hcju/isabelleps huggingface dataset:
mem_existence_ivl_iv_defined_1: fixes t :: "real"   and t0 :: "real"   and x0 :: "real" assumes "t \<in> existence_ivl t0 x0" shows "t0 \<in> T"
assms_3: fixes a :: "'a"   and p :: "'a poly" assumes "P a"   and "Q p" shows "Q ([:a, - (1::'a):] * p)"
id_Rel_components_3: fixes A :: "V" shows "id_Rel A\<lparr>ArrCod\<rparr> = A"
PRelQ: shows "\<forall>y. (P[x::=y], Q[x::=y]) \<in> Rel"
natural_isomorphism_axioms: shows "natural_isomorphism hom.comp hom.comp hom.map GF.map \<phi>.map"
removeAll_filter_not_eq: fixes x :: "'a" shows "removeAll x = filter ((\<noteq>) x)"
card_bind_constant: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd set"   and k :: "nat" assumes "finite S"   and "\<forall>X\<in>S. finite (f X)"   and "disjoint_family_on f S"   and "\<And>x. x \<in> S \<Longrightarrow> card (f x) = k" shows "card (S \<bind> f) = card S * k"
elims_3: fixes x :: "'a mat" assumes "\<not> square_mat x" obtains A :: "'a mat" where "x = A" and "dim_col A \<noteq> dim_row A"
f_def: shows "f \<equiv> \<lambda>x. a + (setdist {x} S / (setdist {x} S + setdist {x} T)) *\<^sub>R (b - a)"
\<A>_3: fixes a :: "'a set" assumes "a \<in> \<A>" shows "open a"
set_less_aux_irrefl: fixes A :: "'a set" shows "\<not> A \<sqsubset>' A"
prj_simps_6: shows "C.dom \<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>0 = HH\<mu>\<nu>H\<pi>\<rho>.chine"
principal_dchar_sum_bound: shows "(\<lambda>x. (\<Sum>p | prime p \<and> real p \<le> x. \<chi>\<^sub>0 p * (Ln (of_nat p) / of_nat p)) - Ln (complex_of_real x)) \<in> O(\<lambda>_. 1)"
hyps: fixes rs :: "'rule stream"   and s :: "'state"   and steps :: "('state \<times> 'rule) stream" assumes "pos rs r < pos rsa__ r"   and "s \<in> S"   and "fair rs"   and "ipath (mkTree rs s) steps"   and "alw (holds (enabledAtStep r)) steps" shows "ev (holds (\<lambda>step. snd step = r)) steps"
lincomb_insert2: fixes S :: "'c set"   and a :: "'c \<Rightarrow> 'a"   and v :: "'c" assumes "finite S"   and "S \<subseteq> carrier V"   and "a \<in> insert v S \<rightarrow> carrier K"   and "v \<notin> S"   and "v \<in> carrier V" shows "lincomb a (insert v S) = a v \<odot>\<^bsub>V\<^esub> v \<oplus>\<^bsub>V\<^esub> lincomb a S"
hyps_2: fixes x :: "'a" shows "list_of (LCons x xs') = x # list_of xs'"
convexD_alt: fixes s :: "'b set"   and a :: "'b"   and b :: "'b"   and u :: "real" assumes "convex s"   and "a \<in> s"   and "b \<in> s"   and "0 \<le> u"   and "u \<le> 1" shows "(1 - u) *\<^sub>R a + u *\<^sub>R b \<in> s"
interleave_unconc: fixes l1 :: "'b list"   and l2 :: "'b list"   and w1 :: "'b list"   and w2 :: "'b list" assumes "l1 @ l2 \<in> w1 \<otimes> w2" shows "\<exists>w11 w12 w21 w22. w1 = w11 @ w12 \<and> w2 = w21 @ w22 \<and> l1 \<in> w11 \<otimes> w21 \<and> l2 \<in> w12 \<otimes> w22"
c_fst_of_c_pair: fixes x :: "nat"   and y :: "nat" shows "c_fst (c_pair x y) = x"
nil_le2: fixes s :: "'c list" shows "(s \<le> []) = (s = [])"
finite_real_of_nat_segment: fixes m :: "nat" shows "finite {n. real n < real m}"
metric_LIMSEQ_I: fixes X :: "nat \<Rightarrow> 'a"   and L :: "'a" assumes "\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r" shows "X \<longlonglongrightarrow> L"
