hcju/isabelleps huggingface dataset:
epm_9: fixes e :: "'b"   and v :: "'b"   and w :: "'b" assumes "e \<le> v \<sqinter> - id v"   and "bounded_distrib_lattice_class.comp_inf.vector v" shows "w \<sqinter> - stone_algebra_class.comp_inf.prim_EP w v e \<sqinter> e = bot"
alphaAll_qSkelAll: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and A :: "('a, 'b, 'varSort, 'var, 'c) qAbs" shows "(\<forall>X'. X #= X' \<longrightarrow> qSkel X = qSkel X') \<and> (\<forall>A'. A $= A' \<longrightarrow> qSkelAbs A = qSkelAbs A')"
left_hom_axioms: shows "left_hom (\<cdot>) (\<star>) \<omega>"
productive_imp_INTERP: fixes N :: "'a literal multiset set"   and C :: "'a literal multiset" assumes "production N C \<noteq> {}" shows "INTERP N \<Turnstile> C"
ass: fixes t' :: "'a" assumes "Join t' \<in> set (ca # cas)" shows "\<not> not_final_thread s t' \<and> t \<noteq> t'"
continuous_mult: fixes F :: "'d filter"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "continuous F f"   and "continuous F g" shows "continuous F (\<lambda>x. f x * g x)"
bigo_const_mult4: fixes c :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "c \<noteq> (0::'a)" shows "O(f) \<subseteq> O(\<lambda>x. c * f x)"
assms_3: shows "vdisjnt (\<D>\<^sub>\<circ> r) (\<D>\<^sub>\<circ> s)"
graph_homomorphism_composes: fixes a :: "('e, 'f) labeled_graph"   and b :: "('e, 'g) labeled_graph"   and x :: "('f \<times> 'g) set"   and c :: "('e, 'h) labeled_graph"   and y :: "('g \<times> 'h) set" assumes "graph_homomorphism a b x"   and "graph_homomorphism b c y" shows "graph_homomorphism a c (x O y)"
cinner_zero_left: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and x :: "'c" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner zero x = 0"
relcompI: fixes r :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'f"   and b :: "'g"   and s :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and c :: "'h" assumes "r a b"   and "s b c" shows "(r OO s) a c"
pf_def: shows "pf \<equiv> prime_factorization_nat n"
d_bot_only: fixes x :: "'a" assumes "d x = bot" shows "x = bot"
Q: shows "Q = (\<lambda>X. if X \<subseteq> B then X else {}) ` Q' - {{}}"
lincomb_closed: fixes S :: "rat vec set"   and a :: "rat vec \<Rightarrow> rat" assumes "S \<subseteq> Rn" shows "gs.lincomb a S \<in> Rn"
point_index_diff: fixes B1 :: "'b set multiset"   and ps :: "'b set"   and B2 :: "'b set multiset" shows "B1 index ps = (B1 + B2) index ps - B2 index ps"
inner_eq_zero_iff: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and x :: "'b" assumes "inner_product_space scale inner" shows "(inner x x = (0::'a)) = (x = (0::'b))"
hyps: fixes n :: "nat" shows "map_idx f xs n = map2 f xs [n..<n + length xs]"
Class_commutes_with_composition: fixes a :: "'a"   and b :: "'a" assumes "a \<in> G"   and "b \<in> G" shows "Class a [\<cdot>] Class b = Class (a \<cdot> b)"
Rep_zet_inject: fixes x :: "'b zet"   and y :: "'b zet" shows "(Rep_zet x = Rep_zet y) = (x = y)"
snapshot_state_unchanged: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and p :: "'a" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'"   and "ps c p \<noteq> None" shows "ps c p = ps c' p"
rtrancl_fold_product1: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd" shows "{(r, s). \<exists>a\<in>A. s = f a r}\<^sup>* = {(r, s). \<exists>a\<in>lists A. s = fold f a r}"
chooseNext_in_allVars: fixes g :: "'g" assumes "redundant g" shows "chooseNext g \<in> allVars g"
