hcju/isabelleps huggingface dataset:
tau_le_c: fixes x :: "'a" shows "\<tau> x \<le> 1\<^sub>\<pi>"
assms_4: shows "t \<le> v * v\<^sup>T \<sqinter> g"
prv_eql_subst: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and \<phi> :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "x \<in> var"   and "y \<in> var"   and "\<phi> \<in> fmla" shows "prv (imp (eql (Var x) (Var y)) (imp \<phi> (subst \<phi> (Var y) x)))"
divide_eq_1_iff: fixes a :: "'a"   and b :: "'a" shows "(a / b = (1::'a)) = (b \<noteq> (0::'a) \<and> a = b)"
pos_minus_divideR_le_eq: fixes c :: "real"   and b :: "'a"   and a :: "'a" assumes "0 < c" shows "(- (b /\<^sub>R c) \<le> a) = (- b \<le> c *\<^sub>R a)"
r2p_inter_hom: fixes R :: "('b \<times> 'b) set"   and S :: "('b \<times> 'b) set" assumes "R \<subseteq> Id"   and "S \<subseteq> Id" shows "r2p (R \<inter> S) = r2p R \<sqinter> r2p S"
Diff_insert_absorb: fixes x :: "'e"   and A :: "'e set" assumes "x \<notin> A" shows "insert x A - {x} = A"
cong_less_unique_int: fixes m :: "int"   and a :: "int" assumes "0 < m" shows "\<exists>!b. 0 \<le> b \<and> b < m \<and> [a = b] (mod m)"
coincides_on_arbitrage: fixes Mkt :: "('a, 'c) discrete_market"   and Mkt2 :: "('a, 'c) discrete_market"   and A :: "'c set"   and p :: "'c \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real" assumes "coincides_on Mkt Mkt2 A"   and "support_set p \<subseteq> A"   and "arbitrage_process Mkt p" shows "arbitrage_process Mkt2 p"
cases: fixes a :: "'h"   and r :: "('h \<times> 'i) set" assumes "a \<in> Domain r" obtains aa :: "'h" and b :: "'i" where "a = aa" and "(aa, b) \<in> r"
monad_reader_optionT: fixes return :: "'a option \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a option \<Rightarrow> 'b) \<Rightarrow> 'b"   and ask :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_reader return bind ask" shows "monad_reader (return_option return) (bind_option return bind) (ask_option ask)"
D_def: fixes P :: "'b process" shows "D P = DIVERGENCES (Rep_process P)"
fun_rel_def: fixes A :: "('a \<times> 'c) set"   and B :: "('b \<times> 'd) set" shows "A \<rightarrow> B \<equiv> {(f, f'). \<forall>(a, a')\<in>A. (f a, f' a') \<in> B}"
dimension_simple_extension: fixes K :: "'a set"   and x :: "'a" assumes "subfield K R"   and "x \<in> carrier R"   and "(algebraic over K) x" shows "dimension (degree (Irr K x)) K (simple_extension K x)"
acyclic_trans': fixes x :: "'a" assumes "is_acyclic x" shows "x\<^sup>\<star> \<le> - x\<^sup>T\<^sup>+"
simps: fixes a1 :: "'a list"   and a2 :: "nat"   and a3 :: "'a" shows "nth_el1 a1 a2 a3 = ((\<exists>x xs. a1 = x # xs \<and> a2 = 0 \<and> a3 = x) \<or> (\<exists>xs i y x. a1 = x # xs \<and> a2 = Suc i \<and> a3 = y \<and> nth_el1 xs i y))"
nonneg_cexpr_Add: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and e2 :: "cexpr"   and V :: "nat set" assumes "\<Gamma> \<turnstile>\<^sub>c e1 : REAL"   and "\<Gamma> \<turnstile>\<^sub>c e2 : REAL"   and "free_vars e1 \<subseteq> V"   and "free_vars e2 \<subseteq> V"   and "nonneg_cexpr V \<Gamma> e1"   and "nonneg_cexpr V \<Gamma> e2" shows "nonneg_cexpr V \<Gamma> (e1 +\<^sub>c e2)"
impl_of_RBT_HM_inverse: fixes x :: "('a, 'b) hashmap" shows "RBT_HM (impl_of_RBT_HM x) = x"
lines_comm: fixes P :: "Points"   and Q :: "Points" shows "lines P Q = lines Q P"
fun_upd_def: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'b" shows "f(a := b) = (\<lambda>x. if x = a then b else f x)"
arc_implies_awalk: fixes e :: "'a \<times> 'a" assumes "e \<in> parcs G" shows "awalk (fst e) [e] (snd e)"
