hcju/isabelleps huggingface dataset:
prv_cnj_mono: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<chi>1 :: "'fmla"   and \<chi>2 :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<chi>1 \<in> fmla"   and "\<chi>2 \<in> fmla"   and "prv (imp \<phi>1 \<chi>1)"   and "prv (imp \<phi>2 \<chi>2)" shows "prv (imp (cnj \<phi>1 \<phi>2) (cnj \<chi>1 \<chi>2))"
B_def: shows "B = foldl Gauss_Jordan_column_k_iarrays_PA (mat_iarray (1::'a) (nrows_iarray (matrix_to_iarray A)), 0, matrix_to_iarray A) [0..<Suc k]"
\<pi>_bigo: shows "primes_pi \<in> O(\<lambda>x. x / ln x)"
mult_scalar_rec_right: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "p \<odot> q = p \<odot> monomial (lookup q v) v + p \<odot> except q {v}"
ik_le_jk: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and a :: "'a"   and b :: "'a"   and g :: "'a \<Rightarrow> 'b"   and j :: "'b" assumes "(f has_integral i) (cbox a b)"   and "(g has_integral j) (cbox a b)"   and "\<forall>x\<in>cbox a b. f x \<bullet> k \<le> g x \<bullet> k" shows "i \<bullet> k \<le> j \<bullet> k"
natural_transformation_\<T>e: shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Y a) F (\<T> (\<E> \<tau>))"
sup2E: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d" assumes "sup A B x y" obtains "A x y"   | "B x y"
cases: fixes a1 :: "'node SDG_node set"   and a2 :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and a3 :: "'node list"   and a4 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a5 :: "'edge list"   and a6 :: "'node list"   and a7 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "a1,a2 \<turnstile> (a3,a4) =a5\<Rightarrow>\<^sub>\<tau> (a6,a7)" obtains ms :: "'node list" and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and S :: "'node SDG_node set" and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind" where "a1 = Sa__" and "a2 = fa__" and "a3 = msa__" and "a4 = sa__" and "a5 = []" and "a6 = msa__" and "a7 = sa__" and "length msa__ = length sa__"   | S :: "'node SDG_node set" and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind" and ms :: "'node list" and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and a :: "'edge" and ms' :: "'node list" and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and as :: "'edge list" and ms'' :: "'node list" and s'' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "a1 = S" and "a2 = f" and "a3 = ms" and "a4 = s" and "a5 = a # as" and "a6 = ms''" and "a7 = s''" and "S,f \<turnstile> (ms,s) -a\<rightarrow>\<^sub>\<tau> (ms',s')" and "S,f \<turnstile> (ms',s') =as\<Rightarrow>\<^sub>\<tau> (ms'',s'')"
sets_restrict_space_cong: fixes M :: "'b measure"   and N :: "'b measure"   and \<Omega> :: "'b set" assumes "sets M = sets N" shows "sets (restrict_space M \<Omega>) = sets (restrict_space N \<Omega>)"
Cons_parallelI1: fixes a :: "'b"   and b :: "'b"   and as :: "'b list"   and bs :: "'b list" assumes "a \<noteq> b" shows "a # as \<parallel> b # bs"
abs_triangle_ineq: fixes abs :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (abs (plus a b)) (plus (abs a) (abs b))"
decr_grading_p_idI': fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and n :: "nat" assumes "hom_grading d"   and "\<And>v. v \<in> keys p \<Longrightarrow> d (pp_of_term v) \<le> n" shows "decr_grading_p d n p = p"
K_def: shows "K = 2 ^ (5 * (degree f div 2) * (degree f div 2)) * \<parallel>f\<parallel>\<^sup>2 ^ (2 * (degree f div 2))"
