hcju/isabelleps huggingface dataset:
finite_field_factorization_explicit: fixes f :: "'a mod_ring poly"   and c :: "'a mod_ring"   and us :: "'a mod_ring poly list" assumes "square_free f"   and "local.finite_field_factorization f = (c, us)" shows "f = Polynomial.smult c (prod_list us) \<and> (\<forall>u\<in>set us. monic u \<and> irreducible u)"
h_2: shows "ident_rel S_Idt (map_graph_fn G' h)"
implements_def: fixes jp :: "'a \<Rightarrow> ('obs, 'aAct, 'd) Protocol" shows "implements jp \<equiv> \<forall>t\<in>jkbpC. set \<circ> actJP jp t = set \<circ> jAction MC t"
unique_idx_subset: fixes bs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data :: "nat \<times> 'h"   and bs' :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list" assumes "unique_idx bs data"   and "set bs' \<subseteq> set bs" shows "unique_idx bs' data"
exec_step_ind_Goto: fixes i :: "int"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" shows "exec_step_ind (StepI (Goto i)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (stk, loc, C\<^sub>0, M\<^sub>0, nat (int pc + i), ics) # frs, sh)"
games_lfp_subset_gfp: shows "games_lfp \<subseteq> games_gfp"
onorm_scaleC_left_lemma: fixes r :: "'a \<Rightarrow> complex"   and f :: "'b" assumes "bounded_clinear r" shows "onorm (\<lambda>x. r x *\<^sub>C f) \<le> onorm r * norm f"
prems_1: shows "subsumes_list (L # Ls) Ks \<sigma>"
list_of_dlist_inject: fixes x :: "'d dlist"   and y :: "'d dlist" shows "(list_of_dlist x = list_of_dlist y) = (x = y)"
c'0: fixes t :: "'a" assumes "t \<in> insert a T'" shows "0 \<le> c' t"
QuasiAmicable_pair_def: fixes m :: "nat"   and n :: "nat" shows "m QAmic n = (m + 1 = aliquot_sum n \<and> n + 1 = aliquot_sum m)"
reset_set1: fixes cs :: "'a list"   and d :: "'b"   and u :: "'a \<Rightarrow> 'b" shows "\<forall>c\<in>set cs. ([cs\<rightarrow>d]u) c = d"
inv_into_f_f: fixes f :: "'f \<Rightarrow> 'g"   and A :: "'f set"   and x :: "'f" assumes "inj_on f A"   and "x \<in> A" shows "inv_into A f (f x) = x"
Join_transient_I1: fixes F :: "'c program"   and A :: "'c set"   and G :: "'c program" assumes "F \<in> transient A" shows "F \<squnion> G \<in> transient A"
assms_2: fixes x :: "'a"   and \<pi> :: "nat \<Rightarrow> nat" assumes "x \<in> X"   and "\<pi> permutes A" shows "mpoly_map_vars \<pi> (f x) = f (g x)"
maxim_greatest: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "x \<in> A"   and "A \<subseteq> Field s" shows "(x, s.maxim A) \<in> s"
of_nat_div: fixes m :: "nat"   and n :: "nat" shows "of_nat (m div n) = (of_nat m - of_nat (m mod n)) / of_nat n"
test_bit_bl: fixes w :: "'a word"   and n :: "nat" shows "w !! n = (rev (to_bl w) ! n \<and> n < size w)"
Max_le_iff: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'e set"   and x :: "'e" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less_eq (linorder.Max less_eq A) x = (\<forall>a\<in>A. less_eq a x)"
quasiconvexI: fixes C :: "real"   and X :: "'b set" assumes "0 \<le> C"   and "\<And>x y. \<lbrakk>x \<in> X; y \<in> X\<rbrakk> \<Longrightarrow> \<exists>G. geodesic_segment_between G x y \<and> (\<forall>z\<in>G. infdist z X \<le> C)" shows "quasiconvex C X"
cart_basis_Rn: shows "cart_basis_Rn \<equiv> cart_basis'"
A6_NSources_L1: shows "\<forall>C\<in>AbstrLevel level1. C \<noteq> sA93 \<and> C \<noteq> sA92 \<and> C \<noteq> sA91 \<and> C \<noteq> sA82 \<and> C \<noteq> sA81 \<and> C \<noteq> sA72 \<and> C \<noteq> sA71 \<longrightarrow> sA6 \<notin> Sources level1 C"
Image_Int_eq: fixes R :: "('b \<times> 'a) set"   and A :: "'b set"   and B :: "'b set" assumes "single_valued (R\<inverse>)" shows "R `` (A \<inter> B) = R `` A \<inter> R `` B"
