hcju/isabelleps huggingface dataset:
s2p_compl_hom: fixes A :: "'b set" shows "s2p (- A) = - s2p A"
rep_eq: fixes x :: "'c regular" shows "Rep_regular (- x) = - Rep_regular x"
rfb_Inf_pres: fixes R :: "('d \<times> 'c) set" shows "Inf_pres (fb\<^sub>\<R> R)"
p_recurrence'': fixes k :: "nat" assumes "n < k" shows "p k = - (\<Sum>i = 1..n. (- 1) ^ i * e i * p (k - i))"
term_pair: fixes v :: "'t" shows "term_of_pair (pair_of_term v) = v"
add_decreasing: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq a zero"   and "less_eq c b" shows "less_eq (plus a c) b"
assm: shows "eval (list_conj (map (fm.Atom \<circ> Eq) (p # A') @ map fm.Atom L @ F)) (xs @ x # \<Gamma>)"
Sigma_Diff_distrib2: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set"   and B :: "'a \<Rightarrow> 'b set" shows "(SIGMA i:I. A i - B i) = Sigma I A - Sigma I B"
ipresIGSwap_termMOD: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'varSort, 'k, 'l, 'var, 'h, 'm, 'n) model_scheme" shows "ipresIGSwap h termMOD MOD = ipresSwap h MOD"
measurable_THE: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and M :: "'e measure"   and I :: "'d set" assumes "\<And>i. Measurable.pred M (P i)"   and "countable I"   and "\<And>i x. \<lbrakk>x \<in> space M; P i x\<rbrakk> \<Longrightarrow> i \<in> I"   and "\<And>x i j. \<lbrakk>x \<in> space M; P i x; P j x\<rbrakk> \<Longrightarrow> i = j" shows "(\<lambda>x. THE i. P i x) \<in> M \<rightarrow>\<^sub>M count_space UNIV"
word_mult_less_cancel: fixes k :: "'b word"   and i :: "'b word"   and j :: "'b word" assumes "0 < k"   and "unat i * unat k < 2 ^ LENGTH('b)"   and "unat j * unat k < 2 ^ LENGTH('b)" shows "(i * k < j * k) = (i < j)"
inst_prod_pcpo: shows "\<bottom> = (\<bottom>, \<bottom>)"
mult_zero_right: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c" assumes "class.mult_zero times zero" shows "times a zero = zero"
hermitean_elems_2: fixes A :: "complex"   and B :: "complex"   and C :: "complex"   and D :: "complex" assumes "hermitean (A, B, C, D)" shows "is_real D"
ABB: shows "vdisjnt B (A -\<^sub>\<circ> B)"
succ_le_iff: fixes i :: "V"   and j :: "V" assumes "Ord i"   and "Ord j" shows "(ZFC_in_HOL.succ i \<le> j) = (i < j)"
measurable_Stream: fixes f :: "'b \<Rightarrow> 'c"   and N :: "'b measure"   and M :: "'c measure"   and g :: "'b \<Rightarrow> 'c stream" assumes "f \<in> N \<rightarrow>\<^sub>M M"   and "g \<in> N \<rightarrow>\<^sub>M stream_space M" shows "(\<lambda>x. f x ## g x) \<in> N \<rightarrow>\<^sub>M stream_space M"
surjective_map_axioms_def: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set" shows "surjective_map_axioms \<alpha> S T \<equiv> \<alpha> ` S = T"
pi: fixes i :: "real" assumes "i \<in> k" shows "p i \<in> path_image p"
lhd_lappend: fixes xs :: "'b llist"   and ys :: "'b llist" shows "lhd (lappend xs ys) = (if lnull xs then lhd ys else lhd xs)"
lookup_update': fixes k :: "'d"   and v :: "'c"   and m :: "('d, 'c) mapping"   and k' :: "'d" shows "Mapping.lookup (Mapping.update k v m) k' = (if k = k' then Some v else Mapping.lookup m k')"
