else 0)"
chart_basis_continuous: fixes b :: "'b" shows "continuous_on (Collect (chart_basis_domainP b)) (chart_basis b)"
bsok_simps_14: fixes e :: "'a expr1"   and D :: "String.literal"   and F :: "String.literal"   and e' :: "'a expr1"   and e'' :: "'a expr1"   and n :: "nat" shows "bsok (e\<bullet>compareAndSwap(D\<bullet>F, e', e'')) n = (bsok e n \<and> bsok e' n \<and> bsok e'' n)"
CALL: fixes pn :: "'a" assumes "wellformed \<Gamma>"   and "P (\<Gamma> pn)" shows "P (call(pn))"
bind_bind: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'b set"   and C :: "'b \<Rightarrow> 'a set" shows "Set.bind (Set.bind A B) C = Set.bind A (\<lambda>x. Set.bind (B x) C)"
simps_2: fixes kstep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and output_f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and s0 :: "'a"   and current :: "'a \<Rightarrow> 'd"   and cswitch :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and kprecondition :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and realistic_execution :: "'b list list \<Rightarrow> bool"   and control :: "'a \<Rightarrow> 'd \<Rightarrow> 'b list list \<Rightarrow> 'b option \<times> 'b list list \<times> 'a"   and kinvolved :: "'b \<Rightarrow> 'd set"   and ifp :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and vpeq :: "'d \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and AS_set :: "'b list set"   and invariant :: "'a \<Rightarrow> bool"   and AS_precondition :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and aborting :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and waiting :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and interrupt :: "nat \<Rightarrow> bool"   and n :: "nat"   and s :: "'a"   and execs :: "'d \<Rightarrow> 'b list list" assumes "Interruptible_Separation_Kernel kstep output_f s0 current cswitch kprecondition realistic_execution control kinvolved ifp vpeq AS_set invariant AS_precondition aborting waiting"   and "interrupt (Suc n)" shows "Interruptible_Separation_Kernel.run_total kstep current cswitch interrupt control (Suc n) s execs = Interruptible_Separation_Kernel.run_total kstep current cswitch interrupt control n (cswitch (Suc n) s) execs"
row_space_row_echelon_eq_phull: fixes ps :: "('d \<Rightarrow>\<^sub>0 'e) list"   and ts :: "'d list" assumes "Keys (set ps) \<subseteq> set ts" shows "row_to_poly ts ` row_space (row_echelon (polys_to_mat ts ps)) = phull (set ps)"
env_restr_below_subset: fixes S :: "'a set"   and S' :: "'a set"   and m1 :: "'a \<Rightarrow> 'b"   and m2 :: "'a \<Rightarrow> 'b" assumes "S \<subseteq> S'"   and "m1 f|` S' \<sqsubseteq> m2 f|` S'" shows "m1 f|` S \<sqsubseteq> m2 f|` S"
induct: fixes x1 :: "trm"   and x2 :: "coname"   and P :: "trm \<Rightarrow> coname \<Rightarrow> bool" assumes "fic x1 x2"   and "\<And>x a. P (Ax x a) a"   and "\<And>a M x. a \<sharp> M \<Longrightarrow> P (NotR x.M a) a"   and "\<And>c a M b N. \<lbrakk>c \<sharp> [a].M; c \<sharp> [b].N\<rbrakk> \<Longrightarrow> P (AndR <a>.M <b>.N c) c"   and "\<And>b a M. b \<sharp> [a].M \<Longrightarrow> P (OrR1 <a>.M b) b"   and "\<And>b a M. b \<sharp> [a].M \<Longrightarrow> P (OrR2 <a>.M b) b"   and "\<And>b a M x. b \<sharp> [a].M \<Longrightarrow> P (ImpR x.<a>.M b) b" shows "P x1 x2"
list_ex_list: fixes A :: "'a set"   and \<phi> :: "'a \<Rightarrow> bool" assumes "finite A" shows "list_ex \<phi> (list A) = (\<exists>a\<in>A. \<phi> a)"
