hcju/isabelleps huggingface dataset:
Smith_1xn_works: fixes Smith_1x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat"   and Smith_2x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat \<times> 'a mat"   and div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'a mat"   and n :: "nat"   and S :: "'a mat"   and Q :: "'a mat" assumes "Smith_Impl Smith_1x2 Smith_2x2 div_op"   and "A \<in> carrier_mat 1 n"   and "(S, Q) = Smith_Impl.Smith_1xn Smith_1x2 A" shows "is_SNF A (1\<^sub>m 1, S, Q)"
invL_FL_bisimilar_is_L_bisimulation: shows "is_L_bisimulation invL_FL_bisimilar"
fpxs_nth_mult: fixes f :: "'a fpxs"   and g :: "'a fpxs"   and r :: "rat" shows "fpxs_nth (f * g) r = (\<Sum>(y, z)\<in>{(y, z). y \<in> fpxs_supp f \<and> z \<in> fpxs_supp g \<and> r = y + z}. fpxs_nth f y * fpxs_nth g z)"
divide_atLeastAtMost_1_absI: fixes x :: "real"   and a :: "real" assumes "\<bar>x\<bar> \<le> \<bar>a\<bar>" shows "x / a \<in> {- 1..1}"
vsubset_eq: fixes A :: "V"   and B :: "V" shows "(A \<subseteq>\<^sub>\<circ> B) = (\<forall>x\<in>elts A. x \<in>\<^sub>\<circ> B)"
cont_nec_fact1_2: fixes v :: "i"   and F :: "\<Pi>\<^sub>1"   and G :: "\<Pi>\<^sub>1" shows "[WeaklyContingent F \<^bold>& \<^bold>\<not>WeaklyContingent G \<^bold>\<rightarrow> F \<^bold>\<noteq> G in v]"
assms: fixes X :: "nat \<Rightarrow> rat" assumes "cauchy X" shows "P (Real X)"
compact_Union: fixes S :: "'a set set" assumes "finite S"   and "\<And>T. T \<in> S \<Longrightarrow> compact T" shows "compact (\<Union> S)"
summable: fixes z :: "real" assumes "norm z < K" shows "summable (\<lambda>n. (a gchoose n) * z ^ n)"
step: shows "\<Gamma>\<turnstile> (c, s) \<rightarrow> (c', s')"
case_distrib: fixes h :: "'c \<Rightarrow> 'b"   and f1 :: "'c"   and f2 :: "'c"   and f3 :: "'c"   and cmp_val :: "cmp_val" shows "h (case cmp_val of LT \<Rightarrow> f1 | EQ \<Rightarrow> f2 | GT \<Rightarrow> f3) = (case cmp_val of LT \<Rightarrow> h f1 | EQ \<Rightarrow> h f2 | GT \<Rightarrow> h f3)"
BisT_MtransT02: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and t' :: "'state" assumes "c \<approx>T d"   and "mustT c s"   and "mustT d t"   and "s \<approx> t"   and "(d, t) \<rightarrow>*t t'" obtains s' :: "'state" where "(c, s) \<rightarrow>*t s'" and "s' \<approx> t'"
insert2_insert_d: fixes x :: "'a"   and t :: "'a tree"   and dl :: "nat"   and t' :: "'a tree"   and n' :: "nat"   and dl' :: "nat" shows "(insert2_d x (t, size t, dl) = (t', n', dl')) = ((t', dl') = insert_d x (t, dl) \<and> n' = size t')"
length_sort: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "length (linorder.sort_key less_eq f xs) = length xs"
pivot_tableau: fixes s :: "('a, 'b) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and v :: "nat \<Rightarrow> 'b" assumes "\<triangle> (\<T> s)"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar_code (\<T> s) x\<^sub>i)" shows "let s' = pivot_code x\<^sub>i x\<^sub>j s in v \<Turnstile>\<^sub>t \<T> s = v \<Turnstile>\<^sub>t \<T> s' \<and> \<triangle> (\<T> s')"
union_eq_empty: fixes M :: "'f multiset"   and N :: "'f multiset" shows "(M + N = {#}) = (M = {#} \<and> N = {#})"
chart_eucl_def: shows "chart_eucl \<equiv> Abs_chart (UNIV, UNIV, \<lambda>x. x, \<lambda>x. x)"
x_f_eq_m_n: shows "insert x F = set_mset M \<union> set_mset N"
pos_le_minus_divideC_eq: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "complex"   and a :: "'b"   and b :: "'b" assumes "class.ordered_complex_vector scaleR scaleC plus zero minus uminus less_eq less"   and "0 < c" shows "less_eq a (uminus (scaleC (inverse c) b)) = less_eq (scaleC c a) (uminus b)"
