hcju/isabelleps huggingface dataset:
Zm_to_M_0: fixes M :: "('e, 'a, 'f) Module_scheme"   and f :: "'g \<Rightarrow> 'e"   and e :: "'g" assumes "R module M"   and "f \<in> mHom R (Zm R e) M" shows "f e = \<zero>\<^bsub>M\<^esub>"
list_equivp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "equivp R" shows "equivp (list_all2 R)"
Plus_eq_empty_conv: fixes A :: "'a set"   and B :: "'b set" shows "(A <+> B = {}) = (A = {} \<and> B = {})"
d_def: shows "d = (case c of LEQ_ns x xa \<Rightarrow> ns_constraint_const c | GEQ_ns linear_poly a \<Rightarrow> - ns_constraint_const c)"
irreducible_linear_poly: fixes b :: "'b"   and a :: "'b" assumes "b \<noteq> (0::'b)"   and "algebraic_semidom_class.coprime a b" shows "irreducible [:a, b:]"
ide_hcomp: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "horizontal_composition V H src trg"   and "partial_magma.ide V \<nu>"   and "partial_magma.ide V \<mu>"   and "src \<nu> = trg \<mu>" shows "partial_magma.ide V (H \<nu> \<mu>)"
a_in_j_xs: shows "to_nat a \<in> set (j # xs)"
is_sublist_def: fixes ls :: "'a list"   and vs :: "'a list" shows "is_sublist ls vs \<equiv> \<exists>as bs. vs = as @ ls @ bs"
int: shows "classes_above P C \<inter> classes_changed P P' = {}"
sorted_cons: fixes x :: "'c"   and xs :: "'c list" assumes "sorted (x # xs)" shows "sorted xs"
poly_map_poly: fixes hom :: "'a \<Rightarrow> 'b"   and p :: "'a poly"   and x :: "'a" assumes "comm_semiring_hom hom" shows "poly (map_poly hom p) (hom x) = hom (poly p x)"
assms_11: shows "indexed_eval_aux (Ps @ [P]) i = indexed_eval_aux (Qs @ [Q]) i"
vfsequence_at_last: fixes i :: "V"   and x :: "V" assumes "i = vcard xs" shows "(xs #\<^sub>\<circ> x)\<lparr>i\<rparr> = x"
is_parallel_solution_for_problem_operator_set: fixes \<Pi> :: "'b strips_problem"   and \<pi> :: "'b strips_operator list list"   and ops :: "'b strips_operator list"   and op :: "'b strips_operator" assumes "is_parallel_solution_for_problem \<Pi> \<pi>"   and "ops \<in> set \<pi>"   and "op \<in> set ops" shows "op \<in> set (\<Pi>\<^sub>\<O>)"
write0_def: fixes a :: "'b"   and P :: "'b process" shows "a \<rightarrow> P \<equiv> Mprefix {a} (\<lambda>x. P)"
homeomorphic_spheres_gen: fixes r :: "real"   and s :: "real"   and a :: "'a"   and b :: "'b" assumes "0 < r"   and "0 < s"   and "DIM('a) = DIM('b)" shows "sphere a r homeomorphic sphere b s"
Ball_set_replicate: fixes n :: "nat"   and a :: "'b"   and P :: "'b \<Rightarrow> bool" shows "(\<forall>x\<in>set (replicate n a). P x) = (P a \<or> n = 0)"
assms: shows "valid_mmsaux args cur (t, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since) auxlist"
step: fixes x :: "'a"   and b :: "'b" assumes "(x, b) \<in> X" shows "(\<exists>b'. (b', b) \<in> r \<and> (x, b') \<in> X) \<or> x \<in> f (fst ` X \<union> gfp f)"
coeffs_pCons_eq_cCons: fixes a :: "'c"   and p :: "'c poly" shows "Polynomial.coeffs (pCons a p) = cCons a (Polynomial.coeffs p)"
coprime_iff_invertible_int: fixes a :: "int"   and m :: "int" shows "coprime a m = (\<exists>x. [a * x = 1] (mod m))"
Q_def: shows "Q = {p. prime p \<and> real p \<in> {real n powr \<alpha><..real n}}"
summable_g: fixes z :: "complex" shows "summable (\<lambda>n. g n * z ^ n)"
