hcju/isabelleps huggingface dataset:
pre_bfs2_def: fixes src :: "nat"   and dst :: "nat" shows "pre_bfs2 src dst \<equiv> init_state src \<bind> (\<lambda>s. while\<^sub>T (\<lambda>(f, PRED, C, N, d). f = False \<and> C \<noteq> {}) (\<lambda>(f, PRED, C, N, d). assert (C \<noteq> {}) \<bind> (\<lambda>_. IICF_Set.op_set_pick C \<bind> (\<lambda>v. let C = C - {v} in assert (v \<in> V) \<bind> (\<lambda>_. succ v \<bind> (\<lambda>sl. inner_loop2 dst sl v PRED N \<bind> (\<lambda>(f, PRED, N). if f then return (f, PRED, C, N, d + 1) else assert (assn1 src dst (f, PRED, C, N, d)) \<bind> (\<lambda>_. if C = {} then let C = N; N = {}; d = d + 1 in return (f, PRED, C, N, d) else return (f, PRED, C, N, d)))))))) s \<bind> (\<lambda>(f, PRED, uu_, uu_, d). if f then return (Some (d, PRED)) else return None))"
labeled_inf_have_prems: fixes \<iota> :: "('f \<times> 'l) inference" assumes "\<iota> \<in> Inf_FL" shows "prems_of \<iota> \<noteq> []"
elims: fixes x :: "'b acom"   and y :: "com" assumes "strip x = y" obtains P :: "'b" where "x = SKIP {P__}" and "y = SKIP"   | xa :: "char list" and e :: "aexp" and P :: "'b" where "x = xa ::= e {Pa__}" and "y = xa ::= e"   | C\<^sub>1 :: "'b acom" and C\<^sub>2 :: "'b acom" where "x = C\<^sub>1__;; C\<^sub>2__" and "y = strip C\<^sub>1__;; strip C\<^sub>2__"   | b :: "bexp" and P\<^sub>1 :: "'b" and C\<^sub>1 :: "'b acom" and P\<^sub>2 :: "'b" and C\<^sub>2 :: "'b acom" and P :: "'b" where "x = IF b__ THEN {P\<^sub>1} C\<^sub>1 ELSE {P\<^sub>2} C\<^sub>2 {Pb__}" and "y = IF b__ THEN strip C\<^sub>1 ELSE strip C\<^sub>2"   | I :: "'b" and b :: "bexp" and P :: "'b" and C :: "'b acom" and Q :: "'b" where "x = {I} WHILE b DO {P} C {Q}" and "y = WHILE b DO strip C"
max_ideal_shapeD_internalized: fixes F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" assumes "class.finite TYPE('c)"   and "generates_max_ideal UNIV F" obtains a :: "'c \<Rightarrow> 'd" where "ideal F = ideal (range (\<lambda>x. monomial (1::'d) (monomial 1 x) - monomial (a x) 0))"
assms_2: shows "while_option fst (rderiv_and_add as) (True, [\<guillemotleft>r\<guillemotright>]) = Some (b, rs)"
min_maxsimpchain_pmaxsimpchain: fixes X :: "'b set set"   and xs :: "'b set list" assumes "SimplicialComplex X"   and "SimplicialComplex.min_maxsimpchain X xs" shows "SimplicialComplex.pmaxsimpchain X xs"
sums_If_finite_set: fixes A :: "nat set"   and f :: "nat \<Rightarrow> 'b" assumes "finite A" shows "(\<lambda>r. if r \<in> A then f r else (0::'b)) sums sum f A"
mult_unit_dvd_iff': fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one" shows "dvd.dvd times (times a b) c = dvd.dvd times b c"
inv_character_principal: fixes G :: "('c, 'd) monoid_scheme" shows "inv_character (principal_char G) = principal_char G"
list_augment_idem: fixes i :: "nat"   and xs :: "'b list" assumes "i < length xs" shows "list_augment xs i (xs ! i) = xs"
no_spoofing_algorithm_executable: fixes iface :: "iface"   and ipassmt :: "iface \<Rightarrow> ('b word \<times> nat) list option"   and rs :: "'b common_primitive rule list"   and allowed :: "'b wordinterval"   and denied :: "'b wordinterval" shows "no_spoofing_algorithm_executable iface ipassmt rs allowed denied = no_spoofing_algorithm iface ipassmt rs (wordinterval_to_set allowed) (wordinterval_to_set denied)"
