hcju/isabelleps huggingface dataset:
collinear: fixes S :: "'a set" shows "collinear S = (\<exists>u. u \<noteq> (0::'a) \<and> (\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R u))"
F_eq: shows "F = {x. \<exists>y\<in>A. x = int p - y * a mod int p \<and> (int p - 1) div 2 < y * a mod int p}"
ideD_3: fixes a :: "'b" assumes "B.ide a" shows "B.cod a = a"
insertI2: fixes a :: "'a"   and B :: "'a set"   and b :: "'a" assumes "a \<in> B" shows "a \<in> insert b B"
measure_pmf_single: fixes M :: "'d pmf"   and x :: "'d" shows "measure_pmf.prob M {x} = pmf M x"
rep_eq: fixes x :: "'c \<Rightarrow>\<^sub>0 'b" shows "deg_pm x = deg_fun (lookup x)"
adj_idem2: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "f \<stileturn> g" shows "g \<circ> f \<circ> (g \<circ> f) = g \<circ> f"
simps_4: fixes v :: "'b"   and va :: "'b"   and vb :: "'c" shows "first_destNet (AllowPortFromTo v va vb) = destNet (AllowPortFromTo v va vb)"
prefix_antisym: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs \<le> ys"   and "ys \<le> xs" shows "xs = ys"
smult_monom: fixes a :: "'a"   and b :: "'a"   and n :: "nat" shows "smult a (monom b n) = monom (a * b) n"
nAct_0: fixes tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "cnf llist" shows "dynamic_component.nAct tCMP active c 0 t = 0"
map_snd_zip: fixes xs :: "'c list"   and ys :: "'d list" assumes "length xs = length ys" shows "map snd (xs || ys) = ys"
iMODb_div_right_strict_mono_on: fixes k :: "nat"   and m :: "nat"   and r :: "nat"   and c :: "nat" assumes "0 < k"   and "k \<le> m" shows "SetInterval2.strict_mono_on (\<lambda>x. x div k) [ r, mod m, c ]"
dependent_raw_def: shows "dependent \<equiv> module.dependent (*\<^sub>R)"
hyps: shows "\<Prod>\<^sub># F ddvd \<Prod>\<^sub># {#f \<in># F. \<not> f dvd (1::'a)#}"
loweq_def: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<sigma> :: "'b \<Rightarrow> 'e"   and \<sigma>' :: "'b \<Rightarrow> 'e" shows "IFC_def.loweq prob \<sigma> \<sigma>' = (\<sigma> \<restriction> (- IFC_def.hvars prob) = \<sigma>' \<restriction> (- IFC_def.hvars prob))"
healthy_bounded_byD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "real"   and P :: "'a \<Rightarrow> real" assumes "healthy t"   and "bounded_by b P"   and "nneg P" shows "bounded_by b (t P)"
"case": fixes f :: "'aq \<Rightarrow> 'ar \<Rightarrow> 'ap"   and x1 :: "'aq"   and x2 :: "'ar" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
c_times_Q: shows "Polynomial.smult (inverse (f (i - 1)) ^ (Suc (length (coeffs (F (i - 2)))) - length (coeffs (F (i - 1))))) (Q i) = fst (pdivmod (F (i - 2)) (F (i - 1)))"
lift_def: fixes t :: "Core.term"   and n :: "nat" shows "lift t n = incr_bv 1 n t"
simps_11: fixes v :: "'b"   and uu :: "'b ltln set" shows "nprop\<^sub>n(v)[uu]\<^sub>\<nu> = nprop\<^sub>n(v)"
conc_fun_def: fixes R :: "('b \<times> 'c) set"   and m :: "'c nres" shows "\<Down> R m \<equiv> case m of FAILi \<Rightarrow> FAIL | RES X \<Rightarrow> RES (R\<inverse> `` X)"
hchop_disj_distr2: fixes \<psi> :: "traffic \<Rightarrow> view \<Rightarrow> bool"   and \<chi> :: "traffic \<Rightarrow> view \<Rightarrow> bool"   and \<phi> :: "traffic \<Rightarrow> view \<Rightarrow> bool" shows "\<forall>ts v. (\<exists>va u. v=va\<parallel>u \<and> (\<psi> ts va \<or> \<chi> ts va) \<and> \<phi> ts u) = ((\<exists>va u. v=va\<parallel>u \<and> \<psi> ts va \<and> \<phi> ts u) \<or> (\<exists>va u. v=va\<parallel>u \<and> \<chi> ts va \<and> \<phi> ts u))"
