hcju/isabelleps huggingface dataset:
cases: fixes x :: "('a, 'b, 'varSort, 'var, 'c) qTerm" obtains xs :: "'varSort" and x :: "'var" where "xa__ = qVar xs x"   | delta :: "'c" and inp :: "'a \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm option" and binp :: "'b \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qAbs option" where "xa__ = qOp delta inp binp"
g': shows "g = fls_shift (int n) (fps_to_fls g')"
min_diff_distrib_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "minus (ord.min less_eq x y) z = ord.min less_eq (minus x z) (minus y z)"
inject: fixes x21 :: "'a"   and x22 :: "'a"   and x23 :: "'b"   and y21 :: "'a"   and y22 :: "'a"   and y23 :: "'b" shows "(MkArr x21 x22 x23 = MkArr y21 y22 y23) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
homeomorphicI: fixes f :: "'d \<Rightarrow> 'c"   and S :: "'d set"   and T :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "f ` S = T"   and "g ` T = S"   and "continuous_on S f"   and "continuous_on T g"   and "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"   and "\<And>y. y \<in> T \<Longrightarrow> f (g y) = y" shows "S homeomorphic T"
simps_3: fixes t :: "'c local.term"   and u :: "'c local.term" shows "(t \<^bold>\<star> u)\<^bold>\<down> = (if Nml (t \<^bold>\<star> u) then t \<^bold>\<star> u else (\<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<Down> \<^bold>\<lfloor>u\<^bold>\<rfloor>) \<^bold>\<cdot> (t\<^bold>\<down> \<^bold>\<star> u\<^bold>\<down>))"
existence_ivl_reverse: fixes t :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "t - t0 \<in> existence_ivl0 x0" shows "t0 - t \<in> existence_ivl0 (flow0 x0 (t - t0))"
filter_mset_mset_set: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "finite A" shows "filter_mset P (mset_set A) = mset_set {x \<in> A. P x}"
3: shows "Map G = \<lbrakk>REP G\<rbrakk>\<^sub>B"
less_1: fixes S :: "'a set"   and p :: "'a \<Rightarrow> 'a" assumes "card S < card Sa__"   and "p permutes S"   and "finite S" shows "cycle_decomp S p"
cClose1: fixes a :: "name"   and x :: "name"   and P' :: "pi"   and y :: "name"   and P'' :: "pi"   and C :: "'a" assumes "P \<longmapsto> a<x> \<prec> P'"   and "!P \<longmapsto> a<\<nu>y> \<prec> P''"   and "y \<sharp> P"   and "x \<sharp> C"   and "y \<sharp> C"   and "\<And>C. F C (!P) (a<\<nu>y> \<prec> P'')" shows "F C (P \<parallel> !P) (\<tau> \<prec> <\<nu>y>(P'[x::=y] \<parallel> P''))"
less_ereal_le: fixes a :: "ereal"   and x :: "real"   and y :: "real" assumes "a < ereal x"   and "x \<le> y" shows "a < ereal y"
algebraic_mult_cancel_right: fixes y :: "'a"   and x :: "'a" assumes "algebraic y"   and "y \<noteq> (0::'a)" shows "algebraic (x * y) = algebraic x"
dis_rews_3: fixes x :: "[:'a:]" shows "(is_snil\<cdot>x = \<bottom>) = (x = \<bottom>)"
inject: fixes x2 :: "'b"   and y2 :: "'b" shows "(Enq x2 = Enq y2) = (x2 = y2)"
set_unset_bit_preserves_len: fixes x :: "nat"   and m :: "nat"   and q :: "nat" assumes "x div 2 ^ m = 2 * q"   and "m < len x" shows "x + 2 ^ m < 2 ^ len x"
poly_mult_one'_2: fixes p :: "'a list" assumes "set p \<subseteq> carrier R" shows "poly_mult p [\<one>] = local.normalize p"