UniverseUnion: fixes u :: "ZF"   and v :: "ZF" assumes "Elem u U"   and "Elem v U" shows "Elem (HOLZF.union u v) U"
to_complex_one_one: shows "to_complex 1\<^sub>h = 1"
upto_append_intsFrom: fixes m :: "int"   and n :: "int" assumes "m \<le> n" shows "upto\<cdot>(MkI\<cdot>m)\<cdot>(MkI\<cdot>n) ++ intsFrom\<cdot>(MkI\<cdot>n + 1) = intsFrom\<cdot>(MkI\<cdot>m)"
neg_length_pos: fixes w :: "bit list" assumes "bv_to_int w < - 1" shows "0 < length w"
assms_9: shows "(hs, data') = add_indices aux (snd data)"
success_changeI: fixes f :: "'a \<Rightarrow> 'a"   and r :: "'a ref"   and h :: "heap" shows "success (Ref.change f r) h"
dvd_0_left_iff: fixes a :: "'a" shows "((0::'a) dvd a) = (a = (0::'a))"
elem_cong_l: fixes S :: "('a, 'b) eq_object_scheme"   and a :: "'a"   and a' :: "'a"   and A :: "'a set" assumes "equivalence S"   and "a \<in> carrier S"   and "a' \<in> carrier S"   and "A \<subseteq> carrier S"   and "a' .=\<^bsub>S\<^esub> a"   and "a .\<in>\<^bsub>S\<^esub> A" shows "a' .\<in>\<^bsub>S\<^esub> A"
implicit_dependent_with: shows "dependent_with (+) (0::'a) (*\<^sub>R) = subspace.dependent"
UINF_true: shows "(\<Sqinter> i \<bullet> true) = true"
least_Upper_above: fixes L :: "('a, 'b) gorder_scheme"   and s :: "'a"   and A :: "'a set"   and x :: "'a" assumes "is_lub L s A"   and "x \<in> A"   and "A \<subseteq> carrier L" shows "x \<sqsubseteq>\<^bsub>L\<^esub> s"
cup_pmatch_list_length_neq: fixes vs :: "v list"   and ps :: "pat list"   and cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and m :: "(char list \<times> v) list match_result" assumes "length vs \<noteq> length ps" shows "Matching.fold2 (\<lambda>p v m. case m of No_match \<Rightarrow> No_match | Match_type_error \<Rightarrow> Match_type_error | Match env \<Rightarrow> cupcake_pmatch cenv p v env) Match_type_error ps vs m = Match_type_error"
k_2: shows "infinite {z \<in> S. (case z of (t, u) \<Rightarrow> t\<^sup>2 - int (u\<^sup>2 * D)) = k}"
bad_sticky2: fixes s1 :: "'s1" assumes "bad1 s1" shows "callee_invariant_on oracle2 (\<lambda>s2. bad2 s2 \<and> X_bad s1 s2) \<I>"
saturate_sound: fixes C :: "'a literal multiset"   and CC :: "'a literal multiset set"   and I :: "'a set" assumes "C \<in> saturate CC"   and "I \<Turnstile>s CC" shows "I \<Turnstile> C"
min_simps_2: fixes b :: "'a"   and a :: "'a" assumes "b < a" shows "min a b = b"
that: fixes g :: "real \<Rightarrow> real" assumes "g \<in> o(\<lambda>x. ln (b * x) powr - (e / 2 + 1))"   and "\<forall>\<^sub>F x in at_top. 1 - ln (b * x + hb * x / ln x powr (1 + e)) powr (- e / 2) \<le> 1 - ln (b * x) powr (- e / 2) - g x"   and "\<forall>\<^sub>F x in at_top. 0 < 1 - ln (b * x) powr (- e / 2) - g x" shows "thesis"
cast_below_imp_below: fixes A :: "'a defl"   and B :: "'a defl" assumes "cast\<cdot>A \<sqsubseteq> cast\<cdot>B" shows "A \<sqsubseteq> B"
int_relprime_power_divisors: fixes n :: "nat"   and a :: "int"   and b :: "int"   and c :: "int" assumes "0 < n"   and "0 \<le> a"   and "0 \<le> b"   and "a * b = c ^ n"   and "coprime a b" shows "\<exists>k. a = k ^ n"
countable_image_inj_eq: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "inj_on f S" shows "countable (f ` S) = countable S"