\<K>p_\<K>_eq: fixes st :: "'s \<times> ('c \<Rightarrow> real) set" shows "\<K>p st = (\<lambda>x. x \<in> \<K> st)"
lexordp_eq_refl: fixes less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'d list" shows "ord.lexordp_eq less xs xs"
reduced_GB_is_GB_Polys: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "i.punit.is_Groebner_basis (i.punit.reduced_GB F)"
override_on_apply_notin: fixes a :: "'a"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "a \<notin> A" shows "(f ++\<^bsub>A\<^esub> g) a = f a"
normal_imp_not_zero: fixes f :: "('e, 'f) IEEE.float" assumes "is_normal f" shows "\<not> is_zero f"
map_indets_in_Polys: fixes f :: "'d \<Rightarrow> 'b"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "map_indets f p \<in> P[f ` indets p]"
exists_path_subpath_to_frontier: fixes g :: "real \<Rightarrow> 'b"   and S :: "'b set" assumes "path g"   and "pathstart g \<in> closure S"   and "pathfinish g \<notin> S" obtains h :: "real \<Rightarrow> 'b" where "path h" and "pathstart h = pathstart g" and "path_image h \<subseteq> path_image g" and "path_image h - {pathfinish h} \<subseteq> interior S" and "pathfinish h \<in> frontier S"
simps: fixes g :: "(location \<times> location) set"   and a :: "location \<Rightarrow> char list set"   and c :: "actor \<Rightarrow> char list set \<times> char list set"   and l :: "location \<Rightarrow> char list \<times> ((actor \<times> actor set) \<times> nat) set" shows "agra (Lgraph g a c l) = a"
convex_rel_interior_closure: fixes S :: "'b set" assumes "convex S" shows "rel_interior (closure S) = rel_interior S"
update_convs_8: fixes getWatch1' :: "(nat \<Rightarrow> Literal option) \<Rightarrow> nat \<Rightarrow> Literal option"   and getSATFlag :: "ExtendedBool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getConflictClause :: "nat"   and getQ :: "Literal list"   and getReason :: "Literal \<Rightarrow> nat option"   and getWatch1 :: "nat \<Rightarrow> Literal option"   and getWatch2 :: "nat \<Rightarrow> Literal option"   and getWatchList :: "Literal \<Rightarrow> nat list"   and getC :: "Literal list"   and getCl :: "Literal"   and getCll :: "Literal"   and getCn :: "nat"   and more :: "'a" shows "getWatch1_update getWatch1' \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr> = \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1' getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr>"
top_clocked_total: fixes env :: "v sem_env"   and s :: "'b state"   and tp :: "top0" shows "\<exists>r. evaluate_top True env s tp r"
finite_has_maximal: fixes A :: "'e list set" assumes "finite A"   and "A \<noteq> {}" shows "\<exists>m\<in>A. \<forall>b\<in>A. prefix m b \<longrightarrow> m = b"
\<omega>consistent_def: shows "B.\<omega>consistent \<equiv> \<forall>\<phi>\<in>fmla. \<forall>x\<in>var. Fvars \<phi> = {x} \<longrightarrow> (\<forall>n\<in>num. bprv (neg (subst \<phi> n x))) \<longrightarrow> \<not> bprv (neg (neg (exi x \<phi>)))"
vec_carrier: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "Matrix.vec n f \<in> carrier_vec n"
