hcju/isabelleps huggingface dataset:
pivotandupdate_tableau_equiv: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and v :: "nat \<Rightarrow> 'a"   and c :: "'a" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "v \<Turnstile>\<^sub>t \<T> s = v \<Turnstile>\<^sub>t \<T> (pivot_and_update x\<^sub>i x\<^sub>j c s)"
alw_alw: fixes \<phi> :: "'c stream \<Rightarrow> bool" shows "alw (alw \<phi>) = alw \<phi>"
wtc_mono: fixes s\<^sub>2 :: "'a"   and s\<^sub>1 :: "'a"   and pc :: "nat" assumes "s\<^sub>2 \<sqsubseteq>\<^bsub>r\<^esub> s\<^sub>1"   and "pc < length \<tau>s"   and "s\<^sub>1 \<in> A"   and "s\<^sub>2 \<in> A" shows "wtc c pc s\<^sub>2 \<sqsubseteq>\<^bsub>r\<^esub> wtc c pc s\<^sub>1"
one_F: fixes g :: "'d \<Rightarrow> 'a"   and A :: "'d set" shows "\<^bold>1 \<^bold>* F g A = F g A"
Polys_closed_plus: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and X :: "'b set"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "p \<in> P[X]"   and "q \<in> P[X]" shows "p + q \<in> P[X]"
Id_fstsnd_eq: shows "Id = {x. fst x = snd x}"
assms: shows "ac \<in>\<^sub>\<circ> r \<circ>\<^sub>\<circ> s"
round_robin_step_invar_None: fixes n0 :: "nat"   and \<sigma> :: "'t list \<times> nat"   and t' :: "'t"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and \<sigma>' :: "'t list \<times> nat" assumes "round_robin_step n0 \<sigma> s t' = \<lfloor>(t, None, \<sigma>')\<rfloor>"   and "round_robin_invar \<sigma> (dom (thr s))" shows "round_robin_invar \<sigma>' (dom (thr s))"
SReal_subset_HFinite: shows "\<real> \<subseteq> HFinite"
A1: shows "(cl, f) \<in> (SIGMA cl:CL. {f. f \<in> pset cl \<rightarrow> pset cl})"
twoCisTypeII_6: shows "g2 piecewise_C1_differentiable_on {a..b}"
involutory: fixes x :: "'a" shows "f \<langle>$\<rangle> (f \<langle>$\<rangle> x) = x"
prod_mono2: fixes B :: "'a set"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite B"   and "A \<subseteq> B"   and "\<And>b. b \<in> B - A \<Longrightarrow> (1::'b) \<le> f b"   and "\<And>a. a \<in> A \<Longrightarrow> (0::'b) \<le> f a" shows "prod f A \<le> prod f B"
later_3: shows "\<forall>x y. x \<in> A \<and> y \<in> A \<and> P x y \<longrightarrow> Q x y"
known_ptr_defs: fixes ptr :: "(_) object_ptr" shows "NodeClass.known_ptr ptr = False"
symmetry_rule: fixes G :: "('b, 'c) labeled_graph"   and l :: "'b" assumes "graph G"   and "maintained (symmetry_rule l) G" shows "sym (getRel l G)"
finite_gpvD: fixes gpv :: "('b, 'out, 'in) gpv"   and out :: "'out"   and c :: "('b, 'out, 'in) rpv"   and input :: "'in" assumes "local.finite_gpv gpv"   and "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<I> out" shows "local.finite_gpv (c input)"
sprod_tau_nu: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> y = \<tau> x + \<nu> x \<cdot> y"
lsorted_lhdD: fixes xs :: "'a llist" assumes "lsorted xs"   and "\<not> lnull xs"   and "\<not> lnull (ltl xs)" shows "lhd xs \<le> lhd (ltl xs)"
icard_infinite_conv: fixes A :: "'a set" shows "(icard A = \<infinity>) = infinite A"
