hcju/isabelleps huggingface dataset:
key1_not_currk: fixes s :: "state"   and r :: "room"   and k' :: "key"   and g :: "guest" assumes "s \<in> reach" shows "(currk s r, k') \<notin> cards s g"
scaleR_prod_def: fixes r :: "real"   and A :: "'a \<times> 'b" shows "r *\<^sub>R A = (r *\<^sub>R fst A, r *\<^sub>R snd A)"
r_pdec2_def: shows "r_pdec2 \<equiv> Cn 1 r_sub [r_maxlt, r_pdec1]"
is_TNil_code: fixes xs :: "unit \<Rightarrow> 'a \<times> ('a, 'b) tllist + 'b" shows "is_TNil (Lazy_tllist xs) = (case xs () of Inl x \<Rightarrow> False | Inr x \<Rightarrow> True)"
trans: fixes b :: "'e"   and a :: "'e"   and c :: "'e" assumes "b \<le> a"   and "c \<le> b" shows "c \<le> a"
rHom_0_0: fixes A :: "('e, 'f) Ring_scheme"   and R :: "('g, 'h) Ring_scheme"   and f :: "'e \<Rightarrow> 'g" assumes "Ring A"   and "Ring R"   and "f \<in> rHom A R" shows "f \<zero>\<^bsub>A\<^esub> = \<zero>\<^bsub>R\<^esub>"
simps_2: fixes n :: "nat"   and l :: "nat" shows "nxt Del (n, l) = (n - 1, if f1 * real l \<le> real (n - 1) then l else if int l0 \<le> \<lfloor>real l / c\<rfloor> then nat \<lfloor>real l / c\<rfloor> else l)"
assms_2: shows "u\<^sub>1 @ v\<^sub>1 =\<^sub>F u\<^sub>2 @ v\<^sub>2"
poly_pCons: fixes a :: "'a"   and p :: "'a poly"   and x :: "'a" shows "poly (pCons a p) x = a + x * poly p x"
N_4: fixes vs :: "nat \<Rightarrow> nat" shows "\<exists>vs'. ((Suc 0, vs), 0, vs') \<in> (step N)\<^sup>* \<and> vs' 0 = vs 0 \<and> vs' 1 = vs 1 + 1 \<and> vs' 2 = h (vs 0) (vs 1)"
simps: fixes A :: "'c ring"   and B :: "'c ring" shows "A \<lesssim> B = (id \<in> ring_hom A B)"
cases: fixes a :: "('a, 'b, 'c) expr" assumes "redex a" obtains x :: "'c" and e :: "('a, 'b, 'c) expr" and v :: "('a, 'b, 'c) val" where "a = Apply (VE (Lambda x e__)) (VE v__)"   | e1 :: "('a, 'b, 'c) expr" and e2 :: "('a, 'b, 'c) expr" where "a = Ite (VE (CV T)) e1__ e2__"   | e1 :: "('a, 'b, 'c) expr" and e2 :: "('a, 'b, 'c) expr" where "a = Ite (VE (CV F)) e1 e2"   | v :: "('a, 'b, 'c) val" where "a = Ref (VE va__)"   | l :: "'b" where "a = Read (VE (Loc l__))"   | l :: "'b" and v :: "('a, 'b, 'c) val" where "a = Assign (VE (Loc l)) (VE v)"   | e :: "('a, 'b, 'c) expr" where "a = Rfork e"   | r :: "'a" where "a = Rjoin (VE (Rid r))"
if_weak_cong: fixes b :: "bool"   and c :: "bool"   and x :: "'e"   and y :: "'e" assumes "b = c" shows "(if b then x else y) = (if c then x else y)"
a_asymptotics: shows "(\<lambda>x. dirichlet_prod' mangoldt (\<lambda>x. real_of_int \<lfloor>x\<rfloor>) x - x * ln x) \<in> O(\<lambda>x. x)"
comp_swap_lassocr: fixes conv1 :: "('i, 'l, 'p, 's) converter"   and conv2 :: "('j, 'm, 'o, 'r) converter"   and conv3 :: "('k, 'n, 'q, 't) converter" shows "(conv1 |\<^sub>= conv2 |\<^sub>= conv3) \<odot> swap_lassocr = swap_lassocr \<odot> (conv2 |\<^sub>= conv1 |\<^sub>= conv3)"
simps_4: fixes x2 :: "'b error_result"   and x1 :: "'a" shows "Rerr x2 \<noteq> Rval x1"
uniformly_convergent_eval_fds_aux': fixes f :: "'a fds"   and s0 :: "'a"   and B :: "'a set" assumes "fds_converges f s0"   and "compact B"   and "\<And>z. z \<in> B \<Longrightarrow> s0 \<bullet> (1::'a) < z \<bullet> (1::'a)" shows "uniformly_convergent_on B (\<lambda>N z. \<Sum>n\<le>N. fds_nth f n / nat_power n z)"
