hcju/isabelleps huggingface dataset:
gt_imp_vars_mset: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "local.kbo_std.gt t s" shows "vars_mset s \<subseteq># vars_mset t"
valid_last_prog: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "last_prog p sb = p"
low_mds_eq_from_conc_to_abs: fixes mds :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem :: "'Var\<^sub>C \<Rightarrow> 'Val"   and mem' :: "'Var\<^sub>C \<Rightarrow> 'Val" assumes "conc.low_mds_eq mds mem mem'" shows "abs.low_mds_eq (mds\<^sub>A_of mds) (mem\<^sub>A_of mem) (mem\<^sub>A_of mem')"
harry_meas_Falses_True: fixes n :: "nat" shows "harry_meas (replicate n False @ [True]) = 2 * n + 1"
measure_eqI_countable: fixes M :: "'d measure"   and A :: "'d set"   and N :: "'d measure" assumes "sets M = Pow A"   and "sets N = Pow A"   and "countable A"   and "\<And>a. a \<in> A \<Longrightarrow> emeasure M {a} = emeasure N {a}" shows "M = N"
vars_of_poly_in_alt_def2: fixes xs :: "('a list \<times> 'b) list"   and \<V> :: "'a set" shows "vars_of_poly_in xs \<V> = fold (\<lambda>(x, uu_) b. b \<and> vars_of_monom_in x \<V>) xs True"
match_4: fixes g :: "('o, 'a) arr"   and f :: "('o, 'a) arr"   and h :: "('o, 'a) arr" assumes "seq g f"   and "seq h g" shows "seq h (COMP g f)"
ord0_less_imp_le: fixes x :: "ord0"   and y :: "ord0" assumes "x < y" shows "x \<le> y"
qTermLess_ingoreFirst3_wf: shows "wf (inv_image qTermLess aux_qSwap_ignoreFirst3)"
surjective: fixes r :: "'a point3d_scheme" shows "r = \<lparr>xc = xc r, yc = yc r, zc = zc r, \<dots> = point3d.more r\<rparr>"
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 "rel_fun A B f g"   and "A x y" shows "B (f x) (g y)"
updEnv_preserves_good: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and x :: "'var"   and xs :: "'varSort" assumes "good X"   and "goodEnv rho" shows "goodEnv (rho [x \<leftarrow> X]_xs)"
sig_red_single_top_lt_rep_list: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> nat"   and term_of_pair :: "'d \<times> nat \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and fs :: "('d \<Rightarrow>\<^sub>0 'e) list"   and sing_reg :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'e"   and q :: "'c \<Rightarrow>\<^sub>0 'e"   and f :: "'c \<Rightarrow>\<^sub>0 'e"   and t :: "'d" assumes "qpm_inf_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "qpm_inf_term.sig_red_single pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict fs sing_reg (=) p q f t"   and "qpm_inf_term.rep_list term_of_pair fs q \<noteq> 0" shows "ord_strict (ordered_term.lt fst ord (qpm_inf_term.rep_list term_of_pair fs q)) (ordered_term.lt fst ord (qpm_inf_term.rep_list term_of_pair fs p))"
vector_compl: fixes x :: "'a" assumes "is_vector x" shows "is_vector (- x)"
filt_equiv_filtration: fixes F :: "'c \<Rightarrow> 'd measure"   and M :: "'d measure"   and N :: "'d measure" assumes "filt_equiv F M N" shows "Filtration.filtration N F"
funion_mono: fixes A :: "'a fset"   and C :: "'a fset"   and B :: "'a fset"   and D :: "'a fset" assumes "A |\<subseteq>| C"   and "B |\<subseteq>| D" shows "A |\<union>| B |\<subseteq>| C |\<union>| D"
hyps_2: shows "G, \<Gamma> \<Rightarrow>\<^sub>n"