simps_1: shows "PX5d D1 = linord_of_list [X5.Xd1, X5.Xd1']"
insert_def: fixes a :: "'d"   and B :: "'d set" shows "Set.insert a B = {x. x = a} \<union> B"
finpref_def: fixes A :: "'b set"   and s :: "'b llist" shows "finpref A s \<equiv> {r \<in> A\<^sup>\<star>. r \<le> s}"
continuous_at_Sup_mono: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "mono f"   and "continuous (at_left (Sup S)) f"   and "S \<noteq> {}"   and "bdd_above S" shows "f (Sup S) = Sup (f ` S)"
ir_While_True: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and c :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_hoare (\<lambda>s s'. P s s' \<and> bval b s) (c;; WHILE b DO c) c' Q" shows "ir_hoare P (WHILE b DO c) c' Q"
cmp_simps_1: fixes f :: "'b"   and g :: "'b" assumes "B.ide f"   and "B.ide g"   and "src\<^sub>B f = trg\<^sub>B g" shows "C.arr (\<Phi>\<^sub>F (f, g))"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Imp p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Iff p q)"   and "\<And>p. P p \<Longrightarrow> P (NOT p)"   and "\<And>a. P (Lt a)"   and "\<And>a. P (Le a)"   and "\<And>a. P (Gt a)"   and "\<And>a. P (Ge a)"   and "\<And>a. P (Eq a)"   and "\<And>a. P (NEq a)"   and "\<And>i a. (i = 0 \<Longrightarrow> P (Eq a)) \<Longrightarrow> P (Dvd i a)"   and "\<And>i a. (i = 0 \<Longrightarrow> P (NEq a)) \<Longrightarrow> P (NDvd i a)"   and "P T"   and "P F"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)"   and "\<And>v. P (Closed v)"   and "\<And>v. P (NClosed v)" shows "P a0"
shuffle_def: fixes xs :: "'a list" shows "shuffle xs = pmf_of_set (permutations_of_multiset (mset xs))"
unilateral_substitutes_onI: fixes A :: "'x set"   and f :: "'x set \<Rightarrow> 'x set" assumes "\<And>B a b. \<lbrakk>B \<subseteq> A; a \<in> A; b \<in> A; Xd b \<notin> Xd ` B; b \<notin> f (insert b B)\<rbrakk> \<Longrightarrow> b \<notin> f (insert a (insert b B))" shows "unilateral_substitutes_on A f"
simps_2: fixes x2 :: "'h" shows "None \<noteq> Some x2"
dim_col_mat_1: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'b" shows "dim_col (Matrix.mat nr nc f) = nc"
IH1_4: fixes j :: "nat" assumes "hi < j"   and "j < length xs'" shows "xs'' ! j = xs' ! j"
simps_2: fixes E :: "char list \<Rightarrow> 'a"   and F :: "char list \<Rightarrow> 'a list \<Rightarrow> 'a"   and f :: "char list"   and ts :: "fterm list" shows "eval\<^sub>t E F (Fun f ts) = F f (eval\<^sub>t\<^sub>s E F ts)"
fls_mult_one_2: fixes f :: "'a fls" shows "f * 1 = f"
span_zero: fixes U :: "'a vec set" shows "0\<^sub>v n \<in> local.span U"
vint_zero: shows "0\<^sub>\<int> = []\<^sub>\<circ>"
monom_closed: fixes R :: "('a, 'b) ring_scheme"   and a :: "'a"   and n :: "nat" assumes "UP_ring R"   and "a \<in> carrier R" shows "monom (UP R) a n \<in> carrier (UP R)"
lderivs_alt: fixes w :: "'a list"   and r :: "'b rexp" shows "\<guillemotleft>lderivs w r\<guillemotright> = fold (\<lambda>a r. \<guillemotleft>lderiv a r\<guillemotright>) w \<guillemotleft>r\<guillemotright>"
valid_subtrace: shows "trace (S t (Suc i)) (take (j - Suc i) (drop (Suc i) t)) (S t j)"
Input: fixes a :: "name"   and x :: "name"   and P :: "pi"   and u :: "name" shows "a<x>.P \<longmapsto> a<u> \<prec> P[x::=u]"
