hcju/isabelleps huggingface dataset:
induct: fixes x1 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and x2 :: "'Var \<Rightarrow> 'BExp set option"   and x3 :: "'Var set \<times> 'Var set"   and x4 :: "'BExp set"   and x5 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and P :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val) \<Rightarrow> ('Var \<Rightarrow> 'BExp set option) \<Rightarrow> 'Var set \<times> 'Var set \<Rightarrow> 'BExp set \<Rightarrow> (('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val) \<Rightarrow> bool" assumes "x1 \<R>\<^sup>3\<^bsub>x2,x3,x4\<^esub> x5"   and "\<And>c\<^sub>1 mds mem\<^sub>1 \<Gamma> \<S> P c\<^sub>2 mem\<^sub>2 c \<Gamma>' \<S>' P'. \<lbrakk>\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>; \<turnstile> \<Gamma>,\<S>,P {c} \<Gamma>',\<S>',P'\<rbrakk> \<Longrightarrow> P \<langle>c\<^sub>1 ;; c, mds, mem\<^sub>1\<rangle> \<Gamma>' \<S>' P' \<langle>c\<^sub>2 ;; c, mds, mem\<^sub>2\<rangle>"   and "\<And>c\<^sub>1 mds mem\<^sub>1 \<Gamma> \<S> P c\<^sub>2 mem\<^sub>2 c \<Gamma>' \<S>' P'. \<lbrakk>\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>; P \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<Gamma> \<S> P \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>; \<turnstile> \<Gamma>,\<S>,P {c} \<Gamma>',\<S>',P'\<rbrakk> \<Longrightarrow> P \<langle>c\<^sub>1 ;; c, mds, mem\<^sub>1\<rangle> \<Gamma>' \<S>' P' \<langle>c\<^sub>2 ;; c, mds, mem\<^sub>2\<rangle>" shows "P x1 x2 x3 x4 x5"
Notes_master_imp_Notes_PMS: fixes A :: "agent"   and s :: "msg"   and B :: "agent"   and PMS :: "nat"   and NA :: "nat"   and NB :: "nat"   and evs :: "event list" assumes "Notes A \<lbrace>s, Agent A, Agent B, Nonce (PRF (PMS, NA, NB))\<rbrace> \<in> set evs"   and "evs \<in> tls" shows "Notes A \<lbrace>Agent B, Nonce PMS\<rbrace> \<in> set evs"
measure_eqI_generator_eq: fixes E :: "'a set set"   and \<Omega> :: "'a set"   and M :: "'a measure"   and N :: "'a measure"   and A :: "nat \<Rightarrow> 'a set" assumes "Int_stable E"   and "E \<subseteq> Pow \<Omega>"   and "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"   and "sets M = sigma_sets \<Omega> E"   and "sets N = sigma_sets \<Omega> E"   and "range A \<subseteq> E"   and "\<Union> (range A) = \<Omega>"   and "\<And>i. emeasure M (A i) \<noteq> \<infinity>" shows "M = N"
\<eta>_leg_simps_4: shows "\<eta>.cod.leg1 = gf.leg1"
r_one: fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a" assumes "Dmonoid prod one" shows "prod x one = x"
On_CTRL_Labels_def: shows "On_CTRL_Labels = {(And (En ''Src'') (In ''CDFull''), defaultguard, defaultaction), (En ''Src'', defaultguard, defaultaction), (En ''CDEject'', defaultguard, defaultaction), (En ''EndOfTitle'', \<lambda>d. (d !P! 0) = (d !P! 1), defaultaction)}"
mem_context_val_9_unchanged: fixes addr1 :: "word36"   and s1 :: "('b, 'c) sparc_state_scheme"   and addr :: "word36"   and val :: "word8" shows "mem_context_val 9 addr1 (mem s1) = mem_context_val 9 addr1 ((mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None)))"
not_cd_1: shows "\<forall>k. \<not> j cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
ex_forward: fixes P' :: "'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" assumes "\<exists>x. P' x"   and "\<And>x. P' x \<Longrightarrow> P x" shows "\<exists>x. P x"