assms_1: shows "constructive_security_obsf' real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common (\<lambda>\<eta>. absorb (\<A> \<eta>) (obsf_converter (1\<^sub>C |\<^sub>= conv \<eta>)))"
rk_singleton_bis: fixes A :: "Points"   and B :: "Points" assumes "A = B" shows "rk {A, B} = 1"
sum_neq_zeroI_1: fixes k :: "real"   and a :: "real"   and b :: "real" assumes "k \<le> \<bar>a\<bar>"   and "\<bar>b\<bar> < k" shows "a + b \<noteq> 0"
tllist_of_llist_LCons: fixes b :: "'b"   and xa :: "'a"   and x :: "'a llist" shows "tllist_of_llist b (LCons xa x) = TCons xa (tllist_of_llist b x)"
intros: fixes n :: "'n"   and h :: "'var" assumes "n \<in> local.nodes"   and "h \<in> local.hvars"   and "h \<in> local.reads n" shows "n \<in> sources"
mcont_eadd1: fixes y :: "enat" shows "mcont Sup (\<le>) Sup (\<le>) (\<lambda>x. x + y)"
mono: fixes c :: "'a"   and a :: "'a"   and d :: "'a"   and b :: "'a" assumes "c \<le> a"   and "d \<le> b" shows "c + d \<le> a + b"
\<theta>_mono: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "\<theta> x \<le> \<theta> y"
filter_interleave: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "xs \<in> ys \<otimes> zs" shows "filter P xs \<in> filter P ys \<otimes> filter P zs"
sum_list_independent_pdevs: fixes xs :: "(real \<times> real) list" shows "sum_list (independent_pdevs xs) = sum_list xs"
is_density_exprD_2: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr"   and t :: "pdf_type"   and e :: "cexpr" assumes "is_density_expr (vs, vs', \<Gamma>, \<delta>) t e" shows "free_vars e \<subseteq> shift_var_set (set vs')"
degen_assoc: fixes a :: "'b"   and b :: "'b"   and p\<^sub>1 :: "'b point"   and p\<^sub>2 :: "'b point"   and p\<^sub>3 :: "'b point" assumes "nonsingular a b"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2"   and "on_curve a b p\<^sub>3"   and "(p\<^sub>1 = Infinity \<or> p\<^sub>2 = Infinity \<or> p\<^sub>3 = Infinity) \<or> (p\<^sub>1 = opp p\<^sub>2 \<or> p\<^sub>2 = opp p\<^sub>3) \<or> opp p\<^sub>1 = ell_field_class.add a p\<^sub>2 p\<^sub>3 \<or> opp p\<^sub>3 = ell_field_class.add a p\<^sub>1 p\<^sub>2" shows "ell_field_class.add a p\<^sub>1 (ell_field_class.add a p\<^sub>2 p\<^sub>3) = ell_field_class.add a (ell_field_class.add a p\<^sub>1 p\<^sub>2) p\<^sub>3"
measurable_P_restriction: fixes P :: "'c \<Rightarrow> bool"   and M :: "'c measure"   and A :: "'c set" assumes "Measurable.pred M P"   and "A \<in> sets M" shows "{x \<in> A. P x} \<in> sets M"
mset_ran_eq_single_conv: fixes a :: "int \<Rightarrow> 'a"   and r :: "int set"   and x :: "'a" shows "(mset_ran a r = {#x#}) = (\<exists>i. r = {i} \<and> x = a i)"
rtrancl_mono_mp: fixes U :: "('c \<times> 'c) set"   and V :: "('c \<times> 'c) set"   and x :: "'c \<times> 'c" assumes "U \<subseteq> V"   and "x \<in> U\<^sup>*" shows "x \<in> V\<^sup>*"
differentiable_componentwise_within: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and S :: "'c set" shows "(f differentiable at a within S) = (\<forall>i\<in>Basis. (\<lambda>x. f x \<bullet> i) differentiable at a within S)"
1: fixes f :: "'c local.term set" assumes "arr f" shows "\<F>\<^sub>SC.arr (D f)"
0: shows "\<guillemotleft>local.comp (eval b c) (HF'.prod h b) : local.prod a b \<rightarrow> c\<guillemotright>"