while_proof': fixes Q :: "('Q, unit) HashMap.hashmap \<times> 'Q list \<times> (('Q, 'L) ta_rule, nat) HashMap.hashmap \<Rightarrow> bool" assumes "\<forall>s. s \<in> wa_invar (det_wa_wa (brc_det_algo (build_rqrm \<delta>) \<delta>)) \<and> s \<notin> wa_cond (det_wa_wa (brc_det_algo (build_rqrm \<delta>) \<delta>)) \<longrightarrow> Q s" shows "Q loop"
spec_ops_sorted: fixes ops :: "('b \<times> 'c) list"   and deps :: "'c \<Rightarrow> 'b set" assumes "spec_ops ops deps" shows "sorted (map fst ops)"
h4: shows "satisfy_equation_R p (qs1 @ qs2) (get_subsets_R (snd (combine_systems_R p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2)))) (get_signs_R (snd (combine_systems_R p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2))))"
isOK_forallM: fixes f :: "'a \<Rightarrow> 'b + unit"   and xs :: "'a list" shows "isOK (forallM f xs) = (\<forall>x\<in>set xs. isOK (f x))"
red_n_to_reds: fixes c :: "com"   and s :: "char list \<Rightarrow> val option"   and n :: "nat"   and c' :: "com"   and s' :: "char list \<Rightarrow> val option" assumes "\<langle>c,s\<rangle> \<rightarrow>\<^bsup>n\<^esup> \<langle>c',s'\<rangle>" shows "\<langle>c,s\<rangle> \<rightarrow>* \<langle>c',s'\<rangle>"
subset: shows "MI.additive.Class ` B \<subseteq> MI.additive.Partition"
a2factor: shows "a\<^sup>2 = (c - b) * (c + b)"
resInputFreeTrans: fixes x :: "name"   and a :: "name"   and y :: "name"   and P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi" assumes "<\<nu>x>a<y>.P \<longmapsto> \<alpha> \<prec> P'" shows "False"
zero_vec_Suc: fixes n :: "nat" shows "0\<^sub>v (Suc n) = vCons (0::'b) (0\<^sub>v n)"
Spy_not_see_encrypted_key: fixes B :: "agent"   and A :: "agent"   and NA :: "msg"   and K :: "nat"   and NB :: "msg"   and evs :: "event list" assumes "Says Server B \<lbrace>Crypt (shrK A) \<lbrace>NA, Agent A, Agent B, Key K\<rbrace>, Crypt (shrK B) \<lbrace>NB, Agent A, Agent B, Key K\<rbrace>\<rbrace> \<in> set evs"   and "Notes Spy \<lbrace>NA, NB, Key K\<rbrace> \<notin> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> otway" shows "Key K \<notin> analz (knows Spy evs)"
srtc_transitive: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "relation.srtc r a b"   and "relation.srtc r b c" shows "relation.srtc r a c"
perm_eq_iff: fixes f :: "'b perm"   and g :: "'b perm" shows "(f = g) = (\<forall>a. f \<langle>$\<rangle> a = g \<langle>$\<rangle> a)"
\<psi>\<^sub>1\<^sub>0_tensor_is_state: fixes n :: "nat" assumes "1 \<le> n" shows "state n (|Deutsch.zero\<rangle> \<otimes>\<^bsup>n\<^esup>)"
\<Pi>_def: shows "\<Pi> = \<lbrace> p \<rbrace> \<cdot> x \<cdot> \<Pi>'"
less_le_not_le: fixes x :: "nat \<Rightarrow> nat"   and y :: "nat \<Rightarrow> nat" shows "(x \<prec> y) = (x \<lesssim> y \<and> \<not> y \<lesssim> x)"
RedTryFail: fixes s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and D :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and C :: "char list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and V :: "char list"   and e\<^sub>2 :: "char list exp" assumes "hp s a = \<lfloor>(D, fs)\<rfloor>"   and "(D, C) \<notin> (subcls1 P)\<^sup>*" shows "P \<turnstile> \<langle>try Throw a catch(C V) e\<^sub>2,s\<rangle> \<rightarrow> \<langle>Throw a,s\<rangle>"
rep_eq: fixes x :: "uint" shows "Rep_uint8 (uint8_of_uint x) = ucast (Rep_uint x)"
FreeUltrafilterNat_nat_gt_real: fixes u :: "real" shows "\<forall>\<^sub>F n in \<U>. u < real n"
