hcju/isabelleps huggingface dataset:
Send: fixes M\<^sub>0 :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S :: "('fun, 'var) extstrand_step list"   and t :: "('fun, 'var) Term.term" assumes "sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> S"   and "ik\<^sub>e\<^sub>s\<^sub>t S \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile>\<^sub>c t \<cdot> \<I>" shows "sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> (S @ [Step (send\<langle>t\<rangle>\<^sub>s\<^sub>t)])"
while_term_hoare_r: fixes p :: "'a upred"   and b :: "'a upred"   and v :: "(nat, 'a) uexpr"   and S :: "'a hrel" assumes "\<And>z. \<lbrace>p \<and> b \<and> v =\<^sub>u \<guillemotleft>z\<guillemotright>\<rbrace> S \<lbrace>p \<and> v <\<^sub>u \<guillemotleft>z\<guillemotright>\<rbrace>\<^sub>u" shows "\<lbrace>p\<rbrace> while\<^sub>\<bottom> b do S od \<lbrace>\<not> b \<and> p\<rbrace>\<^sub>u"
key_rel_def: fixes R :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'f \<times> 'g"   and b :: "'f \<times> 'g" shows "key_rel R a b \<equiv> R (fst a) (fst b)"
odd_two_times_div_two_nat: fixes n :: "nat" assumes "odd n" shows "2 * (n div 2) = n - 1"
powhr_less_cancel: fixes x :: "real star"   and a :: "real star"   and b :: "real star" assumes "x powhr a < x powhr b"   and "1 < x" shows "a < b"
cinner_simps_5: fixes r :: "complex"   and x :: "'b"   and y :: "'b" shows "\<langle>r *\<^sub>C x, y\<rangle> = cnj r * \<langle>x, y\<rangle>"
is_valid_operator_sas_plus_def: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and op :: "('a, 'b) sas_plus_operator" shows "is_valid_operator_sas_plus \<Psi> op \<equiv> let pre = sas_plus_operator.precondition_of op; eff = effect_of op; vs = \<Psi>\<^sub>\<V>\<^sub>+; D = range_of \<Psi> in list_all (\<lambda>(v, a). ListMem v vs) pre \<and> list_all (\<lambda>(v, a). D v \<noteq> None \<and> ListMem a (the (D v))) pre \<and> list_all (\<lambda>(v, a). ListMem v vs) eff \<and> list_all (\<lambda>(v, a). D v \<noteq> None \<and> ListMem a (the (D v))) eff \<and> list_all (\<lambda>(v, a). list_all (\<lambda>(v', a'). v \<noteq> v' \<or> a = a') pre) pre \<and> list_all (\<lambda>(v, a). list_all (\<lambda>(v', a'). v \<noteq> v' \<or> a = a') eff) eff"
ag_add_commute: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier A"   and "b \<in> carrier A" shows "a \<plusminus> b = b \<plusminus> a"
ChiRel_ChiPlus_trans: fixes U :: "'a"   and S :: "'a"   and A :: "('a, 'b, 'c) hierauto"   and T :: "'a" assumes "(U, S) \<in> ChiPlus A"   and "(S, T) \<in> ChiRel A" shows "(U, T) \<in> ChiPlus A"
bangRelSubset: fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set"   and Rel' :: "(pi \<times> pi) set" assumes "(P, Q) \<in> bangRel Rel"   and "\<And>P Q. (P, Q) \<in> Rel \<Longrightarrow> (P, Q) \<in> Rel'" shows "(P, Q) \<in> bangRel Rel'"
comp_inv_arr: fixes f :: "'b"   and g :: "'b" assumes "\<tau>.B.inverse_arrows f g" shows "B g f = \<tau>.B.dom f"
IH_2: fixes xs1 :: "'d list"   and xs2 :: "'d list" assumes "y # ys = map f xs1"   and "ys2 = map f xs2"   and "xs = xs1 @ xs2" shows "thesis"
R: fixes x :: "'a"   and y :: "'b" assumes "(x, y) \<in> set_pmf pq" shows "R x y"