falseE: fixes G :: "'b triple set"   and c :: "com"   and Q :: "'b assn" shows "G|-{\<lambda>Z s. False}. c .{Q}"
trivial_LCoset: fixes H :: "'g set" assumes "order.greater_eq G H" shows "H = LCoset_rel H `` {0::'g}"
T2: shows "\<forall>x\<in>carrier. \<forall>y\<in>carrier. x \<noteq> y \<longrightarrow> (\<exists>u\<in>nhds x. \<exists>v\<in>nhds y. u \<inter> v = {})"
g'_def: shows "g' \<equiv> \<lambda>x. \<Sum>i. f' i x"
transp_combine_per_transp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool" assumes "transp R" shows "transp (combine_per P R)"
open_imp_ENR: fixes S :: "'a set" assumes "open S" shows "ENR S"
lm039: fixes a :: "('c \<times> 'd set) set"   and G :: "'d set"   and N :: "'c set" assumes "a \<in> allAllocations N G"   and "finite G"   and "G \<noteq> {}" shows "finite (pseudoAllocation a)"
braun_height_node_eq: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" assumes "braun \<langle>l, x, r\<rangle>" shows "height \<langle>l, x, r\<rangle> = Suc (height l)"
greatest_le: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and A :: "'a set"   and a :: "'a" assumes "greatest L x A"   and "a \<in> A" shows "le L a x"
IP: fixes \<sigma> :: "'c"   and k :: "'d"   and it :: "'d set"   and v :: "'a" assumes "c \<sigma>"   and "k \<in> it"   and "invar v"   and "m k = Some (\<alpha> v)"   and "it \<subseteq> dom m"   and "I it \<sigma>" shows "I (it - {k}) (f (k, v) \<sigma>)"
avl_split_max: fixes t :: "('b \<times> bal) tree"   and t' :: "('b \<times> bal) tree"   and a :: "'b" assumes "split_max t = (t', a)"   and "avl t"   and "t \<noteq> \<langle>\<rangle>" shows "avl t' \<and> height t = height t' + (if decr t t' then 1 else 0)"
map\<^sub>0_simps_2: fixes a :: "'b" assumes "B.obj a" shows "src\<^sub>C (\<tau>\<^sub>0 a) = \<tau>.F.map\<^sub>0 a"
61158: fixes x :: "'a"   and y :: "'a" shows "- - (- x \<sqinter> y) = - x \<sqinter> - - y"
left_adjoints_compose: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and f' :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.is_left_adjoint f"   and "Span.is_left_adjoint f'"   and "Span.src f = Span.trg f'" shows "Span.is_left_adjoint (f \<circ> f')"
POS_cast: shows "sint (SCAST(32 \<rightarrow> 64) POS_INF) = sint POS_INF"
simps_2: fixes e :: "('f, 'g, 'h) env"   and xs :: "'h list" shows "??.Nested_Environment.lookup_option (Some e) xs = lookup e xs"
assms: shows "initmissing (netgmap sr s) = (\<sigma>, \<zeta>)"
mstep_def: fixes tdb :: "(char list \<times> 'b list) set \<times> nat"   and st :: "'b mstate" shows "mstep tdb st = (let (xs, m) = meval (mstate_n st) (snd tdb) (fst tdb) (mstate_m st) in (\<Union> (set (map (\<lambda>(i, X). Pair i ` X) (List.enumerate (mstate_i st) xs))), \<lparr>mstate_i = mstate_i st + length xs, mstate_m = m, mstate_n = mstate_n st\<rparr>))"
steiner_system_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and grouping :: "int" shows "steiner_system point_set block_collection u_block_size grouping \<equiv> t_design point_set block_collection u_block_size grouping 1"
Quotient_rel_abs: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and T :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'a"   and s :: "'a" assumes "Quotient R Abs Rep T"   and "R r s" shows "Abs r = Abs s"