simps_4: fixes p :: "perm"   and f :: "'d"   and \<alpha> :: "'c"   and t :: "('a, 'b, 'c, 'd) Tree" shows "p \<bullet> tAct f \<alpha> t = tAct (p \<bullet> f) (p \<bullet> \<alpha>) (p \<bullet> t)"
t3_space: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set"   and y :: "'a" assumes "class.t3_space open"   and "topological_space.closed open S"   and "y \<notin> S" shows "\<exists>U V. open U \<and> open V \<and> y \<in> U \<and> S \<subseteq> V \<and> U \<inter> V = {}"
EqSim: fixes S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b" assumes "Eq S T" shows "\<Psi>' \<rhd> S \<leadsto>\<guillemotleft>Rel\<guillemotright> T"
sort_eqv_normalize_sort: fixes cs :: "(String.literal \<times> String.literal) set"   and s :: "String.literal set" assumes "wf_subclass_loc cs"   and "finite s"   and "sort_ex cs s" shows "sort_eqv cs s (normalize_sort cs s)"
T_on_on': fixes s0 :: "'b list"   and qs :: "'b list"   and A :: "('b list \<Rightarrow> 'c) \<times> ('b list \<times> 'c \<Rightarrow> 'b \<Rightarrow> (nat \<times> nat list) \<times> 'c)"   and x :: "'c" shows "T s0 qs (off2 A (s0, x) qs) = T_on' A (s0, x) qs"
power_in_totatives: fixes m :: "nat"   and g :: "nat"   and i :: "nat" assumes "1 < m"   and "coprime m g" shows "g ^ i mod m \<in> totatives m"
closed_segment_subset_domain: fixes T :: "real set"   and t0 :: "real"   and t :: "real" assumes "interval T"   and "t0 \<in> T"   and "t \<in> T" shows "{t0--t} \<subseteq> T"
Des: shows "\<D>s es (\<lfloor>dom l\<rfloor> \<squnion> \<A> e)"
Inf_between_alt: fixes Inf :: "'b inference set"   and N :: "'b set"   and M :: "'b set" shows "inference_system.Inf_between Inf N M = {\<iota> \<in> Inf. \<iota> \<in> inference_system.Inf_from Inf (N \<union> M) \<and> set (prems_of \<iota>) \<inter> M \<noteq> {}}"
max_fun_diff_commute: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "t.max_fun_diff f g = t.max_fun_diff g f"
Nats_add: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.semiring_1 one times plus zero"   and "a \<in> semiring_1.Nats one plus zero"   and "b \<in> semiring_1.Nats one plus zero" shows "plus a b \<in> semiring_1.Nats one plus zero"
x0_hb_bound7': fixes x :: "real"   and i :: "nat" assumes "x\<^sub>1 \<le> x"   and "i < k" shows "1 < bs ! i * x + (hs ! i) x"
sup_comute_2: fixes a :: "'a"   and b :: "'a" shows "a \<squnion> b = b \<squnion> a"
deny_all_fun_def: fixes f :: "'d \<Rightarrow> 'e" shows "D\<^sub>f f \<equiv> D\<^sub>p (Some \<circ> f)"
to_tree_order_parent_child_rel: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and to :: "(_) object_ptr list"   and child :: "(_) object_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.to_tree_order ptr \<rightarrow>\<^sub>r to" shows "((ptr, child) \<in> (Shadow_DOM.parent_child_rel h)\<^sup>*) = (child \<in> set to)"
bex_one_point2: fixes A :: "'g set"   and a :: "'g"   and P :: "'g \<Rightarrow> bool" shows "(\<exists>x\<in>A. a = x \<and> P x) = (a \<in> A \<and> P a)"
