hcju/isabelleps huggingface dataset:
solve_consistent_rref_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" shows "solve_consistent_rref A b = (\<chi>j. if \<exists>i. A $ i $ j = (1::'b) \<and> j = (LEAST n. A $ i $ n \<noteq> (0::'b)) then b $ (THE i. A $ i $ j = (1::'b)) else (0::'b))"
closed_segment_Reals: fixes x :: "complex"   and y :: "complex" assumes "x \<in> \<real>"   and "y \<in> \<real>" shows "closed_segment x y = complex_of_real ` closed_segment (Re x) (Re y)"
subseq_append: fixes xs :: "'c list"   and zs :: "'c list"   and ys :: "'c list" shows "subseq (xs @ zs) (ys @ zs) = subseq xs ys"
pos_mod_bound: fixes l :: "int"   and k :: "int" assumes "0 < l" shows "k mod l < l"
order_iff_strict: fixes a :: "'a"   and b :: "'a" shows "(a \<sqsubseteq> b) = (a \<sqsubseteq> b \<and> a \<noteq> b \<or> a = b)"
substAbs_compose1: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and Y1 :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Y2 :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and y :: "'var" assumes "goodAbs A"   and "good Y1"   and "good Y2" shows "A $[Y1 / y]_ys $[Y2 / y]_ys = A $[Y1 #[Y2 / y]_ys / y]_ys"
simps_1: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d" assumes "class.linorder less_eq less" shows "linorder.insort_key less_eq f x [] = [x]"
lm098: fixes Y :: "'c set"   and X :: "'c set" shows "runiq (X <|| Y)"
fls_base_factor_zero: shows "fls_base_factor 0 = 0"
head_arev: fixes a :: "'a \<times> 'a" assumes "a \<in> arcs (with_proj G)" shows "head (with_proj G) (swap_in (parcs G) a) = tail (with_proj G) a"
induct: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'b"   and a1 :: "'a list" assumes "\<And>f. P f []"   and "\<And>f x. P f [x]"   and "\<And>f v vb vc. \<lbrakk>\<And>va vd x y. \<lbrakk>va = Time_Monad.val (length_tm (v # vb # vc)); vd = Time_Monad.val (split_at_tm (va div 2) (v # vb # vc)); (x, y) = vd\<rbrakk> \<Longrightarrow> P f x; \<And>va vd x y ve. \<lbrakk>va = Time_Monad.val (length_tm (v # vb # vc)); vd = Time_Monad.val (split_at_tm (va div 2) (v # vb # vc)); (x, y) = vd; ve = Time_Monad.val (mergesort_tm f x)\<rbrakk> \<Longrightarrow> P f y\<rbrakk> \<Longrightarrow> P f (v # vb # vc)" shows "P a0 a1"
assms_2: shows "\<Delta> s' vl' s1 vl1"
prod_inversef: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "class.field divide inverse times one plus zero minus uminus" shows "comm_monoid_mult.prod times one (inverse \<circ> f) A = inverse (comm_monoid_mult.prod times one f A)"
eventually_at_right: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and P :: "'c \<Rightarrow> bool" assumes "class.linorder_topology less_eq less open"   and "less x y" shows "eventually P (topological_space.at_within open x (ord.greaterThan less x)) = (\<exists>b. less x b \<and> (\<forall>y. less x y \<longrightarrow> less y b \<longrightarrow> P y))"
log_half: fixes n :: "nat" shows "local.log (n div 2) = local.log n - 1"
dependent_mono: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'b set"   and A :: "'b set" assumes "module scale"   and "module.dependent scale B"   and "B \<subseteq> A" shows "module.dependent scale A"
HEndPhase1_HInv4b_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase1 s s' p"   and "p \<noteq> q"   and "HInv4b s q" shows "HInv4b s' q"
UnionP_Mem2: fixes x :: "tm"   and y :: "tm"   and z :: "tm"   and a :: "tm" shows "{UnionP x y z, a IN y} \<turnstile> a IN z"