aval_eq_if_eq_le: fixes l :: "nat"   and s\<^sub>1 :: "char list \<Rightarrow> int"   and s\<^sub>2 :: "char list \<Rightarrow> int"   and a :: "aexp" assumes "s\<^sub>1 = s\<^sub>2 (\<le> l)"   and "sec a \<le> l" shows "aval a s\<^sub>1 = aval a s\<^sub>2"
exec_meth_take_xt: fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'addr instr list"   and ins' :: "'addr instr list"   and xt' :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'thread_id"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "exec_meth ci P (ins @ ins') (xt' @ shift (length ins) xt) t h (stk, loc, pc, xcp) ta h' s'"   and "pc < length ins" shows "exec_meth ci P ins xt' t h (stk, loc, pc, xcp) ta h' s'"
foldi_fold_conj: fixes P :: "'b \<Rightarrow> bool"   and t :: "('b, 'c) RBT.rbt"   and val :: "bool" shows "RBT.foldi (\<lambda>s. s = True) (\<lambda>k v s. s \<and> P k) t val = fold_keys (\<lambda>k s. s \<and> P k) t val"
tcg_def: fixes x :: "'a"   and y :: "'a" shows "tcg x y = (\<tau> x \<le> \<tau> y \<and> \<tau> y \<le> \<tau> x)"
x: shows "root p (real_of_rat x) = root p (real_of_int a / real_of_int b)"
gf: fixes z :: "complex" assumes "z \<in> ball 0 r" shows "g (f z) = z"
constant_when_zero: fixes v :: "real \<Rightarrow> (real, 'c) vec"   and t0 :: "real"   and i :: "'c"   and x0 :: "real"   and f :: "real \<Rightarrow> (real, 'c) vec \<Rightarrow> (real, 'c) vec"   and T :: "real set"   and S :: "(real, 'c) vec set"   and t :: "real" assumes "v t0 $ i = x0"   and "(v solves_ode f) T S"   and "\<And>s x. s \<in> T \<Longrightarrow> f s x $ i = 0"   and "t0 \<in> T"   and "t \<in> T"   and "convex T" shows "v t $ i = x0"
gt: shows "(\<Union>c. {\<lambda>x. if c < x then f x else 0}) equiintegrable_on {a..b}"
dvd_mult_unit_iff: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "is_unit b" shows "(a dvd c * b) = (a dvd c)"
contour_integral_convex_primitive: fixes S :: "complex set"   and f :: "complex \<Rightarrow> complex" assumes "convex S"   and "continuous_on S f"   and "\<And>a b c. \<lbrakk>a \<in> S; b \<in> S; c \<in> S\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)" obtains g :: "complex \<Rightarrow> complex" where "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative f x) (at x within S)"
simps_2: fixes return :: "int \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> (int \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b"   and E :: "'c \<Rightarrow> 'b"   and i :: "int" shows "exp_base.eval return bind fail E (Const i) = return i"
rel_FGcontra_def: fixes L1 :: "'o \<Rightarrow> 'ab \<Rightarrow> bool"   and Co1 :: "'p \<Rightarrow> 'ac \<Rightarrow> bool"   and Co2 :: "'x \<Rightarrow> 'ak \<Rightarrow> bool"   and Co3 :: "'q \<Rightarrow> 'ad \<Rightarrow> bool"   and Co4 :: "'r \<Rightarrow> 'ae \<Rightarrow> bool"   and Co5 :: "'s \<Rightarrow> 'af \<Rightarrow> bool"   and Contra1 :: "'t \<Rightarrow> 'ag \<Rightarrow> bool"   and Contra2 :: "'u \<Rightarrow> 'ah \<Rightarrow> bool"   and Contra3 :: "'v \<Rightarrow> 'ai \<Rightarrow> bool"   and Contra4 :: "'w \<Rightarrow> 'aj \<Rightarrow> bool"   and Contra5 :: "'z \<Rightarrow> 'al \<Rightarrow> bool" shows "rel_FGcontra L1 Co1 Co2 Co3 Co4 Co5 Contra1 Contra2 Contra3 Contra4 Contra5 = rel_F L1 Co1 Co3 Co1 Co4 Co5 (rel_G Contra1 Contra2 Contra3 Contra4 Co1 Co2) Contra1 Contra5"