lookup_raw_eq_valueI: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and xs :: "('b \<times> 'c) list \<times> 'a"   and k :: "'b"   and v :: "'c" assumes "oalist_raw.oalist_inv rep_key_order xs"   and "(k, v) \<in> set (fst xs)" shows "oalist_raw.lookup_raw rep_key_order xs k = v"
in_homE: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" obtains "arr f" and "local.dom f = a" and "cod f = b"
skip_eval\<^sub>w: fixes mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val" shows "\<langle>Skip, mds, mem\<rangle> \<leadsto> \<langle>Stop, mds, mem\<rangle>"
intros_6: fixes R :: "'a itrm \<Rightarrow> 'a itrm \<Rightarrow> bool"   and x :: "'a itrm"   and y :: "'a itrm"   and z :: "'a itrm" assumes "itrm_cong R x y"   and "itrm_cong R y z" shows "itrm_cong R x z"
integrable_minus: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'e" assumes "integrable M f" shows "integrable M (\<lambda>x. - f x)"
HUnion_def: fixes A :: "hf" shows "\<Squnion>A = (THE z. \<forall>u. (u \<^bold>\<in> z) = (\<exists>y. y \<^bold>\<in> A \<and> u \<^bold>\<in> y))"
ncoplanar_perm_17: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet C D B A"
swap_plus_mat: fixes A :: "'a mat"   and n :: "nat"   and B :: "'a mat"   and C :: "'a mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "C \<in> carrier_mat n n" shows "A + B + C = A + C + B"
AndL: fixes F :: "'b formula"   and G :: "'b formula"   and \<Gamma> :: "'b formula multiset" assumes "F, G, \<Gamma> \<Rightarrow>\<^sub>n" shows "F \<^bold>\<and> G, \<Gamma> \<Rightarrow>\<^sub>n"
e_tuple_nth_def: fixes a :: "nat"   and i :: "nat"   and n :: "nat" shows "e_tuple_nth a i n \<equiv> if n \<le> a then tuple_decode a i ! n else 0"
\<omega>_in_hhom: shows "\<guillemotleft>\<omega> : a \<rightarrow> a\<guillemotright>"
check_OR_def: fixes X :: "'g \<times> 'a"   and A :: "'j \<times> 'd"   and s :: "'e"   and Z :: "('e \<times> 'k) \<times> 'e \<times> 'f" shows "check_OR X A s Z = (s = fst (fst Z) \<oplus> fst (snd Z) \<and> fst (fst Z) \<in> challenge_space \<and> fst (snd Z) \<in> challenge_space \<and> check0 (fst X) (fst A) (fst (fst Z)) (snd (fst Z)) \<and> check1 (snd X) (snd A) (fst (snd Z)) (snd (snd Z)))"
connexI: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> less_eq x y \<or> less_eq y x" shows "connex A less_eq"
simps_18: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and f3 :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and x2 :: "'b" shows "(case Z x2 of Fo x \<Rightarrow> f1 x | Z x \<Rightarrow> f2 x | Less x xa \<Rightarrow> f3 x xa | In x xa \<Rightarrow> f4 x xa) = f2 x2"
simps: fixes a1 :: "'e set"   and a2 :: "'e set"   and R :: "('e \<times> 'e) set" shows "((a1, a2) \<in> max_ext R) = (\<exists>X Y. a1 = X \<and> a2 = Y \<and> finite X \<and> finite Y \<and> Y \<noteq> {} \<and> (\<forall>x. x \<in> X \<longrightarrow> (\<exists>xa\<in>Y. (x, xa) \<in> R)))"
abs_eq: shows "1 = Abs_fls (\<lambda>k. if k = 0 then 1::'a else (0::'a))"
open_discrete_def: fixes U :: "'a discrete set" shows "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U)"
assms_2: shows "(X, Y) \<in> (mult s)\<^sup>="
trans_greater_equiv: fixes cmp :: "'b comparator"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "compare cmp a b = Greater"   and "compare cmp b c = Equiv" shows "compare cmp a c = Greater"
