hcju/isabelleps huggingface dataset:
bezout_coefficients_i: shows "(poly_rel ===> poly_rel ===> rel_prod poly_rel poly_rel) (bezout_coefficients_i ff_ops) bezout_coefficients"
simps_3: fixes n :: "nat"   and x :: "bool"   and xs :: "bool list" shows "flip (Suc n) (x # xs) = x # flip n xs"
heap_heapify: fixes n :: "nat"   and xs :: "'b list" assumes "n \<le> length xs" shows "heap (fst (heapify n xs))"
before: fixes i :: "'b" assumes "\<I> i" shows "BEGIN i \<lless> END i"
succ_in_Limit_iff: fixes \<xi> :: "V"   and \<alpha> :: "V" assumes "Limit \<xi>" shows "(ZFC_in_HOL.succ \<alpha> \<in>\<^sub>\<circ> \<xi>) = (\<alpha> \<in>\<^sub>\<circ> \<xi>)"
angle_def: fixes A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'c" shows "<A>_v \<equiv> A \<and> \<not> unchanged v"
elims: fixes x :: "('f, 'b) Term.term"   and y :: "nat" assumes "weight x = y" obtains x :: "'b" where "xa__ = Var x" and "y = w0"   | f :: "'f" and ts :: "('f, 'b) Term.term list" where "xa__ = Fun f ts" and "y = (let n = length ts; scff = scf (f, n) in w (f, n) + sum_list (map2 (\<lambda>ti i. weight ti * scff i) ts [0..<n]))"
cin_def: shows "(\<in>\<^sub>c) \<equiv> map_fun id (map_fun rcset id) (\<in>)"
HSigma_def: fixes A :: "hf"   and B :: "hf \<Rightarrow> hf" shows "HSigma A B = (\<Squnion>x\<^bold>\<in>A. \<Squnion>y\<^bold>\<in>B x. \<lbrace>\<langle>x, y\<rangle>\<rbrace>)"
mono_imp_derivation_f_refl: shows "\<forall>X x. derivation X \<longrightarrow> x \<in> X \<longrightarrow> f x \<sqsubseteq> f x"
sectionE: fixes f :: "'a" assumes "section f" obtains g :: "'a" where "ide (g \<cdot> f)"
Gauss_Jordan_column_k_def: fixes A' :: "nat \<times> (('a, 'b) vec, 'c) vec"   and k :: "nat" shows "Gauss_Jordan_column_k A' k = (let i = fst A'; A = snd A'; from_nat_i = mod_type_class.from_nat i; from_nat_k = mod_type_class.from_nat k in if (\<forall>m\<ge>from_nat_i. A $ m $ from_nat_k = (0::'a)) \<or> i = nrows A then (i, A) else (i + 1, Gauss_Jordan_in_ij A from_nat_i from_nat_k))"
Inf_apply: fixes A :: "('l \<Rightarrow> 'k) set"   and x :: "'l" shows "Inf A x = (INF f\<in>A. f x)"
assms: shows "local.laplace_integrand s integrable_on {0..}"
IH: fixes x2a :: "('fun, 'var) Term.term" assumes "x2a \<in> set T"   and "s \<sqsubseteq> x2a"   and "wf\<^sub>t\<^sub>r\<^sub>m x2a" shows "\<Gamma> s \<sqsubseteq> \<Gamma> x2a"
SdlHPPB_restricted_on_PP_is_SdlHPPB: fixes d :: "'d"   and PP :: "nat set"   and R' :: "(('exp, 'id) MWLsCom list \<times> ('exp, 'id) MWLsCom list) set"   and V :: "('exp, 'id) MWLsCom list"   and R :: "(('exp, 'id) MWLsCom list \<times> ('exp, 'id) MWLsCom list) set" assumes "SdlHPPB d PP R'"   and "(V, V) \<in> R'"   and "R = {(V', V''). (V', V'') \<in> R' \<and> set (PPV V') \<subseteq> set (PPV V) \<and> set (PPV V'') \<subseteq> set (PPV V)}" shows "SdlHPPB d PP R"
prems_6: shows "InvariantConflictClauseCharacterization (getConflictFlag state') (getConflictClause state') (getF state') (getM state')"
Cons: fixes v :: "'a"   and w :: "'a"   and vs :: "'a list" assumes "v \<rightarrow>\<^bsub>remove_vertex x\<^esub> w"   and "H.walk (w # vs)" shows "H.walk (v # w # vs)"
const_to_constant: fixes a :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp" shows "to_fun (monom (UP Zp) a 0) = \<cc>\<^bsub>Zp\<^esub> a"
