hcju/isabelleps huggingface dataset:
in_obind_eq: fixes f :: "'e \<Rightarrow> 'f option"   and g :: "'f \<Rightarrow> 'e \<Rightarrow> 'd option"   and s :: "'e"   and v :: "'d" shows "((f |>> g) s = Some v) = (\<exists>v'. f s = Some v' \<and> g v' s = Some v)"
coreflexive_linorder_matrix: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_idempotent_semiring.coreflexive f = (\<forall>i j. i \<noteq> j \<longrightarrow> f (i, j) = bot)"
keys_delete: fixes k :: "'a"   and m :: "('a, 'b) mapping" shows "Mapping.keys (Mapping.delete k m) = Mapping.keys m - {k}"
C1: fixes \<Psi>' :: "'b"   and R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi"   and y :: "name" assumes "(\<Psi>', R, S) \<in> Rel"   and "y \<sharp> \<Psi>'" shows "(\<Psi>', \<lparr>\<nu>y\<rparr>R, \<lparr>\<nu>y\<rparr>S) \<in> Rel"
map2_times_proj: fixes xs :: "('a \<Rightarrow>\<^sub>0 'c) list"   and k :: "nat"   and ys :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "map2 (*) xs (map (proj_poly k) ys) = map (proj_poly k \<circ> (\<lambda>(x, y). x \<odot> y)) (zip xs ys)"
vc_sound: fixes C :: "acom"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and LVQ :: "char list set"   and LVE :: "char list set"   and upds :: "(char list \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> char list set) list" assumes "vc C Q LVQ LVE"   and "finite (support Q)"   and "lesvars upds \<inter> varacom C = {}"   and "distinct (map fst upds)"   and "finite (varacom C)"   and "\<forall>l s1 s2. s1 = s2 on LVQ \<longrightarrow> Q l s1 = Q l s2"   and "\<forall>l s1 s2. s1 = s2 on LVE \<longrightarrow> postList upds l s1 = postList upds l s2"   and "\<forall>(a, b, c)\<in>set upds. \<forall>s1 s2. s1 = s2 on c \<longrightarrow> b s1 = b s2"   and "(\<Union>(a, b, c)\<in>set upds. c) \<subseteq> LVE" shows "\<turnstile>\<^sub>1 {\<lambda>l s. pre C Q l s \<and> preList upds C l s} strip C { time C \<Down> \<lambda>l s. Q l s \<and> postList upds l s} \<and> (\<forall>l s. pre C Q l s \<longrightarrow> Q l (postQ C s)) \<and> K4 upds LVQ C Q"
alist_of_remove_min: fixes q :: "('c, 'd) pq" assumes "\<not> is_empty q" shows "alist_of (remove_min q) = tl (alist_of q)"
cases: fixes a1 :: "('g, 'h, 'x, 'i, 'j, 'k) thread_action"   and a2 :: "('g, 'h, 'l, 'i, 'j, 'k) thread_action" assumes "actions_subset a1 a2" obtains ta' :: "('g, 'h, 'x, 'i, 'j, 'k) thread_action" and ta :: "('g, 'h, 'l, 'i, 'j, 'k) thread_action" where "a1 = ta'" and "a2 = ta" and "collect_locks' \<lbrace>ta'\<rbrace>\<^bsub>l\<^esub> \<subseteq> collect_locks \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>" and "collect_cond_actions \<lbrace>ta'\<rbrace>\<^bsub>c\<^esub> \<subseteq> collect_cond_actions \<lbrace>ta\<rbrace>\<^bsub>c\<^esub>" and "collect_interrupts \<lbrace>ta'\<rbrace>\<^bsub>i\<^esub> \<subseteq> collect_interrupts \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>"
relto_funD_6: fixes A :: "('a \<times> 'a) set"   and B :: "('a \<times> 'a) set"   and n :: "nat"   and as :: "nat \<Rightarrow> 'a"   and sel :: "nat \<Rightarrow> bool"   and m :: "nat"   and a :: "'a"   and b :: "'a" assumes "relto_fun A B n as sel m (a, b)" shows "(n = 0) = (m = 0)"