le_TC_imp_VWO: fixes x :: "V"   and y :: "V" assumes "x \<sqsubseteq> y" shows "(x, y) \<in> VWO"
word_or_nth: fixes x :: "'b word"   and y :: "'b word"   and n :: "nat" shows "(x OR y) !! n = (x !! n \<or> y !! n)"
rec: shows "dmu_array fs m dmus' (Suc i) = res"
unicoherent_def: fixes U :: "'b set" shows "unicoherent U \<equiv> \<forall>S T. connected S \<and> connected T \<and> S \<union> T = U \<and> closedin (top_of_set U) S \<and> closedin (top_of_set U) T \<longrightarrow> connected (S \<inter> T)"
path_sum: shows "s \<in>\<^sub>A path_summary loc1 loc2"
gbinomial_int_plus_le: fixes k :: "nat"   and x :: "int"   and y :: "int" assumes "0 < k"   and "0 \<le> x"   and "0 \<le> y" shows "x gchoose k + (y gchoose k) \<le> x + y gchoose k"
cblinfun_eq_on_canonical_basis: fixes f :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c"   and g :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c" assumes "\<And>u. u \<in> set canonical_basis \<Longrightarrow> f *\<^sub>V u = g *\<^sub>V u" shows "f = g"
normbound_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and b :: "real" shows "normbound A b \<equiv> \<forall>x\<in>elements_mat_h A. norm x \<le> b"
intro: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and top :: "'k" assumes "ordering less_eq less"   and "ordering_top_axioms less_eq top" shows "ordering_top less_eq less top"
eq_fin_concat: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and u :: "'a list"   and w\<^sub>1 :: "'a list"   and v :: "'a list"   and w\<^sub>2 :: "'a list" assumes "traces ind" shows "(traces.eq_swap ind)\<^sup>*\<^sup>* (u @ w\<^sub>1 @ v) (u @ w\<^sub>2 @ v) = (traces.eq_swap ind)\<^sup>*\<^sup>* w\<^sub>1 w\<^sub>2"
xor_assoc: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "(x \<oplus> y) \<oplus> z = x \<oplus> y \<oplus> z"
hs_growth: fixes h :: "nat \<Rightarrow> real" assumes "h \<in> set hs" shows "h \<in> O(\<lambda>x. real x / ln (real x) powr (1 + e))"
frame_rule: fixes P :: "assn"   and c :: "'d Heap"   and Q :: "'d \<Rightarrow> assn"   and R :: "assn" assumes "<P> c <Q>" shows "<P * R> c <\<lambda>x. Q x * R>"
assms: shows "g piecewise_C1_differentiable_on {0..1}"
b'': shows "local.prefixes (t + d) j \<down>= b''"
sqrtl_lower: fixes x :: "real"   and n :: "nat" assumes "0 \<le> x" shows "(sqrtl x n)\<^sup>2 \<le> x"
arr_Rel_converse_Rel_comp_Rel: fixes \<alpha> :: "V"   and G :: "V"   and F :: "V" assumes "arr_Rel \<alpha> G"   and "arr_Rel \<alpha> F" shows "(F \<circ>\<^sub>R\<^sub>e\<^sub>l G)\<inverse>\<^sub>R\<^sub>e\<^sub>l = G\<inverse>\<^sub>R\<^sub>e\<^sub>l \<circ>\<^sub>R\<^sub>e\<^sub>l F\<inverse>\<^sub>R\<^sub>e\<^sub>l"
Pi_pmf_of_set: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'c set"   and d :: "'c" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> finite (B x)"   and "\<And>x. x \<in> A \<Longrightarrow> B x \<noteq> {}" shows "Pi_pmf A d (\<lambda>x. pmf_of_set (B x)) = pmf_of_set (PiE_dflt A d B)"
gcd_poly_i_def: fixes ops :: "'b arith_ops_record" shows "gcd_poly_i ops = arith_ops.gcd_eucl_i (poly_ops ops)"
is_path_simps_1: fixes v :: "'v" shows "is_path v [] v = (v \<in> V)"