surj_Compl_image_subset: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "surj f" shows "- f ` A \<subseteq> f ` (- A)"
prime_factorization_prod_mset_primes: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and A :: "'a multiset" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "\<And>p. p \<in># A \<Longrightarrow> normalization_semidom.prime zero times one normalize p" shows "factorial_semiring.prime_factorization zero times one normalize (comm_monoid_mult.prod_mset times one A) = A"
dg_Rel_is_arrI: fixes \<alpha> :: "V"   and S :: "V"   and A :: "V"   and B :: "V" assumes "arr_Rel \<alpha> S"   and "S\<lparr>ArrDom\<rparr> = A"   and "S\<lparr>ArrCod\<rparr> = B" shows "S : A \<mapsto>\<^bsub>dg_Rel \<alpha>\<^esub> B"
smooth_path_imp_reverse: fixes g :: "real \<Rightarrow> 'a" assumes "g C1_differentiable_on {0..1}" shows "reversepath g C1_differentiable_on {0..1}"
image_empty: fixes f :: "'b \<Rightarrow> 'a" shows "f ` {} = {}"
intro: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and k :: "nat set" assumes "Network c s t"   and "Cut c k"   and "NCut_axioms s t k" shows "NCut c s t k"
omega_transfer_domain_rule: shows "Domainp cr_omega = (\<lambda>x. x \<in>\<^sub>\<circ> \<omega>)"
mono1: fixes b :: "'b" shows "monotone orda ordc (\<lambda>a. f (a, b))"
elims: fixes x :: "'c stone_phi_pair"   and y :: "('c regular, 'c dense) lifted_pair" assumes "stone_phi_embed x = y" obtains xa :: "'c stone_phi_pair" where "x = xa" and "y = Abs_lifted_pair (\<lambda>f. if Rep_phi f = stone_phi then Rep_stone_phi_pair xa else triple.pairs_uminus (Rep_phi f) (triple.pairs_uminus (Rep_phi f) (Rep_stone_phi_pair xa)))"
unit_inv_comm: fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and y :: "'a" assumes "Dmonoid prod one"   and "prod x y = one"   and "Dmonoid.unit prod one x"   and "Dmonoid.unit prod one y" shows "prod y x = one"
mem_ball_imp_mem_cball: fixes x :: "'a"   and y :: "'a"   and e :: "real" assumes "x \<in> ball y e" shows "x \<in> cball y e"
sym_iff_eq_refl: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "(x \<sim> y) = (x = y \<and> y \<sqsubseteq> y)"
extreme_boundI: fixes X :: "'a set"   and s :: "'a" assumes "\<And>b. \<lbrakk>bound X (\<sqsubseteq>) b; b \<in> A\<rbrakk> \<Longrightarrow> s \<sqsubseteq> b"   and "\<And>x. x \<in> X \<Longrightarrow> x \<sqsubseteq> s"   and "s \<in> A" shows "local.extreme_bound X s"
N_nonempty: fixes M2 :: "('e, 'f, 'g) FSM"   and V :: "'e list set"   and M1 :: "('e, 'h, 'i) FSM"   and m :: "nat"   and io :: "('e \<times> 'h) list" assumes "is_det_state_cover M2 V"   and "OFSM M1"   and "OFSM M2"   and "asc_fault_domain M2 M1 m"   and "io \<in> L M1" shows "N io M1 V \<noteq> {}"
hyps_free_path_not_in_scope: fixes v :: "'vertex"   and t :: "'vertex"   and pth :: "(('vertex \<times> ('form, 'var) out_port) \<times> 'vertex \<times> ('form, 'var) antecedent) list"   and v' :: "'vertex"   and p' :: "('form, 'var) antecedent"   and p :: "('form, 'var) antecedent" assumes "terminal_path v t pth"   and "(v', p') \<in> snd ` set pth" shows "v' \<notin> scope (v, p)"