same_lab_h_conv_list_all: fixes ils :: "nat list"   and ls :: "t list list"   and ts' :: "t list"   and ts :: "t list" assumes "same_lab_h ils ls ts' = Some ts" shows "list_all (\<lambda>i. i < length ls \<and> ls ! i = ts) ils \<and> ts' = ts"
intro: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "\<And>x y. \<lbrakk>less_eq x y; less_eq y x; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x = y" shows "antisymmetric A less_eq"
snd_d: shows "snd d = (\<SS>, \<alpha>)"
mod_minus_cong: fixes a :: "'a"   and b :: "'a"   and a' :: "'a" assumes "a mod b = a' mod b" shows "- a mod b = - a' mod b"
typ_ok_sig_subst_typ: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and ty :: "typ"   and insts :: "((variable \<times> String.literal set) \<times> typ) list" assumes "is_std_sig \<Sigma>"   and "typ_ok_sig \<Sigma> ty"   and "distinct (map fst insts)"   and "list_all (typ_ok_sig \<Sigma>) (map snd insts)" shows "typ_ok_sig \<Sigma> (subst_typ insts ty)"
bigtheta_sym: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "(f \<in> \<Theta>[F](g)) = (g \<in> \<Theta>[F](f))"
sup_assoc: fixes x :: "'d multiset"   and y :: "'d multiset"   and z :: "'d multiset" shows "x \<union># y \<union># z = x \<union># (y \<union># z)"
i_Exec_Stream_Acc_LocalState_nth: fixes k :: "nat"   and localState :: "'b \<Rightarrow> 'a"   and trans_fun :: "'c message_af \<Rightarrow> 'b \<Rightarrow> 'b"   and input :: "nat \<Rightarrow> 'c message_af"   and c :: "'b"   and n :: "nat" assumes "0 < k" shows "i_Exec_Comp_Stream_Acc_LocalState k localState trans_fun input c n = localState (f_Exec_Comp trans_fun (input \<Down> Suc n \<odot> k) c)"
add_diff_inverse: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "a \<subseteq># b" shows "a + (b - a) = b"
cl_subset_carrier: fixes X :: "'a set" shows "cl X \<subseteq> carrier"
pathsCard_mono': fixes ps :: "Vars.var list set"   and ps' :: "Vars.var list set" assumes "ps \<sqsubseteq> ps'" shows "pathsCard ps \<sqsubseteq> pathsCard ps'"
bequiv_up_to_refl: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp" shows "P \<Turnstile> b <\<sim>> b"
weak_ranking_aux_wf: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "finite_total_preorder_on carrier le"   and "A \<subseteq> carrier" shows "is_weak_ranking (finite_total_preorder_on.weak_ranking_aux carrier le A)"
right_neutral: fixes a :: "'c \<Rightarrow> bool" shows "(a \<and>* \<box>) = a"
FVF_def: fixes \<phi> :: "fml" shows "FVF \<phi> = {x. \<exists>I \<nu> \<omega>. Vagree \<nu> \<omega> (- {x}) \<and> \<nu> \<in> fml_sem I \<phi> \<and> \<omega> \<notin> fml_sem I \<phi>}"
sd_power_prop1: fixes x :: "'a"   and y :: "'b"   and i :: "nat" shows "sd_power (x, y) (Suc i) = (x ^ Suc i, \<Squnion> {\<alpha> (x ^ k) y |k. k \<le> i})"
hyps: fixes w :: "('a, 'b) tm" assumes "list_all (\<lambda>x. x = dir.Left) p"   and "position_of (subst \<rho> w) (p @ [d])"   and "num_args (subst \<rho> w) = num_args w" shows "position_of w (p @ [d])"
snd_safe_distance_iff_nonneg_D2: assumes "a\<^sub>e < a\<^sub>o" shows "(s\<^sub>o - s\<^sub>e \<le> snd_safe_distance) = (0 \<le> D2)"
prod_coprime_left: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'a"   and a :: "'a" assumes "\<And>i. i \<in> A \<Longrightarrow> coprime (f i) a" shows "coprime (prod f A) a"
