hcju/isabelleps huggingface dataset:
y_prop: shows "r < y \<and> (\<forall>(d, e, f)\<in>set neq. \<forall>x\<in>{r<..y}. d * x\<^sup>2 + e * x + f \<noteq> 0)"
match_2: fixes h :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VoV.seq h (VoV.comp g f)"   and "VoV.seq g f" shows "VoV.seq h g"
simple_rotations_inverses: fixes r :: "Vertex \<Rightarrow> Vertex" assumes "r \<in> simple_rotations" shows "\<exists>y\<in>complex_rotations. y \<circ> r = id"
all_eq: shows "\<forall>U\<in>top.neighborhoods \<pp>. \<forall>s\<in>\<O> U. (\<phi> \<circ> st.canonical_fun U) s = local.key_map U s"
vd: fixes x :: "real" assumes "0 < x"   and "x < 1"   and "x \<notin> S" shows "vector_derivative g (at x within {0..1}) = vector_derivative (shiftpath (1 - a) (shiftpath a g)) (at x within {0..1})"
3: shows "\<lbrakk>M; map (\<lambda>d. \<langle>ac: pair (t, s) \<doteq> pair (snd d)\<rangle>\<^sub>s\<^sub>t) D\<rbrakk>\<^sub>d \<I>"
monotone_enat_le_case: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "enat \<Rightarrow> enat \<Rightarrow> 'c"   and bot :: "'c" assumes "monotone (\<le>) ord (\<lambda>x. f x (eSuc x))"   and "\<And>x. ord bot (f x (eSuc x))"   and "ord bot bot" shows "monotone (\<le>) ord (\<lambda>x. case x of 0 \<Rightarrow> bot | eSuc x' \<Rightarrow> f x' x)"
roots3_generic: fixes p :: "'a poly"   and depressed_solver :: "'a \<Rightarrow> 'a \<Rightarrow> 'a list" assumes "degree p = 3"   and "\<And>e f y. (y \<in> set (depressed_solver e f)) = (y ^ 3 + e * y + f = (0::'a))" shows "set (roots3_generic depressed_solver p) = {x. poly p x = (0::'a)}"
blocking_on_imp_not_stable: fixes ds :: "'b set"   and X :: "'a set"   and h :: "'c"   and X'' :: "'a set" assumes "PdXXX.blocking_on ds X h X''" shows "\<not> PdXXX.stable_on ds X"
lconf_hext: fixes P :: "'a prog"   and h :: "heap"   and l :: "sfields"   and E :: "char list \<Rightarrow> ty option"   and h' :: "heap" assumes "P,h \<turnstile> l (:\<le>) E"   and "h \<unlhd> h'" shows "P,h' \<turnstile> l (:\<le>) E"
SN_onI: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" assumes "\<And>f. \<lbrakk>f 0 \<in> A; \<forall>i. (f i, f (Suc i)) \<in> r\<rbrakk> \<Longrightarrow> False" shows "SN_on r A"
not_add_less2: fixes j :: "nat"   and i :: "nat" shows "\<not> j + i < i"
digraph_map_def: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map" shows "digraph_map G M \<equiv> fin_digraph G \<and> bidirected_digraph G (edge_rev M) \<and> digraph_map_axioms G M"
length_output_deep_model: fixes m :: "real mat convnet"   and rs :: "nat list" assumes "remove_weights m = deep_model_l rs" shows "dim_vec (tensors_from_net m) = rs ! 0"
csqrt_of_real_nonpos: fixes x :: "complex" assumes "Im x = 0"   and "Re x \<le> 0" shows "csqrt x = \<i> * complex_of_real (sqrt \<bar>Re x\<bar>)"
splits_at_ex: fixes \<delta> :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and s :: "symbol list" assumes "Derives1 \<delta> i r s" shows "\<exists>\<alpha> N \<beta>. splits_at \<delta> i \<alpha> N \<beta>"
distinct_map_pair: fixes xs :: "('a \<times> 'b) list"   and f :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'c" assumes "distinct (map fst xs)"   and "\<And>a b. fst (f a) = fst (f b) \<Longrightarrow> fst a = fst b" shows "distinct (map fst (map_pair f xs))"