lnth_prepend: fixes xs :: "'b list"   and ys :: "'b llist"   and i :: "nat" shows "lnth (prepend xs ys) i = (if i < length xs then xs ! i else lnth ys (i - length xs))"
less_irrefl: fixes x :: "'d word" shows "\<not> x <s x"
\<ff>_le: fixes s :: "'a list"   and j :: "nat" assumes "s \<noteq> []" shows "\<ff> s j - 1 < length s"
Arg2pi: fixes z :: "complex" shows "0 \<le> Arg2pi z \<and> Arg2pi z < 2 * pi \<and> is_Arg z (Arg2pi z)"
Union_image_insert: fixes f :: "'f \<Rightarrow> 'e set"   and a :: "'f"   and B :: "'f set" shows "\<Union> (f ` insert a B) = f a \<union> \<Union> (f ` B)"
pCons_2: assumes "set (coeffs p) \<subseteq> \<real>" shows "set (coeffs (p * q)) \<subseteq> \<real>"
jkbpT_traces_of_length: fixes T :: "'s Trace set"   and n :: "nat" shows "{t \<in> jkbpT T. tLength t = n} = jkbpT\<^bsub>n\<^esub> T"
stable_constrains_Int: fixes F :: "'d program"   and C :: "'d set"   and A :: "'d set"   and A' :: "'d set" assumes "F \<in> stable C"   and "F \<in> C \<inter> A co A'" shows "F \<in> C \<inter> A co C \<inter> A'"
finite_image_iff: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set" assumes "inj_on f A" shows "finite (f ` A) = finite A"
peek_and_commut: fixes P :: "'b assn"   and p :: "state \<Rightarrow> bool"   and q :: "state \<Rightarrow> bool" shows "(P \<and>. p \<and>. q) = (P \<and>. q \<and>. p)"
scaleC_prod_def: fixes r :: "complex"   and A :: "'a \<times> 'b" shows "r *\<^sub>C A = (r *\<^sub>C fst A, r *\<^sub>C snd A)"
self_finance_succ: fixes Mkt :: "('c, 'd) discrete_market"   and asset :: "'d"   and n :: "nat"   and w :: "'c"   and pf :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and v :: "real" assumes "prices Mkt asset (Suc n) w \<noteq> 0"   and "portfolio pf" shows "val_process Mkt (self_finance Mkt v pf asset) (Suc n) w = prices Mkt asset (Suc n) w * remaining_qty Mkt v pf asset (Suc n) w + cls_val_process Mkt pf (Suc n) w"
MSB_Cons: fixes I :: "nat fset"   and Is :: "nat fset list" shows "MSB (I # Is) = max (if I = {||} then 0 else Suc (fMax I)) (MSB Is)"
coeff_subresultant: fixes J :: "nat"   and F :: "'b poly"   and G :: "'b poly"   and l :: "nat" shows "coeff (subresultant J F G) l = (if degree F - J + (degree G - J) = 0 \<and> l \<noteq> 0 then 0::'b else det (subresultant'_mat J l F G))"
that: fixes f :: "'a \<Rightarrow> 'b" assumes "linear f"   and "f ` S \<subseteq> T"   and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) = norm x" shows "thesis"
int_or_comm: fixes x :: "int"   and y :: "int" shows "x OR y = y OR x"
ET_source_target: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and t :: "'d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd"   and A :: "('d, 'e, 'f) seqauto" assumes "TS \<subseteq> ET ST"   and "t \<in> TS"   and "source t \<in> States A"   and "A \<in> SAs (HA ST)" shows "target t \<in> States A"
univalent_inf_closed: fixes x :: "'a"   and y :: "'a" assumes "univalent x" shows "univalent (x \<sqinter> y)"
col_space_eq_row_space_transpose: fixes A :: "(('b, 'd) vec, 'c) vec" shows "col_space A = row_space (Finite_Cartesian_Product.transpose A)"
