hcju/isabelleps huggingface dataset:
hyps: fixes qs :: "real poly list" assumes "length qs < length qsa__" shows "p \<noteq> 0 \<and> 0 < length qs \<longrightarrow> set (find_consistent_signs_at_roots_R p qs) = set (characterize_consistent_signs_at_roots p qs)"
of_int_closed: fixes i :: "int" shows "\<guillemotleft>i\<guillemotright> \<in> carrier R"
DERIV_arctan: fixes x :: "real" shows "(arctan has_real_derivative inverse (1 + x\<^sup>2)) (at x)"
success_insert_elt: fixes k :: "nat"   and h :: "heap"   and a :: "'b array" assumes "k < Array.length h a" shows "success (insert_elt a k) h"
set_iterator_genord_foldli_conv: fixes iti :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "'e set"   and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" shows "set_iterator_genord iti S R = (\<exists>l0. distinct l0 \<and> S = set l0 \<and> sorted_wrt R l0 \<and> iti = foldli l0)"
has_subprob_densityD_4: fixes M :: "'a measure"   and N :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" assumes "has_subprob_density M N f" shows "subprob_space M"
index_not_p_between: fixes n :: "nat"   and i :: "nat" assumes "index n < i"   and "i < index (Suc n)" shows "\<not> p i"
add_le_imp_le_diff: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and i :: "'k"   and k :: "'k"   and n :: "'k" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq (plus i k) n" shows "less_eq i (minus n k)"
assms_1: shows "q2 \<in> succ A w q1"
associated_polynomials_imp_same_roots: fixes p :: "'a list list"   and q :: "'a list list" assumes "p \<in> carrier (poly_ring (poly_ring R))"   and "q \<in> carrier (poly_ring (poly_ring R))"   and "p \<sim>\<^bsub>poly_ring (poly_ring R)\<^esub> q" shows "UP.roots p = UP.roots q"
assms: shows "a \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>"
succ_nodes: fixes q2 :: "'a"   and A :: "('b, 'c, 'a) FSM"   and w :: "'b \<times> 'c"   and q1 :: "'a" assumes "q2 \<in> succ A w q1"   and "q1 \<in> nodes A" shows "q2 \<in> nodes A"
similar_mat_wit_def: fixes A :: "'b mat"   and B :: "'b mat"   and P :: "'b mat"   and Q :: "'b mat" shows "similar_mat_wit A B P Q = (let n = dim_row A in {A, B, P, Q} \<subseteq> carrier_mat n n \<and> P * Q = 1\<^sub>m n \<and> Q * P = 1\<^sub>m n \<and> A = P * B * Q)"
ysq_dom_y_plus_coeff: fixes b :: "real"   and c :: "real" shows "\<exists>w. \<forall>y<w. b * y + c < y\<^sup>2"
fmap_rel_nat_the_fmlookup: fixes A :: "('a, 'b) fmap"   and B :: "('c, 'd) fmap"   and S :: "('a \<times> 'c) set"   and R :: "('b \<times> 'd) set"   and p :: "'a"   and p' :: "'c" assumes "(A, B) \<in> \<langle>S, R\<rangle>fmap_rel"   and "(p, p') \<in> S"   and "p' \<in># dom_m B" shows "(the (fmlookup A p), the (fmlookup B p')) \<in> R"
cInf_lower: fixes x :: "'k multiset"   and X :: "'k multiset set" assumes "x \<in> X"   and "subset_mset.bdd_below X" shows "Inf X \<subseteq># x"
col_permutation_3: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A B C" shows "Tarski_neutral_dimensionless.Col Bet C B A"
wrap_iso_arcs_def: fixes hom :: "('a, 'b, 'f, 'e) digraph_isomorphism"   and f :: "'b \<Rightarrow> 'b" shows "wrap_iso_arcs hom f = perm_restrict (iso_arcs hom \<circ> f \<circ> iso_arcs (inv_iso hom)) (arcs (app_iso hom G))"
comp_fun_commute: fixes y :: "'a"   and x :: "'a" shows "(f y \<circ>\<circ> f) x = (f x \<circ>\<circ> f) y"