psubst_valid: fixes \<Gamma> :: "(W.var \<times> tyS) list"   and \<theta> :: "(W.tvar \<times> ty) list" assumes "valid \<Gamma>" shows "valid (\<theta><\<Gamma>>)"
dim_span_def: fixes W :: "'a vec set" shows "dim_span W = Max (card ` {V. V \<subseteq> carrier_vec n \<and> V \<subseteq> local.span W \<and> lin_indpt V})"
9: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y = y \<squnion> x"
minus_0_l: fixes x :: "'a" assumes "x \<in> carrier R" shows "\<zero> \<ominus> x = \<ominus> x"
simps_3: fixes t :: "ty"   and e :: "expr" shows "assignsE (Cast t e) = assignsE e"
scaleR1_eq: fixes e :: "real"   and a :: "'a"   and b :: "'a" assumes "0 < e" shows "ccw (0::'a) (e *\<^sub>R a) b = ccw (0::'a) a b"
pair_wf_digraph: shows "pair_wf_digraph (list_digraph G)"
assms: shows "liftAll (\<lambda>X. X \<noteq> ERR) einp"
not_less_iff_gr_or_eq: fixes x :: "'b DBMEntry"   and y :: "'b DBMEntry" shows "(\<not> x \<prec> y) = (y \<prec> x \<or> x = y)"
goodE: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b" assumes "good P f" obtains i :: "nat" and j :: "nat" where "i < j" and "P (f i) (f j)"
length_getVars: fixes \<sigma>l :: "'tp list" shows "length (TE.getVars \<sigma>l) = length \<sigma>l"
bounded_apply_bcontfun: fixes x :: "'d \<Rightarrow>\<^sub>C 'c" shows "bounded (range (apply_bcontfun x))"
closed_segment_closed_segment_subset: fixes s :: "real"   and a :: "real"   and b :: "real"   and c :: "real"   and d :: "real" assumes "s \<in> {a--b}"   and "a \<in> {c--d}"   and "b \<in> {c--d}" shows "s \<in> {c--d}"
int_pow_1: fixes x :: "'a" assumes "x \<in> carrier G" shows "x [^] 1 = x"
assms: shows "\<forall>(loc, t)\<in>#\<^sub>z\<Delta>. \<exists>t'. t' \<in>\<^sub>A frontier (c_imp c0 loc) \<and> t' \<le> t"
assms: shows "p = q + x *\<^sub>R (t - q)"
represented_generic_objects_defined: fixes \<tau> :: "'d state \<times> 'd state"   and pre_post :: "'d state \<times> 'd state \<Rightarrow> 'd state"   and H :: "'d \<Rightarrow> \<langle>'e\<rangle>\<^sub>\<bottom>"   and x :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>'e\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> OclAllInstances_generic pre_post H->includes\<^sub>S\<^sub>e\<^sub>t(x)" shows "\<tau> \<Turnstile> \<delta> OclAllInstances_generic pre_post H \<and> \<tau> \<Turnstile> \<delta> x"
assms: shows "\<forall>v. \<eta>: \<rho>\<cdot>v \<mapsto> \<rho>'\<cdot>v"
rel_compp: fixes R1 :: "'d \<Rightarrow> 'j \<Rightarrow> bool"   and S1 :: "'j \<Rightarrow> 'g \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'k \<Rightarrow> bool"   and S2 :: "'k \<Rightarrow> 'h \<Rightarrow> bool"   and R3 :: "'f \<Rightarrow> 'l \<Rightarrow> bool"   and S3 :: "'l \<Rightarrow> 'i \<Rightarrow> bool" shows "F1rel (R1 OO S1) (R2 OO S2) (R3 OO S3) = F1rel R1 R2 R3 OO F1rel S1 S2 S3"
P1_assert_correct2: fixes \<alpha>0 :: "nat"   and r :: "nat"   and \<alpha>1 :: "nat" shows "(\<^bold>g [^] \<alpha>0) [^] r \<otimes> inv ((\<^bold>g [^] \<alpha>1) [^] r) = (\<^bold>g [^] \<alpha>0 \<otimes> inv (\<^bold>g [^] \<alpha>1)) [^] r"
