hcju/isabelleps huggingface dataset:
f_Exec_Stream_output_causal: fixes n :: "nat"   and xs :: "'d list"   and ys :: "'d list"   and output1 :: "'e list"   and output_fun :: "'f \<Rightarrow> 'e"   and trans_fun :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and c :: "'f"   and output2 :: "'e list" assumes "xs \<down> n = ys \<down> n"   and "output1 = map output_fun (f_Exec_Comp_Stream trans_fun xs c)"   and "output2 = map output_fun (f_Exec_Comp_Stream trans_fun ys c)" shows "output1 \<down> n = output2 \<down> n"
db_defs_5: fixes ic :: "state \<Rightarrow> 'b"   and q :: "state \<Rightarrow> 'b list"   and oc :: "state \<Rightarrow> 'b" shows "ISpec ic q oc \<equiv> BInit ic q oc \<and> \<box>[Nxt ic q oc]_(ic, q, oc) \<and> WF(Deq ic q oc)_(ic, q, oc)"
decr_grading_p_plus: fixes d :: "'a \<Rightarrow> nat"   and n :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'd"   and q :: "'t \<Rightarrow>\<^sub>0 'd" shows "decr_grading_p d n (p + q) = decr_grading_p d n p + decr_grading_p d n q"
a_meet_e: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<sqinter> e_aux a b c = a \<sqinter> (b \<squnion> c)"
M: shows "\<forall>z. M \<le> norm z \<longrightarrow> norm (\<Sum>i\<le>n. c i * z ^ i) \<le> e * norm z ^ Suc n"
filtermap_filtermap: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and F :: "'c filter" shows "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
lcoeff_gbinomial_series: fixes abort :: "bool"   and x :: "real"   and n :: "nat" shows "lcoeff (gbinomial_series abort x) n = x gchoose n"
test1_var: fixes p :: "'a"   and q :: "'a"   and x :: "'a" assumes "test p"   and "test q"   and "x \<cdot> !q = !p \<cdot> x \<cdot> !q" shows "p \<cdot> x = p \<cdot> x \<cdot> q"
degree_pdev_upd: fixes x :: "'b"   and X :: "'b pdevs"   and n :: "nat" assumes "(x = (0::'b)) = (pdevs_apply X n = (0::'b))" shows "degree (pdev_upd X n x) = degree X"
SubstFormP_unique: fixes v :: "tm"   and tm :: "tm"   and x :: "tm"   and y :: "tm"   and y' :: "tm" shows "{SubstFormP v tm x y, SubstFormP v tm x y'} \<turnstile> y' EQ y"
SkolemizedAxiomsSetMcL_axioms: shows "SkolemizedAxiomsSetMcL"
update_cases_kD: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and P :: "(nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option) \<Rightarrow> bool" assumes "(\<pi>\<^sub>2 r = 0) = (\<pi>\<^sub>3 r = unk)"   and "ip \<in> kD rt"   and "sqn rt ip < \<pi>\<^sub>2 r \<Longrightarrow> P (rt(ip \<mapsto> addpre r (\<pi>\<^sub>7 (the (rt ip)))))"   and "\<lbrakk>sqn rt ip = \<pi>\<^sub>2 r; \<pi>\<^sub>5 r < the (dhops rt ip)\<rbrakk> \<Longrightarrow> P (rt(ip \<mapsto> addpre r (\<pi>\<^sub>7 (the (rt ip)))))"   and "\<lbrakk>sqn rt ip = \<pi>\<^sub>2 r; the (flag rt ip) = Aodv_Basic.inv\<rbrakk> \<Longrightarrow> P (rt(ip \<mapsto> addpre r (\<pi>\<^sub>7 (the (rt ip)))))"   and "\<pi>\<^sub>3 r = unk \<Longrightarrow> P (rt(ip \<mapsto> (\<pi>\<^sub>2 (the (rt ip)), \<pi>\<^sub>3 r, \<pi>\<^sub>4 r, \<pi>\<^sub>5 r, \<pi>\<^sub>6 r, \<pi>\<^sub>7 (addpre r (\<pi>\<^sub>7 (the (rt ip)))))))"   and "\<lbrakk>\<pi>\<^sub>2 r \<le> sqn rt ip; \<pi>\<^sub>3 r = kno; sqn rt ip = \<pi>\<^sub>2 r \<Longrightarrow> the (dhops rt ip) \<le> \<pi>\<^sub>5 r \<and> the (flag rt ip) = val\<rbrakk> \<Longrightarrow> P (rt(ip \<mapsto> addpre (the (rt ip)) (\<pi>\<^sub>7 r)))" shows "P (update rt ip r)"
