hcju/isabelleps huggingface dataset:
chain_boundary_diff: fixes p :: "nat"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "chain_boundary p (a - b) = chain_boundary p a - chain_boundary p b"
"1_1": shows "x' \<in> {(s, as) |as. as \<in> valid_plans PROB}"
outstanding_non_volatile_write_not_volatile_read_disj: fixes i :: "nat"   and j :: "nat"   and p\<^sub>i :: "'a"   and is\<^sub>i :: "instr list"   and \<theta>\<^sub>i :: "nat \<Rightarrow> nat option"   and sb\<^sub>i :: "'a memref list"   and \<D>\<^sub>i :: "bool"   and \<O>\<^sub>i :: "nat set"   and \<R>\<^sub>i :: "nat \<Rightarrow> bool option"   and p\<^sub>j :: "'a"   and is\<^sub>j :: "instr list"   and \<theta>\<^sub>j :: "nat \<Rightarrow> nat option"   and sb\<^sub>j :: "'a memref list"   and \<D>\<^sub>j :: "bool"   and \<O>\<^sub>j :: "nat set"   and \<R>\<^sub>j :: "nat \<Rightarrow> bool option" assumes "i < length (t # ts)"   and "j < length (t # ts)"   and "i \<noteq> j"   and "(t # ts) ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i)"   and "(t # ts) ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)" shows "outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>i) \<inter> outstanding_refs (Not \<circ> is_volatile_Read\<^sub>s\<^sub>b) (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j) = {}"
le_cases: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.linorder less_eq less" obtains "less_eq x y"   | "less_eq y x"
Collect_code: fixes P :: "'c \<Rightarrow> bool" shows "Collect P = set (filter P enum)"
domD: fixes a :: "'f"   and m :: "'f \<Rightarrow> 'g option" assumes "a \<in> dom m" shows "\<exists>b. m a = Some b"
part_equiv_def: fixes R :: "('b \<times> 'b) set" shows "part_equiv R \<equiv> sym R \<and> trans R"
IdNatTrans_map: fixes X :: "'d"   and F :: "('d, 'g, 'e, 'h, 'f, 'i) Functor" assumes "X \<in> obj\<^bsub>CatDom F\<^esub>" shows "IdNatTrans F $$ X = id\<^bsub>CatCod F\<^esub> (F @@ X)"
eq_upto_def: fixes B :: "(nat \<Rightarrow> 'a) set"   and f :: "nat \<Rightarrow> 'a"   and i :: "nat" shows "eq_upto B f i = {g \<in> B. \<forall>j<i. f j = g j}"
lucas_weak: fixes n :: "nat"   and a :: "nat" assumes "2 \<le> n"   and "[a ^ (n - 1) = 1] (mod n)"   and "\<forall>m. 0 < m \<and> m < n - 1 \<longrightarrow> [a ^ m \<noteq> 1] (mod n)" shows "prime n"
compose_eq: fixes x :: "'a"   and A :: "'a set"   and g :: "'c \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'c" assumes "x \<in> A" shows "FuncSet.compose A g f x = g (f x)"
hyps: shows "p \<noteq> [] \<longrightarrow> (\<forall>x s. s \<in> set p \<and> x \<in> dom (Cp A) \<longrightarrow> x \<notin> dom (Cp s)) \<longrightarrow> y \<in> dom (Cp A) \<longrightarrow> y \<notin> dom (Cp (list2FWpolicy p))"
\<Z>_sgrp_basis_axioms_def: fixes \<SS> :: "V" shows "\<Z>_sgrp_basis_axioms \<SS> \<equiv> vcard \<SS> = 2\<^sub>\<nat> \<and> binop (\<SS>\<lparr>\<A>\<rparr>) (\<SS>\<lparr>mbinop\<rparr>)"
abs_mult_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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less_eq zero x" shows "times (abs y) x = abs (times y x)"
sum_mono2: fixes B :: "'d set"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'a" assumes "finite B"   and "A \<subseteq> B"   and "\<And>b. b \<in> B - A \<Longrightarrow> (0::'a) \<le> f b" shows "sum_0 f A \<le> sum_0 f B"
