hcju/isabelleps huggingface dataset:
Replace_iff: fixes v :: "hf"   and A :: "hf"   and R :: "hf \<Rightarrow> hf \<Rightarrow> bool" shows "(v \<^bold>\<in> Replace A R) = (\<exists>u. u \<^bold>\<in> A \<and> R u v \<and> (\<forall>y. R u y \<longrightarrow> y = v))"
inv_list_rel_eq: fixes R :: "('e \<times> 'd) set" shows "(\<langle>R\<rangle>list_rel)\<inverse> = \<langle>R\<inverse>\<rangle>list_rel"
lower: fixes x :: "ennreal" assumes "x < B" shows "\<forall>\<^sub>F n in finite_subsets_at_top S. x < sum f n"
Inf_atLeast: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf (ord.atLeast less_eq x) = x"
isLub_cSup: fixes S :: "'a set" assumes "S \<noteq> {}"   and "\<exists>b. S *<= b" shows "isLub UNIV S (Sup S)"
fg_nconst: shows "\<exists>\<^sub>F w in at z. f w * g w \<noteq> 0"
scale_cancel_left: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'a"   and x :: "'b"   and y :: "'b" assumes "vector_space scale" shows "(scale a x = scale a y) = (x = y \<or> a = (0::'a))"
span_is_submodule: fixes R :: "('b, 'c) ring_scheme"   and M :: "('b, 'd, 'e) module_scheme"   and S :: "'d set" assumes "module R M"   and "S \<subseteq> carrier M" shows "LinearCombinations.submodule R (LinearCombinations.module.span R M S) M"
even_take_bit_eq: fixes n :: "nat"   and a :: "'a" shows "even (take_bit n a) = (n = 0 \<or> even a)"
cont_apply: fixes t :: "'d \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "cont t"   and "\<And>x. cont (f x)"   and "\<And>y. cont (\<lambda>x. f x y)" shows "cont (\<lambda>x. f x (t x))"
inv_eq_1_iff: fixes x :: "'a" assumes "x \<in> carrier G" shows "(inv x = \<one>) = (x = \<one>)"
is_readable\<^sub>C': shows "x\<^sub>C \<in> \<C>\<^sub>C \<or> x\<^sub>C \<notin> mds\<^sub>C' AsmNoReadOrWrite"
\<psi>_asymptotics: assumes "prime_number_theorem" shows "\<psi> \<sim>[at_top] (\<lambda>x. x)"
SOCom_iff_strong_reduction_simulation: fixes TRel :: "('procT \<times> 'procT) set" shows "((\<forall>S S'. S \<longmapsto>Source S' \<longrightarrow> (\<exists>T. \<lbrakk>S\<rbrakk> \<longmapsto>Target T \<and> (\<lbrakk>S'\<rbrakk>, T) \<in> TRel\<^sup>*)) \<and> strong_reduction_simulation (TRel\<^sup>+) Target) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> (\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel) \<and> (\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel \<longrightarrow> (T1, T2) \<in> TRel\<^sup>+) \<and> (\<forall>S T. (SourceTerm S, TargetTerm T) \<in> Rel \<longrightarrow> (\<lbrakk>S\<rbrakk>, T) \<in> TRel\<^sup>*) \<and> strong_reduction_simulation Rel (STCal Source Target))"
simps_2: fixes A :: "(('a, 'b) vec, 'c) vec"   and n :: "nat"   and i :: "'c"   and j :: "'b"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "Hermite_reduce_above A (Suc n) i j res = (let i' = mod_type_class.from_nat n; Aij = A $ i $ j; Ai'j = A $ i' $ j in Hermite_reduce_above (row_add A i' i ((res Aij Ai'j - Ai'j) div Aij)) n i j res)"
