hcju/isabelleps huggingface dataset:
assms: shows "subst_range \<delta> \<subseteq> range Var"
actual_desc_3: fixes v :: "i"   and z :: "\<nu>"   and \<phi> :: "\<nu> \<Rightarrow> \<o>" shows "[z\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) \<^bold>\<rightarrow> \<^bold>\<A>\<phi> z in v]"
not_interval_UNIV2_2: fixes a :: "'b"   and b :: "'b" shows "UNIV \<noteq> box a b"
2: shows "fmdom' (fst a) \<union> fmdom' (snd a) \<subseteq> vs"
equiv_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "equiv A r = (refl_on A r \<and> sym r \<and> trans r)"
hd_exp: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and e :: "expr"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal" assumes "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d e \<Rightarrow> f" shows "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d Exp $$ e \<Rightarrow> (\<lambda>\<sigma> x. if 0 < extract_real x then f \<sigma> (lift_RealVal safe_ln x) * ennreal (inverse (extract_real x)) else 0)"
t2_def: shows "approx_tm prec ord I a r env = Some t2"
prj_in_hom_3: shows "\<guillemotleft>gfg.Prj\<^sub>0 : Chn ((g \<star> f) \<star> g) \<rightarrow>\<^sub>C g.apex\<guillemotright>"
euler_mascheroni_lower: fixes n :: "nat" shows "harm (Suc n) - ln (real (n + 2)) + 1 / real (2 * (n + 2)) \<le> euler_mascheroni"
all_termi: shows "\<forall>i. terminate f (args @ [i]) \<and> 0 < rec_exec f (args @ [i])"
gfp_def: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and f :: "'b::type \<Rightarrow> 'b::type" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)" shows "complete_lattice.gfp (Sup::'b::type set \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (f::'b::type \<Rightarrow> 'b::type) = Sup {u::'b::type. less_eq u (f u)}"
igSwapIGAbsSTR_imp_igSwapIGAbs: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('g, 'h, 'd, 'e, 'i, 'j, 'k, 'l, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "igSwapIGAbsSTR MOD" shows "FixSyn.igSwapIGAbs Delta MOD"
that: shows "c \<subseteq>\<^sub>g DynCom C"
iT_Plus_inext_nth: fixes I :: "nat set"   and k :: "nat"   and n :: "nat" assumes "I \<noteq> {}" shows "(I \<oplus> k) \<rightarrow> n = (I \<rightarrow> n) + k"
in_cfls_mono: fixes G :: "('a, 'b) cfg"   and P :: "('a + 'b) list fset"   and w :: "'a list"   and Q :: "('a + 'b) list fset" assumes "in_cfls G P w"   and "P |\<subseteq>| Q" shows "in_cfls G Q w"
lve_edge: fixes valid_edge :: "'g \<Rightarrow> bool"   and a :: "'g"   and src :: "'g \<Rightarrow> 'h"   and E :: "'h"   and trg :: "'g \<Rightarrow> 'h"   and X :: "'h"   and e :: "'h LDCFG_node \<times> ('i, 'j, 'k, 'l) edge_kind \<times> 'h LDCFG_node"   and knd :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind" assumes "valid_edge a"   and "src a \<noteq> E \<or> trg a \<noteq> X"   and "e = (Node (src a), knd a, Node (trg a))" shows "lift_valid_edge valid_edge src trg knd E X e"
a_nonneg: fixes n :: "nat" assumes "0 < n" shows "0 \<le> a n"
pick_mono_inf: fixes S :: "nat set"   and m :: "nat"   and n :: "nat" assumes "infinite S"   and "m < n" shows "pick S m < pick S n"
assms_2: fixes exp :: "exp"   and c :: "'a"   and var :: "Vars.var" assumes "\<And>c. P c exp" shows "P c (App exp var)"