adjust_not_mem: fixes w :: "'a vec"   and us :: "'a vec list" assumes "w \<in> carrier_vec n"   and "set us \<subseteq> carrier_vec n"   and "distinct us"   and "w \<notin> local.span (set us)" shows "adjuster n w us + w \<notin> set us"
lincomb_insert: fixes S :: "'a vec set"   and v :: "'a vec"   and a :: "'a vec \<Rightarrow> 'a" assumes "finite S"   and "S \<subseteq> carrier_vec n"   and "True"   and "v \<notin> S"   and "v \<in> carrier_vec n" shows "lincomb a (S \<union> {v}) = a v \<cdot>\<^sub>v v + lincomb a S"
intros_1: fixes S :: "'d set" shows "\<langle>\<rangle> \<in> trees S"
monoid_axioms: shows "Group_Theory.monoid B addB zeroB"
preserves_ide: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and a :: "'d" assumes "functor A B F"   and "partial_magma.ide A a" shows "partial_magma.ide B (F a)"
empty_fold_graphE: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and x :: "'c" assumes "fold_graph f z {} x" obtains "x = z"
ENFnrSR_offending_set: fixes P :: "'a \<Rightarrow> 'v \<Rightarrow> 'a \<Rightarrow> 'v \<Rightarrow> bool"   and G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a" assumes "sinvar_all_edges_normal_form_not_refl_SR P" shows "set_offending_flows G nP = (if sinvar G nP then {} else {{(e1, e2). (e1, e2) \<in> edges G \<and> e1 \<noteq> e2 \<and> \<not> P (nP e1) e1 (nP e2) e2}})"
Diag_Diagonalize_3: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Arr C t" shows "monoidal_language.Cod C (monoidal_language.Diagonalize C t) = monoidal_language.Diagonalize C (monoidal_language.Cod C t)"
pE_fin': fixes x :: "'v"   and \<sigma> :: "'v set"   and D :: "'v set"   and pE :: "('v \<times> 'v) set" assumes "invar x \<sigma> ([], D, pE)" shows "pE = {}"
hom_dvd_hom_iff: fixes x :: "'a"   and y :: "'a" shows "(hom x dvd hom y) = (x dvd y)"
vpairwiseI: fixes S :: "V"   and R :: "V \<Rightarrow> V \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in>\<^sub>\<circ> S; y \<in>\<^sub>\<circ> S; x \<noteq> y\<rbrakk> \<Longrightarrow> R x y" shows "vpairwise R S"
integral_sin_nx: fixes n :: "int" shows "integral {- pi..pi} (\<lambda>x. sin (x * real_of_int n)) = 0"
assms_12: shows "\<forall>a b c. fmul1 c (fadd2 a b) = fadd2 (fmul1 c a) (fmul1 c b)"
w1Less: shows "sint (SCAST(32 \<rightarrow> 64) w1) < 9223372039002259455"
mult_eq_imp_eq_div: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a * b = c"   and "a \<noteq> (0::'a)" shows "b = c div a"
Lcm_singleton: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "Lcm {a} = normalize a"
\<omega>_in_vhom: shows "S.in_hom \<omega> (w \<star> w) w"
pHom_pol_mem: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and f :: "'a \<Rightarrow> 'g"   and p :: "'a" assumes "PolynRg A B Y"   and "f \<in> pHom R S X, A B Y"   and "p \<in> carrier R"   and "p \<noteq> \<zero>" shows "f p = polyn_expr A Y (deg_n R S X p) (cf_h f (s_cf R S X p))"
top_comp_linorder_matrix: fixes f :: "'d \<times> 'd \<Rightarrow> 'c"   and i :: "'d"   and j :: "'d" shows "(mtop \<odot> f) (i, j) = sup_monoid.sum (\<lambda>k. f (k, j)) {k. True}"