pmult_Nil: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and l2 :: "'b list" assumes "class.semiring_0 plus zero times" shows "semiring_0.pmult plus zero times [] l2 = []"
Levellist_spec_total': fixes \<Theta> :: "(('a globals_scheme, 'b) bdd_state_scheme set \<times> char list \<times> ('a globals_scheme, 'b) bdd_state_scheme set \<times> ('a globals_scheme, 'b) bdd_state_scheme set) set" shows "\<forall>ll \<sigma> t. \<Gamma>,\<Theta>\<turnstile>\<^sub>t \<lbrace>\<sigma>. Dag \<acute>p \<acute>low \<acute>high t \<and> (\<acute>p \<noteq> Null \<longrightarrow> \<acute>p\<rightarrow>\<acute>var < length \<acute>levellist) \<and> ordered t \<acute>var \<and> Levellist \<acute>levellist \<acute>next ll \<and> (\<forall>n\<in>set_of t. if n\<rightarrow>\<acute>mark = \<acute>m then n \<in> set (ll ! (n\<rightarrow>\<acute>var)) \<and> (\<forall>nt p. Dag n \<acute>low \<acute>high nt \<and> p \<in> set_of nt \<longrightarrow> p\<rightarrow>\<acute>mark = \<acute>m) else n \<notin> set (concat ll))\<rbrace> \<acute>levellist :== PROC Levellist(\<acute>p,\<acute>m,\<acute>levellist) \<lbrace>\<exists>ll'. Levellist \<acute>levellist \<acute>next ll' \<and> wf_levellist t ll ll' \<^bsup>\<sigma>\<^esup>var \<and> wf_marking t \<^bsup>\<sigma>\<^esup>mark \<acute>mark \<^bsup>\<sigma>\<^esup>m \<and> (\<forall>p. p \<notin> set_of t \<longrightarrow> \<^bsup>\<sigma>\<^esup>next p = p\<rightarrow>\<acute>next)\<rbrace>"
card_image_Nsetn_Suc: fixes f :: "nat \<Rightarrow> 'e"   and n :: "nat"   and B :: "'e set" assumes "f \<in> {j. j \<le> Suc n} \<rightarrow> B"   and "f (Suc n) \<notin> f ` {j. j \<le> n}" shows "card (f ` {j. j \<le> Suc n}) - Suc 0 = Suc (card (f ` {j. j \<le> n}) - Suc 0)"
scons_eq: fixes a :: "'b"   and v :: "'b stream"   and w :: "'b stream" shows "(a ## v = w) = (shd w = a \<and> stl w = v)"
roots_2: shows "\<rho> v \<^sup>@ l = v"
has_derivative_inner_right: fixes g :: "'a \<Rightarrow> 'b"   and g' :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and a :: "'b" assumes "(g has_derivative g') F" shows "((\<lambda>x. a \<bullet> g x) has_derivative (\<lambda>x. a \<bullet> g' x)) F"
winding_number_exp_integral_2: fixes \<gamma> :: "real \<Rightarrow> complex"   and a :: "real"   and b :: "real"   and z :: "complex" assumes "\<gamma> piecewise_C1_differentiable_on {a..b}"   and "a \<le> b"   and "z \<notin> \<gamma> ` {a..b}" shows "exp (- integral {a..b} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))) * (\<gamma> b - z) = \<gamma> a - z"
dX: fixes X :: "real \<Rightarrow> 'a" assumes " D X = (\<lambda>\<tau>. f \<tau> (X \<tau>)) on U (X t\<^sub>0)" shows " D (\<lambda>\<tau>. \<mu> (X \<tau>) - \<nu> (X \<tau>)) = (*\<^sub>R) 0 on U (X t\<^sub>0)"
poly_coeff_sum: fixes p :: "('d monom \<times> 'c) list" shows "(0::'c) \<le> poly_coeff_sum p"
IH: fixes n :: "nat" assumes "n < length ((x + y) # xs)" shows "psums ((x + y) # xs) ! n = sum ((!) ((x + y) # xs)) {..n}"
gen_upper_bound_decreasing: fixes x :: "real"   and a :: "real"   and fu :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "x \<le> a"   and "\<And>y. \<lbrakk>x \<le> y; y \<le> a\<rbrakk> \<Longrightarrow> ((\<lambda>x. fu x - f x) has_real_derivative g y) (at y)"   and "\<And>y. \<lbrakk>x \<le> y; y \<le> a\<rbrakk> \<Longrightarrow> g y \<le> 0"   and "fu a = f a" shows "f x \<le> fu x"