simps_125: fixes x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list"   and x51 :: "expr"   and x52 :: "bop"   and x53 :: "expr" shows "Call x101 x102 x103 x104 \<noteq> x51 \<guillemotleft>x52\<guillemotright> x53"
p1: fixes t :: "'b"   and x :: "'a" shows "((\<lambda>x. (x, t)) has_derivative (\<lambda>h. (h, 0::'b))) (at x within S)"
mprodR_Suc: fixes R :: "('c, 'd) Ring_scheme"   and e :: "nat \<Rightarrow> nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'c" assumes "Ring R"   and "e \<in> {j. j \<le> Suc n} \<rightarrow> {j. 0 \<le> j}"   and "f \<in> {j. j \<le> Suc n} \<rightarrow> carrier R" shows "mprod_expR R e f (Suc n) = mprod_expR R e f n \<cdot>\<^sub>r\<^bsub>R\<^esub> f (Suc n)^\<^bsup>R e (Suc n)\<^esup>"
unit_simps_1: fixes a :: "'c" assumes "C.obj a" shows "D.arr (F.unit a)"
fam_morphisms_are_maps: fixes X :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and \<FF>' :: "'d set \<Rightarrow> 'f set"   and \<rho>' :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'f \<Rightarrow> 'f"   and b' :: "'f"   and add_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str' :: "'d set \<Rightarrow> 'f"   and one_str' :: "'d set \<Rightarrow> 'f"   and fam_morphisms :: "'d set \<Rightarrow> 'e \<Rightarrow> 'f"   and U :: "'d set" assumes "morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms"   and "is_open U" shows "Set_Theory.map (fam_morphisms U) (\<FF> U) (\<FF>' U)"
distinct_43: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "x41 ;; x42 \<noteq> Stop"
that: fixes C :: "nat \<Rightarrow> 'a set" assumes "pairwise (\<lambda>i j. disjnt (C i) (C j)) UNIV"   and "\<Union> (range C) \<subseteq> A"   and "\<And>i. infinite (C i)" shows "thesis"
that: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "t = s \<cdot> \<sigma>" shows "thesis"
pd_pd_ipd: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and x :: "'a"   and y :: "'a" assumes "IFC prob"   and "x \<in> IFC_def.nodes prob"   and "x \<noteq> IFC_def.return prob"   and "y \<noteq> x"   and "IFC_def.is_pd prob y x" shows "IFC_def.is_pd prob y (IFC_def.ipd prob x)"
minarc_at_most_one: fixes f :: "'d \<times> 'd \<Rightarrow> 'c"   and e :: "'d \<times> 'd"   and d :: "'d \<times> 'd" assumes "(minarc\<^sub>M f) e \<noteq> bot"   and "(minarc\<^sub>M f) d \<noteq> bot" shows "e = d"
good_time_def: shows "good_time = (\<lambda>x. \<Sqinter> {n. (T ^^ n) x \<in> good_set})"
invertible_matrix_inv: fixes A :: "(('d, 'e) vec, 'f) vec" assumes "invertible A" shows "invertible A\<^sup>-\<^sup>1"
idem: fixes a :: "'c multiset" shows "a \<inter># a = a"
ennreal_of_nat_eq_real_of_nat: fixes i :: "nat" shows "of_nat i = ennreal (real i)"
q63_def: shows "q63 = {0, 1, 4, 7, 9, 16, 28, 18, 22, 25, 36, 58, 46, 49, 37, 43}"
complex_inner_1_left: fixes x :: "complex" shows "1 \<bullet>\<^sub>C x = x"
fold_bind_random_permutation_foldl: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c pmf"   and x :: "'d" assumes "finite A" shows "fold_bind_random_permutation f g x A = pmf_of_set (permutations_of_set A) \<bind> (\<lambda>xs. g (foldl (\<lambda>x y. f y x) x xs))"