ts_inv_ok_upd_ts: fixes ts :: "'i \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and t :: "'i"   and x :: "'g \<times> 'h \<Rightarrow>f nat"   and I :: "'i \<Rightarrow> 'j option"   and x' :: "'g \<times> 'h \<Rightarrow>f nat" assumes "ts t = \<lfloor>x\<rfloor>"   and "ts_inv_ok ts I" shows "ts_inv_ok (ts(t \<mapsto> x')) I"
Suc: shows "prod (prob_component pr w) {0..<n} = lprob_comp pr (stake n w)"
set_offending_flows_def: fixes sinvar :: "'b graph \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> bool"   and G :: "'b graph"   and nP :: "'b \<Rightarrow> 'c" shows "SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP = {F. F \<subseteq> edges G \<and> SecurityInvariant_withOffendingFlows.is_offending_flows_min_set sinvar F G nP}"
x_img_2: shows "\<delta> x \<in> subst_range \<delta>"
leg1_composite: shows "\<nu>\<pi>.leg1 = \<nu>.leg1 \<cdot> \<nu>\<pi>.prj\<^sub>1"
hindered: shows "hindered_by (\<Gamma> \<ominus> g) \<epsilon>"
ideal_eq_UNIV_iff_reduced_GB_eq_one_Polys: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "(ideal F = UNIV) = (punit.reduced_GB F = {1})"
exists_linear_eq_matrix': fixes X :: "(('a, 'b) vec, 'b) vec"   and Y :: "(('a, 'c) vec, 'c) vec"   and A :: "(('a, 'b) vec, 'c) vec" assumes "is_basis (set_of_vector X)"   and "is_basis (set_of_vector Y)" shows "\<exists>f. matrix' X Y f = A \<and> linear (*s) (*s) f"
is_binqueue_head: fixes l :: "nat"   and x :: "('c, 'd) bintree option"   and xs :: "('c, 'd) bintree option list" assumes "is_binqueue l (x # xs)" shows "is_binqueue l [x]"
inner2_aform_def: fixes X :: "'b \<times> 'b pdevs"   and a :: "'b"   and b :: "'b" shows "inner2_aform X a b = (inner2 (fst X) a b, pdevs_inner2 (snd X) a b)"
exec_gpv_def: shows "local.exec_gpv \<equiv> curry (spmf.fixp_fun (\<lambda>exec_gpv (c, s). the_gpv c \<bind> case_generat (\<lambda>x. return_spmf (x, s)) (\<lambda>out c. callee s out \<bind> (\<lambda>(x, y). curry exec_gpv (c x) y))))"
small_antisym: fixes h1 :: "'a \<Rightarrow> real"   and h2 :: "'a \<Rightarrow> real" assumes "h1 \<in> G"   and "h2 \<in> G"   and "h1 \<in> o[F](h2)"   and "h2 \<in> o[F](h1)" shows "False"
mor_unfold: fixes s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2" shows "mor UNIV UNIV s1 s2 UNIV UNIV dtor1 dtor2 (unfold1 s1 s2) (unfold2 s1 s2)"
ht_at_phl_cond: fixes u :: "'a"   and v :: "'a"   and x :: "'a"   and z :: "'a"   and w :: "'a"   and y :: "'a" assumes "\<lbrace>\<alpha> u \<cdot> \<tau> v\<rbrace>x\<lbrace>z\<rbrace>"   and "\<lbrace>\<alpha> u \<cdot> \<tau> w\<rbrace>y\<lbrace>z\<rbrace>" shows "\<lbrace>\<alpha> u\<rbrace>\<tau> v \<cdot> x + \<tau> w \<cdot> y\<lbrace>z\<rbrace>"
assms_3: shows "range_vars \<delta> \<subseteq> fv\<^sub>s\<^sub>t S"
gt_imp_ge: fixes x :: "'a"   and y :: "'a" assumes "x \<succ> y" shows "y \<le> x"
assms_1: fixes I :: "'i list" assumes "min_rvar_incdec dir s x\<^sub>i = Inl I"   and "check' dir x\<^sub>i s = set_unsat I s" shows "P (set_unsat I s)"
assms_1: shows "convert_atom var At (xs' @ x # xs) = Some At'"
nb: shows "var \<notin> vars (liftPoly 0 0 b)"