NullObject_def: shows "\<^bold>a\<^sub>\<emptyset> \<equiv> \<^bold>\<iota>x. Null (x\<^sup>P)"
that_1: assumes "(u, w) \<in> E\<^sup>+" shows "thesis"
cong_add_poly: fixes a :: "'c poly"   and b :: "'c poly"   and m :: "'c poly"   and c :: "'c poly"   and d :: "'c poly" assumes "[a = b] (mod m)"   and "[c = d] (mod m)" shows "[a + c = b + d] (mod m)"
itypeI: fixes c :: "'f"   and I :: "interface" shows "c ::\<^sub>i I"
localPrecSS'_def: shows "localPrecSS' = (\<lambda>(u, v) (u', v') x. 0 < u)"
gtrp_env: fixes s :: "'c"   and sl :: "'c"   and c :: "'c multiset"   and e :: "'d"   and s' :: "'c"   and c' :: "'c multiset"   and S :: "(('c \<times> 'c multiset) \<times> 'd \<times> 'c \<times> 'c multiset) set" assumes "((s, add_mset sl c), e, s', add_mset sl c') \<in> S" shows "((sl, add_mset s c), ENV e, sl, add_mset s' c') \<in> gtrp S"
ordinal_le_timesR: fixes y :: "ordinal"   and x :: "ordinal" assumes "0 < y" shows "x \<le> x * y"
trans_converse: fixes r :: "('a \<times> 'a) set" shows "trans (r\<inverse>) = trans r"
bounded_Un: fixes S :: "'c set"   and T :: "'c set" shows "bounded (S \<union> T) = (bounded S \<and> bounded T)"
exp_tends_to_zero: fixes b :: "'a"   and c :: "'a" assumes "(0::'a) < b"   and "b < (1::'a)"   and "(0::'a) < c" shows "\<exists>x. b ^ x \<le> c"
3: shows "(m # ms) \<langle>g\<rangle> [l] \<odot> [l] \<langle>f\<rangle> [l] \<preceq> (m # ms) \<langle>g\<rangle> [l]"
sup_least: fixes y :: "'e multiset"   and x :: "'e multiset"   and z :: "'e multiset" assumes "y \<subseteq># x"   and "z \<subseteq># x" shows "y \<union># z \<subseteq># x"
intro: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd" assumes "functor A B F"   and "dual_category A"   and "dual_category B" shows "dual_functor A B F"
graphI: assumes "symmetric (with_proj G)" shows "graph (with_proj G)"
of_hypnat_eq_0_iff: fixes m :: "nat star" shows "(of_hypnat m = 0) = (m = 0)"
simulation_reaches: fixes a :: "'a"   and a' :: "'a"   and b :: "'b" assumes "A.reaches a a'"   and "a \<sim> b"   and "PA a"   and "PB b" shows "\<exists>b'. B.reaches b b' \<and> a' \<sim> b' \<and> PA a' \<and> PB b'"
proj_along_quasiconvex_contraction': fixes C :: "real"   and G :: "'a set"   and px :: "'a"   and x :: "'a"   and py :: "'a"   and y :: "'a" assumes "quasiconvex C G"   and "px \<in> proj_set x G"   and "py \<in> proj_set y G" shows "dist px py \<le> dist x y + 4 * deltaG TYPE('a) + 2 * C"
omega_sum_drop: fixes ns :: "nat list" shows "omega_sum (dropWhile (\<lambda>n. n = 0) ns) = omega_sum ns"
intro: fixes M :: "'c measure" assumes "emeasure M (space M) \<le> 1"   and "space M \<noteq> {}" shows "subprob_space_axioms M"
tpOf_lOfWax: fixes \<sigma> :: "'tp" shows "TE.tpOf (lOfWax \<sigma>) = \<sigma>"
C_def: shows "C = inv_chart c ` ball (apply_chart c x) e"
mod_add_self1: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and b :: "'b"   and a :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus b a) b = modulo a b"
filtered_seq_def: fixes s :: "nat \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> bool" shows "filtered_seq s P = take_subseq s (seq_filter s P)"
size_22: fixes x111 :: "'b ltln"   and x112 :: "'b ltln" shows "size (x111 M\<^sub>n x112) = size x111 + size x112 + Suc 0"
iT_Div_iUntil_conv: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t1. t1 \<U> t2 I \<oslash> k. Q t2) = (P (t1 div k). t1 \<U> t2 I. Q (t2 div k))"