fake_New: fixes M :: "msg"   and IK :: "msg set"   and c :: "tag"   and A :: "agent"   and bad :: "agent set"   and B :: "agent"   and chan :: "chan set" assumes "M \<in> IK"   and "c = insec \<or> c = confid \<or> A \<in> bad \<or> B \<in> bad" shows "Chan c A B M \<in> fake bad IK chan"
globally_consistent_dom: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and dma :: "'b \<Rightarrow> Sec"   and A :: "'b \<Rightarrow> ('c \<times> 'c) option"   and mds :: "Mode \<Rightarrow> 'b set"   and X :: "'b set" assumes "sifum_security stop eval some_val some_val'"   and "sifum_security.globally_consistent dma A mds"   and "X \<subseteq> dom A" shows "sifum_security.globally_consistent dma (A |` X) mds"
add_mono: fixes a :: "'k multiset"   and b :: "'k multiset"   and c :: "'k multiset"   and d :: "'k multiset" assumes "a \<subseteq># b"   and "c \<subseteq># d" shows "a + c \<subseteq># b + d"
wset_final_okE: fixes ws :: "'d \<Rightarrow> 'e wait_set_status option"   and ts :: "'d \<Rightarrow> ('x \<times> 'f \<Rightarrow>f nat) option"   and t :: "'d"   and w :: "'e wait_set_status" assumes "wset_final_ok ws ts"   and "ws t = \<lfloor>w\<rfloor>" obtains x :: "'x" and ln :: "'f \<Rightarrow>f nat" where "ts t = \<lfloor>(x, ln)\<rfloor>" and "\<not> final x"
Rep_inject: fixes x :: "'b"   and y :: "'b" shows "(Rep x = Rep y) = (x = y)"
proots_rectangle_def: fixes p :: "complex poly"   and lb :: "complex"   and ub :: "complex" shows "proots_rectangle p lb ub = proots_count p (box lb ub)"
args_to_set_alt: fixes gs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list" shows "args_to_set (gs, bs, ps) = fst ` set gs \<union> fst ` set bs \<union> fst ` fst ` set ps \<union> fst ` snd ` set ps"
f6: shows "\<forall>x. \<not> pre x \<or> (x, xa) \<notin> V \<or> (if nofail (REC\<^sub>T body x) then REC\<^sub>T body x \<le> M x else REC\<^sub>T body x \<le> FAIL)"
distinct_12: fixes x31 :: "nat"   and x32 :: "nat" shows "recf.Id x31 x32 \<noteq> S"
writes_subset: fixes S :: "('l, 'm, 'n) prog set"   and f :: "('l, 'm, 'o) prog"   and h :: "'l"   and h' :: "'l"   and S' :: "('l, 'm, 'n) prog set" assumes "writes S f h h'"   and "S \<subseteq> S'" shows "writes S' f h h'"
simps_7: fixes e :: "nat \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a list \<Rightarrow> bool"   and p :: "('b, 'c) form" shows "eval e f g (Neg p) = (\<not> eval e f g p)"
id: shows "(tighten_bounds_3 ^^ Suc n) x = (tighten_bounds_3 ^^ n) (tighten_bounds_3 x)"
assms: shows "A \<subseteq> \<lbrakk>\<H>\<rbrakk>\<^sub>H"
diagvector_def: fixes x :: "'d \<Rightarrow> 'c" shows "diagvector x = (\<chi>i j. if i = j then x i else (0::'c))"
div_poly_eq: fixes x :: "'a poly"   and y :: "'a poly"   and q :: "'a poly"   and r :: "'a poly" assumes "eucl_rel_poly x y (q, r)" shows "x div y = q"
inverse_def: shows "sub.inverse = (\<lambda>u\<in>G. THE v. v \<in> G \<and> u \<cdot> v = \<one> \<and> v \<cdot> u = \<one>)"
rmult_0: fixes R :: "('b, 'c) ring_scheme"   and M :: "('b, 'd, 'e) module_scheme"   and r :: "'b" assumes "Module.module R M"   and "r \<in> carrier R" shows "r \<odot>\<^bsub>M\<^esub> \<zero>\<^bsub>M\<^esub> = \<zero>\<^bsub>M\<^esub>"