DC_mono: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and c :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and d :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "a \<sqsubseteq> b"   and "c \<sqsubseteq> d" shows "(a \<Sqinter> c) \<sqsubseteq> (b \<Sqinter> d)"
appendPath_def: fixes Cs :: "char list list"   and Cs' :: "char list list" shows "Cs @\<^sub>p Cs' \<equiv> if last Cs = hd Cs' then Cs @ tl Cs' else Cs'"
finsum_insert: fixes F :: "'f set"   and a :: "'f"   and f :: "'f \<Rightarrow> 'c" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier M"   and "f a \<in> carrier M" shows "finsum M f (insert a F) = f a \<oplus>\<^bsub>M\<^esub> finsum M f F"
BrnFT_def: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('d, 'e, 'f) cmd list" shows "BrnFT cl dl \<equiv> \<Union>n\<in>theFTOne cl dl. {brnL cl n..<brnL cl n + brn (cl ! n)}"
mbind_def: shows "mbind \<equiv> \<Lambda> f g. case f of Nothing \<Rightarrow> Nothing | Just\<cdot>f' \<Rightarrow> g\<cdot>f'"
and_R2_def: fixes A :: "bool \<times> bool"   and B :: "bool \<times> bool" shows "and_R2 A B = (let (a1, a2) = A; (b1, b2) = B in coin_spmf \<bind> (\<lambda>\<sigma>. let s0 = \<sigma> \<oplus> (a1 \<oplus> False \<and> a2 \<oplus> False); s1 = \<sigma> \<oplus> (a1 \<oplus> False \<and> a2 \<oplus> True); s2 = \<sigma> \<oplus> (a1 \<oplus> True \<and> a2 \<oplus> False); s3 = \<sigma> \<oplus> (a1 \<oplus> True \<and> a2 \<oplus> True) in protocol_14_OT (s0, s1, s2, s3) B \<bind> (\<lambda>(uu_, s). R2_14_OT (s0, s1, s2, s3) B \<bind> (\<lambda>V. return_spmf ((B, V), \<sigma>, s)))))"
select_convs_2: fixes Field1 :: "int \<times> day \<Rightarrow> int"   and Field2 :: "int"   and more :: "'a" shows "Field2 \<lparr>Field1 = Field1, Field2 = Field2, \<dots> = more\<rparr> = Field2"
i_take_i_drop: fixes n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "f \<Up> m \<Down> n = f \<Down> (n + m) \<up> m"
fg: shows "B.is_iso_class F \<and> B.is_iso_class G \<and> f \<in> F \<and> g \<in> G \<and> g \<star> f \<cong>\<^sub>B h"
inorder_baldL: fixes l :: "('b \<times> color) tree"   and a :: "'b"   and r :: "('b \<times> color) tree" shows "Tree2.inorder (baldL l a r) = Tree2.inorder l @ a # Tree2.inorder r"
lm032: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and B :: "'a set" assumes "finite A" shows "sum f A = sum f (A \<inter> B) + sum f (A - B)"
Iod_carrier: fixes D :: "('c, 'd) Order_scheme"   and T :: "'c set" assumes "Order D"   and "T \<subseteq> carrier D" shows "carrier (Iod D T) = T"
assms: shows "<\<nu>x>a<y>.P \<longmapsto> b<\<nu>z> \<prec> P'"
diff_fract_def: fixes q :: "'a fract"   and r :: "'a fract" shows "q - r = q + - r"
simps_1: fixes x :: "arctic" shows "(MinInfty < x) = True"
HeronConf_interp_stepwise_weakly_precedes_cases: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
single_non_qFreshAbs_ordLess_var: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and xs :: "'varSort" assumes "qGoodAbs A" shows "|{x. \<not> qFreshAbs xs x A}| <o |UNIV|"
abs_eq: fixes xa :: "'b"   and x :: "'b" shows "(to_dual_ord xa \<le> to_dual_ord x) = (x \<le> xa)"
hatom_of_fatom_fatom_of_hatom: fixes l :: "char list \<times> hterm list" shows "hatom_of_fatom (fatom_of_hatom l) = l"
nat_filtration_pseudo_proj_True_measurable: fixes n :: "nat" shows "pseudo_proj_True n \<in> nat_filtration n \<rightarrow>\<^sub>M M"
sum_lessThan_telescope: fixes f :: "nat \<Rightarrow> 'b"   and m :: "nat" shows "(\<Sum>n<m. f (Suc n) - f n) = f m - f 0"
