hcju/isabelleps huggingface dataset:
ENV: fixes ss :: "'s"   and ce :: "'s multiset"   and ss' :: "'s"   and csp :: "'s multiset" assumes "c = add_mset ss (add_mset s' ce)"   and "ce' = add_mset ss' (csp + ce)"   and "((ss, add_mset s' ce), e, ss', csp + add_mset s' ce) \<in> gtrs" shows "P"
ideal_S1_and: fixes a1 :: "bool"   and b1 :: "bool"   and a2 :: "bool"   and b2 :: "bool"   and s2 :: "bool" shows "and_secret_sharing.Ideal1 (a1, b1) (a2, b2) s2 = S1_and' (a1, b1) (a2, b2) s2"
eval_norm: fixes t :: "'c local.term" assumes "Arr t" shows "\<lbrace>\<^bold>\<parallel>t\<^bold>\<parallel>\<rbrace> = \<lbrace>t\<rbrace>"
assms: shows "((\<sigma>, p), a, s') \<in> onode_sos S"
suminf_ennreal: fixes f :: "nat \<Rightarrow> real" assumes "\<And>i. 0 \<le> f i"   and "(\<Sum>i. ennreal (f i)) \<noteq> top" shows "(\<Sum>i. ennreal (f i)) = ennreal (\<Sum>i. f i)"
card_b_pos: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i < j"   and "j \<le> k" shows "0 < card (b k (j, i))"
block_closed: fixes S :: "'d set"   and P :: "'d set set"   and a :: "'d"   and A :: "'d set" assumes "Set_Theory.partition S P"   and "a \<in> A"   and "A \<in> P" shows "a \<in> S"
proj2_set_Col_coeff: fixes S :: "proj2 set"   and p :: "proj2"   and q :: "proj2"   and r :: "proj2" assumes "proj2_set_Col S"   and "{p, q, r} \<subseteq> S"   and "p \<noteq> q"   and "r \<noteq> p" shows "r = proj2_abs (proj2_Col_coeff p q r *\<^sub>R proj2_rep p + proj2_rep q)"
Characters_simps_1: fixes G :: "('c, 'd) monoid_scheme" shows "carrier (Characters G) = characters G"
simps_5: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b"   and x2 :: "'c" shows "(case Some x2 of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f2 x2"
abs_eq: fixes xa :: "('form, 'rule) graph_node"   and x :: "('form, 'var) antecedent" shows "hyps_for xa x = Abs_fset (hyps_for' xa x)"
mset_set_set: fixes xs :: "'c list" assumes "distinct xs" shows "mset_set (set xs) = mset xs"
get_root_node_is_strongly_dom_component_safe_step: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and root :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> get_root_node ptr' \<rightarrow>\<^sub>r root" shows "(root \<in> set c) = (ptr' \<in> set c)"
tail_alt: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "ordered_term.tail term_of_pair ord_term p = except p {ordered_term.lt term_of_pair ord_term p}"
ring_homomorphism_axioms: shows "ring_homomorphism g B addB multB zeroB oneB C addC multC zeroC oneC"
ideal_generated_dvd_eq: fixes a :: "'b"   and b :: "'b"   and S :: "'b set" assumes "a dvd b"   and "a \<in> S"   and "a \<noteq> b" shows "ideal_generated S = ideal_generated (S - {b})"
a_join_b_eq_e: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "d_aux a b c \<le> e_aux a b c" shows "a_aux a b c \<squnion> b_aux a b c = e_aux a b c"
S2_def: shows "local.S2 = local.f ` local.S1"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) g"
COMB_init: fixes h :: "nat list"   and init :: "nat list" shows "COMB_init h init = bernoulli_pmf (8 / 10) \<bind> (\<lambda>b. bv (length init) \<bind> (\<lambda>xs. return_pmf (if b then Inl (xs, init) else Inr h)))"