FI_init: fixes P :: "assn"   and Q :: "assn"   and F :: "assn" assumes "FI [] (SLN * P) (SLN * Q) SLN SLN F" shows "FI_QUERY P Q F"
elims_2: fixes x :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.rank_skew_invar x" obtains "x = []"   | t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" where "x = [t__]"   | t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" and t' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" where "x = t # t' # bq" and "SkewBinomialHeapStruc.rank t \<le> SkewBinomialHeapStruc.rank t' \<and> SkewBinomialHeapStruc.rank_invar (t' # bq)"
dbl_inc_def: fixes x :: "'a" shows "neg_numeral_class.dbl_inc x = x + x + (1::'a)"
inj_prime_power: shows "inj_on (\<lambda>x. fst x ^ snd x) ({a. prime a} \<times> {0<..})"
fbox_star_induct: fixes y :: "'a"   and z :: "'a"   and x :: "'a" assumes "d y \<le> d z \<cdot>  |x] y" shows "d y \<le>  |x\<^sup>\<star>] z"
some_third_chamberD_adj: fixes X :: "'b set set"   and C :: "'b set"   and z :: "'b set"   and D :: "'b set" assumes "ThickChamberComplex X"   and "SimplicialComplex.maxsimp X C"   and "z \<lhd> C" shows "C \<sim> ThickChamberComplex.some_third_chamber X C D z"
ffold_finsert2: fixes x :: "val \<times> val"   and A :: "(val \<times> val) fset"   and z :: "bool" assumes "x |\<notin>| A" shows "ffold (good_entry xa__ e \<rho> \<circ> good_prod) z (finsert x A) = ffold (good_entry xa__ e \<rho> \<circ> good_prod) ((good_entry xa__ e \<rho> \<circ> good_prod) x z) A"
routing_rule_sort_key_def: shows "routing_rule_sort_key \<equiv> \<lambda>r. LinordHelper (0 - int (routing_prefix r)) (metric r)"
dord_pm_refl: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and s :: "'a \<Rightarrow>\<^sub>0 'b" assumes "class.linorder less_eq less"   and "ord s s" shows "linorder.dord_pm ord s s"
sorted_wrt2_simps_1: fixes P :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g" shows "sorted_wrt P [x] = True"
untilE: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "(P \<U> Q) \<sigma>" obtains i :: "nat" where "Q (\<sigma> |\<^sub>s i)" and "\<forall>k<i. P (\<sigma> |\<^sub>s k)"
am6_lem: fixes y :: "'a"   and x :: "'a" shows "ar y \<cdot> (x \<cdot> ar (y \<cdot> x)) = x \<cdot> ar (y \<cdot> x)"
OUTfromChCorrect_data22: shows "OUTfromChCorrect data22"
real_pp_np_neg_times: fixes a :: "real"   and f :: "'b \<Rightarrow> real" assumes "a \<le> 0" shows "pp (\<lambda>t. a * f t) = (\<lambda>t. - a * np f t) \<and> np (\<lambda>t. a * f t) = (\<lambda>t. - a * pp f t)"
subst_subst: fixes x :: "name"   and y :: "name"   and u :: "Lambda_Z.term"   and s :: "Lambda_Z.term"   and t :: "Lambda_Z.term" assumes "x \<noteq> y"   and "atom x \<sharp> u" shows "subst y u (subst x s t) = subst x (subst y u s) (subst y u t)"
map_tl: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "map f (tl xs) = tl (map f xs)"
simps_3: fixes \<xi> :: "nat \<Rightarrow> 'a set"   and q :: "'a" shows "\<xi> \<Turnstile>\<^sub>n prop\<^sub>n(q) = (q \<in> \<xi> 0)"
msort_distinct_sorted: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "distinct (linorder.msort less xs) \<and> linorder.sorted less_eq (linorder.msort less xs)"
