hcju/isabelleps huggingface dataset:
lemma2_6_1_list: fixes r :: "('b \<times> 'b) set"   and \<sigma> :: "'b list"   and \<tau> :: "'b list" shows "r \<down>l (\<sigma> @ \<tau>) = r \<down>l \<sigma> \<union> r \<down>l \<tau>"
relpowp_add: fixes m :: "nat"   and n :: "nat"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "P ^^ (m + n) = P ^^ m OO P ^^ n"
simps_4: shows "merge_cstatus CFOUND CSUCCESS = CFOUND"
between_implies_scaled_diff: fixes S :: "'b"   and T :: "'b"   and X :: "'b"   and Y :: "'b" assumes "between (S, T) X"   and "between (S, T) Y"   and "S \<noteq> Y" obtains c :: "real" where "X - Y = c *\<^sub>R (S - Y)"
of_nat_ceiling: fixes r :: "'b" shows "r \<le> of_nat (nat \<lceil>r\<rceil>)"
exp_1: fixes a :: "'a" shows "a \<up> Suc 0 = [a]"
Src_Trg: fixes t :: "'a E.term" assumes "E.Arr t" shows "E.Src (E.Trg t) = E.Trg t"
rel_nabla_def: fixes X :: "('a \<times> 'a) set" shows "rel_nabla X = \<Union> {P. P \<subseteq> relfdia X P}"
exp_cf2_def: shows "exp_cf2 \<equiv> \<lambda>x. (x\<^sup>2 + 6 * x + 12) / (x\<^sup>2 - 6 * x + 12)"
delay_bisimulation_obs_axioms: shows "delay_bisimulation_obs trsys1 trsys2 bisim tlsim (\<lambda>s tl s'. False) (\<lambda>s tl s'. False)"
nb\<^sub>1: shows "(\<phi>\<^sub>S \<Psi> s) ++ map_of (effect_to_assignments op') = \<phi>\<^sub>S \<Psi> s ++ map_of (effect_of (\<phi>\<^sub>O\<inverse> \<Psi> op'))"
br'_rcm_aux3: fixes Q :: "'d set"   and W :: "'d set"   and rcm :: "('d, 'e) ta_rule \<Rightarrow> nat option"   and \<delta> :: "('d, 'e) ta_rule set"   and q :: "'d" assumes "(Q, W, rcm) \<in> br'_invar \<delta>"   and "q \<in> W" shows "br_dsq \<delta> q (Q, W) - Q = {lhs r |r. r \<in> \<delta> \<and> q \<in> set (rhsq r) \<and> the (rcm r) \<le> Suc 0} - Q"
Nil: fixes M\<^sub>0 :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" shows "sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> []"
p_def: shows "p = [:- 1 / 6, - 1 / 2, - 1, 1:]"
simps: fixes a :: "'b llist" shows "lfinite a = (a = LNil \<or> (\<exists>x xs. a = LCons x xs \<and> lfinite xs))"
multiplicity_normalize_right: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "factorial_semiring.multiplicity times one p (normalize x) = factorial_semiring.multiplicity times one p x"
isos_compose: fixes f :: "'a \<times> 'a"   and f' :: "'a \<times> 'a" assumes "VV.iso f"   and "VV.iso f'"   and "VV.seq f' f" shows "VV.iso (VV.comp f' f)"
prime_divisors_induct: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "P (0::'a)"   and "\<And>x. is_unit x \<Longrightarrow> P x"   and "\<And>p x. \<lbrakk>prime p; P x\<rbrakk> \<Longrightarrow> P (p * x)" shows "P x"
mset_le_asym: fixes M :: "'b multiset"   and N :: "'b multiset"   and P :: "bool" assumes "M < N"   and "\<not> P \<Longrightarrow> N < M" shows "P"
defNode_eq: fixes n :: "'node"   and g :: "'g"   and v :: "'val" assumes "n \<in> set (\<alpha>n g)"   and "v \<in> allDefs g n" shows "defNode g v = n"
interaction_any_bounded_converter_start_2: fixes "consider" :: "'h \<Rightarrow> bool"   and gpv :: "('i, 'h, 'j) gpv"   and n :: "enat"   and m :: "enat" assumes "interaction_bounded_by consider gpv n"   and "n \<le> m" shows "interaction_bounded_by consider gpv m"