zmod_helper: fixes n :: "int"   and m :: "int"   and k :: "int"   and a :: "int" assumes "n mod m = k" shows "(n + a) mod m = (k + a) mod m"
domains_comp: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and f :: "'b" assumes "category C"   and "partial_magma.arr C (C g f)" shows "partial_magma.domains C (C g f) = partial_magma.domains C f"
hyps: assumes "k < ncols A" shows "matrix_to_iarray (Gram_Schmidt_upt_k A k) = foldl Gram_Schmidt_column_k_iarrays_efficient (matrix_to_iarray A) [0..<Suc k]"
pochhammer_of_nat_eq_0_lemma': fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "pochhammer (- of_nat n) k \<noteq> (0::'a)"
finite_PiE_dflt: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'c set"   and d :: "'c" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> finite (B x)" shows "finite (PiE_dflt A d B)"
rep_eq: fixes x :: "nat"   and xa :: "float interval"   and xb :: "float interval" shows "bounds_of_interval (mult_float_interval x xa xb) = (case bounds_of_interval xa of (a1, a2) \<Rightarrow> \<lambda>(b1, b2). bnds_mult x a1 a2 b1 b2) (bounds_of_interval xb)"
isMinimalState_def: fixes stateMin :: "State"   and F0 :: "'a"   and decisionVars :: "nat set"   and learnL :: "(Literal list list \<times> Literal list list) set" shows "isMinimalState stateMin F0 decisionVars learnL \<equiv> \<forall>state. (state, stateMin) \<notin> terminationLess F0 decisionVars learnL"
lcm_ge_0_int: fixes x :: "int"   and y :: "int" shows "0 \<le> lcm x y"
ucast_mono_le: fixes x :: "'b word"   and y :: "'b word" assumes "x \<le> y"   and "y < 2 ^ LENGTH('c)" shows "UCAST('b \<rightarrow> 'c) x \<le> UCAST('b \<rightarrow> 'c) y"
type_const_br: fixes i :: "nat"   and lholed :: "Lholed"   and vs :: "e list"   and k :: "nat"   and LI :: "e list"   and \<C> :: "t_context"   and tcs :: "t list"   and \<S> :: "s_context"   and ts :: "t list"   and ts' :: "t list"   and \<C>' :: "t_context" assumes "Lfilled i lholed (vs @ [$Br (i + k)]) LI"   and "k < length (label \<C>)"   and "label \<C> ! k = tcs"   and "length tcs = length vs"   and "\<S>\<bullet>\<C> \<turnstile> LI : ts _> ts'"   and "const_list vs" shows "\<S>\<bullet>\<C>' \<turnstile> vs : [] _> tcs"
sort_upt: fixes m :: "nat"   and n :: "nat" shows "sort [m..<n] = [m..<n]"
rsplit: fixes x :: "real"   and bs :: "real list"   and f :: "int \<Rightarrow> MIR.num \<Rightarrow> fm"   and g :: "MIR.num \<Rightarrow> fm"   and a :: "MIR.num" assumes "0 \<le> x"   and "x < 1"   and "\<forall>a n s. Inum (x # bs) a = Inum (x # bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x # bs) (f n s) = Ifm (x # bs) (g a)" shows "Ifm (x # bs) (rsplit f a) = Ifm (x # bs) (g a)"
isomorphic_trans: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and W :: "'d set"   and smult'' :: "('b, 'a) aezfun \<Rightarrow> 'e \<Rightarrow> 'e"   and X :: "'e set" assumes "FGModule G smult V"   and "\<exists>T. FGModuleIso G smult V smult' T W"   and "\<exists>T. FGModuleIso G smult' W smult'' T X" shows "\<exists>T. FGModuleIso G smult V smult'' T X"
of_nat_nat_take_bit_eq: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and uminus :: "'k \<Rightarrow> 'k"   and n :: "nat"   and k :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "semiring_1.of_nat one plus zero (nat (take_bit n k)) = ring_1.of_int one plus zero minus (take_bit n k)"
