hcju/isabelleps huggingface dataset:
ConstElim1: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and os :: "Val list"   and S :: "(Var \<times> Val) list"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and v :: "Val" assumes "(M, l, (os, S, h), n, ll, t) \<in> Step"   and "get_ins M l = Some (const v)" shows "n = Suc 0 \<and> t = (v # os, S, h) \<and> ll = l + 1"
cong: fixes g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "\<And>a. g a = h a" shows "Sum_any g = Sum_any h"
ac_Zp_p: shows "ac_Zp ([p] \<cdot> \<one>) = \<one>"
open_iff: fixes A :: "'b set"   and m :: "'b llist set" shows "(m open\<^bsub>topo (\<Union>s\<in>A\<^sup>\<star>. {suff A s})\<^esub>) = (m \<in> topo (\<Union>s\<in>A\<^sup>\<star>. {suff A s}))"
R'_def: shows "R' \<equiv> Regions.region \<X> I r"
ar_coset_mem2: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and a :: "'c"   and x :: "'c" assumes "Ring R"   and "ideal R I"   and "a \<in> carrier R"   and "x \<in> a \<uplus>\<^bsub>R\<^esub> I" shows "\<exists>h\<in>I. x = a \<plusminus>\<^bsub>R\<^esub> h"
intros_1: fixes G :: "prog"   and m :: "qtname \<times> memberdecl"   and C :: "qtname" assumes "G\<turnstile> mbr m declared_in C"   and "declclass m = C" shows "G \<turnstile> m member_of C"
simps_5: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and T :: "typ"   and t :: "Core.term" shows "term_ok' \<Sigma> (Abs T t) = (typ_ok_sig \<Sigma> T \<and> term_ok' \<Sigma> t)"
Fst: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<langle>X, Y\<rangle> \<in> analz H" shows "X \<in> analz H"
arr_tensor: fixes f :: "'d"   and g :: "'d" assumes "D.arr f"   and "D.arr g" shows "D.arr (f \<otimes>\<^sub>D g)"
elims_2: fixes x :: "('c, 'd) fmap"   and xa :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" assumes "sat_precond_as x xa" obtains s :: "('c, 'd) fmap" where "x = s__" and "xa = []"   | s :: "('c, 'd) fmap" and a :: "('c, 'd) fmap \<times> ('c, 'd) fmap" and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" where "x = s" and "xa = a # as" and "fst a \<subseteq>\<^sub>f s \<and> sat_precond_as (state_succ s a) as"
mod_mult_self2: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus a (times b c)) b = modulo a b"
equivalentFormulaeAppend: fixes formula1 :: "Literal list list"   and formula1' :: "Literal list list"   and formula2 :: "Literal list list" assumes "equivalentFormulae formula1 formula1'" shows "equivalentFormulae (formula1 @ formula2) (formula1' @ formula2)"
has_witness_two: fixes n :: "nat"   and S :: "'a set" assumes "has n S"   and "2 \<le> n" shows "\<exists>x y. hasw [x, y] S"
rref_condition3_explicit: fixes A :: "(('b, 'c) vec, 'd) vec"   and i :: "'d" assumes "reduced_row_echelon_form A"   and "i < i + (1::'d)"   and "\<not> is_zero_row i A"   and "\<not> is_zero_row (i + (1::'d)) A" shows "(LEAST n. A $ i $ n \<noteq> (0::'b)) < (LEAST n. A $ (i + (1::'d)) $ n \<noteq> (0::'b))"
Lfilled_exact_app_imp_exists_Lfilled: fixes ves :: "e list"   and n :: "nat"   and lholed :: "Lholed"   and es :: "e list"   and LI :: "e list" assumes "const_list ves"   and "Lfilled_exact n lholed (ves @ es) LI" shows "\<exists>lholed'. Lfilled n lholed' es LI"
f'_def: shows "f' = (\<lambda>x. (x + 1) * exp x)"
jprop: shows "j < length cms\<^sub>1 \<and> x \<notin> differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems j"
closed_tree_def: fixes T :: "tree"   and Cs :: "fterm literal set set" shows "closed_tree T Cs = (anybranch T (\<lambda>p. closed_branch p T Cs) \<and> anyinternal T (\<lambda>p. \<not> falsifies\<^sub>c\<^sub>s p Cs))"