proj2_incident_abs: fixes v :: "(real, 3) vec"   and w :: "(real, 3) vec" assumes "v \<noteq> 0"   and "w \<noteq> 0" shows "proj2_incident (proj2_abs v) (proj2_line_abs w) = (v \<bullet> w = 0)"
v_minus_p_orthogonal_complement: fixes S :: "'a set"   and X :: "'a set"   and v :: "'a" assumes "subspace S"   and "independent X"   and "X \<subseteq> S"   and "S \<subseteq> span X"   and "pairwise orthogonal X" shows "v - proj_onto v X \<in> orthogonal_complement S"
405: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (x \<sqinter> y) \<sqinter> (- x \<sqinter> z) = - x \<sqinter> z"
ShiftP_Mem2: fixes u :: "name"   and f :: "tm"   and k :: "tm"   and del :: "tm"   and a :: "tm"   and b :: "tm"   and g :: "tm" assumes "atom u \<sharp> (f, k, del, a, b)" shows "{ShiftP f k del g, HPair a b IN g} \<turnstile> SyntaxN.Ex u (Var u IN k AND HaddP del (Var u) a AND HPair (Var u) b IN f)"
frameIntCompositionSym: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b"   and A\<^sub>F :: "name list"   and \<Psi>\<^sub>F :: "'b" assumes "\<Psi> \<simeq> \<Psi>'" shows "\<langle>A\<^sub>F, \<Psi>\<^sub>F \<otimes> \<Psi>\<rangle> \<simeq>\<^sub>F \<langle>A\<^sub>F, \<Psi>\<^sub>F \<otimes> \<Psi>'\<rangle>"
index_zero_implies_one_group: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and groups :: "'b set set"   and index :: "int"   and ps :: "'b set" assumes "GDD point_set block_collection groups index"   and "ps \<subseteq> point_set"   and "card ps = 2"   and "block_collection index ps = 0" shows "size (filter_mset ((\<subseteq>) ps) (mset_set groups)) = 1"
bounded_clinear_left: fixes prod :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and b :: "'c" assumes "bounded_cbilinear prod" shows "bounded_clinear (\<lambda>a. prod a b)"
strategy_attracts_def: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and A :: "'c set"   and W :: "'c set" assumes "ParityGame G" shows "ParityGame.strategy_attracts G p \<sigma> A W \<equiv> \<forall>v0\<in>A. ParityGame.strategy_attracts_via G p \<sigma> v0 A W"
smult_smult: fixes a :: "'a"   and b :: "'a"   and p :: "'a poly" shows "Polynomial.smult a (Polynomial.smult b p) = Polynomial.smult (a * b) p"
fps_of_poly_div: fixes q :: "'b poly"   and p :: "'b poly" assumes "q dvd p" shows "fps_of_poly (p div q) = fps_of_poly p / fps_of_poly q"
RUN_all_tickfree_traces1: fixes A :: "'a set" shows "T (RUN A) = {s. set s \<subseteq> ev ` A}"
minus_diff_commute: fixes b :: "'a"   and a :: "'a" shows "- b - a = - a - b"
measurable_empty_iff: fixes N :: "'a measure"   and f :: "'b \<Rightarrow> 'a"   and M :: "'b measure" assumes "space N = {}" shows "(f \<in> M \<rightarrow>\<^sub>M N) = (space M = {})"
S2: shows "(\<lambda>x. abs_of_diff (p1 x) (p2 x)) \<in> PrimRec1"
trace_eq_simcl_bind1_scale: fixes p :: "'a spmf"   and S :: "'b spmf \<Rightarrow> 'c spmf \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b spmf"   and q :: "'c spmf" assumes "\<forall>x\<in>set_spmf p. trace_eq_simcl S (f x) q" shows "trace_eq_simcl S (p \<bind> f) (scale_spmf (weight_spmf p) q)"
preorder_inf_closed: fixes x :: "'a"   and y :: "'a" assumes "preorder x"   and "preorder y" shows "preorder (x \<sqinter> y)"
vwelldefined_alt_def_2: fixes C_info :: "(name, dt_def) fmap"   and heads :: "name fset"   and cs :: "(Term_Class.term \<times> sterm) list"   and \<Gamma> :: "(name, value) fmap" shows "pre_constants.vwelldefined C_info heads (Vabs cs \<Gamma>) = (fmpred (\<lambda>_. pre_constants.vwelldefined C_info heads) \<Gamma> \<and> list_all (\<lambda>(uu_, t). consts t |\<subseteq>| pre_constants.all_consts C_info heads) cs)"