mgu_img_consts': fixes s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and z :: "'a"   and c :: "'b" assumes "mgu s t = Some \<delta>"   and "\<delta> z = Fun c []" shows "Fun c [] \<sqsubseteq> s \<or> Fun c [] \<sqsubseteq> t"
left_plus_below_circ: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_kleene_algebra.tc x \<preceq> x\<^sup>\<odot>"
bigo_plus_absorb_lemma1: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> O(g)" shows "f +o O(g) \<subseteq> O(g)"
os_prepend_rule: fixes xs :: "'a list"   and n :: "'a node ref option"   and x :: "'a" shows "<os_list xs n> os_prepend x n <os_list (x # xs)>"
simps_3: fixes x :: "pdf_type" shows "op_type Minus x = (case x of INTEG \<Rightarrow> Some INTEG | REAL \<Rightarrow> Some REAL | PRODUCT pdf_type1 x \<Rightarrow> Map.empty x | _ \<Rightarrow> None)"
chine_hcomp_in_hom: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr \<mu>"   and "Span.arr \<nu>"   and "Span.src \<nu> = Span.trg \<mu>" shows "Maps.in_hom (Span.chine_hcomp \<nu> \<mu>) (Maps.pbdom (Leg0 (Dom \<nu>)) (Leg1 (Dom \<mu>))) (Maps.pbdom (Leg0 (Cod \<nu>)) (Leg1 (Cod \<mu>)))"
diff: fixes x :: "'m"   and y :: "'m" shows "g' (x - y) = g' x - g' y"
retract_ofE: fixes T :: "'a set"   and S :: "'a set" assumes "T retract_of S" obtains r :: "'a \<Rightarrow> 'a" where "T = r ` S" and "r ` S \<subseteq> S" and "continuous_on S r" and "\<And>x. x \<in> S \<Longrightarrow> r (r x) = r x"
S_def: shows "S = Pascal_Property.inter (line C' P) (line R A)"
increasing_measurable_info: fixes f :: "'a \<Rightarrow> 'd"   and s :: "nat"   and N :: "'d measure"   and t :: "nat" assumes "f \<in> F s \<rightarrow>\<^sub>M N"   and "s \<le> t" shows "f \<in> F t \<rightarrow>\<^sub>M N"
vanishes_def: fixes X :: "nat \<Rightarrow> rat" shows "vanishes X = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r)"
comp_eq_dest_lhs: fixes a :: "'d::type \<Rightarrow> 'c::type"   and b :: "'b::type \<Rightarrow> 'd::type"   and c :: "'b::type \<Rightarrow> 'c::type"   and v :: "'b::type" assumes "(a::'d::type \<Rightarrow> 'c::type) \<circ> (b::'b::type \<Rightarrow> 'd::type) = (c::'b::type \<Rightarrow> 'c::type)" shows "(a::'d::type \<Rightarrow> 'c::type) ((b::'b::type \<Rightarrow> 'd::type) (v::'b::type)) = (c::'b::type \<Rightarrow> 'c::type) v"
zero_min: fixes t :: "'x \<Rightarrow>\<^sub>0 nat" shows "lex_pm 0 t"
Cod_memb: fixes f :: "'c local.term set"   and t :: "'c local.term" assumes "arr f"   and "t \<in> f" shows "Cod t = COD f"
packing: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "int (card ps) = \<t>" shows "int (\<B> index ps) \<le> \<Lambda>\<^sub>t"
while_Com: fixes P :: "'a Abs_State.st option acom \<Rightarrow> bool"   and f :: "'a Abs_State.st option acom \<Rightarrow> 'a Abs_State.st option acom"   and c :: "'a Abs_State.st option acom"   and c' :: "'a Abs_State.st option acom"   and X :: "char list set" assumes "while_option P f c = Some c'"   and "\<And>c. strip (f c) = strip c"   and "\<forall>c. c \<in> Com X \<longrightarrow> vars (strip c) \<subseteq> X \<longrightarrow> f c \<in> Com X"   and "c \<in> Com X"   and "vars (strip c) \<subseteq> X" shows "c' \<in> Com X"
termMOD_igFreshIGAbs2: shows "igFreshIGAbs2 termMOD"
l11_51: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "A \<noteq> B"   and "A \<noteq> C"   and "B \<noteq> C"   and "Cong A B A' B'"   and "Cong A C A' C'"   and "Cong B C B' C'" shows "B A C CongA B' A' C' \<and> A B C CongA A' B' C' \<and> B C A CongA B' C' A'"
fac_wrap_def: shows "fac_wrap \<equiv> wrapB\<cdot>fac_work"
meta_allE: fixes P :: "'c \<Rightarrow> prop"   and x :: "'c" assumes "\<And>x. PROP P x" obtains "PROP P x"