leg1_commutes: shows "\<pi>.cod.leg1 \<cdot> \<pi>.chine = \<pi>.dom.leg1"
simps_1: fixes \<theta>1 :: "('b, 'd) trm"   and \<theta>2 :: "('b, 'd) trm"   and \<sigma> :: "'e \<Rightarrow> ('b, 'c, 'd) formula" shows "PFsubst (Geq \<theta>1 \<theta>2) \<sigma> = Geq \<theta>1 \<theta>2"
lspasl_starr_der: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool" assumes "class.heap_sep_algebra plus zero sep_disj"   and "heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0"   and "\<not> sep_algebra.sep_conj plus sep_disj A B h0" shows "heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> \<not> (A h1 \<or> sep_algebra.sep_conj plus sep_disj A B h0) \<and> heap_sep_algebra.starr_applied plus sep_disj h1 h2 h0 (sep_algebra.sep_conj plus sep_disj A B) \<or> heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> \<not> (B h2 \<or> sep_algebra.sep_conj plus sep_disj A B h0) \<and> heap_sep_algebra.starr_applied plus sep_disj h1 h2 h0 (sep_algebra.sep_conj plus sep_disj A B)"
supr_under: fixes a :: "'b" assumes "a \<in> Field s" shows "a = s.supr (s.under a)"
BlockSpec: fixes P :: "'d set"   and init :: "'d \<Rightarrow> 'd"   and P' :: "'e \<Rightarrow> 'd set"   and Q' :: "'e \<Rightarrow> 'd set"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> 'd set"   and A' :: "'e \<Rightarrow> 'd set"   and A :: "'d set"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'g) com option"   and \<Theta> :: "('d set \<times> 'f \<times> 'd set \<times> 'd set) set"   and F :: "'g set"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'f, 'g) com"   and Q :: "'d set"   and bdy :: "('d, 'f, 'g) com" assumes "P \<subseteq> {s. \<exists>Z. init s \<in> P' Z \<and> (\<forall>t. t \<in> Q' Z \<longrightarrow> return s t \<in> R s t) \<and> (\<forall>t. t \<in> A' Z \<longrightarrow> return s t \<in> A)}"   and "\<forall>s t. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(R s t) c s t Q,A"   and "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P' Z) bdy (Q' Z),(A' Z)" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P block init bdy return c Q,A"
dfs_eq_rtrancl: fixes y :: "bool list"   and x :: "bool list" assumes "nfa_is_node A y"   and "nfa_is_node A x" shows "subset_memb y (subset_dfs A x) = ((x, y) \<in> (succsr (subset_succs A))\<^sup>*)"
Err_sup_eq_OK_conv: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and ex :: "'c err"   and ey :: "'d err"   and z :: "'b" shows "(Err.sup f ex ey = OK z) = (\<exists>x y. ex = OK x \<and> ey = OK y \<and> f x y = z)"
maxsimp_map: fixes x :: "'a set" assumes "maxsimp x" shows "SimplicialComplex.maxsimp (f \<turnstile> X) (f ` x)"
lex_half_segments_lowest_vertex: fixes c :: "real \<times> real"   and d :: "real \<times> real"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "(c, d) \<in> set (half_segments_of_aform X)" shows "Counterclockwise_2D_Arbitrary.lex d (lowest_vertex (fst X, nlex_pdevs (snd X)))"
card_Suc_eq: fixes A :: "'a set"   and k :: "nat" shows "(card A = Suc k) = (\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {}))"