transfer: fixes b :: "'g" shows "rel_fun (rel_set (pcr_tllist (=) (=))) (pcr_tllist (=) (=)) (\<lambda>A. (lSup (fst ` A), flat_lub b (snd ` (A \<inter> {(xs, uu_). lfinite xs})))) (tSup b)"
T_Rcd: fixes \<Gamma> :: "binding list"   and fs :: "(char list \<times> trm) list"   and fTs :: "(char list \<times> POPLmarkRecord.type) list" assumes "\<Gamma> \<turnstile> fs [:] fTs" shows "\<Gamma> \<turnstile> Rcd fs : RcdT fTs"
x_before_y_t3: shows "x < cs ! i in s_TS init h ((as @ [x] @ bs @ [x]) @ cs) (length (as @ [x] @ bs @ [x]) + i)"
parallel_composition: fixes V :: "('exp, 'id) MWLfCom list"   and V' :: "('exp, 'id) MWLfCom list"   and d :: "'d" assumes "length V = length V'"   and "\<forall>i<length V. [V ! i] \<approx>\<^bsub>d\<^esub> [V' ! i]" shows "V \<approx>\<^bsub>d\<^esub> V'"
cauchy_lemma: fixes X :: "nat \<Rightarrow> rat" assumes "\<And>n i. n \<le> i \<Longrightarrow> A n \<le> X i \<and> X i \<le> B n" shows "cauchy X"
Domain_fst: fixes r :: "('e \<times> 'f) set" shows "Domain r = fst ` r"
eq: fixes \<rho> :: "complex mat" assumes "\<rho> \<in> density_states" shows "trace (wlp S P * \<rho>) = trace (P * denote S \<rho>) + trace \<rho> - trace (denote S \<rho>)"
assms_1: shows "periodN (u \<cdot> w \<cdot> v) p"
abs_eq: shows "insert_clause' = Abs_comp_fun_idem insert_clause"
idempotent_bot_closed: shows "matrix_idempotent_semiring.idempotent mbot"
m_s2: fixes X :: "char list set"   and S1 :: "char list \<Rightarrow> 'av"   and S2 :: "char list \<Rightarrow> 'av" assumes "finite X"   and "S1 = S2 on - X"   and "S1 < S2" shows "m\<^sub>s S2 X < m\<^sub>s S1 X"
gen_def: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and cs :: "'b list"   and w :: "'c" shows "gen P cs w \<equiv> \<forall>c\<in>set cs. P c w"
assms_3: shows "z \<noteq> y \<or> zs \<noteq> ys"
trans: fixes r :: "'a"   and s :: "'a"   and t :: "'a" assumes "r = s"   and "s = t" shows "r = t"
prod_list_diag_prod: fixes A :: "'a mat" shows "prod_list (diag_mat A) = (\<Prod>i = 0..<dim_row A. A $$ (i, i))"
axioms_2: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" assumes "full_embedding_functor A B F" shows "full_functor A B F"
REFSTEP: fixes it :: "'S set"   and it' :: "'Sa set"   and x' :: "'Sa"   and x :: "'S"   and \<sigma> :: "'\<sigma>"   and \<sigma>' :: "'\<sigma>a" assumes "it \<subseteq> S"   and "it' \<subseteq> S'"   and "x' = \<alpha> x"   and "it' = \<alpha> ` it"   and "((it, \<sigma>), it', \<sigma>') \<in> R"   and "x \<in> it"   and "x' \<in> it'" shows "f x \<sigma> \<le> \<Down> {(\<sigma>, \<sigma>'). ((it - {x}, \<sigma>), it' - {x'}, \<sigma>') \<in> R} (f' x' \<sigma>')"
piecewise_C1_differentiable_cases: fixes f :: "real \<Rightarrow> 'b"   and a :: "real"   and c :: "real"   and g :: "real \<Rightarrow> 'b"   and b :: "real" assumes "f piecewise_C1_differentiable_on {a..c}"   and "g piecewise_C1_differentiable_on {c..b}"   and "a \<le> c"   and "c \<le> b"   and "f c = g c" shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_C1_differentiable_on {a..b}"
square_free_factorizationD_3: fixes p :: "'a poly"   and c :: "'a"   and bs :: "('a poly \<times> nat) list"   and a :: "'a poly"   and i :: "nat"   and b :: "'a poly"   and j :: "nat" assumes "square_free_factorization p (c, bs)"   and "(a, i) \<in> set bs"   and "(b, j) \<in> set bs"   and "(a, i) \<noteq> (b, j)" shows "algebraic_semidom_class.coprime a b"