allocationsUniverseOutsideUnion: fixes a :: "('c \<times> 'd set) set"   and X :: "'c set"   and i :: "'c" assumes "a \<in> allocationsUniverse" shows "Outside' (X \<union> {i}) a \<union> {i} \<times> ({\<Union> (a `` (X \<union> {i}))} - {{}}) \<in> allocationsUniverse \<and> \<Union> (Range (Outside' (X \<union> {i}) a \<union> {i} \<times> ({\<Union> (a `` (X \<union> {i}))} - {{}}))) = \<Union> (Range a)"
iso_sym: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme" assumes "Group.group G"   and "G \<cong> H" shows "H \<cong> G"
simp: shows "mk_all_list (vs @ [v]) B = mk_all idn \<tau> (mk_all_list vs B)"
norm_sgn_le_op_norm: fixes A :: "(('b, 'd) vec, 'c) vec"   and x :: "('b, 'd) vec" shows "\<parallel>A *v sgn x\<parallel> \<le> \<parallel>A\<parallel>\<^sub>o\<^sub>p"
Bex_set_list_ex: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" shows "Bex (set xs) P = list_ex P xs"
HPhase1or2ReadThen_HInv3: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase1or2ReadThen s s' p d q"   and "HInv3 s"   and "p \<noteq> q"   and "Inv2b s" shows "HInv3 s'"
not_null_ang_def_equiv: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool" shows "(QCongAnNull a) = (QCongA a \<and> (\<exists>A B C. a A B C \<and> \<not> B Out A C))"
minus_tree_def: fixes x :: "'a tree"   and y :: "'a tree" shows "x - y = pure (-) \<diamondop> x \<diamondop> y"
antipar_1: shows "trg\<^sub>D g' = src\<^sub>D (F f)"
intros_2: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list"   and y :: "'c" assumes "list_emb P xs ys" shows "list_emb P xs (y # ys)"
ereal_uminus_le_reorder: fixes a :: "ereal"   and b :: "ereal" shows "(- a \<le> b) = (- b \<le> a)"
strict_trans1: fixes b :: "'c word"   and a :: "'c word"   and c :: "'c word" assumes "b \<le>s a"   and "c <s b" shows "c <s a"
Rep_lifted_distrib_lattice_top_inverse: fixes x :: "('c, 'd) lifted_distrib_lattice_top" shows "Abs_lifted_distrib_lattice_top (Rep_lifted_distrib_lattice_top x) = x"
c_refsto_d: shows "(c::'a::refinement_lattice) \<sqsubseteq> (d::'a::refinement_lattice)"
MkIde_Cod': fixes a :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "ide a" shows "MkIde (Cod a) = a"
a: shows "\<exists>as'. exec_plan (state_succ (exec_plan s as1) act) as' = exec_plan (state_succ (exec_plan s as1) act) as2 \<and> subseq as' as2 \<and> length as' \<le> S vs lss PROB (fmrestrict_set vs (state_succ (exec_plan s as1) act))"
bij_extend_act_eq_project_act: fixes h :: "'c \<Rightarrow> 'd" assumes "bij h" shows "extend_act (\<lambda>(x, u). h x) = project_act (\<lambda>(x, u). inv h x)"
While: fixes b :: "bexp"   and A :: "char list set"   and c :: "com"   and A' :: "char list set" assumes "vars b \<subseteq> A"   and "D A c A'" shows "D A (WHILE b DO c) A"
deg_pm_single: fixes x :: "'c"   and k :: "'b" shows "deg_pm (monomial k x) = k"
7: shows "coreflexive (e * (w \<sqinter> - prim_EP w v e)\<^sup>T)"
monom_pow: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and n :: "nat"   and m :: "nat" assumes "UP_cring R"   and "a \<in> carrier R" shows "monom (UP R) a n [^]\<^bsub>UP R\<^esub> m = monom (UP R) (a [^]\<^bsub>R\<^esub> m) (n * m)"
mapping_of_pmf_pmf_of_list': fixes xs :: "('b \<times> real) list" assumes "pmf_of_list_wf xs" shows "mapping_of_pmf (pmf_of_list xs) = Mapping.tabulate (remdups (map fst (filter (\<lambda>z. snd z \<noteq> 0) xs))) (\<lambda>x. sum_list (map snd (filter (\<lambda>z. fst z = x) (filter (\<lambda>z. snd z \<noteq> 0) xs))))"