assms_2: shows "lose t = [True] \<longrightarrow> ack t = [x] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = x"
distinct_63: fixes x8 :: "'a" shows "UnlimitedNatural \<noteq> \<langle>x8\<rangle>\<^sub>\<T>"
subset_spec_ivls_clw_def: fixes M :: "nat"   and X :: "'c set"   and Y :: "'c set" shows "subset_spec_ivls_clw M X Y = split_along_ivls M X Y \<bind> (\<lambda>X. sets_of_coll (sets_of_ivls X) \<bind> (\<lambda>X. FORWEAK X (RETURN True) (\<lambda>X. subset_spec_ivls X Y) (\<lambda>a b. RETURN (a \<and> b))))"
simps_1: fixes e :: "(char list \<Rightarrow> int) \<Rightarrow> nat" shows "Nielson_VCGi.preT SKIP e = e"
assms_1: shows "(\<xi>, p) \<in> reachable A I"
disjoint_demonotonicity: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "P x y"   and "\<not> O z y" shows "\<not> O z x"
sqrt_sgn: fixes a :: "int" shows "sqrt (real_of_int (sgn a)) = real_of_int (sgn a)"
cross_ratio_cltn2: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2"   and s :: "proj2"   and C :: "cltn2" assumes "proj2_set_Col {p, q, r, s}"   and "p \<noteq> q"   and "r \<noteq> p"   and "s \<noteq> p" shows "cross_ratio (apply_cltn2 p C) (apply_cltn2 q C) (apply_cltn2 r C) (apply_cltn2 s C) = cross_ratio p q r s"
as_monic: fixes a :: "'a poly"   and i :: "nat" assumes "(a, i) \<in> as" shows "monic a"
gamma_smooth: shows "\<gamma> piecewise_C1_differentiable_on {0..1}"
set_category_data_axioms: shows "set_category_data (\<cdot>\<acute>)"
rel_with_target_impl_reflC_transC_TRel_is_weak_reduction_bisimulation: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "weak_reduction_bisimulation Rel (STCal Source Target)"   and "\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel"   and "\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel \<longrightarrow> (T1, T2) \<in> TRel\<^sup>*" shows "weak_reduction_bisimulation (TRel\<^sup>*) Target"
subspace_col_space: fixes A :: "(('b, 'd) vec, 'c) vec" shows "vec.subspace (col_space A)"
m_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"
implies_entails: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>s. P s \<Longrightarrow> Q s" shows "\<guillemotleft> P \<guillemotright> \<tturnstile> \<guillemotleft> Q \<guillemotright>"
eq: shows "set (sort_oalist xs) - range (Pair k) = {kv \<in> set xs. snd kv \<noteq> (0::'b)}"
set_plus_mono4: fixes a :: "'c"   and C :: "'c set"   and D :: "'c set" assumes "a \<in> C" shows "a +o D \<subseteq> D + C"
take_bit_take_bit: fixes m :: "nat"   and n :: "nat"   and a :: "'a" shows "take_bit m (take_bit n a) = take_bit (min m n) a"
7: shows "\<forall>x. r\<^sup>T * Rf v (v\<^sup>T) * e\<^sup>T * x = bot"
preserves_arr: fixes f :: "'a" assumes "R'.C.arr f" shows "R'.C.arr (H\<^sub>R a' f)"
better_sred_Abs: fixes r :: "lam"   and r' :: "lam"   and ss :: "lam list"   and ss' :: "lam list"   and x :: "name" assumes "r \<rightarrow>\<^sub>s r'"   and "ss [\<rightarrow>\<^sub>s] ss'" shows "(Lam [x].r) \<degree>\<degree> ss \<rightarrow>\<^sub>s (Lam [x].r') \<degree>\<degree> ss'"