normal: fixes x :: "'x"   and x' :: "'x"   and m' :: "'m"   and ws' :: "'t \<Rightarrow> 'w wait_set_status option" assumes "t \<turnstile> \<langle>x, shr s\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "lock_ok_las (locks s) t \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>"   and "thread_oks (thr s) \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>"   and "cond_action_oks s t \<lbrace>ta\<rbrace>\<^bsub>c\<^esub>"   and "wset_actions_ok (wset s) t \<lbrace>ta\<rbrace>\<^bsub>w\<^esub>"   and "interrupt_actions_ok (interrupts s) \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>"   and "redT_updWs t (wset s) \<lbrace>ta\<rbrace>\<^bsub>w\<^esub> ws'"   and "s' = (redT_updLs (locks s) t \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>, (redT_updTs (thr s) \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>(t \<mapsto> (x', redT_updLns (locks s) t no_wait_locks \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>)), m'), ws', redT_updIs (interrupts s) \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>)" shows "thesis"
right_neutral: fixes a :: "'b list" shows "a \<cdot> \<epsilon> = a"
induct: fixes x :: "('fun, 'var) extstrand_step list"   and \<M> :: "('fun, 'var) Term.term set"   and \<N> :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and P :: "('fun, 'var) extstrand_step list \<Rightarrow> bool" assumes "x \<in> decomps\<^sub>e\<^sub>s\<^sub>t \<M> \<N> \<I>"   and "P []"   and "\<And>\<D> f T K M. \<lbrakk>\<D> \<in> decomps\<^sub>e\<^sub>s\<^sub>t \<M> \<N> \<I>; P \<D>; Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t \<M> \<union> \<N>; Ana (Fun f T) = (K, M); M \<noteq> []; \<M> \<union> ik\<^sub>e\<^sub>s\<^sub>t \<D> \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile>\<^sub>c Fun f T \<cdot> \<I>; \<And>k. k \<in> set K \<Longrightarrow> \<M> \<union> ik\<^sub>e\<^sub>s\<^sub>t \<D> \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile>\<^sub>c k \<cdot> \<I>\<rbrakk> \<Longrightarrow> P (\<D> @ [Decomp (Fun f T)])" shows "P x"
Perp_in_cases: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.PerpAt Bet Cong X A B C D \<or> Tarski_neutral_dimensionless.PerpAt Bet Cong X B A C D \<or> Tarski_neutral_dimensionless.PerpAt Bet Cong X A B D C \<or> Tarski_neutral_dimensionless.PerpAt Bet Cong X B A D C \<or> Tarski_neutral_dimensionless.PerpAt Bet Cong X C D A B \<or> Tarski_neutral_dimensionless.PerpAt Bet Cong X C D B A \<or> Tarski_neutral_dimensionless.PerpAt Bet Cong X D C A B \<or> Tarski_neutral_dimensionless.PerpAt Bet Cong X D C B A" shows "Tarski_neutral_dimensionless.PerpAt Bet Cong X A B C D"
cong_right_commutativity: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Cong A B C D" shows "Cong A B D C"
column_QR_decomposition2: fixes A :: "((real, 'a) vec, 'b) vec"   and k :: "'a" assumes "rank A = ncols A" shows "column k A = (\<Sum>x\<in>{column i (fst (QR_decomposition A)) |i. i \<le> k}. (x \<bullet> column k A) *\<^sub>R x)"
chain_prefix: fixes f :: "nat \<Rightarrow> 'a list"   and n\<^sub>1 :: "nat"   and n\<^sub>2 :: "nat" assumes "wf_infpath f"   and "n\<^sub>1 \<le> n\<^sub>2" shows "\<exists>a. f n\<^sub>1 @ a = f n\<^sub>2"
widen_Class: fixes wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and T :: "ty" assumes "wf_prog wf_md P"   and "is_class P C" shows "P \<turnstile> T \<le> Class C = (T = NT \<or> (\<exists>D. T = Class D \<and> P \<turnstile> Path D to C unique))"