invertible_left_inverse: fixes u :: "'a \<Rightarrow> 'a" assumes "invertible u"   and "u \<in> S \<rightarrow>\<^sub>E S" shows "compose S (inverse u) u = identity S"
assms_2: shows "g \<in> A \<rightarrow>\<^sub>E B"
dom_inv: fixes f :: "'a \<times> 'a" assumes "VV.iso f" shows "VV.dom (VV.inv f) = VV.cod f"
split_conv: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and a :: "'d"   and b :: "'e" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
idinfo_pwd_start: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and s' :: "(agent \<times> msg) set" assumes "s\<^sub>0 \<Turnstile> s"   and "n \<notin> bad_agent"   and "s \<turnstile> s'"   and "\<exists>X. \<langle>n, X\<rangle> \<in> spied s' \<and> X \<noteq> Pwd n"   and "\<nexists>X. \<langle>n, X\<rangle> \<in> spied s \<and> X \<noteq> Pwd n" shows "\<exists>SK. SesKey SK \<in> spied s \<and> ((Owner n, Crypt (SesK SK) (Pwd n)) \<in> s \<or> (Asset n, Crypt (SesK SK) (Num 0)) \<in> s)"
K4_imp_K2: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and evs :: "event list" assumes "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV" shows "\<exists>Ta. Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>\<rbrace>) \<in> set evs"
finsum_empty: fixes f :: "'c \<Rightarrow> 'a" shows "finsum R f {} = \<zero>"
c_snd_at_0: shows "c_snd 0 = 0"
distinct_55: fixes x71 :: "msg"   and x72 :: "msg"   and x81 :: "nat"   and x82 :: "msg" shows "\<lbrace>x71, x72\<rbrace> \<noteq> Crypt x81 x82"
vimage_Compl: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'c set" shows "f -` (- A) = - f -` A"
elims_2: fixes x :: "'d ifex"   and xa :: "('d, bool) mapping" assumes "taut_test_rec x xa" obtains env :: "('d, bool) mapping" where "x = Trueif" and "xa = env__"   | xb :: "'d" and t1 :: "'d ifex" and t2 :: "'d ifex" and env :: "('d, bool) mapping" where "x = IF xb t1 t2" and "xa = env" and "case Mapping.lookup env xb of None \<Rightarrow> taut_test_rec t1 (Mapping.update xb True env) \<and> taut_test_rec t2 (Mapping.update xb False env) | Some b \<Rightarrow> taut_test_rec (if b then t1 else t2) env"
double_compl: fixes minus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus :: "'f \<Rightarrow> 'f"   and inf :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and sup :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and bot :: "'f"   and top :: "'f"   and x :: "'f" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus (uminus x) = x"
perp_per_1: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A B Perp C A" shows "Per B A C"
stateless_callee_def: fixes callee :: "'a \<Rightarrow> ('b, 'd, 'e) gpv"   and s :: "'c" shows "stateless_callee callee s = map_gpv (\<lambda>b. (b, s)) id \<circ> callee"
X_def: shows "X = (\<lambda>i. if \<exists>n\<le>N. i = from_nat n then {z. dist (x i) z < f} else UNIV)"
n\<^sub>1_def: shows "\<forall>x\<ge>n\<^sub>1. token_succeeds x"
llist_all2_lmap1: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'c"   and xs :: "'e llist"   and ys :: "'d llist" shows "llist_all2 P (lmap f xs) ys = llist_all2 (\<lambda>x. P (f x)) xs ys"