gd_inf_term_def: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> nat"   and term_of_pair :: "'b \<times> nat \<Rightarrow> 'a"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ord_strict :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ord_term :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord_term_strict :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "gd_inf_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict \<equiv> gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"
le_add_diff_inverse: fixes plus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and minus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and less_eq :: "'q \<Rightarrow> 'q \<Rightarrow> bool"   and less :: "'q \<Rightarrow> 'q \<Rightarrow> bool"   and zero :: "'q"   and times :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and one :: "'q"   and b :: "'q"   and a :: "'q" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq b a" shows "plus b (minus a b) = a"
mono_above: fixes iv1 :: "ivl"   and iv2 :: "ivl" assumes "iv1 \<le> iv2" shows "Abs_Int2_ivl.above iv1 \<le> Abs_Int2_ivl.above iv2"
IH: fixes k :: "nat" assumes "k \<le> card A" shows "\<exists>B\<subseteq>A. card B = k"
fresh_star_list: fixes xs :: "'a list"   and c :: "'b" shows "xs \<sharp>* c \<equiv> \<forall>x\<in>set xs. x \<sharp> c"
ct: shows "r2.cond_action_ok (ls, (ts2, m2), ws, is) t ct"
mult_ge_prts: fixes a1 :: "'a"   and a :: "'a"   and a2 :: "'a"   and b1 :: "'a"   and b :: "'a"   and b2 :: "'a" assumes "a1 \<le> a"   and "a \<le> a2"   and "b1 \<le> b"   and "b \<le> b2" shows "nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1 \<le> a * b"
distinct_13: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "Skip \<noteq> x31@[x32]"
timpl_closure'_inv''_1: fixes f :: "('a, 'b, 'c) prot_fun"   and S :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and g :: "('a, 'b, 'c) prot_fun"   and T :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and TI :: "('c set \<times> 'c set) set" assumes "(Fun f S, Fun g T) \<in> timpl_closure' TI" shows "length S = length T"
subset_antisym: fixes A :: "'f set"   and B :: "'f set" assumes "A \<subseteq> B"   and "B \<subseteq> A" shows "A = B"
name_upd_incoming_f: fixes x :: "('d, 'e) node_scheme"   and n :: "('d, 'f) node_scheme" shows "name (upd_incoming_f n x) = name x"
sup1: fixes A :: "'c"   and k :: "'c"   and a :: "'c"   and b :: "'c" assumes "A \<equiv> k \<squnion> a" shows "A \<squnion> b \<equiv> k \<squnion> (a \<squnion> b)"
less_sets_weaken2: fixes A :: "'a set"   and B' :: "'a set"   and B :: "'a set" assumes "A \<lless> B'"   and "B \<subseteq> B'" shows "A \<lless> B"
integral: fixes R :: "('e, 'f) ring_scheme"   and a :: "'e"   and b :: "'e" assumes "domain R"   and "a \<otimes>\<^bsub>R\<^esub> b = \<zero>\<^bsub>R\<^esub>"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "a = \<zero>\<^bsub>R\<^esub> \<or> b = \<zero>\<^bsub>R\<^esub>"
sf_le_sfD: fixes x :: "nat"   and y :: "nat" assumes "sf x \<le> sf y" shows "x \<le> y"
gtrp2gtr_s: fixes s :: "'c"   and c :: "'c multiset"   and e :: "'d el_step"   and s' :: "'c"   and c' :: "'c multiset"   and T :: "(('c \<times> 'c multiset) \<times> 'd \<times> 'c \<times> 'c multiset) set" assumes "((s, c), e, s', c') \<in> gtrp T" shows "(add_mset s c, le_rem_s e, add_mset s' c') \<in> gtr T"
ntrancl_onpD_rtrancl_on: fixes F :: "'c set"   and r :: "('c \<times> 'c) set"   and n :: "nat"   and a :: "'c"   and b :: "'c" assumes "ntrancl_onp F r n a b" shows "(a, b) \<in> rtrancl_on F r"