simps: fixes I :: "'b set"   and v :: "nat \<Rightarrow> 'c"   and BL :: "nat \<Rightarrow> 'c option"   and BU :: "nat \<Rightarrow> 'c option"   and IL :: "nat \<Rightarrow> 'b"   and IU :: "nat \<Rightarrow> 'b" shows "(I, v) \<Turnstile>\<^sub>i\<^sub>b ((BL, BU), IL, IU) = ((\<forall>x c. BL x = Some c \<longrightarrow> IL x \<in> I \<longrightarrow> c \<le> v x) \<and> (\<forall>x c. BU x = Some c \<longrightarrow> IU x \<in> I \<longrightarrow> v x \<le> c))"
curry_simp: fixes f1 :: "'a2"   and F :: "'a2 \<times> 'a1 \<Rightarrow> 'b"   and G :: "'a2 \<times> 'a1 \<Rightarrow> 'b"   and \<tau> :: "'a2 \<times> 'a1 \<Rightarrow> 'b" assumes "A2.arr f1" shows "local.curry F G \<tau> f1 = A1_B.MkArr (\<lambda>f2. F (A2.dom f1, f2)) (\<lambda>f2. G (A2.cod f1, f2)) (\<lambda>f2. \<tau> (f1, f2))"
hom_add_eq_zero: fixes x :: "real"   and y :: "real" assumes "x + y = 0" shows "of_real x + of_real y = (0::'a)"
norm_cross7_dot_magnitude: fixes x :: "(real, 7) vec"   and y :: "(real, 7) vec" shows "(norm (x \<times>\<^sub>7 y))\<^sup>2 = (norm x)\<^sup>2 * (norm y)\<^sup>2 - (x \<bullet> y)\<^sup>2"
two_diff: assumes "class.two TYPE('a)" shows "\<exists>x y. x \<noteq> y"
independent_insert: fixes a :: "'c"   and S :: "'c set" shows "vs2.independent (insert a S) = (if a \<in> S then vs2.independent S else vs2.independent S \<and> a \<notin> vs2.span S)"
dist_fst_the_dflt_aux: fixes m :: "('a \<times> 'b) list option" shows "distinct (map fst (the_default [] m)) = (\<forall>xs. m = Some xs \<longrightarrow> distinct (map fst xs))"
lookup_map_default_neq: fixes x :: "'a"   and y :: "'a"   and d :: "'b"   and f :: "'b \<Rightarrow> 'b"   and m :: "('a, 'b) mapping" assumes "x \<noteq> y" shows "Mapping.lookup (Mapping.map_default x d f m) y = Mapping.lookup m y"
eval_oracle_def: fixes c_o :: "'key"   and c_a :: "unit" shows "eval_oracle c_o c_a = (\<lambda>(b, s') guess. map_spmf (\<lambda>b'. ((), b \<or> b', s')) (case guess of (x, h) \<Rightarrow> return_spmf (h = hash c_o x \<and> x \<notin> s')))"
M5exist_var3: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and k :: "'a"   and l :: "'a"   and q :: "'a"   and t :: "'a"   and r :: "'a" assumes "class.interval meets \<I>"   and "meets k l"   and "meets l q"   and "meets q t"   and "meets t r" shows "\<exists>lqt. meets k lqt \<and> meets lqt r"
sequentially_consistentE: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat"   and a :: "nat" assumes "sequentially_consistent P (E, ws)"   and "a \<in> read_actions E" obtains "P,E \<turnstile> a \<leadsto>mrw ws a"
CCexp_pre_simps_3: fixes e :: "exp"   and x :: "Vars.var"   and n :: "Arity" shows "ccExp_syn n (App e x) = ccExp_syn (inc\<cdot>n) e \<squnion> {x} G\<times> insert x (fv e)"
RID\<^sub>GE: fixes r :: "'a"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "r \<in> RID\<^sub>G s" obtains "r \<in> dom s"   | r' :: "'a" and ls :: "('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr" where "s r' = Some ls" and "r \<in> RID\<^sub>L ls"
strict_mono_to_int_mod_ring: shows "strict_mono to_int_mod_ring"
ntrs_step: fixes u :: "'a"   and r :: "'a list"   and ce :: "'a list multiset"   and p :: "'b"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and u' :: "'a"   and w :: "('b, 'c) label list"   and v :: "'a"   and ce' :: "'a list multiset" assumes "((u # r, ce), LCall p, entry fg p # u' # r, ce) \<in> trss fg"   and "(([entry fg p], ce), w, [v], ce') \<in> trcl (trss fg)" shows "((u # r, ce), LCall p # w, v # u' # r, ce') \<in> ntrs fg"