State_ext_def: fixes getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and more :: "'a" shows "\<lparr>getF = getF, getM = getM, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons State_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple getF getM) more"
accepts_trace_step: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and p :: "value list"   and t :: "(String.literal \<times> value list \<times> value list) list" shows "accepts_trace e s r ((l, i, p) # t) = (\<exists>(s', T)|\<in>|possible_steps e s r l i. evaluate_outputs T i r = map Some p \<and> accepts_trace e s' (evaluate_updates T i r) t)"
sat_push_precond_def: fixes f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat" shows "sat_push_precond f l \<equiv> \<lambda>(u, v). (0::'a) < excess f u \<and> cf_of f (u, v) \<le> excess f u \<and> (u, v) \<in> cfE_of f \<and> l u = l v + 1"
indmhomTr2: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "f \<in> mHom R M N" shows "f\<^sup>\<flat>\<^bsub>R M, N\<^esub> \<in> set_mr_cos M (ker\<^bsub>M,N\<^esub> f) \<rightarrow> carrier N"
c1_implies_local_lipschitz: fixes T :: "real set"   and X :: "'b set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and f' :: "real \<times> 'b \<Rightarrow> 'b \<Rightarrow>\<^sub>L 'b" assumes "\<And>t x. \<lbrakk>t \<in> T; x \<in> X\<rbrakk> \<Longrightarrow> (f t has_derivative blinfun_apply (f' (t, x))) (at x)"   and "continuous_on (T \<times> X) f'"   and "open T"   and "open X" shows "local_lipschitz T X f"
bigtheta_trans1: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> L F g"   and "g \<in> \<Theta>[F](h)" shows "f \<in> L F h"
that: fixes V :: "'x set" assumes "V \<subseteq> U"   and "(monomial (1::'a) t, V) \<in> set (snd (split t U S))" shows "thesis"
pathstart_in_path_image: fixes g :: "real \<Rightarrow> 'a" shows "pathstart g \<in> path_image g"
simps_2: fixes n :: "nat"   and rai :: "int poly \<times> rat \<times> rat" shows "sgn_2 (Irrational n rai) = sgn_1 rai"
assms_1: shows "local.hensel_step p q S1 T1 D1 H1 S T D H = (S', T', D', H')"
ptensor_mat_comm: fixes vars1 :: "nat set"   and vars2 :: "nat set"   and m1 :: "complex mat"   and m2 :: "complex mat" assumes "vars1 \<inter> vars2 = {}" shows "partial_state2.ptensor_mat dims vars1 vars2 m1 m2 = partial_state2.ptensor_mat dims vars2 vars1 m2 m1"
tendsto_eval_fds_Re_at_top: fixes f :: "'a fds"   and S :: "'b \<Rightarrow> 'a"   and F :: "'b filter" assumes "conv_abscissa f \<noteq> \<infinity>"   and "LIM x F. S x \<bullet> (1::'a) :> at_top" shows "((\<lambda>x. eval_fds f (S x)) \<longlongrightarrow> fds_nth f 1) F"
wfactors_mult: fixes as :: "'a list"   and a :: "'a"   and bs :: "'a list"   and b :: "'a" assumes "wfactors (Ring_Divisibility.mult_of R) as a"   and "wfactors (Ring_Divisibility.mult_of R) bs b"   and "a \<in> carrier (Ring_Divisibility.mult_of R)"   and "b \<in> carrier (Ring_Divisibility.mult_of R)"   and "set as \<subseteq> carrier (Ring_Divisibility.mult_of R)"   and "set bs \<subseteq> carrier (Ring_Divisibility.mult_of R)" shows "wfactors (Ring_Divisibility.mult_of R) (as @ bs) (a \<otimes> b)"
tame9a_def: fixes g :: "graph" shows "tame9a g \<equiv> \<forall>f\<in>\<F> g. 3 \<le> |vertices f| \<and> |vertices f| \<le> 6"