lc\<^sub>2_def: shows "lc\<^sub>2 \<equiv> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"
h3: shows "\<turnstile> \<box>P \<and> \<box>N \<and> \<box>F \<longrightarrow> \<diamond>Enabled \<langle>A\<rangle>_v"
while_default_def: fixes s :: "'a" shows "local.while_default s = (if \<forall>k. b ((c ^^ k) s) then d else the (while_option b c s))"
discI_1: fixes sum :: "'e + 'f"   and x1 :: "'e" assumes "sum = Inl x1" shows "isl sum"
mult_le: shows "(2 ^ (LENGTH('a) - 1) + 1)\<^sup>2 \<le> 2 ^ (LENGTH('b) - 1)"
nonempty: shows "space (dens_ctxt_measure ({}, V', \<Gamma>, \<lambda>_. 1) \<rho>) \<noteq> {}"
exhaust_sel: fixes prod :: "'c \<times> 'd" obtains "prod = (fst prod, snd prod)"
indep_var_rv1: fixes S :: "'c measure"   and X :: "'b \<Rightarrow> 'c"   and T :: "'c measure"   and Y :: "'b \<Rightarrow> 'c" assumes "X.indep_var S X T Y" shows "X.random_variable S X"
simps_16: fixes \<sigma> :: "(String.literal \<times> event_data list) Trace.trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and I :: "\<I>"   and r :: "formula Regex.regex" shows "sat \<sigma> V v i (MatchP I r) = (\<exists>j\<le>i. mem (\<tau> \<sigma> i - \<tau> \<sigma> j) I \<and> Regex.match (sat \<sigma> V v) r j i)"
BlockRedNone: fixes E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and T :: "ty"   and e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" assumes "P,E(V \<mapsto> T) \<turnstile> \<langle>e,(h, l(V := None))\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>"   and "l' V = None"   and "\<not> assigned V e" shows "P,E \<turnstile> \<langle>{V:T; e},(h, l)\<rangle> \<rightarrow> \<langle>{V:T; e'},(h', l'(V := l V))\<rangle>"
pnu_unfold: fixes f :: "'a \<Rightarrow> 'a" assumes "has_greatest_postfixpoint f" shows "p\<nu> f \<le> f (p\<nu> f)"
inext_nth_card_less_Max: fixes n :: "nat"   and I :: "nat set" assumes "Suc n < card I" shows "I \<rightarrow> n < Max I"
zle_int: fixes m :: "nat"   and n :: "nat" shows "(int m \<le> int n) = (m \<le> n)"
less_eq_def: fixes s :: "val option"   and s' :: "val option" shows "s \<preceq> s' \<equiv> \<exists>rs. s' = s \<star> rs"
map_of_constant_assignments_defined_if: fixes l :: "('c \<times> 'd) list"   and v :: "'c"   and a :: "'d" assumes "\<forall>(v, a)\<in>set l. \<forall>(v', a')\<in>set l. v \<noteq> v' \<or> a = a'"   and "(v, a) \<in> set l" shows "map_of l v = Some a"
clauseEntailsLiteralThenFormulaEntailsLiteral: fixes clause :: "Literal list"   and formula :: "Literal list list"   and literal :: "Literal" assumes "clause el formula"   and "clauseEntailsLiteral clause literal" shows "formulaEntailsLiteral formula literal"
in_basis_finmapI: fixes I :: "'a set"   and S :: "'a \<Rightarrow> 'b set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> S i \<in> local.basis_proj" shows "Pi' I S \<in> local.basis_finmap"
convex_onD: fixes A :: "'a set"   and f :: "'a \<Rightarrow> real"   and t :: "real"   and x :: "'a"   and y :: "'a" assumes "convex_on A f"   and "0 \<le> t"   and "t \<le> 1"   and "x \<in> A"   and "y \<in> A" shows "f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