concurrent_direct_execution_simulates_store_buffer_execution: fixes ts\<^sub>s\<^sub>b :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m\<^sub>s\<^sub>b :: "nat \<Rightarrow> nat"   and x :: "'c"   and ts\<^sub>s\<^sub>b' :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m\<^sub>s\<^sub>b' :: "nat \<Rightarrow> nat"   and x' :: "'c"   and \<S>\<^sub>s\<^sub>b :: "nat \<Rightarrow> bool option"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option" assumes "(ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, x) \<Rightarrow>\<^sub>s\<^sub>b\<^sup>* (ts\<^sub>s\<^sub>b', m\<^sub>s\<^sub>b', x')"   and "initial\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b \<S>\<^sub>s\<^sub>b"   and "valid ts\<^sub>s\<^sub>b"   and "(ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) \<sim> (ts, m, \<S>)"   and "safe_reach_direct safe_free_flowing (ts, m, \<S>)" shows "\<exists>ts\<^sub>h' \<S>\<^sub>h' ts' m' \<S>'. (ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) \<Rightarrow>\<^sub>s\<^sub>b\<^sub>h\<^sup>* (ts\<^sub>h', m\<^sub>s\<^sub>b', \<S>\<^sub>h') \<and> ts\<^sub>s\<^sub>b' \<sim>\<^sub>h ts\<^sub>h'  \<and> (ts, m, \<S>) \<Rightarrow>\<^sub>d\<^sup>* (ts', m', \<S>') \<and> (ts\<^sub>h', m\<^sub>s\<^sub>b', \<S>\<^sub>h') \<sim> (ts', m', \<S>')"
weakenR: fixes \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset"   and F :: "'a formula" assumes "\<Gamma> \<Rightarrow> \<Delta>" shows "\<Gamma> \<Rightarrow> F, \<Delta>"
fl_subdg_is_fl_dghm_inc: shows "dghm_inc \<BB> \<CC> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>u\<^sub>l\<^sub>l\<^bsub>\<alpha>\<^esub> \<CC>"
rquot_empt_def: fixes M :: "nat bdd list \<times> bool list" shows "rquot_empt M = replicate (length (fst M)) False"
T_LAM: fixes x :: "W.var"   and \<Gamma> :: "(W.var \<times> tyS) list"   and T\<^sub>1 :: "ty"   and t :: "trm"   and T\<^sub>2 :: "ty" assumes "x \<sharp> \<Gamma>"   and " (x, Ty T\<^sub>1) # \<Gamma> \<turnstile> t : T\<^sub>2 " shows " \<Gamma> \<turnstile> Lam [x].t : T\<^sub>1\<rightarrow>T\<^sub>2 "
neg_equal_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "(uminus a = a) = (a = zero)"
separating_hyperplane_closed_compact: fixes S :: "'b set"   and T :: "'b set" assumes "convex S"   and "closed S"   and "convex T"   and "compact T"   and "T \<noteq> {}"   and "S \<inter> T = {}" shows "\<exists>a b. (\<forall>x\<in>S. a \<bullet> x < b) \<and> (\<forall>x\<in>T. b < a \<bullet> x)"
rel_gpv''_Pause: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and C :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and out :: "'f"   and c :: "('d, 'f, 'h) rpv"   and out' :: "'g"   and c' :: "('e, 'g, 'i) rpv" shows "rel_gpv'' A C R (Generative_Probabilistic_Value.Pause out c) (Generative_Probabilistic_Value.Pause out' c') = (C out out' \<and> (\<forall>x x'. R x x' \<longrightarrow> rel_gpv'' A C R (c x) (c' x')))"
fps_inverse_eq_0': fixes f :: "'a fps" assumes "inverse (fps_nth f 0) = (0::'a)" shows "inverse f = 0"
type_progress: fixes e :: "Syntax.term"   and \<tau> :: "ty" assumes "{$$} \<turnstile>\<^sub>W e : \<tau>" shows "value e \<or> (\<exists>e'. \<lless>[], e\<ggreater> I\<rightarrow> \<lless>[], e'\<ggreater>)"
dom_h': fixes x :: "nat"   and y :: "nat" assumes "0 < h' x y" shows "(x, y) \<in> R'"
cnf_of_encode_initial_state_set_2: fixes \<Pi> :: "'b strips_problem"   and v :: "'b" assumes "is_valid_problem_strips \<Pi>"   and "v \<in> dom (\<Pi>\<^sub>I)" shows "(\<Pi>\<^sub>I) v = Some False \<longrightarrow> (\<exists>!C. C \<in> cnf (\<Phi>\<^sub>I \<Pi>) \<and> C = {(State 0 (index (\<Pi>\<^sub>\<V>) v))\<inverse>})"
vfsequence_vdomain_in_omega: fixes xs :: "V" assumes "vfsequence xs" shows "\<D>\<^sub>\<circ> xs \<in>\<^sub>\<circ> \<omega>"
countable_card_of_nat: fixes A :: "'b set" shows "countable A = (|A| \<le>o |UNIV|)"