sets_extend_measure: fixes G :: "'e \<Rightarrow> 'd set"   and I :: "'e set"   and \<Omega> :: "'d set"   and \<mu> :: "'e \<Rightarrow> ennreal" assumes "G ` I \<subseteq> Pow \<Omega>" shows "sets (extend_measure \<Omega> I G \<mu>) = sigma_sets \<Omega> (G ` I)"
Cod_in_Obj: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "('a, 'b) concrete_category.arr" assumes "concrete_category Obj Hom Id Comp"   and "partial_magma.arr (concrete_category.COMP Obj Hom Comp) f" shows "concrete_category.Cod f \<in> Obj"
sorted_merge: fixes cmp :: "'b comparator"   and xs :: "'b list"   and ys :: "'b list" assumes "Sorting_Algorithms.sorted cmp xs"   and "Sorting_Algorithms.sorted cmp ys" shows "Sorting_Algorithms.sorted cmp (Sorting_Algorithms.merge cmp xs ys)"
isomorphic_to_local_is_local: fixes B :: "'c set"   and addB :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multB :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zeroB :: "'c"   and oneB :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and addA :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multA :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zeroA :: "'d"   and oneA :: "'d" assumes "local_ring B addB multB zeroB oneB"   and "ring_isomorphism f A addA multA zeroA oneA B addB multB zeroB oneB" shows "local_ring A addA multA zeroA oneA"
Diff_not_in: fixes a :: "'c"   and A :: "'c set" shows "a \<notin> A - {a}"
discr0_transC: fixes c :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "discr0 c"   and "mustT c s"   and "(c, s) \<rightarrow>c (c', s')" shows "discr0 c'"
rSim: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "(\<Psi>, P, Q) \<in> X" shows "\<Psi> \<rhd> P \<leadsto><{(\<Psi>, P, Q) |\<Psi> P Q. \<exists>P' Q'. \<Psi> \<rhd> P \<sim> P' \<and> (\<Psi>, P', Q') \<in> X \<and> \<Psi> \<rhd> Q' \<sim> Q}> Q"
intro: fixes lookup :: "'a \<Rightarrow> ('b, 'c option) state"   and update :: "'a \<Rightarrow> 'c \<Rightarrow> ('b, unit) state"   and P :: "'b \<Rightarrow> bool" assumes "mem_correct lookup update P" shows "dp_consistency lookup update P"
assms_2: fixes x :: "real" assumes "a \<le> x"   and "x \<le> b" shows "(F has_vector_derivative f x) (at x within {a..b})"
equivR_simp: fixes \<Delta> :: "('sf, 'sc, 'sz) formula list"   and p :: "('sf, 'sc, 'sz) formula"   and q :: "('sf, 'sc, 'sz) formula"   and \<Gamma> :: "('sf, 'sc, 'sz) formula list"   and SS :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" assumes "\<Delta> ! j = (p \<leftrightarrow> q)"   and "(\<Gamma>, \<Delta>) = SS" shows "Rrule_result EquivR j SS = [(p # \<Gamma>, q # closeI \<Delta> j), (q # \<Gamma>, p # closeI \<Delta> j)]"
simps: fixes x :: "'a"   and v :: "'a \<Rightarrow> 'b"   and lb :: "'a \<Rightarrow> 'b option"   and ub :: "'a \<Rightarrow> 'b option" shows "in_bounds x v (lb, ub) = (v x \<ge>\<^sub>l\<^sub>b lb x \<and> v x \<le>\<^sub>u\<^sub>b ub x)"
ln_less: shows "ln (real p) \<le> 2 * ln (real n) + 2 * ln (12 / exp 1)"
rel_interior: fixes S :: "'b set" shows "rel_interior S = {x \<in> S. \<exists>T. open T \<and> x \<in> T \<and> T \<inter> affine hull S \<subseteq> S}"