hcomp_in_vhom': fixes \<nu> :: "'a"   and \<mu> :: "'a"   and f :: "'a"   and h :: "'a"   and g :: "'a"   and k :: "'a" assumes "hseq \<nu> \<mu>"   and "local.dom \<mu> = f"   and "local.dom \<nu> = h"   and "cod \<mu> = g"   and "cod \<nu> = k"   and "\<guillemotleft>\<mu> : f \<Rightarrow> g\<guillemotright>"   and "\<guillemotleft>\<nu> : h \<Rightarrow> k\<guillemotright>"   and "src h = trg f" shows "\<guillemotleft>\<nu> \<star> \<mu> : h \<star> f \<Rightarrow> k \<star> g\<guillemotright>"
mgu_sound: fixes AAA :: "'a set set"   and \<sigma> :: "'s" assumes "finite AAA"   and "\<forall>AA\<in>AAA. finite AA"   and "mgu AAA = Some \<sigma>" shows "is_mgu \<sigma> AAA"
abs_eq: fixes xa :: "'c"   and x :: "'c \<Rightarrow> 'd option" shows "Mapping.delete xa (mapping.Mapping x) = mapping.Mapping (x(xa := None))"
u: shows "u = (t, s) \<and> x \<in> bin (Gen P) k \<and> next_symbol x = Some t \<and> is_terminal t \<and> l \<in> Lex t Doc k \<and> s = take l (drop k Doc)"
lunit_simps_1: fixes f :: "'a" assumes "ide f" shows "arr \<l>[f]"
X_carrier: shows "X \<subseteq> carrier_vec (n + 1)"
jkbpCSn_jkbpCS_subset: fixes n :: "nat" shows "Clock.jkbpCSn n \<subseteq> Clock.jkbpCS"
monoI1: fixes X :: "nat \<Rightarrow> 'b" assumes "\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n" shows "monoseq X"
simple_firewall_without_interfaces_def: fixes rs :: "'a simple_rule list" shows "simple_firewall_without_interfaces rs \<equiv> \<forall>m\<in>match_sel ` set rs. iiface m = ifaceAny \<and> oiface m = ifaceAny"
q2_def: shows "q2 \<equiv> [:\<i>, 1:]"
embed_inj_on: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b" assumes "Well_order r"   and "embed r r' f" shows "inj_on f (Field r)"
ln_upper_11_neg: fixes x :: "real" assumes "0 < x"   and "x \<le> 1" shows "ln x \<le> ln_upper_11 x"
default_val_no_Ref: fixes T :: "ty"   and a :: "nat"   and Cs :: "char list list" assumes "default_val T = Ref (a, Cs)" shows "False"
flatten_sigma_2_correct: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi>[\<G>\<F> \<phi> w]\<^sub>\<Sigma>\<^sub>2 = w \<Turnstile>\<^sub>n \<phi>"
arr_char: fixes f :: "'a" shows "hom.arr f = \<guillemotleft>f : trg e \<rightarrow> trg (trg \<mu>)\<guillemotright>"
picard_lindeloef_def: fixes f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "real set"   and S :: "'b set"   and t\<^sub>0 :: "real" shows "picard_lindeloef f T S t\<^sub>0 \<equiv> (open T \<and> open S \<and> is_interval T) \<and> t\<^sub>0 \<in> T \<and> (\<forall>s\<in>S. continuous_on T (\<lambda>t. f t s)) \<and> local_lipschitz T S f"
cases: fixes x :: "'a \<times> 'a \<times> nat" obtains xa :: "'a" and y :: "'a" and n :: "nat" where "x = (xa, y, n)"
silent_move_preserves_stacks: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and m :: "'node"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a :: "'edge"   and m' :: "'node"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and cs :: "'edge list"   and rs :: "'edge list" assumes "S,f \<turnstile> (m # ms,s) -a\<rightarrow>\<^sub>\<tau> (m' # ms',s')"   and "valid_call_list cs m"   and "\<forall>i<length rs. rs ! i \<in> get_return_edges (cs ! i)"   and "valid_return_list rs m"   and "length rs = length cs"   and "ms = targetnodes rs" obtains cs' :: "'edge list" and rs' :: "'edge list" where "valid_node m'" and "valid_call_list cs' m'" and "\<forall>i<length rs'. rs' ! i \<in> get_return_edges (cs' ! i)" and "valid_return_list rs' m'" and "length rs' = length cs'" and "ms' = targetnodes rs'" and "upd_cs cs [a] = cs'"