wtC_append: fixes wtFsym :: "'b \<Rightarrow> bool"   and wtPsym :: "'c \<Rightarrow> bool"   and arOf :: "'b \<Rightarrow> 'a list"   and resOf :: "'b \<Rightarrow> 'a"   and parOf :: "'c \<Rightarrow> 'a list"   and c1 :: "('b, 'c) lit list"   and c2 :: "('b, 'c) lit list" assumes "Signature TYPE('a) wtFsym wtPsym" shows "Signature.wtC wtFsym wtPsym arOf resOf parOf (c1 @ c2) = (Signature.wtC wtFsym wtPsym arOf resOf parOf c1 \<and> Signature.wtC wtFsym wtPsym arOf resOf parOf c2)"
set_measurable_continuous_on_ivl: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real" assumes "continuous_on {a..b} f" shows "set_borel_measurable borel {a..b} f"
distinct_1: fixes x1 :: "'b"   and x2 :: "'b" shows "Pos x1 \<noteq> Neg x2"
il: shows "A \<TTurnstile> onl \<Gamma> P"
h_def: shows "h = (\<lambda>Z. if Z \<in> X then 0 else Suc 0)"
step_4: shows "list.set \<beta>s \<subseteq> ON"
mk_poly_sub_corresp: fixes A :: "'b mat"   and l :: "nat"   and pv :: "'b poly vec"   and j :: "nat" assumes "dim_col A = Suc l"   and "dim_vec pv = dim_row A"   and "j < dim_col A" shows "pv + col (mk_poly_sub (coeff_lift_hom.mat_hom A) l j) l = mk_poly2_col A pv (Suc j)"
extensible_refl_concr: fixes e\<^sub>1 :: "('b \<times> 'c \<times> 'c) set"   and v :: "'c set"   and G :: "('b, 'd) labeled_graph"   and f :: "('c \<times> 'd) set"   and e\<^sub>2 :: "('b \<times> 'c \<times> 'c) set" assumes "graph_homomorphism (LG e\<^sub>1 v) G f" shows "extensible (LG e\<^sub>1 v, LG e\<^sub>2 v) G f = graph_homomorphism (LG e\<^sub>2 v) G f"
lookup_zero: fixes k :: "'b" shows "poly_mapping.lookup 0 k = (0::'a)"
prod_emb_empty: fixes M :: "'b set"   and L :: "'b \<Rightarrow> 'c measure"   and K :: "'b set" shows "prod_emb M L K {} = {}"
prime_elem_mult_dvdI: fixes p :: "'a"   and c :: "'a"   and b :: "'a" assumes "prime_elem p"   and "p dvd c"   and "b dvd c"   and "\<not> p dvd b" shows "p * b dvd c"
assms: shows "\<not> change_on_neither (block i t) (gap i t)"
MkIde_Cod': fixes a :: "(bool, unit) arr" assumes "ide a" shows "MkIde (Cod a) = a"
subsetI: fixes A :: "'x set"   and B :: "'x set" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<in> B" shows "A \<subseteq> B"
CH_range': fixes c :: "'a"   and B :: "'a set" assumes "c \<in> Singleton_for_d.CH B" shows "c \<in> B"
concurrent_direct_steps_simulates_store_buffer_history_steps: fixes ts\<^sub>s\<^sub>b :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m\<^sub>s\<^sub>b :: "nat \<Rightarrow> nat"   and \<S>\<^sub>s\<^sub>b :: "nat \<Rightarrow> bool option"   and ts\<^sub>s\<^sub>b' :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m\<^sub>s\<^sub>b' :: "nat \<Rightarrow> nat"   and \<S>\<^sub>s\<^sub>b' :: "nat \<Rightarrow> bool option"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option" assumes "(ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) \<Rightarrow>\<^sub>s\<^sub>b\<^sub>h\<^sup>* (ts\<^sub>s\<^sub>b', m\<^sub>s\<^sub>b', \<S>\<^sub>s\<^sub>b')"   and "valid_ownership \<S>\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b"   and "valid_reads m\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b"   and "valid_history program_step ts\<^sub>s\<^sub>b"   and "valid_sharing \<S>\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b"   and "tmps_distinct ts\<^sub>s\<^sub>b"   and "valid_sops ts\<^sub>s\<^sub>b"   and "valid_data_dependency ts\<^sub>s\<^sub>b"   and "load_tmps_fresh ts\<^sub>s\<^sub>b"   and "enough_flushs ts\<^sub>s\<^sub>b"   and "valid_program_history ts\<^sub>s\<^sub>b"   and "valid ts\<^sub>s\<^sub>b"   and "(ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) \<sim> (ts, m, \<S>)"   and "safe_reach_direct safe_delayed (ts, m, \<S>)" shows "valid_ownership \<S>\<^sub>s\<^sub>b' ts\<^sub>s\<^sub>b' \<and> valid_reads m\<^sub>s\<^sub>b' ts\<^sub>s\<^sub>b' \<and> valid_history program_step ts\<^sub>s\<^sub>b' \<and> valid_sharing \<S>\<^sub>s\<^sub>b' ts\<^sub>s\<^sub>b' \<and> tmps_distinct ts\<^sub>s\<^sub>b' \<and> valid_data_dependency ts\<^sub>s\<^sub>b' \<and> valid_sops ts\<^sub>s\<^sub>b' \<and> load_tmps_fresh ts\<^sub>s\<^sub>b' \<and> enough_flushs ts\<^sub>s\<^sub>b' \<and> valid_program_history ts\<^sub>s\<^sub>b' \<and> valid ts\<^sub>s\<^sub>b' \<and> (\<exists>ts' m' \<S>'. (ts, m, \<S>) \<Rightarrow>\<^sub>d\<^sup>* (ts', m', \<S>') \<and> (ts\<^sub>s\<^sub>b', m\<^sub>s\<^sub>b', \<S>\<^sub>s\<^sub>b') \<sim> (ts', m', \<S>'))"
assms: shows "f `\<^sub>\<bullet> A = g `\<^sub>\<bullet> B"
Cong3_def: 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 A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.Cong3 Cong A B C A' B' C' \<equiv> Cong A B A' B' \<and> Cong A C A' C' \<and> Cong B C B' C'"
real_sqrt_pow2_iff: fixes x :: "real" shows "((sqrt x)\<^sup>2 = x) = (0 \<le> x)"
carrier_eq: shows "alts' = {x. Pareto R' x x}"
unique_expression1: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and H :: "'e set"   and s :: "nat \<Rightarrow> 'f"   and n :: "nat"   and m :: "nat \<Rightarrow> 'e" assumes "R module M"   and "H \<subseteq> carrier M"   and "free_generator R M H"   and "s \<in> {j. j \<le> n} \<rightarrow> carrier R"   and "m \<in> {j. j \<le> n} \<rightarrow> H"   and "inj_on m {j. j \<le> n}"   and "l_comb R M n s m = \<zero>\<^bsub>M\<^esub>" shows "\<forall>j\<in>{j. j \<le> n}. s j = \<zero>\<^bsub>R\<^esub>"
Span_replace_hd: fixes n :: "'v"   and ms :: "'v list"   and m :: "'v" assumes "n \<in> V"   and "set ms \<subseteq> V"   and "m \<in> RSpan (n # ms)" shows "RSpan (m # ms) \<subseteq> RSpan (n # ms)"
vifunion_vintersection_index_right: fixes B :: "V"   and A :: "V" shows "\<Union>\<^sub>\<circ> (VLambda B ((\<inter>\<^sub>\<circ>) A) `\<^sub>\<circ> B) = A \<inter>\<^sub>\<circ> \<Union>\<^sub>\<circ> B"
left_inverse: fixes a :: "'b" shows "- a + a = (0::'b)"
list_induct2: fixes xs :: "'g list"   and ys :: "'h list"   and P :: "'g list \<Rightarrow> 'h list \<Rightarrow> bool" assumes "length xs = length ys"   and "P [] []"   and "\<And>x xs y ys. \<lbrakk>length xs = length ys; P xs ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P xs ys"
ccw'_sortedP_convex_rotate_aux: fixes zs :: "(real \<times> real) list"   and x :: "real \<times> real" assumes "ccw'.sortedP 0 zs"   and "ccw'.sortedP x (map snd (polychain_of x zs))" shows "ccw'.sortedP (snd (last (polychain_of x zs))) (map snd (butlast (polychain_of x zs)))"