normalize_preserves_generators: fixes l :: "(bool \<times> 'c) list"   and gens :: "'c set" assumes "l \<in> lists (UNIV \<times> gens)" shows "Cancelation.normalize l \<in> lists (UNIV \<times> gens)"
T_take_eq: fixes n :: "nat"   and xs :: "'a list" shows "T_take n xs = min n (length xs) + 1"
WeakPDGBackwardSliceCorrect: fixes prog :: "cmd" shows "BackwardSlice sourcenode targetnode kind (valid_edge prog) (_Entry_) (Defs prog) (Uses prog) id (WWeakControlDependence.PDG_BS_w prog)"
bounded_pos: fixes S :: "'b set" shows "bounded S = (\<exists>b>0. \<forall>x\<in>S. norm x \<le> b)"
simps_4: fixes a1 :: "'a aexp"   and a2 :: "'a aexp"   and s :: "'a \<Rightarrow> value option" shows "gval (Eq a1 a2) s = value_eq (aval a1 s) (aval a2 s)"
in_inv_image_betw: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and B :: "'b set"   and x :: "'c"   and y :: "'c" shows "inv_image_betw P f A B x y = (x \<in> A \<and> y \<in> A \<and> f x \<in> B \<and> f y \<in> B \<and> P (f x) (f y))"
PER': shows "part_equiv (per_union (Restr (uconnected F) verts) x y)"
normalize_keep_eigenvector: fixes A :: "complex mat"   and v :: "complex vec"   and e :: "complex"   and n :: "nat" assumes "eigenvector A v e"   and "A \<in> carrier_mat n n"   and "v \<in> carrier_vec n" shows "eigenvector A (vec_normalize v) e"
subdag_dag: fixes p :: "ref"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref"   and t :: "dag"   and st :: "dag" assumes "Dag p low high t"   and "st \<le> t" shows "\<exists>stp. Dag stp low high st"
valid_graph_rep_def: fixes l :: "('c \<times> 'd \<times> 'e) list" shows "valid_graph_rep l \<equiv> (\<forall>u d d' v. (u, v, d) \<in> set l \<and> (u, v, d') \<in> set l \<longrightarrow> d = d') \<and> distinct l"
span_sum: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and S :: "'d set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> ideal S" shows "sum f A \<in> ideal S"
hull_def: fixes S :: "'a set \<Rightarrow> bool"   and s :: "'a set" shows "S hull s = \<Inter> {t. S t \<and> s \<subseteq> t}"
lex_prodp_wfP: fixes r1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and r2 :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "wfP r1"   and "wfP r2" shows "wfP (lex_prodp r1 r2)"
simps_8: fixes uv :: "Arr" shows "local.comp Null uv = Null"
sum_of_2_squares_nat_prod: fixes A :: "'a set"   and f :: "'a \<Rightarrow> nat" assumes "\<And>x. x \<in> A \<Longrightarrow> sum_of_2_squares_nat (f x)" shows "sum_of_2_squares_nat (prod f A)"
nn_integral_indicator_singleton1: fixes y :: "'h"   and M :: "'h measure"   and f :: "'h \<Rightarrow> ennreal" assumes "{y} \<in> sets M" shows "\<integral>\<^sup>+ x. indicator {y} x * f x \<partial>M = emeasure M {y} * f y"
norm_nat_power: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and nat_power :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and real_power :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and y :: "'b" assumes "class.nat_power_normed_field one times plus zero nat_power divide inverse minus uminus dist norm scaleR sgn uniformity open inner real_power"   and "0 < n" shows "norm (nat_power n y) = real n powr inner y one"