pred_cong: fixes x :: "'c list"   and ya :: "'c list"   and P :: "'c \<Rightarrow> bool"   and Pa :: "'c \<Rightarrow> bool" assumes "x = ya"   and "\<And>z. z \<in> set ya \<Longrightarrow> P z = Pa z" shows "list_all P x = list_all Pa ya"
ctr: fixes a :: "real" shows "cis a = Complex (cos a) (sin a)"
simps_2: fixes s :: "'b"   and p :: "'c \<times> 'b"   and ps :: "('c \<times> 'b) list" shows "last_state (s, Cons p ps) = snd p"
has_locks_Suc_has_lock: fixes l :: "('b \<times> nat) option"   and t :: "'b"   and n :: "nat" assumes "has_locks l t = Suc n" shows "has_lock l t"
m_closed: fixes x :: "'a list"   and y :: "'a list" assumes "x \<in> carrier (K [X])"   and "y \<in> carrier (K [X])" shows "x \<otimes>\<^bsub>K [X]\<^esub> y \<in> carrier (K [X])"
card_of_ordLeqI: fixes f :: "'f \<Rightarrow> 'g"   and A :: "'f set"   and B :: "'g set" assumes "inj_on f A"   and "\<And>a. a \<in> A \<Longrightarrow> f a \<in> B" shows "|A| \<le>o |B|"
vec_upd_eq: fixes s :: "('b, 'c) vec"   and i :: "'c"   and a :: "'b" shows "HS_VC_Spartan.vec_upd s i a = (\<chi>j. if j = i then a else s $ j)"
conv_normal_density_zero_mean: fixes \<sigma> :: "real"   and \<tau> :: "real" assumes "0 < \<sigma>"   and "0 < \<tau>" shows "(\<lambda>x. \<integral>\<^sup>+ y. ennreal (normal_density 0 \<sigma> (x - y) * normal_density 0 \<tau> y) \<partial>lborel) = (\<lambda>x. ennreal (normal_density 0 (sqrt (\<sigma>\<^sup>2 + \<tau>\<^sup>2)) x))"
point_not_equal_aux: fixes p :: "'a"   and q :: "'a" assumes "is_point p"   and "is_point q" shows "(p \<noteq> q) = (p \<cdot> - q \<noteq> 0)"
distinct_map: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "distinct (map f xs) = (distinct xs \<and> inj_on f (set xs))"
elims: fixes x :: "'b list"   and y :: "'b list" assumes "remdups_adj x = y" obtains "x = []" and "y = []"   | xa :: "'b" where "x = [xa__]" and "y = [xa__]"   | xa :: "'b" and ya :: "'b" and xs :: "'b list" where "x = xa # ya # xs" and "y = (if xa = ya then remdups_adj (xa # xs) else xa # remdups_adj (ya # xs))"
P2: shows "Bet A C D' \<and> Cong C D' C D"
suminf_finite: fixes N :: "nat set"   and f :: "nat \<Rightarrow> 'a" assumes "finite N"   and "\<And>n. n \<notin> N \<Longrightarrow> f n = (0::'a)" shows "suminf f = sum f N"
memoization_def: fixes return :: "'b \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and get :: "(('d \<Rightarrow> 'b option) \<Rightarrow> 'c) \<Rightarrow> 'c"   and put :: "('d \<Rightarrow> 'b option) \<Rightarrow> 'c \<Rightarrow> 'c" shows "memoization return bind get put \<equiv> monad_state return bind get put"
ip_cidr_set_less: fixes r1 :: "nat"   and r2 :: "nat"   and i :: "'b word" assumes "r1 \<le> r2" shows "ip_cidr_set i r2 \<subseteq> ip_cidr_set i r1"
imp_regular: fixes M :: "'b nfa" assumes "nfa M" shows "regular (nfa.language M)"
map_sel_1: fixes a :: "('a, 'b) tllist"   and f1 :: "'a \<Rightarrow> 'd"   and f2 :: "'b \<Rightarrow> 'c" assumes "is_TNil a" shows "terminal (tmap f1 f2 a) = f2 (terminal a)"
trunc_bound_eucl_def: fixes p :: "nat"   and s :: "'a" shows "trunc_bound_eucl p s = (let d = eucl_truncate_down p s; ed = \<bar>d - s\<bar> in (d, eucl_truncate_up p ed))"