leI: fixes x :: "'c word"   and y :: "'c word" assumes "\<not> x <s y" shows "y \<le>s x"
seq_def: shows "seq = (\<lambda>i. let j = SOME k. i < length (seq' k) in rev (seq' j) ! i)"
M'_def: shows "M' \<equiv> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l' \<A>))"
of_int_less_numeral_power_cancel_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "int"   and x :: "num"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (ring_1.of_int one plus zero minus a) (power.power one times (numeral.numeral one plus x) n) = (a < numeral x ^ n)"
A: shows "(\<lambda>n. ereal (root n (norm (f n * z ^ n)))) = (\<lambda>n. ereal (root n (norm (f n))) * ereal (norm z))"
f1: shows "(Ratreal p)\<^sup>2 = real 2"
submonoid_inverse_closed: fixes u :: "'a" assumes "sub.invertible u"   and "u \<in> H" shows "inverse u \<in> H"
real_of_float_inject: fixes x :: "float"   and y :: "float" shows "(real_of_float x = real_of_float y) = (x = y)"
AE_distr_iff: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure"   and P :: "'d \<Rightarrow> bool" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "{x \<in> space N. P x} \<in> sets N" shows "almost_everywhere (distr M N f) P = (AE x in M. P (f x))"
strict_order_iff: fixes a :: "'a"   and b :: "'a" shows "(a < b) = (a = a \<sqinter> b \<and> a \<noteq> b)"
from_F_def: fixes \<iota> :: "'f inference" shows "from_F \<iota> = {\<iota>'' \<in> Inf_FL. to_F \<iota>'' = \<iota>}"
split_asm: fixes P :: "'d \<Rightarrow> bool"   and f1 :: "'e option \<Rightarrow> 'd"   and f2 :: "'f \<Rightarrow> ('g \<Rightarrow> ('e, 'f, 'g) Resumption.resumption) \<Rightarrow> 'd"   and resumption :: "('e, 'f, 'g) Resumption.resumption" shows "P (case resumption of Resumption.resumption.Done x \<Rightarrow> f1 x | Resumption.resumption.Pause x xa \<Rightarrow> f2 x xa) = (\<not> ((\<exists>x1. resumption = Resumption.resumption.Done x1 \<and> \<not> P (f1 x1)) \<or> (\<exists>x21 x22. resumption = Resumption.resumption.Pause x21 x22 \<and> \<not> P (f2 x21 x22))))"
acyclic_reachable_points: fixes p :: "'a"   and q :: "'a"   and x :: "'a" assumes "is_point p"   and "is_point q"   and "p \<le> x ; q"   and "is_acyclic x" shows "p \<noteq> q"
triangular_eq_diagonal_1x2: shows "(\<forall>A\<in>carrier_mat 1 2. admits_triangular_reduction A) = (\<forall>A\<in>carrier_mat 1 2. admits_diagonal_reduction A)"
length_fixed_length_sublist: fixes i :: "nat"   and l :: "nat"   and xs :: "'b list" assumes "Suc i * l \<le> length xs" shows "length (fixed_length_sublist xs l i) = l"
dnorm_par: fixes Tr\<^sub>1 :: "'b \<Rightarrow> 'a set"   and Up\<^sub>1 :: "'b \<Rightarrow> 'a \<Rightarrow> 'b"   and s\<^sub>1 :: "'b"   and Tr\<^sub>2 :: "'c \<Rightarrow> 'a set"   and Up\<^sub>2 :: "'c \<Rightarrow> 'a \<Rightarrow> 'c"   and s\<^sub>2 :: "'c" shows "((P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>1,Up\<^sub>1\<rbrakk>) s\<^sub>1||(P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>2,Up\<^sub>2\<rbrakk>) s\<^sub>2) = (P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>\<lambda>s. case s of (s\<^sub>1, s\<^sub>2) \<Rightarrow> Tr\<^sub>1 s\<^sub>1 \<inter> Tr\<^sub>2 s\<^sub>2,\<lambda>s. case s of (s\<^sub>1, s\<^sub>2) \<Rightarrow> \<lambda>e. (Up\<^sub>1 s\<^sub>1 e, Up\<^sub>2 s\<^sub>2 e)\<rbrakk>) (s\<^sub>1, s\<^sub>2)"