orthogonal_complement_orthoI': fixes x :: "'a"   and M :: "'a set"   and y :: "'a" assumes "x \<in> M"   and "y \<in> orthogonal_complement M" shows "is_orthogonal x y"
case_call: fixes m :: "'a match_expr"   and y :: "char list" assumes "r = Rule m (Call y)" shows "\<exists>t. \<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m (Call y)], Undecided\<rangle> \<Rightarrow> t"
IFmap_unique: fixes u :: "'d IF1 \<Rightarrow> 'e IF1"   and f :: "'d \<Rightarrow> 'e"   and v :: "'d IF2 \<Rightarrow> 'e IF2" assumes "u \<circ> ctor1 = ctor1 \<circ> F1map f u v"   and "v \<circ> ctor2 = ctor2 \<circ> F2map f u v" shows "u = IF1map f \<and> v = IF2map f"
filter_fset: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b fset" shows "fset (ffilter P xs) = Collect P \<inter> fset xs"
pp_inf_p: fixes x :: "'a" shows "- - x \<squnion> - x = top"
linear_orderable_8_implies_7: fixes x :: "'a" assumes "asymmetric (x\<^sup>+) \<and> strict_linear (x\<^sup>+)" shows "x\<^sup>\<star> \<sqinter> x\<^sup>\<star>\<^sup>T = (1::'a) \<and> linear (x\<^sup>\<star>)"
p_1: fixes i :: "nat" shows "coeff p i \<in> \<int>"
prime_factorization_normalize: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "factorial_semiring.prime_factorization zero times one normalize (normalize x) = factorial_semiring.prime_factorization zero times one normalize x"
Sup_pres_comp: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'c" assumes "Sup_pres f"   and "Sup_pres g" shows "Sup_pres (f \<circ> g)"
subterms\<^sub>s\<^sub>e\<^sub>t_mono: fixes M :: "('a, 'b) Term.term set"   and N :: "('a, 'b) Term.term set" assumes "M \<subseteq> N" shows "subterms\<^sub>s\<^sub>e\<^sub>t M \<subseteq> subterms\<^sub>s\<^sub>e\<^sub>t N"
cos_Taylor_1_lower_bound: fixes x :: "real" assumes "0 \<le> x" shows "1 - x\<^sup>2 / 2 \<le> cos x"
matrix_mul_rid: fixes A :: "(('d, 'e) vec, 'f) vec" shows "A ** mat (1::'d) = A"
prems_1: shows "z \<sharp> Lam [u].N1"
asymmetric_irreflexive_inf: fixes x :: "'a" assumes "asymmetric x" shows "irreflexive_inf x"
nprv_bexiI: fixes F :: "'fmla set"   and \<phi> :: "'fmla"   and t1 :: "'trm"   and x :: "'var"   and t :: "'trm" assumes "nprv F (subst \<phi> t1 x)"   and "nprv F (LLq t1 t)"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "t \<in> atrm"   and "t1 \<in> atrm"   and "x \<in> var"   and "x \<notin> FvarsT t" shows "nprv F (bexi x t \<phi>)"
complement_blocks_wf: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and bl :: "'b set" assumes "incidence_system point_set block_collection"   and "bl \<in># incidence_system.complement_blocks point_set block_collection" shows "bl \<subseteq> point_set"
nres_rel_def: fixes R :: "('c \<times> 'd) set" shows "\<langle>R\<rangle>nres_rel \<equiv> {(c, a). c \<le> \<Down> R a}"
faithful_functor_def: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e" shows "faithful_functor A B F \<equiv> functor A B F \<and> faithful_functor_axioms A F"
L: shows "zimage Abs_game (zexplode l) = left_options g"
l10_4: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and P' :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Reflect Bet Cong P P' A B" shows "Tarski_neutral_dimensionless.Reflect Bet Cong P' P A B"
sndI: fixes x :: "'b \<times> 'c"   and y :: "'b"   and z :: "'c" assumes "x = (y, z)" shows "snd x = z"
match_list_to_match_expr_not_has_disc: fixes disc :: "'c \<Rightarrow> bool"   and X :: "'b \<Rightarrow> 'c"   and ls :: "'b list" assumes "\<forall>a. \<not> disc (X a)" shows "\<not> has_disc disc (match_list_to_match_expr (map (Match \<circ> X) ls))"