add_right: fixes a :: "'d \<Rightarrow>\<^sub>L 'c"   and b :: "'d"   and b' :: "'d" shows "a *\<^sub>v (b + b') = a *\<^sub>v b + a *\<^sub>v b'"
Bot_delR: fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> \<Delta>" shows "\<Gamma> \<Rightarrow> \<Delta> - (\<bottom>, {#})"
poly_IVT_pos: fixes a :: "real"   and b :: "real"   and p :: "real poly" assumes "a < b"   and "poly p a < 0"   and "0 < poly p b" shows "\<exists>x>a. x < b \<and> poly p x = 0"
shiftr_over_and_dist: fixes a :: "'b word"   and b :: "'b word"   and c :: "nat" shows "a AND b >> c = (a >> c) AND (b >> c)"
pairs_le_eq_Sigma: fixes m :: "nat" shows "{(i, j). i + j \<le> m} = (SIGMA r:{..m}. {..m - r})"
latticeof_def: fixes r :: "('b \<times> 'b) set" shows "latticeof r \<equiv> {X. \<forall>s t. s \<in> X \<and> (s, t) \<in> r \<longrightarrow> t \<in> X}"
Morf: shows "f \<in> mor\<^bsub>C\<^esub>"
ms_of_greek_def: fixes as :: "(accent \<times> 'a) list" shows "ms_of_greek as = mset (map (\<lambda>x. case x of (xs, y, zs) \<Rightarrow> adj_msog xs zs (y, [])) (list_splits as))"
tconcat_TNil: fixes b :: "'b"   and b' :: "'b" shows "tconcat b (TNil b') = TNil b'"
prodag_sameTr1: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme"   and B :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)"   and "\<forall>k\<in>I. A k = B k" shows "carr_prodag I A = carr_prodag I B"
SUP_insert: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and f :: "'e \<Rightarrow> 'd"   and a :: "'e"   and A :: "'e set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (f ` insert a A) = sup (f a) (Sup (f ` A))"
eq_iff: fixes x :: "'b list"   and y :: "'b list" shows "(x = y) = (subseq x y \<and> subseq y x)"
pinduct: fixes a0 :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list \<times> ((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set"   and P :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list \<times> ((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set \<Rightarrow> bool" assumes "shift1_dom a0"   and "\<And>qs B. \<lbrakk>shift1_dom (qs, B); \<And>x xa y xb. \<lbrakk>1 < card B; x = (SOME b. b \<in> B \<and> card (snd b) = Suc m); (xa, y) = x; xb = (SOME ya. ya \<in> y)\<rbrakk> \<Longrightarrow> P (shift_list (xa, y) xb qs, B - {(xa, y)})\<rbrakk> \<Longrightarrow> P (qs, B)" shows "P a0"
allUses_def: fixes u :: "'g \<Rightarrow> ('node, 'val set) mapping"   and p :: "'g \<Rightarrow> ('node \<times> 'val, 'val list) mapping"   and g :: "'g"   and n :: "'node" shows "uninst_code.ssa.allUses u p g n \<equiv> (usesOf \<circ> u) g n \<union> uninst_code.ssa.phiUses p g n"
presAbs_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'e"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'f"   and MOD :: "('index, 'bindex, 'varSort, 'g, 'opSym, 'var, 'e, 'f, 'h) Recursion.model_scheme" shows "presAbs h hA MOD \<equiv> \<forall>xs x s X. isInBar (xs, s) \<and> wls s X \<longrightarrow> hA (Abs xs x X) = gAbs MOD xs x X (h X)"
char_poly_four_block_mat_lower_left_zero: fixes A :: "'a mat"   and B :: "'a mat"   and C :: "'a mat"   and m :: "nat"   and n :: "nat"   and D :: "'a mat" assumes "A = four_block_mat B C (0\<^sub>m m n) D"   and "B \<in> carrier_mat n n"   and "C \<in> carrier_mat n m"   and "D \<in> carrier_mat m m" shows "char_poly A = char_poly B * char_poly D"