skipD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = []" shows "s = t"
seq: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and c\<^sub>1 :: "('a, 'c, 'd) Stmt"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c\<^sub>1' :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b"   and c\<^sub>2 :: "('a, 'c, 'd) Stmt" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "(((c\<^sub>1, mds), mem), (c\<^sub>1', mds'), mem') \<in> sifum_lang_no_dma.eval\<^sub>w eval\<^sub>A eval\<^sub>B" shows "(((c\<^sub>1 ;; c\<^sub>2, mds), mem), (c\<^sub>1' ;; c\<^sub>2, mds'), mem') \<in> sifum_lang_no_dma.eval\<^sub>w eval\<^sub>A eval\<^sub>B"
crel_vs_execute_Some: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e"   and b :: "['f]"   and heap :: "heap" assumes "crel_vs R a b"   and "P heap"   and "Q heap"   and "state_dp_consistency.cmem heap" obtains x :: "'f" and heap' :: "heap" where "execute b heap = Some (x, heap')" and "P heap'" and "Q heap'"
evContainsAll: fixes f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and i :: "nat"   and body :: "formula"   and gamma :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list" assumes "EV (contains f (i, FAll Pos body))"   and "branch subs gamma f"   and "\<forall>n. \<not> proofTree (tree subs (f n))" shows "\<exists>v. EV (contains f (0, instanceF v body))"
case_2: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c"   and x2 :: "'d" shows "(case Some x2 of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f2 x2"
localize_def: fixes v :: "'c \<Rightarrow> 'd"   and F :: "'c program" shows "localize v F \<equiv> mk_program (Init F, Acts F, AllowedActs F \<inter> \<Union> (Acts ` preserves v))"
shadow_root_ptr_kinds_M_eq: fixes h :: "(_) heap"   and h' :: "(_) heap" assumes "|h \<turnstile> object_ptr_kinds_M|\<^sub>r = |h' \<turnstile> object_ptr_kinds_M|\<^sub>r" shows "|h \<turnstile> shadow_root_ptr_kinds_M|\<^sub>r = |h' \<turnstile> shadow_root_ptr_kinds_M|\<^sub>r"
distinct_1: fixes x2 :: "'c" shows "MinInfty_delta \<noteq> Num_arc_delta x2"
emp_UNpart01: fixes cl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and dl :: "('a, 'b, 'c) cmd list" assumes "\<And>n. n < length cl \<Longrightarrow> {} \<notin> P n" shows "{} \<notin> UNpart01 cl dl P - {BrnFT cl dl}"
finally_strong_release: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n F\<^sub>n (\<phi> M\<^sub>n \<psi>) = w \<Turnstile>\<^sub>n F\<^sub>n (\<phi> and\<^sub>n \<psi>)"
g_def: shows "g \<equiv> \<lambda>i. [i] \<odot> 0\<^sup>\<infinity>"
rep_eq: fixes x :: "'a fls"   and xa :: "'a fls" shows "fls_nth (x - xa) = (\<lambda>n. fls_nth x n - fls_nth xa n)"
cont_nec_fact1_1: fixes v :: "i"   and F :: "\<Pi>\<^sub>1" shows "[WeaklyContingent F \<^bold>\<equiv> WeaklyContingent (F\<^sup>-) in v]"
cases: fixes a1 :: "'g LDCFG_node"   and a2 :: "'h"   and Def :: "'g \<Rightarrow> 'h set"   and E :: "'g"   and X :: "'g"   and H :: "'h set"   and L :: "'h set" assumes "(a1, a2) \<in> lift_Def_set Def E X H L" obtains V :: "'h" and n :: "'g" where "a1 = Node n" and "a2 = V__" and "V__ \<in> Def n"   | V :: "'h" where "a1 = Node E" and "a2 = V" and "V \<in> H"
set_2_atLeast0: shows "{0..<2} = {0, 1}"
exec_1I: fixes P :: "jvm_prog"   and \<sigma> :: "jvm_state"   and \<sigma>' :: "jvm_state" assumes "exec (P, \<sigma>) = \<lfloor>\<sigma>'\<rfloor>" shows "P \<turnstile> \<sigma> -jvm\<rightarrow>\<^sub>1 \<sigma>'"