content_decompose_fract: fixes p :: "'a fract poly" obtains c :: "'a fract" and p' :: "'a poly" where "p = smult c (map_poly to_fract p')" and "content p' = (1::'a)"
invar_tree_root_min: fixes t :: "'b tree"   and x :: "'b" assumes "invar_tree t"   and "x \<in># mset_tree t" shows "tree.root t \<le> x"
existence_ivl_initial_time_iff: fixes T :: "real set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and t0 :: "real"   and x0 :: "'b" assumes "ll_on_open_it T f X" shows "(t0 \<in> ll_on_open.existence_ivl T f X t0 x0) = (t0 \<in> T \<and> x0 \<in> X)"
dsjI1: fixes A :: "fmla"   and B :: "fmla" shows "A IMP A OR B \<in> boolean_axioms"
bind_map_pmf: fixes f :: "'d \<Rightarrow> 'c"   and M :: "'d pmf"   and g :: "'c \<Rightarrow> 'b pmf" shows "map_pmf f M \<bind> g = M \<bind> (\<lambda>x. g (f x))"
hom_boundary: shows "((\<forall>p X S c. c \<notin> carrier (relative_homology_group p X S) \<longrightarrow> hom_boundary p X S c = \<one>\<^bsub>homology_group (p - 1) (subtopology X S)\<^esub>) \<and> (\<forall>p X S. hom_boundary p X S \<in> hom (relative_homology_group p X S) (homology_group (p - 1) (subtopology X S))) \<and> (\<forall>p X S c. singular_relcycle p X S c \<and> 1 \<le> p \<longrightarrow> hom_boundary (int p) X S (homologous_rel_set p X S c) = homologous_rel_set (p - Suc 0) (subtopology X S) {} (chain_boundary p c)) \<and> (\<forall>p X S. hom_boundary p X S = hom_boundary p X (topspace X \<inter> S))) \<and> (\<forall>p X S c. hom_boundary p X S c \<in> carrier (homology_group (p - 1) (subtopology X S))) \<and> (\<forall>p\<le>0. hom_boundary p = (\<lambda>q r t. undefined))"
compow_id_point: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and c :: "nat" assumes "f x = x" shows "(f ^^ c) x = x"
action_dom_def: fixes s1 :: "('b, 'c) fmap"   and s2 :: "('b, 'd) fmap" shows "action_dom s1 s2 \<equiv> fmdom' s1 \<union> fmdom' s2"
sin_theta_div_2: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool" assumes "grover_state n f" shows "sin (grover_state.\<theta> n f / 2) = sqrt (real (grover_state.M n f) / real (grover_state.N n))"
match_dom: fixes p :: "term"   and t :: "'a"   and env :: "(name, 'a) fmap" assumes "match p t = Some env" shows "fmdom env = frees p"
cr_fmap_def: shows "cr_fmap \<equiv> \<lambda>x y. x = fmlookup y"
split_max_bal: fixes t :: "'a btree"   and k :: "nat" assumes "bal t"   and "t \<noteq> Leaf"   and "nonempty_lasttreebal t" shows "bal (fst (split_max k t))"
has_integral_null: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "content (cbox a b) = 0" shows "(f has_integral (0::'b)) (cbox a b)"
rule_states_simp: fixes x :: "('a, 'b) ta_rule" shows "rule_states x = (case x of q \<rightarrow> l qs \<Rightarrow> insert q (set qs))"
median_in_set: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "median xs \<in> set xs"
rpCons: fixes b :: "'b"   and q :: "'b poly" assumes "b \<noteq> (0::'b) \<or> q \<noteq> 0" shows "P 0 (pCons b q)"
cospan_\<nu>\<pi>: shows "C.cospan \<nu>.dom.leg0 \<pi>.dom.leg1"
construct_def: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and B :: "'e set"   and g :: "'e \<Rightarrow> 'f"   and v :: "'e" assumes "vector_space_pair s1 s2" shows "vector_space_pair.construct s1 s2 B g v = (\<Sum>b | module.representation s1 (vector_space.extend_basis s1 B) v b \<noteq> (0::'d). s2 (module.representation s1 (vector_space.extend_basis s1 B) v b) (if b \<in> B then g b else (0::'f)))"