surjective: fixes r :: "'a session_scheme" shows "r = \<lparr>NonceS = NonceS r, IntMapK = IntMapK r, ExtMapK = ExtMapK r, IntAgrK = IntAgrK r, ExtAgrK = ExtAgrK r, \<dots> = more r\<rparr>"
dim_eq_card_independent: fixes B :: "'b set" assumes "local.independent B" shows "local.dim B = card B"
uac: shows "iso_tuple_update_accessor_cong_assist upd ac"
simps_1: fixes cfg1 :: "('p, 'v, 's) configuration"   and p :: "'p"   and v :: "bool"   and cfg2 :: "('p, 'v, 's) configuration" shows "cfg1 \<turnstile> <p, inM v> \<mapsto> cfg2 = ((\<forall>s. (s = p \<longrightarrow> states cfg2 p = trans p (states cfg1 p) (Bool v)) \<and> (s \<noteq> p \<longrightarrow> states cfg2 s = states cfg1 s)) \<and> enabled cfg1 <p, inM v> \<and> msgs cfg2 = sends p (states cfg1 p) (Bool v) \<union># msgs cfg1 -# <p, inM v>)"
module_ow_axioms_def: fixes U\<^sub>M :: "'d set"   and plus\<^sub>M :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd" shows "module_ow_axioms U\<^sub>M plus\<^sub>M scale \<equiv> ((\<forall>x a. x \<in> U\<^sub>M \<longrightarrow> scale a x \<in> U\<^sub>M) \<and> (\<forall>x y a. x \<in> U\<^sub>M \<longrightarrow> y \<in> U\<^sub>M \<longrightarrow> scale a (plus\<^sub>M x y) = plus\<^sub>M (scale a x) (scale a y))) \<and> (\<forall>x a b. x \<in> U\<^sub>M \<longrightarrow> scale (a + b) x = plus\<^sub>M (scale a x) (scale b x)) \<and> (\<forall>x a b. x \<in> U\<^sub>M \<longrightarrow> scale a (scale b x) = scale (a * b) x) \<and> (\<forall>x. x \<in> U\<^sub>M \<longrightarrow> scale (1::'e) x = x)"
fresh_12: fixes a :: "atom"   and "term" :: "Syntax.term" shows "a \<sharp> Prj2 term = a \<sharp> term"
P_sup_iterate: fixes P :: "'s stream \<Rightarrow> bool"   and s :: "'s" assumes "Measurable.pred St P" shows "P_sup s P = (\<Squnion>D\<in>K s. \<integral>\<^sup>+ t. P_sup t (\<lambda>\<omega>. P (t ## \<omega>)) \<partial>measure_pmf D)"
fls_base_factor_to_fps_X: shows "fls_base_factor_to_fps fls_X = 1"
bangRelBisimE: fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "(P, Q) \<in> bangRel Rel"   and "\<And>P Q. (P, Q) \<in> Rel \<Longrightarrow> (Q, P) \<in> Rel" shows "(Q, P) \<in> bangRel Rel"
asEnv_pickE: fixes rho :: "'d \<Rightarrow> 'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm set option"   and xs :: "'d"   and x :: "'e" assumes "goodEnv rho" shows "asEnv (pickE rho) xs x = rho xs x"
dghm_ArrMap_vdomain: shows "\<D>\<^sub>\<circ> (\<phi> i\<lparr>ArrMap\<rparr>) = \<CC>\<lparr>Arr\<rparr>"
identity_arrow_of_spans_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" shows "identity_arrow_of_spans C \<mu> \<equiv> arrow_of_spans C \<mu> \<and> identity_arrow_of_spans_axioms C \<mu>"
tree_recurse_simps_3: fixes l :: "'b \<Rightarrow> 'b"   and r :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "right (tree_recurse l r x) = map_tree r (tree_recurse l r x)"
snoc_1: fixes c' :: "('p, 's, 'm) configuration" assumes "trace c t c'"   and "\<And>c. P c [] c"   and "\<And>c t c' ev c''. \<lbrakk>trace c t c'; P c t c'; c' \<turnstile> ev \<mapsto> c''\<rbrakk> \<Longrightarrow> P c (t @ [ev]) c''" shows "P c t c'"
Lambert_W_less_iff: fixes x :: "real"   and y :: "real" assumes "- exp (- 1) \<le> x"   and "- exp (- 1) \<le> y" shows "(Lambert_W x < Lambert_W y) = (x < y)"
axioms_1: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set" assumes "rational_preference carrier relation" shows "preference carrier relation"