flip_blinfun_apply: fixes f :: "'f \<Rightarrow>\<^sub>L 'g \<Rightarrow>\<^sub>L 'e"   and a :: "'g"   and b :: "'f" shows "blinfun_apply (blinfun_apply (flip_blinfun f) a) b = blinfun_apply (blinfun_apply f b) a"
cases: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "Inf.inf r a" obtains x :: "'b" and y :: "'b" where "a = x" and "r x y" and "Inf.inf r y"
onorm_pos_le: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_linear f" shows "0 \<le> onorm f"
coerceQuant_eq_iff2: fixes y :: "'c['b, 'd]"   and x :: "'c['a, 'd]" assumes "QD('a) = QD('b)"   and "y = QCOERCE['b] x" shows "x \<cong>\<^sub>Q y"
prems_2: shows "I (set (x # S)) \<sigma>0"
map_fun_apply: fixes f :: "'f \<Rightarrow> 'g"   and g :: "'h \<Rightarrow> 'e"   and h :: "'g \<Rightarrow> 'h"   and x :: "'f" shows "(f ---> g) h x = g (h (f x))"
Qcarr: shows "convex_hull Q \<subseteq> carrier_vec n"
bell01_is_state: shows "state 2 |\<beta>\<^sub>0\<^sub>1\<rangle>"
add_diff_inverse: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "\<not> less a b" shows "plus b (minus a b) = a"
pp_of_term_splus: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and t :: "'b"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.pp_of_term pair_of_term (term_powerprod.splus pair_of_term term_of_pair t v) = t + term_powerprod.pp_of_term pair_of_term v"
add_pred_def: fixes P :: "'BExp set"   and \<S> :: "'Var set \<times> 'Var set"   and e :: "'BExp" shows "P +\<^sub>\<S> e \<equiv> if \<forall>x\<in>bexp_vars e. stable \<S> x then P \<union> {e} else P"
setmult_rcos_assoc: fixes H :: "'a set"   and K :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier G"   and "K \<subseteq> carrier G"   and "x \<in> carrier G" shows "H <#>\<^bsub>add_monoid G\<^esub> (K #>\<^bsub>add_monoid G\<^esub> x) = H <#>\<^bsub>add_monoid G\<^esub> K #>\<^bsub>add_monoid G\<^esub> x"
subset_trans: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "order.greater_eq B A"   and "order.greater_eq C B" shows "order.greater_eq C A"
c1: shows "w \<Turnstile>\<^sub>n \<phi>[set ms]\<^sub>\<Sigma>\<^sub>2"
\<iota>_in_hom: shows "\<guillemotleft>\<iota>\<^sub>C : \<I>\<^sub>C \<otimes>\<^sub>C \<I>\<^sub>C \<rightarrow>\<^sub>C \<I>\<^sub>C\<guillemotright>"
asm: shows "x\<^sub>C \<in> (mdss ! i) AsmNoReadOrWrite"
openin_subopen: fixes U :: "'a topology"   and S :: "'a set" shows "openin U S = (\<forall>x\<in>S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)"
ringI: fixes R :: "('a, 'b) ring_scheme" assumes "abelian_group R"   and "Group.monoid R"   and "\<And>x y z. \<lbrakk>x \<in> carrier R; y \<in> carrier R; z \<in> carrier R\<rbrakk> \<Longrightarrow> (x \<oplus>\<^bsub>R\<^esub> y) \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> z \<oplus>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z"   and "\<And>x y z. \<lbrakk>x \<in> carrier R; y \<in> carrier R; z \<in> carrier R\<rbrakk> \<Longrightarrow> z \<otimes>\<^bsub>R\<^esub> (x \<oplus>\<^bsub>R\<^esub> y) = z \<otimes>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> z \<otimes>\<^bsub>R\<^esub> y" shows "ring R"
seq_inf_distrib: fixes c :: "'a::refinement_lattice"   and d\<^sub>0 :: "'a::refinement_lattice"   and d\<^sub>1 :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice) ; ((d\<^sub>0::'a::refinement_lattice) \<sqinter> (d\<^sub>1::'a::refinement_lattice)) = c ; d\<^sub>0 \<sqinter> c ; d\<^sub>1"
reduced_word_forI_compare: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and A :: "'b set"   and a :: "'b"   and as :: "'b list"   and bs :: "'b list" assumes "class.monoid_add plus zero"   and "monoid_add.reduced_word_for plus zero A a as"   and "bs \<in> lists A"   and "monoid_add.sum_list plus zero bs = a"   and "length bs = length as" shows "monoid_add.reduced_word_for plus zero A a bs"