4: fixes x :: "'a"   and y :: "'a" assumes "fst (rep_nat_term x) \<noteq> fst (rep_nat_term y)"   and "lex_pp (fst (rep_nat_term x)) (fst (rep_nat_term y))" shows "x < y"
safe_lpd_fv_regex_le: fixes r :: "'a regex"   and s :: "'a regex"   and test :: "'d \<Rightarrow> 'a \<Rightarrow> bool"   and i :: "'d" assumes "local.safe_regex Futu Strict r"   and "s \<in> lpd test i r" shows "fv_regex fv r \<subseteq> fv_regex fv s"
set_pmf_Int_Some: fixes p :: "'c spmf"   and A :: "'c set" shows "(set_pmf p \<inter> Some ` A = {}) = (set_spmf p \<inter> A = {})"
inf_pres_conj: fixes x :: "'b"   and y :: "'b" assumes "x \<in> conjunctive"   and "y \<in> conjunctive" shows "x \<sqinter> y \<in> conjunctive"
decode_plan_set_is: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat" shows "set (\<Phi>\<inverse> \<Pi> \<A> t) = (\<Union>k<t. {decode_plan' \<Pi> \<A> k})"
simps_981: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "tf \<Rightarrow> b_e list \<Rightarrow> 'a"   and f6 :: "tf \<Rightarrow> b_e list \<Rightarrow> 'a"   and f7 :: "tf \<Rightarrow> b_e list \<Rightarrow> b_e list \<Rightarrow> 'a"   and f8 :: "nat \<Rightarrow> 'a"   and f9 :: "nat \<Rightarrow> 'a"   and f10 :: "nat list \<Rightarrow> nat \<Rightarrow> 'a"   and f11 :: "'a"   and f12 :: "nat \<Rightarrow> 'a"   and f13 :: "nat \<Rightarrow> 'a"   and f14 :: "nat \<Rightarrow> 'a"   and f15 :: "nat \<Rightarrow> 'a"   and f16 :: "nat \<Rightarrow> 'a"   and f17 :: "nat \<Rightarrow> 'a"   and f18 :: "nat \<Rightarrow> 'a"   and f19 :: "t \<Rightarrow> (tp \<times> sx) option \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f20 :: "t \<Rightarrow> tp option \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f21 :: "'a"   and f22 :: "'a"   and f23 :: "v \<Rightarrow> 'a"   and f24 :: "t \<Rightarrow> unop_i \<Rightarrow> 'a"   and f25 :: "t \<Rightarrow> unop_f \<Rightarrow> 'a"   and f26 :: "t \<Rightarrow> binop_i \<Rightarrow> 'a"   and f27 :: "t \<Rightarrow> binop_f \<Rightarrow> 'a"   and f28 :: "t \<Rightarrow> testop \<Rightarrow> 'a"   and f29 :: "t \<Rightarrow> relop_i \<Rightarrow> 'a"   and f30 :: "t \<Rightarrow> relop_f \<Rightarrow> 'a"   and f31 :: "t \<Rightarrow> cvtop \<Rightarrow> t \<Rightarrow> sx option \<Rightarrow> 'a"   and x271 :: "t"   and x272 :: "binop_f" shows "(case Binop_f x271 x272 of Unreachable \<Rightarrow> f1 | Nop \<Rightarrow> f2 | Drop \<Rightarrow> f3 | Select \<Rightarrow> f4 | Block x xa \<Rightarrow> f5 x xa | Loop x xa \<Rightarrow> f6 x xa | b_e.If x xa xb \<Rightarrow> f7 x xa xb | Br x \<Rightarrow> f8 x | Br_if x \<Rightarrow> f9 x | Br_table x xa \<Rightarrow> f10 x xa | Return \<Rightarrow> f11 | Call x \<Rightarrow> f12 x | Call_indirect x \<Rightarrow> f13 x | Get_local x \<Rightarrow> f14 x | Set_local x \<Rightarrow> f15 x | Tee_local x \<Rightarrow> f16 x | Get_global x \<Rightarrow> f17 x | Set_global x \<Rightarrow> f18 x | Load x xa xb xc \<Rightarrow> f19 x xa xb xc | Store x xa xb xc \<Rightarrow> f20 x xa xb xc | Current_memory \<Rightarrow> f21 | Grow_memory \<Rightarrow> f22 | C x \<Rightarrow> f23 x | Unop_i x xa \<Rightarrow> f24 x xa | Unop_f x xa \<Rightarrow> f25 x xa | Binop_i x xa \<Rightarrow> f26 x xa | Binop_f x xa \<Rightarrow> f27 x xa | Testop x xa \<Rightarrow> f28 x xa | Relop_i x xa \<Rightarrow> f29 x xa | Relop_f x xa \<Rightarrow> f30 x xa | Cvtop x xa xb xc \<Rightarrow> f31 x xa xb xc) = f27 x271 x272"