tdghm_NTMap_app_in_Arr: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "\<NN>\<lparr>NTMap\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>"
SeqReds2: fixes e\<^sub>1 :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and v\<^sub>1 :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (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"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>2 :: "expr"   and e\<^sub>2' :: "expr"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,E \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<rightarrow>* \<langle>e\<^sub>2',s\<^sub>2\<rangle>" shows "P,E \<turnstile> \<langle>e\<^sub>1;; e\<^sub>2,s\<^sub>0\<rangle> \<rightarrow>* \<langle>e\<^sub>2',s\<^sub>2\<rangle>"
lfilter_LNil: fixes P :: "'a \<Rightarrow> bool" shows "lfilter P []\<^sub>l = []\<^sub>l"
butterfly_add_left: fixes a :: "'b"   and a' :: "'b"   and b :: "'a" shows "butterfly (a + a') b = butterfly a b + butterfly a' b"
assms_4: shows "w \<in> range (pseudo_proj_True n)"
ball_scale: fixes a :: "real"   and c :: "'c"   and r :: "real" assumes "a \<noteq> 0" shows "(*\<^sub>R) a ` ball c r = ball (a *\<^sub>R c) (\<bar>a\<bar> * r)"
partial_vector_derivative_works: fixes F :: "'a \<Rightarrow> 'b"   and base_vec :: "'a"   and a :: "'a" assumes "partially_vector_differentiable F base_vec a" shows "has_partial_vector_derivative F base_vec (partial_vector_derivative F base_vec a) a"
empty_not_add_mset: fixes a :: "'f"   and A :: "'f multiset" shows "{#} \<noteq> add_mset a A"
cases: fixes a1 :: "char list"   and a2 :: "char list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list" assumes "P \<turnstile> a1 \<prec>\<^sup>1 a2" obtains C :: "char list" and D :: "char list" and rest :: "(char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list" where "a1 = C" and "a2 = D" and "class P C = \<lfloor>(D, rest)\<rfloor>" and "C \<noteq> Object"
commute: fixes a :: "'c"   and b :: "'c" shows "a \<cdot> b = b \<cdot> a"
card_of_ordLeq: fixes A :: "'f set"   and B :: "'g set" shows "(\<exists>f. inj_on f A \<and> f ` A \<subseteq> B) = (|A| \<le>o |B|)"
is_lifting_def: fixes f :: "'a \<Rightarrow> 'b"   and basis :: "(real \<Rightarrow> real) list"   and basis' :: "(real \<Rightarrow> real) list" shows "is_lifting f basis basis' = (\<forall>C. eval (f C) = eval C \<and> (is_expansion C basis \<longrightarrow> is_expansion (f C) basis') \<and> trimmed (f C) = trimmed C \<and> fst (dominant_term (f C)) = fst (dominant_term C))"
statement55: fixes p :: "proj2"   and q :: "proj2"   and C :: "cltn2"   and l :: "proj2_line"   and r :: "proj2" assumes "p \<noteq> q"   and "apply_cltn2 p C = q"   and "apply_cltn2 q C = p"   and "proj2_incident p l"   and "proj2_incident q l"   and "proj2_incident r l" shows "apply_cltn2 (apply_cltn2 r C) C = r"
take_bit_set_bit_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and n :: "nat"   and m :: "nat"   and a :: "'c" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "take_bit n (ring_bit_operations.set_bit one push_bit or m a) = (if n \<le> m then take_bit n a else ring_bit_operations.set_bit one push_bit or m (take_bit n a))"
simps: fixes p :: "'a Point" shows "positionVector p = \<lparr>tdir = tval p, xdir = xval p, ydir = yval p, zdir = zval p\<rparr>"
lsl_lsu_prop_eq: shows "\<nu> \<circ> \<nu>\<^sup>\<natural> = \<nu>\<^sup>\<natural>"