set_eq_fold: fixes xs :: "'a list" shows "lcm_list xs = fold lcm xs (1::'a)"
rotate_conv_mod': fixes n :: "nat"   and ls :: "'c list" shows "rotate (n mod |ls|) ls = rotate n ls"
append_self_conv2: fixes xs :: "'k list"   and ys :: "'k list" shows "(xs @ ys = ys) = (xs = [])"
add_zero: fixes x :: "'a" shows "x \<oplus> (0::'a) = Some x"
eq: fixes U' :: "'b set" assumes "f ` topspace X = topspace X'"   and "U' \<subseteq> topspace X'" shows "topspace X - {x \<in> topspace X. f x \<in> U'} = {x \<in> topspace X. f x \<in> topspace X' \<and> f x \<notin> U'}"
order_antisym_conv: fixes y :: "'b"   and x :: "'b" assumes "y \<le> x" shows "(x \<le> y) = (x = y)"
divmod_numeral_simps_18: fixes m :: "num"   and n :: "num" shows "(num.Bit0 m \<le> num.Bit1 n) = (m \<le> n)"
tp_mult: fixes f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "trigpoly f"   and "trigpoly g" shows "trigpoly (\<lambda>x. f x * g x)"
Abs_data_inverse: fixes y :: "'a list \<times> 'a dataspace" assumes "y \<in> data" shows "Rep_data (Abs_data y) = y"
rep_eq: shows "(\<langle>$\<rangle>) 1 = id"
C1: shows "f (nat (padic_val p f)) = \<zero>\<^bsub>residue_ring (p ^ nat (padic_val p f))\<^esub>"
simps_1: fixes f :: "'a" shows "\<^bold>\<lfloor>\<^bold>\<langle>f\<^bold>\<rangle>\<^bold>\<rfloor> = \<^bold>\<langle>f\<^bold>\<rangle>"
hyps_1: fixes Z :: "'b" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> {s. s = Z \<and> \<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F) \<and> \<Gamma>\<turnstile>the (\<Gamma> p) \<down> Normal \<sigma> \<and> (\<exists>cs css. \<Gamma>\<turnstile> ([the (\<Gamma> p)], [], Normal \<sigma>) \<rightarrow>\<^sup>* (c\<^sub>1 # cs, css, Normal s))} c\<^sub>1 {t. \<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal Z\<rangle> \<Rightarrow> Abrupt t}"
transpE: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "transp r"   and "r x y"   and "r y z" obtains "r x z"
fmupd_reorder_neq: fixes a :: "'c"   and b :: "'c"   and x :: "'d"   and y :: "'d"   and m :: "('c, 'd) fmap" assumes "a \<noteq> b" shows "m(b $$:= y, a $$:= x) = m(a $$:= x, b $$:= y)"
intro: fixes act :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and G :: "('b, 'c) monoid_scheme" assumes "\<And>b. act b \<one>\<^bsub>G\<^esub> = b"   and "\<And>g h b. g \<in> carrier G \<and> h \<in> carrier G \<longrightarrow> act (act b g) h = act b (g \<otimes>\<^bsub>G\<^esub> h)" shows "action_axioms G act"
eq_iff: fixes x :: "'a Word.word"   and y :: "'a Word.word" shows "(x = y) = (x \<le>s y \<and> y \<le>s x)"
Di'_def: fixes i :: "nat"   and y :: "nat list" shows "Di' i y = {dij i (j + n - length y) - 1 |j. j < length y \<and> eij i (j + n - length y) \<le> y ! j}"
icds_3: shows "i' icd\<^bsup>\<pi>'\<^esup>\<rightarrow> k'"
a3: fixes S' :: "atom set" assumes "finite S'"   and "S' supports x" shows "S \<subseteq> S'"
simps_2: fixes f :: "'a"   and t :: "'a"   and ts :: "'a list" shows "list_comb f (t # ts) = list_comb (app f t) ts"
valid_unconc_2: fixes fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and x :: "'a list"   and c' :: "'a list multiset" shows "valid fg (add_mset x c') = (valid fg c' \<and> mon_s fg x \<inter> mon_c fg c' = {})"