cases: fixes x :: "(real \<times> real list) \<times> atom" obtains r :: "real" and cs :: "real list" and s :: "real" and d :: "real" and ds :: "real list" where "x = ((r__, cs), Less s (d__ # ds__))"   | rcs :: "real \<times> real list" and r :: "real" and d :: "real" and ds :: "real list" where "x = (rcs__, Eq r (d # ds))"   | rcs :: "real \<times> real list" and v :: "real" where "x = (rcsa__, Less v__ [])"   | rcs :: "real \<times> real list" and v :: "real" where "x = (rcs, Eq v [])"
assms_2: shows "proj_poly_syz n p \<noteq> 0"
reachablep_reachable_eq: fixes A :: "('e, 'f) automaton"   and I :: "'f \<Rightarrow> bool" shows "reachablep A I = (\<lambda>x. x \<in> reachable A I)"
assoc: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" shows "a \<inter># b \<inter># c = a \<inter># (b \<inter># c)"
strongly_connected_imp_induce_subgraph_strongly_connected: fixes H :: "('a, 'b) pre_digraph" assumes "subgraph H G"   and "strongly_connected H" shows "strongly_connected (G \<restriction> verts H)"
add_neg_numeral_special_9: shows "- (1::'a) + - (1::'a) = - (2::'a)"
axioms_1: fixes hom :: "'a \<Rightarrow> 'b" assumes "inj_monoid_mult_hom hom" shows "monoid_mult_hom hom"
orthogonal_sumlist: fixes S :: "'a vec list"   and v :: "'a vec" assumes "\<And>x. x \<in> set S \<Longrightarrow> v \<bullet> x = (0::'a)"   and "set S \<subseteq> carrier_vec n"   and "v \<in> carrier_vec n" shows "v \<bullet> M.sumlist S = (0::'a)"
comp_permute: fixes f :: "'a"   and g :: "'a"   and k :: "'a"   and l :: "'a"   and h :: "'a" assumes "f \<cdot> g = k \<cdot> l"   and "seq f g"   and "seq g h" shows "f \<cdot> g \<cdot> h = k \<cdot> l \<cdot> h"
false_propag_on_create_graph: shows "create_graph \<phi> \<le> SPEC (\<lambda>nds. \<forall>nd\<in>nds. false\<^sub>r \<notin> old nd)"
max_min_distrib2: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.linorder less_eq less" shows "ord.max less_eq a (ord.min less_eq b c) = ord.min less_eq (ord.max less_eq a b) (ord.max less_eq a c)"
2: shows "mat (row_length (A1 \<circ> A2 \<otimes> B1 \<circ> B2)) (length (A1 \<circ> A2 \<otimes> B1 \<circ> B2)) (A1 \<circ> A2 \<otimes> B1 \<circ> B2)"
cp\<^sub>P\<^sub>O\<^sub>S\<^sub>T: fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and a1 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1"   and a2 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>2"   and res :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'res"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "POST self a1 a2 res \<tau> = POST (\<lambda>_. self \<tau>) (\<lambda>_. a1 \<tau>) (\<lambda>_. a2 \<tau>) (\<lambda>_. res \<tau>) \<tau>"
h4: fixes i :: "nat"   and j :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "i < dim_row (take_rows A [0..<length (pivot_positions A)]) + (dim_row A - length (pivot_positions A))"
proj_stoch_proc_eq_snth: fixes f :: "nat \<Rightarrow> 'c \<Rightarrow> 'b"   and n :: "nat"   and x :: "'c"   and y :: "'c"   and k :: "nat" assumes "proj_stoch_proc f n x = proj_stoch_proc f n y"   and "k \<le> n" shows "f k x = f k y"
isomorphicI: fixes f :: "'a" assumes "Right_a.iso f" shows "Right_a.isomorphic (Right_a.dom f) (Right_a.cod f)"