is_natural_transformation: shows "natural_transformation A B F G \<tau>'.map"
multisetUnionLeMono1: fixes r :: "('b \<times> 'b) set"   and a1 :: "'b multiset"   and a2 :: "'b multiset"   and b :: "'b multiset" assumes "trans r"   and "multiset_le a1 a2 r" shows "multiset_le (a1 + b) (a2 + b) r"
flat_below_iff: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.flat below" shows "below x y = (x = pcpo.bottom below \<or> x = y)"
pheap_pass2: fixes h :: "'b tree" assumes "pheap h" shows "pheap (pass\<^sub>2 h)"
notIn_Un_bij_betw3: fixes b :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and A' :: "'c set" assumes "b \<notin> A"   and "f b \<notin> A'" shows "bij_betw f A A' = bij_betw f (A \<union> {b}) (A' \<union> {f b})"
m_red_to_comp: fixes A :: "nat set"   and B :: "nat set" assumes "many_reducible_to A B"   and "computable B" shows "computable A"
UNIF1_singleton_Var_right: fixes x :: "'c"   and t :: "('d, 'c) Term.term" assumes "x \<notin> vars_term t" shows "UNIF1 (subst x t) {#(t, Var x)#} {#}"
antisym_conv: fixes y :: "'b list"   and x :: "'b list" assumes "y \<le>p x" shows "(x \<le>p y) = (x = y)"
path_weightp_def: fixes l1 :: "'loc"   and l2 :: "'loc"   and s :: "'sum" shows "path_weightp l1 l2 s \<equiv> \<exists>xs. path l1 l2 xs \<and> s = sum_weights (map (\<lambda>(s, l, t). l) xs)"
deriv_stirling_integral_real_bound: fixes m :: "nat"   and j :: "nat" assumes "0 < m" shows "(deriv ^^ j) (stirling_integral m) \<in> O(\<lambda>x. 1 / x ^ (m + j))"
recurrent_subset_infty_def: fixes A :: "'a set" shows "recurrent_subset_infty A = A - (\<Union>n. (T ^^ n) -` (A - recurrent_subset A))"
s2': shows "s2' = (ls2', (ts2', m2'), ws2', is2')"
prems: fixes x :: "'a"   and y :: "'a" assumes "(x, y) \<in> set (xy # xys)" shows "y < x"
abs_eq: fixes x :: "'b \<Rightarrow> 'c" assumes "eq_onp (\<lambda>f. finite {x. f x \<noteq> (0::'c)}) x x" shows "keys (Abs_poly_mapping x) = {k. x k \<noteq> (0::'c)}"
id_assn_list: fixes xs :: "'a list"   and ys :: "'a list"   and h :: "heap \<times> nat set" assumes "h \<Turnstile> list_assn id_assn xs ys" shows "xs = ys"
disjE5: fixes A :: "bool"   and B :: "bool"   and C :: "bool"   and D :: "bool"   and E :: "bool" assumes "A \<or> B \<or> C \<or> D \<or> E" obtains "A"   | "B"   | "C"   | "D"   | "E"
br'_invar_initial: fixes \<delta> :: "('d, 'e) ta_rule set" shows "br'_initial \<delta> \<subseteq> br'_invar_add \<delta>"
piecewise_C1_differentiable_diff: fixes f :: "real \<Rightarrow> 'a"   and S :: "real set"   and g :: "real \<Rightarrow> 'a" assumes "f piecewise_C1_differentiable_on S"   and "g piecewise_C1_differentiable_on S" shows "(\<lambda>x. f x - g x) piecewise_C1_differentiable_on S"
lcsTr7: fixes H :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "b \<in> carrier G"   and "\<rho> a \<cdot> b \<in> H" shows "a \<diamondsuit> H \<subseteq> b \<diamondsuit> H"
matrix_inv_unique: fixes A :: "(('a, 'c) vec, 'b) vec"   and B :: "(('a, 'b) vec, 'c) vec" assumes "A ** B = mat (1::'a)"   and "B ** A = mat (1::'a)" shows "matrix_inv A = B"
arc_to_block_to_ends_swap: fixes e :: "'b" shows "{snd (arc_to_ends G e), fst (arc_to_ends G e)} = arc_to_block e"