H_from_kat: fixes p :: "'a \<Rightarrow> bool"   and x :: "('a \<times> 'a) set"   and q :: "'a \<Rightarrow> bool" shows "PRE p x POST q = (\<lceil>p\<rceil> \<subseteq> wp x \<lceil>q\<rceil>)"
qring_zero: fixes R :: "('d, 'e) Ring_scheme"   and I :: "'d set" assumes "Ring R"   and "ideal R I" shows "\<zero>\<^bsub>R /\<^sub>r I\<^esub> = I"
correctCompositionDiffLevelsA71: shows "correctCompositionDiffLevels sA71"
u: shows "(\<Sum>v\<in>S. u v *s v) = (0::'b)"
p_def: shows "p = \<lceil>y * real q\<rceil> - 1"
le_sup_iff: fixes sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type"   and y :: "'b::type"   and z :: "'b::type" assumes "class.semilattice_sup (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) (y::'b::type)) (z::'b::type) = (less_eq x z \<and> less_eq y z)"
last_prog_idem: fixes p :: "'b"   and sb :: "'b memref list" shows "last_prog (last_prog p sb) sb = last_prog p sb"
lr_fps_denominator'_nz: fixes cs :: "'b list" assumes "last cs \<noteq> (0::'b)"   and "cs \<noteq> []" shows "lr_fps_denominator' cs \<noteq> 0"
subset_eq: fixes A :: "'c set"   and B :: "'c set" shows "(A \<subseteq> B) = (\<forall>x\<in>A. x \<in> B)"
funpow_add_app: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'b"   and m :: "nat"   and x :: "'b" shows "(f ^^ n) ((f ^^ m) x) = (f ^^ (n + m)) x"
stalk_axioms: shows "stalk Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y {U. is_open\<^sub>Y U \<and> f x \<in> U} (f x)"
clause_saturated_def: fixes S :: "'a eclause set" shows "clause_saturated S = (\<forall>C P \<sigma> C' D \<theta> \<eta>. derivable C P S \<sigma> Ground C' \<longrightarrow> ground_clause (cl_ecl C) \<longrightarrow> derivable D P S \<theta> FirstOrder C' \<longrightarrow> trms_subsumes D C \<eta> \<longrightarrow> \<sigma> \<doteq> \<theta> \<lozenge> \<eta> \<longrightarrow> redundant_clause (subst_ecl D \<eta>) S \<sigma> C')"
colossless_restrict_gpvD: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv" assumes "colossless_gpv \<I> (restrict_gpv \<I> gpv)"   and "\<I> \<turnstile>g gpv \<surd>" shows "colossless_gpv \<I> gpv"
G_def: shows "G = (\<lambda>n x. norm (f x) + norm (F n x) - norm (F n x - f x))"
psimps: fixes t :: "'x \<Rightarrow>\<^sub>0 nat"   and U :: "'x set"   and S :: "('x \<Rightarrow>\<^sub>0 nat) set" assumes "split_dom TYPE('b) (t, U, S)" shows "split t U S = (if 0 \<in> S then ([(monomial (1::'b) t, U)], []) else if S \<inter> .[U] = {} then ([], [(monomial (1::'b) t, U)]) else let x = SOME x'. x' \<in> U - max_subset U (\<lambda>V. S \<inter> .[V] = {}); (ps0, qs0) = split t (U - {x}) S; (ps1, qs1) = split (monomial 1 x + t) U ((\<lambda>f. f - monomial 1 x) ` S) in (ps0 @ ps1, qs0 @ qs1))"
awalk_cost_Cons: fixes f :: "'b \<Rightarrow> real"   and x :: "'b"   and xs :: "'b list" shows "awalk_cost f (x # xs) = f x + awalk_cost f xs"
SV_inv1_invariant: shows "reach sv_TS \<subseteq> Vinv1"