kernel_padr: fixes a :: "'a vec"   and A :: "'a mat"   and nr1 :: "nat"   and nc1 :: "nat"   and B :: "'a mat"   and nc2 :: "nat"   and D :: "'a mat"   and nr2 :: "nat" assumes "a \<in> mat_kernel A"   and "A \<in> carrier_mat nr1 nc1"   and "B \<in> carrier_mat nr1 nc2"   and "D \<in> carrier_mat nr2 nc2" shows "a @\<^sub>v 0\<^sub>v nc2 \<in> mat_kernel (four_block_mat A B (0\<^sub>m nr2 nc1) D)"
ij_le2_2: shows "j < card {i. i < dim_col (submatrix A UNIV J) \<and> i \<in> UNIV}"
vneq: shows "(\<sigma>\<^bsup>k\<^esup>) v \<noteq> (\<sigma>'\<^bsup>k'\<^esup>) v"
game_sem_loop_iterate: fixes I :: "interp"   and \<alpha> :: "game" shows "game_sem I \<alpha>** = (\<lambda>X. X \<union> game_sem I \<alpha> (game_sem I \<alpha>** X))"
lin_data_in': shows "linearize data_in' = linearize data_in @ move"
gtrp2gtr: fixes s :: "'f"   and c :: "'f multiset"   and w :: "'g el_step list"   and s' :: "'f"   and c' :: "'f multiset"   and T :: "(('f \<times> 'f multiset) \<times> 'g \<times> 'f \<times> 'f multiset) set" assumes "((s, c), w, s', c') \<in> trcl (gtrp T)" shows "(add_mset s c, map le_rem_s w, add_mset s' c') \<in> trcl (gtr T)"
init_fin_\<tau>move_simps_6: fixes xm :: "(status \<times> 'x2) \<times> 'm2"   and ta :: "('l, 't, status \<times> 'x2, 'm2, 'w, 'o action) thread_action"   and x' :: "'x2"   and m' :: "'m2" shows "r2.init_fin_\<tau>move xm ta ((Finished, x'), m') = False"
simps_2: fixes l :: "'b tree"   and uu :: "'b"   and r :: "'b tree" shows "lrh \<langle>l, uu, r\<rangle> = rh l r + lrh l"
that: fixes e' :: "nat \<Rightarrow> real" assumes "pdevs_val e X = pdevs_val e' (pdevs_of_list (inl X))"   and "e' \<in> UNIV \<rightarrow> {- 1..1}" shows "thesis"
simps_1: fixes f :: "'d \<Rightarrow> 'd option" shows "option_comp_pow f 0 = Map.empty"
those_def: fixes xq :: "'a option lazy_sequence" shows "Lazy_Sequence.those xq = map_option lazy_sequence_of_list (those (list_of_lazy_sequence xq))"
rot_circle_cube_boundary_to_polarcircle_def: fixes \<gamma> :: "int \<times> (real \<Rightarrow> real \<times> real)" shows "rot_circle_cube_boundary_to_polarcircle \<gamma> \<equiv> if \<gamma> = rot_circle_left_edge then {subcube (1 / 4) (1 / 2) (1, circle_polar), subcube (1 / 2) (3 / 4) (1, circle_polar)} else if \<gamma> = rot_circle_right_edge then {subcube (3 / 4) 1 (1, circle_polar), subcube 0 (1 / 4) (1, circle_polar)} else {}"
has_typ1_weaken_Ts: fixes Ts :: "typ list"   and t :: "Core.term"   and rT :: "typ"   and T :: "typ" assumes "Ts \<turnstile>\<^sub>\<tau> t : rT" shows "Ts @ [T] \<turnstile>\<^sub>\<tau> t : rT"
mcp_ex: fixes W :: "'e list set"   and z :: "'e list" assumes "[] \<in> W"   and "finite W" obtains p :: "'e list" where "mcp z W p"
p_3: fixes q :: "nat" assumes "prime q"   and "m < q" shows "p \<le> q"
DISJNT: shows "P\<^sub>1 \<inter> (P\<^sub>2 \<union> wrap B\<^sub>2) = {}"
relation_in_carrier: fixes x :: "'a pmf"   and y :: "'a pmf" assumes "x \<succeq>[\<R>] y" shows "x \<in> local.\<P> \<and> y \<in> local.\<P>"
max_Var_floatariths_map_times: fixes fa1 :: "'e \<Rightarrow> floatarith"   and fa2 :: "'e \<Rightarrow> floatarith"   and xs :: "'e list" shows "max_Var_floatariths (map (\<lambda>i. fa1 i * fa2 i) xs) = max (max_Var_floatariths (map fa1 xs)) (max_Var_floatariths (map fa2 xs))"