finite_orbit: fixes s :: "'b"   and f :: "'b \<Rightarrow> 'b" assumes "s \<in> orbit f s" shows "finite (orbit f s)"
node_ptr_casts_commute2: fixes node_ptr :: "(_) node_ptr" shows "cast (cast node_ptr) = Some node_ptr"
semiring_hom_axioms: shows "semiring_hom (\<lambda>a. [:a:])"
inf_compl_bot_left1: fixes x :: "'a"   and y :: "'a" shows "- x \<cdot> (x \<cdot> y) = 0"
max_length_finite_set: fixes F :: "'a set"   and n :: "nat" assumes "finite F" shows "finite {xs. set xs \<subseteq> F \<and> length xs = n}"
elims: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and x :: "'d \<Rightarrow> nat option"   and xa :: "'c"   and xb :: "'d"   and y :: "nat set" assumes "semi_mojmir_def.pre_ranks \<Sigma> \<delta> q\<^sub>0 x xa xb = y" obtains r :: "'d \<Rightarrow> nat option" and \<nu> :: "'c" and q :: "'d" where "x = r" and "xa = \<nu>" and "xb = q" and "y = {i. \<exists>q'. r q' = Some i \<and> q = \<delta> q' \<nu>} \<union> (if q = q\<^sub>0 then {semi_mojmir_def.max_rank \<Sigma> \<delta> q\<^sub>0} else {})"
contract\<Delta>\<Delta>: fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and \<Delta>' :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> \<Delta> + \<Delta>'"   and "\<Delta>' \<subseteq># \<Delta>" shows "\<Gamma> \<Rightarrow> \<Delta>"
iT_Div_imp_mem: fixes k :: "nat"   and x :: "nat"   and I :: "nat set" assumes "0 < k"   and "x \<in> I" shows "x div k \<in> I \<oslash> k"
pd_pd_ipd: fixes x :: "'n"   and y :: "'n" assumes "x \<in> local.nodes"   and "x \<noteq> local.return"   and "y \<noteq> x"   and "y pd\<rightarrow> x" shows "y pd\<rightarrow> ipd x"
iWeakSince_def: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "iWeakSince I P Q \<equiv> (\<box> t I. P t) \<or> (\<diamond> t I. Q t \<and> (\<box> t' I \<down>> t. P t'))"
weakly_bisimilar_def: fixes P :: "'state"   and Q :: "'state" shows "P \<approx>\<cdot>\<^sub>S Q \<equiv> \<exists>R. S_transform.is_weak_bisimulation R \<and> R P Q"
R2: fixes v :: "'a set" assumes "v \<in> nhds y" shows "x \<in> v"
nt_mult_mm: fixes a :: "'a"   and m :: "int"   and n :: "int" assumes "a \<in> carrier A"   and "m < 0"   and "n < 0" shows "m\<triangleright>n\<triangleright>a\<^bsub>A\<^esub>\<^bsub>A\<^esub> = (m * n)\<triangleright>a\<^bsub>A\<^esub>"
return_time_some: fixes P :: "'a set"   and x :: "'a" assumes "returns_to P x" shows "return_time P x = (SOME t. 0 < t \<and> t \<in> existence_ivl0 x \<and> flow0 x t \<in> P \<and> (\<forall>s\<in>{0<..<t}. flow0 x s \<notin> P))"
VSigmaI: fixes a :: "V"   and A :: "V"   and b :: "V"   and B :: "V \<Rightarrow> V" assumes "a \<in> elts A"   and "b \<in> elts (B a)" shows "\<langle>a, b\<rangle> \<in> elts (VSigma A B)"
Quotient_transaction\<^sub>m: shows "Quotient (=) Transaction\<^sub>m the_Transaction\<^sub>m transaction_merkle.cr_transaction\<^sub>m"
IH: assumes "\<And>x. x \<in># A \<times># {#b#} \<Longrightarrow> P x = Q (fst x)" shows "filter_mset P (A \<times># {#b#}) = filter_mset Q A \<times># {#b#}"
set_union: fixes xs :: "'c list"   and ys :: "'c list" shows "set (List.union xs ys) = set xs \<union> set ys"