complete2: fixes M :: "'b measure"   and A :: "'b set"   and B :: "'b set" assumes "complete_measure M"   and "A \<subseteq> B"   and "B \<in> null_sets M" shows "A \<in> null_sets M"
plossless_gpv_ContD: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and out :: "'a"   and c :: "('c, 'a, 'b) rpv"   and input :: "'b" assumes "plossless_gpv \<I> gpv"   and "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<I> out"   and "\<I> \<turnstile>g gpv \<surd>" shows "plossless_gpv \<I> (c input)"
\<psi>_nat_cancel: fixes x :: "int" shows "\<psi> (real (nat x)) = \<psi> (real_of_int x)"
C_eq_compile: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and l :: "(int \<times> int \<times> protocol) set set set list" assumes "DenyAll \<in> set (policy2list p)"   and "all_in_list (policy2list p) l"   and "allNetsDistinct (policy2list p)" shows "Cp (list2FWpolicy (removeAllDuplicates (insertDenies (separate (FWNormalisationCore.sort (removeShadowRules2 (remdups (rm_MT_rules Cp (insertDeny (removeShadowRules1 (policy2list p)))))) l))))) = Cp p"
dense_ge: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and z :: "'d"   and y :: "'d" assumes "class.dense_linorder less_eq less"   and "\<And>x. less z x \<Longrightarrow> less_eq y x" shows "less_eq y z"
filtermap_tan_at_left: fixes a :: "real" assumes "cos a \<noteq> 0" shows "filtermap tan (at_left a) = at_left (tan a)"
pmf_eqI: fixes M :: "'a pmf"   and N :: "'a pmf" assumes "\<And>i. pmf M i = pmf N i" shows "M = N"
J1_def: shows "J1 = flow_to_path x t1 t2"
IH1_3: assumes "\<not> M \<Turnstile> F" shows "\<not> M(n := True) \<Turnstile> F"
fmlookup_default_fmmap_keys: fixes d :: "'a"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and M :: "('b, 'a) fmap"   and x :: "'b" shows "fmlookup_default d (fmmap_keys f M) x = (if x \<in> fmdom' M then f x (fmlookup_default d M x) else d)"
GreatestI: fixes P :: "'a \<Rightarrow> bool"   and k :: "'a" assumes "P k" shows "P (GREATEST x. P x)"
unity_def: shows "\<F>C.unity \<equiv> \<F>C.cod \<F>C.\<iota>\<^sub>F\<^sub>M\<^sub>C"
set_sel_2: fixes a :: "('a, 'b) tllist"   and x :: "'a" assumes "\<not> is_TNil a"   and "x \<in> tset (ttl a)" shows "x \<in> tset a"
add_is_1: fixes m :: "nat"   and n :: "nat" shows "(m + n = Suc 0) = (m = Suc 0 \<and> n = 0 \<or> m = 0 \<and> n = Suc 0)"
sel: fixes x2 :: "'d" shows "the (Some x2) = x2"
sets_Sup_sigma: fixes M :: "'c set set set"   and \<Omega> :: "'c set" assumes "M \<noteq> {}"   and "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>" shows "sets (Sup (sigma \<Omega> ` M)) = sets (sigma \<Omega> (\<Union> M))"
mustT_def: fixes c :: "('test, 'atom) com"   and s :: "'state" shows "mustT c s \<equiv> \<not> mayDiverge c s"
hta_lookup_sf_def: fixes q :: "'a"   and f :: "'b"   and H :: "('a, 'b, 'c) hashedTa_scheme" shows "hta_lookup_sf q f H \<equiv> hll_idx.lookup (q, f) (the (hta_idx_sf H))"
powr_smallo_iff: fixes g :: "'a \<Rightarrow> real"   and F :: "'a filter"   and p :: "real"   and q :: "real" assumes "filterlim g at_top F"   and "F \<noteq> bot" shows "((\<lambda>x. g x powr p) \<in> o[F](\<lambda>x. g x powr q)) = (p < q)"
assms: shows "T : b \<mapsto>\<^bsub>dg_Rel \<alpha>\<^esub> a"
Max_X_in_XY: shows "Max X \<in> X \<union> Y"