in_orthogonal_complement_imp_orthogonal: fixes y :: "'b"   and S :: "'b set"   and x :: "'b" assumes "y \<in> S"   and "x \<in> orthogonal_complement S" shows "orthogonal x y"
AxPh: fixes m :: "Body" assumes "IOb m" shows "\<exists>v. cTest m v"
simps_5: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" shows "SC_BisT (c1 | c2) = (SC_BisT c1 \<and> SC_BisT c2 \<or> SC_discr (c1 | c2) \<or> SC_WbisT (c1 | c2))"
image_ring_zero: fixes f :: "'d \<Rightarrow> 'c"   and R :: "('d, 'e) ring_scheme" shows "\<zero>\<^bsub>image_ring f R\<^esub> = f \<zero>\<^bsub>R\<^esub>"
appendl_correct_1: fixes s :: "'queue"   and x :: "'t" assumes "queue_invar s" shows "queue_\<alpha> (queue_push x s) = x # queue_\<alpha> s"
null_eqI: fixes n :: "'a" assumes "\<And>f. Left_a.comp n f = n \<and> Left_a.comp f n = n" shows "n = Left_a.null"
distinct_36: fixes x71 :: "key"   and x72 :: "msg"   and x4 :: "key" shows "Crypt x71 x72 \<noteq> Key x4"
qFreshAbs_imp_ex_qAFreshAbs: fixes V :: "'var set"   and A :: "('e, 'f, 'varSort, 'var, 'g) qAbs"   and fZs :: "'var \<Rightarrow> 'varSort set" assumes "finite V"   and "qGoodAbs A"   and "\<forall>z\<in>V. \<forall>zs\<in>fZs z. qFreshAbs zs z A" shows "\<exists>A'. qGoodAbs A' \<and> A $= A' \<and> (\<forall>z\<in>V. \<forall>zs\<in>fZs z. qAFreshAbs zs z A')"
enum_ind_nth: fixes i :: "nat" assumes "i < CARD('b)" shows "enum_ind (enum_class.enum ! i) = i"
get_proc_get_return_edges: fixes a :: "'edge"   and a' :: "'edge" assumes "valid_edge a"   and "a' \<in> get_return_edges a" shows "get_proc (targetnode a) = get_proc (sourcenode a')"
quant_eqI: fixes x :: "'e['f, 'g]"   and y :: "'e['f, 'g]" assumes "\<lbrakk>x\<rbrakk>\<^sub>Q = \<lbrakk>y\<rbrakk>\<^sub>Q" shows "x = y"
comp_assoc: fixes h :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "local.comp (local.comp h g) f = local.comp h (local.comp g f)"
lincomb_must_include: fixes T :: "'c set"   and S :: "'c set"   and A :: "'c set"   and b :: "'c \<Rightarrow> 'a"   and v :: "'c" assumes "T \<subseteq> carrier V"   and "lin_indpt S"   and "S \<subseteq> T"   and "A \<subseteq> T"   and "finite A"   and "b \<in> A \<rightarrow> carrier K"   and "lincomb b A = \<zero>\<^bsub>V\<^esub>"   and "v \<in> A"   and "b v \<noteq> \<zero>\<^bsub>K\<^esub>" shows "\<exists>w\<in>A - S. b w \<noteq> \<zero>\<^bsub>K\<^esub>"
bits_one_mod_two_eq_one: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "modulo one (numeral.numeral one plus (num.Bit0 num.One)) = one"
IH: fixes f :: "frame" assumes "P,\<Phi> |- (None, h, f # frs', sh) [ok]"   and "curr_method f \<noteq> clinit" shows "P,\<Phi> |- find_handler P xcp h frs' sh [ok]"
finsum_singleton_set: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b" shows "sum f {a} = f a"
unique_decomposition: fixes K :: "'a set"   and Us :: "'a list"   and a :: "'a" assumes "subfield K R"   and "local.independent K Us"   and "a \<in> Span K Us" shows "\<exists>!Ks. set Ks \<subseteq> K \<and> length Ks = length Us \<and> a = combine Ks Us"
ts''_def_1: shows "\<S>\<bullet>\<C> \<turnstile> vs : ts _> ts''"