scast_ucast_high_bits: shows "(SCAST('c \<rightarrow> 'a) (UCAST('a \<rightarrow> 'c) w) = w) = (\<forall>i\<in>{LENGTH('c)..<size w}. w !! i = w !! (LENGTH('c) - 1))"
fact_le_power: fixes n :: "nat" shows "fact n \<le> of_nat (n ^ n)"
binary_code_def: fixes u\<^sub>0 :: "'b list"   and u\<^sub>1 :: "'b list" shows "binary_code u\<^sub>0 u\<^sub>1 \<equiv> u\<^sub>0 \<cdot> u\<^sub>1 \<noteq> u\<^sub>1 \<cdot> u\<^sub>0"
bunch_of_lemmata_about_matches_1: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and m1 :: "'b match_expr"   and m2 :: "'b match_expr"   and a :: "action"   and p :: "'c" shows "matches \<gamma> (MatchAnd m1 m2) a p = (matches \<gamma> m1 a p \<and> matches \<gamma> m2 a p)"
linear_diff: fixes f :: "('c, 'd) vec \<Rightarrow> ('c, 'e) vec"   and x :: "('c, 'd) vec"   and y :: "('c, 'd) vec" assumes "Vector_Spaces.linear (*s) (*s) f" shows "f (x - y) = f x - f y"
create_graph_nofail: fixes \<phi> :: "'b ltlr" shows "nofail (create_graph \<phi>)"
antisym: fixes b :: "'c"   and a :: "'c" assumes "b \<le> a"   and "a \<le> b" shows "a = b"
monoE: fixes g :: "'b"   and f :: "'b"   and f' :: "'b" assumes "\<phi>.B.mono g"   and "\<phi>.B.seq g f"   and "\<phi>.B.seq g f'"   and "B g f = B g f'" shows "f' = f"
isl_map_sum: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and s :: "'c + 'd" shows "isl (map_sum f g s) = isl s"
eq_iff: fixes x :: "'b multiset"   and y :: "'b multiset" shows "(x = y) = (x \<subseteq># y \<and> y \<subseteq># x)"
nn_integral_0_iff_AE: fixes u :: "'d \<Rightarrow> ennreal"   and M :: "'d measure" assumes "u \<in> borel_measurable M" shows "(integral\<^sup>N M u = 0) = (AE x in M. u x = 0)"
WT_map_converter_idD: fixes \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('j, 'k) \<I>"   and f :: "'h \<Rightarrow> 'j"   and g :: "'k \<Rightarrow> 'i" assumes "\<I>, \<I>' \<turnstile>\<^sub>C map_converter id id f g 1\<^sub>C \<surd>" shows "\<I> \<le> map_\<I> f g \<I>'"
inf1D1: fixes A :: "'c \<Rightarrow> bool"   and B :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "inf A B x" shows "A x"
asym: fixes b :: "'a"   and a :: "'a" assumes "b < a"   and "a < b" shows "False"
adopt_node_writes: fixes h :: "(_) heap"   and node :: "(_) node_ptr"   and document_ptr :: "(_) document_ptr"   and h' :: "(_) heap" shows "writes (adopt_node_locs |h \<turnstile> get_parent node|\<^sub>r |h \<turnstile> get_owner_document (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)|\<^sub>r document_ptr) (adopt_node document_ptr node) h h'"
write_actionsI: fixes a :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist" assumes "a \<in> actions E"   and "is_write_action (action_obs E a)" shows "a \<in> write_actions E"
case_RESUMPTION_def: fixes t :: "'h \<Rightarrow> 'j"   and l :: "'g \<Rightarrow> ('f + 'g + 'h, 'i) Old_Datatype.node set \<Rightarrow> 'j"   and br :: "'f \<Rightarrow> ('i \<Rightarrow> ('f + 'g + 'h, 'i) Old_Datatype.node set) \<Rightarrow> 'j" shows "case_RESUMPTION t l br = Old_Datatype.Case (t \<circ> inv (Old_Datatype.Leaf \<circ> Inr \<circ> Inr)) (Old_Datatype.Case (Old_Datatype.Split (\<lambda>x. l (inv (Old_Datatype.Leaf \<circ> Inr \<circ> Inl) x))) (Old_Datatype.Split (\<lambda>x. DTBranch (br (inv (Old_Datatype.Leaf \<circ> Inl) x)))))"