simps_1: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "insort_key f x [] = [x]"
ord_resolve_obtain_clauses: fixes S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and M :: "'a literal multiset set"   and CAs :: "'a literal multiset list"   and DA :: "'a literal multiset"   and AAs :: "'a multiset list"   and As :: "'a list"   and \<sigma> :: "'s"   and E :: "'a literal multiset"   and n :: "nat"   and D :: "'a literal multiset"   and Cs :: "'a literal multiset list" assumes "ord_resolve (S_M S M) CAs DA AAs As \<sigma> E"   and "selection S"   and "{DA} \<union> set CAs \<subseteq> grounding_of_clss M"   and "length CAs = n"   and "DA = D + negs (mset As)"   and "\<forall>i<n. CAs ! i = Cs ! i + poss (AAs ! i)"   and "length Cs = n"   and "length AAs = n" obtains DA0 :: "'a literal multiset" and \<eta>0 :: "'s" and CAs0 :: "'a literal multiset list" and \<eta>s0 :: "'s list" and As0 :: "'a list" and AAs0 :: "'a multiset list" and D0 :: "'a literal multiset" and Cs0 :: "'a literal multiset list" where "length CAs0 = n" and "length \<eta>s0 = n" and "DA0 \<in> M" and "DA0 \<cdot> \<eta>0 = DA" and "S DA0 \<cdot> \<eta>0 = S_M S M DA" and "\<forall>CA0\<in>set CAs0. CA0 \<in> M" and "CAs0 \<cdot>\<cdot>cl \<eta>s0 = CAs" and "map S CAs0 \<cdot>\<cdot>cl \<eta>s0 = map (S_M S M) CAs" and "is_ground_subst \<eta>0" and "is_ground_subst_list \<eta>s0" and "As0 \<cdot>al \<eta>0 = As" and "AAs0 \<cdot>\<cdot>aml \<eta>s0 = AAs" and "length As0 = n" and "D0 \<cdot> \<eta>0 = D" and "DA0 = D0 + negs (mset As0)" and "S_M S M (D + negs (mset As)) \<noteq> {#} \<Longrightarrow> negs (mset As0) = S DA0" and "length Cs0 = n" and "Cs0 \<cdot>\<cdot>cl \<eta>s0 = Cs" and "\<forall>i<n. CAs0 ! i = Cs0 ! i + poss (AAs0 ! i)" and "length AAs0 = n"
representation_def: fixes basis :: "'d set"   and v :: "'d" shows "crepresentation basis v = (if cindependent basis \<and> v \<in> cspan basis then SOME f. (\<forall>v. f v \<noteq> 0 \<longrightarrow> v \<in> basis) \<and> finite {v. f v \<noteq> 0} \<and> (\<Sum>v | f v \<noteq> 0. f v *\<^sub>C v) = v else (\<lambda>b. 0))"
power_commuting_commutes: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x + y = y + x" shows "x +^ n + y = y + x +^ n"
Units_closed: fixes x :: "'a" assumes "x \<in> Units (Ring_Divisibility.mult_of R)" shows "x \<in> carrier (Ring_Divisibility.mult_of R)"
infinite_solution_set_no_homogeneous_eq: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" shows "infinite (solution_set A b) = (consistent A b \<and> 0 < vec.dim (solution_set A 0))"
new_vars_private_def: fixes \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set" shows "new_vars_private \<R> \<equiv> \<forall>c\<^sub>1\<^sub>A mds\<^sub>A mem\<^sub>1\<^sub>A c\<^sub>1\<^sub>C mds\<^sub>C mem\<^sub>1\<^sub>C. (\<langle>c\<^sub>1\<^sub>A, mds\<^sub>A, mem\<^sub>1\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<longrightarrow> (\<forall>c\<^sub>1\<^sub>C' mds\<^sub>C' mem\<^sub>1\<^sub>C'. \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C \<leadsto>\<^sub>C \<langle>c\<^sub>1\<^sub>C', mds\<^sub>C', mem\<^sub>1\<^sub>C'\<rangle>\<^sub>C \<longrightarrow> (\<forall>v\<^sub>C. (mem\<^sub>1\<^sub>C' v\<^sub>C \<noteq> mem\<^sub>1\<^sub>C v\<^sub>C \<or> dma\<^sub>C mem\<^sub>1\<^sub>C' v\<^sub>C < dma\<^sub>C mem\<^sub>1\<^sub>C v\<^sub>C) \<and> v\<^sub>C \<notin> range var\<^sub>C_of \<longrightarrow> v\<^sub>C \<in> mds\<^sub>C' AsmNoReadOrWrite) \<and> mds\<^sub>C AsmNoReadOrWrite - range var\<^sub>C_of \<subseteq> mds\<^sub>C' AsmNoReadOrWrite - range var\<^sub>C_of)"