sum_heights_measure_def: fixes l :: "nat \<Rightarrow> nat" shows "sum_heights_measure l \<equiv> \<Sum>v\<in>V. 2 * card V - l v"
aux: fixes s :: "'s" assumes "gen_rwof s"   and "gen_cond s" shows "gen_step s \<le> SPEC (\<lambda>s'. (s, s') \<in> rwof_rel (gds_init gds) gen_cond gen_step)"
set_sorted_list_of_set: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "class.linorder less_eq less"   and "finite A" shows "set (linorder.sorted_list_of_set less_eq A) = A"
lt: shows "t + punit.lt (rep_list f) = punit.lt (rep_list p)"
coeff_list: fixes p :: "'a list" shows "map (local.coeff p) (rev [0..<length p]) = p"
max_ex_gr: fixes X :: "'a set"   and k :: "'a" assumes "\<exists>x\<in>X. k < x"   and "finite X"   and "X \<noteq> {}" shows "k < Max X"
index_mat_four_block_2: fixes A :: "'b mat"   and B :: "'b mat"   and C :: "'b mat"   and D :: "'b mat" shows "dim_row (four_block_mat A B C D) = dim_row A + dim_row D"
assms_2: shows "\<S>\<bullet>\<C> \<turnstile> vs @ [$Block (tn _> tm) es] : ts _> ts'"
sim: fixes s :: "bool list option"   and \<eta> :: "nat" assumes "s \<noteq> None \<longrightarrow> length (the s) = \<eta>" shows "\<I>_real \<eta>, \<I>_ideal \<eta> \<turnstile>\<^sub>C CNV sim s \<surd>"
arr_cod_iff_arr: fixes f :: "'a" shows "Right_a.arr (Right_a.cod f) = Right_a.arr f"
bit_xor_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and n :: "nat" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "bit (xor a b) n = (bit a n \<noteq> bit b n)"
K_incl: fixes N :: "nat" shows "K N \<subseteq> A"
ord_p_lt: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.ord_p p q" shows "punit.lt p \<preceq> punit.lt q"
simps_2: fixes x :: "nat" shows "Suc x! = Suc x * x!"
f_Max\<^sub>0_correct_2: fixes m :: "nat"   and T :: "nat \<Rightarrow> nat" assumes "0 < m" shows "\<exists>x\<in>{1..m}. T x = f_Max\<^sub>0 T m"
oalist_inv_raw_induct: fixes xs :: "('a \<times> 'd) list"   and P :: "('a \<times> 'd) list \<Rightarrow> bool" assumes "oalist_inv_raw xs"   and "P []"   and "\<And>k v xs. \<lbrakk>oalist_inv_raw ((k, v) # xs); oalist_inv_raw xs; v \<noteq> (0::'d); \<And>k'. k' \<in> fst ` set xs \<Longrightarrow> lt k k'; P xs\<rbrakk> \<Longrightarrow> P ((k, v) # xs)" shows "P xs"
lsu_inf_closed_var: fixes x :: "'b"   and y :: "'b" shows "\<nu>\<^sup>\<natural> (\<nu>\<^sup>\<natural> x \<sqinter> \<nu>\<^sup>\<natural> y) = \<nu>\<^sup>\<natural> x \<sqinter> \<nu>\<^sup>\<natural> y"
poly_neg_number_rootat: fixes p :: "rat poly" shows "poly_neg_inf (real_of_rat_poly p) = real_of_rat (poly_neg_number_rootat p)"
\<delta>_symbols_finite: fixes \<delta> :: "('a, 'b) ta_rule set" assumes "finite \<delta>" shows "finite (\<delta>_symbols \<delta>)"
guarantees_INT_right: fixes X :: "'b program set"   and Y :: "'c \<Rightarrow> 'b program set"   and I :: "'c set" shows "X guarantees \<Inter> (Y ` I) = (\<Inter>i\<in>I. X guarantees Y i)"