ex1: shows "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> P1 x = P1'"
f5: fixes r :: "real"   and n :: "nat" shows "r + - frac r = real n \<or> \<not> r < real (n + 1) \<or> \<not> real n < r"
Aizerman_on_idem_on_consistency_on: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" assumes "Aizerman_on A f"   and "\<forall>B\<subseteq>A. f (f B) = f B" shows "consistency_on A f"
Cantor_sum_Nil2: fixes \<alpha> :: "V"   and \<alpha>s :: "V list" shows "Cantor_sum (\<alpha> # \<alpha>s) [] = 0"
mset_remdups_remdups_mset: fixes D :: "'c list" shows "mset (remdups D) = remdups_mset (mset D)"
append: fixes xs :: "'a list"   and ys :: "'a list" shows "prod_list (xs @ ys) = prod_list xs * prod_list ys"
ex_n_lists_iff: fixes cenum :: "'a list"   and P :: "'a list \<Rightarrow> bool"   and n :: "nat" shows "ex_n_lists (Bex (set cenum)) P n = (\<exists>xs\<in>set (List.n_lists n cenum). P xs)"
z'_def: shows "z' = (THE (x, y). x \<le> y \<and> x\<^sup>2 + y\<^sup>2 = p)"
cut_le_image: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and I :: "'a set"   and n :: "'a" assumes "SetInterval2.strict_mono_on f A"   and "I \<subseteq> A"   and "n \<in> A" shows "(f ` I) \<down>\<le> f n = f ` I \<down>\<le> n"
DERIV_ident: fixes F :: "'e filter" shows "((\<lambda>x. x) has_field_derivative (1::'e)) F"
t2: fixes t :: "real" assumes "t * 2 = 1" shows "t = 1 / 2"
max_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" shows "ord.max less_eq a b = (if less_eq a b then b else a)"
test_def: fixes p :: "'a" shows "test p \<equiv> t p = p"
linear_simps_2: fixes t\<^sub>1 :: "term"   and t\<^sub>2 :: "term" shows "linear (app t\<^sub>1 t\<^sub>2) = (linear t\<^sub>1 \<and> linear t\<^sub>2 \<and> \<not> is_free t\<^sub>1 \<and> fdisjnt (frees t\<^sub>1) (frees t\<^sub>2))"
take_Suc: fixes xs :: "'a list"   and n :: "nat" assumes "xs \<noteq> []" shows "take (Suc n) xs = hd xs # take n (tl xs)"
simps_2: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and x :: "variable"   and \<theta> :: "trm" shows "usubstappp \<sigma> U (x := \<theta>) = (U \<union> {x}, Assigno x (usubstappt \<sigma> U \<theta>))"
Sup_subset_mono: fixes B :: "V set"   and A :: "V set" assumes "small B"   and "A \<subseteq> B" shows "\<Squnion> A \<subseteq>\<^sub>\<circ> \<Squnion> B"
rel_mono_strong: fixes R1 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and R2 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R3 :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and x :: "('g, 'i, 'k) generat"   and y :: "('h, 'j, 'l) generat"   and R1a :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and R2a :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R3a :: "'k \<Rightarrow> 'l \<Rightarrow> bool" assumes "rel_generat R1 R2 R3 x y"   and "\<And>z1 y1. \<lbrakk>z1 \<in> generat_pures x; y1 \<in> generat_pures y; R1 z1 y1\<rbrakk> \<Longrightarrow> R1a z1 y1"   and "\<And>z2 y2. \<lbrakk>z2 \<in> generat_outs x; y2 \<in> generat_outs y; R2 z2 y2\<rbrakk> \<Longrightarrow> R2a z2 y2"   and "\<And>z3 y3. \<lbrakk>z3 \<in> generat_conts x; y3 \<in> generat_conts y; R3 z3 y3\<rbrakk> \<Longrightarrow> R3a z3 y3" shows "rel_generat R1a R2a R3a x y"
sharing_consistent_append: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and xs :: "'b memref list"   and ys :: "'b memref list" shows "sharing_consistent \<S> \<O> (xs @ ys) = (sharing_consistent \<S> \<O> xs \<and> sharing_consistent (share xs \<S>) (acquired True xs \<O>) ys)"