comp_1_def: fixes p2 :: "'f \<Rightarrow> 'e decision option"   and p1 :: "'d \<Rightarrow> 'f decision option" shows "p2 \<circ>\<^sub>1 p1 \<equiv> map_option flat_1 \<circ> (lift p2 \<circ>\<^sub>m p1)"
octo_of_real_minus: fixes x :: "real" shows "octo_of_real (- x) = - octo_of_real x"
real_sqrt_pow2: fixes x :: "real" assumes "0 \<le> x" shows "(sqrt x)\<^sup>2 = x"
UP_DOWN: fixes t :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "t \<in> SC.Univ" shows "SC.UP (SC.DOWN t) = t"
bv_prepend_split: fixes x :: "nat"   and b :: "'a"   and left :: "'a list"   and right :: "'a list" shows "bv_prepend x b (left @ right) = bv_prepend x b left @ right"
valid_Conj: fixes xset :: "('a, 'pred, 'act) formula set['a]"   and P :: "'state" assumes "finite (supp xset)" shows "P \<Turnstile> Conj xset = (\<forall>x\<in>set_bset xset. P \<Turnstile> x)"
F: fixes x :: "'d" assumes "x \<in> s" shows "norm (f x) \<le> F"
Suc: fixes x :: "real" shows "tan (x + real n * pi) = tan x"
matrix_scalar_mult_def: fixes k :: "'a"   and A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "k *k A \<equiv> \<chi>i j. k * A $h i $h j"
inject_2: fixes x2 :: "('a \<Rightarrow> 'b option) \<Rightarrow> bool"   and y2 :: "('a \<Rightarrow> 'b option) \<Rightarrow> bool" shows "((x2)\<^sub>\<surd> = (y2)\<^sub>\<surd>) = (x2 = y2)"
groupWIs_def: fixes c :: "parts_connection"   and rs :: "'a simple_rule list" shows "groupWIs c rs = (let W = getParts rs; f = \<lambda>wi. (map (\<lambda>d. runFw (getOneIp wi) d c rs) (map getOneIp W), map (\<lambda>s. runFw s (getOneIp wi) c rs) (map getOneIp W)) in groupF f W)"
finite_cartesian_product_iff: fixes A :: "'c set"   and B :: "'d set" shows "finite (A \<times> B) = (A = {} \<or> B = {} \<or> finite A \<and> finite B)"
abs: fixes X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and x :: "'var"   and xs :: "'varSort" assumes "good X"   and "x \<notin> vars xs"   and "\<And>Y. Y \<in> terms \<Longrightarrow> fresh xs x Y"   and "\<And>A. A \<in> abs \<Longrightarrow> freshAbs xs x A"   and "\<And>rho. rho \<in> envs \<Longrightarrow> freshEnv xs x rho"   and "\<And>Y. (X, Y) \<in> rel \<Longrightarrow> phi Y" shows "phiAbs (Abs xs x X)"
Conj_cong: fixes H :: "fm set"   and A :: "fm"   and A' :: "fm"   and B :: "fm"   and B' :: "fm" assumes "H \<turnstile> A IFF A'"   and "H \<turnstile> B IFF B'" shows "H \<turnstile> A AND B IFF A' AND B'"
sum_mset_distrib_div_if_dvd: fixes M :: "'a multiset"   and k :: "'b"   and f :: "'a \<Rightarrow> 'b" assumes "\<forall>m\<in>#M. k dvd f m" shows "\<Sum>\<^sub># (image_mset f M) div k = (\<Sum>m\<in>#M. f m div k)"
word_minus_def: fixes a :: "'b word" shows "- a = word_of_int (- uint a)"
circ_decompose_7: fixes x :: "'b"   and y :: "'b" shows "(x \<squnion> y)\<^sup>\<star> = x\<^sup>\<star> * y\<^sup>\<star> * (x \<squnion> y)\<^sup>\<star>"
inj: fixes f :: "'a \<Rightarrow> 'a" assumes "bijection f" shows "inj f"