in_hom_char: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright> = (arr a \<and> arr b \<and> arr f \<and> \<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>)"
assms_2: fixes y :: "'a vec" assumes "y \<in> S" shows "x \<bullet> c \<le> y \<bullet> c"
card_1_Collect_ex1: fixes P :: "'a \<Rightarrow> bool" assumes "card (Collect P) = 1" shows "\<exists>!x. P x"
map_of_zip_nth: fixes xs :: "'g list"   and ys :: "'h list"   and i :: "nat" assumes "length xs = length ys"   and "distinct xs"   and "i < length ys" shows "map_of (xs || ys) (xs ! i) = Some (ys ! i)"
pos_mod_sign: fixes b :: "'d"   and a :: "'d" assumes "(0::'d) < b" shows "(0::'d) \<le> a mod b"
higher_differentiable_on_subset: fixes T :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and n :: "nat"   and S :: "'a set" assumes "higher_differentiable_on T f n"   and "S \<subseteq> T" shows "higher_differentiable_on S f n"
ring_mat_def: fixes ty :: "'a itself"   and n :: "nat"   and b :: "'b" shows "ring_mat ty n b \<equiv> \<lparr>carrier = carrier_mat n n, monoid.mult = (*), one = 1\<^sub>m n, zero = 0\<^sub>m n n, add = (+), \<dots> = b\<rparr>"
m: fixes p :: "'b"   and q :: "'b" shows "((p, q) \<in> m) = p \<parallel> q"
alphaBinp_imp_asBinp_equal: fixes qbinp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qAbs option"   and qbinp' :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qAbs option" assumes "qGoodBinp qbinp"   and "(qbinp, qbinp') \<in> alphaBinp" shows "asBinp qbinp = asBinp qbinp'"
unfold_simps_2: fixes A :: "'d set"   and B :: "'d set" shows "(A \<subseteq> B) = (\<forall>t. t \<in> A \<longrightarrow> t \<in> B)"
basic_trans_rules_31: fixes A :: "'e set"   and B :: "'e set"   and c :: "'e" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
is_leveln_ftree_def: shows "is_leveln_ftree \<equiv> \<lambda>uu uua. rec_FingerTreeStruc (\<lambda>n. True) (\<lambda>nd n. is_leveln_node n nd) (\<lambda>_ l t r ta n. is_leveln_digit n l \<and> is_leveln_digit n r \<and> ta (Suc n)) uua uu"
rel_in_carr: fixes A :: "'a set"   and r :: "'a \<Rightarrow> int" assumes "A \<subseteq> carrier G"   and "r \<in> relations A" shows "(\<lambda>a. a [^] r a) \<in> A \<rightarrow> carrier G"
homeomorphic_refl: fixes s :: "'a set" shows "s homeomorphic s"
simps: fixes ante :: "('a, 'b, 'c) formula list"   and succ :: "('a, 'b, 'c) formula list" shows "seq2fml (ante, succ) = (foldr (&&) ante TT \<rightarrow> foldr (||) succ FF)"
TensorDiag_preserves_Ide: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Ide t"   and "Ide u"   and "Diag t"   and "Diag u" shows "Ide (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u)"
"1_2": shows "w\<^sub>1 \<preceq>\<^sub>F v\<^sub>2"
Eq\<^sub>0S: fixes v :: "i"   and F :: "\<o>"   and G :: "\<o>" shows "[F \<^bold>=\<^sub>0 G in v] = (F = G)"
homeomorphism_on_empty: fixes f :: "'c \<Rightarrow> 'd" shows "homeomorphism_on {} f"
pmf_le_1: fixes p :: "'d pmf"   and x :: "'d" shows "pmf p x \<le> 1"
verit_sko_ex': fixes P :: "'c \<Rightarrow> bool"   and A :: "bool" assumes "P (SOME x. P x) = A" shows "(\<exists>x. P x) = A"
add_idem': fixes x :: "'a" shows "x \<squnion> x = x"