pw_leq_lstep: fixes x :: "nat \<times> nat"   and y :: "nat \<times> nat" assumes "(x, y) \<in> pair_leq" shows "pw_leq {#x#} {#y#}"
take_bit_minus_one_eq_mask: fixes n :: "nat" shows "take_bit n (- (1::'a)) = mask n"
assms_3: fixes a :: "'b"   and p :: "'b"   and q :: "'a" assumes "a \<in> span B"   and "P a"   and "p \<in> B"   and "q \<noteq> (0::'a)" shows "P (a + q *s p)"
unat_lt2p: shows "unat a < 2 ^ LENGTH(1)"
ord_canc: fixes s :: "'x \<Rightarrow>\<^sub>0 nat"   and u :: "'x \<Rightarrow>\<^sub>0 nat"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "s + u \<preceq> t + u" shows "s \<preceq> t"
finite_zcount_pos: fixes M :: "'a zmultiset" shows "finite {t. 0 < zcount M t}"
simps_10: fixes f :: "nat \<Rightarrow> atom \<Rightarrow> atom fm"   and i :: "nat"   and \<phi> :: "atom fm"   and var :: "nat" shows "liftmap f (ExN i \<phi>) var = ExN i (liftmap f \<phi> (var + i))"
arev_in_arcs: fixes a :: "'b" assumes "a \<in> arcs G" shows "arev a \<in> arcs G"
lang_dual_final_dual: fixes r :: "'b rexp_dual"   and n :: "nat" shows "final_dual r = ([] \<in> lang_dual n r)"
abs_ltln\<^sub>P_def: shows "abs_ltln\<^sub>P \<equiv> quot_type.abs (\<sim>\<^sub>P) Abs_ltln\<^sub>P"
leg1_simps_3: shows "src u\<^sub>1 = src u\<^sub>0"
degree_def': fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and p :: "'c list" assumes "ring R"   and "polynomial\<^bsub>R\<^esub> K p" shows "degree p = (LEAST n. \<forall>i>n. ring.coeff R p i = \<zero>\<^bsub>R\<^esub>)"
f_def: shows "f = (\<lambda>x. - ln x * x powr e)"
intro: fixes L :: "('c, 'd) ring_scheme"   and K :: "'c set" assumes "\<And>x. x \<in> carrier L \<Longrightarrow> ((\<lambda>K x. \<not> ring.transcendental L K x) over K) x"   and "\<And>P. P \<in> carrier (K [X]\<^bsub>L\<^esub>) \<Longrightarrow> ring.splitted L P" shows "algebraic_closure_axioms L K"
Cinfinite_ordLess_cexp: fixes r :: "('c \<times> 'c) set" assumes "Cinfinite r" shows "r <o r ^c r"
edge_weight_def: fixes G :: "('b, 'a) graph" shows "edge_weight G \<equiv> sum (fst \<circ> snd) (edges G)"
perp_proj: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Perp C D"   and "\<not> Col A C D" shows "\<exists>X. Col A B X \<and> A X Perp C D"
fmfilter_alt_defs_5: fixes B :: "'e fset" shows "fmrestrict_fset B = fmfilter (\<lambda>a. a |\<in>| B)"
L_def: shows "L = [0..<MPoly_Type.degree p var + 1]"
llist_all2_llengthD: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist" assumes "llist_all2 P xs ys" shows "llength xs = llength ys"
framed_linkrel_inv: fixes w1 :: "wall"   and w2 :: "wall" assumes "framed_linkrel w1 w2" shows "kauff_mat w1 = kauff_mat w2"
cases: fixes x :: "'b list \<times> 'b list" obtains ys :: "'b list" where "xa__ = ([], ys__)"   | xs :: "'b list" where "xa__ = (xsa__, [])"   | x :: "'b" and xs :: "'b list" and y :: "'b" and ys :: "'b list" where "xa__ = (x # xs, y # ys)"