Unifier_comp': fixes t :: "('a, 'b) Term.term"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and \<delta> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and u :: "('a, 'b) Term.term" assumes "Unifier \<delta> (t \<cdot> \<theta>) (u \<cdot> \<theta>)" shows "Unifier (\<theta> \<circ>\<^sub>s \<delta>) t u"
induct: fixes P :: "'c fm \<Rightarrow> bool"   and a0 :: "'c fm" assumes "\<And>a. P (Atom a)"   and "P TrueF"   and "P FalseF"   and "\<And>\<phi>\<^sub>1 \<phi>\<^sub>2. \<lbrakk>P \<phi>\<^sub>1; P \<phi>\<^sub>2\<rbrakk> \<Longrightarrow> P (And \<phi>\<^sub>1 \<phi>\<^sub>2)"   and "\<And>\<phi>\<^sub>1 \<phi>\<^sub>2. \<lbrakk>P \<phi>\<^sub>1; P \<phi>\<^sub>2\<rbrakk> \<Longrightarrow> P (Or \<phi>\<^sub>1 \<phi>\<^sub>2)"   and "\<And>v. P (Neg v)"   and "\<And>v. P (ExQ v)" shows "P a0"
simps_3: fixes p1 :: "('a, 'b, 'c, 'd) seqp"   and p2 :: "('a, 'b, 'c, 'd) seqp" shows "subterms (p1 \<oplus> p2) = {p1 \<oplus> p2} \<union> subterms p1 \<union> subterms p2"
perm_remove: fixes x :: "'b"   and ys :: "'b list" assumes "x \<in> set ys" shows "ys <~~> x # remove1 x ys"
set_mset_def: fixes M :: "'c multiset" shows "set_mset M = {x. 0 < count M x}"
subseqs_length_simple_False: fixes b :: "'c list"   and xs :: "'c list" assumes "b \<in> set (subseqs xs)"   and "length xs < length b" shows "False"
simps_1: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "x # xs \<and>\<^sub>p y # ys = (if x = y then x # (xs \<and>\<^sub>p ys) else \<epsilon>)"
one: shows "S O S\<^sup>* O NS\<^sup>* \<subseteq> (NS \<union> S)\<^sup>* O S O (NS \<union> S)\<^sup>*"
L: shows "((\<lambda>x. ln (eval C x) + e * ln (b x)) expands_to L) basis"
2: fixes S :: "'b set"   and a :: "'b" assumes "card S \<le> 1" shows "card (a \<triangleright> S) \<le> 2"
V0_altdef: shows "V\<^sub>0 = {[i] \<odot> f |i f. f \<in> \<R> \<and> \<phi> i = [i] \<odot> f}"
hyps_1: shows "P \<turnstile> \<langle>e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>throw a,(h', l', sh')\<rangle>"
msubsteq2_nb: fixes t :: "tm"   and a :: "poly"   and r :: "tm"   and c :: "poly" assumes "tmbound0 t"   and "islin (Eq (CNP 0 a r))" shows "bound0 (msubsteq2 c t a r)"
simplex_in_max: fixes y :: "'a set" assumes "y \<in> X" shows "\<exists>x. folding_g.chamber x \<and> order.greater_eq x y"
inf1: fixes A :: "'b::semilattice_inf"   and k :: "'b::semilattice_inf"   and a :: "'b::semilattice_inf"   and b :: "'b::semilattice_inf" assumes "A::'b::semilattice_inf \<equiv> (k::'b::semilattice_inf) \<sqinter> (a::'b::semilattice_inf)" shows "(A::'b::semilattice_inf) \<sqinter> (b::'b::semilattice_inf) \<equiv> (k::'b::semilattice_inf) \<sqinter> ((a::'b::semilattice_inf) \<sqinter> b)"
poly_rel_karatsuba_main: fixes ops :: "'b arith_ops_record"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x1 :: "'b list"   and x2 :: "'c list"   and y1 :: "'b list"   and y2 :: "'c list"   and n :: "nat"   and m :: "nat" assumes "ring_ops ops R"   and "list_all2 R x1 x2"   and "list_all2 R y1 y2" shows "ring_ops.poly_rel R (karatsuba_main_i ops x1 n y1 m) (karatsuba_main x2 n y2 m)"
strict_equal_def: fixes m :: "'a option"   and a :: "'a" shows "strict_equal m a \<equiv> case m of None \<Rightarrow> False | Some a' \<Rightarrow> a' = a"
Range_converse: fixes r :: "('e \<times> 'f) set" shows "Range (r\<inverse>) = Domain r"
split_sum_ex: fixes P :: "'b + 'c \<Rightarrow> bool" shows "(\<exists>x. P x) = ((\<exists>x. P (Inl x)) \<or> (\<exists>x. P (Inr x)))"