signed_def: shows "signed \<equiv> map_fun Word.rep id (of_int \<circ> signed_take_bit (LENGTH('c) - Suc 0))"
allDefs_def: fixes p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g"   and n :: "'node" shows "uninst.allDefs p g n \<equiv> defs g n \<union> uninst.phiDefs p g n"
3: shows "\<forall>x. x |\<in>| |dom|f \<longrightarrow> f |@| x |\<in>| |cod|f"
getRel_dom_2: fixes G :: "('d, 'e) labeled_graph"   and a :: "'e"   and b :: "'e"   and l :: "'d" assumes "graph G"   and "(a, b) \<in> getRel l G" shows "b \<in> vertices G"
cases: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "'c"   and a2 :: "'c" assumes "r\<^sup>+\<^sup>+ a1 a2" obtains a :: "'c" and b :: "'c" where "a1 = a__" and "a2 = b__" and "r a__ b__"   | a :: "'c" and b :: "'c" and c :: "'c" where "a1 = a" and "a2 = c" and "r\<^sup>+\<^sup>+ a b" and "r b c"
rev_image_eqI: fixes x :: "'g"   and A :: "'g set"   and b :: "'h"   and f :: "'g \<Rightarrow> 'h" assumes "x \<in> A"   and "b = f x" shows "b \<in> f ` A"
J_2: fixes i :: "nat" assumes "i \<in> J" shows "F' i \<in> sets M"
is_morphism_of_sheaves: shows "morphism_sheaves_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str local.im_sheaf im_sheaf_morphisms b add_im_sheaf mult_im_sheaf zero_im_sheaf one_im_sheaf (\<lambda>U. identity (\<FF> U))"
alg_dvd_mult: fixes a :: "'b"   and c :: "'b"   and b :: "'b" assumes "a alg_dvd c"   and "algebraic_int b" shows "a alg_dvd b * c"
is_weak_ranking_aux: fixes A :: "'b set"   and xs :: "'b set list" shows "is_weak_ranking_aux A xs = (A \<inter> \<Union> (set xs) = {} \<and> is_weak_ranking xs)"
permutes_def: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set" shows "(p permutes S) = ((\<forall>x. x \<notin> S \<longrightarrow> p x = x) \<and> (\<forall>y. \<exists>!x. p x = y))"
sum_wt_Par: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and s :: "'state" assumes "properL cl"   and "n < length cl" shows "sum (wt (Par cl) s) {brnL cl n..<brnL cl n + brn (cl ! n)} = 1 / real (length cl) * sum (wt (cl ! n) s) {..<brn (cl ! n)}"
induced_automorphism_selfcomp_halftrivial_g: shows "fixespointwise (\
 \<circ> \
) (\<Union> (g \<turnstile> X))"
d_s_prod_eq_meet: fixes x :: "'a"   and y :: "'a" shows "d x \<cdot> d y = d x \<sqinter> d y"
simps_2: fixes x :: "'c"   and y :: "'d"   and l :: "(('c \<times> 'd) \<times> nat) tree"   and a :: "'c"   and b :: "'d"   and h :: "nat"   and r :: "(('c \<times> 'd) \<times> nat) tree" shows "update x y \<langle>l, ((a, b), h), r\<rangle> = (case cmp x a of LT \<Rightarrow> balL (update x y l) (a, b) r | EQ \<Rightarrow> \<langle>l, ((x, y), h), r\<rangle> | GT \<Rightarrow> balR l (a, b) (update x y r))"
dualization_3: fixes A :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<turnstile> \<not> \<box>[A]_v = \<diamond>\<langle>\<not> A\<rangle>_v"
hyps_2: fixes P :: "'c multiset \<Rightarrow> 'a" assumes "P \<in> carrier L" shows "carrier_coeff P"
cancel_sym_neg: fixes a :: "bool \<times> 'e"   and b :: "bool \<times> 'e" assumes "\<not> canceling a b" shows "\<not> canceling b a"
SUP_eq_top_iff: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "(\<Squnion> (f ` A) = \<top>) = (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