OPT_zerlegen: fixes init :: "'c list"   and qs :: "'c list" assumes "distinct init"   and "init \<noteq> []"   and "set qs \<subseteq> set init" shows "(\<Sum>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init \<and> x < y}. T\<^sub>p_opt (Lxy init {x, y}) (Lxy qs {x, y})) \<le> T\<^sub>p_opt init qs"
val_Bool_simp2_2: shows "\<lbrakk>FF\<rbrakk> = False"
empty_rr_hlp_def: fixes pm :: "'b prefix_match" shows "empty_rr_hlp pm = routing_rule.make pm default_metric (routing_action.make [] None)"
K2_def: shows "hyp2 \<equiv> {p. conic_sgn p < 0}"
Suc_div_unat_helper: fixes sz :: "nat"   and us :: "nat" assumes "sz < LENGTH('b)"   and "us \<le> sz" shows "2 ^ (sz - us) = Suc (unat ((2 ^ sz - 1) div 2 ^ us))"
ncc_def: fixes X :: "'c set" shows "ncc X = (X \<noteq> {} \<and> compact X \<and> convex X)"
jkbpC_traces_of_length: fixes n :: "nat" shows "{t \<in> jkbpC. tLength t = n} = jkbpC\<^bsub>n\<^esub>"
all_less_two: fixes P :: "nat \<Rightarrow> bool" shows "(\<forall>i<Suc (Suc 0). P i) = (P 0 \<and> P (Suc 0))"
RPSpec_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and p :: "PrIds" shows "RPSpec ch mm rs p \<equiv> Init PInit rs p \<and> \<box>[RNext ch mm rs p]_(rtrner ch!p, rs!p) \<and> WF(RNext ch mm rs p)_(rtrner ch!p, rs!p) \<and> WF(MemReturn ch rs p)_(rtrner ch!p, rs!p)"
stopping_timeD: fixes F :: "'b \<Rightarrow> 'c measure"   and T :: "'c \<Rightarrow> 'b"   and t :: "'b" assumes "stopping_time F T" shows "Measurable.pred (F t) (\<lambda>x. T x \<le> t)"
equal_foldr: assumes "Some (r, m) = foldr (div_int_poly_step q) (coeffs p) (Some (0, 0))" shows "(case (r, m) of (a, b) \<Rightarrow> (of_int_poly a, of_int_poly b)) = foldr (div_rat_poly_step (of_int_poly q)) (map rat_of_int (coeffs p)) (case (0, 0) of (a, b) \<Rightarrow> (of_int_poly a, of_int_poly b))"
l12_22_aux: fixes P :: "'p"   and A :: "'p"   and C :: "'p"   and B :: "'p"   and D :: "'p" assumes "P \<noteq> A"   and "A \<noteq> C"   and "Bet P A C"   and "P A OS B D"   and "B A P CongA D C P" shows "A B Par C D"
disjoint_from_second_factor: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "P x y \<and> \<not> O x (y \<otimes> z)" shows "\<not> O x z"
less_imp_le: fixes x :: "'b word"   and y :: "'b word" assumes "x <s y" shows "x \<le>s y"
elims: fixes x :: "('a, 'b, 'c) unboxed"   and y :: "'a option" assumes "cast_Dyn x = y" obtains d :: "'a" where "x = OpDyn d" and "y = Some d"   | v :: "'b" where "x = OpUbx1 v__" and "y = None"   | v :: "'c" where "x = OpUbx2 v" and "y = None"
unl_pref_imp: fixes F :: "'b \<Rightarrow> bool"   and G :: "'b \<Rightarrow> bool"   and w :: "'b" assumes "|~ F \<longrightarrow> G"   and "F w" shows "G w"
final1_simulation: fixes s1 :: "'s1"   and s2 :: "'s2" assumes "s1 \<approx> s2"   and "final1 s1" shows "\<exists>s2'. s2 -\<tau>2\<rightarrow>* s2' \<and> s1 \<approx> s2' \<and> final2 s2'"
ncc_def: fixes X :: "'b set" shows "ncc X = (X \<noteq> {} \<and> compact X \<and> convex X)"
S_def: fixes h :: "'a"   and e :: "'e" shows "\<Sigma>1.S h e = map_spmf (\<lambda>(a, z). (a, e, z)) (S1_raw h e)"
has_derivative_zero_unique: fixes F :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "((\<lambda>x. 0::'b) has_derivative F) (at x)" shows "F = (\<lambda>h. 0::'b)"
strict_implies_not_eq: fixes b :: "'c multiset"   and a :: "'c multiset" assumes "b \<subset># a" shows "a \<noteq> b"