create_opt_sol_in_LP_4: fixes X :: "(nat, rat) mapping"   and x :: "rat vec"   and y :: "rat vec"   and i :: "nat" assumes "create_optimal_solutions A b c = Inr X"   and "(x, y) = split_n_m_x (dim_vec c) (dim_vec b) X"   and "i < dim_vec y" shows "0 \<le> y $ i"
finsum_cong: fixes A :: "'e set"   and B :: "'e set"   and f :: "'e \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'c" assumes "A = B"   and "(f \<in> B \<rightarrow> carrier S) = True"   and "\<And>i. i \<in> B =simp=> f i = g i" shows "finsum S f A = finsum S g B"
err_mono: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "err A \<subseteq> err B"
path_join_path_ends: fixes g1 :: "real \<Rightarrow> 'c"   and g2 :: "real \<Rightarrow> 'c" assumes "path (g1 +++ g2)"   and "path g2" shows "pathfinish g1 = pathstart g2"
Inf_level: fixes L :: "level set" shows "\<Sqinter> L \<equiv> if Lo \<in> L then Lo else Hi"
mem_Collect_eq: fixes a :: "'k"   and P :: "'k \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
trans: fixes a :: "'b word"   and b :: "'b word"   and c :: "'b word" assumes "a \<le>s b"   and "b \<le>s c" shows "a \<le>s c"
integral_FTC_Icc_real: fixes a :: "real"   and b :: "real"   and F :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> real" assumes "a \<le> b"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> (F has_real_derivative f x) (at x within {a..b})"   and "continuous_on {a..b} f" shows "LBINT x. f x * indicat_real {a..b} x = F b - F a"
all_subset_image: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and P :: "'c set \<Rightarrow> bool" shows "(\<forall>B\<subseteq>f ` A. P B) = (\<forall>B\<subseteq>A. P (f ` B))"
addconst: fixes a :: "nat"   and b :: "nat"   and c :: "nat" assumes "a = b" shows "c + a = c + b"
write_sops_le: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'a"   and \<O> :: "'b" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "\<forall>i\<in>\<Union> (fst ` write_sops sb). i < t"
padic_add_def: fixes p :: "int"   and f :: "nat \<Rightarrow> int"   and g :: "nat \<Rightarrow> int" shows "padic_add p f g \<equiv> \<lambda>n. f n \<oplus>\<^bsub>residue_ring (p ^ n)\<^esub> g n"
3: shows "Left.in_hom \<l>[Left.cod \<mu>] (trg \<mu> \<star> cod \<mu>) (cod \<mu>)"
inject_1: fixes x1 :: "'b equation"   and y1 :: "'b equation" shows "(Pos x1 = Pos y1) = (x1 = y1)"
atS_def: fixes p :: "'i"   and ls :: "'j set"   and s :: "('k, 'j, 'i, 'l, 'm) system_state" shows "atS p ls s = (\<exists>l\<in>ls. at p l s)"
antisym: fixes x :: "'b"   and y :: "'b" assumes "x \<le> y"   and "y \<le> x" shows "x = y"
writes_union_right_I: fixes S' :: "('x, 'y, 'z) prog set"   and f :: "('x, 'y, 'aa) prog"   and h :: "'x"   and h' :: "'x"   and S :: "('x, 'y, 'z) prog set" assumes "writes S' f h h'" shows "writes (S \<union> S') f h h'"
latestAct_def: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "dynamic_component.latestAct active c t n = (GREATEST n'. n' < n \<and> active c (t n'))"
iMODb_iIN_eq_conv: fixes d :: "nat"   and n :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" assumes "0 < d" shows "([n\<dots>,d] = [ r, mod m, c ]) = (n = r \<and> m = Suc 0 \<and> c = d)"
scale_cancel_right: fixes a :: "'a"   and x :: "('a, 'b) vec"   and b :: "'a" shows "(a *s x = b *s x) = (a = b \<or> x = 0)"