binary_meet_exists: fixes x :: "'a"   and y :: "'a" shows "\<exists>l. l \<sqsubseteq> x \<and> l \<sqsubseteq> y \<and> (\<forall>z. z \<sqsubseteq> x \<longrightarrow> z \<sqsubseteq> y \<longrightarrow> z \<sqsubseteq> l)"
htchLocSet_def: fixes PP :: "nat set" shows "htchLocSet PP \<equiv> \<Union> {h. \<exists>\<iota>\<in>PP. h = htchLoc \<iota>}"
right_impl_ded: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b * a l\<rightarrow> c = b l\<rightarrow> a l\<rightarrow> c"
flrestriction_vsubset_fpairs: fixes r :: "V"   and A :: "V" shows "r \<restriction>\<^sup>l\<^sub>\<bullet> A \<subseteq>\<^sub>\<circ> fpairs r"
that: fixes C :: "real" assumes "lb \<le> C"   and "\<And>n. norm (f n) \<le> C" shows "thesis"
infsetsum_cnj: fixes f :: "'c \<Rightarrow> complex"   and M :: "'c set" shows "(\<Sum>\<^sub>ax\<in>M. cnj (f x)) = cnj (infsetsum f M)"
sp_append: fixes g :: "('d \<times> nat, 'e) rgraph_scheme"   and v1 :: "'d \<times> nat"   and es1 :: "('d \<times> nat) edge list"   and es2 :: "('d \<times> nat) edge list"   and v2 :: "'d \<times> nat"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set" shows "SubRel.subpath g v1 (es1 @ es2) v2 subs = (\<exists>v. SubRel.subpath g v1 es1 v subs \<and> SubRel.subpath g v es2 v2 subs)"
agree2': shows "Vagree \<nu> \<nu>' (FVT t2)"
assms_2: shows "f = set {\<langle>c, f\<lparr>c\<rparr>\<rangle>}"
emeasure_bind_pmf: fixes M :: "'i pmf"   and N :: "'i \<Rightarrow> 'h pmf"   and X :: "'h set" shows "emeasure (measure_pmf (M \<bind> N)) X = \<integral>\<^sup>+ x. emeasure (measure_pmf (N x)) X \<partial>measure_pmf M"
sumAssoc: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" shows "(P \<oplus> Q) \<oplus> R \<sim> P \<oplus> Q \<oplus> R"
that: fixes g :: "'a \<Rightarrow> 'b" assumes "continuous_on (U - L) g"   and "g ` (U - L) \<subseteq> T"   and "\<And>x. x \<in> S \<Longrightarrow> g x = f x" shows "thesis"
f_unique: fixes f' :: "nat \<times> nat \<Rightarrow> 'a" assumes "NPreflow c s t f'"   and "cf = cf_of f'" shows "f' = f"
fresh_18: fixes a :: "coname"   and x2 :: "coname"   and x3 :: "trm"   and x1 :: "coname" shows "a \<sharp> OrR2 <x2>.x3 x1 = (a \<sharp> [x2].x3 \<and> a \<sharp> x1)"
finite_Inf_fun: fixes A :: "('a \<Rightarrow> 'b) set" shows "Inf A = Finite_Set.fold inf top A"
negligible_iff_null_sets: fixes S :: "'c set" shows "negligible S = (S \<in> null_sets lebesgue)"
expectation_gpv_restrict_gpv: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and fail :: "ennreal"   and f :: "'c \<Rightarrow> ennreal" assumes "\<I> \<turnstile>g gpv \<surd>" shows "expectation_gpv fail \<I> f (restrict_gpv \<I> gpv) = expectation_gpv fail \<I> f gpv"
p: shows "p = ((1 - s1 - s2) / (1 - s1)) *\<^sub>R q + (s2 / (1 - s1)) *\<^sub>R r"
left_inverse: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and inverse :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "group f z inverse" shows "f (inverse a) a = z"
path_appendE: fixes l1 :: "'vtx"   and l3 :: "'vtx"   and xs :: "('vtx \<times> 'lbl \<times> 'vtx) list"   and ys :: "('vtx \<times> 'lbl \<times> 'vtx) list" assumes "path l1 l3 (xs @ ys)" shows "\<exists>l2. path l2 l3 ys \<and> path l1 l2 xs"