end_idhom: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set" assumes "VectorSpace smult V" shows "VectorSpaceEnd smult V (id \<down> V)"
cnf_of_encode_interfering_operator_exclusion_is_iii: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "cnf ` set (concat (map (\<lambda>(op\<^sub>1, op\<^sub>2). map (\<lambda>k. encode_interfering_operator_pair_exclusion \<Pi> k op\<^sub>1 op\<^sub>2) [0..<t]) (filter (\<lambda>(op\<^sub>1, op\<^sub>2). index (\<Pi>\<^sub>\<O>) op\<^sub>1 \<noteq> index (\<Pi>\<^sub>\<O>) op\<^sub>2 \<and> are_operators_interfering op\<^sub>1 op\<^sub>2) (List.product (\<Pi>\<^sub>\<O>) (\<Pi>\<^sub>\<O>))))) = (\<Union>(op\<^sub>1, op\<^sub>2)\<in>{(op\<^sub>1, op\<^sub>2). (op\<^sub>1, op\<^sub>2) \<in> set (\<Pi>\<^sub>\<O>) \<times> set (\<Pi>\<^sub>\<O>) \<and> index (\<Pi>\<^sub>\<O>) op\<^sub>1 \<noteq> index (\<Pi>\<^sub>\<O>) op\<^sub>2 \<and> are_operators_interfering op\<^sub>1 op\<^sub>2}. {{{(Operator k (index (\<Pi>\<^sub>\<O>) op\<^sub>1))\<inverse>, (Operator k (index (\<Pi>\<^sub>\<O>) op\<^sub>2))\<inverse>}} |k. k \<in> {0..<t}})"
e0: fixes x :: "real" assumes "x \<noteq> 0"   and "\<bar>x\<bar> < e0" shows "\<bar>sin (x / 2) / x - 1 / 2\<bar> < 1 / 4"
set_eq_insert_aux: fixes A :: "'a set"   and x :: "'a"   and x' :: "'a"   and y :: "'a" assumes "A \<subseteq> carrier S"   and "x \<in> carrier S"   and "x' \<in> carrier S"   and "x .= x'"   and "y \<in> insert x A" shows "y .\<in> insert x' A"
finite_field_ops64_def: fixes p :: "uint64" shows "finite_field_ops64 p \<equiv> Arith_Ops_Record 0 1 (plus_p64 p) (mult_p64 p) (minus_p64 p) (uminus_p64 p) (divide_p64 p) (inverse_p64 p) (\<lambda>x y. if y = 0 then x else 0) (\<lambda>x. if x = 0 then 0 else 1) (\<lambda>x. x) uint64_of_int int_of_uint64 (\<lambda>x. 0 \<le> x \<and> x < p)"
find_fund_sol_def: fixes D :: "nat" shows "find_fund_sol D = (if square_test D then (0, 0) else projr (while isl (find_fund_sol_step D) (Inl (1, 1 + D))))"
sum_character: fixes x :: "i"   and y :: "i" shows "\<forall>w. O w (x \<oplus> y) = (O w x \<or> O w y)"
nn_integral_add: fixes f :: "'h \<Rightarrow> ennreal"   and M :: "'h measure"   and g :: "'h \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. f x + g x \<partial>M = integral\<^sup>N M f + integral\<^sup>N M g"
bool_to_ternary_simp3: fixes X :: "bool" shows "(eval_ternary_Not (bool_to_ternary X) = TernaryTrue) = (\<not> X)"
vsv_valeq_v11I: fixes r :: "V" assumes "vsv r"   and "\<And>x y. \<lbrakk>x \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r; y \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r; r\<lparr>x\<rparr> = r\<lparr>y\<rparr>\<rbrakk> \<Longrightarrow> x = y" shows "v11 r"
domains_comp: fixes g :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data" assumes "V.seq g f" shows "V.domains (g \<bullet> f) = V.domains f"
Rep_Ainteg: fixes x :: "ant" shows "Rep_Ainteg x \<in> Ainteg"
a: shows "take x (take x Strat @ (fst (Strat ! x), take l (snd (Strat ! x)) @ drop (Suc l) (snd (Strat ! x))) # drop (Suc x) Strat) = take x Strat"
Nil_notin_lex: fixes ys :: "'b list"   and r :: "('b \<times> 'b) set" shows "([], ys) \<notin> lex r"
msteps_msteps_stateless: fixes w :: "(String.literal \<times> event_data list) prefix"   and st :: "('a, 'b) mstate" shows "fst (msteps w st) = msteps_stateless w st"