in_set_clearjunk_iff_map_of_eq_Some: fixes a :: "'c"   and b :: "'d"   and xs :: "('c \<times> 'd) list" shows "((a, b) \<in> set (AList.clearjunk xs)) = (map_of xs a = Some b)"
While: fixes P :: "nat \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and b :: "bexp" assumes "\<And>n. \<turnstile>\<^sub>t {P (Suc n)} c {P n}"   and "\<forall>n s. P (Suc n) s \<longrightarrow> bval b s"   and "\<forall>s. P 0 s \<longrightarrow> \<not> bval b s" shows "\<turnstile>\<^sub>t {\<lambda>s. \<exists>n. P n s} WHILE b DO c {P 0}"
invertible_inverse_invertible: fixes u :: "'a \<Rightarrow> 'a" assumes "invertible u"   and "u \<in> S \<rightarrow>\<^sub>E S" shows "invertible (inverse u)"
vimage_UNIV: fixes f :: "'d \<Rightarrow> 'e" shows "f -` UNIV = UNIV"
him: shows "h ` ({0..1} \<times> {0..1}) \<subseteq> S"
length_induct: fixes P :: "'c list \<Rightarrow> bool"   and xs :: "'c list" assumes "\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs" shows "P xs"
12407: fixes x :: "'a"   and y :: "'a" shows "- - x \<sqinter> - - (x \<squnion> y) = - - x"
production_unfold: fixes C :: "'a literal multiset" shows "production C = {A. C \<in> N \<and> C \<noteq> {#} \<and> Max_mset C = Pos A \<and> \<not> interp C \<Turnstile> C \<and> S C = {#}}"
image_add_0: fixes S :: "'g set" shows "(+) (0::'g) ` S = S"
times_fds_def: fixes f :: "'a fds"   and g :: "'a fds" shows "f * g = fds (dirichlet_prod (fds_nth f) (fds_nth g))"
le_less_linear: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.linorder less_eq less" shows "less_eq x y \<or> less y x"
a0i_defs_2: shows "a0i_init \<equiv> {s. \<exists>ds. s = \<lparr>signals = \<lambda>s. 0, corrupted = ds\<rparr>}"
timpl_closure_set_is_timpl_closure_union: fixes M :: "(('a, 'b, 'c) prot_fun, 'd) Term.term set"   and TI :: "('c set \<times> 'c set) set" shows "timpl_closure_set M TI = (\<Union>t\<in>M. timpl_closure t TI)"
simps_5: fixes r :: "mregex"   and s :: "mregex"   and \<phi>s :: "Formula.formula list" shows "from_mregex (MTimes r s) \<phi>s = regex.Times (from_mregex r \<phi>s) (from_mregex s \<phi>s)"
inv: fixes h :: "'c"   and H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "h \<in> H" shows "inv\<^bsub>G\<^esub> h \<in> generate G H"
inf_def: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate" shows "AlternativeSmallStep.inf \<Gamma> cs css s = (\<exists>f. f 0 = (cs, css, s) \<and> (\<forall>i. \<Gamma>\<turnstile> f i \<rightarrow> f (Suc i)))"
is_unit_gcd_iff: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times (gcd a b) one = (gcd a b = one)"
simps_2: fixes T :: "POPLmarkRecord.type" shows "type_ofB (TVarB T) = T"
gf_U_eq_U: fixes x :: "'s"   and y :: "'s" shows "gf_U x y 1 = U x y"
closed_singleton: fixes "open" :: "'b set \<Rightarrow> bool"   and a :: "'b" assumes "class.t1_space open" shows "topological_space.closed open {a}"
s_mono: fixes a :: "real"   and v :: "real"   and u :: "real"   and t :: "real"   and s0 :: "real" assumes "braking_movement a v"   and "u \<le> t"   and "0 \<le> u" shows "movement.s a v s0 u \<le> movement.s a v s0 t"