norm_real_power: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and nat_power :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and real_power :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "real"   and c :: "'a" 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 < x" shows "norm (real_power x c) = x powr inner c one"
continuous_on_exp: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. exp (f x))"
absolutely_integrable_on_subinterval: fixes S :: "real set"   and f :: "real \<Rightarrow> 'b"   and a :: "real"   and b :: "real" assumes "f absolutely_integrable_on S"   and "{a..b} \<subseteq> S" shows "f absolutely_integrable_on {a..b}"
induces_monoidal_category: shows "monoidal_category (\<cdot>) T \<alpha> \<iota>"
get_signs_R_def: fixes data :: "rat mat \<times> (nat list \<times> nat list) list \<times> rat list list" shows "get_signs_R data = snd (snd data)"
dfk: fixes k :: "nat" assumes "k < Suc (Suc n)" shows "f k \<in> carrier_mat d d"
borel_measurable_indicator: fixes A :: "'c set"   and M :: "'c measure" assumes "A \<in> sets M" shows "indicator A \<in> borel_measurable M"
c: shows "\<bar>y\<bar> \<noteq> \<infinity>"
Int_interval_mixed_eq_empty: fixes c :: "'a"   and d :: "'a"   and a :: "'a"   and b :: "'a" assumes "box c d \<noteq> {}" shows "(box a b \<inter> cbox c d = {}) = (box a b \<inter> box c d = {})"
nonsat_push_alt: fixes f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and u :: "nat"   and v :: "nat" assumes "nonsat_push_precond f l (u, v)" shows "push_effect f (u, v) = augment_edge f (u, v) (excess f u)"
valid_plan_then_is_serial_sol_encoded: fixes \<pi>s :: "char list list"   and h :: "nat" assumes "valid_plan \<pi>s"   and "length \<pi>s \<le> h" shows "\<exists>\<A>. \<A> \<Turnstile> \<Phi>\<^sub>\<forall> \<phi> prob_with_noop abs_prob  h"
qSwapBinp_preserves_qGoodBinp: fixes qbinp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qAbs option"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var" assumes "qGoodBinp qbinp" shows "qGoodBinp (qSwapBinp xs x1 x2 qbinp)"
triple_inj_eq: fixes P :: "'b assn"   and t :: "Term.term"   and Q :: "'b assn"   and P' :: "'b assn"   and t' :: "Term.term"   and Q' :: "'b assn" shows "({P} t\<succ> {Q} = {P'} t'\<succ> {Q'}) = (P = P' \<and> t = t' \<and> Q = Q')"
satPB_def: fixes wtFsym :: "'e \<Rightarrow> bool"   and wtPsym :: "'f \<Rightarrow> bool"   and arOf :: "'e \<Rightarrow> 'g list"   and resOf :: "'e \<Rightarrow> 'g"   and intT :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and intF :: "'e \<Rightarrow> 'h list \<Rightarrow> 'h"   and intP :: "'f \<Rightarrow> 'h list \<Rightarrow> bool"   and \<xi> :: "Preliminaries.var \<Rightarrow> 'h"   and \<Phi> :: "('e, 'f) lit list set" assumes "Struct wtFsym wtPsym arOf resOf intT intF intP" shows "Struct.satPB intF intP \<xi> \<Phi> \<equiv> \<forall>c\<in>\<Phi>. Struct.satC intF intP \<xi> c"
top_greatest: fixes x :: "'c" shows "comp_inf.coreflexive x"
has_disc_alist_and': fixes disc :: "'b \<Rightarrow> bool"   and as :: "'b negation_type list" shows "has_disc disc (alist_and' as) = (\<exists>a\<in>set as. has_disc disc (negation_type_to_match_expr a))"
std_axioms: shows "std F_empty F_get F_add F_to_list heap_empty heap_get heap_add heap_to_list is_true is_false \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy>"
P_lnth_Suc: fixes n :: "nat" shows "P' $ Suc n = ltl P' $ n"
char_matrix_closed: fixes A :: "'b mat"   and n :: "nat"   and e :: "'b" assumes "A \<in> carrier_mat n n" shows "char_matrix A e \<in> carrier_mat n n"