homeomorphic_maps_map: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" shows "homeomorphic_maps X Y f g = (homeomorphic_map X Y f \<and> homeomorphic_map Y X g \<and> (\<forall>x\<in>topspace X. g (f x) = x) \<and> (\<forall>y\<in>topspace Y. f (g y) = y))"
subst_Var: fixes i :: "nat"   and s :: "dB"   and k :: "nat" shows "Var i[s/k] = (if k < i then Var (pred i) else if i = k then s else Var i)"
GEN_OP_D: fixes c :: "'b"   and a :: "'c"   and R :: "('b \<times> 'c) set" assumes "GEN_OP c a R" shows "(c, a) \<in> R"
AA29: fixes N :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b"   and A :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<box>[N]_v \<and> \<box>\<diamond>\<langle>A\<rangle>_v \<longrightarrow> \<box>\<diamond>\<langle>N \<and> A\<rangle>_v"
deriv_prod_rule_times_monom: fixes a :: "'a"   and b :: "'a"   and q :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "q \<in> carrier P" shows "deriv (monom P a n \<otimes>\<^bsub>P\<^esub> q) b = deriv (monom P a n) b \<otimes> to_fun q b \<oplus> to_fun (monom P a n) b \<otimes> deriv q b"
ide_implies_incl: fixes a :: "'s" assumes "S.ide a" shows "S.incl a"
chine_is_induced_map: shows "r.is_induced_map r.chine"
minus: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "additive f" shows "f (- x) = - f x"
drop_upto_enum: fixes n :: "'b word"   and m :: "'b word" shows "drop (unat n) [0 .e. m] = [n .e. m]"
comparator_pot_comp: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "comparator cmp" shows "comparator (pot_comp cmp)"
map_is_Nil_conv: fixes f :: "'h \<Rightarrow> 'g"   and xs :: "'h list" shows "(map f xs = []) = (xs = [])"
coplanar_perm_6: 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"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "Tarski_neutral_dimensionless.Coplanar Bet B A C D"
CQRightSubst_def: shows "CQRightSubst = \<lparr>SFunctions = Map.empty, SPredicates = \<lambda>p. if p = vid1 then Some (Geq ($' vid1) ($f (Inr vid1) local.empty)) else None, SContexts = Map.empty, SPrograms = Map.empty, SODEs = Map.empty\<rparr>"
sel_3: fixes x21 :: "'a"   and x22 :: "'a llist" shows "ltl (LCons x21 x22) = x22"
simps_3: fixes c1 :: "com"   and c2 :: "com"   and t :: "char list \<Rightarrow> int option" shows "Fold.fold (c1;; c2) t = Fold.fold c1 t;; Fold.fold c2 (defs c1 t)"
verit_eq_simplify_23: fixes a :: "'c" shows "(a = - a) = (a = (0::'c))"
prime: shows "pr_ideal R x (+) (\<cdot>) \<zero> \<one>"
less_divide_eq_1_pos: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and 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 less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a" shows "less one (divide b a) = less a b"
axioms_2: fixes \<alpha> :: "V"   and \<SS> :: "V" assumes "\<Z>_sgrp \<alpha> \<SS>" shows "\<Z>_sgrp_axioms \<SS>"
role_comp_inv: fixes x :: "role_t" shows "role_comp (role_comp x) = x"
tms: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm" shows "tms t1 t2 \<in> trm"
is_tm_tdghm_op: shows "op_tdghm \<NN> : op_dghm \<GG> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y op_dghm \<FF> : op_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> op_dg \<BB>"
Cartan_is_linear: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set" assumes "f holomorphic_on s"   and "open s"   and "connected s"   and "0 \<in> s"   and "\<And>u z. \<lbrakk>cmod u = 1; z \<in> s\<rbrakk> \<Longrightarrow> u * z \<in> s"   and "\<And>u z. \<lbrakk>cmod u = 1; z \<in> s\<rbrakk> \<Longrightarrow> f (u * z) = u * f z" shows "\<exists>c. \<forall>z\<in>s. f z = c * z"