set_sum_eq: shows "(\<Sum>s\<in>set signs \<inter> consistent_sign_vec_copr qs ` set (characterize_root_list_p p). z I s * rat_of_nat (card {x. poly p x = 0 \<and> consistent_sign_vec_copr qs x = s})) = (\<Sum>s\<in>set (remdups (map (consistent_sign_vec_copr qs) (characterize_root_list_p p))). z I s * rat_of_nat (card {x. poly p x = 0 \<and> consistent_sign_vec_copr qs x = s}))"
f_p_2: shows "\<forall>i<n. CAs0f i \<cdot> \<eta>s0f i = CAs ! i"
elims: fixes x :: "'a bro"   and y :: "'a bro" assumes "tree x = y" obtains a :: "'a" where "x = L2 a" and "y = N2 N0 a N0"   | t1 :: "'a bro" and a1 :: "'a" and t2 :: "'a bro" and a2 :: "'a" and t3 :: "'a bro" where "x = N3 t1 a1 t2 a2 t3" and "y = N2 (N2 t1 a1 t2) a2 (N1 t3)"   | "x = N0" and "y = N0"   | v :: "'a bro" where "x = N1 v__" and "y = N1 v__"   | v :: "'a bro" and va :: "'a" and vb :: "'a bro" where "x = N2 v va vb" and "y = N2 v va vb"
FSwImorph_igWlsBinp: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'h, 'i, 'j, 'k) model_scheme"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 'j option"   and h :: "'i \<Rightarrow> 'l"   and hA :: "'j \<Rightarrow> 'm"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'h, 'l, 'm, 'n) model_scheme" assumes "igWlsBinp MOD delta binp"   and "FSwImorph h hA MOD MOD'" shows "igWlsBinp MOD' delta (lift hA binp)"
disj_conj_distrib: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<^bold>\<squnion> y \<^bold>\<sqinter> z = (x \<^bold>\<squnion> y) \<^bold>\<sqinter> (x \<^bold>\<squnion> z)"
intro: fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" assumes "outstanding_non_volatile_refs_owned_or_read_only \<S> ts"   and "outstanding_volatile_writes_unowned_by_others ts"   and "read_only_reads_unowned ts"   and "ownership_distinct ts" shows "valid_ownership \<S> ts"
f3: shows "x' \<in> (r' |\<^bsub>rel\<^esub> s')"
edc_inv: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr"   and e :: "expr"   and f :: "cexpr" assumes "(vs, vs', \<Gamma>, \<delta>) \<turnstile>\<^sub>c e \<Rightarrow> f" shows "(vs, vs', \<Gamma>, \<delta>) \<turnstile>\<^sub>c Inverse $$ e \<Rightarrow> (f \<circ>\<^sub>c inverse\<^sub>c (CVar 0)) *\<^sub>c inverse\<^sub>c (CVar 0) ^\<^sub>c CInt 2"
linear_surj_adj_imp_inj: fixes f :: "'a \<Rightarrow> 'b" assumes "linear f"   and "surj (adjoint f)" shows "inj f"
mcont_mono: fixes luba :: "'b set \<Rightarrow> 'b"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and lubb :: "'c set \<Rightarrow> 'c"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" assumes "mcont luba orda lubb ordb f" shows "monotone orda ordb f"
Infinitesimal_HFinite_mult2: fixes x :: "'b star"   and y :: "'b star" assumes "x \<in> Infinitesimal"   and "y \<in> HFinite" shows "y * x \<in> Infinitesimal"
prveqlPT_prv_instInpP_instInp: fixes \<phi> :: "'fmla"   and \<tau> :: "'fmla"   and t :: "'trm" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> \<subseteq> {inp}"   and "\<tau> \<in> ptrm 0"   and "t \<in> trm"   and "FvarsT t = {}"   and "prveqlPT \<tau> t" shows "prv (imp (instInp \<phi> t) (instInpP \<phi> 0 \<tau>))"
Bell_recursive_eq: fixes n :: "nat" shows "Bell (n + 1) = (\<Sum>k\<le>n. (n choose k) * Bell k)"
foundation16: fixes \<tau> :: "'a state \<times> 'a state"   and X :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "(\<tau> \<Turnstile> \<delta> X) = (X \<tau> \<noteq> \<bottom> \<and> X \<tau> \<noteq> null)"
