hcju/isabelleps huggingface dataset:
ivl_less_eq: fixes x :: "'a ivl"   and y :: "'a ivl" shows "(x \<le> y) = (low x < low y \<or> low x = low y \<and> high x \<le> high y)"
vars_def: fixes e :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b" shows "Aexp.vars e = {v. \<exists>\<sigma> val. e (\<sigma>(v := val)) \<noteq> e \<sigma>}"
distinct_25: shows "ToLowerCaseOp \<noteq> ToBooleanOp"
surj_tnam': fixes n :: "tnam" shows "\<exists>m. n = tnam' m"
tendsto_mult_left: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and F :: "'a filter"   and c :: "'b" assumes "(f \<longlongrightarrow> l) F" shows "((\<lambda>x. c * f x) \<longlongrightarrow> c * l) F"
UNIV_not_singleton: fixes x :: "'a" shows "UNIV \<noteq> {x}"
CC_def: shows "CC = set (side_prems_of \<iota>)"
nths_assoc_three_C: fixes A :: "nat set"   and B :: "nat set"   and C :: "nat set"   and xs :: "'b list" assumes "A \<inter> B = {}"   and "(A \<union> B) \<inter> C = {}" shows "nths (nths xs (ind_in_set (A \<union> B \<union> C) ` (B \<union> C))) (ind_in_set (B \<union> C) ` C) = nths xs (ind_in_set (A \<union> B \<union> C) ` C)"
leg0_simps_2: shows "hseq u\<^sub>0 t\<^sub>0u\<^sub>1.p\<^sub>0"
times_divide_eq_left: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and b :: "'c"   and c :: "'c"   and a :: "'c" assumes "class.field divide inverse times one plus zero minus uminus" shows "times (divide b c) a = divide (times b a) c"
bernoulli_cond_exp: fixes N :: "bool stream measure"   and q :: "real"   and X :: "bool stream \<Rightarrow> real"   and n :: "nat"   and w :: "bool stream" assumes "N = bernoulli_stream q"   and "0 < q"   and "q < 1"   and "integrable N X" shows "real_cond_exp N (stoch_proc_filt N geom_proc borel n) X w = expl_cond_expect N (proj_stoch_proc geom_proc n) X w"
dens'_1: fixes x :: "nat \<Rightarrow> val" assumes "x \<in> space (state_measure (V \<union> V') \<Gamma>)" shows "F x = density (stock_measure t) (f x)"
assms_3: fixes n :: "nat" assumes "Q n" shows "R (Suc n)"
ifm: shows "ifmi v ni1 ni2 s = (ni, s')"
item_def: shows "x = Item (N, \<alpha> @ \<beta>) (length \<alpha>) (charslength (take r p)) k'"
in_filtermap_iff: fixes f' :: "'b \<Rightarrow> 'c"   and h' :: "'d \<Rightarrow> 'b"   and F' :: "'d filter"   and g' :: "'b \<Rightarrow> 'c" shows "(f' \<in> O[filtermap h' F'](g')) = ((\<lambda>x. f' (h' x)) \<in> O[F'](\<lambda>x. g' (h' x)))"
SigmaE: fixes c :: "'c \<times> 'd"   and A :: "'c set"   and B :: "'c \<Rightarrow> 'd set" assumes "c \<in> Sigma A B" obtains x :: "'c" and y :: "'d" where "x \<in> A" and "y \<in> B x" and "c = (x, y)"
cases: fixes x :: "'c \<times> nat" obtains q :: "'c" and n :: "nat" where "x = (q, n)"
free_module_mem: fixes M :: "('g, 'a, 'h) Module_scheme"   and N :: "('i, 'a, 'h) Module_scheme"   and m :: "'g"   and n :: "'i"   and P :: "('g \<times> 'i, 'a, 'h) Module_scheme" assumes "R module M"   and "R module N"   and "m \<in> carrier M"   and "n \<in> carrier N"   and "FM\<^bsub>R\<^esub> P M N" shows "(m, n) \<in> carrier P"
lookup_HSem_other: fixes y :: "Vars.var"   and h :: "(Vars.var \<times> exp) list"   and \<rho> :: "Vars.var \<Rightarrow> Value" assumes "y \<notin> domA h" shows "(\<lbrace>h\<rbrace>\<rho>) y = \<rho> y"
bin_lcp_def: shows "\<alpha> = u\<^sub>0 \<cdot> u\<^sub>1 \<and>\<^sub>p u\<^sub>1 \<cdot> u\<^sub>0"
ide_cod: fixes f :: "('o, 'a) arr" assumes "arr f" shows "ide (cod f)"
simps_4: fixes f1 :: "'f"   and f2 :: "nat \<Rightarrow> 'f" shows "(case 0 of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = f1"
insertE: fixes a :: "'g"   and b :: "'g"   and A :: "'g set" assumes "a \<in> insert b A" obtains "a = b"   | "a \<in> A"