index_mat_swapcols_1: fixes i :: "nat"   and A :: "'b mat"   and j :: "nat"   and k :: "nat"   and l :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "swapcols k l A $$ (i, j) = (if k = j then A $$ (i, l) else if l = j then A $$ (i, k) else A $$ (i, j))"
inj_monoid_add_hom_axioms: shows "inj_monoid_add_hom (inv hom)"
Nil: fixes r :: "('c \<times> 'd) set" shows "([], []) \<in> listrel r"
square_mat: fixes n :: "nat"   and A :: "complex mat" assumes "gate n A" shows "square_mat A"
dependent_live_vars_ddep_dependent_live_vars: fixes n'' :: "'node"   and V :: "'var"   and as'' :: "'edge list"   and n' :: "'node"   and V' :: "'var"   and as' :: "'edge list"   and as :: "'edge list" assumes "n'' -{V}as''\<rightarrow>\<^sub>d\<^sub>d n'"   and "(V', as', as) \<in> dependent_live_vars n''" shows "(V', as', as @ as'') \<in> dependent_live_vars n'"
random_oracle_def: fixes \<sigma> :: "'c \<Rightarrow> 'a option"   and x :: "'c" shows "random_oracle \<sigma> x = (case \<sigma> x of None \<Rightarrow> p \<bind> (\<lambda>y. return_spmf (y, \<sigma>(x \<mapsto> y))) | Some y \<Rightarrow> return_spmf (y, \<sigma>))"
sub: fixes a :: "'a \<Rightarrow> 'a" assumes "a \<in> M" shows "a \<in> S \<rightarrow>\<^sub>E S"
gets_def: fixes v :: "char list"   and e :: "(char list \<Rightarrow> 'a) \<Rightarrow> 'a" shows "v ::= e = {(s, s(v := e s)) |s. True}"
rel_mono_strong: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "('d, 'f) nondetT"   and y :: "('e, 'g) nondetT"   and R1a :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2a :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "rel_nondetT' R1 R2 x y"   and "\<And>z1 y1. \<lbrakk>z1 \<in> phantom_nondetT x; y1 \<in> phantom_nondetT y; R1 z1 y1\<rbrakk> \<Longrightarrow> R1a z1 y1"   and "\<And>z2 y2. \<lbrakk>z2 \<in> set_nondetT x; y2 \<in> set_nondetT y; R2 z2 y2\<rbrakk> \<Longrightarrow> R2a z2 y2" shows "rel_nondetT' R1a R2a x y"
exec_catch: fixes P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and h :: "'heap"   and xcp :: "'addr"   and pc :: "nat"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and pc' :: "nat"   and d :: "nat"   and stk :: "'addr val list"   and ci :: "('addr, 'heap) check_instr"   and ins :: "'addr instr list"   and t :: "'thread_id"   and loc :: "'addr val list" assumes "match_ex_table P (cname_of h xcp) pc xt = \<lfloor>(pc', d)\<rfloor>"   and "d \<le> length stk" shows "exec_meth ci P ins xt t h (stk, loc, pc, \<lfloor>xcp\<rfloor>) \<lbrace>\<rbrace> h (Addr xcp # drop (length stk - d) stk, loc, pc', None)"
comp_char: fixes g :: "(bool, 'arr list) concrete_category.arr"   and f :: "(bool, 'arr list) concrete_category.arr" shows "g \<cdot> f = (if seq g f then MkArr (Dom f) (Cod g) (Map f @ Map g) else null)"
dx2: fixes \<rho> :: "complex mat" assumes "\<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>" shows "positive (denote x2 \<rho>) \<and> trace (denote x2 \<rho>) \<le> trace \<rho> \<and> denote x2 \<rho> \<in> carrier_mat d d"
bdd_above_norm_f: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_linear f" shows "bdd_above {norm (f x) |x. norm x = 1}"
inv_approx_def: fixes S :: "'a set"   and X :: "'a set"   and r :: "nat" shows "inv_approx S X r = (inv_iv S X \<and> card X \<le> card S * r)"