domains_def: fixes f :: "'c" shows "C.domains f \<equiv> {a. C.ide a \<and> C f a \<noteq> C.null}"
1: shows "ladder_\<alpha> \<alpha> D L 1 = Derive \<alpha> (take (ladder_n L 0) D)"
e_inverse: fixes e :: "'a \<rightarrow> 'b"   and p :: "'b \<rightarrow> 'a"   and x :: "'a" assumes "ep_pair e p" shows "p\<cdot>(e\<cdot>x) = x"
equal_def: fixes equal :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "equal equal \<equiv> equal = (=)"
simps_2: fixes M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list" shows "\<lbrakk>M; send\<langle>t\<rangle>\<^sub>s\<^sub>t # S\<rbrakk>\<^sub>d' = (\<lambda>\<I>. M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile> t \<cdot> \<I> \<and> \<lbrakk>M; S\<rbrakk>\<^sub>d' \<I>)"
assms_5: fixes n :: "'node"   and v :: "'var" assumes "(n, v) \<in> Mapping.keys p" shows "length (predecessors g n) \<noteq> 1"
freshs_floatariths_def: fixes fas :: "floatarith list"   and xs :: "nat list" shows "freshs_floatariths fas xs = (\<forall>fa\<in>set fas. freshs_floatarith fa xs)"
sndI: fixes x :: "'c \<times> 'd"   and y :: "'c"   and z :: "'d" assumes "x = (y, z)" shows "snd x = z"
LT1: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> F \<leadsto> F"
inorder_preserve_set: fixes t :: "interval_tree" shows "tree_set t = set (in_traverse t)"
lg_lemma': fixes x :: "nat"   and y :: "nat" assumes "Suc 0 < x"   and "Suc 0 < y" shows "rec_exec rec_lg [x, y] = lg x y"
flow_initial_time: fixes t0 :: "real"   and x0a :: "'a"   and x0 :: "'a \<Rightarrow>\<^sub>L 'a" assumes "t0 \<in> Y.existence_ivl0 x0a"   and "x0 \<in> UNIV" shows "Y.mvar.flow x0a t0 x0 t0 = x0"
le_lt_convs_1: fixes comp :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b"   and y :: "'b" assumes "comparator comp" shows "(comp x y \<noteq> Lt) = le_of_comp comp y x"
mat0_col: fixes i :: "nat"   and nc :: "nat"   and ze :: "'c"   and nr :: "nat" assumes "i < nc" shows "col (mat0I ze nr nc) i = vec0I ze nr"
simps_1: fixes i :: "nat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and P :: "'b mat"   and A :: "'b mat"   and Q :: "'b mat" shows "diagonal_to_Smith_i_PQ_JNF [] i bezout (P, A, Q) = (P, A, Q)"
alt_group_one: fixes n :: "nat" shows "\<one>\<^bsub>alt_group n\<^esub> = id"
set_drop_subset: fixes n :: "nat"   and xs :: "'g list" shows "set (drop n xs) \<subseteq> set xs"
elims: fixes x :: "'d decision decision"   and y :: "'d decision" assumes "flat_2 x = y" obtains ya :: "'d" where "x = allow (allow ya__)" and "y = allow ya__"   | ya :: "'d" where "x = allow (deny yaa__)" and "y = deny yaa__"   | ya :: "'d" where "x = deny (allow yab__)" and "y = allow yab__"   | ya :: "'d" where "x = deny (deny ya)" and "y = deny ya"
coeff_Poly_eq: fixes xs :: "'a list" shows "poly.coeff (Poly xs) = nth_default (0::'a) xs"
assms_7: fixes p :: "'s_core spmf"   and q :: "'s_core' spmf"   and iu :: "'iusr_core"   and s1 :: "'s_core"   and s2 :: "'s_core'"   and ou :: "'ousr_core"   and s1' :: "'s_core"   and s2' :: "'s_core'" assumes "S p q"   and "iu \<in> IU"   and "s1 \<in> set_spmf p"   and "s2 \<in> set_spmf q"   and "(ou, s1') \<in> set_spmf (cfunc_usr core1 s1 iu)"   and "(ou, s2') \<in> set_spmf (cfunc_usr core2 s2 iu)" shows "trace_eq_simcl S (cond_spmf_fst (p \<bind> (\<lambda>s1. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (q \<bind> (\<lambda>s2. cfunc_usr core2 s2 iu)) ou)"