ex_min_if_finite: fixes S :: "'d set" assumes "finite S"   and "S \<noteq> {}" shows "\<exists>m\<in>S. \<not> (\<exists>x\<in>S. x < m)"
bij_betw_from_nat_into_finite: fixes S :: "'a set" assumes "finite S" shows "bij_betw (from_nat_into S) {..<card S} S"
approx_eq_imp: fixes a :: "'c star"   and b :: "'c star" assumes "a = b" shows "a \<approx> b"
assoc_in_hom: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "\<guillemotleft>assoc a b c : local.prod (local.prod a b) c \<rightarrow> local.prod a (local.prod b c)\<guillemotright>"
base: assumes "cs\<^bsup>\<pi>\<^esup> i = [\<pi> i]"   and "\<forall>k. \<not> i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k" shows "thesis"
phl_cond: fixes u :: "'a"   and v :: "'a"   and w :: "'a"   and x :: "'a"   and z :: "'a"   and y :: "'a" assumes "u \<cdot> v \<le> v \<cdot> u \<cdot> v"   and "u \<cdot> w \<le> w \<cdot> u \<cdot> w"   and "\<And>x y. u \<cdot> (x + y) \<le> u \<cdot> x + u \<cdot> y"   and "\<lbrace>u \<cdot> v\<rbrace>x\<lbrace>z\<rbrace>"   and "\<lbrace>u \<cdot> w\<rbrace>y\<lbrace>z\<rbrace>" shows "\<lbrace>u\<rbrace>v \<cdot> x + w \<cdot> y\<lbrace>z\<rbrace>"
assms_2: shows "\<langle>\<V> s\<rangle> \<Turnstile>\<^sub>t \<T> s"
intros_1: fixes trn :: "('state, 'act, 'out) trans" assumes "validTrans trn" shows "valid [trn]"
bounded_clinear_scaleC_const: fixes g :: "'a \<Rightarrow> complex"   and x :: "'b" assumes "bounded_clinear g" shows "bounded_clinear (\<lambda>xa. g xa *\<^sub>C x)"
Guard: fixes s :: "'d"   and g :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and t :: "('d, 'f) xstate"   and f :: "'f" assumes "s \<in> g"   and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Guard f g c,Normal s\<rangle> \<Rightarrow> t"
intrinsic_border_less: fixes w :: "'a list" assumes "w \<noteq> []" shows "length (intrinsic_border w) < length w"
del_UnEdge_even_odd: fixes G :: "('a, 'b) graph"   and v :: "'a"   and w :: "'b"   and v' :: "'a" assumes "valid_unMultigraph G"   and "finite (edges G)"   and "finite (nodes G)"   and "(v, w, v') \<in> edges G"   and "even (degree v G)"   and "odd (degree v' G)" shows "num_of_odd_nodes (del_unEdge v w v' G) = num_of_odd_nodes G"
runit_tensor: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "\<r>[a \<otimes> b] = (a \<otimes> \<r>[b]) \<cdot> \<a>[a, b, \<I>]"
acomplete_Node_if_wbal1: fixes l :: "'a tree"   and r :: "'a tree"   and x :: "'a" assumes "acomplete l"   and "acomplete r"   and "size l = size r + 1" shows "acomplete \<langle>l, x, r\<rangle>"
useNodes_of_def: fixes u :: "'g \<Rightarrow> ('node, 'val set) mapping"   and g :: "'g" shows "uninst_code.ssa.useNodes_of u g = fold (uninst_code.ssa.addN u g) (\<alpha>n g) Mapping.empty"
changes_poly_at_def: fixes ps :: "'a poly list"   and a :: "'a" shows "changes_poly_at ps a = changes (map (\<lambda>p. poly p a) ps)"
term_ok'_incr_bv: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and t :: "Core.term"   and inc :: "nat"   and lev :: "nat" assumes "term_ok' \<Sigma> t" shows "term_ok' \<Sigma> (incr_bv inc lev t)"
setA: fixes L :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "\<And>y. y \<in> set L \<Longrightarrow> P y" shows "list_all P L"
divide_eq_minus_1_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.field divide inverse times one plus zero minus uminus" shows "(divide a b = uminus one) = (b \<noteq> zero \<and> a = uminus b)"
InitConf_HAStates: fixes A :: "('a, 'b, 'c) hierauto" shows "InitConf A \<subseteq> HAStates A"
asprod_pos_pos: fixes x :: "ant"   and n :: "nat" assumes "0 \<le> x" shows "0 \<le> int n *\<^sub>a x"