toy_trans: fixes i :: "nat" shows "automaton.trans (ptoy i) = seqp_sos \<Gamma>\<^sub>T\<^sub>O\<^sub>Y"
llist_all2_lsetD2: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist"   and y :: "'d" assumes "llist_all2 P xs ys"   and "y \<in> lset ys" shows "\<exists>x\<in>lset xs. P x y"
mset_s_mul_ext: fixes xs :: "'b list"   and ys :: "'b list"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool" shows "((mset xs, mset ys) \<in> s_mul_ext {(x, y). snd (f x y)} {(x, y). fst (f x y)}) = fst (mul_ext f xs ys)"
image_Int_subset: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and B :: "'d set" shows "f ` (A \<inter> B) \<subseteq> f ` A \<inter> f ` B"
AQ: shows "lower_triangular (Matrix.mat 1 2 (\<lambda>(i, j). if i = 0 \<and> j = 0 then a else b) * Q)"
cprank_max_plus: fixes i :: "nat"   and A :: "'b tensor"   and j :: "nat"   and B :: "'b tensor" assumes "cprank_max i A"   and "cprank_max j B"   and "dims A = dims B" shows "cprank_max (i + j) (A + B)"
T2: shows "c_assoc_value (pr_gr upb1) (c_pair (c_snd (c_snd n)) x) = univ_for_pr (c_pair (c_snd (c_snd n)) x)"
list_update_beyond: fixes xs :: "'c list"   and i :: "nat"   and x :: "'c" assumes "length xs \<le> i" shows "xs[i := x] = xs"
inf_continuous_def: fixes F :: "'a \<Rightarrow> 'b" shows "inf_continuous F = (\<forall>M. decseq M \<longrightarrow> F (\<Sqinter> range M) = (\<Sqinter>i. F (M i)))"
singular_boundary_imp_chain: fixes p :: "nat"   and X :: "'b topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "singular_relboundary p X {} c" shows "singular_chain p X c"
casts_prim: fixes T :: "ty"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and v :: "val" assumes "\<forall>C. T \<noteq> Class C" shows "P \<turnstile> T casts v to v "
5: shows "case case s of (x, y) \<Rightarrow> (real x, real y) of (n, l) \<Rightarrow> if l = 0 then n = 0 else n \<le> l \<and> b / a * l \<le> n"
limI: fixes X :: "nat \<Rightarrow> 'd"   and L :: "'d" assumes "X \<longlonglongrightarrow> L" shows "lim X = L"
1: shows "[\<^bold>\<forall>y. y\<^sup>P \<^bold>= (\<^bold>\<iota>x. TAO_8_Definitions.Universal (x\<^sup>P)) \<^bold>\<rightarrow> TAO_8_Definitions.Universal (y\<^sup>P) in v]"
strengthen_pre: fixes P' :: "'a \<Rightarrow> state \<Rightarrow> bool"   and P :: "'a \<Rightarrow> state \<Rightarrow> bool"   and C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and c :: "com"   and Q :: "'a \<Rightarrow> state \<Rightarrow> bool" assumes "\<forall>z s. P' z s \<longrightarrow> P z s"   and "C \<turnstile> {P} c {Q}" shows "C \<turnstile> {P'} c {Q}"
algebraic_add_cancel_right: fixes y :: "'a"   and x :: "'a" assumes "algebraic y" shows "algebraic (x + y) = algebraic x"
asymp_equiv_filtermap_iff: fixes f :: "'a \<Rightarrow> 'b"   and h :: "'c \<Rightarrow> 'a"   and F :: "'c filter"   and g :: "'a \<Rightarrow> 'b" shows "(f \<sim>[filtermap h F] g) = ((\<lambda>x. f (h x)) \<sim>[F] (\<lambda>x. g (h x)))"
diffE: fixes x :: "'a" assumes "x \<in> src.carrier" obtains c1 :: "('a, 'e) chart" and c2 :: "('b, 'f) chart" where "c1 \<in> src.atlas" and "c2 \<in> dest.atlas" and "x \<in> domain c1" and "f ` domain c1 \<subseteq> domain c2" and "k-smooth_on (codomain c1) (apply_chart c2 \<circ> f \<circ> inv_chart c1)"
free_tv_ML_scheme: fixes n :: "nat"   and sch :: "type_scheme" shows "(n \<in> free_tv sch) = (n \<in> set (free_tv_ML sch))"