\<phi>_simps_5: shows "cod fg\<^sub>0h\<^sub>1.\<phi> = tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0"
M: fixes n :: "nat" assumes "M \<le> n" shows "f n \<noteq> (0::'a)"
diff_eq_diff_eq: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.group_add minus plus zero uminus"   and "minus a b = minus c d" shows "(a = b) = (c = d)"
exhaust: fixes y :: "k" obtains "y = kno"   | "y = unk"
type_less_eq_x_Optional: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a basic_type" assumes "\<tau> \<le> \<sigma>[?]" obtains \<rho> :: "'a basic_type" where "\<tau> = \<rho>__[1]" and "\<rho>__ \<le> \<sigma>"   | \<rho> :: "'a basic_type" where "\<tau> = \<rho>[?]" and "\<rho> \<le> \<sigma>"
take_bit_minus: fixes n :: "nat"   and k :: "int" shows "take_bit n (- take_bit n k) = take_bit n (- k)"
UN_constant_eq: fixes a :: "'a"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b set"   and c :: "'b set" assumes "a \<in> A"   and "\<forall>y\<in>A. f y = c" shows "\<Union> (f ` A) = c"
sec_pubk_less_msg: fixes m :: "nat"   and m' :: "nat"   and n :: "nat" assumes "{m, m'} \<subseteq> bad_sec_prik"   and "(n, m) \<notin> bad_id"   and "(n, m') \<in> bad_id" shows "{Sec_PriKey m, Sec_PriKey m', PubKey {Tok_PriK n, Rev_PriK}, ID n (Sec_PubKey m), Hash (ID n (Sec_PubKey m)), \<langle>n, ID n (Sec_PubKey m')\<rangle>} \<subseteq> spied (sec_pubk_less n) \<and> {\<langle>n, PubKey {Tok_PriK n, Rev_PriK}\<rangle>, \<langle>n, ID n (Sec_PubKey m)\<rangle>, \<langle>n, Hash (ID n (Sec_PubKey m))\<rangle>} \<inter> spied (sec_pubk_less n) = {}"
sorted_descending_alt: fixes xs :: "'b list" shows "sorted_descending xs = sorted (rev xs)"
dbproj_subseq_1: fixes Di :: "('d \<times> 'e) list"   and k :: "'d"   and D :: "('d \<times> 'e) list" assumes "Di \<in> set (subseqs (dbproj k D))" shows "dbproj k Di = Di"
Quotient_rep_reflp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and T :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "Quotient R Abs Rep T" shows "R (Rep a) (Rep a)"
fds_abs_converges_altdef': fixes f :: "'b fds"   and s :: "'b" shows "fds_abs_converges f s = ((\<lambda>n. fds_nth f n / nat_power n s) abs_summable_on UNIV)"
monoB: shows "monotone_on B (\<sqsubseteq>) (\<sqsubseteq>) f"
maxGon_def: fixes p :: "nat" shows "maxGon p \<equiv> p + 3"
best_def: shows "local.best = {x \<in> local.\<P>. \<forall>y\<in>local.\<P>. x \<succeq>[\<R>] y}"
multi_drop_mem_not_eq: fixes c :: "'k"   and B :: "'k multiset" assumes "c \<in># B" shows "B - {#c#} \<noteq> B"
Ann_2: shows "\<turnstile>\<^sub>! (p \<^bold>\<longleftrightarrow>\<^sub>! reduce p)"
fds_converges_cmult_left: fixes f :: "'a fds"   and s :: "'a"   and c :: "'a" assumes "fds_converges f s" shows "fds_converges (fds_const c * f) s"
length_butlast: fixes xs :: "'b list" shows "length (butlast xs) = length xs - 1"
a: shows "Esigma (a * u * q) = Esigma (a * u) * Esigma q"
closedin_locally_finite_Union: fixes \<A> :: "'b set set"   and X :: "'b topology" assumes "\<And>S. S \<in> \<A> \<Longrightarrow> closedin X S"   and "locally_finite_in X \<A>" shows "closedin X (\<Union> \<A>)"
term_variants_pred_inv_2: fixes P :: "'a \<Rightarrow> 'a list"   and f :: "'a"   and T :: "('a, 'b) Term.term list"   and h :: "'a"   and S :: "('a, 'b) Term.term list"   and i :: "nat" assumes "term_variants_pred P (Fun f T) (Fun h S)"   and "i < length T" shows "term_variants_pred P (T ! i) (S ! i)"