avg_proj_eq_id_on_right: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and W :: "'c set"   and v :: "'c" assumes "FinGroupRepresentation G smult V"   and "VectorSpace (aezfun_scalar_mult.fsmult smult) W"   and "add_independentS [W, V]"   and "v \<in> V" shows "(aezfun_scalar_mult.fsmult smult ((1::'b) / of_nat (card G)) \<circ> (\<Sum>g\<in>G. aezfun_scalar_mult.Gmult smult (- g) \<circ> (\<lambda>a. if a \<in> (\<Oplus>G\<leftarrow>[W, V]. G) then W\<oplus>V\<leftarrow> a ! 1 else (0::'c)) \<circ> aezfun_scalar_mult.Gmult smult g)) v = v"
simps_1: fixes F :: "'b set \<Rightarrow> 'a"   and M :: "'b acom set" shows "lift F com.SKIP M = SKIP {F (post ` M)}"
lookup_pair_tl': fixes x :: "'o"   and xs :: "('a \<times> 'c) list"   and k :: "'a" assumes "oalist_inv_raw x xs" shows "lookup_pair x (tl xs) k = (if k = fst (hd xs) then 0::'c else lookup_pair x xs k)"
inverse_design_isomorphism: shows "design_isomorphism \<V>' \<B>' \<V> \<B> (inv_into \<V> \<pi>)"
op_dg_def: fixes \<CC> :: "V" shows "op_dg \<CC> = [\<CC>\<lparr>Obj\<rparr>, \<CC>\<lparr>Arr\<rparr>, \<CC>\<lparr>Cod\<rparr>, \<CC>\<lparr>Dom\<rparr>]\<^sub>\<circ>"
fresh_6: fixes a :: "coname"   and x2 :: "name"   and x3 :: "trm"   and x1 :: "coname" shows "a \<sharp> NotR x2.x3 x1 = (a \<sharp> [x2].x3 \<and> a \<sharp> x1)"
evalOP_inv_7: fixes C :: "db"   and T :: "db"   and E :: "db"   and v :: "db" assumes "DBCond C T E \<Down> v" obtains "C \<Down> DBtt" and "T \<Down> v"   | "C \<Down> DBff" and "E \<Down> v"
Class_def: shows "orbit.Class = (\<lambda>a\<in>G. {b \<in> G. (b, a) \<in> Congruence})"
mult_scalar_assoc: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c" shows "(p * q) \<odot> r = p \<odot> (q \<odot> r)"
no_loops: fixes G :: "('c, 'd) pre_digraph"   and e :: "'d" assumes "loopfree_digraph G"   and "e \<in> arcs G" shows "tail G e \<noteq> head G e"
simps: fixes a1 :: "('c, 'd) conf"   and a2 :: "('c, 'd) label list"   and a3 :: "('c, 'd) conf" shows "se_star a1 a2 a3 = ((\<exists>c. a1 = c \<and> a2 = [] \<and> a3 = c) \<or> (\<exists>c1 l c2 ls c3. a1 = c1 \<and> a2 = l # ls \<and> a3 = c3 \<and> se c1 l c2 \<and> se_star c2 ls c3))"
HSem_fresh_cong_below: fixes S :: "Vars.var set"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue"   and \<rho>' :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" assumes "\<rho> f|` (S \<union> fv \<Gamma> - domA \<Gamma>) \<sqsubseteq> \<rho>' f|` (S \<union> fv \<Gamma> - domA \<Gamma>)" shows "(\<N>\<lbrace>\<Gamma>\<rbrace>\<rho>) f|` S \<sqsubseteq> (\<N>\<lbrace>\<Gamma>\<rbrace>\<rho>') f|` S"
Rep_uexpr: fixes x :: "('f, 'e) uexpr" shows "\<lbrakk>x\<rbrakk>\<^sub>e \<in> UNIV"
atLeast0_lessThan_Suc: fixes g :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "sum g {0..<Suc n} = sum g {0..<n} + g n"
ifthenelse_def: fixes P :: "'c \<Rightarrow> bool"   and X :: "'c \<Rightarrow> 'd set"   and Y :: "'c \<Rightarrow> 'd set" shows "IF P THEN X ELSE Y \<equiv> \<lambda>s. if P s then X s else Y s"
cycle_remove_edge_3: fixes x :: "'a"   and s :: "'a"   and e :: "'a" assumes "cycle x"   and "point s"   and "point e"   and "e ; s\<^sup>T \<le> x" shows "end_points (x \<cdot> - (e ; s\<^sup>T)) \<le> e"
right_lesseq: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a l\<rightarrow> b = (1::'a))"