arg_min_on_def: fixes less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and S :: "'c set" shows "ord.arg_min_on less f S = ord.arg_min less f (\<lambda>x. x \<in> S)"
absorb1: fixes b :: "'c"   and a :: "'c" assumes "b \<le> a" shows "sup a b = a"
empty: fixes g :: "'c \<Rightarrow> 'b" shows "prod g {} = (1::'b)"
cinner_minus_right: fixes x :: "'a"   and y :: "'a" shows "cinner x (- y) = - cinner x y"
hcomp_in_vhomE: fixes \<nu> :: "'a"   and \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<nu> \<star>\<^sub>C \<mu> : f \<Rightarrow>\<^sub>C g\<guillemotright>" obtains "C.arr \<mu>" and "C.arr \<nu>" and "src\<^sub>C \<nu> = trg\<^sub>C \<mu>" and "src\<^sub>C \<mu> = src\<^sub>C f" and "src\<^sub>C \<mu> = src\<^sub>C g" and "trg\<^sub>C \<nu> = trg\<^sub>C f" and "trg\<^sub>C \<nu> = trg\<^sub>C g"
simps: fixes \<Sigma> :: "'a set set" shows "\<A>.\<delta>\<^sub>\<A> \<Sigma> = \<up>af \<times> \<Delta>\<^sub>\<times> (semi_mojmir_def.step \<Sigma> \<up>af\<^sub>G \<circ> Abs \<circ> theG)"
p2c_mono_c2p: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and n' :: "nat" assumes "\<langle>c \<and> t\<rangle> \<le> n"   and "\<^bsub>c\<^esub>\<down>\<^bsub>t\<^esub>n \<le> n'" shows "n \<le> \<^bsub>c\<^esub>\<up>\<^bsub>t\<^esub>n'"
nnegI: fixes P :: "'c \<Rightarrow> 'd" assumes "\<And>x. (0::'d) \<le> P x" shows "nneg P"
rbt_lookup_rbt_bulkload: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "('b \<times> 'c) list" assumes "class.linorder less_eq less" shows "ord.rbt_lookup less (ord.rbt_bulkload less xs) = map_of xs"
substInp_def: fixes xs :: "'varSort"   and X :: "('b, 'c, 'varSort, 'var, 'd) qTerm set"   and x :: "'var" shows "substInp xs X x \<equiv> psubstInp (idEnv [x \<leftarrow> X]_xs)"
sparse_row_vector_cons: fixes a :: "nat \<times> 'a"   and arr :: "(nat \<times> 'a) list" shows "sparse_row_vector (a # arr) = singleton_matrix 0 (fst a) (snd a) + sparse_row_vector arr"
inorder_delete: fixes t :: "'a bro"   and h :: "nat"   and x :: "'a" assumes "t \<in> T h"   and "sorted (inorder t)" shows "inorder (delete x t) = del_list x (inorder t)"
bottomD_2: fixes P :: "'b set"   and x :: "'b" assumes "codomain.has_bottom P"   and "x \<in> P" shows "codomain.greater_eq x (codomain.bottom P)"
eq: shows "{x \<in> topspace X. (g \<circ> f) x \<in> U} = {x \<in> topspace X. f x \<in> {y \<in> topspace X'. g y \<in> U}}"
PNT2_imp_PNT1_1: assumes "(\<lambda>x. \<pi> x * ln (\<pi> x)) \<sim>[at_top] (\<lambda>x. x)" shows "(\<lambda>x. ln (\<pi> x)) \<sim>[at_top] ln"
unit_div_mult_swap: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "is_unit c" shows "a * (b div c) = a * b div c"
mat1_index: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and ze :: "'c"   and on :: "'c" assumes "i < n"   and "j < n" shows "mat1I ze on n ! i ! j = (if i = j then on else ze)"
all_Suc_le_conv: fixes n :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<forall>i\<le>Suc n. P i) = (P 0 \<and> (\<forall>i\<le>n. P (Suc i)))"
equivalent_objects_def: fixes a :: "'a"   and b :: "'a" shows "equivalent_objects a b \<equiv> \<exists>f. \<guillemotleft>f : a \<rightarrow> b\<guillemotright> \<and> equivalence_map f"