extensional0_outside: fixes x :: "'c"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "x \<notin> S"   and "extensional0 S f" shows "f x = (0::'d)"
ceiling_diff_of_int: fixes x :: "'b"   and z :: "int" shows "\<lceil>x - of_int z\<rceil> = \<lceil>x\<rceil> - z"
simps_1: fixes uu :: "unit" shows "BsSkewBinomialHeapStruc.bs_invar (Inl uu) = True"
nDAcharn: fixes p :: "('b, 'c) Combinators list" shows "noDenyAll p = (\<forall>r\<in>set p. \<not> member DenyAll r)"
correct_fin_inf_append1: fixes s1 :: "nat \<Rightarrow> 'd"   and x :: "'d"   and s :: "nat \<Rightarrow> 'd"   and i :: "nat" assumes "s1 = fin_inf_append [x] s" shows "s1 (Suc i) = s i"
l9_8_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and Q :: "'a"   and A :: "'a"   and C :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.TS Bet P Q A C"   and "Tarski_neutral_dimensionless.TS Bet P Q B C" shows "Tarski_neutral_dimensionless.OS Bet P Q A B"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and p :: "'psym"   and Tl :: "'fsym trm list" shows "satA \<xi> (Pr p Tl) = intP p (map (int \<xi>) Tl)"
wf_envs_progress_le_2: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and P' :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and db :: "(String.literal, event_data list set list) mapping"   and \<phi> :: "Formula.formula" assumes "wf_envs \<sigma> j P P' V db" shows "Monitor.progress \<sigma> P' \<phi> (Suc j) \<le> Suc j"
negate_def: fixes P :: "'a \<Rightarrow> bool" shows "\<N> P = (\<lambda>s. \<not> P s)"
a: fixes n :: "nat"   and e :: "(ZF, ZF) Expression"   and d :: "(ZF, ZF) Expression" assumes "n = ZFDepth (Expr2ZF e)"   and "Expr2ZF e = Expr2ZF d" shows "e = d"
map_iterator_abs_trivial: shows "map_iterator_abs id (\<lambda>_. True) = map_iterator"
distinct_3: fixes x11 :: "nat set"   and x12 :: "'a"   and x31 :: "nat set"   and x32 :: "nat set"   and x33 :: "'a" shows "x11:*cast(x12) \<noteq> x31\<not>x32:arrive(x33)"
hp0: shows "\<turnstile>\<^sub>p {adjoint U2 * P * U2} Utrans U2 {P}"
bijI': fixes f :: "'b \<Rightarrow> 'c" assumes "\<And>x y. (f x = f y) = (x = y)"   and "\<And>y. \<exists>x. y = f x" shows "bij f"
pderiv_smult: fixes a :: "'b"   and p :: "'b poly" shows "pderiv (smult a p) = smult a (pderiv p)"
assms_3: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"
lastbits_BC: fixes xs :: "bit list"   and ys :: "bit list" shows "BC = show_rightmost_bits (xs @ ys @ BC) 8"
infinite_typed_consts: fixes a :: "'atom" shows "infinite {c. \<Gamma> (Fun c []) = Var a \<and> public c}"
subst_atms_id_subst: fixes AA :: "'a set" shows "AA \<cdot>as id_subst = AA"
Q_hit_decomp: shows "Q_hit = Q_hit_pre @ y # Q_hit_post"
nth_rows: fixes i :: "nat"   and A :: "'a mat" assumes "i < dim_row A" shows "rows A ! i = row A i"
fst_not_one_snd: fixes a :: "('c, 'd) OrdSum" assumes "fst (Rep_OrdSum a) \<noteq> (1::'c)" shows "snd (Rep_OrdSum a) = (1::'d)"
neutral_const: fixes A :: "'b set" shows "(\<Prod>_\<in>A. (1::'a)) = (1::'a)"
ts_invI: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and I :: "'f \<Rightarrow> 'i option"   and P :: "'i \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'j \<Rightarrow> bool"   and m :: "'j" assumes "\<And>t x ln. ts t = \<lfloor>(x, ln)\<rfloor> \<Longrightarrow> \<exists>i. I t = \<lfloor>i\<rfloor> \<and> P i t x m" shows "ts_inv P I ts m"
try_spmf_bind_out: fixes p :: "'b spmf"   and f :: "'b \<Rightarrow> 'c spmf"   and q :: "'c spmf" assumes "lossless_spmf p" shows "p \<bind> (\<lambda>x. TRY f x ELSE q) = TRY p \<bind> f ELSE q"