cfg'_eq: fixes \<Gamma> :: "(String.literal \<Rightarrow> unit option) \<times> (String.literal \<Rightarrow> unit option)"   and c :: "cmd"   and a :: "action + brk_ctd"   and c' :: "cmd" assumes "ty_cmd (\<Gamma>, False) c" shows "cfg c a c' = (\<exists>aa. a = Inl aa \<and> cfg' c aa c')"
min_maxsimpchain_betw_CCons_reduce: fixes w :: "'a set"   and x :: "'a set"   and ys :: "'a set list"   and z :: "'a set" assumes "min_gallery (w # x # ys @ [z])" shows "min_gallery (x # ys @ [z])"
or_def: fixes \<phi>\<^sub>1 :: "'a fm"   and \<phi>\<^sub>2 :: "'a fm" shows "or \<phi>\<^sub>1 \<phi>\<^sub>2 = (if \<phi>\<^sub>1 = FalseF then \<phi>\<^sub>2 else if \<phi>\<^sub>2 = FalseF then \<phi>\<^sub>1 else if \<phi>\<^sub>1 = TrueF \<or> \<phi>\<^sub>2 = TrueF then TrueF else Or \<phi>\<^sub>1 \<phi>\<^sub>2)"
compl_eq_compl_iff: fixes x :: "'b upred"   and y :: "'b upred" shows "((\<not> x) = (\<not> y)) = (x = y)"
WT_oracle2: fixes s2 :: "'a" shows "\<I> \<turnstile>c oracle2 s2 \<surd>"
commute: fixes a :: "'h"   and b :: "'h" shows "a + b = b + a"
subtuple_acyclic: fixes xm :: "'e \<rightharpoonup>\<^sub>f 'd"   and P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ym :: "'e \<rightharpoonup>\<^sub>f 'd" assumes "acyclicP_on (fmran' xm) P"   and "subtuple (\<lambda>x y. P x y \<or> x = y)\<^sup>+\<^sup>+ xm ym"   and "subtuple (\<lambda>x y. P x y \<or> x = y) ym xm" shows "xm = ym"
rep_eq: fixes x :: "nat"   and xa :: "nat"   and xb :: "(nat \<Rightarrow>\<^sub>0 nat, 'b) fmap" shows "fmlookup (higherPoly\<^sub>f x xa xb) = (\<lambda>mon. if \<exists>j\<in>{x..<x + xa}. 0 < lookup mon j then None else fmlookup xb (higherPowers x xa mon))"
push: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat"   and e :: "nat \<times> nat" assumes "Height_Bounded_Labeling c s t f l"   and "push_precond f l e" shows "((push_effect f e, l), f, l) \<in> pr_algo_rel"
pfp_def: fixes f :: "'b \<Rightarrow> 'b" shows "pfp f = while_option (\<lambda>x. \<not> f x \<sqsubseteq> x) f"
intros: fixes pc1 :: "nat"   and len :: "nat"   and pc2 :: "nat"   and xcp1 :: "'a option"   and xcp2 :: "'a option" assumes "pc1 \<le> len"   and "pc2 \<le> len"   and "xcp1 \<noteq> None \<and> xcp2 = None \<and> pc1 = pc2 \<or> xcp1 = None \<and> pc2 < pc1" shows "sim21_size_aux len (pc1, xcp1) (pc2, xcp2)"
minimal_in_def: fixes P :: "'a set"   and x :: "'a" shows "minimal_in P x \<equiv> x \<in> P \<and> (\<forall>z\<in>P. \<not> x \<succ> z)"
update_def: fixes f :: "'c \<Rightarrow> 'c" shows "update f = get \<bind> local.set \<circ> f"
matrix_left_invertible_span_rows: fixes A :: "((real, 'b) vec, 'a) vec" shows "(\<exists>B. B ** A = mat 1) = (span (rows A) = UNIV)"
list_of_vec_plus: fixes v1 :: "complex vec"   and v2 :: "complex vec" assumes "dim_vec v1 = dim_vec v2" shows "list_of_vec (v1 + v2) = map2 (+) (list_of_vec v1) (list_of_vec v2)"
UN_empty2: fixes A :: "'b set" shows "(\<Union>x\<in>A. {}) = {}"
dgrad_p_set_closed_snd_crit_pair: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and m :: "nat"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d"   and "p \<in> dgrad_p_set d m"   and "q \<in> dgrad_p_set d m" shows "snd (crit_pair p q) \<in> dgrad_p_set d m"