Trg_Inv: fixes t :: "'a local.term" assumes "Can t" shows "Trg (Inv t) = Trg t"
assms_3: shows "a \<otimes> p (deg R p) \<noteq> \<zero>"
induct: fixes P :: "'a tree list \<Rightarrow> bool"   and a0 :: "'a tree list" assumes "\<And>t. P [t]"   and "\<And>t\<^sub>1 t\<^sub>2 ts. P (insortTree (uniteTrees t\<^sub>1 t\<^sub>2) ts) \<Longrightarrow> P (t\<^sub>1 # t\<^sub>2 # ts)"   and "P []" shows "P a0"
norm_fst_le: fixes x :: "'a"   and y :: "'b" shows "norm x \<le> norm (x, y)"
dom_comp: fixes g :: "('o, 'a) arr"   and f :: "('o, 'a) arr" assumes "seq g f" shows "local.dom (COMP g f) = local.dom f"
inv_join2: fixes l :: "('a \<times> 'b) tree"   and r :: "('a \<times> 'b) tree" assumes "inv l"   and "inv r" shows "inv (join2 l r)"
measurable_enat_coinduct: fixes R :: "('a \<Rightarrow> enat) \<Rightarrow> bool"   and f :: "'a \<Rightarrow> enat"   and M :: "'a measure" assumes "R f"   and "\<And>f. R f \<Longrightarrow> \<exists>g h i P. R g \<and> f = (\<lambda>x. if P x then h x else eSuc (g (i x))) \<and> Measurable.pred M P \<and> i \<in> M \<rightarrow>\<^sub>M M \<and> h \<in> M \<rightarrow>\<^sub>M count_space UNIV" shows "f \<in> M \<rightarrow>\<^sub>M count_space UNIV"
side_prems_of_def: fixes inference :: "'b inference" shows "side_prems_of inference = (case inference of Infer x1 x2 x3 \<Rightarrow> x1)"
degree_aforms_real_def: shows "degree_aforms_real = degree_aforms"
empty_inside_imp_simply_connected: assumes "connected S"   and "inside S = {}" shows "simply_connected S"
that_2: shows "\<I> \<turnstile>g \<A> \<surd>"
hyps: fixes k :: "nat"   and cs :: "('b, 'a, 'c) com list" assumes "f k = (cs, css, Fault m)" shows "False"
t'_def: shows "t' = t /\<^sub>C complex_of_real (norm t)"
unit_resolvent_is_resolvent: shows "less_restrictive resolvent unit_resolvent"
dist_eq_dist_code_abs_lt: fixes c :: "int"   and p\<^sub>0 :: "int \<times> int"   and p\<^sub>1 :: "int \<times> int" shows "(\<bar>real_of_int c\<bar> < dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) = (c\<^sup>2 < dist_code p\<^sub>0 p\<^sub>1)"
composite_simps_2: shows "src ((f \<star> g) \<star> f) = src f"
length_norm_stack: fixes Finca_empty :: "'a"   and Finca_get :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'f) Inca.instr fundef option"   and Finca_add :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'f) Inca.instr fundef \<Rightarrow> 'a"   and Finca_to_list :: "'a \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e, 'f) Inca.instr fundef) list"   and Fubx_empty :: "'g"   and Fubx_get :: "'g \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'f, 'h, 'i, 'j) Ubx.instr fundef option"   and Fubx_add :: "'g \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'f, 'h, 'i, 'j) Ubx.instr fundef \<Rightarrow> 'g"   and Fubx_to_list :: "'g \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e, 'f, 'h, 'i, 'j) Ubx.instr fundef) list"   and heap_empty :: "'k"   and heap_get :: "'k \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c option"   and heap_add :: "'k \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c \<Rightarrow> 'k"   and heap_to_list :: "'k \<Rightarrow> (('d \<times> 'c) \<times> 'c) list"   and is_true :: "'c \<Rightarrow> bool"   and is_false :: "'c \<Rightarrow> bool"   and box_ubx1 :: "'i \<Rightarrow> 'c"   and unbox_ubx1 :: "'c \<Rightarrow> 'i option"   and box_ubx2 :: "'j \<Rightarrow> 'c"   and unbox_ubx2 :: "'c \<Rightarrow> 'j option"   and \<OO>\<pp> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<AA>\<rr>\<ii>\<tt>\<yy> :: "'e \<Rightarrow> nat"   and \<II>\<nn>\<ll>\<OO>\<pp> :: "'f \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<II>\<nn>\<ll> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'f option"   and \<II>\<ss>\<II>\<nn>\<ll> :: "'f \<Rightarrow> 'c list \<Rightarrow> bool"   and \<DD>\<ee>\<II>\<nn>\<ll> :: "'f \<Rightarrow> 'e"   and \<UU>\<bb>\<xx>\<OO>\<pp> :: "'h \<Rightarrow> ('c, 'i, 'j) unboxed list \<Rightarrow> ('c, 'i, 'j) unboxed option"   and \<UU>\<bb>\<xx> :: "'f \<Rightarrow> Unboxed.type option list \<Rightarrow> 'h option"   and \<BB>\<oo>\<xx> :: "'h \<Rightarrow> 'f"   and \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> :: "'h \<Rightarrow> Unboxed.type option list \<times> Unboxed.type option"   and xs :: "('c, 'i, 'j) unboxed list" assumes "inca_to_ubx_simulation Finca_empty Finca_get Finca_add Finca_to_list Fubx_empty Fubx_get Fubx_add Fubx_to_list heap_empty heap_get heap_add heap_to_list is_true is_false box_ubx1 unbox_ubx1 box_ubx2 unbox_ubx2 \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll> \<UU>\<bb>\<xx>\<OO>\<pp> \<UU>\<bb>\<xx> \<BB>\<oo>\<xx> \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp>" shows "length (inca_to_ubx_simulation.norm_stack box_ubx1 box_ubx2 xs) = length xs"
intro\<^sub>2: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> Sec option"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> Sec option" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>2\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"   and "\<turnstile> \<Gamma> {c} \<Gamma>'" shows "\<langle>c\<^sub>1 ;; c, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>'\<^esub> \<langle>c\<^sub>2 ;; c, mds, mem\<^sub>2\<rangle>"
D_neg: shows "\<integral>\<^sup>+ x. ennreal (- D x) \<partial>M = 0"
wt_elim_cases_13: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and va :: "Term.var"   and v :: "expr"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (va:=v)\<Colon>T" obtains T :: "ty" and T' :: "ty" where "Tb__ = Inl T'" and "E,dt\<Turnstile>va\<Colon>=T" and "va \<noteq> LVar This" and "E,dt\<Turnstile>v\<Colon>-T'" and "prg E\<turnstile>T'\<preceq>T"
lexordp_eq_refl: fixes xs :: "'a list" shows "lexordp_eq xs xs"
continuous_within_compose2: fixes x :: "'a"   and s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c" assumes "continuous (at x within s) f"   and "continuous (at (f x) within f ` s) g" shows "continuous (at x within s) (\<lambda>x. g (f x))"
simps_2: fixes prec :: "nat"   and t :: "nat"   and lf :: "floatarith"   and rt :: "floatarith"   and s :: "nat"   and ivl :: "float interval"   and ivl' :: "float interval" shows "approx_tse_concl prec t (LessEqual lf rt) s ivl ivl' = approx_tse_form' prec t (Add rt (Minus lf)) s (sup ivl ivl') (\<lambda>ivl. 0 \<le> lower ivl)"
sum_Cauchy_positive: fixes a :: "nat \<Rightarrow> real" assumes "\<And>n. 0 \<le> a n"   and "\<exists>K. \<forall>n. sum a {0..n} \<le> K" shows "Cauchy (\<lambda>n. sum a {0..n})"
assms_3: shows "\<beta> = \<phi> $$ (1, 0)"
intros_4: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c) Formula.Tree\<^sub>\<alpha>"   and \<alpha> :: "'c" assumes "Formula.hereditarily_fs t\<^sub>\<alpha>" shows "Formula.hereditarily_fs (Formula.Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha>)"
t_utm_def: shows "t_utm \<equiv> F_tprog @ shift (mopup (Suc (Suc 0))) (length F_tprog div 2)"
prefix_length_less: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs <p ys" shows "\<^bold>|xs\<^bold>| < \<^bold>|ys\<^bold>|"
x_def: shows "xs = vinsert \<langle>vcard xs', x\<rangle> xs'"
L3_5v_lemma3: fixes S :: "'b set" shows "SEM (\<Omega>) S = {}"
ra0: shows "r k (a $ 0) ^ k = a $ 0"