ide_char: fixes a :: "'a" shows "Fop.Bop.ide a = Cop.ide a"
preciseI: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn" assumes "\<And>a a' h p F F'. h \<Turnstile> R a p * F \<and>\<^sub>A R a' p * F' \<Longrightarrow> a = a'" shows "precise R"
incl_Rel_components_3: fixes A :: "V"   and B :: "V" shows "incl_Rel A B\<lparr>ArrCod\<rparr> = B"
union_grouped_by_fst: fixes ys :: "('b \<times> 'c) list" shows "\<Union> (sset (grouped_by_fst ys)) = set ys"
simps_1: 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 x :: "Preliminaries.var" assumes "ProblemIk wtFsym wtPsym arOf resOf parOf \<Phi> infTp" shows "ProblemIk.nvT (Var x) = {x}"
true_cls_mset_singleton: fixes I :: "'a set"   and C :: "'a literal multiset" shows "(I \<Turnstile>m {#C#}) = (I \<Turnstile> C)"
impl_of_empty: shows "impl_of RBT.empty = rbt.Empty"
not_None_eq: fixes x :: "'b option" shows "(x \<noteq> None) = (\<exists>y. x = Some y)"
self_codomain_iff_ide: fixes a :: "'a \<times> 'a" shows "(a \<in> VV.codomains a) = VV.ide a"
assms_4: shows "(i, Create i e1) \<in> set (node_deliver_messages xs)"
IH: assumes "v \<in> fv\<^sub>s\<^sub>t S" shows "\<exists>S' x S''. S = S' @ x # S'' \<and> v \<in> fv\<^sub>s\<^sub>t\<^sub>p x"
gcd_poly_decompose: fixes p :: "'a poly"   and q :: "'a poly" shows "gcd p q = smult (gcd (content p) (content q)) (gcd (primitive_part p) (primitive_part q))"
supd_def2: fixes f :: "st \<Rightarrow> st"   and x :: "abopt"   and s :: "st" shows "supd f (x, s) = (x, f s)"
le_sub_flip_aux: fixes n :: "nat" shows "\<forall>v v' t t'. n = val_size v + val_size v' + fun_size t + fun_size t' \<longrightarrow> (v \<sqsubseteq> v' \<longrightarrow> T v' <: T v) \<and> (t \<lesssim> t' \<longrightarrow> Tf t' <:: Tf t)"
ord_iso_the_inv: fixes f :: "'b \<Rightarrow> 'c" assumes "ord_iso f" shows "mono (the_inv f)"
cr_envT_stateTD: fixes s :: "'b"   and M :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m1 :: "('b, 'c) envT"   and m2 :: "('b, 'd) stateT" assumes "cr_envT_stateT s M m1 m2" shows "M (run_env m1 s) (run_state m2 s)"
ideal_generated_pair_exists: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and d :: "'b" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "ring.ideal_generated plus zero uminus times {a, b} = ring.ideal_generated plus zero uminus times {d}" shows "\<exists>p q. plus (times p a) (times q b) = d"
with_proj_simps_1: fixes G :: "'a pair_pre_digraph" shows "verts (with_proj G) = pverts G"
rep_eq: fixes x :: "i"   and xa :: "\<o>" shows "[xa in x] = eval\<o> xa dj x"
eq_imp_rtranclp: fixes x :: "'c"   and y :: "'c"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "x = y" shows "r\<^sup>*\<^sup>* x y"
Sym: fixes R :: "'a language \<Rightarrow> 'a language \<Rightarrow> bool"   and x :: "'a language"   and y :: "'a language" assumes "Plus_cong R x y" shows "Plus_cong R y x"