product_nn_integral_prod: fixes I :: "'b set"   and f :: "'b \<Rightarrow> ennreal \<Rightarrow> ennreal" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> Y.random_variable i borel (f i)" shows "\<integral>\<^sup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^sub>M I (\<lambda>i. distr M borel (Y i)) = (\<Prod>i\<in>I. integral\<^sup>N (distr M borel (Y i)) (f i))"
cases: fixes x :: "'d"   and \<Gamma> :: "('e, 'f) \<I>"   and a :: "('d, 'e, 'f) gpv" assumes "x \<in> results_gpv \<Gamma> a" obtains gpv :: "('d, 'e, 'f) gpv" where "a = gpva__" and "Pure x \<in> set_spmf (the_gpv gpva__)"   | out :: "'e" and c :: "('d, 'e, 'f) rpv" and gpv :: "('d, 'e, 'f) gpv" and input :: "'f" where "a = gpv" and "IO out c \<in> set_spmf (the_gpv gpv)" and "input \<in> responses_\<I> \<Gamma> out" and "x \<in> results_gpv \<Gamma> (c input)"
imirror_card: fixes I :: "nat set" assumes "finite I" shows "card (imirror I) = card I"
quasi_inverses_def: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.quasi_inverses f g \<equiv> \<exists>\<phi> \<psi>. Span.internal_equivalence f g \<phi> \<psi>"
in_prime_factors_imp_dvd: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and p :: "'c"   and x :: "'c" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "p \<in># factorial_semiring.prime_factorization zero times one normalize x" shows "dvd.dvd times p x"
br'_invar_add_def: fixes \<delta> :: "('d, 'e) ta_rule set" shows "br'_invar_add \<delta> \<equiv> {(Q, W, rcm). (\<forall>r\<in>\<delta>. rcm r = Some (card (set (rhsq r) - (Q - W)))) \<and> {lhs r |r. r \<in> \<delta> \<and> the (rcm r) = 0} \<subseteq> Q}"
frac_le_self: fixes x :: "'a" assumes "(0::'a) \<le> x" shows "frac x \<le> x"
remove_child_preserves_type_wf: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> remove_child ptr child \<rightarrow>\<^sub>h h'" shows "type_wf h'"
spn_simps_4: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "trg (spn \<mu>) = src (tab\<^sub>0 (cod \<mu>))"
S_def: shows "S \<equiv> \<lambda>r. {x \<in> topspace (Euclidean_space n). enorm x = r}"
2: shows "morphism_presheaves_of_rings X is_open \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' \<FF>'' \<rho>'' b'' add_str'' mult_str'' zero_str'' one_str'' \<phi>'"
cos_sin_eq: fixes x :: "'b" shows "cos x = sin (of_real pi / (2::'b) - x)"
arr_MkArr: fixes F :: "'b \<Rightarrow> 'a"   and G :: "'b \<Rightarrow> 'a"   and \<tau> :: "'b \<Rightarrow> 'a" shows "J_C.arr (J_C.MkArr F G \<tau>) = natural_transformation J (\<cdot>) F G \<tau>"
sgn_zero: shows "sgn (0::'b) = (0::'b)"
usubst_condr: fixes \<sigma> :: "'b \<Rightarrow> 'c"   and b :: "'c upred"   and P :: "('a, 'c) uexpr"   and Q :: "('a, 'c) uexpr" shows "\<sigma> \<dagger> (P \<triangleleft> b \<triangleright> Q) = \<sigma> \<dagger> P \<triangleleft> \<sigma> \<dagger> b \<triangleright> \<sigma> \<dagger> Q"
pos_divide_le_eq: 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 c :: "'c"   and b :: "'c"   and a :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero c" shows "less_eq (divide b c) a = less_eq b (times a c)"
none: fixes p :: "real"   and Da :: "'e pmf"   and Db :: "'f pmf" assumes "0 < p"   and "p < 1" shows "set_pmf (bernoulli_pmf p \<bind> (\<lambda>b. if b then map_pmf Inl Da else map_pmf Inr Db)) \<inter> {f. \<exists>e. Inl e = f} \<noteq> {}"
prefs_wf'_2: fixes agents :: "'c set"   and alts :: "'d set"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "'c" assumes "pref_profile_wf agents alts R"   and "i \<in> agents" shows "Order_Predicates.preorder_on alts (R i)"