nat_pow_def: fixes G :: "('e, 'f) monoid_scheme"   and a :: "'e"   and n :: "nat" shows "a [^]\<^bsub>G\<^esub> n = rec_nat \<one>\<^bsub>G\<^esub> (\<lambda>u b. b \<otimes>\<^bsub>G\<^esub> a) n"
poly_of_pm_prod: fixes x :: "'b"   and f :: "'c \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and I :: "'c set" shows "poly_of_pm x (prod f I) = (\<Prod>i\<in>I. poly_of_pm x (f i))"
standard_control_dependence_def: fixes n :: "'node"   and n' :: "'node" shows "n controls\<^sub>s n' = (\<exists>a a' as. n' \<notin> set (sourcenodes (a # as)) \<and> n -a # as\<rightarrow>* n' \<and> n' postdominates targetnode a \<and> valid_edge a' \<and> sourcenode a' = n \<and> \<not> n' postdominates targetnode a')"
min_add_distrib_left: fixes x :: "'k multiset"   and y :: "'k multiset"   and z :: "'k multiset" shows "subset_mset.min x y + z = subset_mset.min (x + z) (y + z)"
wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t_def: fixes \<sigma> :: "('fun, 'atom) Term.term \<times> nat \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<sigma> \<equiv> \<forall>v. \<Gamma> (Var v) = \<Gamma> (\<sigma> v)"
measure_map_pmf: fixes f :: "'d \<Rightarrow> 'c"   and M :: "'d pmf"   and X :: "'c set" shows "measure_pmf.prob (map_pmf f M) X = measure_pmf.prob M (f -` X)"
ccBindsExtra_simp: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and i :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls" shows "ccBindsExtra \<Gamma>\<cdot>i = snd i \<squnion> ccBinds \<Gamma>\<cdot>i \<squnion> (\<Squnion>x\<mapsto>e\<in>map_of \<Gamma>. fv e G\<times> ccNeighbors x (snd i))"
eval_Fields_conv: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list" shows "pred.eval (Fields_i_i_o P C) = Fields P C"
kth_state_suc: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<sigma> :: "'b \<Rightarrow> 'c"   and k :: "nat" assumes "IFC prob" shows "IFC_def.kth_state prob \<sigma> (Suc k) = snd (IFC_def.step prob (IFC_def.path prob \<sigma> k, IFC_def.kth_state prob \<sigma> k))"
memory_write_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and addr :: "word32"   and bm :: "word4"   and data :: "word32"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> Some t1 = memory_write 10 addr bm data s1 \<and> Some t2 = memory_write 10 addr bm data s2" shows "low_equal t1 t2"
smult_vec_nonneg_eq: fixes c :: "'b"   and x :: "'b vec"   and y :: "'b vec" assumes "c \<noteq> (0::'b)" shows "(c \<cdot>\<^sub>v x = c \<cdot>\<^sub>v y) = (x = y)"
vdisjnt_vempty1: fixes A :: "V" shows "[]\<^sub>\<circ> \<inter>\<^sub>\<circ> A = []\<^sub>\<circ>"
Par1F: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and Q :: "pi" assumes "P \<longmapsto> \<alpha> \<prec> P'" shows "P \<parallel> Q \<longmapsto> \<alpha> \<prec> P' \<parallel> Q"
simps_5: fixes v :: "variable"   and va :: "typ"   and uu :: "nat"   and uv :: "Core.term" shows "subst_bv2 (Fv v va) uu uv = Fv v va"
assms_2: shows "geodesic_segment_between Gxz x z"
iMin_le_Max: fixes A :: "'b set" assumes "finite A"   and "A \<noteq> {}" shows "iMin A \<le> Max A"
rel_conversep: fixes R :: "'e \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_bset R\<inverse>\<inverse> = (rel_bset R)\<inverse>\<inverse>"
mset_neg_supset: fixes A :: "'a zmultiset" shows "- A \<subseteq>#\<^sub>z zmset_of (mset_neg A)"