representation_eqI: fixes basis :: "'d set"   and v :: "'d"   and f :: "'d \<Rightarrow> real" assumes "independent basis"   and "v \<in> span basis"   and "\<And>b. f b \<noteq> 0 \<Longrightarrow> b \<in> basis"   and "finite {b. f b \<noteq> 0}"   and "(\<Sum>b | f b \<noteq> 0. f b *\<^sub>R b) = v" shows "representation basis v = f"
that: fixes f :: "V" assumes "f \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"   and "g = \<FF>\<lparr>ArrMap\<rparr>\<lparr>f\<rparr>" shows "thesis"
H_def: fixes s :: "'s"   and t :: "'s" shows "H s t = T.prob s {\<omega> \<in> space (T s). alw (ev (HLD {t})) \<omega>}"
soundness_strat_helper: fixes D :: "sdiagram"   and P :: "interface"   and Q :: "interface"   and \<rho> :: "cell \<times> interface"   and \<gamma> :: "cell" shows "(prov_sdia D P Q \<longrightarrow> prov_triple (asn P, com_sdia D, asn Q)) \<and> (prov_row \<rho> P Q \<longrightarrow> prov_triple (asn P, com_cell (fst \<rho>), asn Q)) \<and> (prov_cell \<gamma> P Q \<longrightarrow> prov_triple (asn P, com_cell \<gamma>, asn Q))"
inside_arc_empty: fixes \<gamma> :: "real \<Rightarrow> 'a" assumes "arc \<gamma>" shows "inside (path_image \<gamma>) = {}"
card1_Tr2: fixes A :: "'c set" assumes "finite A"   and "card A = Suc 0" shows "\<exists>f. f \<in> {0} \<rightarrow> A \<and> surj_to f {0} A"
th0: shows "f ` b \<subseteq> f ` insert a b"
height_balR2: fixes l :: "('b \<times> nat) tree"   and r :: "('b \<times> nat) tree"   and a :: "'b" assumes "hbt l"   and "hbt r"   and "height r \<noteq> height l + m + 1" shows "height (balR l a r) = 1 + max (height l) (height r)"
t: shows "t = monomial (poly_deg p - poly_deg q) x + t'"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"   and "l_new_element_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M" shows "l_new_element_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
not_out_bet: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Col A B C"   and "\<not> B Out A C" shows "Bet A B C"
get_component_separates_tree_order: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and to :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and c' :: "(_) object_ptr list" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r to"   and "h \<turnstile> get_component ptr' \<rightarrow>\<^sub>r c'"   and "ptr' \<notin> set c" shows "set to \<inter> set c' = {}"
cases: fixes a1 :: "'c net_state"   and a2 :: "'d node_action"   and a3 :: "'c net_state"   and S :: "('c \<times> 'd seq_action \<times> 'c) set" assumes "(a1, a2, a3) \<in> node_sos S" obtains s :: "'c" and m :: "'d" and s' :: "'c" and i :: "nat" and R :: "nat set" where "a1 = NodeS i__ sa__ R__" and "a2 = R__:*cast(ma__)" and "a3 = NodeS i__ s'a__ R__" and "(sa__, broadcast ma__, s'a__) \<in> S"   | s :: "'c" and D :: "nat set" and m :: "'d" and s' :: "'c" and i :: "nat" and R :: "nat set" where "a1 = NodeS ia__ sb__ Ra__" and "a2 = (Ra__ \<inter> D):*cast(mb__)" and "a3 = NodeS ia__ s'b__ Ra__" and "(sb__, groupcast D mb__, s'b__) \<in> S"   | s :: "'c" and d :: "nat" and m :: "'d" and