fold_congs_1: fixes r :: "('b, 'c) state_scheme"   and r' :: "('b, 'c) state_scheme"   and v' :: "nat"   and f :: "nat \<Rightarrow> nat"   and f' :: "nat \<Rightarrow> nat" assumes "r = r'"   and "m r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "m_update f r = m_update f' r'"
Seq_def: fixes f :: "unit \<Rightarrow> 'c Predicate.seq" shows "Predicate.Seq f = Predicate.pred_of_seq (f ())"
f: assumes "x \<in> s \<union> closure s \<inter> closure t" shows "(f \<longlongrightarrow> l x) (at x within s \<union> closure s \<inter> closure t)"
path_first_vertex: fixes x :: "'a"   and xs :: "'a list" assumes "path (x # xs)" shows "x \<notin> set xs"
ereal_add_diff_cancel: fixes b :: "ereal"   and a :: "ereal" assumes "\<bar>b\<bar> \<noteq> \<infinity>" shows "a + b - b = a"
maxim_greatest: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<in> A"   and "A \<subseteq> Field r" shows "(x, maxim A) \<in> r"
mult_imp_less_div_pos: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and y :: "'c"   and z :: "'c"   and x :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero y"   and "less (times z y) x" shows "less z (divide x y)"
beukers_nn_integral1_same_1: fixes s :: "nat"   and r :: "nat" assumes "s \<le> r"   and "r = s" shows "beukers_nn_integral1 r s = ennreal (2 * (Re (zeta 3) - (\<Sum>k = 1..r. 1 / real (k ^ 3))))"
get_root_node_si_no_parent: fixes h :: "(_) heap"   and node_ptr :: "(_) node_ptr" assumes "h \<turnstile> Shadow_DOM.get_parent node_ptr \<rightarrow>\<^sub>r None" shows "h \<turnstile> get_root_node_si (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr) \<rightarrow>\<^sub>r cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr"
add_independentS_Cons_conv_sum_right: fixes A :: "'a set"   and As :: "'a set list" shows "add_independentS (A # As) = (add_independentS [A, \<Sum>B\<leftarrow>As. B] \<and> add_independentS As)"
lm048: fixes P :: "('d \<times> 'c) set"   and Q :: "('d \<times> 'c) set" assumes "Range P \<inter> Range Q = {}"   and "runiq (P\<inverse>)"   and "runiq (Q\<inverse>)" shows "runiq ((P +< Q)\<inverse>)"
pj_mem: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and x :: "'c" assumes "Ring R"   and "ideal R I"   and "x \<in> carrier R" shows "pj R I x = x \<uplus>\<^bsub>R\<^esub> I"
less_eq_sum_def: fixes x :: "'a + 'b"   and y :: "'a + 'b" shows "(x \<le> y) = sum_le (\<le>) (\<le>) x y"
4: shows "F \<^bsub>\<a>\<^esub> (Id A) = Id\<^bsub>Set\<^esub> F \<^bsub>\<o>\<^esub> A"
unfold2': fixes E :: "('\<AA> state \<times> '\<AA> state \<Rightarrow> 'res) \<Rightarrow> '\<AA> state \<times> '\<AA> state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "'\<AA> state \<times> '\<AA> state"   and self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and POST' :: "('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0) \<Rightarrow> 'a \<Rightarrow> '\<AA> state \<times> '\<AA> state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and a1 :: "'a"   and BODY :: "('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0) \<Rightarrow> 'a \<Rightarrow> '\<AA> state \<times> '\<AA> state \<Rightarrow> 'res" assumes "cp E"   and "\<tau> \<Turnstile> \<delta> self \<and> True"   and "\<tau> \<Turnstile> PRE self"   and "\<tau> \<Turnstile> POST' self a1"   and "\<And>res. POST self res = (POST' self a1 and (res \<triangleq> BODY self a1))" shows "(\<tau> \<Turnstile> E (f self)) = (\<tau> \<Turnstile> E (BODY self a1))"
fref_cons': fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'd"   and P :: "'c \<Rightarrow> bool"   and R :: "('a \<times> 'c) set"   and S :: "('b \<times> 'd) set"   and Q :: "'c \<Rightarrow> bool" assumes "(f, g) \<in> [P]\<^sub>f R \<rightarrow> S"   and "\<And>c a. \<lbrakk>(c, a) \<in> R; Q a\<rbrakk> \<Longrightarrow> P a" shows "(f, g) \<in> [Q]\<^sub>f R \<rightarrow> S"