implodeable_Ext_trancl: fixes R :: "(ZF \<times> ZF) set" assumes "set_like R" shows "set_like (R\<^sup>+)"
ComposeC: fixes T :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term list"   and f :: "('fun, 'atom, 'sets) prot_fun"   and M :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term set" assumes "length T = arity f"   and "public f"   and "\<And>t. t \<in> set T \<Longrightarrow> M \<turnstile>\<^sub>c t" shows "M \<turnstile>\<^sub>c Fun f T"
difference_intro: fixes z :: "i"   and x :: "i"   and y :: "i" assumes "\<forall>w. P w z = (P w x \<and> \<not> O w y)" shows "x \<ominus> y = z"
errMOD_igSubstIPresIGWlsSTR: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) model" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.igSubstIPresIGWls Delta MOD"   and "igWlsDisj MOD" shows "FixSyn.igSubstIPresIGWlsSTR Delta (FixSyn.errMOD Delta MOD)"
idx_iteratei_aux_simps_3: fixes i :: "nat"   and c :: "'c \<Rightarrow> bool"   and \<sigma> :: "'c"   and get :: "'d \<Rightarrow> nat \<Rightarrow> 'e"   and sz :: "nat"   and l :: "'d"   and f :: "'e \<Rightarrow> 'c \<Rightarrow> 'c" assumes "i \<noteq> 0"   and "c \<sigma>" shows "idx_iteratei_aux get sz i l c f \<sigma> = idx_iteratei_aux get sz (i - 1) l c f (f (get l (sz - i)) \<sigma>)"
simps: fixes a :: "'e"   and g :: "'e \<Rightarrow> ('f, 'e) step" shows "(a \<in> terminates_on g) = ((\<exists>s. a = s \<and> g s = Done) \<or> (\<exists>s s'. a = s \<and> g s = Skip s' \<and> s' \<in> terminates_on g) \<or> (\<exists>s aa s'. a = s \<and> g s = Yield aa s' \<and> s' \<in> terminates_on g))"
fg_eq: fixes x :: "real" assumes "a < x"   and "x < b"   and "x \<notin> s" shows "f x = g x"
assms_1: shows "(\<box>(\<lambda>s. P s \<longrightarrow> (Q \<U> R) s)) \<sigma>"
fin2: shows "finite (set ps - set (ps\<^sub>+))"
disjoint_def: fixes a :: "'b"   and b :: "'b" shows "total_partial.disjoint a b \<equiv> Sep_Eq.add2 a b \<noteq> None"
prod_induct_gen: fixes f :: "'b \<Rightarrow> 'c"   and h :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and lst :: "'a list" assumes "\<And>a r. f (h a * r) = f (g a * r)" shows "f (prod_list (map h lst)) = f (prod_list (map g lst))"
composite_simps_2: shows "trg (h \<star> k) = trg h"
join_def: shows "join \<equiv> \<Lambda> m. m \<bind> (\<Lambda> x. x)"
one_lane_notfree: shows "\<forall>ts v. |lan v| = 1 \<and> 0 < \<parallel>ext v\<parallel> \<and> \<not> (0 < \<parallel>ext v\<parallel> \<and> |lan v| = 1 \<and> (\<forall>c. \<parallel>len v ts c\<parallel> = 0 \<or> restrict v (clm ts) c = bot \<and> restrict v (res ts) c = bot)) \<longrightarrow> (\<exists>va u. v=va\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (\<exists>x. 0 < \<parallel>ext v\<parallel> \<and> len v ts x = ext v \<and> restrict v (res ts) x = lan v \<and> |lan v| = 1 \<or> 0 < \<parallel>ext v\<parallel> \<and> len v ts x = ext v \<and> restrict v (clm ts) x = lan v \<and> |lan v| = 1) \<and> True))"
nat_filtration_pseudo_proj_True_measurable: fixes p :: "real"   and M :: "bool stream measure"   and n :: "nat" assumes "infinite_coin_toss_space p M" shows "pseudo_proj_True n \<in> infinite_coin_toss_space.nat_filtration M n \<rightarrow>\<^sub>M M"
simps_1: fixes varT :: "variable \<Rightarrow> 'a"   and objT :: "object \<Rightarrow> 'a"   and v :: "variable" shows "ty_term varT objT (term.VAR v) = varT v"
gne: fixes w :: "complex" assumes "w \<in> ball \<xi> r" shows "g w \<noteq> 0"