set_generate: fixes A :: "nat"   and B :: "nat"   and m :: "nat"   and n :: "nat" shows "set (generate A B m n) = {(x, y). (x, y) \<noteq> (zeroes m, zeroes n) \<and> (x, y) \<in> set (gen2 A B m n)}"
hom_induced_compose: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'b set"   and Z :: "'c topology"   and g :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and p :: "int" assumes "continuous_map X Y f"   and "f ` S \<subseteq> T"   and "continuous_map Y Z g"   and "g ` T \<subseteq> U" shows "hom_induced p X S Z U (g \<circ> f) = hom_induced p Y T Z U g \<circ> hom_induced p X S Y T f"
mset_ran_insert: fixes r :: "int set"   and i :: "int"   and a :: "int \<Rightarrow> 'b" assumes "finite r"   and "i \<notin> r" shows "mset_ran a (insert i r) = add_mset (a i) (mset_ran a r)"
obj_at_field_on_heap_no_pending_stores: fixes m :: "'h"   and r :: "'g"   and f :: "'e"   and s :: "'h process_name \<Rightarrow> ('e, 'h, 'f, 'g) Model.local_state"   and opt_r' :: "'g option" assumes "sys_load (mutator m) (mr_Ref r f) (s sys) = mv_Ref opt_r'"   and "\<forall>opt_r'. mw_Mutate r f opt_r' \<notin> set (sys_mem_store_buffers (mutator m) s)"   and "obj_at (\<lambda>s. True) r s" shows "obj_at_field_on_heap (\<lambda>r. opt_r' = Some r) r f s"
not_ex: fixes P :: "'a \<Rightarrow> bool" shows "(\<nexists>x. P x) = (\<forall>x. \<not> P x)"
B_def: shows "B = inv_into A (\<lambda>a. generate G {a}) ` (\<lambda>a. generate G {a}) ` A"
decomp_nonnil: fixes t :: "('fun, 'var) Term.term" shows "decomp t \<noteq> []"
map_lookup: fixes g :: "'d \<Rightarrow> 'c"   and f :: "'e \<Rightarrow>\<^sub>0 'd"   and x :: "'e" assumes "g (0::'d) = (0::'c)" shows "lookup (Poly_Mapping.map g f) x = g (lookup f x)"
least_UpperI: fixes A :: "'a set"   and L :: "('a, 'b) gorder_scheme"   and s :: "'a" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq>\<^bsub>L\<^esub> s"   and "\<And>y. y \<in> Upper L A \<Longrightarrow> s \<sqsubseteq>\<^bsub>L\<^esub> y"   and "A \<subseteq> carrier L"   and "s \<in> carrier L" shows "is_lub L s A"
prime_def: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "normalization_semidom.prime zero times one normalize p = (comm_semiring_1.prime_elem times one zero p \<and> normalize p = p)"
open_Compl: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set" assumes "class.topological_space open"   and "topological_space.closed open S" shows "open (- S)"
ext_inject: fixes set_dom :: "'b set"   and set_func :: "'b \<Rightarrow> 'b"   and set_cod :: "'b set"   and more :: "'c"   and set_dom' :: "'b set"   and set_func' :: "'b \<Rightarrow> 'b"   and set_cod' :: "'b set"   and more' :: "'c" shows "(\<lparr>set_dom = set_dom, set_func = set_func, set_cod = set_cod, \<dots> = more\<rparr> = \<lparr>set_dom = set_dom', set_func = set_func', set_cod = set_cod', \<dots> = more'\<rparr>) = (set_dom = set_dom' \<and> set_func = set_func' \<and> set_cod = set_cod' \<and> more = more')"
simps_2: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and n :: "nat"   and c :: "'d"   and cs :: "'d list"   and v :: "'d \<Rightarrow> nat"   and d :: "'c" shows "reset' M n (c # cs) v d = reset (reset' M n cs v d) n (v c) d"