word_oa_dist2: fixes x :: "'b word"   and y :: "'b word"   and z :: "'b word" shows "x || y && z = (x || y) && (x || z)"
less_list_def: fixes s :: "'b list"   and t :: "'b list" shows "(s < t) = (s \<le> t \<and> s \<noteq> t)"
merge_on_axioms_def: fixes A :: "'c set"   and h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "'c \<Rightarrow> 'c \<Rightarrow> 'c option" shows "merge_on_axioms A h bo m \<equiv> (\<forall>a b. h a = h b \<longrightarrow> a \<in> A \<longrightarrow> (\<exists>ab. m a b = Some ab \<and> bo a ab \<and> bo b ab \<and> (\<forall>u. bo a u \<longrightarrow> bo b u \<longrightarrow> bo ab u))) \<and> (\<forall>a b. h a \<noteq> h b \<longrightarrow> a \<in> A \<longrightarrow> m a b = None)"
relpow_seq: fixes x :: "'b"   and y :: "'b"   and n :: "nat"   and r :: "('b \<times> 'b) set" assumes "(x, y) \<in> r ^^ n" shows "\<exists>f. f 0 = x \<and> f n = y \<and> (\<forall>i<n. (f i, f (Suc i)) \<in> r)"
abs_eq: fixes x :: "'b" shows "Const x = MPoly (Const\<^sub>0 x)"
count_violations_swap: fixes t :: "('a, 'b, 'c) event list"   and j :: "nat"   and i :: "nat"   and n :: "nat" assumes "prerecording_event t j"   and "postrecording_event t i"   and "i < j"   and "j < length t"   and "count_violations t = Suc n"   and "\<forall>k. i < k \<and> k < j \<longrightarrow> \<not> regular_event (t ! k)"   and "count_violations t = Suc n"   and "trace init t final" shows "count_violations (swap_events i j t) = n"
cos_one_2pi_int: fixes x :: "real" shows "(cos x = 1) = (\<exists>xa. x = real_of_int xa * 2 * pi)"
cproper_interval: assumes "ID ccompare \<noteq> None"   and "finite UNIV" shows "class.proper_interval cless cproper_interval"
unit_factor_gcd: 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 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"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "unit_factor (gcd a b) = (if a = zero \<and> b = zero then zero else one)"
ereal_times_divide_eq_left: fixes b :: "ereal"   and c :: "ereal"   and a :: "ereal" shows "b / c * a = b * a / c"
hyps: fixes vs :: "'a poly list" shows "sum_list (drop (length (fold remove1 ws vs) div 2) (sort (map degree (fold remove1 ws vs)))) \<le> sum_list (drop (length vs div 2) (sort (map degree vs)))"
bind_spmf_def: fixes x :: "'d spmf"   and f :: "'d \<Rightarrow> 'b spmf" shows "x \<bind> f = x \<bind> (\<lambda>a. case a of None \<Rightarrow> return_pmf None | Some a' \<Rightarrow> f a')"
point_linorder_matrix: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.point f = (\<exists>i. \<forall>j. f (i, j) = top \<and> (\<forall>k. i \<noteq> k \<longrightarrow> f (k, j) = bot))"
fmrestrict_fset_dom: fixes m :: "('c, 'd) fmap" shows "fmrestrict_fset (fmdom m) m = m"
Markov_Decision_Process_Invariant_def: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set" shows "Markov_Decision_Process_Invariant K S \<equiv> Markov_Decision_Process K \<and> Markov_Decision_Process_Invariant_axioms K S"
ine_exprChannelSet_def: fixes sP :: "specID"   and chSet :: "chanID set"   and E :: "Expression" shows "ine_exprChannelSet sP chSet E \<equiv> (\<forall>x. x \<in> chSet \<longrightarrow> x \<in> ins sP \<and> exprChannel x E) \<and> (\<forall>x. x \<notin> chSet \<and> x \<in> ins sP \<longrightarrow> \<not> exprChannel x E)"
fmset_ee: fixes as :: "'a list"   and bs :: "'a list" assumes "fmset G as = fmset G bs"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "essentially_equal G as bs"