simps: fixes fs :: "face list"   and n :: "nat"   and f :: "face list list"   and h :: "nat list" shows "heights (Graph fs n f h) = h"
tMap_is_tStep_conv: fixes f :: "'c \<Rightarrow> 'b"   and t :: "'c Trace"   and tp :: "'b Trace"   and s :: "'b" shows "(tMap f t = tp \<leadsto> s) = (\<exists>tp' s'. t = tp' \<leadsto> s' \<and> s = f s' \<and> tMap f tp' = tp)"
l_def: shows "set l = s - {z} \<and> sorted_wrt (to_ord r) l \<and> distinct l"
perfect_hiding: fixes \<A> :: "('a \<times> nat \<Rightarrow> ((nat \<times> nat) \<times> 'c) spmf) \<times> ('a \<Rightarrow> 'c \<Rightarrow> bool spmf)" shows "Schnorr_\<Sigma>_commit.abstract_com.perfect_hiding_ind_cpa \<A>"
vfsequence_cases: fixes xs :: "V" assumes "vfsequence xs" obtains "xs = []\<^sub>\<circ>"   | xs' :: "V" and x :: "V" where "xs = xs' #\<^sub>\<circ> x" and "vfsequence xs'"
intro: fixes x :: "'d \<Longrightarrow> 'e" assumes "weak_lens x"   and "pbij_lens_axioms x" shows "pbij_lens x"
assms_3: fixes S :: "'a set" assumes "S \<in> B" shows "open S"
sup_assoc: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "x \<sqinter> y \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
wf_ts_def: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and j :: "nat"   and \<phi> :: "'a MFOTL.formula"   and \<psi> :: "'a MFOTL.formula"   and ts :: "nat list" shows "wf_ts \<sigma> j \<phi> \<psi> ts = list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [min (Monitor.progress \<sigma> \<phi> j) (Monitor.progress \<sigma> \<psi> j)..<j] ts"
elims: fixes x :: "nat"   and xa :: "'b"   and xb :: "nat"   and y :: "'b btree" assumes "full_node x xa xb = y" obtains k :: "nat" and c :: "'b" where "x = ka__" and "xa = c__" and "xb = 0" and "y = Leaf"   | k :: "nat" and c :: "'b" and n :: "nat" where "x = k" and "xa = c" and "xb = Suc n" and "y = Node (replicate (2 * k) (full_node k c n, c)) (full_node k c n)"
open_vimage: fixes s :: "'b set"   and f :: "'c \<Rightarrow> 'b" assumes "open s"   and "continuous_on UNIV f" shows "open (f -` s)"
assms: shows "(embP Q \<and>* $ n) = (embP P \<and>* $ m)"
hyps_2: assumes "0 < degree p" shows "filterlim (poly p) at_infinity at_infinity"
cod_object: fixes CC :: "('d, 'e, 'f) category_scheme"   and f :: "'e" assumes "category CC"   and "f \<in> Ar\<^bsub>CC\<^esub>" shows "Cod\<^bsub>CC\<^esub> f \<in> Ob\<^bsub>CC\<^esub>"
process_prop_2_def: fixes P :: "('e list \<times> 'e set) set \<times> 'e list set" shows "process_prop_2 P \<equiv> \<forall>xs x X. (xs @ [x], X) \<in> fst P \<longrightarrow> (xs, {}) \<in> fst P"
list_eq_map_lookup1: fixes t :: "'b tree" assumes "braun t" shows "list t = map (lookup1 t) [1..<size t + 1]"
assms: shows "(Fun (Val n) [], Fun (Set s) []) \<in> set D"
cs_1: shows "(fst (f a), cs) \<in> group_by f as"
unity_root_diff: fixes k :: "nat"   and m :: "int"   and n :: "int" shows "unity_root k (m - n) = unity_root k m * cnj (unity_root k n)"
length_replicate: fixes n :: "nat"   and x :: "'c" shows "length (replicate n x) = n"
eucl_of_list_image_Joints: fixes x :: "(real \<times> real pdevs) list" assumes "length x = DIM('a)" shows "eucl_of_list ` Joints x = Affine (eucl_of_list_aform x)"
prj0_simps_arr: fixes f :: "'a"   and g :: "'a" shows "arr \<p>\<^sub>0[f, g] = cospan f g"
PUQU: fixes s :: "'a"   and s' :: "'a" assumes "PU s s'" shows "QU s s'"
time_take_tm: fixes n :: "nat"   and xs :: "'a list" shows "time (take_tm n xs) = min n (length xs) + 1"