basic: shows "\<forall>p q v a. from p to q = a ** v \<longrightarrow> from q to p = (- a) ** v"
that: fixes y :: "'a" assumes "periodic_orbit y"   and "range (flow0 y) \<subseteq> K" shows "thesis"
82: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<squnion> x \<sqinter> (y \<squnion> z) = x \<sqinter> (y \<squnion> z)"
moebius_mu_Suc_0: shows "moebius_mu (Suc 0) = (1::'b)"
wt_subst_compose: fixes \<theta> :: "('fun, 'atom) Term.term \<times> nat \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and \<delta> :: "('fun, 'atom) Term.term \<times> nat \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>" shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t (\<theta> \<circ>\<^sub>s \<delta>)"
isomorphic_implies_equicomposable_1: fixes f :: "'a"   and g :: "'a"   and \<tau> :: "'a" assumes "f \<cong> g" shows "(\<tau> \<star> f \<noteq> null) = (\<tau> \<star> g \<noteq> null)"
neg_less_pos: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less (uminus a) a = less zero a"
i_take_take_eq1: fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'd" assumes "m \<le> n" shows "f \<Down> n \<down> m = f \<Down> m"
minus_set_def: fixes i :: "'e \<Rightarrow> 'e"   and A :: "'e set" shows "minus_set i A = {x. \<exists>y\<in>A. x = \<^sub>i- y}"
pdevs_applys_def: fixes X :: "'b pdevs list"   and i :: "nat" shows "pdevs_applys X i = map (\<lambda>x. pdevs_apply x i) X"
1: shows "v\<^sup>T \<le> r\<^sup>T * (t \<squnion> e)\<^sup>\<star>"
lspace_max: fixes f :: "'a \<Rightarrow> real"   and p :: "real"   and S :: "'a set"   and g :: "'a \<Rightarrow> real" assumes "f \<in> lspace (lebesgue_on S) (ennreal p)"   and "g \<in> lspace (lebesgue_on S) (ennreal p)"   and "0 < p" shows "(\<lambda>x. max (f x) (g x)) \<in> lspace (lebesgue_on S) (ennreal p)"
simps: fixes a :: "'val"   and g :: "'g"   and \<phi> :: "'val" shows "(a \<in> reachable g \<phi>) = (a = \<phi> \<and> unnecessaryPhi g \<phi> \<or> (\<exists>\<phi>'' \<phi>'''. a = \<phi>''' \<and> \<phi>'' \<in> reachable g \<phi> \<and> phiArg g \<phi>'' \<phi>''' \<and> unnecessaryPhi g \<phi>'''))"
Int_ac_3: fixes A :: "'b set"   and B :: "'b set" shows "A \<inter> B = B \<inter> A"
geq0: fixes i :: "nat" assumes "i < n" shows "0 \<le> B $$ (i, i)"
source_stability: fixes x :: "'c \<Longrightarrow> 'd"   and \<sigma> :: "'d" assumes "wb_lens x" shows "\<exists>v. put\<^bsub>x\<^esub> \<sigma> v = \<sigma>"
order_trans: fixes x :: "'e word"   and y :: "'e word"   and z :: "'e word" assumes "x \<le>s y"   and "y \<le>s z" shows "x \<le>s z"
disc_2: fixes x2 :: "'b" shows "\<not> is_Prim\<^sub>0 \<^bold>\<langle>x2\<^bold>\<rangle>"
ahm_invar_ahm_empty: shows "ArrayHashMap_Impl.ahm_invar (ArrayHashMap_Impl.ahm_empty ())"
dec_word_def: fixes s :: "('a \<times> 'b) stream" shows "dec_word s = cut_same any (smap fst s)"
exhaust: fixes y :: "('c, 'd) tm" obtains x1 :: "('c, 'd) hd" where "y = Hd x1"   | x21 :: "('c, 'd) tm" and x22 :: "('c, 'd) tm" where "y = App x21 x22"
prems_2: shows "convert_term (app t\<^sub>1 t\<^sub>2) = convert_term t'"
callee_invariant: fixes y :: "'b"   and s' :: "'s1"   and s :: "'s1"   and x :: "'a" assumes "(y, s') \<in> set_spmf (oracle1 s x)"   and "I1 s"   and "x \<in> outs_\<I> \<I>" shows "I1 s'"