fmap_ext: fixes m :: "('b, 'c) fmap"   and n :: "('b, 'c) fmap" assumes "\<And>x. fmlookup m x = fmlookup n x" shows "m = n"
unwind_secure: fixes \<Delta> :: "'state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool" assumes "\<And>vl vl1. B vl vl1 \<Longrightarrow> \<Delta> istate vl istate vl1"   and "unwind \<Delta>" shows "secure"
assms: fixes x :: "'a" assumes "x \<noteq> \<bottom>" shows "f\<cdot>x \<noteq> \<bottom>"
path2_simple_loop: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and n' :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>n"   and "n' \<in> set ns" obtains ns' :: "'node list" where "g \<turnstile> n-ns'\<rightarrow>n" and "n' \<in> set ns'" and "n \<notin> set (tl (butlast ns'))" and "set ns' \<subseteq> set ns"
folding': fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and C :: "'b set" assumes "ChamberComplexFolding X f"   and "C \<in> f \<turnstile> ChamberComplex.chamber_system X" shows "\<exists>!D. D \<in> ChamberComplex.chamber_system X - f \<turnstile> ChamberComplex.chamber_system X \<and> f ` D = C"
P: fixes K :: "ennreal" assumes "K < \<infinity>" shows "AE x\<in>A in M. K \<le> (\<Sum>n. (nn_transfer_operator ^^ n) f x)"
continuous_real_polymonial_function: fixes f :: "'c \<Rightarrow> real"   and x :: "'c" assumes "real_polynomial_function f" shows "isCont f x"
Span_is_subalgebra: fixes K :: "'a set"   and Us :: "'a list" assumes "subfield K R"   and "set Us \<subseteq> carrier R" shows "subalgebra K (Span K Us) R"
f_nxt_upd_empty: fixes f :: "'b \<Rightarrow> 'b ttree"   and x' :: "'b"   and T :: "'b set"   and x :: "'b" shows "f_nxt (f(x' := TTree.empty)) T x = (f_nxt f T x)(x' := TTree.empty)"
Extra: fixes H :: "fm set" shows "H \<turnstile> extra_axiom"
two_comparisons_into_case_order_7: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a"   and Q :: "'b"   and P :: "'b"   and R :: "'b" assumes "comparator comp" shows "(if lt_of_comp comp x y then Q else if x = y then P else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
assms: shows "\<angle> z1 z2 \<noteq> pi"
assms: shows "D y \<notin> sources I D u (y # zs)"
open_cover_of_open_subset_from_target_to_source: fixes U :: "'c set"   and I :: "real set"   and C :: "real \<Rightarrow> 'c set" assumes "open_cover_of_open_subset S' is_open' U I C" shows "open_cover_of_open_subset S is_open (f \<^sup>\<inverse> S U) I (\<lambda>i. f \<^sup>\<inverse> S (C i))"
lim_Array_alloc: fixes x :: "'a list"   and h :: "heap" shows "lim (snd (Array.alloc x h)) = Suc (lim h)"
fps_lcm: fixes f :: "'a fps"   and g :: "'a fps" assumes "f \<noteq> 0"   and "g \<noteq> 0" shows "lcm f g = fps_X ^ max (subdegree f) (subdegree g)"
singleton_insert_inj_eq': fixes a :: "'d"   and A :: "'d set"   and b :: "'d" shows "(insert a A = {b}) = (a = b \<and> A \<subseteq> {b})"
Pi_pmf_default_swap: fixes A :: "'a set"   and dflt' :: "'b"   and dflt :: "'b"   and p :: "'a \<Rightarrow> 'b pmf" assumes "finite A" shows "map_pmf (\<lambda>f x. if x \<in> A then f x else dflt') (Pi_pmf A dflt p) = Pi_pmf A dflt' p"