bet2_out_out: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a"   and A :: "'a"   and B' :: "'a"   and C :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "B \<noteq> A"   and "B' \<noteq> A"   and "Tarski_neutral_dimensionless.Out Bet A C C'"   and "Bet A B C"   and "Bet A B' C'" shows "Tarski_neutral_dimensionless.Out Bet A B B'"
distPath_def: fixes h :: "'c \<Rightarrow> 'c ref"   and x :: "'c ref"   and as :: "'c list"   and y :: "'c ref" shows "distPath h x as y = (Path h x as y \<and> distinct as)"
res_i3: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "residuated f" shows "f (residual f (f (residual f x))) = f (residual f x)"
obsI: fixes n' :: "'node"   and n :: "'node"   and S :: "'node set"   and nsx' :: "'node list"   and ns :: "'node list"   and nsx :: "'node list" assumes "n' \<in> obs_intra n S"   and "\<forall>nx\<in>set nsx'. \<exists>nx'. call_of_return_node nx nx' \<and> nx' \<in> S"   and "ns = nsx @ n # nsx'"   and "\<forall>xs x xs'. nsx = xs @ x # xs' \<and> obs_intra x S \<noteq> {} \<longrightarrow> (\<exists>x''\<in>set (xs' @ [n]). \<exists>nx. call_of_return_node x'' nx \<and> nx \<notin> S)" shows "n' # nsx' \<in> obs ns S"
simple_match_and_correct: fixes m1 :: "'a simple_match"   and p :: "('a, 'b) simple_packet_scheme"   and m2 :: "'a simple_match" shows "(simple_matches m1 p \<and> simple_matches m2 p) = (case simple_match_and m1 m2 of None \<Rightarrow> False | Some m \<Rightarrow> simple_matches m p)"
is_extensional: fixes f :: "'b" assumes "\<not> \<sigma>.B.arr f" shows "\<tau> f = \<tau>.B.null"
closure_dyadic_rationals: shows "closure (\<Union>k. \<Union>f\<in>Basis \<rightarrow> \<int>. {\<Sum>i\<in>Basis. (f i / 2 ^ k) *\<^sub>R i}) = UNIV"
lossless2: fixes s2 :: "'s2"   and x :: "'a" assumes "bad2 s2"   and "x \<in> outs_\<I> \<I>" shows "lossless_spmf (oracle2 s2 x)"
assms_2: fixes x :: "'a" assumes "\<And>y. y \<in> set (coeffs f) \<Longrightarrow> x dvd y"   and "\<not> x dvd (1::'a)" shows "thesis"
power_eq_iff_eq_base: 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 n :: "nat"   and a :: "'b"   and b :: "'b" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "0 < n"   and "less_eq zero a"   and "less_eq zero b" shows "(power.power one times a n = power.power one times b n) = (a = b)"
sinh_square_eq: shows "(sinh x)\<^sup>2 = (cosh x)\<^sup>2 - (1::'a)"
pfa_init_def: fixes cap :: "nat"   and v :: "'b"   and n :: "'c" shows "pfa_init cap v n \<equiv> Array.new cap v \<bind> (\<lambda>a. return (a, n))"
dim_row_of_dagger: fixes M :: "complex mat" shows "dim_row M\<^sup>\<dagger> = dim_col M"
bezout_add_strong_nat: fixes a :: "nat"   and b :: "nat" assumes "a \<noteq> 0" shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
abf': shows "wfactors G (as @ bs) (a' \<otimes> b')"
wf_UN: fixes I :: "'g set"   and r :: "'g \<Rightarrow> ('h \<times> 'h) set" assumes "\<And>i. i \<in> I \<Longrightarrow> wf (r i)"   and "\<And>i j. \<lbrakk>i \<in> I; j \<in> I; r i \<noteq> r j\<rbrakk> \<Longrightarrow> Domain (r i) \<inter> Range (r j) = {}" shows "wf (\<Union> (r ` I))"