assms: fixes i :: "'a" assumes "i \<in> I" shows "g i \<in> ideal B"
continuous_imp_closed_map: fixes S :: "'a set"   and U :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set" assumes "closedin (top_of_set S) U"   and "continuous_on S f"   and "f ` S = T"   and "compact S" shows "closedin (top_of_set T) (f ` U)"
eucl_Sup: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and abs :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and X :: "'c set" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less" shows "Sup X = comm_monoid_add.sum plus zero (\<lambda>i. scaleR (SUP x\<in>X. inner x i) i) Basis"
seqI: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "VV.arr f"   and "VV.arr g"   and "VV.dom g = VV.cod f" shows "VV.seq g f"
inject_1: fixes x11 :: "exp"   and x12 :: "exp"   and y11 :: "exp"   and y12 :: "exp" shows "(Alts x11 x12 = Alts y11 y12) = (x11 = y11 \<and> x12 = y12)"
simps_1: fixes t1 :: "ternaryformula"   and t2 :: "ternaryformula" shows "ternary_ternary_eval (TernaryAnd t1 t2) = eval_ternary_And (ternary_ternary_eval t1) (ternary_ternary_eval t2)"
ask_option_def: fixes ask :: "('f \<Rightarrow> 'e) \<Rightarrow> 'e"   and f :: "'f \<Rightarrow> ('d, 'e) optionT" shows "ask_option ask f = OptionT (ask (\<lambda>r. run_option (f r)))"
prems_2: shows "set ys \<subseteq> set (a # xs)"
hyps: shows "snd (snd (snd (foldl echelon_form_of_column_k_det_iarrays (n, A, 0, bezout) [0..<k]))) = bezout"
simps: fixes cms :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) list"   and mem :: "'Var\<^sub>A \<Rightarrow> 'Val" shows "abs.sound_mode_use (cms, mem) = (list_all (\<lambda>cm. abs.locally_sound_mode_use (cm, mem)) cms \<and> abs.globally_sound_mode_use (cms, mem))"
hom_zero: shows "map_poly complex_of_real 0 = 0"
genby_genset_closed: fixes s :: "'b"   and S :: "'b set" assumes "s \<in> S" shows "s \<in> \<langle>S\<rangle>"
scene_compat_top: fixes X :: "'a scene" assumes "idem_scene X" shows "X ##\<^sub>S \<top>\<^sub>S"
gamma2_vec_diffable: fixes x :: "real" assumes "x \<in> {0..1} - (\<phi> -` (s' \<inter> {0..1}) \<union> s \<inter> {0..1})" shows "(\<gamma>2 \<circ> \<phi> has_vector_derivative vector_derivative (\<gamma>2 \<circ> \<phi>) (at x)) (at x)"
chkchop_def: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and gt :: "('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool"   and t :: "('b, 'a) tm"   and s :: "('b, 'a) tm" assumes "epo ground_heads_var gt_sym arity_sym arity_var extf" shows "epo.chkchop gt t s = (is_Hd s \<or> gt t (chop s))"