fwi_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "nat"   and n :: "nat"   and j :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and k :: "nat"   and i' :: "nat"   and j' :: "nat" assumes "class.linordered_ab_monoid_add plus zero less_eq less"   and "i \<le> n"   and "j \<le> n" shows "less_eq (linordered_ab_monoid_add.fwi plus less_eq m n k i' j' i j) (m i j)"
simps_3: fixes var :: "nat"   and p :: "real mpoly"   and xs :: "real list" shows "convert_atom var (Leq p) xs = map_option LeqUni (convert_poly var p xs)"
Ord_def: fixes x :: "V" shows "Ord x \<equiv> Transset x \<and> (\<forall>y\<in>elts x. Transset y)"
divides_refl: fixes a :: "'a" assumes "a \<in> carrier G" shows "a divides a"
pidgc_\<alpha>_def: fixes pgc :: "('f, 'g, 'h) pid_global_config" shows "pidgc_\<alpha> pgc \<equiv> \<lparr>global_config.processes = mset (pid_global_config.processes pgc), state = pid_global_config.state pgc\<rparr>"
incomplete_design_axioms: shows "incomplete_design \<V> \<B> \<k>"
trivial_group_alt: shows "trivial_group (Z 1) = (\<exists>a. carrier (Z 1) \<subseteq> {a})"
intro\<^sub>1: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and lc' :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "(case lc of (x, xa) \<Rightarrow> (case x of (x, xa) \<Rightarrow> \<lambda>xaa ((xaaa, xaaaa), xaaaaa). \<langle>x, xa, xaa\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>xaaa, xaaaa, xaaaaa\<rangle>) xa) lc'" shows "lc \<R>\<^sup>u\<^bsub>\<Gamma>,\<S>,P\<^esub> lc'"
fds_nth_mult_const_right: fixes f :: "'b fds"   and c :: "'b"   and n :: "nat" shows "fds_nth (f * fds_const c) n = fds_nth f n * c"
id_2: shows "cn = pderiv p div gcd p (pderiv p)"
get_parent_is_l_get_parent: shows "l_get_parent ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs"
hyps: fixes w11 :: "'a list"   and w21 :: "'a list"   and l2 :: "'a list"   and w12 :: "'a list"   and w22 :: "'a list" assumes "l1' \<in> w11 \<otimes> w21"   and "l2 \<in> w12 \<otimes> w22" shows "l1' @ l2 \<in> w11 @ w12 \<otimes> w21 @ w22"
get_root_node_no_parent: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and h :: "(_) heap"   and node_ptr :: "(_) node_ptr" assumes "l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs"   and "h \<turnstile> get_parent node_ptr \<rightarrow>\<^sub>r None" shows "h \<turnstile> get_root_node (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr) \<rightarrow>\<^sub>r cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr"
totient_mult_coprime: fixes m :: "nat"   and n :: "nat" assumes "coprime m n" shows "totient (m * n) = totient m * totient n"
nu_par_aux2: fixes x :: "'a"   and y :: "'a" shows "\<nu> (\<nu> x \<parallel> \<nu> y) = \<nu> x \<parallel> \<nu> y"
return_time_le: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and P :: "'b set"   and x :: "'b"   and t :: "real" assumes "auto_ll_on_open f X"   and "auto_ll_on_open.returns_to f X P x"   and "closed P"   and "auto_ll_on_open.flow0 f X x t \<in> P"   and "0 < t" shows "auto_ll_on_open.return_time f X P x \<le> t"
oseq_broadcastTE: fixes \<sigma> :: "nat \<Rightarrow> 'e"   and l :: "'h"   and s\<^sub>m\<^sub>s\<^sub>g :: "'e \<Rightarrow> 'f"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and i :: "nat" assumes "((\<sigma>, {l}broadcast(s\<^sub>m\<^sub>s\<^sub>g) . p), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i" obtains "a = broadcast (s\<^sub>m\<^sub>s\<^sub>g (\<sigma> i))" and "\<sigma>' i = \<sigma> i" and "q = p"