rev_fimage_eqI: fixes x :: "'m"   and A :: "'m fset"   and b :: "'n"   and f :: "'m \<Rightarrow> 'n" assumes "x |\<in>| A"   and "b = f x" shows "b |\<in>| f |`| A"
next_preserves_inv_imps_work_sum: fixes s :: "('loc, 't) configuration stream" assumes "next s"   and "holds inv_imps_work_sum s" shows "nxt (holds inv_imps_work_sum) s"
subsequent_half_segments: fixes i :: "nat"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "Suc i < length (half_segments_of_aform X)" shows "snd (half_segments_of_aform X ! i) = fst (half_segments_of_aform X ! Suc i)"
prems_1: shows "H \<turnstile> A(i::=Zero)"
Collect_empty_eq: fixes P :: "'c \<Rightarrow> bool" shows "(Collect P = {}) = (\<forall>x. \<not> P x)"
repl_zero: fixes zero :: "'a"   and succ :: "'a \<Rightarrow> 'a"   and x :: "'b" assumes "peano zero succ" shows "peano.repl zero succ zero x = []"
proper_part_asymmetry: fixes x :: "i"   and y :: "i" assumes "PP x y" shows "\<not> PP y x"
size_word_rsplit_rcat_size: fixes ws :: "'d word list"   and frcw :: "'c word" assumes "word_rcat ws = frcw"   and "size frcw = length ws * LENGTH('d)" shows "length (word_rsplit frcw) = length ws"
nonneg: fixes x :: "real" assumes "0 \<le> x" shows "1 \<le> x + exp (- x)"
exhaust: fixes y :: "('d, 'e) ta_rule" obtains x1 :: "'d" and x2 :: "'e" and x3 :: "'d list" where "y = x1 \<rightarrow> x2 x3"
p_not_Leaf: shows "\<not> isLeaf_pt p low high"
sub: shows "root_cond (p, l', r') (real_of_1 (p, l, r))"
list_order_def: fixes ops :: "('a \<times> 'a option) list"   and x :: "'a"   and y :: "'a" shows "list_order ops x y \<equiv> \<exists>xs ys zs. interp_ins ops = xs @ [x] @ ys @ [y] @ zs"
maybe_not_invalid: fixes x :: "trilean" shows "(\<not>? x = invalid) = (x = invalid)"
eps_induct: fixes P :: "V \<Rightarrow> bool"   and a :: "V" assumes "\<And>x. (\<And>y. y \<in>\<^sub>\<circ> x \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
image2pE: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'g"   and R :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and fx :: "'e"   and gy :: "'g" assumes "BNF_Greatest_Fixpoint.image2p f g R fx gy" obtains x :: "'d" and y :: "'f" where "fx = f x" and "gy = g y" and "R x y"
assms_1: fixes A :: "'a set" assumes "A \<in> sets F" shows "set_lebesgue_integral M A f = set_lebesgue_integral M A g"
cont: shows "continuous_on S (\<lambda>x. \<Sum>i\<in>B. (f x \<bullet> i) *\<^sub>R i)"
ord_spmf_eqD_emeasure: fixes p :: "'a spmf"   and q :: "'a spmf"   and A :: "'a set" assumes "ord_spmf (=) p q" shows "emeasure (measure_spmf p) A \<le> emeasure (measure_spmf q) A"
nat_term_order_eq_DEG_dg: fixes to1 :: "'a nat_term_order"   and to2 :: "'a nat_term_order"   and ps :: "bool" shows "nat_term_order_eq (DEG to1) to2 True ps = nat_term_order_eq to1 to2 True ps"
liveness_dense_iff: fixes P :: "'a llist set" assumes "P \<subseteq> A\<^sup>\<infinity>" shows "liveness A P = (P dense)"
finite_UnI: fixes F :: "'f set"   and G :: "'f set" assumes "finite F"   and "finite G" shows "finite (F \<union> G)"
Prf_imp_proved: fixes s :: "hf"   and k :: "hf"   and x :: "hf"   and e :: "(name, hf) finfun" assumes "Prf s k x" shows "\<exists>A. x = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e \<and> {} \<turnstile> A"