simps_3: fixes n :: "'n"   and \<phi> :: "('a, 'k) aformula" shows "local.wf n (FNot \<phi>) = local.wf n \<phi>"
continuous_at_left_imp_sup_continuous: fixes f :: "'a \<Rightarrow> 'b" assumes "mono f"   and "\<And>x. continuous (at_left x) f" shows "sup_continuous f"
finite_dimensional_vector_space_pair_with_def: fixes plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and basis\<^sub>1 :: "'d set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and basis\<^sub>2 :: "'f set" shows "finite_dimensional_vector_space_pair_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 basis\<^sub>2 \<equiv> finite_dimensional_vector_space_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 \<and> finite_dimensional_vector_space_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 basis\<^sub>2"
ord_of_nat_csucc: fixes a :: "nat" shows "csucc (a\<^sub>\<nat>) = ZFC_in_HOL.succ (a\<^sub>\<nat>)"
ID_ccompare': fixes c :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "ID ccompare = Some c" shows "comparator c"
extensibleI: fixes R2 :: "('a, 'b) labeled_graph"   and G :: "('a, 'c) labeled_graph"   and g :: "('b \<times> 'c) set"   and R1 :: "('a, 'b) labeled_graph"   and f :: "('b \<times> 'c) set" assumes "graph_homomorphism R2 G g"   and "agree_on R1 f g" shows "extensible (R1, R2) G f"
lrevT: fixes xs :: "'b llist"   and A :: "'b set" assumes "xs \<in> A\<^sup>\<star>" shows "lrev xs \<in> A\<^sup>\<star>"
lex_pp_antisym: fixes s :: "('b, 'c) pp"   and t :: "('b, 'c) pp" assumes "lex_pp s t"   and "lex_pp t s" shows "s = t"
less_def: fixes x :: "nat"   and y :: "nat" shows "nat_dvd.less x y = (x dvd y \<and> x \<noteq> y)"
writes_union_left_I: fixes S :: "('x, 'y, 'z) prog set"   and f :: "('x, 'y, 'aa) prog"   and h :: "'x"   and h' :: "'x"   and S' :: "('x, 'y, 'z) prog set" assumes "writes S f h h'" shows "writes (S \<union> S') f h h'"
induct: fixes P :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'b list"   and a1 :: "'b list" assumes "\<And>b. P [] b"   and "\<And>a. P a []"   and "\<And>a l b r. \<lbrakk>P l (b # r); P (a # l) r\<rbrakk> \<Longrightarrow> P (a # l) (b # r)" shows "P a0 a1"
weak_complete_latticeI: fixes L :: "('a, 'b) gorder_scheme" assumes "weak_partial_order L"   and "\<And>A. A \<subseteq> carrier L \<Longrightarrow> \<exists>s. is_lub L s A"   and "\<And>A. A \<subseteq> carrier L \<Longrightarrow> \<exists>i. is_glb L i A" shows "weak_complete_lattice L"
succ_def: fixes g :: "'m"   and v :: "'v" shows "succ g v = the_default [] (M_lookup g v)"
that: fixes t :: "('b, 'a) Term.term" assumes "receive\<langle>t\<rangle>\<^sub>s\<^sub>t \<in> set S"   and "v \<in> fv t" shows "thesis"
multiplicity_gauss_cnj: fixes a :: "gauss_int"   and b :: "gauss_int" shows "multiplicity (gauss_cnj a) (gauss_cnj b) = multiplicity a b"
cobounded2: fixes a :: "'a"   and b :: "'a" shows "inf a b \<le> b"
le_divide_eq_1_neg: fixes a :: "'a"   and b :: "'a" assumes "a < (0::'a)" shows "((1::'a) \<le> b / a) = (b \<le> a)"
nat_to_bv_non_empty: fixes n :: "nat" assumes "0 < n" shows "nat_to_bv n \<noteq> []"
post_in_annos: fixes c :: "'a acom" shows "post c \<in> set (annos c)"
q_has_vector_derivative: fixes v :: "real"   and s0 :: "real"   and t :: "real"   and u :: "real set" shows "(movement.q v s0 has_vector_derivative movement.q' v t) (at t within u)"