assm_3: fixes i :: "'a set"   and x :: "'a"   and K :: "'a set" assumes "i \<in> I"   and "(x, K) \<in> pfn i" shows "K \<subseteq> i"
basis_in_indep_in_card: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and \<E> :: "'b set"   and X :: "'b set"   and B :: "'b set" assumes "matroid carrier indep"   and "\<E> \<subseteq> carrier"   and "indep_system.indep_in indep \<E> X"   and "indep_system.basis_in indep \<E> B" shows "card X \<le> card B"
Mg: shows "g \<in> M \<rightarrow>\<^sub>M subprob_algebra R"
morphic_prop_def: fixes R :: "('c, 'd) ring_scheme"   and P :: "'c \<Rightarrow> bool" shows "morphic_prop R P = (P \<one>\<^bsub>R\<^esub> \<and> (\<forall>r\<in>carrier R. P r) \<and> (\<forall>r1\<in>carrier R. \<forall>r2\<in>carrier R. P (r1 \<otimes>\<^bsub>R\<^esub> r2)) \<and> (\<forall>r1\<in>carrier R. \<forall>r2\<in>carrier R. P (r1 \<oplus>\<^bsub>R\<^esub> r2)))"
sup_left_idem: fixes sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.semilattice_sup sup less_eq less" shows "sup x (sup x y) = sup x y"
Plg_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" shows "Plg A B C D \<equiv> (A \<noteq> C \<or> B \<noteq> D) \<and> (\<exists>M. M Midpoint A C \<and> M Midpoint B D)"
div_by_Suc_0: fixes m :: "nat" shows "m div Suc 0 = m"
between_trivial2: fixes A :: "'p"   and B :: "'p" shows "Bet A A B"
det_swap_initial_rows: fixes A :: "'b mat"   and m :: "nat"   and k :: "nat"   and n :: "nat" assumes "A \<in> carrier_mat m m"   and "k + n \<le> m" shows "det A = (- (1::'b)) ^ (k * n) * det (mat m m (\<lambda>(i, j). A $$ (if i < n then i + k else if i < k + n then i - n else i, j)))"
funion_upper1: fixes A :: "'b fset"   and B :: "'b fset" shows "A |\<subseteq>| A |\<union>| B"
callD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and m :: "'c match_expr"   and chain :: "char list"   and rs :: "'c rule list" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = [Rule m (Call chain)]"   and "s = Undecided"   and "matches \<gamma> m p"   and "\<Gamma> chain = Some rs" obtains "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, s\<rangle> \<Rightarrow> t"   | rs\<^sub>1 :: "'c rule list" and rs\<^sub>2 :: "'c rule list" and m' :: "'c match_expr" where "rs = rs\<^sub>1 @ Rule m' Return # rs\<^sub>2" and "matches \<gamma> m' p" and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> Undecided" and "no_matching_Goto \<gamma> p rs\<^sub>1" and "t = Undecided"
affine_parallel_commut: fixes A :: "'b set"   and B :: "'b set" assumes "affine_parallel A B" shows "affine_parallel B A"
assms_2: shows "(f \<longlongrightarrow> a) F"
SeqStTermP_imp_OrdP: fixes v :: "tm"   and i :: "tm"   and t :: "tm"   and u :: "tm"   and s :: "tm"   and k :: "tm" shows "{SeqStTermP v i t u s k} \<turnstile> OrdP k"
scalar_product: fixes v :: "('c, 'd) Finite_Cartesian_Product.vec"   and w :: "('c, 'd) Finite_Cartesian_Product.vec" shows "from_hma\<^sub>v v \<bullet> from_hma\<^sub>v w = scalar_product v w"
min_gallery_betw_uniform_length: fixes X :: "'b set set"   and x :: "'b set"   and xs :: "'b set list"   and y :: "'b set"   and ys :: "'b set list" assumes "ChamberComplex X"   and "SimplicialComplex.min_maxsimpchain X (x # xs @ [y])"   and "SimplicialComplex.min_maxsimpchain X (x # ys @ [y])" shows "length xs = length ys"