eval_const: fixes s :: "'c"   and r :: "'a" assumes "s \<in> carrier S"   and "r \<in> carrier R" shows "eval R S h s (monom P r 0) = h r"
C2r: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and star :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.C2r_algebra plus times one zero less_eq less star"   and "x = times x y" shows "x = times x (star y)"
dvd_power_iff: fixes x :: "'a"   and m :: "nat"   and n :: "nat" assumes "x \<noteq> (0::'a)" shows "(x ^ m dvd x ^ n) = (is_unit x \<or> m \<le> n)"
finite_permutations_of_multiset: fixes A :: "'a multiset" shows "finite (permutations_of_multiset A)"
less_llist_conv_lstrict_prefix: shows "(<) = lstrict_prefix"
square_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and T :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "square R S T U = (\<forall>x y. R x y \<longrightarrow> (\<forall>z. S x z \<longrightarrow> (\<exists>u. T y u \<and> U z u)))"
priority_def: fixes q :: "('c, 'd) pq" shows "PQ.priority q = map_of (pq.alist_of q)"
unit_simps_4: fixes a :: "'c" assumes "C.obj a" shows "D.dom (F.unit a) = F.map\<^sub>0 a"
is_wt_instance_of_condD': fixes t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and s :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m t"   and "wf\<^sub>t\<^sub>r\<^sub>m s"   and "is_wt_instance_of_cond \<Gamma> t s" shows "\<exists>\<delta>. wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> t = s \<cdot> \<delta>"
right_idem: fixes a :: "'c"   and b :: "'c" shows "sup (sup a b) b = sup a b"
intros_23: shows "\<lparr>[$Nop]\<rparr> \<leadsto> \<lparr>[]\<rparr>"
n: shows "\<Gamma> + image_mset \<^bold>\<not> \<Delta> \<Rightarrow> F, {#}"
x_rat: shows "\<bar>x\<bar> = real m / real n"
PolysI_alt: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and X :: "'b set" assumes "indets p \<subseteq> X" shows "p \<in> P[X]"
dim_col_take_rows: fixes A :: "'b mat"   and inds :: "nat list" shows "dim_col (take_rows A inds) = dim_col A"
pmf_map: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'c pmf"   and x :: "'b" shows "pmf (map_pmf f p) x = measure_pmf.prob p (f -` {x})"
eval\<^sub>G_And_map: fixes S :: "'a ltl set"   and xs :: "'a ltl list" shows "eval\<^sub>G S (And xs) = And (map (eval\<^sub>G S) xs)"
subgraph_refl: fixes G :: "('e, 'f) labeled_graph" shows "subgraph G G = graph G"
in_carrier: shows "[\<one>, \<ominus> a] \<in> carrier (poly_ring R)"
Suc_mod_le_divisor: fixes m :: "nat"   and n :: "nat" assumes "0 < m" shows "Suc (n mod m) \<le> m"
cong_transitivity: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "Cong A B C D"   and "Cong C D E F" shows "Cong A B E F"
select_correct_factor_int_poly: fixes bnd_get :: "'a \<Rightarrow> rat \<times> rat"   and bnd_update :: "'a \<Rightarrow> 'a"   and init :: "'a"   and x :: "real"   and p :: "int poly"   and rai :: "real_alg_2" assumes "converges_to (\<lambda>i. bnd_get ((bnd_update ^^ i) init)) x"   and "select_correct_factor_int_poly bnd_update bnd_get init p = rai"   and "poly (real_of_int_poly p) x = 0"   and "p \<noteq> 0" shows "invariant_2 rai \<and> real_of_2 rai = x"
set_pmf_subset_singleton: fixes p :: "'c pmf"   and x :: "'c" shows "(set_pmf p \<subseteq> {x}) = (p = return_pmf x)"
rel_funE: fixes A :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and B :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and f :: "'p \<Rightarrow> 'r"   and g :: "'q \<Rightarrow> 's"   and x :: "'p"   and y :: "'q" assumes "rel_fun A B f g"   and "A x y" obtains "B (f x) (g y)"