unallocatedBlocksInvariant1: shows "unallocatedBlocksInvariant makeCF"
add_mono_thms_linordered_field_1: fixes i :: "'a"   and j :: "'a"   and k :: "'a"   and l :: "'a" assumes "i < j \<and> k = l" shows "i + k < j + l"
elims: fixes x :: "'d \<Rightarrow> 'e \<Rightarrow> 'f option"   and xa :: "'e"   and xb :: "'d list"   and y :: "'d option" assumes "applied_rule x xa xb = y" obtains C :: "'d \<Rightarrow> 'e \<Rightarrow> 'f option" and a :: "'e" and xc :: "'d" and xs :: "'d list" where "x = C__" and "xa = a__" and "xb = xc # xs" and "y = (if a__ \<in> dom (C__ xc) then \<lfloor>xc\<rfloor> else applied_rule C__ a__ xs)"   | C :: "'d \<Rightarrow> 'e \<Rightarrow> 'f option" and a :: "'e" where "x = C" and "xa = a" and "xb = []" and "y = \<bottom>"
frameImpResCommRight': fixes xvec :: "name list"   and x :: "name"   and F :: "'b frame" shows "\<lparr>\<nu>*xvec\<rparr>\<lparr>\<nu>x\<rparr>F \<hookrightarrow>\<^sub>F \<lparr>\<nu>x\<rparr>\<lparr>\<nu>*xvec\<rparr>F"
BRRes: fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set"   and a :: "name" assumes "(P, Q) \<in> bangRel Rel" shows "(<\<nu>a>P, <\<nu>a>Q) \<in> bangRel Rel"
M_eq: fixes a :: "real" shows "emeasure M {x \<in> space M. X x \<le> a} = ennreal (g a)"
Iod_Order: fixes D :: "('c, 'd) Order_scheme"   and T :: "'c set" assumes "Order D"   and "T \<subseteq> carrier D" shows "Order (Iod D T)"
numeral_times_numeral: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and m :: "num"   and n :: "num" assumes "class.semiring_numeral one times plus" shows "times (numeral.numeral one plus m) (numeral.numeral one plus n) = numeral.numeral one plus (m * n)"
Least_equality: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "class.order less_eq less"   and "P x"   and "\<And>y. P y \<Longrightarrow> less_eq x y" shows "ord.Least less_eq P = x"
xor_zero_left: fixes x :: "'a" shows "\<zero> \<oplus> x = x"
Eq_Eats_sf: fixes t :: "tm"   and u :: "tm"   and i :: "name" assumes "\<And>j. Sigma_fm (Var j EQ t)"   and "\<And>k. Sigma_fm (Var k EQ u)" shows "Sigma_fm (Var i EQ Eats t u)"
iaction_def: fixes \<Delta> :: "'a \<Rightarrow> 'b \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool"   and s :: "'a"   and vl :: "'b"   and s1 :: "'state"   and vl1 :: "'value list" shows "iaction \<Delta> s vl s1 vl1 \<equiv> \<exists>a1 ou1 s1' vl1'. let trn1 = Trans s1 a1 ou1 s1' in validTrans trn1 \<and> \<phi> trn1 \<and> consume trn1 vl1 vl1' \<and> \<not> \<gamma> trn1 \<and> \<Delta> s vl s1' vl1'"
norm_mult_less: fixes x :: "'a"   and r :: "real"   and y :: "'a"   and s :: "real" assumes "norm x < r"   and "norm y < s" shows "norm (x * y) < r * s"
faithful_functor_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" shows "faithful_functor A B F \<equiv> functor A B F \<and> faithful_functor_axioms A F"
card_of_Un_ordLess_infinite_Field: fixes r :: "('b \<times> 'b) set"   and A :: "'c set"   and B :: "'c set" assumes "infinite (Field r)"   and "Card_order r"   and "|A| <o r"   and "|B| <o r" shows "|A \<union> B| <o r"
rep_eq: fixes x :: "('e \<Rightarrow> 'd) af"   and xa :: "'e af" shows "??.Abstract_AF.af.Rep_af (Applicative.ap x xa) = ??.Abstract_AF.af.Rep_af x (??.Abstract_AF.af.Rep_af xa)"