simI: fixes M :: "('b, 'c, 'd) KripkeStructure"   and M' :: "('b, 'c, 'e) KripkeStructure"   and f :: "'d \<Rightarrow> 'e" assumes "sim_range M M' f"   and "sim_val M M' f"   and "sim_f M M' f"   and "sim_r M M' f" shows "sim M M' f"
tmi'_def: fixes s :: "bdd" shows "tmi' s \<equiv> oassert (bdd_sane s) \<bind> (\<lambda>_. Some (tmi s))"
iter_mono_emeasure1: fixes n :: "nat"   and s :: "'a" shows "emeasure (measure_spmf (iter n s)) {s. \<not> guard s} \<le> emeasure (measure_spmf (iter (Suc n) s)) {s. \<not> guard s}"
neg_def: fixes \<gamma> :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool" shows "\<not>\<^sup>b \<gamma> \<equiv> \<lambda>t n. \<not> \<gamma> t n"
open_iff: fixes X :: "'b set"   and t :: "'b set" shows "(t open\<^bsub>discrete_top X\<^esub>) = (t \<in> Pow carrier\<^bsub>discrete_top X\<^esub>)"
altc_cUNION: fixes f :: "'c \<Rightarrow> 'c cset"   and C :: "'c cset"   and g :: "'c \<Rightarrow> 'm" shows "altc (cUnion (cimage f C)) g = altc C (\<lambda>x. altc (f x) g)"
tranclp_induct4: fixes r :: "'a \<times> 'b \<times> 'c \<times> 'd \<Rightarrow> 'a \<times> 'b \<times> 'c \<times> 'd \<Rightarrow> bool"   and ax :: "'a"   and ay :: "'b"   and az :: "'c"   and aw :: "'d"   and bx :: "'a"   and "by" :: "'b"   and bz :: "'c"   and bw :: "'d"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "r\<^sup>+\<^sup>+ (ax, ay, az, aw) (bx, by, bz, bw)"   and "\<And>a aa ab b. r (ax, ay, az, aw) (a, aa, ab, b) \<Longrightarrow> P a aa ab b"   and "\<And>a aa ab b ac ad ae ba. \<lbrakk>r\<^sup>+\<^sup>+ (ax, ay, az, aw) (a, aa, ab, b); r (a, aa, ab, b) (ac, ad, ae, ba); P a aa ab b\<rbrakk> \<Longrightarrow> P ac ad ae ba" shows "P bx by bz bw"
step_ivl_def: shows "step_ivl \<equiv> Abs_Int1.step' num_ivl plus_ivl in_ivl filter_plus_ivl filter_less_ivl"
chop_empty1: fixes r :: "real_int"   and s :: "real_int"   and t :: "real_int" shows "R_Chop(r,s,t) \<and> \<parallel>s\<parallel> = 0 \<longrightarrow> r = t"
S_def: shows "S = (\<lambda>b. {x. P x \<and> a < x \<and> x < b})"
distance0_2_def: shows "distance0_2 = v\<^sub>e * \<delta> + 1 / 2 * v\<^sub>o\<^sup>2 / a\<^sub>o"
vintersection_vsingleton: fixes a :: "V" shows "a \<in>\<^sub>\<circ> ZFC_in_HOL.set {a}"
Rep_mod_ring_inject: fixes x :: "'b mod_ring"   and y :: "'b mod_ring" shows "(Rep_mod_ring x = Rep_mod_ring y) = (x = y)"
reflpD: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d" assumes "reflp r" shows "r x x"
rel_witness_fun_def: fixes A :: "'c \<Rightarrow> 'g \<Rightarrow> bool"   and A' :: "'g \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_witness_fun A A' = (\<lambda>(f, g) b. (f (THE a. A a b), g (THE c. A' b c)))"
elims_1: fixes x :: "real_max_bot"   and xa :: "real_max_bot"   and y :: "bool" assumes "(x < xa) = y" obtains uu_ :: "real_max_bot" where "xa__ = uu_" and "xa = MInfty" and "\<not> ya__"   | uv_ :: "real" where "xa__ = MInfty" and "xa = R uv_" and "ya__"   | x :: "real" and y :: "real" where "xa__ = R x" and "xa = R y" and "ya__ = (x < y)"
density_def: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" shows "density M f = measure_of (space M) (sets M) (\<lambda>A. set_nn_integral M A f)"