WT_converterD': fixes \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('j, 'k) \<I>"   and conv :: "('h, 'i, 'j, 'k) converter"   and q :: "'h" assumes "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>"   and "q \<in> outs_\<I> \<I>" shows "\<I>' \<turnstile>g run_converter conv q \<surd> \<and> (\<forall>(r, conv')\<in>results_gpv \<I>' (run_converter conv q). r \<in> responses_\<I> \<I> q \<and> \<I>, \<I>' \<turnstile>\<^sub>C conv' \<surd>)"
diagonal_to_Smith_PQ_JNF_canceled_both: fixes n :: "nat"   and m :: "nat"   and A :: "'b mat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and P :: "'b mat"   and S :: "'b mat"   and Q :: "'b mat" assumes "{0..<int n} \<noteq> {}"   and "{0..<int m} \<noteq> {}"   and "1 < m"   and "1 < n"   and "isDiagonal_mat A"   and "is_bezout_ext bezout"   and "A \<in> carrier_mat m n"   and "(P, S, Q) = diagonal_to_Smith_PQ_JNF A bezout" shows "S = P * A * Q \<and> invertible_mat P \<and> invertible_mat Q \<and> Smith_normal_form_mat S \<and> P \<in> carrier_mat m m \<and> S \<in> carrier_mat m n \<and> Q \<in> carrier_mat n n"
n_s_def: fixes S :: "'av st"   and X :: "char list set" shows "n\<^sub>s S X = (\<Sum>x\<in>X. n (fun S x))"
szip_sset_snd: fixes u :: "'d stream"   and v :: "'c stream" shows "snd ` sset (u ||| v) = sset v"
strip_all_vars_step: fixes t :: "Core.term"   and T :: "typ" assumes "strip_all_single_var t = Some T" shows "T # strip_all_vars (strip_all_single_body t) = strip_all_vars t"
in_GSMP_in_proj: fixes t :: "('fun, 'var) Term.term"   and A :: "('b strand_label \<times> ('fun, 'var) strand_step) list" assumes "t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t A)" shows "\<exists>n. t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj n A))"
fac_work_final_def: shows "fac_work_final \<equiv> fix\<cdot>fac_body_final"
model_to_dimacs_model_all_vars: fixes f :: "nat formula"   and n_vars :: "nat"   and M :: "nat \<Rightarrow> bool" assumes "\<forall>v\<in>atoms f. 0 < v \<and> v < n_vars"   and "is_cnf f"   and "M \<Turnstile> f" shows "\<forall>n<n_vars. 0 < n \<longrightarrow> int n \<in> set (model_to_dimacs_model M [1..<n_vars]) \<or> - int n \<in> set (model_to_dimacs_model M [1..<n_vars])"
last_replicate: fixes n :: "nat"   and x :: "'c" assumes "n \<noteq> 0" shows "last (replicate n x) = x"
euclidean_eq_iff: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and x :: "'a"   and y :: "'a" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "(x = y) = (\<forall>b\<in>Basis. inner x b = inner y b)"
split_indicator_asm: fixes P :: "'b \<Rightarrow> bool"   and S :: "'c set"   and x :: "'c" shows "P (indicator S x) = (\<not> (x \<in> S \<and> \<not> P (1::'b) \<or> x \<notin> S \<and> \<not> P (0::'b)))"
below_ccTTreeI: fixes t :: "Vars.var ttree"   and S :: "Vars.var set"   and G :: "CoCalls" assumes "carrier t \<subseteq> S"   and "ccApprox t \<sqsubseteq> G" shows "t \<sqsubseteq> ccTTree S G"
globEANow: fixes n :: "nat"   and i :: "nat"   and c :: "'kid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'KS"   and \<gamma> :: "(nat \<Rightarrow> 'KS) \<Rightarrow> nat \<Rightarrow> bool" assumes "n \<le> i"   and "\<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "sb.eval c t t' n (\<box>\<^sub>b\<gamma>)" shows "sb.eval c t t' i \<gamma>"
cfE_of_ss_VxV: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity" shows "cfE_of f \<subseteq> V \<times> V"