s' :: "'c" and R :: "nat set" and i :: "nat" where "a1 = NodeS ib__ sc__ Rb__" and "a2 = {d__}:*cast(mc__)" and "a3 = NodeS ib__ s'c__ Rb__" and "(sc__, unicast d__ mc__, s'c__) \<in> S" and "d__ \<in> Rb__"   | s :: "'c" and d :: "nat" and s' :: "'c" and R :: "nat set" and i :: "nat" where "a1 = NodeS ic__ sd__ Rc__" and "a2 = \<tau>" and "a3 = NodeS ic__ s'd__ Rc__" and "(sd__, \<not>unicast da__, s'd__) \<in> S" and "da__ \<notin> Rc__"   | s :: "'c" and d :: "nat" and s' :: "'c" and i :: "nat" and R :: "nat set" where "a1 = NodeS id__ se__ Rd__" and "a2 = id__:deliver(d)" and "a3 = NodeS id__ s'e__ Rd__" and "(se__, deliver d, s'e__) \<in> S"   | s :: "'c" and m :: "'d" and s' :: "'c" and i :: "nat" and R :: "nat set" where "a1 = NodeS ie__ sf__ Re__" and "a2 = {ie__}\<not>{}:arrive(md__)" and "a3 = NodeS ie__ s'f__ Re__" and "(sf__, receive md__, s'f__) \<in> S"   | s :: "'c" and s' :: "'c" and i :: "nat" and R :: "nat set" where "a1 = NodeS if__ sg__ Rf__" and "a2 = \<tau>" and "a3 = NodeS if__ s' Rf__" and "(sg__, \<tau>, s') \<in> S"   | i :: "nat" and s :: "'c" and R :: "nat set" and m :: "'d" where "a1 = NodeS ig__ sh__ Rg__" and "a2 = {}\<not>{ig__}:arrive(m)" and "a3 = NodeS ig__ sh__ Rg__"   | i :: "nat" and s :: "'c" and R :: "nat set" and i' :: "nat" where "a1 = NodeS ih__ si__ Rh__" and "a2 = connect(ih__, i'__)" and "a3 = NodeS ih__ si__ (Rh__ \<union> {i'__})"   | i :: "nat" and s :: "'c" and R :: "nat set" and i' :: "nat" where "a1 = NodeS ij__ sj__ Ri__" and "a2 = connect(i'a__, ij__)" and "a3 = NodeS ij__ sj__ (Ri__ \<union> {i'a__})"   | i :: "nat" and s :: "'c" and R :: "nat set" and i' :: "nat" where "a1 = NodeS ik__ sk__ Rj__" and "a2 = disconnect(ik__, i'b__)" and "a3 = NodeS ik__ sk__ (Rj__ - {i'b__})"   | i :: "nat" and s :: "'c" and R :: "nat set" and i' :: "nat" where "a1 = NodeS il__ sl__ Rk__" and "a2 = disconnect(i'c__, il__)" and "a3 = NodeS il__ sl__ (Rk__ - {i'c__})"   | i :: "nat" and i' :: "nat" and i'' :: "nat" and s :: "'c" and R :: "nat set" where "a1 = NodeS im__ sm__ Rl__" and "a2 = connect(i'd__, i''__)" and "a3 = NodeS im__ sm__ Rl__" and "im__ \<noteq> i'd__" and "im__ \<noteq> i''__"   | i :: "nat" and i' :: "nat" and i'' :: "nat" and s :: "'c" and R :: "nat set" where "a1 = NodeS i s R" and "a2 = disconnect(i', i'')" and "a3 = NodeS i s R" and "i \<noteq> i'" and "i \<noteq> i''"
simps_11: fixes v :: "'b formula"   and va :: "'b formula" shows "is_lit_plus (v \<^bold>\<rightarrow> va) = False"
funpow_0: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "(f ^^ 0) x = x"
YFtor'_def: fixes C :: "('d, 'e, 'f) LSCategory_scheme" shows "YFtor' C \<equiv> \<lparr>CatDom = C, CatCod = CatExp (Op C) SET, MapM = YFtorNT C\<rparr>"
ediff_le_self: fixes x :: "enat"   and y :: "enat" shows "x - y \<le> x"
Solve_exec_ns_axioms: shows "Solve_exec_ns solve_exec_ns"
assms: fixes m :: "'a"   and n :: "'a" assumes "m \<in> S"   and "n \<in> S"   and "m \<noteq> n" shows "A m \<inter> A n = {}"