set_of_mul_inc_left: fixes A :: "'b interval"   and A' :: "'b interval"   and B :: "'b interval" assumes "set_of A \<subseteq> set_of A'" shows "set_of (A * B) \<subseteq> set_of (A' * B)"
angle_action_def: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<diamond>\<langle>P\<rangle>_v \<equiv> \<not> \<box>[\<not> P]_v"
assms_2: shows "algebraic_semidom_class.coprime a c"
unique_connecting_path_unique: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and xs :: "'c list"   and w :: "'c" assumes "Tree G"   and "v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w" shows "xs = v \<leadsto>\<^bsub>G\<^esub> w"
cod_simps_1: shows "gf.cod.apex = gf.apex"
LeftDerives1_skip_suffix: fixes i :: "nat"   and a :: "symbol list"   and c :: "symbol list"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "i < length a"   and "LeftDerives1 (a @ c) i r (b @ c)" shows "LeftDerives1 a i r b"
no_lb: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.dlo less_eq less" shows "\<exists>l. less l x"
rbt_ord_props_2: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "'b"   and t :: "('b, 'c) RBT_Impl.rbt" shows "ord.rbt_greater less k t = (\<forall>x\<in>set (RBT_Impl.keys t). less k x)"
Inl_eqvt: fixes p :: "perm"   and x :: "'a" shows "p \<bullet> Inl x = Inl (p \<bullet> x)"
cod_simp: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>C.cod f = \<F>C.mkarr (\<F>C.COD f)"
simps: fixes a1 :: "'form \<times> nat list"   and a2 :: "('form, 'var) antecedent"   and a3 :: "'form \<times> nat list" shows "scope' a1 a2 a3 = (\<exists>c is' is i ip. a1 = (c, is) \<and> a2 = ip \<and> a3 = (c, is') \<and> c \<in> set conclusions \<and> is' \<in> (#) 0 ` it_paths (it' c) \<and> prefix (is @ [i]) is' \<and> ip = in_port_at (c, is) i)"
strict_g: shows "\<forall>i>M. g (Suc i) <\<cdot> g i"
compI_3: fixes CC :: "('b, 'c, 'd) category_scheme"   and f :: "'c"   and g :: "'c" assumes "category CC"   and "f \<in> Ar\<^bsub>CC\<^esub>"   and "g \<in> Ar\<^bsub>CC\<^esub>"   and "Cod\<^bsub>CC\<^esub> f = Dom\<^bsub>CC\<^esub> g" shows "Cod\<^bsub>CC\<^esub> (g \<bullet>\<^bsub>CC\<^esub> f) = Cod\<^bsub>CC\<^esub> g"
freeVarsFL_def: fixes gamma :: "formula list" shows "freeVarsFL gamma = \<Union> (freeVarsF ` set gamma)"
power_sum: fixes c :: "'a"   and f :: "'b \<Rightarrow> nat"   and A :: "'b set" shows "c ^ sum f A = (\<Prod>a\<in>A. c ^ f a)"
out_distinct: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Out Bet A B C" shows "B \<noteq> A \<and> C \<noteq> A"
wf_lex: fixes r :: "('g \<times> 'g) set" assumes "wf r" shows "wf (lex r)"
Deriv_def: fixes x :: "'a"   and A :: "'a list set" shows "Deriv x A = {xs. x # xs \<in> A}"
fun_upd_None_restrict: fixes m :: "'c \<Rightarrow> 'd option"   and D :: "'c set"   and x :: "'c" shows "(m |` D)(x := None) = (if x \<in> D then m |` (D - {x}) else m |` D)"
sqrt_inverse_power2: fixes n :: "nat" shows "Discrete.sqrt (n\<^sup>2) = n"
assms: shows "a_consistent (ae, a, as) (\<Gamma>, scrut ? e1 : e2, S)"
ord_Zp_pow: fixes x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "x \<in> nonzero Zp" shows "ord_Zp (x [^] n) = int n * ord_Zp x"
induced_left_coset_equality: fixes a :: "'a" assumes "a \<in> G" shows "induced (a \<cdot>| L) = \<eta> a"
absorb2: fixes b :: "'b word"   and a :: "'b word" assumes "b \<le>s a" shows "signed.min a b = b"