top_closed: fixes L :: "('a, 'b) gorder_scheme" assumes "weak_partial_order_top L" shows "\<top>\<^bsub>L\<^esub> \<in> carrier L"
snoc: shows "set as' \<Turnstile>\<^sub>i \<B>\<I> (assert_all_state t as')"
Main_no_return_source: fixes sourcenode :: "'g \<Rightarrow> 'h"   and targetnode :: "'g \<Rightarrow> 'h"   and kind :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and valid_edge :: "'g \<Rightarrow> bool"   and Entry :: "'h"   and get_proc :: "'h \<Rightarrow> 'l"   and get_return_edges :: "'g \<Rightarrow> 'g set"   and procs :: "('l \<times> 'i list \<times> 'i list) list"   and Main :: "'l"   and a :: "'g"   and Q' :: "('i \<Rightarrow> 'j option) \<times> 'k \<Rightarrow> bool"   and f' :: "('i \<Rightarrow> 'j option) \<Rightarrow> ('i \<Rightarrow> 'j option) \<Rightarrow> 'i \<Rightarrow> 'j option" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "kind a = Q'\<hookleftarrow>\<^bsub>Main\<^esub>f'" shows "False"
inf_SUP: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and a :: "'b"   and f :: "'c \<Rightarrow> 'b"   and B :: "'c set" assumes "class.complete_distrib_lattice Inf Sup inf less_eq less sup bot top" shows "inf a (Sup (f ` B)) = Sup ((\<lambda>b. inf a (f b)) ` B)"
eval_equality_simplify: fixes i :: "char list \<Rightarrow> tv"   and p :: "fm"   and q :: "fm" shows "eval i (Eql p q) = Det (eval i p = eval i q)"
p: fixes p :: "nat"   and pm :: "nat" assumes "fixed_root p pm" shows "p = Suc pm"
isom_gch_units_transpTr8_2: fixes E :: "('e, 'f) Group_scheme"   and n :: "nat"   and g :: "nat \<Rightarrow> ('g set, 'h) Group_scheme"   and i :: "nat"   and j :: "nat" assumes "Ugp E"   and "Gchain n g"   and "i \<le> n"   and "j \<le> n"   and "\<not> g i \<cong> E"   and "\<not> g j \<cong> E" shows "{i. i \<le> n \<and> g i \<cong> E} = {i. i \<le> n \<and> g i \<cong> E} - {i, j}"
abs_neg: fixes a :: "'a" assumes "a \<le> (0::'a)" shows "\<bar>a\<bar> = - a"
ge_refl: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.non_strict_order less_eq" shows "less_eq x x"
simple_def: fixes Pm :: "'a \<Rightarrow> ('b \<times> 'b) set"   and Pw :: "'b \<Rightarrow> ('a \<times> 'a) set"   and \<mu> :: "('a \<times> 'b) set" assumes "StableMarriage Pm Pw" shows "StableMarriage.simple Pm Pw \<mu> = (StableMarriage.match Pm Pw \<mu> \<and> (\<forall>m w. StableMarriage.blocks Pm Pw m w \<mu> \<longrightarrow> w \<notin> Range \<mu>))"
flush_append: fixes m :: "nat \<Rightarrow> nat"   and sb :: "'b memref list"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option"   and m' :: "nat \<Rightarrow> nat"   and sb' :: "'b memref list"   and \<O>' :: "nat set"   and \<R>' :: "nat \<Rightarrow> bool option"   and \<S>' :: "nat \<Rightarrow> bool option"   and xs :: "'b memref list" assumes "(m, sb, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>f\<^sup>* (m', sb', \<O>', \<R>', \<S>')" shows "(m, sb @ xs, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>f\<^sup>* (m', sb' @ xs, \<O>', \<R>', \<S>')"
monitor_def: fixes \<phi> :: "'b MFOTL.formula"   and \<pi> :: "(char list \<times> 'b list) prefix" shows "Monitor.monitor \<phi> \<pi> = msteps_stateless \<pi> (minit_safe \<phi>)"
