hcju/isabelleps huggingface dataset:
hcomp_ide_cod: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b" assumes "natural_transformation A B F G \<tau>" shows "identity_functor.map B \<circ> \<tau> = \<tau>"
some_GB_props_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "finite (component_of_term ` Keys F)"   and "F \<subseteq> dgrad_p_set d m" shows "finite (some_GB F) \<and> is_Groebner_basis (some_GB F) \<and> pmdl (some_GB F) = pmdl F"
i: shows "(i < row_length A1 * row_length B1) = (i < row_length (A1 \<circ> A2 \<otimes> B1 \<circ> B2))"
Trg_VcompNml: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Nml t"   and "Nml u"   and "Dom t = Cod u" shows "Trg (t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u) = Trg u"
cont_def: fixes tree :: "'d tree" shows "cont tree = (case tree of Node x1 x2 \<Rightarrow> x2)"
real_less_ereal_iff: fixes y :: "ereal"   and x :: "real" shows "(real_of_ereal y < x) = ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y < ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 < x))"
jkbpCSt_refl: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and agents :: "'b odlist"   and t :: "'e Trace"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'f list"   and envTrans :: "'f \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'g" assumes "FiniteLinorderEnvironment jkbp agents"   and "t \<in> PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal (Environment.clock_jview envObs)" shows "Environment.clock_sim jkbp envInit envAction envTrans envVal envObs t \<in> Environment.clock_sim jkbp envInit envAction envTrans envVal envObs ` PreEnvironmentJView.jkbpCn jkbp envInit envAction envTrans envVal (Environment.clock_jview envObs) (tLength t)"
game0_def: shows "game0 = TRY lift_spmf key_gen \<bind> (\<lambda>(pk, uu_). lift_spmf coin_spmf \<bind> (\<lambda>b. inline hash_oracle' (\<A>1 pk) {} \<bind> (\<lambda>(((msg1, msg2), \<sigma>), s). assert_gpv (valid_plains msg1 msg2) \<bind> (\<lambda>_. aencrypt pk (if b then msg1 else msg2) \<bind> (\<lambda>cipher. inline hash_oracle' (\<A>2 cipher \<sigma>) s \<bind> (\<lambda>(guess, s'). Generative_Probabilistic_Value.Done (guess = b))))))) ELSE lift_spmf coin_spmf"
decomp_vars_4: fixes t :: "('fun, 'var) Term.term" shows "fv\<^sub>s\<^sub>t (decomp t) = fv t"
inext_neq_iMin_not_card_1: fixes I :: "nat set"   and n :: "nat" assumes "I \<noteq> {}"   and "card I \<noteq> Suc 0" shows "inext n I \<noteq> iMin I"
simps_4: fixes X :: "'d list"   and F :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<cdot>\<^sub>s\<^sub>t\<^sub>p \<theta> = \<forall>X\<langle>\<or>\<noteq>: (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s rm_vars (set X) \<theta>)\<rangle>\<^sub>s\<^sub>t"
not_extf_gt_nil_singleton_if_\<delta>\<^sub>h_eq_\<epsilon>\<^sub>h: fixes s :: "('s, 'v) tm"   and f :: "'s" assumes "wary s"   and "\<delta>\<^sub>h = \<epsilon>\<^sub>h" shows "\<not> extf f (>\<^sub>t) [] [s]"
hyps_2: fixes t\<^sub>2 :: "'a tree"   and t :: "'a tree" assumes "subtract r t\<^sub>2 = Some t"   and "all_distinct t\<^sub>2" shows "all_distinct t"
simps_6: fixes K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and i :: "nat" shows "\<lbrakk> K\<^sub>1 weakly precedes K\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> i\<^esup> = {\<rho>. \<forall>n\<ge>i. #\<^sub>\<le> \<rho> K\<^sub>2 n \<le> #\<^sub>\<le> \<rho> K\<^sub>1 n}"
red_preserves_shconf: fixes e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "P \<turnstile> \<langle>e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>e',(h', l', sh'),b'\<rangle>"   and "P,E,h,sh \<turnstile> e : T"   and "P,h \<turnstile>\<^sub>s sh \<surd>" shows "P,h' \<turnstile>\<^sub>s sh' \<surd>"
insertI1: fixes a :: "'a"   and B :: "'a set" shows "a \<in> {a} \<union> B"
invariant_1_sub_interval_2: fixes p :: "int poly"   and l :: "rat"   and r :: "rat"   and l' :: "rat"   and r' :: "rat" assumes "invariant_1 (p, l, r)"   and "root_cond (p, l', r') (real_of_1 (p, l, r))"   and "l \<le> l'"   and "r' \<le> r" shows "real_of_1 (p, l', r') = real_of_1 (p, l, r)"