weakly_legal_read_value_typeable: fixes f :: "String.literal \<Rightarrow> String.literal \<Rightarrow> ty list \<Rightarrow> ty \<Rightarrow> 'md \<Rightarrow> 'addr val list \<Rightarrow> 'x"   and C :: "String.literal"   and M :: "String.literal"   and vs :: "'addr val list"   and status :: "status"   and E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat"   and a :: "nat"   and ad :: "'addr"   and al :: "addr_loc"   and v :: "'addr val" assumes "ts_ok wfx (thr (h.start_state f P C M vs)) h.start_heap"   and "wf_syscls P"   and "weakly_legal_execution P (h.\<E>_start f P C M vs status) (E, ws)"   and "enat a < llength E"   and "action_obs E a = NormalAction (ReadMem ad al v)" shows "\<exists>T. P \<turnstile> ad@al : T \<and> P \<turnstile> v :\<le> T"
interval_union_commute: fixes A :: "'b interval"   and B :: "'b interval" shows "sup A B = sup B A"
wf_ruleset_normalize_match: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and m :: "'b match_expr"   and a :: "action" assumes "wf_ruleset \<gamma> p [Rule m a]" shows "wf_ruleset \<gamma> p (map (\<lambda>m. Rule m a) (normalize_match m))"
injective_closed_imp_embedding_map: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_map X Y f"   and "closed_map X Y f"   and "inj_on f (topspace X)" shows "embedding_map X Y f"
cfE_ss_invE: shows "l'.cf.E \<subseteq> E \<union> E\<inverse>"
truncate_ivl_correct: fixes A :: "real interval"   and prec :: "nat" shows "set_of A \<subseteq> set_of (truncate_ivl prec A)"
birkhoff_sum_def: fixes f :: "'a \<Rightarrow> 'b"   and n :: "nat"   and x :: "'a" shows "A.birkhoff_sum f n x = (\<Sum>i<n. f ((TA ^^ i) x))"
JJ: shows "JJ \<in> UNpart1 dl cl (\<lambda>n. F n ` P n)"
as: shows "mset (ah @ a # at) = mset (ah @ at) + {#a#}"
term_subst_eq_rev: fixes t :: "('a, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('a, 'b) Term.term"   and \<tau> :: "'c \<Rightarrow> ('a, 'b) Term.term" assumes "t \<cdot> \<sigma> = t \<cdot> \<tau>" shows "\<forall>x\<in>vars_term t. \<sigma> x = \<tau> x"
simps_8: fixes \<A> :: "'b ltl set"   and v :: "'b ltl" shows "\<A> \<Turnstile>\<^sub>P G v = (G v \<in> \<A>)"
smallo_powr: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real"   and p :: "real" assumes "f \<in> o[F](g)"   and "0 < p" shows "(\<lambda>x. \<bar>f x\<bar> powr p) \<in> o[F](\<lambda>x. \<bar>g x\<bar> powr p)"
left_commute: fixes b :: "'b multiset"   and a :: "'b multiset"   and c :: "'b multiset" shows "b \<inter># (a \<inter># c) = a \<inter># (b \<inter># c)"
igVarIPresIGWls_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gVarPresGWls MOD" shows "igVarIPresIGWls (fromMOD MOD)"
local_unique_solution: fixes T :: "real set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and t0 :: "real"   and x0 :: "'b" assumes "ll_on_open_it T f X"   and "t0 \<in> T"   and "x0 \<in> X" obtains et :: "real" and ex :: "real" and B :: "real" and L :: "real" where "0 < et" and "0 < ex" and "cball t0 et \<subseteq> T" and "cball x0 ex \<subseteq> X" and "unique_on_cylinder t0 (cball t0 et) x0 ex f B L"
minus_Box_def: fixes x :: "'b Box"   and y :: "'b Box" shows "x - y \<equiv> bliftM2 (\<Lambda> a b. a - b)\<cdot>x\<cdot>y"