bi_total_def: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "bi_total R = ((\<forall>x. \<exists>y. R x y) \<and> (\<forall>y. \<exists>x. R x y))"
empty_subsetI: fixes A :: "'b set" shows "{} \<subseteq> A"
wf_Int2: fixes r :: "('c \<times> 'c) set"   and r' :: "('c \<times> 'c) set" assumes "wf r" shows "wf (r' \<inter> r)"
functor_comp: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and G :: "'b \<Rightarrow> 'd" assumes "functor A B F"   and "functor B C G" shows "functor A C (G \<circ> F)"
class_add_is_relevant_class_rev: fixes i :: "instr"   and C :: "char list"   and cdec :: "jvm_method class"   and P :: "jvm_prog"   and C\<^sub>0 :: "char list" assumes "is_relevant_class i (class_add P (C, cdec)) C\<^sub>0"   and "\<And>cd D'. cd \<in> set P \<Longrightarrow> (fst cd, C) \<notin> (subcls1 P)\<^sup>*"   and "wf_syscls P" shows "is_relevant_class i P C\<^sub>0"
quotients_def: shows "quotients = map (\<lambda>i. G\<lparr>carrier := \<GG> ! (i + 1)\<rparr> Mod \<GG> ! i) [0..<length \<GG> - 1]"
select_convs_1: fixes carrier :: "'b set"   and more :: "'c" shows "carrier \<lparr>carrier = carrier, \<dots> = more\<rparr> = carrier"
sum_indicator_mult: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite A" shows "(\<Sum>x\<in>A. indicator B x * f x) = sum f (A \<inter> B)"
mkM_kripke: fixes T :: "('a, 'es, 'as) BEState Trace set" shows "kripke (SPR.mkM T)"
phl_seq: fixes p :: "'a"   and x :: "'a"   and r :: "'a"   and y :: "'a"   and q :: "'a" assumes "\<lbrace>p\<rbrace>x\<lbrace>r\<rbrace>"   and "\<lbrace>r\<rbrace>y\<lbrace>q\<rbrace>" shows "\<lbrace>p\<rbrace>x \<cdot> y\<lbrace>q\<rbrace>"
d_eq: fixes k :: "nat" shows "d k = fst (grab (nxt (M k) (enum N (Suc (2 * Suc k)))) (Suc k))"
sameDom_def: fixes inp1 :: "'f \<Rightarrow> 'g option"   and inp2 :: "'f \<Rightarrow> 'h option" shows "sameDom inp1 inp2 \<equiv> \<forall>i. (inp1 i = None) = (inp2 i = None)"
before_in_swap: fixes xs :: "'b list"   and ys :: "'b list"   and n :: "nat"   and x :: "'b"   and y :: "'b" assumes "dist_perm xs ys"   and "Suc n < length xs" shows "x < y in swap n xs = (x < y in xs \<and> \<not> (x = xs ! n \<and> y = xs ! Suc n) \<or> x = xs ! Suc n \<and> y = xs ! n)"
simps_7: fixes I :: "interp"   and \<theta> :: "trm" shows "term_sem I (Differential \<theta>) = (\<lambda>\<omega>. \<Sum>x\<in>allidents. \<omega> (DVar x) * deriv (\<lambda>X. term_sem I \<theta> (repv \<omega> (RVar x) X)) (\<omega> (RVar x)))"
phantom_rel_union: fixes un :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "('c \<times> 'd set) set" assumes "GEN_OP un (\<union>) (A \<rightarrow> A \<rightarrow> A)" shows "(\<lambda>a b. a \<bind> (\<lambda>a. b \<bind> (\<lambda>b. Some (un a b))), op_union_phantom) \<in> \<langle>A\<rangle>phantom_rel \<rightarrow> \<langle>A\<rangle>phantom_rel \<rightarrow> \<langle>A\<rangle>phantom_rel"
uint_0_iff: fixes x :: "'a Word.word" shows "(uint x = 0) = (x = 0)"