simps_3: fixes cmp :: "'b comparator"   and x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "Sorting_Algorithms.merge cmp (x # xs) (y # ys) = (if compare cmp x y = Greater then y # Sorting_Algorithms.merge cmp (x # xs) ys else x # Sorting_Algorithms.merge cmp xs (y # ys))"
ineq_3: shows "Suc l < (if Suc l = mm then Suc j else Suc (Suc l))"
simps_3: fixes v :: "bool"   and va :: "bool list" shows "v # va \<preceq>\<^sub>b [] = False"
Vinv1_event_pres: fixes r :: "nat"   and v_f :: "process \<Rightarrow> val option"   and d_f :: "process \<Rightarrow> val option" shows "{Vinv1} v_round r v_f d_f {> Vinv1}"
v11_app_vconverse_app: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vrat_inverse" shows "(vrat_inverse\<inverse>\<^sub>\<circ>\<lparr>a\<rparr>)\<inverse>\<^sub>\<rat> = a"
sup_continuous_mult_right_ennreal: fixes f :: "'a \<Rightarrow> ennreal"   and c :: "ennreal" assumes "sup_continuous f" shows "sup_continuous (\<lambda>x. f x * c)"
splitAtRec_distinct_snd: fixes vs :: "'c list"   and s :: "'c list"   and ram1 :: "'c" assumes "distinct vs"   and "distinct s"   and "set s \<inter> set vs = {}" shows "distinct (snd (splitAtRec ram1 s vs))"
minus_rat: fixes a :: "int"   and b :: "int" shows "- Fract a b = Fract (- a) b"
prefix_proj_2: fixes A :: "('a strand_label \<times> 'b) list"   and B :: "('a strand_label \<times> 'b) list"   and n :: "'a" assumes "prefix A B" shows "prefix (proj n A) (proj n B)"
cdf_nonneg: fixes x :: "real" shows "0 \<le> M.C x"
init_sim1: fixes a0 :: "('c, 'ls, 'gs) global_config" assumes "a0 \<in> init1" shows "\<exists>b0. b0 \<in> init2 \<and> rel_gc a0 b0"
map_upd_Some_expand: fixes tab :: "'b \<Rightarrow> 'a option"   and vn :: "'b"   and z :: "'a"   and x :: "'b"   and y :: "'a" assumes "tab vn = Some z" shows "\<exists>z. (tab(x \<mapsto> y)) vn = Some z"
assms: fixes x :: "'a" assumes "x \<in> set (dropWhile P xs)" shows "\<not> P x"
\<Upsilon>_l: fixes p :: "fm" assumes "isrlfm p" shows "\<forall>(t, k)\<in>set (\<Upsilon> p). numbound0 t \<and> 0 < k"
iso_is_image_block: shows "\<B>' = blocks_image \<B> \<pi>"
simps_1: fixes f :: "'d \<Rightarrow> 'c" shows "bt_map f Lf = Lf"
edges_of_walk_empty_1: shows "H.edges_of_walk [] = {}"
is_minimal_basis_def: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "is_minimal_basis B = (0 \<notin> B \<and> (\<forall>p q. p \<in> B \<longrightarrow> q \<in> B \<longrightarrow> p \<noteq> q \<longrightarrow> \<not> lt p adds\<^sub>t lt q))"
sqrt_tc: fixes x :: "int" shows "[0 \<le> x]   r := 0; WHILE (r + 1) * (r + 1) \<le> x INV {r * r \<le> x}  VAR {\<lambda>r. nat (x - r)}  DO r := r + 1 OD   [r * r \<le> x \<and> x < (r + 1) * (r + 1)]"
monoid_add_ow_axioms: shows "monoid_add_ow U (+\<^sub>o\<^sub>w) 0\<^sub>o\<^sub>w"
set_attribute_is_l_set_attribute: shows "l_set_attribute DocumentClass.type_wf set_attribute set_attribute_locs"
semiring_normalization_rules_21: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "plus (plus a b) c = plus a (plus b c)"
diff: fixes x :: "'b"   and y :: "'b" shows "f (x - y) = f x - f y"
n_val_n_val: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "n_val K (n_val K v) = n_val K v"
UNIV: fixes S :: "'a set set" shows "generate_topology S UNIV"