select_convs_2: fixes xc :: "int"   and yc :: "int"   and more :: "'a" shows "yc \<lparr>xc = xc, yc = yc, \<dots> = more\<rparr> = yc"
irreducible_imp_prime_poly: fixes p :: "'b poly" assumes "irreducible p" shows "prime_elem p"
range_eqI: fixes b :: "'b"   and f :: "'c \<Rightarrow> 'b"   and x :: "'c" assumes "b = f x" shows "b \<in> range f"
awalk_simps_2: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and e :: "'d"   and es :: "'d list"   and w :: "'c" assumes "wf_digraph G" shows "pre_digraph.awalk G u (e # es) w = (e \<in> arcs G \<and> u = tail G e \<and> pre_digraph.awalk G (head G e) es w)"
distinct_2: fixes x2 :: "'b" shows "OK x2 \<noteq> Err"
update_convs_1: fixes sccs' :: "'c set set \<Rightarrow> 'c set set"   and counter :: "nat"   and discovered :: "'c \<Rightarrow> nat option"   and finished :: "'c \<Rightarrow> nat option"   and pending :: "('c \<times> 'c) set"   and stack :: "'c list"   and tree_edges :: "('c \<times> 'c) set"   and back_edges :: "('c \<times> 'c) set"   and cross_edges :: "('c \<times> 'c) set"   and sccs :: "'c set set"   and lowlink :: "'c \<Rightarrow> nat option"   and tj_stack :: "'c list"   and more :: "'d" shows "sccs_update sccs' \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, sccs = sccs, lowlink = lowlink, tj_stack = tj_stack, \<dots> = more\<rparr> = \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, sccs = sccs' sccs, lowlink = lowlink, tj_stack = tj_stack, \<dots> = more\<rparr>"
blocksRedsFinal: fixes Vs :: "char list list"   and Ts :: "ty list"   and vs :: "val list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and l'' :: "sfields" assumes "length Vs = length Ts"   and "length vs = length Ts"   and "distinct Vs"   and "P \<turnstile> \<langle>e,(h, l(Vs [\<mapsto>] vs), sh),b\<rangle> \<rightarrow>* \<langle>e',(h', l', sh'),b'\<rangle>"   and "final e'"   and "l'' = override_on l' l (set Vs)" shows "P \<turnstile> \<langle>blocks (Vs, Ts, vs, e),(h, l, sh),b\<rangle> \<rightarrow>* \<langle>e',(h', l'', sh'),b'\<rangle>"
ListMem_iff: fixes x :: "'g"   and xs :: "'g list" shows "ListMem x xs = (x \<in> set xs)"
rep_eq: fixes x :: "nat" shows "pdevs_apply (coord_pdevs x) = (\<lambda>i. if i = x then 1 else 0)"
card_euler_liars_cosets_limit_2: assumes "\<not> prime n"   and "2 < n" shows "card H < (n - 1) div 2"
Exit_successor_of_Low: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and backward_slice :: "'b set \<Rightarrow> 'b set"   and Exit :: "'b"   and H :: "'d set"   and L :: "'d set"   and High :: "'b"   and Low :: "'b"   and a :: "'a" assumes "NonInterferenceIntraGraph sourcenode targetnode kind valid_edge Entry Def Use state_val backward_slice Exit H L High Low"   and "valid_edge a"   and "sourcenode a = Low" shows "targetnode a = Exit"
hermitian_def: fixes A :: "'a mat" shows "hermitian A = (adjoint A = A)"
Inf_aform_def: fixes X :: "'a \<times> 'a pdevs" shows "Inf_aform X = fst X - Radius X"
M_neq_0: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool" assumes "grover_state n f" shows "grover_state.M n f \<noteq> 0"
bot_def: fixes b :: "alt"   and p :: "alt \<Rightarrow> real" shows "b \<cdot>< p \<equiv> \<forall>a. a \<noteq> b \<longrightarrow> p b < p a"