zorder_exist_zero: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set"   and z :: "complex" assumes "f holomorphic_on s"   and "open s"   and "connected s"   and "z \<in> s"   and "\<exists>w\<in>s. f w \<noteq> 0" shows "(if f z = 0 then 0 < zorder f z else zorder f z = 0) \<and> (\<exists>r>0. cball z r \<subseteq> s \<and> zor_poly f z holomorphic_on cball z r \<and> (\<forall>w\<in>cball z r. f w = zor_poly f z w * (w - z) ^ nat (zorder f z) \<and> zor_poly f z w \<noteq> 0))"
vrat_def: shows "\<rat>\<^sub>\<circ> = ZFC_in_HOL.set (range vrat_of_rat)"
simps_5: fixes c :: "int"   and e :: "MIR.num" shows "isrlfm (Lt (CN 0 c e)) = (0 < c \<and> numbound0 e)"
asymptotics2: fixes c :: "real" shows "\<forall>\<^sub>F x in at_top. (1 + c * inverse b * ln x powr - (1 + e)) powr p * (1 - ln (b * x + hb * x / ln x powr (1 + e)) powr (- e / 2)) \<le> 1 - ln x powr (- e / 2)"
1: fixes x :: "real" assumes "0 \<le> x"   and "x \<le> 1" shows "x \<in> (\<lambda>x. x * 2) ` {0..1 / 2}"
ex_distrib_and: fixes P :: "'a \<Rightarrow> assn"   and Q :: "assn" shows "(\<exists>\<^sub>Ax. P x \<and>\<^sub>A Q) = (\<exists>\<^sub>Ax. P x) \<and>\<^sub>A Q"
run_get_option: fixes f :: "'s \<Rightarrow> ('a, 'm) optionT" shows "run_option (local.get_option f) = get (\<lambda>s. run_option (f s))"
undefg_equiv: fixes \<alpha> :: "game option" shows "(\<alpha> \<noteq> undefg) = (\<exists>g. \<alpha> = Agame g)"
HEndPhase0_HInv4d: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase0 s s' p"   and "HInv4d s q"   and "Inv2c s"   and "Inv1 s" shows "HInv4d s' q"
det_four_block_mat_lower_left_zero_col: fixes A1 :: "'b mat"   and A2 :: "'b mat"   and n :: "nat"   and A3 :: "'b mat"   and A4 :: "'b mat" assumes "A1 \<in> carrier_mat 1 1"   and "A2 \<in> carrier_mat 1 n"   and "A3 = 0\<^sub>m n 1"   and "A4 \<in> carrier_mat n n" shows "det (four_block_mat A1 A2 A3 A4) = det A1 * det A4"
subst_3: fixes x :: "name"   and y :: "name"   and t' :: "trm"   and t :: "trm" assumes "x \<sharp> (y, t')" shows "Lam [x].t[y::=t'] = Lam [x].t[y::=t']"
Cons_4: shows "schur_decomposition A (e # es) = (C, P, Q)"
simps: fixes x :: "nat"   and n :: "nat" shows "rank x n = (if x \<le> n \<and> \<not> sink (token_run x n) then Some (card (older_seniors x n)) else None)"
continuous_map_eq: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "continuous_map X X' f"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f x = g x" shows "continuous_map X X' g"
ne_some_unit_eq: fixes x :: "unit option" shows "(x \<noteq> Some ()) = (x = None)"
cases: fixes a :: "intv" assumes "isGreater a" obtains uu_ :: "nat" where "a = Greater uu_"
nact_exists: fixes n :: "nat"   and c :: "'nid"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>" shows "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub> \<and> \<not> (\<exists>k\<ge>n. latestAct_cond c t i k)"
cong_iff_dvd_diff: 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 uminus :: "'b \<Rightarrow> 'b"   and division_segment :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and m :: "'b" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment" shows "unique_euclidean_semiring.cong modulo a b m = dvd.dvd times m (minus a b)"