Lcm_eq_0_I: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "zero \<in> A" shows "Lcm A = zero"
FAssThrow2: fixes P :: "(char list list \<times> char list exp) prog"   and e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and v :: "val"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and e\<^sub>2 :: "char list exp"   and e' :: "char list exp"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>"
one_closed: shows "\<one>\<^bsub>S\<^esub> \<in> carrier S"
simps_2: fixes f :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'f"   and xs :: "'f list" shows "foldr f (x # xs) = f x \<circ> foldr f xs"
assms_14: shows "InvariantNoDecisionsWhenConflict (butlast (getF state)) (getM state) (currentLevel (getM state))"
squarefreeD: fixes n :: "'b"   and x :: "'b" assumes "squarefree n"   and "x\<^sup>2 dvd n" shows "x dvd (1::'b)"
remove_all_def: fixes x :: "'b"   and xs :: "'b list" shows "remove_all x xs = (if x \<in> set xs then tl (remove_cycles xs x []) else xs)"
cInf_lower2: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and X :: "'a set"   and y :: "'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "x \<in> X"   and "less_eq x y"   and "preorder.bdd_below less_eq X" shows "less_eq (Inf X) y"
set_integrable_cong: fixes M :: "'c measure"   and M' :: "'c measure"   and A :: "'c set"   and A' :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd" assumes "M = M'"   and "A = A'"   and "\<And>x. x \<in> A \<Longrightarrow> f x = f' x" shows "set_integrable M A f = set_integrable M' A' f'"
under_ofilter: fixes a :: "'a" shows "r.ofilter (r.under a)"
moves: shows "s1 -\<tau>1\<rightarrow>+ s1'"
trapped_backward_def: fixes x :: "'a"   and K :: "'a set" shows "rev.trapped_backward x K = (rev.flow0 x ` (rev.existence_ivl0 x \<inter> {..0}) \<subseteq> K)"
zero_less_one: fixes one :: "'d"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "class.zero_less_one one zero less_eq less" shows "less zero one"
in_round_intervalI: fixes x :: "real"   and A :: "float interval"   and prec :: "nat" assumes "x \<in>\<^sub>r A" shows "x \<in>\<^sub>r round_interval prec A"
mult_le_0_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less_eq (times a b) zero = (less_eq zero a \<and> less_eq b zero \<or> less_eq a zero \<and> less_eq zero b)"
append_rows_mult_left_id2: fixes A :: "'a mat"   and a :: "nat"   and n :: "nat"   and B :: "'a mat"   and b :: "nat"   and C :: "'a mat"   and D :: "'a mat" assumes "A \<in> carrier_mat a n"   and "B \<in> carrier_mat b n"   and "C = four_block_mat D (0\<^sub>m a b) (0\<^sub>m b a) (1\<^sub>m b)"   and "D \<in> carrier_mat a a" shows "C * (A @\<^sub>r B) = D * A @\<^sub>r B"
nrows_iarray_def: fixes A :: "'a iarray iarray" shows "nrows_iarray A = IArray.length A"
preallocated_start: fixes P :: "'a prog" shows "preallocated (start_heap P)"
scaleR_fun_def: fixes r :: "real"   and f :: "'a \<Rightarrow> 'b" shows "r *\<^sub>R f = (\<lambda>x. r *\<^sub>R f x)"
simps_4: fixes x2 :: "'e"   and x1 :: "'d" shows "Fault x2 \<noteq> Normal x1"
rtrancl_refl: fixes a :: "'a"   and r :: "('a \<times> 'a) set" shows "(a, a) \<in> r\<^sup>*"
fold_congs_3: fixes r :: "('b, 'c) state_scheme"   and r' :: "('b, 'c) state_scheme"   and v' :: "nat"   and f :: "nat \<Rightarrow> nat"   and f' :: "nat \<Rightarrow> nat" assumes "r = r'"   and "carrots r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "carrots_update f r = carrots_update f' r'"