ipoly_poly_compose: fixes p :: "int poly"   and q :: "int poly"   and x :: "'a" shows "ipoly (p \<circ>\<^sub>p q) x = ipoly p (ipoly q x)"
pred_set: fixes P :: "'c \<Rightarrow> bool" shows "pred_dual P = (\<lambda>x. Ball (set_dual x) P)"
mreq_end3: fixes C :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and x :: "'b"   and p :: "'a list"   and b :: "'a" assumes "applied_rule_rev C x p \<noteq> \<bottom>" shows "applied_rule_rev C x (b # p) = applied_rule_rev C x p"
g_orbital_def: fixes f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> bool"   and U :: "'b \<Rightarrow> real set"   and S :: "'b set"   and t\<^sub>0 :: "real"   and s :: "'b" shows "x\<acute>=f & G on U S @ t\<^sub>0 s = \<mu> {g_orbit X G (U s) |X. X \<in> Sols f U S t\<^sub>0 s}"
h: shows "u \<in> V (T\<^sub>1\<rightarrow>T\<^sub>2)"
that: fixes m :: "'node" assumes "m \<in> set (predecessors g n)"   and "\<not> dominates g n m" shows "thesis"
disj_elim: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "(P \<or> Q) x" obtains "P x"   | "Q x"
scale_closed: fixes x :: "'vs"   and a :: "'f" assumes "x \<in> U\<^sub>V\<^sub>S" shows "a *s\<^sub>o\<^sub>w x \<in> U\<^sub>V\<^sub>S"
test_sub_identity: fixes x :: "'a" assumes "test x" shows "x \<le> top"
leading_coeff_0_iff: fixes p :: "'c poly" shows "(lead_coeff p = (0::'c)) = (p = 0)"
Hii: shows "Hermite_of_row_i ass res A i $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b)) = M $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))"
consistency_def: fixes C :: "('a, 'b) form set set" shows "consistency C = (\<forall>S. S \<in> C \<longrightarrow> (\<forall>p ts. \<not> (Pred p ts \<in> S \<and> Neg (Pred p ts) \<in> S)) \<and> FF \<notin> S \<and> Neg TT \<notin> S \<and> (\<forall>Z. Neg (Neg Z) \<in> S \<longrightarrow> S \<union> {Z} \<in> C) \<and> (\<forall>A B. And A B \<in> S \<longrightarrow> S \<union> {A, B} \<in> C) \<and> (\<forall>A B. Neg (Or A B) \<in> S \<longrightarrow> S \<union> {Neg A, Neg B} \<in> C) \<and> (\<forall>A B. Or A B \<in> S \<longrightarrow> S \<union> {A} \<in> C \<or> S \<union> {B} \<in> C) \<and> (\<forall>A B. Neg (And A B) \<in> S \<longrightarrow> S \<union> {Neg A} \<in> C \<or> S \<union> {Neg B} \<in> C) \<and> (\<forall>A B. Impl A B \<in> S \<longrightarrow> S \<union> {Neg A} \<in> C \<or> S \<union> {B} \<in> C) \<and> (\<forall>A B. Neg (Impl A B) \<in> S \<longrightarrow> S \<union> {A, Neg B} \<in> C) \<and> (\<forall>P t. closedt 0 t \<longrightarrow> Forall P \<in> S \<longrightarrow> S \<union> {P[t/0]} \<in> C) \<and> (\<forall>P t. closedt 0 t \<longrightarrow> Neg (Exists P) \<in> S \<longrightarrow> S \<union> {Neg (P[t/0])} \<in> C) \<and> (\<forall>P. Exists P \<in> S \<longrightarrow> (\<exists>x. S \<union> {P[App x []/0]} \<in> C)) \<and> (\<forall>P. Neg (Forall P) \<in> S \<longrightarrow> (\<exists>x. S \<union> {Neg (P[App x []/0])} \<in> C)))"
ins_inv1_inv2_1: fixes t :: "('a, 'c) rbt"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'a"   and x :: "'c" assumes "inv1 t"   and "inv2 t" shows "inv2 (rbt_ins f k x t)"
sets_M_M: shows "sets (M \<Otimes>\<^sub>M M) = sets (borel \<Otimes>\<^sub>M borel)"