distinct_1: fixes \<Delta> :: "('a, 'b) network_scheme"   and x2 :: "'c" assumes "flow_attainability \<Delta>" shows "SOURCE \<noteq> Inner x2"
Suc: shows "higher_differentiable_on {..<0} f k \<and> (\<forall>t<0. nth_derivative k f t 1 = 0)"
min_group_size: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and groups :: "'b set set"   and G :: "'b set" assumes "group_design point_set block_collection groups"   and "G \<in> groups" shows "1 \<le> card G"
intP_def: shows "intP \<equiv> ProblemIkTpartG_GEModel.intP eintP"
fold_empty: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a" shows "Finite_Set.fold f z {} = z"
trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "R x y"   and "R y z"   and "x \<in> S"   and "y \<in> S"   and "z \<in> S" shows "R x z"
V_def: shows "V \<equiv> {v. \<forall>x\<in>X. 0 \<le> v x}"
boundedBySuc': fixes N :: "nat"   and A :: "(nat \<times> 'b) set" assumes "boundedBy N A" shows "boundedBy (Suc N) A"
intro: fixes G :: "('c, 'd) graph_rec_scheme" assumes "param_DFS G" shows "Tarjan G"
qTerm_induct: fixes phi :: "('c, 'd, 'a, 'b, 'e) qTerm \<Rightarrow> bool"   and phiAbs :: "('c, 'd, 'a, 'b, 'e) qAbs \<Rightarrow> bool"   and X :: "('c, 'd, 'a, 'b, 'e) qTerm"   and A :: "('c, 'd, 'a, 'b, 'e) qAbs" assumes "\<And>xs x. phi (qVar xs x)"   and "\<And>delta inp binp. \<lbrakk>liftAll phi inp; liftAll phiAbs binp\<rbrakk> \<Longrightarrow> phi (qOp delta inp binp)"   and "\<And>xs x X. \<lbrakk>\<And>Y. (X, Y) \<in> qSwapped \<Longrightarrow> phi Y; \<And>Y. qSkel Y = qSkel X \<Longrightarrow> phi Y\<rbrakk> \<Longrightarrow> phiAbs (qAbs xs x X)" shows "phi X \<and> phiAbs A"
bdd_above_Icc: fixes a :: "'a"   and b :: "'a" shows "bdd_above {a..b}"
r3: fixes x :: "name"   and s :: "trm"   and t :: "trm" assumes "x \<sharp> s" shows "App (\<Lambda> x . t) s \<mapsto> t[x::=s]"
assms: shows "prime P \<and> odd P \<and> P dvd A \<and> P ^ n = p\<^sup>2 + N * q\<^sup>2 \<and> A ^ n = a\<^sup>2 + N * b\<^sup>2 \<and> coprime a b \<and> coprime p (N * q) \<and> 0 < n"
attractor_set_base: fixes W :: "'a set"   and p :: "Player" shows "W \<subseteq> attractor p W"
eq_resource_on_UNIV_D: fixes res :: "('a, 'b) resource"   and res' :: "('a, 'b) resource" assumes "UNIV \<turnstile>\<^sub>R res \<sim> res'" shows "res = res'"
hom_1_iff: fixes x :: "rat" shows "(of_rat x = (1::'a)) = (x = 1)"
imp_ab_rational: assumes "\<exists>B>0. \<exists>c. \<forall>\<^sub>F n in sequentially. B * b n = c n * a n - c (n + 1) \<and> real_of_int \<bar>c (n + 1)\<bar> < real_of_int (a n) / 2" shows "(\<Sum>n. real_of_int (b n) / real_of_int (prod a {..n})) \<in> \<rat>"
sum_bounded_above: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and K :: "'b" assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<le> K" shows "sum f A \<le> of_nat (card A) * K"
group_l_invI: assumes "\<And>x. x \<in> carrier (G\<lparr>carrier := S\<rparr>) \<Longrightarrow> \<exists>y\<in>carrier (G\<lparr>carrier := S\<rparr>). y \<otimes>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> x = \<one>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub>" shows "Group.group (G\<lparr>carrier := S\<rparr>)"