is_true_and_is_false_implies_False: fixes d :: "'dyn" assumes "is_true d"   and "is_false d" shows "False"
emeasure_eq_1_AE: fixes M :: "'d measure"   and S :: "'d set" assumes "prob_space M"   and "S \<in> sets M"   and "AE x in M. x \<in> S" shows "emeasure M S = 1"
intro: fixes hom :: "'d \<Rightarrow> 'c" assumes "hom (0::'d) = (0::'c)" shows "zero_hom hom"
Pair: fixes X :: "msg"   and H :: "msg set"   and Y :: "msg" assumes "X \<in> synth H"   and "Y \<in> synth H" shows "\<langle>X, Y\<rangle> \<in> synth H"
countable_INT': fixes X :: "'b set"   and A :: "'b \<Rightarrow> 'c set"   and M :: "'c measure" assumes "countable X"   and "X \<noteq> {}"   and "A ` X \<subseteq> sets M" shows "\<Inter> (A ` X) \<in> sets M"
has_integral_affinity': fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and a :: "'a"   and b :: "'a"   and m :: "real"   and c :: "'a" assumes "(f has_integral i) (cbox a b)"   and "0 < m" shows "((\<lambda>x. f (m *\<^sub>R x + c)) has_integral i /\<^sub>R m ^ DIM('a)) (cbox ((a - c) /\<^sub>R m) ((b - c) /\<^sub>R m))"
bqtran_iso: fixes x :: "'b" shows "mono \<lfloor>x\<rfloor>"
polys_def: shows "polys = map (\<lambda>q. (q, root_info q)) qs"
ordinate: fixes x :: "real"   and y :: "real" shows "ordinate (Point x y) = y"
while_if1: fixes c' :: "com"   and s :: "loc \<Rightarrow> nat"   and t :: "loc \<Rightarrow> nat"   and e :: "exp"   and c :: "com" assumes "(c', s) -[eval]-> t"   and "c' = WHILE e DO c" shows "(IF e THEN c;;c' ELSE SKIP, s) -[eval]-> t"
union_subtract_elt1: fixes x :: "'a"   and B :: "'a set" assumes "x \<notin> B" shows "{x} \<union> B - {x} = B"
convex_Inter: fixes f :: "'a set set" assumes "\<And>s. s \<in> f \<Longrightarrow> convex s" shows "convex (\<Inter> f)"
phi_in_P2: shows "\<phi> \<in> \<P>\<^sup>2"
cases: fixes rha :: "'e \<Rightarrow> 'f"   and boa :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and rhb :: "'g \<Rightarrow> 'h"   and bob :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a1 :: "('e, 'f, 'g, 'h) G\<^sub>m"   and a2 :: "('e, 'f, 'g, 'h) G\<^sub>m" assumes "blinding_of_G rha boa rhb bob a1 a2" obtains x :: "(('e, 'f) T\<^sub>m, 'f T\<^sub>h, 'g, 'h) F\<^sub>m" and y :: "(('e, 'f) T\<^sub>m, 'f T\<^sub>h, 'g, 'h) F\<^sub>m" where "a1 = G\<^sub>m x" and "a2 = G\<^sub>m y" and "blinding_of_F (root_hash_T rha) (blinding_of_T rha boa) rhb bob x y"
vsubst_ident: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and zs :: "'varSort"   and z :: "'var" assumes "good X" shows "X #[z // z]_zs = X"
simps: fixes t\<^sub>1 :: "term"   and t\<^sub>2 :: "'a"   and u :: "'a" shows "rewrite_step (t\<^sub>1, t\<^sub>2) u = map_option (subst t\<^sub>2) (match t\<^sub>1 u)"
lenext_map_strong: fixes ys :: "'b list"   and xs :: "'b list"   and gts :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b" assumes "\<lbrakk>length ys = length xs; gts ys xs\<rbrakk> \<Longrightarrow> gts (map f ys) (map f xs)"   and "lenext gts ys xs" shows "lenext gts (map f ys) (map f xs)"
2: fixes q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "q \<in> pmdl F"   and "q \<noteq> 0" shows "is_red G q"
set_zip: fixes xs :: "'a list"   and ys :: "'b list" shows "list.set (zip xs ys) = {(xs ! i, ys ! i) |i. i < min (length xs) (length ys)}"