simps_2: fixes x :: "'b message_af"   and xs :: "'b message_af list" shows "last_message (x # xs) = (if last_message xs = NoMsg then x else last_message xs)"
prems_1: shows "sorted_wrt (\<lambda>x y. y < x) (\<alpha> # a # \<alpha>s)"
upper_asymptotic_density_subset: fixes A :: "nat set"   and B :: "nat set" assumes "A \<subseteq> B" shows "upper_asymptotic_density A \<le> upper_asymptotic_density B"
preserves_reflects_arr: fixes f :: "'a" shows "B.arr (B.\<rr> f) = B.arr f"
converged_cs_single: fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and l :: "nat"   and m :: "nat"   and m' :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "l < m"   and "cs\<^bsup>\<pi>\<^esup> m = cs\<^bsup>\<pi>'\<^esup> m'"   and "\<pi> 0 = \<pi>' 0"   and "cs\<^bsup>\<pi>\<^esup> l = [\<pi> l]" shows "\<exists>l'. cs\<^bsup>\<pi>\<^esup> l = cs\<^bsup>\<pi>'\<^esup> l'"
take: shows "take (\<^bold>|w\<^bold>| - \<^bold>|x\<^bold>|) w = w\<^sup><\<inverse>x "
hyps_8: shows "Nom a' at b in (ps, a) # branch"
N0: fixes A :: "nat set" assumes "A \<in> [N0]\<^bsup>4\<^esup>" shows "f0 A = j0"
encode_all_operator_effects_def: fixes \<Pi> :: "'a strips_problem"   and ops :: "'a strips_operator list"   and t :: "nat" shows "encode_all_operator_effects \<Pi> ops t \<equiv> let l = List.product [0..<t] ops in foldr (\<^bold>\<and>) (map (\<lambda>(t, op). encode_operator_effect \<Pi> t op) l) (\<^bold>\<not> \<bottom>)"
rk_ABA': fixes A :: "Points"   and B :: "Points"   and P :: "Points"   and A' :: "Points" assumes "rk {A, B, P} = 3"   and "rk {A, A'} = 2"   and "rk {A, A', P} = 2" shows "rk {A, B, A'} = 3"
simps_1: fixes f :: "'a \<Rightarrow> 'a fmUni"   and a :: "'a" shows "map_atomUni f (AtomUni a) = f a"
FlexRayArch_CorrectSheaf: fixes n :: "nat"   and nReturn :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nC :: "nat \<Rightarrow> Config"   and nStore :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "nat \<Rightarrow> nat \<Rightarrow> nat list" assumes "FlexRayArch n nReturn nC nStore nGet" shows "CorrectSheaf n"
map_of_SomeI: fixes l :: "('a \<times> 'b) list"   and k :: "'a"   and x :: "'b" assumes "unique l"   and "(k, x) \<in> set l" shows "table_of l k = Some x"
list_emb_Cons2: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and xs :: "'c list"   and ys :: "'c list" assumes "P x y"   and "list_emb P xs ys" shows "list_emb P (x # xs) (y # ys)"
weak_group_morphism_def: fixes f :: "'d \<Rightarrow> 'e"   and H :: "'d set"   and G :: "('d, 'f) monoid_scheme" shows "weak_group_morphism f H G \<equiv> H \<lhd> G \<and> weak_group_morphism_axioms f H G"
find_first_insert_nth_neq: fixes x :: "'a"   and y :: "'a"   and n :: "nat"   and xs :: "'a list" assumes "x \<noteq> y" shows "find_first x (insert_nth n y xs) = map_option (\<lambda>i. if i < n then i else Suc i) (find_first x xs)"
simps: fixes a1 :: "nat"   and a2 :: "s"   and a3 :: "v list"   and a4 :: "e list"   and a5 :: "t list" shows "\<turnstile>_ a1 a2;a3;a4 : a5 = (\<exists>s \<S> i vs es ts. a1 = i \<and> a2 = s \<and> a3 = vs \<and> a4 = es \<and> a5 = ts \<and> store_typing s \<S> \<and> \<S>\<bullet>None \<tturnstile>_ i vs;es : ts)"