iAnnot_globalize: fixes is' :: "nat list"   and "is" :: "nat list"   and f :: "'var \<Rightarrow> 'var"   and t :: "('form, 'rule, 'subst, 'var) itnode rose_tree" assumes "is' \<in> it_paths (globalize is f t)" shows "iAnnot (tree_at (globalize is f t) is') = isidx (is @ is')"
bisimilar_simulation_step: fixes P :: "'state"   and Q :: "'state"   and \<alpha> :: "('act, 'pred) S_action"   and P' :: "'state" assumes "P \<sim>\<cdot>\<^sub>S Q"   and "bn \<alpha> \<sharp>* Q"   and "P \<rightarrow>\<^sub>S \<langle>\<alpha>,P'\<rangle>" obtains Q' :: "'state" where "Q \<rightarrow>\<^sub>S \<langle>\<alpha>,Q'\<rangle>" and "P' \<sim>\<cdot>\<^sub>S Q'"
span_breakdown: fixes b :: "'b"   and S :: "'b set"   and a :: "'b" assumes "b \<in> S"   and "a \<in> cspan S" shows "\<exists>k. a - k *\<^sub>C b \<in> cspan (S - {b})"
intros_3: fixes x :: "'c"   and G :: "('c, 'd) monoid_scheme"   and gens :: "'c set" assumes "x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>" shows "inv\<^bsub>G\<^esub> x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>"
rel_endofun_def: shows "rel_endofun \<equiv> (id ---> rep_endofun ---> rep_endofun ---> id) rel_endofun'"
that: fixes b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow> c\<guillemotright>" shows "thesis"
def_homo: fixes x :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'a"   and y :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'b" shows "\<delta> f x y = (d\<^sub>x x and d\<^sub>y y)"
arith_extra_simps_19: fixes a :: "'d" shows "a * (0::'d) = (0::'d)"
elims: fixes x :: "xmldoc"   and y :: "char list \<Rightarrow> char list" assumes "shows_xmldoc x = y" obtains h :: "char list list" and x :: "xml" where "xa__ = XMLDOC h x" and "y = shows_lines h \<circ> shows_nl \<circ> shows x"
insert: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and top :: "'a"   and bot :: "'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and A :: "'a set" assumes "bounded_quasi_semilattice_set f top bot normalize" shows "bounded_quasi_semilattice_set.F f top bot (insert a A) = f a (bounded_quasi_semilattice_set.F f top bot A)"
fps_one_nth: fixes n :: "nat" shows "fps_nth 1 n = (if n = 0 then 1::'a else (0::'a))"
assms: fixes m :: "'a" assumes "invar m" shows "map_iterator (iti m) (\<alpha> m)"
functor_axioms: shows "functor A C (H \<circ> F)"
ennreal_inverse_mult': fixes b :: "ennreal"   and a :: "ennreal" assumes "(0 < b \<or> a < top) \<and> (0 < a \<or> b < top)" shows "inverse (a * b) = inverse a * inverse b"
inner_sum_left: fixes f :: "'c \<Rightarrow> 'a"   and A :: "'c set"   and y :: "'a" shows "inner (sum f A) y = (\<Sum>x\<in>A. inner (f x) y)"
hyps_2: shows "poly_y_x (poly_y_x p) = p"
nat_to_bv_length: fixes n :: "nat"   and l :: "nat" shows "nat_to_bv_length n l = (if length (nat_to_bv n) \<le> l then bv_extend l \<zero> (nat_to_bv n) else [])"
has_int: shows "((\<lambda>x. vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1}) *\<^sub>R (F ((\<gamma> x \<bullet> i) *\<^sub>R i + g (\<gamma> x \<bullet> i)) \<bullet> i)) has_integral integral {\<gamma> 0 \<bullet> i..\<gamma> 1 \<bullet> i} (\<lambda>f_var. F (f_var *\<^sub>R i + g f_var) \<bullet> i)) {0..1}"
simps_5: fixes r :: "'a set"   and b :: "'a set"   and i :: "'a set"   and c :: "'a ann_com" shows "atomics (AnnWhile r b i c) = atomics c"