rfr_step: fixes r :: "real"   and n :: "nat"   and w :: "'b" shows "disc_rfr_proc r (Suc n) w = (1 + r) * disc_rfr_proc r n w"
pred_of_succ_def: fixes s :: "'b \<Rightarrow> 'd set" shows "pred_of_succ s \<equiv> \<lambda>a b. b \<in> s a"
countably_additiveI: fixes M :: "'c set set"   and f :: "'c set \<Rightarrow> ennreal" assumes "\<And>A. \<lbrakk>range A \<subseteq> M; disjoint_family A; \<Union> (range A) \<in> M\<rbrakk> \<Longrightarrow> (\<Sum>i. f (A i)) = f (\<Union> (range A))" shows "countably_additive M f"
glb_reflclp: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set"   and x :: "'b" shows "glb P\<^sup>=\<^sup>= B x = glb P B x"
simps_1: fixes h :: "nat"   and n_ops :: "nat"   and t :: "nat"   and k :: "nat" shows "cnf_to_dimacs.var_to_dimacs h n_ops (Operator t k) = 1 + t + k * h"
m_comm: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp" shows "x \<otimes> y = y \<otimes> x"
simps_8: fixes \<phi> :: "'a MFOTL.formula"   and I :: "\<I>"   and \<psi> :: "'a MFOTL.formula" shows "MFOTL.future_reach (formula.Since \<phi> I \<psi>) = max (MFOTL.future_reach \<phi>) (MFOTL.future_reach \<psi> - enat (left I))"
PFadjoint_def: fixes I :: "('sf, 'sc, 'sz) interp"   and \<sigma> :: "'b \<Rightarrow> ('sf, 'sc, 'sz) formula" shows "PFadjoint I \<sigma> = \<lparr>Functions = Functions I, Predicates = Predicates I, Contexts = \<lambda>f. case f of Inl f' \<Rightarrow> Contexts I f' | Inr f' \<Rightarrow> \<lambda>_. fml_sem I (\<sigma> f'), Programs = Programs I, ODEs = ODEs I, ODEBV = ODEBV I\<rparr>"
wp2WhileFalse': fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com" assumes "\<not> bval b s" shows "Q s + 1 = wp\<^sub>2 (WHILE b DO c) Q s"
assms_2: shows "h \<in> O(\<lambda>x. real x / ln (real x) powr (1 + e))"
Disjoint_commute_I: fixes H :: "fm set"   and t :: "tm"   and u :: "tm" assumes "H \<turnstile> Disjoint t u" shows "H \<turnstile> Disjoint u t"
simps_4: fixes A :: "hfm" shows "hfm_to_fm (HNeg A) = Neg (hfm_to_fm A)"
compact_attains_inf: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set" assumes "class.linorder_topology less_eq less open"   and "topological_space.compact open S"   and "S \<noteq> {}" shows "\<exists>s\<in>S. \<forall>t\<in>S. less_eq s t"
join_raw_Cons: fixes f :: "'b \<Rightarrow> 'c \<times> 'c \<Rightarrow> 'c"   and xs :: "('b \<times> 'c) list"   and k :: "'b"   and v :: "'c"   and ys :: "('b \<times> 'c) list" shows "join_raw f xs ((k, v) # ys) = map_default k v (\<lambda>v'. f k (v', v)) (join_raw f xs ys)"
1: shows "(\<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star> snd \<mu>\<nu> \<noteq> null) = (\<lambda>\<mu>\<nu>. arr (fst \<mu>\<nu>) \<and> arr (snd \<mu>\<nu>) \<and> src (fst \<mu>\<nu>) = trg (snd \<mu>\<nu>))"
orderE: fixes a :: "'d upred"   and b :: "'d upred" assumes "b \<sqsubseteq> a" obtains "a = (a \<and> b)"
has_field_derivative_transform_within_open: fixes f :: "'a \<Rightarrow> 'a"   and f' :: "'a"   and a :: "'a"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'a" assumes "(f has_field_derivative f') (at a)"   and "open S"   and "a \<in> S"   and "\<And>x. x \<in> S \<Longrightarrow> f x = g x" shows "(g has_field_derivative f') (at a)"
dist_perm_eta_equiv: fixes n :: "nat"   and vs :: "nat list"   and s :: "dB"   and t :: "dB" assumes "perm_vars n vs"   and "\<forall>i\<in>frees s. n \<le> i"   and "\<forall>i\<in>frees t. n \<le> i"   and "(Abs ^^ n) (var_dist vs s) \<leftrightarrow> (Abs ^^ n) (var_dist vs t)" shows "strip_context n s 0 \<leftrightarrow> strip_context n t 0"