C_def: shows "C = {\<psi> + \<phi> |\<psi> \<phi>. \<psi> \<in> A \<and> \<phi> \<in> B}"
inorder_del: fixes t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and x :: "'c" assumes "sorted1 (Tree2.inorder t)" shows "Tree2.inorder (del x t) = AList_Upd_Del.del_list x (Tree2.inorder t)"
Max_ne_def: fixes X :: "'b set" shows "Max_ne X = FOLD_bin_ne max X"
distinct_3: fixes x1 :: "label" shows "Break x1 \<noteq> Ret"
distinct_1: fixes x11 :: "nat"   and x12 :: "nat"   and x13 :: "nat"   and x14 :: "k"   and x15 :: "nat"   and x16 :: "nat"   and x17 :: "nat"   and x21 :: "nat"   and x22 :: "nat"   and x23 :: "nat"   and x24 :: "nat"   and x25 :: "nat" shows "Rreq x11 x12 x13 x14 x15 x16 x17 \<noteq> Rrep x21 x22 x23 x24 x25"
inverse_arrows_lunit: fixes a :: "'a" assumes "ide a" shows "inverse_arrows (lunit a) (lunit' a)"
elims_2: fixes x :: "'b \<Rightarrow> bool"   and xa :: "bool"   and xb :: "'b match_expr" assumes "has_disc_negated x xa xb" obtains disc :: "'b \<Rightarrow> bool" and neg :: "bool" and a :: "'b" where "x = disc__" and "xa = neg__" and "xb = Match a" and "if disc__ a then neg__ else False"   | disc :: "'b \<Rightarrow> bool" and neg :: "bool" and m :: "'b match_expr" where "x = disca__" and "xa = nega__" and "xb = MatchNot m" and "has_disc_negated disca__ (\<not> nega__) m"   | disc :: "'b \<Rightarrow> bool" and neg :: "bool" and m1 :: "'b match_expr" and m2 :: "'b match_expr" where "x = disc" and "xa = neg" and "xb = MatchAnd m1 m2" and "has_disc_negated disc neg m1 \<or> has_disc_negated disc neg m2"
part_def: fixes r :: "nat"   and s :: "nat"   and Y :: "'b set"   and f :: "'b set \<Rightarrow> nat" shows "part r s Y f = (\<forall>X. X \<subseteq> Y \<and> finite X \<and> card X = r \<longrightarrow> f X < s)"
update_cases_kD: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat"   and ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and P :: "(nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option) \<Rightarrow> bool" assumes "(\<pi>\<^sub>2 r = 0) = (\<pi>\<^sub>3 r = unk)"   and "ip \<in> kD rt"   and "sqn rt ip < \<pi>\<^sub>2 r \<Longrightarrow> P (rt(ip \<mapsto> r))"   and "\<lbrakk>sqn rt ip = \<pi>\<^sub>2 r; \<pi>\<^sub>5 r < the (dhops rt ip)\<rbrakk> \<Longrightarrow> P (rt(ip \<mapsto> r))"   and "\<lbrakk>sqn rt ip = \<pi>\<^sub>2 r; the (flag rt ip) = Aodv_Basic.inv\<rbrakk> \<Longrightarrow> P (rt(ip \<mapsto> r))"   and "\<pi>\<^sub>3 r = unk \<Longrightarrow> P (rt(ip \<mapsto> (\<pi>\<^sub>2 (the (rt ip)), \<pi>\<^sub>3 r, \<pi>\<^sub>4 r, \<pi>\<^sub>5 r, \<pi>\<^sub>6 r)))"   and "\<lbrakk>\<pi>\<^sub>2 r \<le> sqn rt ip; \<pi>\<^sub>3 r = kno; sqn rt ip = \<pi>\<^sub>2 r \<Longrightarrow> the (dhops rt ip) \<le> \<pi>\<^sub>5 r \<and> the (flag rt ip) = val\<rbrakk> \<Longrightarrow> P (rt(ip \<mapsto> the (rt ip)))" shows "P (update rt ip r)"
wf_trm_Var: fixes x :: "'a" shows "wf\<^sub>t\<^sub>r\<^sub>m (Var x)"
promela_language_def: fixes ast :: "AST.module list" shows "promela_language ast \<equiv> {promela_props \<circ> r |r. promela_is_run ast r}"
assms: fixes xs :: "'a list" assumes "xs \<in> set xss" shows "i \<le> length xs"