component_leaks_def: fixes n :: "'lbl"   and \<A> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set" shows "component_leaks n \<A> Sec \<equiv> \<exists>\<A>' \<I>\<^sub>\<tau>. (subst_domain \<I>\<^sub>\<tau> = \<V> \<and> ground (subst_range \<I>\<^sub>\<tau>)) \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<I>\<^sub>\<tau> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<I>\<^sub>\<tau>) \<and> prefix \<A>' \<A> \<and> (\<exists>t\<in>Sec - declassified\<^sub>l\<^sub>s\<^sub>t \<A>' \<I>\<^sub>\<tau>. \<I>\<^sub>\<tau> \<Turnstile> \<langle>proj_unl n \<A>' @ [send\<langle>t\<rangle>\<^sub>s\<^sub>t]\<rangle>)"
negs_as_le_d: shows "negs (mset As) \<subseteq># DA"
fold1_ctor: shows "fold1 ctor1 ctor2 = id"
assms_3: shows "b \<notin>\<^sub>\<circ> B"
abs_prob_def: shows "abs_prob = \<lparr>variables_of = abs_ast_variable_section, operators_of = abs_ast_operator_section, initial_of = abs_ast_initial_state, goal_of = abs_ast_goal, range_of = abs_range_map\<rparr>"
GO: fixes F :: "'a formula"   and G :: "'a formula"   and S :: "'a formula multiset" assumes "F, G, S \<Rightarrow>\<^sub>n" shows "G, F, S \<Rightarrow>\<^sub>n"
local_help: fixes vs :: "'c list"   and a :: "'c list"   and r :: "'c"   and b :: "'c list"   and c :: "'c list"   and d :: "'c list" assumes "vs = a @ r # b"   and "vs = c @ r # d"   and "r \<notin> set a"   and "r \<notin> set b" shows "a = c \<and> b = d"
not_prfOf_Pf: fixes "prf" :: "'proof"   and \<phi> :: "'fmla" shows "let PPf = \<lambda>t1 t2. psubst Pf [(t1, yy), (t2, xx)] in prf \<in> proof \<and> \<phi> \<in> fmla \<and> Fvars \<phi> = {} \<longrightarrow> \<not> prfOf prf \<phi> \<longrightarrow> bprv (neg (PPf (encPf prf) \<langle>\<phi>\<rangle>))"
has_domain_iff_arr: fixes f :: "'b" shows "(B.domains f \<noteq> {}) = B.arr f"
prime_divides: fixes a :: "'a"   and b :: "'a"   and p :: "'a" assumes "a \<in> carrier (Ring_Divisibility.mult_of R)"   and "b \<in> carrier (Ring_Divisibility.mult_of R)"   and "prime (Ring_Divisibility.mult_of R) p"   and "p divides\<^bsub>Ring_Divisibility.mult_of R\<^esub> a \<otimes> b" shows "p divides\<^bsub>Ring_Divisibility.mult_of R\<^esub> a \<or> p divides\<^bsub>Ring_Divisibility.mult_of R\<^esub> b"
Approximation1_5Tr1: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and a :: "'a"   and x :: "'a" assumes "vals_nonequiv K (Suc n) vv"   and "n_val K (vv 0) = vv 0"   and "a \<in> carrier K"   and "vv 0 a = 1"   and "x \<in> carrier K"   and "Ostrowski_elem K (Suc n) vv x" shows "\<forall>m\<ge>2. vv 0 ((1\<^sub>r \<plusminus> -\<^sub>a x)^\<^bsup>K m\<^esup> \<plusminus> a \<cdot>\<^sub>r x^\<^bsup>K m\<^esup>) = 1"
edges_irrefl: fixes g :: "'a ugraph" shows "irrefl (edges g)"
hyps: fixes x2a :: "('b, 'a) Term.term" assumes "x2a \<in> set ss" shows "vars_term (scf_term scf x2a) \<subseteq> vars_term x2a"
EM_remainder_converges_cong: fixes a :: "int"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and n :: "nat"   and n' :: "nat"   and a' :: "int" assumes "\<And>x. real_of_int a \<le> x \<Longrightarrow> f x = g x"   and "n = n'"   and "a = a'" shows "EM_remainder_converges n f a = EM_remainder_converges n' g a'"
all_eqI: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "(\<forall>x. P x) = (\<forall>x. Q x)"