fid_onE: fixes x :: "V"   and A :: "V" assumes "x \<in>\<^sub>\<circ> fid_on A" obtains "\<exists>a\<in>elts A. x = [a, a]\<^sub>\<circ>"
full_normalize_poly_normalize_poly_p: fixes p :: "(char list list \<times> int) list"   and p' :: "(char list multiset \<times> int) multiset" assumes "(p, p') \<in> fully_unsorted_poly_rel" shows "full_normalize_poly p \<le> \<Down> sorted_poly_rel (SPEC (normalize_poly_p\<^sup>*\<^sup>* p'))"
abs_of_nonneg: fixes abs :: "'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less_eq zero a" shows "abs a = a"
sigma_sets_UNION: fixes B :: "'c set set"   and X :: "'c set"   and A :: "'c set set" assumes "countable B"   and "\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets X A" shows "\<Union> B \<in> sigma_sets X A"
spec2: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and y :: "'c" assumes "\<forall>x y. P x y" shows "P x y"
Sard_lemma2: fixes B :: "real"   and S :: "(real, 'a) vec set"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'b) vec"   and f' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'b) vec" assumes "CARD('a) \<le> CARD('b)"   and "0 < B"   and "bounded S"   and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"   and "\<And>x. x \<in> S \<Longrightarrow> rank (matrix (f' x)) < CARD('b)"   and "\<And>x. x \<in> S \<Longrightarrow> onorm (f' x) \<le> B" shows "negligible (f ` S)"
case_1: fixes f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'd" shows "(case \<bottom> of \<bottom> \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = f1"
case_2: fixes f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'e \<Rightarrow> 'c"   and x2 :: "'e" shows "(case Inr x2 of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = f2 x2"
oStrictLimit_def: fixes f :: "nat \<Rightarrow> ordinal" shows "oStrictLimit f = Abs_ordinal (ord0rel `` {ord0_Lim (\<lambda>n. SOME x. x \<in> Rep_ordinal (f n))})"
gpv_coinduct_bind: fixes R :: "('a, 'b, 'c) gpv \<Rightarrow> ('a, 'b, 'c) gpv \<Rightarrow> bool"   and gpv :: "('a, 'b, 'c) gpv"   and gpv' :: "('a, 'b, 'c) gpv" assumes "R gpv gpv'"   and "\<And>gpv gpv'. R gpv gpv' \<Longrightarrow> rel_spmf (rel_generat (=) (=) (rel_fun (=) (\<lambda>gpv gpv'. R gpv gpv' \<or> gpv = gpv' \<or> (\<exists>gpv2 gpv2' f f'. gpv = gpv2 \<bind> f \<and> gpv' = gpv2' \<bind> f' \<and> rel_gpv (\<lambda>x y. R (f x) (f' y)) (=) gpv2 gpv2')))) (the_gpv gpv) (the_gpv gpv')" shows "gpv = gpv'"
SubstAtomicP_graph_aux_def: shows "SubstAtomicP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>SubstAtomicP_aux t v tm y y' t' u u'. x1 = (v, tm, y, y') \<and> x2 = SyntaxN.Ex t (SyntaxN.Ex u (SyntaxN.Ex t' (SyntaxN.Ex u' (SubstTermP v tm (Var t) (Var t') AND SubstTermP v tm (Var u) (Var u') AND (y EQ Q_Eq (Var t) (Var u) AND y' EQ Q_Eq (Var t') (Var u') OR y EQ Q_Mem (Var t) (Var u) AND y' EQ Q_Mem (Var t') (Var u')))))) \<and> atom t \<sharp> (v, tm, y, y', t', u, u') \<and> atom t' \<sharp> (v, tm, y, y', u, u') \<and> atom u \<sharp> (v, tm, y, y', u') \<and> atom u' \<sharp> (v, tm, y, y'))"
ih_7: shows "InvariantWatchCharacterization (getF state') (getWatch1 state') (getWatch2 state') (getM state')"
nv2L_cOfFax: fixes \<sigma> :: "'fsym" shows "nv2L (Pos (cOfFax \<sigma>)) = {}"