sup_aci_3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
wordinterval_to_set_ipcidr_tuple_to_wordinterval: fixes b :: "'a word"   and m :: "nat" shows "wordinterval_to_set (ipcidr_tuple_to_wordinterval (b, m)) = ipset_from_cidr b m"
has_real_derivative_imp_continuous_on: fixes A :: "real set"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real" assumes "\<And>x. x \<in> A \<Longrightarrow> (f has_real_derivative f' x) (at x)" shows "continuous_on A f"
rk_A'B'PQ: fixes A :: "Points"   and A' :: "Points"   and B :: "Points"   and C :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and Q :: "Points" assumes "rk {A, A'} = 2"   and "rk {A, B, C, A', B', C'} = 3"   and "rk {A, A', P} = 2"   and "rk {A, B, P} = 3"   and "rk {B, B', P} = 2"   and "rk {A', P} = 2"   and "rk {B', P} = 2"   and "4 \<le> rk {A, B, C, A', B', C', P, Q}" shows "rk {A', B', P, Q} = 4"
log2_is_x: shows "log2 (u + 2 ^ x) = x"
set_zip_map_fst_snd2: fixes u :: "'a"   and x :: "'b"   and us :: "'a list"   and txs :: "('c \<times> 'b) list"   and t :: "'c" assumes "(u, x) \<in> set (zip us (map snd txs))"   and "(t, x) \<in> set txs"   and "distinct (map snd txs)"   and "distinct us"   and "length us = length txs" shows "(t, u) \<in> set (zip (map fst txs) us)"
f: shows "f \<in> N \<rightarrow>\<^sub>M M"
mult2_s_one_step': fixes ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and X :: "'a multiset"   and Y :: "'a multiset" assumes "ns O s \<subseteq> s"   and "refl ns"   and "irrefl s"   and "trans s" shows "((X, Y) \<in> mult2_s ns s) = (\<exists>X1 X2 Y1 Y2. X = X1 + X2 \<and> Y = Y1 + Y2 \<and> (X1, Y1) \<in> multpw ns \<and> (X2, Y2) \<in> mult s)"
copy: fixes rel :: "(nat \<times> nat) set"   and n :: "nat"   and x :: "nat"   and y :: "nat" assumes "Steps rel n (x, y)" shows "Steps rel (Suc n) (x, y)"
assms_1: fixes i :: "nat" assumes "i < length T" shows "(T ! i, S ! i) \<in> timpl_closure' c"
R_lin_independentI_all_scalars: fixes ms :: "'m list" assumes "set ms \<subseteq> M"   and "\<forall>rs. set rs \<subseteq> R \<and> length rs = length ms \<and> rs \<bullet>\<cdot> ms = (0::'m) \<longrightarrow> set rs \<subseteq> 0" shows "R_lin_independent ms"
TC_rank_mem: fixes x :: "V"   and y :: "V" assumes "x \<sqsubset> y" shows "rank x \<in> elts (rank y)"
S5_live: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> \<box>(ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p)) \<and> WF(RPCNext crCh rmCh rst p)_r p \<longrightarrow> (S5 rmhist p \<leadsto> S6 rmhist p)"
countable_card_of_nat: fixes A :: "'c set" shows "countable A = (|A| \<le>o |UNIV|)"
berlekamp_resulting_mat_closed_2: fixes u :: "'b mod_ring poly" shows "dim_row (berlekamp_resulting_mat u) = degree u"
lin_dependent_dependence_relation: fixes ms :: "'b list" assumes "set ms \<subseteq> V"   and "\<not> R_lin_independent ms" shows "\<exists>rs. set rs \<subseteq> UNIV \<and> set rs \<noteq> 0 \<and> length rs = length ms \<and> rs \<bullet>\<cdot> ms = (0::'b)"
alt_state_def: fixes alt :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and m1 :: "('d, 'e) stateT"   and m2 :: "('d, 'e) stateT" shows "alt_state alt m1 m2 = StateT (\<lambda>s. alt (run_state m1 s) (run_state m2 s))"