t_int: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and T :: "'fsym trm" shows "TE.int \<xi> (tT T) = Ik.int \<xi> T"
i: shows "(E2, E2 + {#a#}) \<in> mult1 r"
splitAt_subset_ab: fixes a :: "'c list"   and b :: "'c list"   and ram :: "'c"   and vs :: "'c list" assumes "(a, b) = splitAt ram vs" shows "set a \<subseteq> set vs \<and> set b \<subseteq> set vs"
cases: fixes x :: "('a, 'e, 'f, 'g) SLin_state_scheme \<times> ('a, 'h, 'i, 'j) SLin_state_scheme" obtains s1 :: "('a, 'e, 'f, 'g) SLin_state_scheme" and s2 :: "('a, 'h, 'i, 'j) SLin_state_scheme" where "x = (s1, s2)"
IH: fixes y :: "nat" assumes "y < n" shows "eheight_exp y \<le> real (y + 3 choose 3) / 4"
some_basis_is_basis: fixes V :: "'b set" shows "is_basis V (some_basis V)"
axioms_2: fixes f :: "'d \<Rightarrow> 'e" assumes "bounded_clinear f" shows "bounded_clinear_axioms f"
simps_1: fixes v :: "'c" shows "removeLeaf (T v E E) = (v, E)"
styping_def: fixes \<Gamma> :: "char list \<Rightarrow> ty"   and s :: "char list \<Rightarrow> val" shows "(\<Gamma> \<turnstile> s) = (\<forall>x. type (s x) = \<Gamma> x)"
reduced_word_forI_compare: fixes A :: "'a set"   and a :: "'a"   and as :: "'a list"   and bs :: "'a list" assumes "reduced_word_for A a as"   and "bs \<in> lists A"   and "sum_list bs = a"   and "length bs = length as" shows "reduced_word_for A a bs"
finer_def: fixes P :: "'b set set"   and Q :: "'b set set" shows "finer P Q \<equiv> (\<forall>J\<in>Q. J = \<Union> {I \<in> P. I \<subseteq> J}) \<and> (P \<noteq> {} \<longrightarrow> Q \<noteq> {})"
distinct_107: fixes x91 :: "'a ltln"   and x92 :: "'a ltln"   and x111 :: "'a ltln"   and x112 :: "'a ltln" shows "x91 R\<^sub>n x92 \<noteq> x111 M\<^sub>n x112"
zero_in_circline_set: fixes H :: "circline" shows "(0\<^sub>h \<in> circline_set H) = circline_D0 H"
order_iff_strict: fixes a :: "'b list"   and b :: "'b list" shows "prefix a b = (strict_prefix a b \<or> a = b)"
unit_r_inv: fixes x :: "'a" assumes "prod.unit x" shows "x ** prod.inv x = one"
measure: shows "\<forall>\<sigma>. cond \<sigma> \<and> P \<sigma> \<longrightarrow> M \<sigma> = None \<or> f (snd (the (M \<sigma>))) < f \<sigma>"
edge_as_set_def: fixes e :: "'b \<times> 'b" shows "edge_as_set e = {fst e, snd e}"
characteristic_iso_4: fixes f :: "'a" assumes "Right.ide f" shows "Right.iso (Right.comp (f \<star> \<i>[src f]) \<a>[f, src f, src f])"
induct: fixes x :: "('d, 'e) trm"   and P :: "('d, 'e) trm \<Rightarrow> bool" assumes "dfree x"   and "\<And>i. P (Var i)"   and "\<And>r. P (Const r)"   and "\<And>args i. \<lbrakk>\<And>i. dfree (args i); \<And>i. P (args i)\<rbrakk> \<Longrightarrow> P ($f i args)"   and "\<And>\<theta>\<^sub>1 \<theta>\<^sub>2. \<lbrakk>dfree \<theta>\<^sub>1; P \<theta>\<^sub>1; dfree \<theta>\<^sub>2; P \<theta>\<^sub>2\<rbrakk> \<Longrightarrow> P (Plus \<theta>\<^sub>1 \<theta>\<^sub>2)"   and "\<And>\<theta>\<^sub>1 \<theta>\<^sub>2. \<lbrakk>dfree \<theta>\<^sub>1; P \<theta>\<^sub>1; dfree \<theta>\<^sub>2; P \<theta>\<^sub>2\<rbrakk> \<Longrightarrow> P (Times \<theta>\<^sub>1 \<theta>\<^sub>2)" shows "P x"