pair_map_spmf: fixes f :: "'e \<Rightarrow> 'c"   and p :: "'e spmf"   and g :: "'f \<Rightarrow> 'd"   and q :: "'f spmf" shows "pair_spmf (map_spmf f p) (map_spmf g q) = map_spmf (map_prod f g) (pair_spmf p q)"
gfinprod_closed: fixes f :: "'f \<Rightarrow> 'a"   and A :: "'f set" assumes "f \<in> A \<rightarrow> carrier G" shows "gfinprod G f A \<in> carrier G"
case_1: fixes f1 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f2 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f3 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f4 :: "'e list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> 'c"   and x1 :: "('d, 'e) Term.term" shows "(case send\<langle>x1\<rangle>\<^sub>s\<^sub>t of send\<langle>x\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f1 x | receive\<langle>x\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f2 x | \<langle>x: xa \<doteq> xb\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f3 x xa xb | \<forall>x\<langle>\<or>\<noteq>: xa\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f4 x xa) = f1 x1"
residue_simple_pole: fixes f :: "complex \<Rightarrow> complex"   and z0 :: "complex" assumes "isolated_singularity_at f z0"   and "is_pole f z0"   and "zorder f z0 = - 1" shows "residue f z0 = zor_poly f z0 z0"
step: shows "(is, \<O>, \<D>, \<theta>, sb, \<S>, m) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is', \<O>', \<D>', \<theta>', sb', \<S>', m')"
l_def: shows "l \<equiv> limsup (\<lambda>n. ereal (p n * f n / f (Suc n) - p (Suc n)))"
map_comp: fixes g :: "'y \<Rightarrow> 'x"   and f :: "'z \<Rightarrow> 'y"   and v :: "'z fset" shows "g |`| f |`| v = (g \<circ> f) |`| v"
image_insert: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d"   and B :: "'d set" shows "f ` ??.Set.insert a B = ??.Set.insert (f a) (f ` B)"
A''_1: shows "A'' \<in> set (tr\<^sub>p\<^sub>c A D)"
strategyproofness: shows "(pmf (sds R19) d < pmf (sds R17) d \<or> pmf (sds R19) d + (pmf (sds R19) b + pmf (sds R19) a) < pmf (sds R17) d + (pmf (sds R17) b + pmf (sds R17) a)) \<or> pmf (sds R19) d = pmf (sds R17) d \<and> pmf (sds R19) d + (pmf (sds R19) b + pmf (sds R19) a) = pmf (sds R17) d + (pmf (sds R17) b + pmf (sds R17) a)"
non_distinct_low_rank: fixes A :: "'a mat" assumes "A \<in> carrier_mat n n"   and "\<not> distinct (cols A)" shows "rank A < n"
index_take: fixes i :: "nat"   and xs :: "'c list"   and x :: "'c" assumes "i \<le> index xs x" shows "x \<notin> set (take i xs)"
C_ss_VIS: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and src :: "nat"   and dst :: "nat"   and PRED :: "nat \<Rightarrow> nat option"   and C :: "nat set"   and N :: "nat set"   and d :: "nat"   and v :: "nat" assumes "nf_invar' c src dst PRED C N d"   and "v \<in> C" shows "v \<in> dom PRED"
simps_8: fixes P :: "pi"   and Q :: "pi" shows "depth (P \<parallel> Q) = depth P + depth Q"
list_encode_decode: fixes n :: "nat" shows "Partial_Recursive.list_encode (Partial_Recursive.list_decode n) = n"
path_from_to_last: fixes v :: "'a"   and xs :: "'a list"   and w :: "'a" assumes "v \<leadsto>xs\<leadsto>\<^bsub>H\<^esub> w" shows "w \<notin> set (butlast xs)"
iv2: shows "size (tree_to_multiset (Node e2 a2 r ts2)) \<le> 2 ^ Suc r - 1"
in_lex_prod: fixes a :: "'e"   and b :: "'f"   and a' :: "'e"   and b' :: "'f"   and r :: "('e \<times> 'e) set"   and s :: "('f \<times> 'f) set" shows "(((a, b), a', b') \<in> r <*lex*> s) = ((a, a') \<in> r \<or> a = a' \<and> (b, b') \<in> s)"
search_nonempty_eq0_1: fixes b :: "nat"   and i :: "nat"   and y :: "nat" assumes "s b \<down>= i"   and "\<phi> i y \<down>"   and "e_length b \<le> y" shows "search b \<down>"