rep_eq: fixes x :: "'b ugraph" shows "edges_internal x = snd (Rep_ugraph x)"
lcons_def: fixes a :: "'c \<times> 'd"   and t :: "('c, 'd) FingerTreeStruc" shows "a \<lhd> t = nlcons (Tip (fst a) (snd a)) t"
funpow_in_Chains: shows "{(f ^^ k) a |k. True} \<in> Chains leq"
inject_2: fixes x31 :: "com"   and x32 :: "com"   and y31 :: "com"   and y32 :: "com" shows "(x31;; x32 = y31;; y32) = (x31 = y31 \<and> x32 = y32)"
eq_\<rho>: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and U :: "'c set"   and V :: "'c set"   and W :: "'c set"   and s :: "'d"   and t :: "'d"   and W' :: "'c set" assumes "presheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"   and "is_open U"   and "is_open V"   and "is_open W"   and "W \<subseteq> U \<inter> V"   and "s \<in> \<FF> U"   and "t \<in> \<FF> V"   and "\<rho> U W s = \<rho> V W t"   and "is_open W'"   and "W' \<subseteq> W" shows "\<rho> U W' s = \<rho> V W' t"
comp_char: fixes g :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "g \<cdot> f = (if seq g f then MkArr (Dom f) (Cod g) (compose (Dom f) (Map g) (Map f)) else null)"
star_back_loop_is_least_fixpoint: fixes y :: "'a"   and z :: "'a" shows "is_least_fixpoint (\<lambda>x. x * y \<squnion> z) (z * y\<^sup>\<star>)"
eval_ge_len: fixes P :: "'v"   and p :: "nat" assumes "len P \<le> p" shows "\<not> eval P p"
is_type_int: fixes P :: "'b prog" shows "is_type P Integer = True"
map_steps: fixes g :: "'a \<Rightarrow> 'b"   and vs :: "'a \<Rightarrow> nat"   and ws :: "'b \<Rightarrow> nat"   and s :: "'c"   and t :: "'c"   and vs' :: "'a \<Rightarrow> nat"   and M :: "('c \<times> ('c, 'a) Op) set"   and f :: "'c \<Rightarrow> 'd" assumes "inj g"   and "vs = ws \<circ> g"   and "((s, vs), t, vs') \<in> (step M)\<^sup>*" shows "((f s, ws), f t, \<lambda>x. if x \<in> range g then vs' (inv g x) else ws x) \<in> (step (map_minsky f g M))\<^sup>*"
frrestriction_fconst_on: fixes c :: "V"   and B :: "V"   and A :: "V" assumes "c \<in>\<^sub>\<circ> B" shows "fconst_on A c \<restriction>\<^sup>r\<^sub>\<bullet> B = fconst_on A c"
Suc_4: shows "\<forall>j\<in>{k..<l}. v \<notin> local.writes (path \<sigma> j)"
coupling_init: fixes s :: "'a" shows "D.coupling (Dijkstra.initQ s) D.initV (Dijkstra.initD s) D.initS"
no_small_separations: fixes S :: "'a set" assumes "Separation G v0 v1 S" shows "Suc sep_size \<le> card S"
abelian_monoidI: fixes R :: "('c, 'd) ring_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"   and "\<zero>\<^bsub>R\<^esub> \<in> carrier 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 \<oplus>\<^bsub>R\<^esub> z = x \<oplus>\<^bsub>R\<^esub> (y \<oplus>\<^bsub>R\<^esub> z)"   and "\<And>x. x \<in> carrier R \<Longrightarrow> \<zero>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> x = x"   and "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> x \<oplus>\<^bsub>R\<^esub> y = y \<oplus>\<^bsub>R\<^esub> x" shows "abelian_monoid R"
inverse_real_alg: fixes x :: "real_alg" shows "inverse (real_of x) = real_of (inverse x)"
ins0_neq_Unbal: fixes t :: "'a tree"   and n :: "nat"   and a :: "'a"   and t' :: "'a tree" assumes "size t \<le> n" shows "ins n 0 a t \<noteq> Unbal t'"