wx_def: fixes X :: "'b program set" shows "wx X \<equiv> \<Union> {Y. Y \<subseteq> X \<and> ex_prop Y}"
starfun_exp_HFinite: fixes x :: "real star" assumes "x \<in> HFinite" shows "(*f* exp) x \<in> HFinite"
trivial_subalg_cond_expect_eq: fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> real" assumes "prob_space M"   and "subalgebra M N"   and "sets N = {{}, space M}"   and "integrable M f" shows "\<forall>x\<in>space M. real_cond_exp M N f x = integral\<^sup>L M f"
implication_implies_pointstamp: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and t :: "'c"   and c :: "('a, 'c) configuration"   and loc :: "'a" assumes "dataflow_topology summary results_in"   and "t \<in>\<^sub>A frontier (c_imp c loc)"   and "dataflow_topology.inv_imps_work_sum summary results_in c" shows "\<exists>t' loc' s. s \<in>\<^sub>A graph.path_weight summary loc' loc \<and> results_in t' s \<le> t \<and> (t' \<in>\<^sub>A frontier (c_pts c loc') \<or> zcount (c_work c loc') t' < 0)"
llength_less_llength_lselect_less: fixes i :: "nat"   and s :: "nat set"   and w :: "'a llist" shows "(enat i < esize s \<and> enat (nth_least s i) < llength w) = (enat i < llength (lselect s w))"
not_less_eq_is_greater_Sec: fixes a :: "Sec"   and b :: "Sec" shows "(\<not> a \<le> b) = (b < a)"
des1_is_proper: assumes "\<B> \<noteq> {#}" shows "proper_design \<V> \<B>"
rel_frontier_translation: fixes a :: "'a"   and S :: "'a set" shows "rel_frontier ((+) a ` S) = (+) a ` rel_frontier S"
simps_4: fixes var :: "nat"   and p :: "real mpoly"   and as :: "atom list" shows "find_eq var (Leq p # as) = (let (A, B) = find_eq var as in (A, Leq p # B))"
cos_float_intervalI: fixes x :: "real"   and X :: "float interval"   and p :: "nat" assumes "x \<in>\<^sub>r X" shows "cos x \<in>\<^sub>r cos_float_interval p X"
index_simps_1: fixes l :: "'a list"   and n :: "nat" assumes "length l \<le> n" shows "index l n = None"
exact_3: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat" assumes "valid_decomp X qs"   and "standard_decomp k qs" shows "exact_decomp 0 (exact k qs)"
pigeonhole_infinite: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "infinite A"   and "finite (f ` A)" shows "\<exists>a0\<in>A. infinite {a \<in> A. f a = f a0}"
ListAif1: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and upds :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> 'b) list"   and C1 :: "acom"   and C2 :: "acom"   and l :: "'a \<Rightarrow> nat" assumes "bval b s" shows "preList upds (IF b THEN C1 ELSE C2) l s = preList upds C1 l s"
weakCongRepl: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs" assumes "P \<parallel> !P \<Longrightarrow>\<alpha> \<prec> P'" shows "!P \<Longrightarrow>\<alpha> \<prec> P'"
rank_invar_to_set: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list" assumes "BinomialHeapStruc.rank_invar (t # bq)" shows "\<forall>t'\<in>set bq. BinomialHeapStruc.rank t < BinomialHeapStruc.rank t'"