module: fixes R :: "('e, 'f) ring_scheme"   and N :: "'g set"   and M :: "('e, 'g, 'h) module_scheme" assumes "LinearCombinations.submodule R N M" shows "module R M"
span_eq_iff: fixes s :: "'b set" shows "(ideal s = s) = ideal.subspace s"
inv_dghm_def: fixes \<FF> :: "V" shows "inv_dghm \<FF> = [(\<FF>\<lparr>ObjMap\<rparr>)\<inverse>\<^sub>\<circ>, (\<FF>\<lparr>ArrMap\<rparr>)\<inverse>\<^sub>\<circ>, \<FF>\<lparr>HomCod\<rparr>, \<FF>\<lparr>HomDom\<rparr>]\<^sub>\<circ>"
total2E: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and x :: "nat"   and y :: "nat" assumes "total2 f" shows "f x y \<down>"
IH: assumes "Sorting_Algorithms.sorted cmp xs" shows "P xs"
mangoldt_sum: fixes n :: "nat" assumes "n \<noteq> 0" shows "(\<Sum>d | d dvd n. mangoldt d) = of_real (ln (real n))"
hom_carrier: fixes h :: "'e \<Rightarrow> 'f"   and G :: "('e, 'g) monoid_scheme"   and H :: "('f, 'h) monoid_scheme" assumes "h \<in> hom G H" shows "h ` carrier G \<subseteq> carrier H"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list" assumes "\<And>x xs y ys. \<lbrakk>key y < key x \<Longrightarrow> P (x # xs) ys; \<not> key y < key x \<Longrightarrow> P xs (y # ys)\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)"   and "\<And>ys. P [] ys"   and "\<And>v va. P (v # va) []" shows "P a0 a1"
simps_16: fixes j :: "nat"   and T :: "ty"   and vo :: "'a val option"   and e :: "'a expr1"   and i :: "nat" shows "\<B> {j:T=vo; e} i = (i = j \<and> \<B> e (i + 1))"
domI: fixes m :: "'f \<Rightarrow> 'e option"   and a :: "'f"   and b :: "'e" assumes "m a = Some b" shows "a \<in> dom m"
abs_one: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs one = one"
lm016: fixes bids :: "'a \<times> 'b set \<Rightarrow> 'c"   and N :: "'a set"   and G :: "'b set" shows "summedBidVectorRel bids N G = {summedBid bids pair |pair. pair \<in> N \<times> (Pow G - {{}})}"
simps_2: fixes clause :: "Literal list"   and formula :: "Literal list list" shows "varsFormula (clause # formula) = varsClause clause \<union> varsFormula formula"
sum1: shows "(\<lambda>x. if x = Suc 0 then l * g 1 else fds_nth f x * g x * mangoldt x /\<^sub>R ln (real x) / nat_power x s) abs_summable_on Collect primepow"
Ord_ex_eqpoll_vdisjnt: fixes \<alpha> :: "V"   and X :: "V"   and Y :: "V" assumes "Ord \<alpha>"   and "X \<subseteq>\<^sub>\<circ> Y"   and "Y \<in>\<^sub>\<circ> Vset \<alpha>" obtains Z :: "V" where "X \<approx>\<^sub>\<circ> Z" and "Z \<inter>\<^sub>\<circ> Y = []\<^sub>\<circ>" and "Z \<subseteq>\<^sub>\<circ> Vset (ZFC_in_HOL.succ \<alpha>)"
sound_mode_use_def: shows "conc.sound_mode_use \<equiv> \<lambda>uu. case uu of (cms, mem) \<Rightarrow> list_all (\<lambda>cm. conc.locally_sound_mode_use (cm, mem)) cms \<and> conc.globally_sound_mode_use (cms, mem)"
ifields_def: fixes P :: "'b prog"   and C :: "char list" shows "ifields P C \<equiv> filter (\<lambda>((F, D), b, T). b = NonStatic) (fields P C)"
bijI: fixes f :: "'a \<Rightarrow> 'b" assumes "inj f"   and "top_pres (\<P> f)" shows "bij f"
transfer_prover_start: fixes x :: "'b"   and x' :: "'b"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c" assumes "x = x'"   and "Transfer.Rel R x' y" shows "Transfer.Rel R x y"
cyclic_order_is_ord: fixes G :: "('c, 'd) monoid_scheme"   and g :: "'c" assumes "Group.group G"   and "g \<in> carrier G" shows "group.ord G g = order (subgroup_generated G {g})"
intros_2: fixes c :: "real" shows "real_polynomial_function (\<lambda>x. c)"
simps_6: fixes b :: "'d set"   and c :: "('d, 'e, 'f) com" shows "merge_guards (While b c) = While b (merge_guards c)"