Nil: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" shows "P \<turnstile> \<langle>[],s\<rangle> [\<Rightarrow>] \<langle>[],s\<rangle>"
cr_spmf_gpv_def: fixes p :: "'d spmf"   and gpv :: "('d, 'e, 'f) gpv" shows "cr_spmf_gpv p gpv = (gpv = lift_spmf p)"
eq_statesE: fixes s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and s' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "s \<approx> s'" obtains \<alpha> :: "'a \<Rightarrow> 'a" and \<beta> :: "'b \<Rightarrow> 'b" where "\<R>\<^sub>G \<alpha> \<beta> s = s'" and "bij \<alpha>" and "bij \<beta>"
cindex_poly_changes_itv_mods: fixes a :: "real"   and b :: "real"   and p :: "real poly"   and q :: "real poly" assumes "a < b"   and "poly p a \<noteq> 0"   and "poly p b \<noteq> 0" shows "cindex_poly a b q p = changes_itv_smods a b p q"
integral: fixes a :: "'a list"   and b :: "'a list" assumes "a \<otimes>\<^bsub>K [X]\<^esub> b = \<zero>\<^bsub>K [X]\<^esub>"   and "a \<in> carrier (K [X])"   and "b \<in> carrier (K [X])" shows "a = \<zero>\<^bsub>K [X]\<^esub> \<or> b = \<zero>\<^bsub>K [X]\<^esub>"
Le_enat_imp_ile_enat_k: fixes m :: "enat"   and n :: "enat"   and l :: "nat" assumes "(m, n) \<in> Le_enat"   and "n < enat l" shows "m < enat l"
pmf_of_set: fixes S :: "'a set"   and x :: "'a" assumes "S \<noteq> {}"   and "finite S" shows "pmf (pmf_of_set S) x = indicat_real S x / real (card S)"
norm_le: fixes x :: "'a"   and y :: "'b" shows "(\<parallel>x\<parallel> \<le> \<parallel>y\<parallel>) = (x \<bullet> x \<le> y \<bullet> y)"
mult_sm_neutral: fixes A :: "complex \<times> complex \<times> complex \<times> complex" shows "1 *\<^sub>s\<^sub>m A = A"
ctr_2: fixes p :: "'c \<Rightarrow> bool"   and a :: "'c"   and g21 :: "'c \<Rightarrow> 'd"   and g22 :: "'c \<Rightarrow> 'c" assumes "\<not> p a" shows "unfold_llist p g21 g22 a = LCons (g21 a) (unfold_llist p g21 g22 (g22 a))"
intros: fixes x :: "'a"   and y :: "'a" shows "foo x y"
less_fract_def: fixes z :: "'a fract"   and w :: "'a fract" shows "(z < w) = (z \<le> w \<and> \<not> w \<le> z)"
suffix_to_prefix: fixes xs :: "'a list"   and ys :: "'a list" shows "suffix xs ys = prefix (rev xs) (rev ys)"
StepExec_determ_Aux: fixes n :: "nat" shows "(\<forall>n1 l M s l1 t. n1 \<le> n \<longrightarrow> (M, l, s, n1, l1, t) \<in> Step \<longrightarrow> (\<forall>n2 l2 r. (M, l, s, n2, l2, r) \<in> Step \<longrightarrow> n1 = n2 \<and> t = r \<and> l1 = l2)) \<and> (\<forall>n1 l M s h v. n1 \<le> n \<longrightarrow> (M, l, s, n1, h, v) \<in> Exec \<longrightarrow> (\<forall>n2 k w. (M, l, s, n2, k, w) \<in> Exec \<longrightarrow> n1 = n2 \<and> h = k \<and> v = w))"
hyps_3: fixes a :: "complex"   and s :: "complex set"   and thesis :: "bool" assumes "\<And>g. \<lbrakk>valid_path g; pathstart g = a; pathfinish g = b; path_image g \<subseteq> s - pts; f contour_integrable_on g\<rbrakk> \<Longrightarrow> thesis"   and "open s"   and "connected (s - pts)"   and "finite pts"   and "f holomorphic_on s - pts"   and "a \<in> s - pts"   and "b \<in> s - pts" shows "thesis"
basis_enum_eq_vec_of_basis_enumI: fixes a :: "'b"   and b :: "'b" assumes "vec_of_basis_enum a = vec_of_basis_enum b" shows "a = b"