assms_4: shows "\<forall>x. cmod x = 1 \<longrightarrow> order x (map_poly complex_of_real (char_poly A)) \<le> d"
assms_2: fixes b :: "'a set" assumes "b \<in> B" shows "partition_on b (P b)"
uminus_not_id': fixes x :: "'b literal" shows "- x \<noteq> x"
round_unique': fixes x :: "'b"   and n :: "int" assumes "\<bar>x - of_int n\<bar> < (1::'b) / (2::'b)" shows "round x = n"
bounded_linear_right: fixes a :: "'b \<Rightarrow>\<^sub>L 'c" shows "bounded_linear ((*\<^sub>v) a)"
pivot_tableau'_def: fixes x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and t :: "(nat \<times> linear_poly) list" shows "pivot_tableau' x\<^sub>i x\<^sub>j t \<equiv> let x\<^sub>i_idx = eq_idx_for_lvar t x\<^sub>i; eqa = t ! x\<^sub>i_idx; eq' = pivot_eq eqa x\<^sub>j in map (\<lambda>idx. if idx = x\<^sub>i_idx then eq' else subst_var_eq x\<^sub>j (rhs eq') (t ! idx)) [0..<length t]"
cases: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and p :: "'d pmf"   and B :: "('e, 'e cset Monomorphic_Monad.id) nondetT" assumes "cr_prob_ndi A p B" obtains "rel_set A (set_pmf p) (rcset (extract (run_nondet B)))"
integrable_on_def: fixes f :: "'b \<Rightarrow> 'c"   and i :: "'b set" shows "(f integrable_on i) = (\<exists>y. (f has_integral y) i)"
lift_streams_append: fixes xs :: "'d stream list"   and ys :: "'d stream list" shows "lift_streams (xs @ ys) = smap2 (@) (lift_streams xs) (lift_streams ys)"
get_shadow_root_ptr_simp5: fixes character_data_ptr :: "(_) character_data_ptr"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and f :: "('p, 'o, 'n, 'w option) RShadowRoot_scheme"   and h :: "(_) heap" shows "get character_data_ptr (put shadow_root_ptr f h) = get character_data_ptr h"
inv_SOME: fixes a :: "nat option"   and b :: "bool list" assumes "(a, b) \<in> dc_crypto" shows "inversion (SOME x. inversion x = inversion (a, b) \<and> x \<in> dc_crypto) = inversion (a, b)"
interp_imp_general: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and C :: "'b literal multiset"   and D :: "'b literal multiset"   and D' :: "'b literal multiset"   and N :: "'b literal multiset set"   and CC :: "'b literal multiset set" assumes "ground_resolution_with_selection S"   and "C \<le> D"   and "D \<le> D'"   and "ground_resolution_with_selection.interp S N D \<Turnstile> C"   and "ground_resolution_with_selection.interp S N D' \<subseteq> \<Union> (ground_resolution_with_selection.production S N ` CC)" shows "\<Union> (ground_resolution_with_selection.production S N ` CC) \<Turnstile> C"
eq_fin_concat_end: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w :: "'a list"   and w\<^sub>2 :: "'a list" assumes "traces ind" shows "(traces.eq_swap ind)\<^sup>*\<^sup>* (w\<^sub>1 @ w) (w\<^sub>2 @ w) = (traces.eq_swap ind)\<^sup>*\<^sup>* w\<^sub>1 w\<^sub>2"
runit_hcomp_1: fixes f :: "'a"   and g :: "'a" assumes "ide f"   and "ide g"   and "src f = trg g" shows "\<r>[f \<star> g] = (f \<star> \<r>[g]) \<cdot> \<a>[f, g, src g]"
inv2: shows "\<exists>hs. rev qs @ h = [y, y] @ hs"
assms_1: shows "monoid_homomorphism \<eta> A multA oneA B multB oneB"
1: fixes s :: "real"   and A :: "real set" shows "((\<lambda>s. complex_of_real (- ((x - s) ^ Suc n / real (Suc n))) * iexp s) has_vector_derivative complex_of_real ((x - s) ^ n) * iexp s + \<i> * iexp s * complex_of_real (- ((x - s) ^ Suc n / real (Suc n)))) (at s within A)"