leftmost_cons_nonterminal: fixes x :: "symbol"   and xs :: "symbol list" assumes "is_nonterminal x" shows "leftmost 0 (x # xs)"
KGinormGSi: shows "K \<inter> \<GG> ! i \<lhd> G\<lparr>carrier := \<GG> ! (i + 1)\<rparr>"
IfTrue: fixes b :: "bexp"   and s :: "char list \<Rightarrow> val"   and c1 :: "com"   and c2 :: "com" assumes "tbval b s True" shows "(IF b THEN c1 ELSE c2, s) \<rightarrow> (c1, s)"
A: shows "\<forall>\<^sub>F n in sequentially. 3 * alpha < Sigma_Algebra.measure (distr M borel (\<lambda>x. birkhoff_sum f n x / B n)) {x. a < \<bar>x\<bar>}"
simps: fixes f :: "'c \<Rightarrow> 'b option"   and h :: "'c" shows "Map f\<langle>h\<rangle> = f h"
lemma_2: fixes c_k :: "int \<Rightarrow> int \<Rightarrow> int \<Rightarrow> nat"   and c :: "int \<Rightarrow> int \<Rightarrow> nat"   and w :: "int \<Rightarrow> int \<Rightarrow> nat"   and l :: "nat"   and j' :: "int"   and i :: "int"   and i' :: "int"   and j :: "int" assumes "QI c_k c w"   and "l = nat (j' - i)"   and "i \<le> i'"   and "i' \<le> j"   and "j \<le> j'" shows "c i j + c i' j' \<le> c i j' + c i' j"
abelian_groupE_1: fixes R :: "('e, 'f) ring_scheme"   and x :: "'e"   and y :: "'e" assumes "abelian_group R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"
elims_3: fixes x :: "('var, 'val, 'ret, 'pname) edge_kind list"   and xa :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "\<not> preds x xa" obtains et :: "('var, 'val, 'ret, 'pname) edge_kind" and ets :: "('var, 'val, 'ret, 'pname) edge_kind list" and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = et # ets" and "xa = s" and "\<not> (pred et s \<and> preds ets (transfer et s))"
elims_2: fixes x :: "Preliminaries.var \<Rightarrow> univ"   and xa :: "('fsym, 'psym) lit" assumes "M.satL x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> univ" and a :: "('fsym, 'psym) atm" where "x = \<xi>'__" and "xa = Pos a__" and "M.satA \<xi>'__ a__"   | \<xi> :: "Preliminaries.var \<Rightarrow> univ" and a :: "('fsym, 'psym) atm" where "x = \<xi>" and "xa = Neg a" and "\<not> M.satA \<xi> a"
WF1_general: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and N :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and A :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b"   and w :: "'a \<Rightarrow> 'c" assumes "|~ P \<and> N \<longrightarrow> \<circle>P \<or> \<circle>Q"   and "|~ P \<and> N \<and> \<langle>A\<rangle>_v \<longrightarrow> \<circle>Q"   and "\<turnstile> P \<and> N \<longrightarrow> Enabled \<langle>A\<rangle>_v"   and "|~ P \<and> Unchanged w \<longrightarrow> \<circle>P" shows "\<turnstile> \<box>N \<and> WF(A)_v \<longrightarrow> P \<leadsto> Q"
independent_imp_finite: fixes B :: "'c set" assumes "independent B" shows "finite B"
zs: shows "zs = snd (solve_factored_ratfps p cs)"
"1_3": shows "\<not> Suc (snd (Strat ! x) ! l) < length init"
conj_subset_def: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "(A \<subseteq> {x. P x \<and> Q x}) = (A \<subseteq> {x. P x} \<and> A \<subseteq> {x. Q x})"
PDP_in: fixes x :: "'a"   and d :: "'a \<Rightarrow>\<^sub>L 'a" assumes "(x, d) \<in> PDP" shows "\<forall>\<^sub>F x in at x within plane n c. x \<in> R"
H_var: fixes p :: "'a"   and x :: "'a"   and q :: "'a" shows "H p x q = (\<tt>\<tt> p \<cdot> x \<cdot> !q = (0::'a))"
cond_spmf_def: fixes p :: "'h spmf"   and A :: "'h set" shows "cond_spmf p A = (if set_spmf p \<inter> A = {} then return_pmf None else cond_pmf p (Some ` A))"