map_gpv'_sel: fixes f :: "'j \<Rightarrow> 'g"   and g :: "'k \<Rightarrow> 'h"   and h :: "'i \<Rightarrow> 'l"   and gpv :: "('j, 'k, 'l) gpv" shows "the_gpv (map_gpv' f g h gpv) = map_spmf (map_generat f g (h ---> map_gpv' f g h)) (the_gpv gpv)"
sel_2: fixes x1 :: "'c extended"   and x2 :: "'c"   and x3 :: "'c extended"   and x4 :: "'d" shows "I (InsertMessage x1 x2 x3 x4) = x2"
le_SUP_iff: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "less_eq x (Sup (f ` A)) = (\<forall>y. less y x \<longrightarrow> (\<exists>i\<in>A. less y (f i)))"
vsv_vcard_vdomain: shows "vcard (\<AA>\<lparr>Arr\<rparr>) = vcard (\<FF>\<lparr>ArrMap\<rparr>)"
diamond_y_gen_def: shows "diamond_y_gen \<equiv> \<lambda>t. 1 / 2 - \<bar>t\<bar>"
assms: shows "h \<turnstile> CD.a_set_tag_name element_ptr tag \<rightarrow>\<^sub>h h'"
gen_gbg_impl_rel_eext_def: fixes Rm :: "('k \<times> 'm) set"   and Rf :: "('j \<times> 'l set set) set" shows "\<langle>Rm, Rf\<rangle>gen_gbg_impl_rel_eext = {(\<lparr>gbgi_F = Fi, \<dots> = mi\<rparr>, \<lparr>gbg_F = F, \<dots> = m\<rparr>) |Fi mi F m. (Fi, F) \<in> Rf \<and> (mi, m) \<in> Rm}"
exec_n_drop_left: fixes P :: "instr list"   and P' :: "instr list"   and i :: "int"   and s :: "char list \<Rightarrow> int"   and stk :: "int list"   and k :: "nat"   and n :: "int"   and s' :: "char list \<Rightarrow> int"   and stk' :: "int list" assumes "P @ P' \<turnstile> (i, s, stk) \<rightarrow>^k (n, s', stk')"   and "size P \<le> i"   and "exits P' \<subseteq> {0..}" shows "P' \<turnstile> (i - size P, s, stk) \<rightarrow>^k (n - size P, s', stk')"
v_S\<^sub>e: fixes cfg :: "'s cfg" assumes "cfg \<in> valid_cfg"   and "state cfg \<in> S\<^sub>e" shows "v cfg = 0"
effect_to_assignments_iii_b: fixes v :: "'c"   and op :: "'c strips_operator"   and as :: "('c \<times> bool) list" assumes "v \<in> set (delete_effects_of op)"   and "as = effect_to_assignments op" obtains a :: "'c \<times> bool" where "a \<in> set as" and "a = (v, False)"
degree: fixes x :: "'a pdevs"   and j :: "nat" assumes "degree x \<le> j" shows "pdevs_apply x j = (0::'a)"
inject_2: fixes x21 :: "'c"   and x22 :: "('b, 'c) pform list"   and y21 :: "'c"   and y22 :: "('b, 'c) pform list" shows "(Fun x21 x22 = Fun y21 y22) = (x21 = y21 \<and> x22 = y22)"
se_succs_states: fixes c :: "('c, 'd) conf"   and l :: "('c, 'd) label"   and c1 :: "('c, 'd) conf"   and c2 :: "('c, 'd) conf" assumes "se c l c1"   and "se c l c2" shows "states c1 = states c2"
constT_def: shows "constT \<equiv> map_of (consts D)"
pdiv_2: fixes K :: "'a set" assumes "K \<in> p" shows "K \<subseteq> \<Union> p"
rep_eq: fixes x :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and xa :: "'d set!['f]"   and xb :: "'e set!['f]" shows "rel_nebset x xa xb = rel_set x (set_nebset xa) (set_nebset xb)"
one_Measurement_System_ext_def: shows "1 = \<lparr>unit_sys = unit, \<dots> = 1::'b\<rparr>"
elims_3: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xa :: "'a option"   and xb :: "'a option" assumes "\<not> option_le x xa xb" obtains P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and xc :: "'a" where "x = P__" and "xa = Some xc__" and "xb = None"   | P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and xc :: "'a" and y :: "'a" where "x = P" and "xa = Some xc" and "xb = Some y" and "\<not> P xc y"