cosh_0: shows "cosh (0::'a) = (1::'a)"
FL_valid_Act: fixes P :: "'state"   and f :: "'effect"   and \<alpha> :: "'act"   and x :: "('c, 'pred, 'act, 'effect) formula" shows "P \<Turnstile> Act f \<alpha> x = (\<exists>\<alpha>'' x' P'. Act f \<alpha> x = Act f \<alpha>'' x' \<and> \<langle>f\<rangle>P \<rightarrow> \<langle>\<alpha>'',P'\<rangle> \<and> P' \<Turnstile> x')"
free_compose: shows "\<forall>ts v. (\<exists>va u. v=va\<parallel>u \<and> (0 < \<parallel>ext va\<parallel> \<and> |lan va| = 1 \<and> (\<forall>c. \<parallel>len va ts c\<parallel> = 0 \<or> restrict va (clm ts) c = bot \<and> restrict va (res ts) c = bot)) \<and> 0 < \<parallel>ext u\<parallel> \<and> |lan u| = 1 \<and> (\<forall>c. \<parallel>len u ts c\<parallel> = 0 \<or> restrict u (clm ts) c = bot \<and> restrict u (res ts) c = bot)) \<longrightarrow> 0 < \<parallel>ext v\<parallel> \<and> |lan v| = 1 \<and> (\<forall>c. \<parallel>len v ts c\<parallel> = 0 \<or> restrict v (clm ts) c = bot \<and> restrict v (res ts) c = bot)"
conversion_join_or_peak_or_cliff: fixes ds :: "accent list" obtains as :: "accent list" and bs :: "accent list" and cs :: "accent list" where "set as__ \<subseteq> {Grave}" and "set bs__ \<subseteq> {Macron}" and "set cs \<subseteq> {Acute}" and "ds = as__ @ bs__ @ cs"   | as :: "accent list" and bs :: "accent list" where "ds = asa__ @ ([Acute] @ [Grave]) @ bsa__"   | as :: "accent list" and bs :: "accent list" where "ds = asb__ @ ([Acute] @ [Macron]) @ bsb__"   | as :: "accent list" and bs :: "accent list" where "ds = as @ ([Macron] @ [Grave]) @ bs"
ae4: shows "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Pyz (snd x) = 0 \<longrightarrow> Pxyz x = 0"
HMA_M_def: shows "HMA_M = (\<lambda>a b. a = from_hma\<^sub>m b)"
bound_plus: fixes a :: "'a"   and b :: "'a" shows "bound (a + b) \<le> bound a + bound b"
le_less: fixes x :: "nat list"   and y :: "nat list" shows "(x \<le>\<^sub>v y) = (x <\<^sub>v y \<or> x = y)"
subset_Compl_self_eq: fixes A :: "'c set" shows "(A \<subseteq> - A) = (A = {})"
inject_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(Terminates x1 = Terminates y1) = (x1 = y1)"
r_badblock_prim: shows "recfn 2 r_badblock"
intros_1: fixes p :: "'state" shows "p \<in> reachable p"
set_s_A': fixes n :: "nat" shows "set (s_A' n) = set init"
that: fixes x :: "'b" assumes "x \<in> S" shows "norm (f x) \<le> g x"
h_def: shows "h \<equiv> \<lambda>u. f (t + u) + f (t - u) - (l + r)"
le_numeral_iff: fixes m :: "nat"   and n :: "num" shows "(m \<le> numeral n) = (m = numeral n \<or> m \<le> pred_numeral n)"
gcd_greatest: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times c a"   and "dvd.dvd times c b" shows "dvd.dvd times c (gcd a b)"
inject_1: fixes x1 :: "int"   and y1 :: "int" shows "(Pc x1 = Pc y1) = (x1 = y1)"
ideduct_mono_eq: fixes M :: "('fun, 'var) Term.term set"   and M' :: "('fun, 'var) Term.term set"   and N :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term" assumes "\<forall>t. (M \<turnstile> t) = (M' \<turnstile> t)" shows "(M \<union> N \<turnstile> t) = (M' \<union> N \<turnstile> t)"
assms_1: shows "composition_series (G\<lparr>carrier := H\<rparr>) \<HH>"