that: fixes a :: "real" assumes "A \<le> a"   and "\<forall>a'\<ge>a. a' \<in> \<nat> \<longrightarrow> f \<in> \<Omega>(\<lambda>x. real x powr p * (1 + integral (\<lambda>u. g' u / u powr (p + 1)) a' (real x)))" shows "thesis"
dg_Dom_vdomain: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "\<D>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) = \<CC>\<lparr>Arr\<rparr>"
open_closed_step_invariant: fixes A :: "(('e \<Rightarrow> 'f) \<times> 'g, 'h) automaton"   and I :: "'h \<Rightarrow> bool"   and P :: "(('e \<Rightarrow> 'f) \<times> 'g) \<times> 'h \<times> ('e \<Rightarrow> 'f) \<times> 'g \<Rightarrow> bool"   and J :: "'e set"   and U :: "('e \<Rightarrow> 'f) \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> bool" assumes "A \<TTurnstile>\<^sub>A (I \<rightarrow>) P"   and "local_steps (automaton.trans A) J"   and "other_steps U J"   and "\<And>\<sigma> \<zeta> a \<sigma>' \<zeta>' s s'. \<lbrakk>\<forall>j\<in>J. \<sigma> j = \<zeta> j; \<forall>j\<in>J. \<sigma>' j = \<zeta>' j; P ((\<sigma>, s), a, \<sigma>', s')\<rbrakk> \<Longrightarrow> P ((\<zeta>, s), a, \<zeta>', s')" shows "A \<Turnstile>\<^sub>A (act I, U \<rightarrow>) P"
Suc_diff_Suc: fixes n :: "nat"   and m :: "nat" assumes "order.greater m n" shows "Suc (m - Suc n) = m - n"
sets_uniform_count_measure_count_space: fixes A :: "'b set" shows "sets (uniform_count_measure A) = sets (count_space A)"
remove_elt_pairs_sorted: fixes ys :: "('c \<times> 'd) list"   and x :: "'c" assumes "Lists_Ex.strict_sorted (map fst ys)" shows "Lists_Ex.strict_sorted (map fst (remove_elt_pairs x ys))"
orbit_cyclic_eq3: fixes f :: "'e \<Rightarrow> 'e"   and S :: "'e set"   and y :: "'e" assumes "cyclic_on f S"   and "y \<in> S" shows "orbit f y = S"
select_result_I2: fixes h :: "'l"   and f :: "('l, 'm, 'n) prog"   and x :: "'n" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x" shows "|h \<turnstile> f|\<^sub>r = x"
greedy_path_deadend_v': fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a"   and v0 :: "'a"   and v :: "'a" assumes "greedy_conforming_path p \<sigma> \<sigma>' v0 = LCons v LNil" shows "deadend v"
multiset_filter_mono2: fixes P :: "'b \<Rightarrow> bool"   and A :: "'b multiset"   and Q :: "'b \<Rightarrow> bool" shows "(filter_mset P A \<subseteq># filter_mset Q A) = (\<forall>a\<in>#A. P a \<longrightarrow> Q a)"
senior_same_state: fixes x :: "nat"   and n :: "nat" shows "token_run (senior x n) n = token_run x n"
BS3: fixes f :: "'a"   and f' :: "'a"   and \<mu> :: "'a"   and \<mu>' :: "'a" assumes "C.is_left_adjoint f"   and "C.is_left_adjoint f'"   and "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>C f'\<guillemotright>"   and "\<guillemotleft>\<mu>' : f \<Rightarrow>\<^sub>C f'\<guillemotright>" shows "C.iso \<mu> \<and> C.iso \<mu>' \<and> \<mu> = \<mu>'"
Bij_def: fixes S :: "'c set" shows "Bij S = extensional S \<inter> {f. bij_betw f S S}"
fresh_freshAbs_substAbs: fixes Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and ys :: "'varSort"   and y :: "'var" assumes "good Y"   and "goodAbs A"   and "fresh ys y Y" shows "freshAbs ys y (A $[Y / y]_ys)"
fst_edge_from: fixes c :: "'form"   and "is" :: "nat list" shows "fst (edge_from c is) = (c, 0 # is)"