cfun_eqI: fixes f :: "'b \<rightarrow> 'c"   and g :: "'b \<rightarrow> 'c" assumes "\<And>x. f\<cdot>x = g\<cdot>x" shows "f = g"
neutl: fixes x :: "'c DBMEntry" shows "dbm_add (Le (0::'c)) x = x"
ideal_asubg: fixes I :: "'a set" assumes "ideal R I" shows "R +> I"
simps_6: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "\<xi> \<Turnstile>\<^sub>n \<phi> or\<^sub>n \<psi> = (\<xi> \<Turnstile>\<^sub>n \<phi> \<or> \<xi> \<Turnstile>\<^sub>n \<psi>)"
tendsto_lowerbound: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'b"   and x :: "'b"   and F :: "'d filter"   and a :: "'b" assumes "class.linorder_topology less_eq less open"   and "filterlim f (topological_space.nhds open x) F"   and "\<forall>\<^sub>F i in F. less_eq a (f i)"   and "F \<noteq> bot" shows "less_eq a x"
arr_dom: fixes f :: "'a2" assumes "C2.arr f" shows "C2.arr (C2.dom f)"
Full_step_simulation: shows "|~ _actrans FullNxt vars \<longrightarrow> _actrans (Nxt inp qc out) (inp, qc, out)"
transp_sequence: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and seq :: "nat \<Rightarrow> 'd"   and i :: "nat"   and j :: "nat" assumes "transp r"   and "\<And>i. r (seq (Suc i)) (seq i)"   and "i < j" shows "r (seq j) (seq i)"
prems: shows "k < length (trace_parallel_plan_sas_plus I (a # \<psi>))"
vt_e': fixes \<rho> :: "nat \<Rightarrow> val" assumes "\<rho> \<in> space (state_measure V' \<Gamma>)" shows "val_type (expr_sem_rf \<rho> e') = INTEG"
xxs_i_ge_a: shows "\<forall>i<length (x # xs). a \<le> (x # xs) ! i"
iMOD_iIN_iMODb_conv: fixes r :: "nat"   and m :: "nat"   and d :: "nat" shows "[ r, mod m ] \<inter> [r\<dots>,d] = [ r, mod m, d div m ]"
step_reflects_SN_on: fixes a :: "'b"   and r :: "('b \<times> 'b) set" assumes "\<And>b. (a, b) \<in> r \<Longrightarrow> SN_on r {b}" shows "SN_on r {a}"
"3_3": fixes c :: "poscheckvariant"   and t :: "('d, 'c) Term.term"   and s :: "('d, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p (\<langle>c: t \<doteq> s\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) = fv (t \<cdot> \<theta>) \<union> fv (s \<cdot> \<theta>)"
assms_2: fixes j :: "nat" assumes "j < n" shows "polyfun N (\<lambda>x. v x $ j)"
four_block_mat_def: fixes A :: "'b mat"   and B :: "'b mat"   and C :: "'b mat"   and D :: "'b mat" shows "four_block_mat A B C D = (let nra = dim_row A; nrd = dim_row D; nca = dim_col A; ncd = dim_col D in mat (nra + nrd) (nca + ncd) (\<lambda>(i, j). if i < nra then if j < nca then A $$ (i, j) else B $$ (i, j - nca) else if j < nca then C $$ (i - nra, j) else D $$ (i - nra, j - nca)))"
S1S: shows "poly_mod.Mp p S1 = poly_mod.Mp p S"
execn_strip_to_execn: fixes F :: "'f set"   and \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "strip F \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t" shows "\<exists>t'. \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t' \<and> (isFault t \<longrightarrow> isFault t') \<and> (t' \<in> Fault ` (- F) \<longrightarrow> t' = t) \<and> (\<not> isFault t' \<longrightarrow> t' = t)"
q_cons_rel: shows "\<forall>q\<in>Q. consequence_relation Bot (entails_q q)"