assms: shows "p \<in> set (comp_red_monic_basis xs)"
seqs_Suc': fixes n :: "nat" shows "seqs (Suc n) = (\<lambda>xs. xs @ [True]) ` seqs n \<union> (\<lambda>xs. xs @ [False]) ` seqs n"
Bseq_realpow: fixes x :: "real" assumes "0 \<le> x"   and "x \<le> 1" shows "Bseq ((^) x)"
inj_on_finite_image_iff: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A" shows "finite (f ` A) = finite A"
lsimulation_trans: fixes R1 :: "('h \<times> 'i) set"   and A :: "('h, 'j, 'k) sa_rec_scheme"   and B :: "('i, 'j, 'l) sa_rec_scheme"   and R2 :: "('i \<times> 'm) set"   and C :: "('m, 'j, 'n) sa_rec_scheme" assumes "lsimulation R1 A B"   and "lsimulation R2 B C" shows "lsimulation (R1 O R2) A C"
bdd_belowI: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and m :: "'b" assumes "class.preorder less_eq less"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq m x" shows "preorder.bdd_below less_eq A"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "comm_ring_isom hom"   and "inj_idom_hom hom" shows "idom_isom hom"
poly_strict_mono_def: fixes p :: "('c monom \<times> 'a) list"   and v :: "'c" shows "poly_strict_mono p v \<equiv> \<forall>\<alpha> \<beta>. (\<forall>x. v \<noteq> x \<longrightarrow> \<alpha> x = \<beta> x) \<longrightarrow> pos_assign \<beta> \<longrightarrow> \<alpha> v \<succ> \<beta> v \<longrightarrow> eval_poly \<alpha> p \<succ> eval_poly \<beta> p"
compare_real_of_rat: fixes x :: "rat"   and y :: "rat" shows "compare (real_of_rat x) (real_of_rat y) = compare x y"
simps_5: fixes vc :: "String.literal"   and \<alpha> :: "formula"   and \<beta> :: "formula" shows "past_only (formula.Let vc \<alpha> \<beta>) = (past_only \<alpha> \<and> past_only \<beta>)"
intros_1: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "('d, 'e, 'f) configuration" assumes "distributed_system channel" shows "distributed_system.trace channel trans send recv c [] c"
Field_restrict_rel: fixes R :: "('b \<times> 'b) set"   and A :: "'b set" assumes "Refl R" shows "Field (restrict_rel A R) = A \<inter> Field R"
add_assoc: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and c :: "nat \<Rightarrow> int" shows "a \<oplus> b \<oplus> c = a \<oplus> (b \<oplus> c)"
Arr_implies_Ide_Dom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "monoidal_language C"   and "monoidal_language.Arr C t" shows "monoidal_language.Ide C (monoidal_language.Dom C t)"
roots4_generic: fixes p :: "'a poly"   and depressed_solver :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a list" assumes "degree p = 4"   and "\<And>p q r y. (y \<in> set (depressed_solver p q r)) = (y ^ 4 + p * y\<^sup>2 + q * y + r = (0::'a))" shows "set (roots4_generic depressed_solver p) = {x. poly p x = (0::'a)}"
Bad0_def: shows "Bad0 = (\<lambda>n. {x \<in> space M. local_time B n x < k})"
assms_2: shows "other.s' \<delta> < v\<^sub>e"
ih2: fixes x :: "name"   and M :: "trm" assumes "x:M \<in> \<parallel>(A)\<parallel>" shows "SNa M"
wp_is_pre: fixes c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "\<turnstile> {wp c Q} c {Q}"
bdd_below_insert: fixes a :: "'b multiset"   and A :: "'b multiset set" shows "subset_mset.bdd_below (insert a A) = subset_mset.bdd_below A"
next_tame0_comp: fixes p :: "nat"   and g :: "graph" assumes "Seed\<^bsub>p\<^esub> [next_plane\<^bsub>p\<^esub>]\<rightarrow>* g"   and "final g"   and "tame g" shows "Seed\<^bsub>p\<^esub> [next_tame0\<^bsub>p\<^esub>]\<rightarrow>* g"
sets_Collect_5: fixes M :: "'b measure"   and Pb :: "bool" shows "{x \<in> space M. Pb} \<in> sets M"