fmeasurableI2: fixes A :: "'b set"   and M :: "'b measure"   and B :: "'b set" assumes "A \<in> fmeasurable M"   and "B \<subseteq> A"   and "B \<in> sets M" shows "B \<in> fmeasurable M"
ntr_is_tr: fixes c :: "'f list multiset"   and ww :: "('g, 'h) label list list"   and c' :: "'f list multiset"   and fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme" assumes "(c, ww, c') \<in> trcl (ntr fg)" shows "(c, foldl (@) [] ww, c') \<in> trcl (tr fg)"
ex_ex_eq_hint: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "(\<exists>x. (\<exists>xs ys. x = f xs ys \<and> P xs ys) \<and> Q x) = (\<exists>xs ys. Q (f xs ys) \<and> P xs ys)"
free_monoidal_category_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "free_monoidal_category C \<equiv> monoidal_language C"
absch: fixes s :: "char list \<Rightarrow> int"   and k :: "nat" assumes "1 + s ''x'' = 2 ^ k" shows "5 * k \<le> 96 + 100 * Discrete.log (nat (s ''x''))"
that: fixes x :: "'b" assumes "x |\<in>| S"   and "f x = Some y" shows "thesis"
exhaust: fixes y :: "staticb" obtains "y = Static"   | "y = NonStatic"
wp_under_wlp_def: fixes prog :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "wp_under_wlp prog \<equiv> \<forall>P. unitary P \<longrightarrow> wp prog P \<tturnstile> wlp prog P"
scast_def: fixes w :: "'b Word.word" shows "SCAST('b \<rightarrow> 'a) w = word_of_int (sint w)"
comp_inv_arr: fixes f :: "'d"   and g :: "'d" assumes "D.inverse_arrows f g" shows "g \<cdot>\<^sub>D f = D.dom f"
t_lApp: fixes \<Gamma> :: "(name \<times> ty) list"   and t1 :: "llam"   and T1 :: "ty"   and T2 :: "ty"   and t2 :: "llam" assumes " \<Gamma> \<turnstile> t1 : T1 \<rightarrow> T2 "   and " \<Gamma> \<turnstile> t2 : T1 " shows " \<Gamma> \<turnstile> lApp t1 t2 : T2 "
bisimilar_symp: shows "symp (\<sim>\<cdot>\<^sub>L)"
C: shows "Crypt (sesK (c * f)) \<lbrace>pubAK (c * (s + a * b')), Auth_Data g b', pubAK g, Crypt (priSK CA) (Hash (pubAK g))\<rbrace> \<in> synth (analz (A \<union> spies evsFR4))"
Ge_def: 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" shows "Tarski_neutral_dimensionless.Ge Bet Cong A B C D \<equiv> Tarski_neutral_dimensionless.Le Bet Cong C D A B"
enn2ereal_cases: fixes y :: "ereal" assumes "y \<in> {x. 0 \<le> x}" obtains x :: "ennreal" where "y = enn2ereal x"
natural_transformation_axioms: shows "natural_transformation J (\<cdot>) \<chi>f.A.map D' \<tau>o\<chi>f.map"
insert_3: shows "finite {M. vars (elements M) = V' \<and> uniq (elements M) \<and> consistent (elements M)}"
lunit'_simps_4: fixes f :: "'a" assumes "ide f" shows "local.dom \<l>\<^sup>-\<^sup>1[f] = f"
assms_1: shows "\<forall>c. strip (f c) = strip c"
prv_zer_dsj_suc: fixes t :: "'trm"   and x :: "'var" assumes "t \<in> atrm"   and "x \<in> var"   and "x \<notin> FvarsT t" shows "prv (dsj (eql t zer) (exi x (eql t (suc (Var x)))))"
Grp_def: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "BNF_Def.Grp A f = (\<lambda>a b. b = f a \<and> a \<in> A)"
eln_simps_1: shows "eln 0 = - \<infinity>"
hyps_1: shows "\<Gamma>\<turnstile> (c, s) \<rightarrow> (c', s')"