atLeast_iff: fixes i :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'a" shows "(i \<in> ord.atLeast less_eq k) = less_eq k i"
hyps: shows "isOK (check_pairwise c xs) = (\<forall>j<length xs. \<forall>i<j. isOK (c (xs ! i) (xs ! j)))"
a4: shows "\<forall>nP e1 e2. \<not> P (nP e1) (nP e2) \<longrightarrow> \<not> P (nP e1) \<bottom>"
g_Zorn_lemma3: fixes D :: "('h, 'i) Order_scheme" assumes "Order D"   and "inductive_set D" shows "\<exists>m\<in>carrier D. maximal\<^bsub>D\<^esub> m"
wp2_Seq: fixes c\<^sub>1 :: "com"   and c\<^sub>2 :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "wp\<^sub>2 (c\<^sub>1;; c\<^sub>2) Q = wp\<^sub>2 c\<^sub>1 (wp\<^sub>2 c\<^sub>2 Q)"
elims: fixes x :: "('a, 'b) Term.term"   and y :: "nat" assumes "num_syms x = y" obtains xa :: "'b" where "x = Var xa" and "y = 1"   | f :: "'a" and ts :: "('a, 'b) Term.term list" where "x = Fun f ts" and "y = Suc (sum_list (map num_syms ts))"
invertibleE: fixes u :: "'a"   and thesis :: "bool" assumes "IA.additive.sub.invertible u"   and "\<And>v. \<lbrakk>addA u v = zeroA \<and> addA v u = zeroA; v \<in> I\<rbrakk> \<Longrightarrow> thesis"   and "u \<in> I" shows "thesis"
prv_imp_refl2: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c"   and \<psi> :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi> \<in> fmla"   and "\<psi> \<in> fmla"   and "\<phi> = \<psi>" shows "prv (imp \<phi> \<psi>)"
lprefix_lappend: fixes xs :: "'c llist"   and ys :: "'c llist" shows "xs \<sqsubseteq> lappend xs ys"
heq: fixes z :: "real \<times> 'c" assumes "z \<in> {0..1} \<times> U" shows "h z = p (k z)"
small_imp_big: fixes \<pi> :: "char list \<Rightarrow> com option"   and cs\<^sub>1 :: "com \<times> (char list \<Rightarrow> int \<Rightarrow> int)"   and s\<^sub>2 :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "small_steps \<pi> cs\<^sub>1 (Some (SKIP, s\<^sub>2))" shows "\<pi>: cs\<^sub>1 \<Rightarrow> s\<^sub>2"
naturality: fixes f :: "'a \<times> 'a \<times> 'a" assumes "VVV.arr f" shows "\<a>[fst (VVV.cod f), fst (snd (VVV.cod f)), snd (snd (VVV.cod f))] \<cdot> HoHV f = HoVH f \<cdot> \<a>[fst (VVV.dom f), fst (snd (VVV.dom f)), snd (snd (VVV.dom f))]"
RedLAss: fixes V :: "char list"   and e :: "expr"   and s :: "char list \<Rightarrow> val option" shows "\<langle>V:=e,s\<rangle> \<rightarrow> \<langle>Skip,s(V := interpret e s)\<rangle>"
bij_betw_insert: fixes b :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and A' :: "'d set" assumes "b \<notin> A"   and "f b \<notin> A'"   and "bij_betw f A A'" shows "bij_betw f (insert b A) (insert (f b) A')"
valid_plan_valid_head: fixes h :: "'d"   and as :: "'d list"   and PROB :: "'d set" assumes "h # as \<in> valid_plans PROB" shows "h \<in> PROB"
vimage_vsubset: fixes r :: "V"   and A :: "V"   and B :: "V"   and C :: "V" assumes "r \<subseteq>\<^sub>\<circ> A \<times>\<^sub>\<circ> B" shows "r `\<^sub>\<circ> C \<subseteq>\<^sub>\<circ> B"
Qg_i: fixes N :: "'a set" assumes "G \<triangleright> N" shows "\<forall>x\<in>set_rcs G N. c_top G N (c_iop G N x) x = N"
chain_Field: fixes M :: "('v \<times> 'v \<Rightarrow> ennreal) set" assumes "M \<in> Chains leq"   and "M \<noteq> {}" shows "\<Sqinter> M \<in> F"