sturm_aux_rat: fixes p :: "rat poly"   and q :: "rat poly" shows "sturm_aux (real_of_rat_poly p) (real_of_rat_poly q) = map real_of_rat_poly (sturm_aux_rat p q)"
IH: shows "stake n ` streams UNIV = gener_bool_list n"
interior_lessThanAtMost: fixes a :: "real"   and b :: "real" shows "interior {a<..b} = {a<..<b}"
emeasure_PiM_emb: fixes J :: "'i set"   and X :: "'i \<Rightarrow> 'b set" assumes "J \<subseteq> I"   and "finite J"   and "\<And>i. i \<in> J \<Longrightarrow> X i \<in> D'.events i" shows "emeasure (Pi\<^sub>M I (\<lambda>i. distr M (M' i) (X i))) (P.emb I J (Pi\<^sub>E J X)) = (\<Prod>i\<in>J. emeasure (distr M (M' i) (X i)) (X i))"
sum_conjugate: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite X" shows "conjugate (sum f X) = (\<Sum>x\<in>X. conjugate (f x))"
lQuot_rQuot: fixes a :: "'c"   and b :: "'c"   and A :: "'c list set" shows "lQuot a (rQuot b A) = rQuot b (lQuot a A)"
assms_4: fixes x :: "'b" assumes "x \<in> space S" shows "0 \<le> Px x"
elims: fixes x :: "'c monom \<times> 'd"   and xa :: "('c monom \<times> 'd) list"   and y :: "('c monom \<times> 'd) list" assumes "monom_mult_poly x xa = y" obtains uu_ :: "'c monom \<times> 'd" where "x = uu_" and "xa = []" and "y = []"   | m :: "'c monom" and c :: "'d" and m' :: "'c monom" and d :: "'d" and p :: "('c monom \<times> 'd) list" where "x = (m, c)" and "xa = (m', d) # p" and "y = (if c * d = (0::'d) then monom_mult_poly (m, c) p else (m * m', c * d) # monom_mult_poly (m, c) p)"
distinct_disc_7: fixes event :: "('a, 'b, 'c) event" assumes "isTrans event" shows "\<not> isRecvMarker event"
Discrete_sqrt_square_is_square: fixes n :: "nat" assumes "is_square n" shows "(Discrete.sqrt n)\<^sup>2 = n"
A_2: shows "\<forall>k'\<le>n. \<one> \<le> fw m n k n n k' k'"
hom_grading_fun_def: fixes d :: "'b \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> 'b" shows "hom_grading_fun d f = (\<forall>n. (\<forall>s t. f n (s + t) = f n s + f n t) \<and> (\<forall>t. d (f n t) \<le> n \<and> (d t \<le> n \<longrightarrow> f n t = t)))"
wf\<^sub>l\<^sub>s\<^sub>t\<^sub>s'_def: fixes \<S> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list set"   and \<A> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list" shows "wf\<^sub>l\<^sub>s\<^sub>t\<^sub>s' \<S> \<A> \<equiv> (\<forall>\<A>'\<in>\<S>. wf\<^sub>l\<^sub>s\<^sub>t (wfrestrictedvars\<^sub>l\<^sub>s\<^sub>t \<A>) \<A>') \<and> (\<forall>\<A>'\<in>\<S>. \<forall>\<A>''\<in>\<S>. fv\<^sub>l\<^sub>s\<^sub>t \<A>' \<inter> bvars\<^sub>l\<^sub>s\<^sub>t \<A>'' = {}) \<and> (\<forall>\<A>'\<in>\<S>. fv\<^sub>l\<^sub>s\<^sub>t \<A>' \<inter> bvars\<^sub>l\<^sub>s\<^sub>t \<A> = {}) \<and> (\<forall>\<A>'\<in>\<S>. fv\<^sub>l\<^sub>s\<^sub>t \<A> \<inter> bvars\<^sub>l\<^sub>s\<^sub>t \<A>' = {})"
extract_IPSets: fixes rs :: "'a simple_rule list" shows "set (extract_IPSets rs) = set (extract_IPSets_generic0 src rs) \<union> set (extract_IPSets_generic0 dst rs)"
trunc_simps_2: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "f \<ominus>\<^bsub>P\<^esub> Cring_Poly.truncate R f = monom P (f (deg R f)) (deg R f)"
CauchyD: fixes X :: "nat \<Rightarrow> 'e"   and e :: "real" assumes "Cauchy X"   and "0 < e" shows "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"