splitDigit_inv: fixes p :: "'b \<Rightarrow> bool"   and i :: "'b"   and d :: "('c, 'b) Digit"   and n :: "nat"   and l :: "('c, 'b) Node list"   and nd :: "('c, 'b) Node"   and r :: "('c, 'b) Node list" assumes "\<And>a b. p a \<Longrightarrow> p (a + b)"   and "\<not> p i"   and "is_measured_digit d"   and "is_leveln_digit n d"   and "p (i + gmd d)"   and "splitDigit p i d = (l, nd, r)" shows "\<not> p (i + gmnl l) \<and> p (i + gmnl l + gmn nd) \<and> (\<forall>x\<in>set l. is_measured_node x \<and> is_leveln_node n x) \<and> (\<forall>x\<in>set r. is_measured_node x \<and> is_leveln_node n x) \<and> is_measured_node nd \<and> is_leveln_node n nd"
dg_is_arrD_2: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" shows "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
R1_14_def: fixes msgs :: "bool \<times> bool \<times> bool \<times> bool"   and choice :: "bool \<times> bool" shows "R1_14 msgs choice = (let (m00, m01, m10, m11) = msgs; (c0, c1) = choice in coin_spmf \<bind> (\<lambda>S0. coin_spmf \<bind> (\<lambda>S1. coin_spmf \<bind> (\<lambda>S2. coin_spmf \<bind> (\<lambda>S3. coin_spmf \<bind> (\<lambda>S4. coin_spmf \<bind> (\<lambda>S5. R1_OT12 (S0, S1) c0 \<bind> (\<lambda>a. R1_OT12 (S2, S3) c1 \<bind> (\<lambda>b. R1_OT12 (S4, S5) c1 \<bind> (\<lambda>c. return_spmf (msgs, (S0, S1, S2, S3, S4, S5), a, b, c)))))))))))"
implies_curry_flip: fixes z :: "'a"   and y :: "'a"   and x :: "'a" shows "z - y - x = z - x - y"
amalgamation_total: fixes i :: "nat"   and j :: "nat" assumes "total1 (\<phi> i) \<or> total1 (\<phi> j)" shows "total1 (amalgamation i j)"
valid2_Cons: fixes tr :: "('state, 'act, 'out) trans list"   and trn :: "('state, 'act, 'out) trans" assumes "valid2 tr"   and "tgtOf trn = srcOf (hd tr)"   and "validTrans trn" shows "valid2 (trn # tr)"
add_nonpos_nonpos: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq a zero"   and "less_eq b zero" shows "less_eq (plus a b) zero"
p_def: fixes s :: "'a"   and s' :: "'a"   and t :: "real" shows "p s s' t = L.prob (0, s) {\<omega> \<in> space (K.lim_stream (0, s)). trace_in {s'} t s \<omega>}"
pred_envirD: fixes p :: "'d \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd"   and x :: "'e" assumes "pred_envir p f" shows "p (f x)"
assms: fixes k :: "nat" shows "\<exists>v. v \<le>\<^sub>F\<^sub>I w \<and> k < length v \<and> path v p"
block_size_lt_v: shows "\<k> \<le> \<v>"
singleton_insert_inj_eq': fixes a :: "'c"   and A :: "'c set"   and b :: "'c" shows "(insert a A = {b}) = (a = b \<and> A \<subseteq> {b})"
retraction: fixes v :: "'a" assumes "v \<in> \<Union> X" shows "g (g v) = g v"
diff: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "f \<in> R"   and "g \<in> R" shows "(\<lambda>x. f x - g x) \<in> R"
aux: fixes b :: "'a option" shows "(\<bottom> \<noteq> b) = (\<exists>y. b = \<lfloor>y\<rfloor>)"
nn_integral_density: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and g :: "'b \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "integral\<^sup>N (density M f) g = \<integral>\<^sup>+ x. f x * g x \<partial>M"
intros_7: fixes uu :: "'c \<Rightarrow> 'd"   and uv :: "'c option"   and uw :: "'c option" shows "dbm_entry_val uu uv uw \<infinity>"
ChamberComplex_\<Sigma>: fixes S :: "'b set" assumes "CoxeterComplex S" shows "ChamberComplex (CoxeterComplex.TheComplex S)"