sum_var_monom_mult: fixes m :: "'b monom"   and n :: "'b monom"   and x :: "'b" shows "sum_var (m * n) x = sum_var m x + sum_var n x"
branch_nominals_def: fixes branch :: "(('e, 'd) fm list \<times> 'd) list" shows "branch_nominals branch \<equiv> \<Union> (block_nominals ` set branch)"
1: shows "{y \<in> A. (y, x) \<in> r} = A \<inter> {y. (y, x) \<in> r}"
free_const_distinct: fixes name\<^sub>1 :: "name"   and name\<^sub>2 :: "name" shows "free name\<^sub>1 \<noteq> const name\<^sub>2"
Red1ALengthNull: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>null\<bullet>length,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer),s\<rangle>"
card_all_edges: fixes A :: "nat set" assumes "finite A" shows "card (all_edges A) = card A choose 2"
combine_in_carrier: fixes R :: "('a, 'b) ring_scheme"   and Ks :: "'a list"   and Us :: "'a list" assumes "ring R"   and "set Ks \<subseteq> carrier R"   and "set Us \<subseteq> carrier R" shows "ring.combine R Ks Us \<in> carrier R"
finite: shows "(\<Sum>\<^sup>+ x. ennreal (pmf p x) * ennreal (measure_pmf.prob (f x) A)) < \<top>"
Abs_multiset_inverse: fixes y :: "'b \<Rightarrow> nat" assumes "y \<in> multiset" shows "count (Abs_multiset y) = y"
set_integrable_abs: fixes M :: "'c measure"   and A :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "set_integrable M A f" shows "set_integrable M A (\<lambda>x. \<bar>f x\<bar>)"
33: shows "bprv (all yy' (imp (NN \<langle>\<phi>\<rangle> (Var yy')) (eql \<langle>neg \<phi>\<rangle> (Var yy'))))"
maps_ide_in_hom: fixes a :: "'d" assumes "D.ide a" shows "\<guillemotleft>\<eta>o a : D.map a \<rightarrow>\<^sub>D FG.map a\<guillemotright>"
gen_elim: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and cs :: "'c list"   and w :: "'d" assumes "gen P cs w" obtains "\<And>c. c \<in> set cs \<Longrightarrow> P c w"
oppositeLiteralListRemove: fixes literal :: "Literal"   and literalList :: "Literal list" shows "oppositeLiteralList (removeAll literal literalList) = removeAll (opposite literal) (oppositeLiteralList literalList)"
dist_in_open_segment: fixes x :: "'b"   and a :: "'b"   and b :: "'b" assumes "x \<in> open_segment a b" shows "dist x a < dist a b \<and> dist x b < dist a b"
CCexp_eq: fixes cccExp :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and e :: "exp"   and a :: "Arity" shows "CoCallArityAnalysis.CCexp cccExp e\<cdot>a = snd (cccExp e\<cdot>a)"
a: shows "valid (\<Delta> @ [(x, T)] @ \<Gamma>)"
False: shows "\<not> (a = (0::'a) \<or> b = (0::'a))"
A_def: shows "A \<equiv> {c. \<Gamma> (Fun c []) = Var a \<and> public c}"
case_prod_beta: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and p :: "'c \<times> 'd" shows "(case p of (x, xa) \<Rightarrow> f x xa) = f (fst p) (snd p)"
SUP_inf: 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 f :: "'c \<Rightarrow> 'b"   and B :: "'c set"   and a :: "'b" assumes "class.complete_distrib_lattice Inf Sup inf less_eq less sup bot top" shows "inf (Sup (f ` B)) a = Sup ((\<lambda>b. inf (f b) a) ` B)"