reduced_word_for_minimal: fixes A :: "'a set"   and a :: "'a"   and as :: "'a list"   and bs :: "'a list" assumes "reduced_word_for A a as"   and "bs \<in> lists A"   and "sum_list bs = a" shows "order.greater_eq (length bs) (length as)"
a_adopt_node_locs_def: fixes parent :: "(_) object_ptr option"   and owner_document :: "(_) document_ptr"   and document_ptr :: "(_) document_ptr" shows "local.a_adopt_node_locs parent owner_document document_ptr = (if parent = None then {} else remove_child_locs (the parent) owner_document) \<union> set_disconnected_nodes_locs document_ptr \<union> set_disconnected_nodes_locs owner_document"
sr: shows "s \<in> reachable (pnet np (p1 \<parallel> p2)) TT"
pre1: fixes S :: "'x set"   and \<sigma> :: "'\<sigma>" assumes "S \<subseteq> S0"   and "S \<noteq> {}"   and "\<not> c \<sigma>"   and "I (S0 - S) \<sigma>"   and "\<forall>x\<in>S. \<forall>y\<in>S0 - S. R y x" shows "P \<sigma>"
mod_pos_pos_trivial: fixes k :: "int"   and l :: "int" assumes "0 \<le> k"   and "k < l" shows "k mod l = k"
db\<^sub>s\<^sub>s\<^sub>t_append: fixes A :: "('a, 'b) stateful_strand_step list"   and B :: "('a, 'b) stateful_strand_step list"   and I :: "'b \<Rightarrow> ('a, 'b) Term.term"   and D :: "(('a, 'b) Term.term \<times> ('a, 'b) Term.term) list" shows "db'\<^sub>s\<^sub>s\<^sub>t (A @ B) I D = db'\<^sub>s\<^sub>s\<^sub>t B I (db'\<^sub>s\<^sub>s\<^sub>t A I D)"
in_quotient_imp_non_empty: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and X :: "'b set" assumes "equiv A r"   and "X \<in> A // r" shows "X \<noteq> {}"
da_elim_cases_12: fixes Env :: "env"   and B :: "lname set"   and l :: "label"   and e :: "expr"   and c :: "stmt"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In1r (l\<bullet> While(e) c)\<guillemotright> A" obtains E :: "assigned" and C :: "assigned" where "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E" and "Env\<turnstile> B \<union> assigns_if True e \<guillemotright>\<langle>c\<rangle>\<guillemotright> C" and "nrm A = nrm C \<inter> (B \<union> assigns_if False e)" and "brk A = brk C"
fv_eq_FunI: fixes T :: "('c, 'd) Term.term list"   and S :: "('e, 'd) Term.term list"   and f :: "'c"   and g :: "'e" assumes "length T = length S"   and "\<And>i. i < length T \<Longrightarrow> fv (T ! i) = fv (S ! i)" shows "fv (Fun f T) = fv (Fun g S)"
perm_closed: fixes as :: "'a list"   and bs :: "'a list" assumes "as <~~> bs"   and "set as \<subseteq> carrier G" shows "set bs \<subseteq> carrier G"
loose_bvar_leq: fixes p :: "nat"   and k :: "nat"   and t :: "Core.term" assumes "p \<le> k"   and "loose_bvar t k" shows "loose_bvar t p"
CauchysMeanTheorem_Less: fixes z :: "real list" assumes "pos z"   and "0 < het z" shows "gmean z < mean z"
is_slim_if_subdivide: fixes e :: "'a \<times> 'a"   and w :: "'a"   and V :: "'a set" assumes "pair_pre_digraph.is_slim (subdivide G' e w) V"   and "e \<in> parcs G'"   and "w \<notin> pverts G'"   and "w \<notin> V" shows "pd_G'.is_slim V"
f1_def: fixes uu :: "bool"   and uv :: "bool"   and uw :: "bool"   and ux :: "nat llist" shows "f1 uu uv uw ux = ()"