substT_num1: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and t :: "'b"   and y :: "'a"   and n :: "'b" assumes "Syntax_with_Numerals var trm fmla Var FvarsT substT Fvars subst num"   and "t \<in> trm"   and "y \<in> var"   and "n \<in> num" shows "substT n t y = n"
is_up_def: shows "is_up \<equiv> map_fun (map_fun Word.Word Word.rep) id (\<lambda>_. LENGTH('d) \<le> LENGTH('e))"
Act_eqvt: fixes p :: "perm"   and \<alpha> :: "'c"   and x :: "('a, 'b, 'c) formula" shows "p \<bullet> Act \<alpha> x = Act (p \<bullet> \<alpha>) (p \<bullet> x)"
induct: fixes A :: "('c, 'd) ioa"   and x :: "'c"   and P :: "'c \<Rightarrow> bool" assumes "reachable A x"   and "\<And>s. s \<in> start A \<Longrightarrow> P s"   and "\<And>s a t. \<lbrakk>reachable A s; P s; s \<midarrow>a\<midarrow>A\<longrightarrow> t\<rbrakk> \<Longrightarrow> P t" shows "P x"
stieltjes_gamma_aux2: fixes k :: "nat" shows "(deriv ^^ k) g 1 = fact k * G $ k"
steps_appendD1: fixes xs :: "'a list"   and ys :: "'a list" assumes "steps (xs @ ys)"   and "xs \<noteq> []" shows "steps xs"
special_sg_e: shows "G \<guillemotright> {\<one>} "
fetch_t_wcode_14_Oc: shows "fetch t_wcode_main (14 + length t_twice div 2 + t_fourtimes_len) Oc = (L, Suc 0)"
elims_3: fixes x :: "'a"   and xa :: "'a \<Rightarrow> 'b"   and xb :: "('a \<Rightarrow> 'b option) \<times> ('a \<Rightarrow> 'b option)" assumes "\<not> in_bounds x xa xb" obtains xc :: "'a" and v :: "'a \<Rightarrow> 'b" and lb :: "'a \<Rightarrow> 'b option" and ub :: "'a \<Rightarrow> 'b option" where "x = xc" and "xa = v" and "xb = (lb, ub)" and "\<not> (v xc \<ge>\<^sub>l\<^sub>b lb xc \<and> v xc \<le>\<^sub>u\<^sub>b ub xc)"
opnet_net_ips_net_tree_ips: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and ns :: "'e net_state"   and np :: "nat \<Rightarrow> ((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and p :: "net_tree"   and S :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> 'f node_action \<Rightarrow> bool"   and U :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> bool" assumes "(\<sigma>, ns) \<in> oreachable (opnet np p) S U" shows "net_ips ns = net_tree_ips p"
field_differentiable_within_Arctan: fixes z :: "complex"   and s :: "complex set" assumes "Re z = 0 \<Longrightarrow> \<bar>Im z\<bar> < 1" shows "Arctan field_differentiable at z within s"
assms_2: shows "sys \<Turnstile>\<^bsub>pre\<^esub> Q"
X: fixes y :: "'a" shows "\<exists>x. perm_restrict f (S - A) x = y"
ZFSubsetRangeExplode: fixes X :: "ZF set"   and Y :: "ZF set" assumes "X \<in> range explode"   and "Y \<subseteq> X" shows "Y \<in> range explode"
csj: shows "cs\<^bsup>\<pi>\<^esup> j = cs\<^bsup>\<pi>'\<^esup> j'"
is_expansion_ms_def: shows "is_expansion \<equiv> rec_ms is_expansion_aux"
is_span: shows "C.span \<nu>\<mu>.dom.leg0 \<nu>\<mu>.dom.leg1"
Strongly_Secure_Skip: fixes d :: "'d" shows "[skip] \<approx>\<^bsub>d\<^esub> [skip]"
Can_Nmlize_Can: fixes t :: "'a local.term" assumes "Can t" shows "Can \<^bold>\<lfloor>t\<^bold>\<rfloor>"
maxR_def: fixes x :: "'a"   and y :: "'a" shows "maxR x y = (if (x, y) \<in> r then y else x)"
CompDiag_Diag_Dom: fixes t :: "'c local.term" assumes "Diag t" shows "t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> Dom t = t"
to_nat_plus_one_less_card': fixes a :: "'a" assumes "a + (1::'a) \<noteq> (0::'a)" shows "mod_type_class.to_nat a + 1 < CARD('a)"
h: shows "\<forall>a. (i, a) \<notin> setOf t"