assms_1: shows "\<lless>\<pi>, Syntax.Let e\<^sub>1 x e\<^sub>2\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>"
hyps: fixes ms :: "mregex" assumes "ok m mr"   and "ms \<in> RPD mr" shows "ok m ms"
heap_first_el: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and heap :: "int \<Rightarrow> 'a"   and l :: "int"   and r :: "int"   and x :: "int" assumes "is_heap cmp heap l r"   and "transp cmp"   and "l < x"   and "x < r" shows "cmp (heap l) (heap x)"
yFresh: shows "qAFresh xs y X \<and> qAFresh xs y X'"
intros: fixes t :: "real"   and t' :: "real"   and \<omega> :: "(real \<times> 'a) stream"   and s' :: "'a" assumes "t < t'"   and "wf_times t' \<omega>" shows "wf_times t ((t', s') ## \<omega>)"
bags_exist: fixes G :: "('e, 'f) Graph_scheme"   and T :: "('g, 'h) Graph_scheme"   and bag :: "'g \<Rightarrow> 'e set"   and v :: "'e" assumes "TreeDecomposition G T bag"   and "v \<in> V\<^bsub>G\<^esub>" shows "\<exists>t\<in>V\<^bsub>T\<^esub>. v \<in> bag t"
set_coeffs_subset_singleton_0_iff: fixes p :: "'b poly" shows "(set (coeffs p) \<subseteq> {0::'b}) = (p = 0)"
wb_lens_axioms_def: fixes x :: "'e \<Longrightarrow> 'f" shows "wb_lens_axioms x \<equiv> \<forall>\<sigma>. put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<sigma>) = \<sigma>"
deg_of_inverse: fixes w :: "real" shows "rad_of (deg_of w) = w"
coeff_map_poly_of_int: fixes p :: "int poly"   and n :: "nat" shows "poly.coeff (of_int_poly p) n = of_int (poly.coeff p n)"
map_disc_iff: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b llist" shows "lnull (lmap f a) = lnull a"
while_red_n_induct: fixes b :: "expr"   and cx :: "com"   and s :: "char list \<Rightarrow> val option"   and n :: "nat"   and s' :: "char list \<Rightarrow> val option"   and P :: "(char list \<Rightarrow> val option) \<Rightarrow> (char list \<Rightarrow> val option) \<Rightarrow> bool" assumes "\<langle>while (b) cx,s\<rangle> \<rightarrow>\<^bsup>n\<^esup> \<langle>Skip,s'\<rangle>"   and "\<And>s. \<lbrakk>b\<rbrakk>s = Some false \<Longrightarrow> P s s"   and "\<And>s i j s''. \<lbrakk>\<lbrakk>b\<rbrakk>s = Some true; \<langle>cx,s\<rangle> \<rightarrow>\<^bsup>i\<^esup> \<langle>Skip,s''\<rangle>; \<langle>while (b) cx,s''\<rangle> \<rightarrow>\<^bsup>j\<^esup> \<langle>Skip,s'\<rangle>; P s'' s'\<rbrakk> \<Longrightarrow> P s s'" shows "P s s'"
mult_strict_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a < b"   and "c < d"   and "(0::'a) < b"   and "(0::'a) \<le> c" shows "a * c < b * d"
assms_4: fixes k :: "'a" assumes "k \<in> K" shows "h [k] = g [k]"
outputCases': fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "Early_Semantics.residual" assumes "a{b}.P \<longmapsto>\<^sub>e Rs" obtains aa :: "name" and ba :: "name" and Pa :: "pi" where "a = aa \<and> b = ba \<and> P = Pa" and "Rs = aa[ba] \<prec>\<^sub>e Pa"
assms: fixes t :: "('a, 'b) Term.term" assumes "\<forall>s\<lhd>t. P s" shows "P t"
HARootCarAudioSystem: shows "HARoot CarAudioSystem = Root_CTRL"
rotate_conv_mod: fixes n :: "nat"   and xs :: "'d list" shows "rotate n xs = rotate (n mod length xs) xs"
At0: shows "(case (x0, y0, z0) of (x, y, z) \<Rightarrow> At x y z) 0"
assms: fixes x :: "'a" assumes "x \<in> S" shows "m1 x \<sqsubseteq> m2 x"
star_fps_def: shows "star \<equiv> map_fun ($) Abs_fps star_fps_rep"
"4_1": shows "u =\<^sub>F s\<^sub>1 @ s\<^sub>2"
fls_residue_nonzero_ex_antiderivative: fixes f :: "'b fls" assumes "fls_residue f = (0::'b)" shows "\<exists>F. fls_deriv F = f"