ProblemIkTpart_TEModel_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and eintT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and eintF :: "('a, 'c) efsym \<Rightarrow> univ list \<Rightarrow> univ"   and eintP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" shows "ProblemIkTpart_TEModel wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw eintT eintF eintP \<equiv> ProblemIkTpart wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw \<and> CM.Model (ProblemIkTpart.TE_wtFsym wtFsym resOf) wtPsym (ProblemIkTpart.TE_arOf arOf) (ProblemIkTpart.TE_resOf resOf) parOf (ProblemIkTpart.tPB wtFsym arOf resOf \<Phi> prot protFw) eintT eintF eintP"
h2: shows "MPoly (higherPoly\<^sub>0 w z (mapping_of a + mapping_of b)) = MPoly (higherPoly\<^sub>0 w z (mapping_of a)) + MPoly (higherPoly\<^sub>0 w z (mapping_of b))"
1: fixes ars :: "String.literal \<Rightarrow> String.literal set list option"   and ss :: "String.literal set list"   and s :: "String.literal set" assumes "ars \<in> ran (map_of (map (apsnd map_of) arss))"   and "ss \<in> ran ars"   and "s \<in> set ss" shows "wf_sort (set cs) s"
a_mul_d: fixes x :: "'a" shows "ad x \<cdot> ans_d x = (zero_class.zero::'a)"
simps_3: fixes x :: "'a" shows "extB.Val x + extB.Bot = extB.Val x"
simps_1: shows "eval_ternary_Not TernaryTrue = TernaryFalse"
subsums_imp_feasible: fixes ls :: "('d, 'e) label list"   and c1 :: "('d, 'e) conf"   and c2 :: "('d, 'e) conf" assumes "finite_labels ls"   and "finite (pred c1)"   and "finite (pred c2)"   and "\<forall>e\<in>pred c1. finite (Bexp.vars e)"   and "\<forall>e\<in>pred c2. finite (Bexp.vars e)"   and "c2 \<sqsubseteq> c1"   and "feasible c2 ls" shows "feasible c1 ls"
l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_dom_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog" shows "l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms to_tree_order get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node \<equiv> get_dom_component = l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_dom_component get_root_node to_tree_order \<and> is_strongly_dom_component_safe = l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_strongly_dom_component_safe get_root_node to_tree_order \<and> is_weakly_dom_component_safe = l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_weakly_dom_component_safe get_root_node to_tree_order"
subst_restrict': fixes t :: "'a"   and env :: "(name, 'a) fmap" shows "subst t (fmrestrict_fset (frees t) env) = subst t env"
single_diff: fixes k :: "'a"   and a :: "'b"   and b :: "'b" shows "Poly_Mapping.single k (a - b) = Poly_Mapping.single k a - Poly_Mapping.single k b"
False: shows "\<not> (0 \<le> x \<or> x \<le> - 1)"
lcm_proj2_if_dvd_int: fixes x :: "int"   and y :: "int" assumes "x dvd y" shows "lcm x y = \<bar>y\<bar>"
that: fixes g :: "'a \<Rightarrow> complex" assumes "continuous_on S g"   and "\<And>x. x \<in> S \<Longrightarrow> f x = exp (g x)" shows "thesis"
ltl_ldrop: fixes n :: "enat"   and xs :: "'c llist" shows "ltl (ldrop n xs) = ldrop n (ltl xs)"
is_min2_min2_insI: fixes y :: "'d \<times> 'e"   and ys :: "('d \<times> 'e) set"   and x :: "'d \<times> 'e" assumes "is_min2 y ys" shows "is_min2 (min2 x y) (insert x ys)"
OCC2: shows "snd ` TI \<subseteq> OCC"
simps_4: fixes f1 :: "'a"   and f2 :: "'a" shows "(case EV_SIGNAL_FINISH of EV_SIGNAL_PREP \<Rightarrow> f1 | EV_SIGNAL_FINISH \<Rightarrow> f2) = f2"
lmap_lappend_distrib: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b llist"   and ys :: "'b llist" shows "lmap f (lappend xs ys) = lappend (lmap f xs) (lmap f ys)"
arrow_of_tabulations_in_maps_axioms: shows "arrow_of_tabulations_in_maps (\<cdot>) (\<star>) \<a> \<i> src trg s \<sigma> s\<^sub>0 s\<^sub>1 u \<mu> u\<^sub>0 u\<^sub>1 \<chi>"
pos_ord_irrefl: shows "\<forall>x y. irrefl (pos_ord x y)"