spr_simInit_def: fixes envInit :: "('b, 'e, 'f) BEState list"   and envObsC :: "'e \<Rightarrow> 'c"   and envObs :: "'b \<Rightarrow> ('b, 'e, 'f) BEState \<Rightarrow> 'c \<times> 'd" shows "SPRViewDet.spr_simInit envInit envObsC envObs \<equiv> \<lambda>a iobs. (fromList (concat (map (\<lambda>s. if envObsC (es s) = fst iobs then [(s, s)] else []) envInit)), fromList (concat (map (\<lambda>s. if envObs a s = iobs then [(s, s)] else []) envInit)))"
u_simps_5: shows "cod s\<^sub>0 = s\<^sub>0"
l0f2f1c: shows "f2 / (f2 - f1 * c) \<le> real l0"
or_eq_0_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "(or a b = zero) = (a = zero \<and> b = zero)"
card_idx_nat: shows "|UNIV| <o |UNIV|"
cexpr_sem_branch_prob: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr"   and \<rho> :: "nat \<Rightarrow> val" assumes "cdens_ctxt_invar vs vs' \<Gamma> \<delta>"   and "\<rho> \<in> space (state_measure (set vs') \<Gamma>)" shows "ennreal (extract_real (cexpr_sem \<rho> (branch_prob_cexpr (vs, vs', \<Gamma>, \<delta>)))) = branch_prob (dens_ctxt_\<alpha> (vs, vs', \<Gamma>, \<delta>)) \<rho>"
new_f21_norm: shows "normFace f21 \<notin> set (normFaces (faces g))"
wt_jvm_prog_impl_wt_start: fixes \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and ins :: "instr list"   and xt :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = (mxs, mxl\<^sub>0, ins, xt) in C" shows "0 < length ins \<and> wt_start P C Ts mxl\<^sub>0 (\<Phi> C M)"
SINK_def: shows "MFMC_Network.SINK \<equiv> \<lambda>f. {x. SINKp f x}"
le_zero_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "'b" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq n zero = (n = zero)"
assms_3: fixes i :: "'b" assumes "i \<in> I" shows "0 \<le> u i \<and> (u i = 0 \<or> f i \<in> S)"
ur: shows "\<exists>!x. root_cond (p, l, r) x"
PTrans: shows "P \<Longrightarrow>\<^sub>l \<tau> \<prec> P'"
smooth_on_imp_continuous_on: fixes k :: "enat"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "k-smooth_on S f" shows "continuous_on S f"
thetaZOSeqD_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis" shows "PL_Indis.thetaZOSeqD aval tval cval indis \<equiv> {(c1 ;; c2, d1 ;; d2) |c1 c2 d1 d2. proper c1 \<and> proper d1 \<and> proper c2 \<and> proper d2 \<and> PL_Indis.discr aval tval cval indis c2 \<and> PL_Indis.discr aval tval cval indis d2 \<and> (c1, d1) \<in> PL_Indis.ZObis aval tval cval indis}"
inj_term_simps_4: fixes es :: "expr list" shows "\<langle>es\<rangle> = In3 es"
deutsch_algo_result: shows "deutsch_algo = Tensor.mat_of_cols_list 4 (map (map complex_of_real) [[real (1 - f 0) / (2 * sqrt 2) - real (f 0) / (2 * sqrt 2) + real (1 - f 1) / (2 * sqrt 2) - real (f 1) / (2 * sqrt 2), real (f 0) / (2 * sqrt 2) - real (1 - f 0) / (2 * sqrt 2) + (real (f 1) / (2 * sqrt 2) - real (1 - f 1) / (2 * sqrt 2)), real (1 - f 0) / (2 * sqrt 2) - real (f 0) / (2 * sqrt 2) - real (1 - f 1) / (2 * sqrt 2) + real (f 1) / (2 * sqrt 2), real (f 0) / (2 * sqrt 2) - real (1 - f 0) / (2 * sqrt 2) - real (f 1) / (2 * sqrt 2) + real (1 - f 1) / (2 * sqrt 2)]])"
rep_eq: fixes x :: "nat"   and xa :: "float interval" shows "bounds_of_interval (cos_float_interval x xa) = (case bounds_of_interval xa of (lx, ux) \<Rightarrow> bnds_cos x lx ux)"
one_eq_mult_iff: fixes m :: "nat"   and n :: "nat" shows "(Suc 0 = m * n) = (m = Suc 0 \<and> n = Suc 0)"
oft_def: shows "oft = pack_OF_entries ifs (map (apfst word_of_nat) (annotate_rlen (lr_of_tran_fbs rt fw ifs)))"