FOREACH_rule: fixes S :: "'b set"   and I :: "'b set \<Rightarrow> 'c \<Rightarrow> bool"   and \<sigma>0 :: "'c"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c nres"   and P :: "'c \<Rightarrow> bool" assumes "finite S"   and "I S \<sigma>0"   and "\<And>x it \<sigma>. \<lbrakk>x \<in> it; it \<subseteq> S; I it \<sigma>\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> SPEC (I (it - {x}))"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>" shows "FOREACH S f \<sigma>0 \<le> SPEC P"
bintrunc_numeral_simps_5: fixes k :: "num" shows "take_bit (numeral k) 1 = 1"
Pair_le: fixes a :: "'a"   and b :: "'b"   and c :: "'a"   and d :: "'b" shows "((a, b) \<le> (c, d)) = (a \<le> c \<and> b \<le> d)"
assms_4: fixes s :: "real" assumes "s \<in> existence_ivl0 x \<inter> {0..}" shows "V' (flow0 x s) (f (flow0 x s)) \<le> g (flow0 x s) * V (flow0 x s)"
lnull_ldropn: fixes n :: "nat"   and xs :: "'b llist" shows "lnull (ldropn n xs) = (llength xs \<le> enat n)"
K: shows "K \<in> S \<rightarrow>\<^sub>M prob_algebra S"
char_to_digit_simps_6: shows "char_to_digit CHR ''5'' = 5"
fresh_floatarith_fold_const_fa: fixes fa :: "floatarith"   and i :: "nat" assumes "fresh_floatarith fa i" shows "fresh_floatarith (fold_const_fa fa) i"
simps_1: fixes t :: "'b btree" shows "bal_up\<^sub>i (T\<^sub>i t) = bal t"
subseteq_mset_def: fixes A :: "'a multiset"   and B :: "'a multiset" shows "(A \<subseteq># B) = (\<forall>a. count A a \<le> count B a)"
\<I>\<^sub>\<tau>_2: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<I>\<^sub>\<tau>"
ineq_rec: fixes a :: "'a Bonk_Schramm_extension_unfolded" shows "\<forall>x y z t. x \<in> wo.under a \<inter> extended_distance_set \<longrightarrow> y \<in> wo.under a \<inter> extended_distance_set \<longrightarrow> z \<in> wo.under a \<inter> extended_distance_set \<longrightarrow> t \<in> wo.under a \<inter> extended_distance_set \<longrightarrow> extended_distance x y + extended_distance z t \<le> max (extended_distance x z + extended_distance y t) (extended_distance x t + extended_distance y z) + 2 * deltaG TYPE('a)"
two_comparisons_into_case_order_11: fixes comp :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b"   and y :: "'b"   and P :: "'c"   and R :: "'c"   and Q :: "'c" assumes "comparator comp" shows "(if x = y then P else if le_of_comp comp y x then R else Q) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
dr_Skip: fixes \<phi> :: "'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> bool" shows "drefines \<phi> G Skip Skip"
add: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and y :: "'d" assumes "additive f" shows "f (x + y) = f x + f y"
homotopic_loops_imp_homotopic_paths_null: fixes s :: "'c set"   and p :: "real \<Rightarrow> 'c"   and a :: "'c" assumes "homotopic_loops s p (linepath a a)" shows "homotopic_paths s p (linepath (pathstart p) (pathstart p))"
Interleaves_length: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "xs \<cong> {ys, zs, P}" shows "length xs = length ys + length zs"
assms: shows "a \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>)"
assms_2: shows "g \<in> Y \<rightarrow> space M"
lists_fc_succ_permutes: shows "lists_fc_succ xss permutes \<Union> (sset xss)"
part_paval_aval: fixes s :: "char list \<Rightarrow> int"   and x :: "char list"   and a :: "aexp" shows "part (s(x := aval a s)) = part s(x \<mapsto> paval a (part s))"