P_mod_diffxxx1: fixes t :: "'a"   and R' :: "('a set, 'e) Ring_scheme"   and Y :: "'a set"   and f :: "'a"   and g :: "'a"   and h :: "'a"   and u :: "'a"   and v :: "'a"   and ra :: "'a"   and m :: "nat" assumes "Idomain S"   and "t \<in> carrier S"   and "t \<noteq> \<zero>\<^bsub>S\<^esub>"   and "maximal_ideal S (S \<diamondsuit>\<^sub>p t)"   and "PolynRg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y"   and "f \<in> carrier R"   and "g \<in> carrier R"   and "h \<in> carrier R"   and "f \<noteq> \<zero>"   and "g \<noteq> \<zero>"   and "h \<noteq> \<zero>"   and "u \<in> carrier R"   and "v \<in> carrier R"   and "erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) g \<noteq> \<zero>\<^bsub>R'\<^esub>"   and "erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) h \<noteq> \<zero>\<^bsub>R'\<^esub>"   and "ra \<in> carrier R"   and "f \<plusminus> -\<^sub>a g \<cdot>\<^sub>r h = t^\<^bsup>S m\<^esup> \<cdot>\<^sub>r ra"   and "0 < m"   and "erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) u \<cdot>\<^sub>r\<^bsub>R'\<^esub> erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) g \<plusminus>\<^bsub>R'\<^esub> erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) v \<cdot>\<^sub>r\<^bsub>R'\<^esub> erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) h = erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) ra" shows "P_mod R S X (S \<diamondsuit>\<^sub>p (t^\<^bsup>S Suc m\<^esup>)) (f \<plusminus> -\<^sub>a (g \<plusminus> t^\<^bsup>S m\<^esup> \<cdot>\<^sub>r v) \<cdot>\<^sub>r (h \<plusminus> t^\<^bsup>S m\<^esup> \<cdot>\<^sub>r u))"
brc_initial_invar: fixes \<delta> :: "('a, 'b) ta_rule dlist" assumes "ls.invar \<delta>" shows "brc_initial \<delta> \<in> brc_invar_add"
functions_of_number_partition_of: fixes A :: "'c set"   and B :: "'d set"   and F :: "('c \<Rightarrow> 'd) set" assumes "finite A"   and "finite B"   and "F \<in> (A \<rightarrow>\<^sub>E B) // domain_and_range_permutation A B" shows "Twelvefold_Way_Entry10.functions_of A B (number_partition_of A B F) = F"
step_term2: fixes s :: "trm"   and t :: "trm"   and u :: "trm" assumes "s \<longlonglongrightarrow>\<^sup>* t"   and "t \<longlonglongrightarrow> u" shows "s \<longlonglongrightarrow>\<^sup>* u"
supp_mono: fixes B :: "'b set"   and A :: "'b set" assumes "finite B"   and "A \<subseteq> B" shows "supp A \<subseteq> supp B"
boundedE: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "A \<noteq> {}"   and "x \<le> Min A" obtains "\<And>a. a \<in> A \<Longrightarrow> x \<le> a"
defAss_dominating: fixes n :: "'node"   and g :: "'g"   and v :: "'val" assumes "n \<in> set (\<alpha>n g)" shows "defAss g n v = (\<exists>m\<in>set (\<alpha>n g). old.dominates g m n \<and> v \<in> allDefs g m)"
FAddSA_States: fixes S :: "'d"   and SA :: "('d, 'e, 'f) seqauto"   and A :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option" assumes "S \<in> States SA" shows "(A [f+] (S, SA)) = A"
progress_mono: fixes \<pi> :: "'a prefix"   and \<pi>' :: "'a prefix" assumes "\<pi> \<le> \<pi>'" shows "progress \<pi> \<le> progress \<pi>'"
hom_add: fixes x :: "'a"   and y :: "'a"   and d :: "nat" shows "monom (x + y) d = monom x d + monom y d"