asymp_equiv_uminus_iff: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" shows "((\<lambda>x. - f x) \<sim>[F] g) = (f \<sim>[F] (\<lambda>x. - g x))"
frac_from_carrier: fixes X :: "('a \<times> 'a) set" assumes "X \<in> local.carrier_quotient_ring" obtains r :: "'a" and s :: "'a" where "r \<in> R" and "s \<in> R\<setminus>I" and "X = equivalence.Class (R \<times> (R\<setminus>I)) {(x, y). (x, y) \<in> (R \<times> (R\<setminus>I)) \<times> R \<times> (R\<setminus>I) \<and> local.rel x y} (r, s)"
STL1: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "\<turnstile> F" shows "\<turnstile> \<box>F"
CHAOS_DF_refine_F: fixes A :: "'d set" shows "CHAOS A \<sqsubseteq>\<^sub>F DF A"
inf_aci_3: fixes x :: "'b upred"   and y :: "'b upred"   and z :: "'b upred" shows "(x \<and> y \<and> z) = (y \<and> x \<and> z)"
sorted_wrt_gen2: fixes A :: "nat"   and B :: "nat"   and a :: "nat list"   and b :: "nat list" shows "sorted_wrt (<\<^sub>r\<^sub>l\<^sub>e\<^sub>x\<^sub>2) (Algorithm.gen2 A B a b)"
prems_1: fixes xs' :: "'a list" assumes "rtrancl_path r x xs' y"   and "distinct (x # xs')"   and "set xs' \<subseteq> set xs" shows "thesis"
inverse_transformation_def: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" shows "inverse_transformation A B F G \<tau> \<equiv> (category A \<and> category B) \<and> functor A B F \<and> functor A B G \<and> natural_isomorphism A B F G \<tau>"
fls_as_fps': fixes f :: "'b fls"   and n :: "int" assumes "- fls_subdegree f \<le> n" shows "\<exists>f'. f = fls_shift n (fps_to_fls f')"
c_def: shows "c = (\<lambda>i. [:cofactor (sylvester_mat p q) i (m + n - 1):])"
deg_smult_decr: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and f :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "a \<in> carrier R"   and "f \<in> carrier (UP R)" shows "deg R (a \<odot>\<^bsub>UP R\<^esub> f) \<le> deg R f"
\<aa>: shows "ideal \<aa> R (+) (\<cdot>) \<zero> \<one>"
o_inv_o_cancel: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" assumes "inj f" shows "g \<circ> inv' f \<circ> f = g"
simps_1: fixes i :: "nat"   and j :: "nat" shows "is_Less (Less i j) = True"
relabel: fixes f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and u :: "nat" assumes "Height_Bounded_Labeling c s t f l"   and "relabel_precond f l u" shows "((f, relabel_effect f l u), f, l) \<in> pr_algo_rel"
cSum2: fixes Q :: "pi"   and a :: "name"   and x :: "name"   and Q' :: "pi"   and C :: "'a"   and P :: "pi" assumes "Q \<longmapsto> a<x> \<prec> Q'"   and "\<And>C. F C Q a x Q'" shows "F C (P \<oplus> Q) a x Q'"
times_hom_axioms: shows "times_hom (inv (inv (inv hom)))"
T_def: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg" shows "R_G.T cfg = distr (MDP.MC.T cfg) R_G.St (smap state)"
hyps_exist': fixes c :: "'form"   and "is" :: "nat list"   and i :: "nat"   and s :: "'subst"   and ants :: "('form, 'rule, 'subst, 'var) itnode rose_tree list" assumes "c \<in> set conclusions"   and "is \<in> it_paths (it' c)"   and "tree_at (it' c) is = HNode i s ants" shows "subst s (freshen i anyP) \<in> hyps_along (it' c) is"