valid_regions_distinct: fixes X :: "'d set"   and k :: "'d \<Rightarrow> nat"   and I :: "'d \<Rightarrow> intv"   and r :: "('d \<times> 'd) set"   and I' :: "'d \<Rightarrow> intv"   and r' :: "('d \<times> 'd) set"   and v :: "'d \<Rightarrow> real" assumes "valid_region X k I r"   and "valid_region X k I' r'"   and "v \<in> region X I r"   and "v \<in> region X I' r'" shows "region X I r = region X I' r'"
keys_row_to_poly: fixes ts :: "'c list"   and r :: "'d vec" shows "keys (row_to_poly ts r) \<subseteq> set ts"
lappend_ltake_enat_ldropn: fixes n :: "nat"   and xs :: "'c llist" shows "lappend (ltake (enat n) xs) (ldropn n xs) = xs"
poincare_collinear_zero_iff: fixes y' :: "complex"   and z' :: "complex" assumes "of_complex y' \<in> unit_disc"   and "of_complex z' \<in> unit_disc"   and "y' \<noteq> z'"   and "y' \<noteq> 0"   and "z' \<noteq> 0" shows "poincare_collinear {0\<^sub>h, of_complex y', of_complex z'} = (y' * cnj z' = cnj y' * z')"
AA2: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<box>[P]_v \<and> \<diamond>\<langle>Q\<rangle>_v \<longrightarrow> \<diamond>\<langle>P \<and> Q\<rangle>_v"
submatrix_def: fixes A :: "'a mat"   and I :: "nat set"   and J :: "nat set" shows "submatrix A I J = Matrix.mat (card {i. i < dim_row A \<and> i \<in> I}) (card {j. j < dim_col A \<and> j \<in> J}) (\<lambda>(i, j). A $$ (pick I i, pick J j))"
A_chambers_1: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and B :: "'b set set"   and B' :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set" assumes "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E z" shows "SimplicialComplex.maxsimp A C"
preserves_src: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and src\<^sub>C :: "'d \<Rightarrow> 'd"   and trg\<^sub>C :: "'d \<Rightarrow> 'd"   and D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and src\<^sub>D :: "'e \<Rightarrow> 'e"   and trg\<^sub>D :: "'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and \<mu> :: "'d" assumes "weak_arrow_of_homs C src\<^sub>C trg\<^sub>C D src\<^sub>D trg\<^sub>D F"   and "partial_magma.arr C \<mu>" shows "src\<^sub>D (F \<mu>) = weak_arrow_of_homs.map\<^sub>0 src\<^sub>D F (src\<^sub>C \<mu>)"
trd_def: fixes cmp_term :: "(('a, 'b) pp \<times> nat) nat_term_order"   and fs :: "(('a, 'b) pp \<times> nat \<Rightarrow>\<^sub>0 'c) list"   and p :: "('a, 'b) pp \<times> nat \<Rightarrow>\<^sub>0 'c" shows "trd_pprod cmp_term fs p = trd_aux_pprod cmp_term fs p 0"
F_eq: fixes x :: "real" shows "ereal (F x) = (INF n\<in>{n. x < r n}. ereal (lim (\<lambda>k. f (d k) (r n))))"
approx_tm_range: fixes a :: "float interval list"   and I :: "float interval list"   and prec :: "nat"   and ord :: "nat"   and f :: "floatarith"   and env :: "taylor_model list"   and t :: "taylor_model"   and xs :: "real list"   and e :: "nat \<Rightarrow> real" assumes "a all_subset I"   and "approx_tm prec ord I a f env = Some t"   and "xs all_in\<^sub>i range_tms e env"   and "develops_at_within e a I"   and "\<And>tm. tm \<in> set env \<Longrightarrow> num_params (tm_poly tm) \<le> length I" shows "interpret_floatarith f xs \<in>\<^sub>i range_tm e t"