permutes_swap_id: fixes a :: "'a"   and S :: "'a set"   and b :: "'a" assumes "a \<in> S"   and "b \<in> S" shows "Fun.swap a b id permutes S"
cond_spmf_alt: fixes p :: "'c spmf"   and A :: "'c set" shows "cond_spmf p A = mk_lossless (p \<upharpoonleft> A)"
arg_cong: fixes x :: "'b"   and y :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "x = y" shows "f x = f y"
simps_5: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and l :: "floatarith"   and r :: "floatarith"   and env :: "taylor_model list" shows "approx_tm prec ord I a (Mult l r) env = (case (approx_tm prec ord I a l env, approx_tm prec ord I a r env) of (None, b) \<Rightarrow> None | (Some t1, None) \<Rightarrow> None | (Some t1, Some t2) \<Rightarrow> Some (tm_mul prec ord I a t1 t2))"
C1: fixes \<Psi>' :: "'b"   and R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi"   and \<Psi>'' :: "'b" assumes "(\<Psi>', R, S) \<in> Rel"   and "\<Psi>' \<simeq> \<Psi>''" shows "(\<Psi>'', R, S) \<in> Rel'"
closed_sum_def: fixes A :: "'d set"   and B :: "'d set" shows "A +\<^sub>M B = closure (A + B)"
IH: fixes c :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int" assumes "simplicial_chain p (standard_simplex s) c" shows "(\<forall>r g. simplicial_simplex s (standard_simplex r) g \<longrightarrow> chain_map (Suc p) g (subd p c) = subd p (chain_map p g c)) \<and> simplicial_chain (Suc p) (standard_simplex s) (subd p c) \<and> chain_boundary (Suc p) (subd p c) + subd (p - Suc 0) (chain_boundary p c) = simplicial_subdivision p c - c"
has_derivative_within_alt: fixes f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and x :: "'b"   and s :: "'b set" shows "(f has_derivative f') (at x within s) = (bounded_linear f' \<and> (\<forall>e>0. \<exists>d>0. \<forall>y\<in>s. norm (y - x) < d \<longrightarrow> norm (f y - f x - f' (y - x)) \<le> e * norm (y - x)))"
T1: fixes u :: "'a"   and \<omega> :: "'a" assumes "ide u"   and "\<guillemotleft>\<omega> : local.dom \<omega> \<Rightarrow> r \<star> u\<guillemotright>" shows "\<exists>w \<theta> \<nu>. ide w \<and> \<guillemotleft>\<theta> : f \<star> w \<Rightarrow> u\<guillemotright> \<and> \<guillemotleft>\<nu> : local.dom \<omega> \<Rightarrow> g \<star> w\<guillemotright> \<and> local.iso \<nu> \<and> ((r \<star> \<theta>) \<cdot> \<a>[r, f, w] \<cdot> (\<rho> \<star> w)) \<cdot> \<nu> = \<omega>"
atLeastLessThan_injective: fixes m :: "nat"   and n :: "nat" assumes "{0..<m} = {0..<n}" shows "m = n"
Field_restrict_rel_subset: fixes A :: "'b set"   and R :: "('b \<times> 'b) set" shows "Field (restrict_rel A R) \<subseteq> A \<inter> Field R"
nc_finished_eq_reachable: assumes "\<not> cond s"   and "\<not> is_break tarjan_params s" shows "dom (finished s) = reachable"
leftderives1_def: fixes u :: "symbol list"   and v :: "symbol list" shows "leftderives1 u v = (\<exists>x y N \<alpha>. u = x @ [N] @ y \<and> v = x @ \<alpha> @ y \<and> is_word x \<and> is_sentence y \<and> (N, \<alpha>) \<in> \<RR>)"
intros: fixes a :: "'c"   and b :: "'d"   and r :: "('c \<times> 'd) set"   and c :: "'e"   and s :: "('d \<times> 'e) set" assumes "(a, b) \<in> r"   and "(b, c) \<in> s" shows "(a, c) \<in> r O s"