simps_2: fixes xs :: "'d list" shows "stl (cycle xs) = cycle (tl xs @ [hd xs])"
I2: assumes "Arr u" shows "\<guillemotleft>\<lbrace>u\<rbrace> : \<lbrace>Dom u\<rbrace> \<rightarrow> \<lbrace>Cod u\<rbrace>\<guillemotright>"
set_map_1: fixes f1 :: "'f \<Rightarrow> 'c"   and f2 :: "'g \<Rightarrow> 'd"   and f3 :: "'h \<Rightarrow> 'e"   and v :: "('f, 'g, 'h) F2" shows "F2set1 (F2map f1 f2 f3 v) = f1 ` F2set1 v"
interp_imp_Interp: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and C :: "'b literal multiset"   and D :: "'b literal multiset"   and D' :: "'b literal multiset"   and N :: "'b literal multiset set" assumes "ground_resolution_with_selection S"   and "C \<le> D"   and "D \<le> D'"   and "ground_resolution_with_selection.interp S N D \<Turnstile> C" shows "ground_resolution_with_selection.Interp S N D' \<Turnstile> C"
rel_spmf_reflI: fixes p :: "'b spmf"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> P x x" shows "rel_spmf P p p"
lcm_neg2_int: fixes x :: "int"   and y :: "int" shows "lcm x (- y) = lcm x y"
DBM_val_bounded_len_1'_aux: fixes v :: "'c \<Rightarrow> nat"   and u :: "'c \<Rightarrow> 'd"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and n :: "nat"   and c :: "'c"   and vs :: "nat list" assumes "u \<turnstile>\<^bsub>v,n\<^esub> m"   and "v c \<le> n"   and "\<forall>k\<in>set vs. 0 < k \<and> k \<le> n \<and> (\<exists>c. v c = k)" shows "dbm_entry_val u (Some c) None (len m (v c) 0 vs)"
chain_union_closed': fixes f :: "nat \<Rightarrow> 'b set"   and C :: "'b set set"   and S' :: "'b set" assumes "is_chain f"   and "\<forall>n. f n \<in> C"   and "\<forall>S'\<in>C. \<forall>S\<subseteq>S'. S \<in> C"   and "finite S'"   and "S' \<subseteq> \<Union> (range f)" shows "S' \<in> C"
to_woot_character_keeps_i_lifted: fixes M :: "('a, 'b) message set"   and X :: "('a, 'b) insert_message set" shows "I ` to_woot_character M ` X = I ` X"
hd_nonzero_imp_sign_changes_const: fixes x\<^sub>0 :: "real" assumes "poly (hd ps) x\<^sub>0 \<noteq> 0" shows "\<forall>\<^sub>F x in at x\<^sub>0. sign_changes ps x = sign_changes ps x\<^sub>0"
A_rw: shows "(\<chi>i. if i = a then A $ a else A $ i) = A"
ltrm_of_X_plus: fixes a :: "'a" assumes "a \<in> carrier R"   and "\<one> \<noteq> \<zero>" shows "monom P (X_poly_plus R a (deg R (X_poly_plus R a))) (deg R (X_poly_plus R a)) = X_poly R"
that: fixes z :: "complex" assumes "z \<in> inside (path_image (p +++ linepath (a - complex_of_real e) (a + complex_of_real e)))"   and "cmod (winding_number (p +++ linepath (a - complex_of_real e) (a + complex_of_real e)) z) = 1" shows "thesis"
ProofStepInductOptimalPaths_axioms_def: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and P_new :: "'c list"   and sep_size :: "nat" shows "ProofStepInductOptimalPaths_axioms G v0 v1 P_new sep_size \<equiv> \<forall>paths' P_new'. ProofStepInduct G v0 v1 paths' P_new' sep_size \<longrightarrow> Digraph.distance (Digraph.remove_vertex G v0) (last P_new) v1 \<le> Digraph.distance (Digraph.remove_vertex G v0) (last P_new') v1"
simps_2: fixes x :: "int poly fract list"   and xs :: "int poly fract list list" shows "rat_poly.reduct (x # xs) = xs"