SimIncrEnvironment_axioms_def: fixes jkbp :: "'a \<Rightarrow> ('a, 'b, 'c) GC list"   and envInit :: "'d list"   and envAction :: "'d \<Rightarrow> 'e list"   and envTrans :: "'e \<Rightarrow> ('a \<Rightarrow> 'c) \<Rightarrow> 'd \<Rightarrow> 'd"   and envVal :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and jview :: "'a \<Rightarrow> 'd Trace \<Rightarrow> 'f"   and simf :: "'d Trace \<Rightarrow> 'g"   and simRels :: "'a \<Rightarrow> ('g \<times> 'g) set"   and simVal :: "'g \<Rightarrow> 'b \<Rightarrow> bool" shows "SimIncrEnvironment_axioms jkbp envInit envAction envTrans envVal jview simf simRels simVal \<equiv> sim (PreEnvironmentJView.mkM envVal jview (PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview)) (mkKripke (simf ` PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview) simRels simVal) simf"
seqI: fixes f :: "'a"   and g :: "'a" assumes "arr f"   and "arr g"   and "local.dom g = cod f" shows "seq g f"
fa: shows "(\<lambda>u. a u \<odot>\<^bsub>V\<^esub> u) \<in> U \<rightarrow> carrier V"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and finite_3 :: "finite_3" shows "P (case finite_3 of a\<^sub>1 \<Rightarrow> f1 | a\<^sub>2 \<Rightarrow> f2 | a\<^sub>3 \<Rightarrow> f3) = (\<not> (finite_3 = a\<^sub>1 \<and> \<not> P f1 \<or> finite_3 = a\<^sub>2 \<and> \<not> P f2 \<or> finite_3 = a\<^sub>3 \<and> \<not> P f3))"
bdia_while: fixes x :: "'a"   and t :: "'a"   and p :: "'a" assumes "\<langle>x| (d t \<cdot> r p) \<le> r p" shows "\<langle>while t do x od| p \<le> r p \<cdot> ad t"
subtype_complete: fixes P :: "'BExp set"   and t :: "'BExp set"   and t' :: "'BExp set" assumes "\<And>mem. pred P mem \<Longrightarrow> type_max t mem \<le> type_max t' mem" shows "t \<le>:\<^sub>P t'"
directed_alt: fixes X :: "'a set" shows "ord.directed (\<lambda>x y. y \<le> x) X = (X \<noteq> {} \<and> (\<forall>x\<in>X. \<forall>y\<in>X. \<exists>z\<in>X. z \<le> x \<and> z \<le> y))"
in_gamma_inf: fixes x :: "int"   and a1 :: "'av"   and a2 :: "'av" assumes "x \<in> \<gamma> a1"   and "x \<in> \<gamma> a2" shows "x \<in> \<gamma> (a1 \<sqinter> a2)"
orthogonal_fst_QR_decomposition: fixes A :: "(('a, 'c) vec, 'b) vec" shows "pairwise orthogonal (columns (fst (QR_decomposition A)))"
disj_of_clause_simps_2: fixes F :: "'b literal"   and FF :: "'b literal list" shows "disj_of_clause (F # FF) = form_of_lit F \<^bold>\<or> disj_of_clause FF"
effect_changeI: fixes h' :: "heap"   and r :: "'a ref"   and f :: "'a \<Rightarrow> 'a"   and h :: "heap"   and x :: "'a" assumes "h' = Ref.set r (f (Ref.get h r)) h"   and "x = f (Ref.get h r)" shows "effect (Ref.change f r) h h' x"
match_def: fixes s :: "'a rexp"   and w :: "'a list" shows "match s w = final (fold delta w (init s))"
greatest_def: fixes r :: "('a \<times> 'a) set"   and X :: "'a set" shows "greatest r X = {x \<in> X. \<forall>y\<in>X. (y, x) \<in> r}"
False: shows "\<not> (a = 0 \<and> b = 0 \<and> c = 0)"
image_Liminf_llist_subset: fixes f :: "'c \<Rightarrow> 'b"   and Ns :: "'c set llist" shows "f ` Liminf_llist Ns \<subseteq> Liminf_llist (lmap ((`) f) Ns)"
NO2: shows "OadmitFO \<sigma> ODE2 U"
defNode_1: fixes v :: "'val"   and g :: "'g" assumes "v \<in> uninst_code.ssa.allVars uses phis g" shows "uninst_code.ssa.defNode phis g v \<in> set (\<alpha>n g)"