lcf_monom_1: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "monom P a n (deg R (monom P a n)) = a"
mset_seqeuences: fixes xs :: "'a list" shows "mset (concat (local.sequences xs)) = mset xs"
conway_c4_var: fixes x :: "'a"   and y :: "'a" shows "(x\<^sup>\<star> \<cdot> y\<^sup>\<star>)\<^sup>\<star> = x\<^sup>\<star> \<cdot> y\<^sup>\<star>"
one_relative_homology_group: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set" shows "\<one>\<^bsub>relative_homology_group (int p) X S\<^esub> = singular_relboundary_set p X S"
contr: assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> Stuck" shows "False"
res: fixes i :: "nat"   and f :: "'a mod_ring poly" assumes "(i, f) \<in> set res" shows "i \<noteq> 0 \<and> degree f \<noteq> 0 \<and> monic f \<and> (\<forall>g. irreducible g \<longrightarrow> g dvd f \<longrightarrow> degree g = i)"
ideal_generated_dvd2: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and d :: "'b"   and d' :: "'b" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "ring.ideal_generated plus zero uminus times S = ring.ideal_generated plus zero uminus times {d}"   and "finite S"   and "\<forall>x\<in>S. dvd.dvd times d' x" shows "dvd.dvd times d' d"
degree_eqI: fixes x :: "'d pdevs"   and d :: "nat" assumes "pdevs_apply x d \<noteq> (0::'d)"   and "\<And>j. d < j \<Longrightarrow> pdevs_apply x j = (0::'d)" shows "degree x = Suc d"
free_group_is_group: fixes gens :: "'c set" shows "Group.group \<F>\<^bsub>gens\<^esub>"
dismantle_cases: fixes t :: "trm"   and k :: "stack"   and r :: "trm" assumes "t \<star> k \<mapsto> r" obtains t' :: "trm" where "t \<mapsto> t'" and "r = t' \<star> k"   | k' :: "stack" where " k \<mapsto> k' " and "r = t \<star> k'"   | s :: "trm" and y :: "name" and n :: "trm" and l :: "stack" where "t = [s]" and "k = [y__]n__\<ggreater>l__" and "r = n__[y__::=s] \<star> l__"   | u :: "trm" and x :: "name" and v :: "trm" and y :: "name" and n :: "trm" and l :: "stack" where "x \<sharp> y" and "x \<sharp> n" and "t = u to x in v" and "k = [y]n\<ggreater>l" and "r = (u to x in v to y in n) \<star> l"
jj_squared: shows "\<j>\<^sup>2 = - 1"
flush_all_empty: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and m :: "nat \<Rightarrow> nat" assumes "\<forall>i p is xs sb \<D> \<O> \<R>. i < length ts \<longrightarrow> ts ! i = (p, is, xs, sb, \<D>, \<O>, \<R>) \<longrightarrow> sb = []" shows "flush_all_until_volatile_write ts m = m"
foldr_snd_zip: fixes ys :: "'a list"   and xs :: "'b list"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c" assumes "length ys \<le> length xs" shows "foldr (\<lambda>(x, y). f y) (zip xs ys) b = foldr f ys b"
1: shows "det_fold_map ordR c' f' \<sigma>' result"
continuous_at_open: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'd" shows "isCont f x = (\<forall>t. open t \<and> f x \<in> t \<longrightarrow> (\<exists>s. open s \<and> x \<in> s \<and> (\<forall>x'\<in>s. f x' \<in> t)))"
numer_cf4_def: shows "numer_cf4 \<equiv> \<lambda>x. x ^ 4 + 20 * x ^ 3 + 180 * x\<^sup>2 + 840 * x + 1680"
HoHV_eq_ToTC: shows "H.HoHV = T.ToTC"
compactin_Un: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "compactin X S"   and "compactin X T" shows "compactin X (S \<union> T)"