VLambda_def: fixes A :: "V"   and b :: "V \<Rightarrow> V" shows "VLambda A b \<equiv> set ((\<lambda>x. \<langle>x, b x\<rangle>) ` elts A)"
clm_compose: fixes c :: "cars" shows "\<forall>ts v. (\<exists>va u. v=va\<parallel>u \<and> (0 < \<parallel>ext va\<parallel> \<and> len va ts c = ext va \<and> restrict va (clm ts) c = lan va \<and> |lan va| = 1) \<and> 0 < \<parallel>ext u\<parallel> \<and> len u ts c = ext u \<and> restrict u (clm ts) c = lan u \<and> |lan u| = 1) \<longrightarrow> 0 < \<parallel>ext v\<parallel> \<and> len v ts c = ext v \<and> restrict v (clm ts) c = lan v \<and> |lan v| = 1"
uniqueConclusion_def: fixes R :: "('a sequent list \<times> 'a sequent) set" shows "uniqueConclusion R \<equiv> \<forall>r1\<in>R. \<forall>r2\<in>R. snd r1 = snd r2 \<longrightarrow> r1 = r2"
bilateral_substitutes_on_def2: fixes A :: "'x set"   and f :: "'x set \<Rightarrow> 'x set" shows "bilateral_substitutes_on A f = (\<forall>B\<subseteq>A. \<forall>a\<in>A. \<forall>b\<in>A. Xd a \<notin> Xd ` B \<and> Xd b \<notin> Xd ` B \<and> b \<notin> f (B \<union> {b}) \<longrightarrow> b \<notin> f (B \<union> {a, b}))"
oDeriv_increasing: fixes F :: "ordinal \<Rightarrow> ordinal"   and x :: "ordinal" assumes "continuous F" shows "F x \<le> oDeriv F x"
bconf_RI: fixes P :: "'a prog"   and sh :: "sheap"   and C :: "char list"   and e :: "'b exp"   and Cs :: "char list list"   and e' :: "'b exp"   and b :: "bool" shows "(P,sh \<turnstile>\<^sub>b (RI (C,e) ; Cs \<leftarrow> e',b) \<surd>) = (P,sh \<turnstile>\<^sub>b (e,b) \<surd>)"
wf_mdecl_def: fixes wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'a \<Rightarrow> bool"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list" shows "wf_mdecl wf_md P C \<equiv> \<lambda>(M, Ts, T, mb). (\<forall>T\<in>set Ts. is_type P T) \<and> is_type P T \<and> wf_md P C (M, Ts, T, mb)"
E2: fixes x2 :: "'b"   and x2a :: "'d" assumes "e = Inr x2"   and "e' = Inr x2a" shows "hn_refine (hn_ctxt B x2 x2a * INVe * F) (f2' x2a) (hn_ctxt B' x2 x2a * hn_ctxt XX2 e e' * \<Gamma>2') R (f2 x2)"
dim_mult_mat_vec: fixes A :: "'c mat"   and v :: "'c vec" shows "dim_vec (A *\<^sub>v v) = dim_row A"
sp_from_de_empty: fixes v1 :: "'c \<times> nat"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set"   and g :: "('c \<times> nat, 'd) rgraph_scheme"   and es :: "('c \<times> nat) edge list"   and v2 :: "'c \<times> nat" assumes "v1 \<notin> subsumees subs"   and "out_edges g v1 = {}"   and "SubRel.subpath g v1 es v2 subs" shows "es = []"
mod_0_imp_dvd: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_modulo plus minus zero times one divide modulo"   and "modulo a b = zero" shows "dvd.dvd times b a"
reduced_mkIF: fixes x :: "'d"   and X :: "'d set"   and t1 :: "'d ifex"   and t2 :: "'d ifex" assumes "x \<notin> X"   and "reduced t1 (insert x X)"   and "reduced t2 (insert x X)" shows "reduced (mkIF x t1 t2) X"
sum_le_zero: fixes s :: "'lbl" shows "(s \<le> (0::'lbl)) = (s = (0::'lbl))"
p'_def: shows "p' = p \<circ>\<^sub>p [:0, \<i>:] * p \<circ>\<^sub>p [:0, - \<i>:]"
signed_divide_int_def: fixes k :: "int"   and l :: "int" shows "k sdiv l = sgn k * sgn l * (\<bar>k\<bar> div \<bar>l\<bar>)"