differing_vars_neg: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and x :: "'a"   and mem1 :: "'a \<Rightarrow> 'b"   and mem2 :: "'a \<Rightarrow> 'b"   and mems :: "(('a \<Rightarrow> 'b) \<times> ('a \<Rightarrow> 'b)) list"   and i :: "nat" assumes "sifum_security_init dma \<C>_vars eval"   and "x \<notin> sifum_security_init.differing_vars_lists mem1 mem2 mems i" shows "fst (mems ! i) x = mem1 x \<and> snd (mems ! i) x = mem2 x"
app\<^sub>i_mono: fixes p :: "'c prog \<Rightarrow> String.literal \<Rightarrow> 'c mdecl \<Rightarrow> bool"   and P :: "'c prog"   and \<tau> :: "ty list \<times> ty err list"   and \<tau>' :: "ty list \<times> ty err list"   and i :: "'d instr"   and mxs :: "nat"   and mpc :: "nat"   and rT :: "ty" assumes "wf_prog p P"   and "P \<turnstile> \<tau> \<le>\<^sub>i \<tau>'"   and "app\<^sub>i (i, P, mxs, mpc, rT, \<tau>')" shows "app\<^sub>i (i, P, mxs, mpc, rT, \<tau>)"
simps_2: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and xs :: "'c Trace"   and x :: "'c"   and ys :: "'d Trace"   and y :: "'d" shows "tZip f (xs \<leadsto> x) (ys \<leadsto> y) = tZip f xs ys \<leadsto> f x y"
is_CB_def: shows "is_CB \<equiv> CValue_case\<cdot>(\<Lambda> cfun. FF)\<cdot>(\<Lambda> discr. TT)"
Span_Snorm: shows "ccspan (set Snorm) = ccspan (set S)"
root_int_floor_pos_def: fixes p :: "nat"   and x :: "int" shows "root_int_floor_pos p x = (if p = 0 then 0 else fst (root_int_main p x))"
adds_fun_iff: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "(f adds g) = (\<forall>x. f x adds g x)"
qmpt_factor_projection: fixes f :: "'a \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "sigma_finite_measure (distr M borel (\<lambda>x n. f ((T ^^ n) x)))" shows "qmpt_factor (\<lambda>x n. f ((T ^^ n) x)) (distr M borel (\<lambda>x n. f ((T ^^ n) x))) nat_left_shift"
1: shows "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& \<^bold>\<not>(\<^bold>\<exists>F. \<lbrace>a\<^sup>P,F\<rbrace>) in v]"
gauss_jordan_i: shows "(mat_rel R ===> mat_rel R) (gauss_jordan_single_i ff_ops) gauss_jordan_single"
step_codem_def: fixes u :: "'g \<Rightarrow> ('node, 'val set) mapping"   and p :: "'g \<Rightarrow> ('node \<times> 'val, 'val list) mapping"   and g :: "'g"   and "next" :: "'node \<times> 'val"   and next' :: "'val"   and nodes_of_uses :: "('val, 'node set) mapping"   and nodes_of_phis :: "('val, ('node \<times> 'val) set) mapping" shows "uninst_code.step_codem u p g next next' nodes_of_uses nodes_of_phis = (uninst_code.uses'_codem u g next next' nodes_of_uses, uninst_code.phis'_codem p g next next' nodes_of_phis)"
cases: fixes r :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e" assumes "Domainp r a" obtains aa :: "'e" and b :: "'f" where "a = aa" and "r aa b"
sscan_snth: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and xs :: "'d stream"   and a :: "'c"   and i :: "nat" shows "sscan f xs a !! i = fold f (stake (Suc i) xs) a"
induct: fixes P :: "nat list \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a) \<Rightarrow> (('a, 'b) vec, 'b) vec \<times> (('a, 'c) vec, 'b) vec \<times> (('a, 'c) vec, 'c) vec \<Rightarrow> bool"   and a0 :: "nat list"   and a1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and a2 :: "(('a, 'b) vec, 'b) vec \<times> (('a, 'c) vec, 'b) vec \<times> (('a, 'c) vec, 'c) vec" assumes "\<And>bezout Pa A Q. P [] bezout (Pa, A, Q)"   and "\<And>i xs bezout Pa A Q. P xs bezout (Diagonal_to_Smith_row_i_PQ i bezout (Pa, A, Q)) \<Longrightarrow> P (i # xs) bezout (Pa, A, Q)" shows "P a0 a1 a2"
distinct_29: shows "cospan_shape.AT \<noteq> cospan_shape.BT"