user_reg_mod_privilege: fixes s' :: "'b sparc_state"   and d :: "word32"   and w :: "'b word"   and r :: "word5"   and s :: "'b sparc_state" assumes "s' = user_reg_mod d w r s \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
MaxS_1: fixes n :: "nat" assumes "n \<in> S" shows "n \<le> Max S"
completely_multiplicative_function'_axioms: shows "completely_multiplicative_function' f f_prime"
precedes_def: fixes a :: "'b"   and b :: "'b"   and l :: "'b list" shows "precedes a b l \<equiv> a \<in> set l \<and> b \<in> set l \<and> firstPos a l \<le> firstPos b l"
append_eq_appendI: fixes xs :: "'g list"   and xs1 :: "'g list"   and zs :: "'g list"   and ys :: "'g list"   and us :: "'g list" assumes "xs @ xs1 = zs"   and "ys = xs1 @ us" shows "xs @ ys = zs @ us"
P3: shows "Bet E D' D0 \<and> Cong D' D0 B A'"
op_hs_empty_def: shows "op_hs_empty \<equiv> op_set_empty"
complete_hb_ta_hb_consistent: fixes s :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set"   and E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) list" assumes "hb_completion s E"   and "\<And>E t ln. ta_hb_consistent P E (llist_of (map (Pair t) (convert_RA ln)))" shows "ta_hb_consistent P E (lconcat (lmap (\<lambda>(t, ta). llist_of (map (Pair t) \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)) (complete_hb s E)))"
eq_swap_remove1: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list"   and c :: "'item" assumes "w\<^sub>1 =\<^sub>S w\<^sub>2" obtains "remove1 c w\<^sub>1 = remove1 c w\<^sub>2"   | "remove1 c w\<^sub>1 =\<^sub>S remove1 c w\<^sub>2"
snd_transfer: fixes A :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "(rel_prod A B ===> B) snd snd"
mem_cball_leI: fixes x :: "'a"   and y :: "'a"   and e :: "real"   and f :: "real" assumes "x \<in> cball y e"   and "e \<le> f" shows "x \<in> cball y f"
saturated_upto_complete_if: fixes N :: "'a literal multiset set" assumes "saturated_upto N"   and "\<not> satisfiable N" shows "{#} \<in> N"
R: shows "Chn (\<lparr>Chn = \<mu>_\<nu>_\<pi>\<rho>.chine_assoc, Dom = Dom ((\<mu> \<star> \<nu>) \<star> \<pi> \<star> \<rho>), Cod = Cod (\<mu> \<star> \<nu> \<star> \<pi> \<star> \<rho>)\<rparr> \<bullet> \<lparr>Chn = \<mu>\<nu>_\<pi>_\<rho>.chine_assoc, Dom = Dom (((\<mu> \<star> \<nu>) \<star> \<pi>) \<star> \<rho>), Cod = Cod ((\<mu> \<star> \<nu>) \<star> \<pi> \<star> \<rho>)\<rparr>) = \<mu>_\<nu>_\<pi>\<rho>.chine_assoc \<cdot> \<mu>\<nu>_\<pi>_\<rho>.chine_assoc"
valid_sops: shows "valid_sops ts\<^sub>s\<^sub>b"
igOpIPresIGWlsSTR_imp_igOpIPresIGWls: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.igOpIPresIGWlsSTR Delta MOD" shows "FixSyn.igOpIPresIGWls Delta MOD"
sum_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and K :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "\<And>i. i \<in> K \<Longrightarrow> less_eq (f i) (g i)" shows "less_eq (comm_monoid_add.sum plus zero f K) (comm_monoid_add.sum plus zero g K)"
pirreducible_roots: fixes p :: "'a list" assumes "p \<in> carrier (poly_ring R)"   and "pirreducible (carrier R) p"   and "degree p \<noteq> 1" shows "roots p = {#}"
sup_ge1_classes1: fixes c :: "classes1"   and d :: "classes1" shows "c \<le> c \<squnion> d"
insert: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "Max (insert x A) = max x (Max A)"