cases: fixes a1 :: "'d list"   and a2 :: "'d list" assumes "pcancel1 a1 a2" obtains xss :: "'d list list" and yss :: "'d list list" where "a1 = concat xss" and "a2 = concat yss" and "list_all2 (\<lambda>xs ys. xs = ys \<or> xs = ys @ ys) xss yss"
int_validate_cl_with_pos_max: fixes C :: "'at Literal set"   and A :: "'at"   and S :: "'at Literal set set" assumes "strictly_maximal_literal C (Pos A)"   and "C \<in> S" shows "canonic_int_ordered S \<Turnstile> C"
that: fixes a :: "int" assumes "coprime a (int (bernoulli_denom m))"   and "bernoulli m = real_of_int a / real (bernoulli_denom m)" shows "thesis"
braking_movement_def: fixes a :: "real"   and v :: "real" shows "braking_movement a v \<equiv> a < 0 \<and> 0 \<le> v"
Least_strict_mono: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "strict_mono f"   and "\<exists>x\<in>S. \<forall>y\<in>S. x \<le> y" shows "(LEAST y. y \<in> f ` S) = f (LEAST x. x \<in> S)"
Units_bij_betwI: fixes \<alpha> :: "'c \<Rightarrow> 'c"   and S :: "'c set" assumes "\<alpha> \<in> monoid.Units (S \<rightarrow>\<^sub>E S) (compose S) (identity S)" shows "bij_betw \<alpha> S S"
simps_3: fixes uv :: "'b list"   and zs :: "'b list" shows "l_sort_inv_1 ([], uv, zs) = l_sorted zs"
gt_ex: fixes x :: "'b" shows "\<exists>y. x < y"
walk_middle_edge: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list"   and v :: "'c"   and w :: "'c"   and ys :: "'c list" assumes "Graph G"   and "Graph.walk G (xs @ v # w # ys)" shows "v \<rightarrow>\<^bsub>G\<^esub> w"
hom_frag_add: fixes h :: "('b \<Rightarrow>\<^sub>0 int) \<Rightarrow> 'c \<Rightarrow>\<^sub>0 int"   and S :: "'b set"   and T :: "'c set"   and a :: "'b \<Rightarrow>\<^sub>0 int"   and b :: "'b \<Rightarrow>\<^sub>0 int" assumes "h \<in> hom (free_Abelian_group S) (free_Abelian_group T)"   and "Poly_Mapping.keys a \<subseteq> S"   and "Poly_Mapping.keys b \<subseteq> S" shows "h (a + b) = h a + h b"
Suc_not_Zero: fixes m :: "nat" shows "Suc m \<noteq> 0"
HPhase1or2ReadThen_Inv6: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase1or2ReadThen s s' p d q"   and "HInv6 s"   and "Inv2c s" shows "HInv6 s'"
nt: shows "nt = \<tau> \<sigma> (i + length (fst st))"
consequence_relation_def: fixes Bot :: "'a set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" shows "consequence_relation Bot entails \<equiv> (Bot \<noteq> {} \<and> (\<forall>B N1. B \<in> Bot \<longrightarrow> entails {B} N1)) \<and> (\<forall>N2 N1. N2 \<subseteq> N1 \<longrightarrow> entails N1 N2) \<and> (\<forall>N2 N1. (\<forall>C\<in>N2. entails N1 {C}) \<longrightarrow> entails N1 N2) \<and> (\<forall>N1 N2 N3. entails N1 N2 \<longrightarrow> entails N2 N3 \<longrightarrow> entails N1 N3)"
S: shows "{(a, b). a \<subseteq> A \<and> card a = n \<and> b \<subseteq> f a \<and> card b = g a} = (SIGMA a:{a. a \<subseteq> A \<and> card a = n}. {b. b \<subseteq> f a \<and> card b = g a})"
coprime_id_is_unit: fixes a :: "'a" shows "comm_monoid_mult_class.coprime a a = is_unit a"
SBReadBuffered: fixes sb :: "'c memref list"   and a :: "nat"   and v :: "nat"   and volatile :: "bool"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "'d"   and \<O> :: "'e"   and \<R> :: "'f"   and \<S> :: "'g" assumes "buffered_val sb a = Some v" shows "(Read volatile a t # is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b (is, \<theta>(t \<mapsto> v), sb, m, \<D>, \<O>, \<R>, \<S>)"