directional_derivative_def: fixes I :: "('e, 'f, 'd) interp"   and t :: "('e, 'd) trm" shows "directional_derivative I t = (\<lambda>v. frechet I t (fst v) (snd v))"
has_ticket_simps_3: fixes k :: "nat"   and k' :: "nat" shows "has_ticket (HOLD k) k' = (k' = k)"
that: fixes i :: "nat"   and j :: "nat" assumes "i < j"   and "seq i adds seq j" shows "thesis"
hyp: shows "x\<^bsub>0\<^esub>\<^bsup>m\<^esup> \<cdot> x\<^bsub>0\<^esub>\<^bsup>Suc n\<^esup> = x\<^bsub>0\<^esub>\<^bsup>Suc (m + n)\<^esup>"
subset_insert_iff: fixes A :: "'a set"   and x :: "'a"   and B :: "'a set" shows "(A \<subseteq> insert x B) = (if x \<in> A then A - {x} \<subseteq> B else A \<subseteq> B)"
continuous_map_product_then_coordinatewise_1: fixes T1 :: "'a topology"   and T :: "'b \<Rightarrow> 'c topology"   and I :: "'b set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and i :: "'b" assumes "continuous_map T1 (product_topology T I) f"   and "i \<in> I" shows "continuous_map T1 (T i) (\<lambda>x. f x i)"
ih_6: shows "InvariantClCharacterization (getCl state') (getC state') (getM state')"
projection_on_subset2: fixes l :: "'b list"   and L :: "'b set"   and X' :: "'b set"   and X :: "'b set" assumes "set l \<subseteq> L"   and "l \<upharpoonleft> X' = []"   and "X \<inter> L \<subseteq> X'" shows "l \<upharpoonleft> X = []"
exec_mark: shows "\<Gamma>\<turnstile> \<langle>mark_guards f c,s\<rangle> \<Rightarrow> t"
INF_top_conv_2: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and B :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(top = Inf (B ` A)) = (\<forall>x\<in>A. B x = top)"
False_False_in_steps_atom: fixes a :: "'b"   and w :: "'b list" shows "(([False], [False]) \<in> steps (atom a) w) = (w = [])"
dom_char: fixes f :: "'b \<times> 'b" shows "B.VxV.dom f = (if B.arr (fst f) \<and> B.arr (snd f) then (B.dom (fst f), B.dom (snd f)) else (B.null, B.null))"
is_shortest_path_on_def: fixes G :: "graph"   and m :: "nat"   and n :: "nat"   and p :: "nat list"   and V :: "nat set" shows "is_shortest_path_on G m n p V = (p \<in> path_set_on G m n V \<and> (\<forall>p'\<in>path_set_on G m n V. path_weight G p \<le> path_weight G p'))"
rel_funD: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f"   and x :: "'c"   and y :: "'d" assumes "(A ===> B) f g"   and "A x y" shows "B (f x) (g y)"
sgn_minus_rat: fixes x :: "rat" shows "sgn (- x) = - sgn x"
src_def: fixes \<mu> :: "'a arrow_of_spans_data" shows "src \<mu> \<equiv> if V.arr \<mu> then mkObj (C.cod (Leg0 (Dom \<mu>))) else V.null"
EQ: fixes x :: "'c"   and y :: "'c" shows "(cmp x y = EQ) = (x = y)"
b00: shows "prv (PPf (encPf prf) \<langle>\<phi>R\<rangle>)"
CondTrue: fixes s :: "'a"   and b :: "'a set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c\<^sub>1 :: "('a, 'b, 'c) com"   and c\<^sub>2 :: "('a, 'b, 'c) com" assumes "s \<in> b" shows "\<Gamma>\<turnstile> (Cond b c\<^sub>1 c\<^sub>2, Normal s) \<rightarrow> (c\<^sub>1, Normal s)"
measure_scale_measure: fixes r :: "real"   and M :: "'b measure"   and A :: "'b set" assumes "0 \<le> r" shows "Sigma_Algebra.measure (scale_measure (ennreal r) M) A = r * Sigma_Algebra.measure M A"
ConI: fixes p :: "'b pfm"   and q :: "'b pfm" assumes "\<turnstile>\<^sub>! p"   and "\<turnstile>\<^sub>! q" shows "\<turnstile>\<^sub>! (p \<^bold>\<and>\<^sub>! q)"
smult_power: fixes x :: "'b"   and n :: "nat"   and p :: "'b poly" shows "smult (x ^ n) (p ^ n) = smult x p ^ n"
assms_2: shows "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"