points_to_Mutate: fixes y :: "'g"   and x :: "'g"   and s :: "'h process_name \<Rightarrow> ('e, 'h, 'f, 'g) Model.local_state"   and p :: "'h process_name"   and ws :: "('e, 'f, 'g) mem_store_action list"   and r :: "'g"   and f :: "'e"   and opt_r' :: "'g option" shows "(x points_to y) (s(sys := s sys\<lparr>heap := (sys_heap s)(r := map_option (\<lambda>obj. obj\<lparr>obj_fields := (obj_fields obj)(f := opt_r')\<rparr>) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p := ws)\<rparr>)) = (r \<noteq> x \<and> (x points_to y) s \<or> r = x \<and> obj_at (\<lambda>s. True) r s \<and> (opt_r' = Some y \<or> (x points_to y) s \<and> obj_at (\<lambda>obj. \<exists>f'. obj_fields obj f' = Some y \<and> f \<noteq> f') r s))"
ascP_def: fixes f :: "'b list \<Rightarrow> 'b list" shows "ascP f = (\<forall>xs. f xs = f [] @ xs)"
conj_idem: fixes P :: "'a upred" shows "(P \<and> P) = P"
fg: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b" assumes "three_composable_identities_in_bicategory_of_spans V H \<a> \<i> src trg f g h" shows "src f = trg g"
Dp_def: fixes f :: "int poly" shows "Dp f = map_poly (\<lambda>a. a div m) f"
discriminant_def: fixes s\<^sub>e :: "real"   and v\<^sub>e :: "real"   and a\<^sub>e :: "real"   and s\<^sub>o :: "real"   and v\<^sub>o :: "real"   and a\<^sub>o :: "real" shows "discriminant s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o \<equiv> (v\<^sub>o - v\<^sub>e)\<^sup>2 - 2 * (a\<^sub>o - a\<^sub>e) * (s\<^sub>o - s\<^sub>e)"
u'_def: shows "u' = (\<lambda>x. sqrt (4 - x))"
weighted_majority_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" shows "weighted_majority R x y = int (card (supporters R x y)) - int (card (supporters R y x))"
Drop_neq_Accept_unknown_match_tacs_2: shows "in_doubt_deny Drop \<noteq> in_doubt_deny Accept"
Liminf_state_def: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist" shows "Liminf_state Sts = (Liminf_llist (lmap N_of_state Sts), Liminf_llist (lmap P_of_state Sts), Liminf_llist (lmap Q_of_state Sts))"
ket_injective: fixes i :: "'b"   and j :: "'b" shows "(ket i = ket j) = (i = j)"
elims_1: fixes x :: "bool list"   and xa :: "tree"   and y :: "bool" assumes "path x xa = y" obtains T :: "tree" where "x = []" and "xa = T" and "y"   | d :: "bool" and ds :: "bool list" and T1 :: "tree" and T2 :: "tree" where "x = d # ds" and "xa = Branching T1 T2" and "y = (if d then path ds T1 else path ds T2)"   | v :: "bool" and va :: "bool list" where "x = v # va" and "xa = Leaf" and "\<not> y"
ext_inject: fixes worlds :: "'g set"   and relations :: "'e \<Rightarrow> ('g \<times> 'g) set"   and valuation :: "'g \<Rightarrow> 'f \<Rightarrow> bool"   and more :: "'h"   and worlds' :: "'g set"   and relations' :: "'e \<Rightarrow> ('g \<times> 'g) set"   and valuation' :: "'g \<Rightarrow> 'f \<Rightarrow> bool"   and more' :: "'h" shows "(\<lparr>worlds = worlds, relations = relations, valuation = valuation, \<dots> = more\<rparr> = \<lparr>worlds = worlds', relations = relations', valuation = valuation', \<dots> = more'\<rparr>) = (worlds = worlds' \<and> relations = relations' \<and> valuation = valuation' \<and> more = more')"
for_g: fixes y :: "'a" assumes "norm y < \<delta> / 2" shows "\<exists>!x. norm x < \<delta> \<and> f x = y"
T_nth_eq: fixes xs :: "'a list"   and n :: "nat" shows "T_nth xs n = min n (length xs) + 1"
boundedI: fixes b :: "'d multiset"   and a :: "'d multiset"   and c :: "'d multiset" assumes "b \<subseteq># a"   and "c \<subseteq># a" shows "b \<union># c \<subseteq># a"