homeomorphism_UNIV_imp_open_map: fixes p :: "'c \<Rightarrow> 'd"   and p' :: "'d \<Rightarrow> 'c"   and f' :: "'c set" assumes "homeomorphism UNIV UNIV p p'"   and "open f'" shows "open (p ` f')"
rtrancl_path_Range: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list"   and y :: "'a"   and z :: "'a" assumes "rtrancl_path R x xs y"   and "z \<in> set xs" shows "Rangep R z"
\<pi>k': shows "(\<pi> \<guillemotleft> k) k' = x"
\<sigma>_2: fixes \<sigma>' :: "'a \<Rightarrow> 'a" assumes "(\<sigma>', \<sigma>) \<in> r - Id" shows "\<sigma>' \<notin> good v"
density_context_if_dens_det: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and e :: "expr"   and b :: "bool" assumes "density_context V V' \<Gamma> \<delta>"   and "\<Gamma> \<turnstile> e : BOOL"   and "randomfree e"   and "free_vars e \<subseteq> V \<union> V'" shows "density_context V V' \<Gamma> (if_dens_det \<delta> e b)"
commute: fixes a :: "'b DBMEntry"   and b :: "'b DBMEntry" shows "dbm_add a b = dbm_add b a"
iteration_subdist: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<infinity> \<le> (x + y)\<^sup>\<infinity>"
LI_no_infinite_chain: shows "\<nexists>f. \<forall>i. f i \<leadsto>\<^sup>+ f (Suc i)"
poly_root_order_induct: fixes P :: "'b poly \<Rightarrow> bool"   and p :: "'b poly" assumes "P 0"   and "\<And>p. (\<And>x. poly p x \<noteq> (0::'b)) \<Longrightarrow> P p"   and "\<And>p x n. \<lbrakk>0 < n; poly p x \<noteq> (0::'b); P p\<rbrakk> \<Longrightarrow> P ([:- x, 1::'b:] ^ n * p)" shows "P p"
cnj_EH_1: fixes A :: "fmla"   and B :: "fmla"   and H :: "fmla set"   and C :: "fmla" assumes "insert A (insert B H) \<turnstile> C" shows "insert (A AND B) H \<turnstile> C"
Act_def: shows "Formula.Act \<equiv> (id ---> Rep_formula ---> Abs_formula) Act\<^sub>\<alpha>"
regular_event_implies_same_channel_snapshot_Recv: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "\<not> isRecv ev"   and "regular_event ev"   and "isRecv ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "cs e i = cs e' i"
zero_le_dist: fixes x :: "'a"   and y :: "'a" shows "0 \<le> dist x y"
cInf_ge: fixes S :: "'a set"   and b :: "'a" assumes "S \<noteq> {}"   and "b <=* S" shows "b \<le> Inf S"
zero_smallest: fixes j :: "'a" assumes "j \<in> Field r" shows "(local.zero, j) \<in> r"
closed_subspace_closed_sum: fixes A :: "'b set"   and B :: "'b set" assumes "csubspace A"   and "csubspace B" shows "closed_csubspace (A +\<^sub>M B)"
assms_1: shows "a \<in>\<^sub>\<circ> \<Inter>\<^sub>\<circ> A"
Un_fTo_Chain_mem_fTo: fixes CC :: "'a set set" assumes "Chain fTo D CC" shows "\<Union> CC \<in> carrier fTo D"
hmsetmset_inject: fixes A :: "hmultiset"   and B :: "hmultiset" shows "(hmsetmset A = hmsetmset B) = (A = B)"
agg_sum_distrib: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c" shows "aggregation.sum_0 (\<lambda>k. f k + g k) {k. True} = aggregation.sum_0 f {k. True} + aggregation.sum_0 g {k. True}"
right_edge_explicit: shows "(\<lambda>y. twoC (1, y)) = (\<lambda>x. (g1 (a + (b - a) * x), a + (b - a) * x))"
correct: fixes \<G> :: "'a cyclic_group"   and m :: "nat" assumes "pedersen \<G>" shows "spmf (abstract_commitment.correct_game (pedersen_base.key_gen \<G>) (pedersen_base.commit \<G>) (pedersen_base.verify \<G>) m) True = 1"
v11_vconverse_app_in_vdomain: fixes y :: "V" assumes "y \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "(\<FF>\<lparr>ObjMap\<rparr>)\<inverse>\<^sub>\<circ>\<lparr>y\<rparr> \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"