field_differentiable_def: fixes f :: "'a \<Rightarrow> 'a"   and F :: "'a filter" shows "f field_differentiable F \<equiv> \<exists>f'. (f has_field_derivative f') F"
induces_horizontal_composition: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b" assumes "prebicategory_with_homs V H \<a> src trg" shows "horizontal_composition V H src trg"
map_prod_vimage: fixes f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set" shows "map_prod f g -` (A \<times> B) = f -` A \<times> g -` B"
normal_coeff_nonneg: fixes p :: "'a poly"   and i :: "nat" assumes "normal_poly p" shows "(0::'a) \<le> coeff p i"
quorum_is_quorum_of_some_slice: fixes p :: "'node"   and Q :: "'node set" assumes "quorum_of p Q"   and "p \<in> W" obtains S :: "'node set" where "S \<in> slices p" and "S \<subseteq> Q" and "\<And>p'. p' \<in> S \<inter> W \<Longrightarrow> quorum_of p' Q"
assms: shows "S : A \<mapsto>\<^bsub>dg_Par \<alpha>\<^esub> B"
deMorgan1: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and Y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "not (X and Y) = (not X or not Y)"
project_def: shows "project \<equiv> \<lambda>A ts. \<lparr>security_level = card (A \<inter> (taints ts - untaints ts)), trusted = A \<inter> untaints ts \<noteq> {}\<rparr>"
select_convs_1: fixes states :: "'a \<Rightarrow> 'b"   and msgs :: "nat \<Rightarrow> 'c message list"   and process_snapshot :: "'a \<Rightarrow> 'b option"   and channel_snapshot :: "nat \<Rightarrow> 'c list \<times> recording_state"   and more :: "'d" shows "states \<lparr>states = states, msgs = msgs, process_snapshot = process_snapshot, channel_snapshot = channel_snapshot, \<dots> = more\<rparr> = states"
valid_typeI_div: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and two_chain :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and s :: "(real \<times> real) set" assumes "green_typeI_chain i j F two_chain s" shows "valid_typeI_division s two_chain"
upd_elim': fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and upd :: "'Var ModeUpd"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mem' :: "'Var \<Rightarrow> 'Val"   and P :: "bool" assumes "eval\<^sub>w_simple_abv (c@[upd], mem) (c', mem')" shows "P"
assms_2: shows "FieldProj (New C0 es) fi \<in> subexprs(e)"
semiring_norm_3: fixes n :: "Num.num" shows "num.One + num.Bit0 n = num.Bit1 n"
SemTrue: fixes b :: "BExpr"   and s :: "Var \<Rightarrow> Val"   and c1 :: "IMP"   and n :: "nat"   and t :: "Var \<Rightarrow> Val"   and c2 :: "IMP" assumes "evalB b s"   and " s , c1 \<rightarrow>\<^sub>n  t " shows " s , Iff b c1 c2 \<rightarrow>\<^sub>n + 1  t "
replicate_eqI: fixes xs :: "'g list"   and n :: "nat"   and x :: "'g" assumes "length xs = n"   and "\<And>y. y \<in> set xs \<Longrightarrow> y = x" shows "xs = replicate n x"
sig_red_zero_filter: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and r :: "'t \<Rightarrow>\<^sub>0 'b"   and j :: "nat" assumes "sig_red_zero (\<preceq>\<^sub>t) (set bs) r"   and "component_of_term (lt r) < j" shows "sig_red_zero (\<preceq>\<^sub>t) (set (filter (\<lambda>b. component_of_term (lt b) < j) bs)) r"
poly_all_0_iff_0: fixes p :: "'b poly" shows "(\<forall>x. poly p x = (0::'b)) = (p = 0)"
state_def: fixes n :: "nat"   and v :: "complex mat" shows "state n v \<equiv> dim_col v = 1 \<and> dim_row v = 2 ^ n \<and> \<parallel>col v 0\<parallel> = 1"
param_case_nat: fixes Ra :: "('a \<times> 'b) set" shows "(case_nat, case_nat) \<in> Ra \<rightarrow> (nat_rel \<rightarrow> Ra) \<rightarrow> nat_rel \<rightarrow> Ra"
vdisjnt_self_iff_vempty: fixes S :: "V" shows "(S \<inter>\<^sub>\<circ> S = []\<^sub>\<circ>) = (S = []\<^sub>\<circ>)"
assms_2: shows "compl_fam (\<lambda>g. generate G {g}) gs"