rtrancl_NS_O_S_1: shows "NS\<^sup>* O S = S"
simps_4: fixes v :: "'c"   and va :: "'c"   and vb :: "'c Tree"   and vc :: "'c Tree"   and vd :: "'c"   and ve :: "'c Tree"   and vf :: "'c Tree" shows "removeLeaf (T v (T va vb vc) (T vd ve vf)) = (fst (removeLeaf (T va vb vc)), T v (snd (removeLeaf (T va vb vc))) (T vd ve vf))"
exhaust: fixes y :: "Enum.finite_2" obtains "y = a\<^sub>1"   | "y = a\<^sub>2"
elims: fixes x :: "Unboxed.type"   and y :: "('b, 'ubx1, 'ubx2) unboxed \<Rightarrow> 'dyn option" assumes "cast_and_box x = y" obtains "x = Ubx1" and "y = map_option box_ubx1 \<circ> cast_Ubx1"   | "x = Ubx2" and "y = map_option box_ubx2 \<circ> cast_Ubx2"
inf_step_table_imp_inf_step: fixes s :: "'s2"   and stls :: "('s2 \<times> 'tl2 \<times> 's2) llist" assumes "s -2-stls\<rightarrow>*t \<infinity>" shows "s -2-lmap (fst \<circ> snd) stls\<rightarrow>* \<infinity>"
Upair: fixes x :: "ZF"   and a :: "ZF"   and b :: "ZF" shows "Elem x (Upair a b) = (x = a \<or> x = b)"
simps_1: fixes M :: "(('fun, 'b, 'sets) prot_fun, 'c) Term.term list"   and TI :: "('sets set \<times> 'sets set) list"   and x :: "'c" shows "intruder_synth_mod_timpls M TI (Var x) = List.member M (Var x)"
ordering_def: fixes A :: "ordinal set" shows "OrdinalVeblen.ordering A = ordinal_rec (LEAST z. z \<in> A) (\<lambda>p x. LEAST z. z \<in> A \<and> x < z)"
SN_on_trancl: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "SN_on r A" shows "SN_on (r\<^sup>+) A"
le_supI: fixes a :: "'b multiset"   and x :: "'b multiset"   and b :: "'b multiset" assumes "a \<subseteq># x"   and "b \<subseteq># x" shows "a \<union># b \<subseteq># x"
l2_inv7_step4: fixes Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and Ni :: "msg"   and gnx :: "msg" shows "{l2_inv7 \<inter> l2_inv1 \<inter> l2_inv5 \<inter> l2_inv6 \<inter> l2_inv6'} l2_step4 Rb A B Ni gnx {> l2_inv7}"
alpha_l_real: fixes l :: "nat" shows "alpha_l l \<in> \<real>"
rhs_def: shows "rhs \<equiv> let n = CARD('a) in if n = 0 then False else let xs' = remdups xs; ys' = remdups ys in length xs' + length ys' = n \<and> (\<forall>x\<in>set xs'. x \<notin> set ys') \<and> (\<forall>y\<in>set ys'. y \<notin> set xs')"
columns_Gram_Schmidt_matrix_rw: fixes A :: "(('d, 'f) vec, 'e) vec" shows "{column i (Gram_Schmidt_matrix A) |i. i \<in> UNIV} = {column i (Gram_Schmidt_upt_k A (ncols A - 1)) |i. mod_type_class.to_nat i \<le> ncols A - 1}"
timpl_closure'_timpls_rtrancl_supset: fixes c :: "('g set \<times> 'g set) set" shows "timpl_closure' c \<subseteq> timpl_closure' (c\<^sup>*)"
simps_1: fixes \<sigma> :: "char list \<Rightarrow> nat option"   and c :: "nat" shows "eval \<sigma> (Const c) = Some c"
raw_db_subst_def: fixes i :: "nat"   and x :: "'a" shows "raw_db_subst i x = (\<lambda>y. Hd (if y = x then Sym (db i) else Var y))"
commutative: fixes x :: "'a"   and y :: "'a" assumes "x \<in> G"   and "y \<in> G" shows "x \<cdot> y = y \<cdot> x"
B: shows "g2 * h2 \<in> o[F](g1 * h1)"
k_def: shows "k = Max {the (threshold \<psi> w \<G>) |\<psi>. G \<psi> \<in> \<G>}"
irreducibleE: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c" assumes "irreducible G a" obtains "a \<notin> Units G" and "\<forall>b. b \<in> carrier G \<and> properfactor G b a \<longrightarrow> b \<in> Units G"
class_les_irrefl: fixes c :: "String.literal" shows "\<not> class_les cs c c"