lateEarlyBisim: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim>\<^sub>l Q" shows "P \<sim>\<^sub>e Q"
negligible_differentiable_image_negligible: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "DIM('c) \<le> DIM('d)"   and "negligible S"   and "f differentiable_on S" shows "negligible (f ` S)"
preserves_ide: fixes a :: "'a \<times> 'a" assumes "VV.ide a" shows "ide (fst a \<star> snd a)"
1: fixes \<U> :: "'a set set" assumes "\<U> \<subseteq> {S. \<exists>\<V>\<subseteq>Collect P. \<Union> \<V> = S}" shows "\<exists>\<U>'\<subseteq>Collect P. \<Union> \<U>' = \<Union> \<U>"
iso_assoc: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "B.ide f"   and "B.ide g"   and "B.ide h"   and "src\<^sub>B f = trg\<^sub>B g"   and "src\<^sub>B g = trg\<^sub>B h" shows "B.iso \<a>\<^sub>B[f, g, h]"
proj_psi_dim: shows "proj_psi \<in> carrier_mat N N"
h': shows "h3 \<turnstile> set_disconnected_nodes document_ptr (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr # disc_nodes) \<rightarrow>\<^sub>h h'"
VectorSpaceEndSet_is_VectorSpaceHomSet: shows "VectorSpaceHomSet (\<cdot>) V (\<cdot>) V = VectorSpaceEndSet"
zmult_zless_mono2: fixes i :: "Int.int"   and j :: "Int.int"   and k :: "Int.int" assumes "i < j"   and "0 < k" shows "k * i < k * j"
carrier_vs_is_self: fixes W :: "'c set" shows "carrier (V.vs W) = W"
a1: shows "\<exists>y'>r. \<forall>(d, e, f)\<in>set b. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f < 0"
f_max: fixes f' :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "f' \<in> F" shows "poly_deg f' \<le> poly_deg f"
f_def: fixes x :: "nat"   and y :: "nat" shows "f x y \<equiv> 3 * x + 2 * y"
distinct_1: fixes x11 :: "vrs"   and x12 :: "ty"   and x21 :: "tyvrs"   and x22 :: "ty" shows "VarB_Rep x11 x12 \<noteq> TVarB_Rep x21 x22"
FAssNull: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>2 :: "char list exp"   and v :: "val"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NullPointer,s\<^sub>2\<rangle>"
has_integral_integral: fixes f :: "'a \<Rightarrow> 'b"   and s :: "'a set" shows "(f integrable_on s) = (f has_integral integral s f) s"
elims_2: fixes x :: "vname"   and xa :: "vname" assumes "x < xa" obtains n1 :: "nat" and n2 :: "nat" where "x = I n1__" and "xa = R n2__"   | n1 :: "nat" and n2 :: "nat" where "x = I n1a__" and "xa = I n2a__" and "n1a__ < n2a__"   | n1 :: "nat" and n2 :: "nat" where "x = R n1" and "xa = R n2" and "n1 < n2"
not_final_thread_iff: fixes final :: "'a \<Rightarrow> bool"   and s :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set"   and t :: "'c" shows "final_thread.not_final_thread final s t = (\<exists>x ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> (\<not> final x \<or> ln \<noteq> no_wait_locks \<or> (\<exists>w. wset s t = \<lfloor>w\<rfloor>)))"
g_INV: fixes proga :: "program"   and decls :: "varDecl list"   and pre_procs :: "('a \<times> nat \<times> String.literal \<times> 'b) list" assumes "program_inv proga" shows "gState_inv proga (foldl (\<lambda>g (uu_, a, name, uu_). foldl (\<lambda>g name. fst (runProc name [] proga g emptyProc)) g (replicate a name)) (foldl (\<lambda>g d. fst (mkVarChannel d (apfst \<circ> gState.vars_update) g emptyProc)) \<lparr>gState.vars = lm.sng STR ''__assert__'' (Var (VTBounded 0 1) 0), channels = [InvChannel], timeout = False, procs = []\<rparr> decls) pre_procs)"
Pair_inject: fixes a :: "'m"   and b :: "'n"   and a' :: "'m"   and b' :: "'n" assumes "(a, b) = (a', b')" obtains "a = a'" and "b = b'"
FieldI1: fixes i :: "'c"   and j :: "'c"   and R :: "('c \<times> 'c) set" assumes "(i, j) \<in> R" shows "i \<in> Field R"
ring_simprules_2: shows "(0::'a) \<in> UNIV"
mult_poly_0_right: fixes p :: "'b poly" shows "p * 0 = 0"
sum_distrib_left: fixes r :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "r * sup_monoid.sum f A = sup_monoid.sum (\<lambda>n. r * f n) A"
case_eq_if: fixes f1 :: "'a"   and f2 :: "'a"   and bool :: "bool" shows "(case bool of True \<Rightarrow> f1 | False \<Rightarrow> f2) = (if bool = True then f1 else f2)"
tmps_le: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'a"   and \<O> :: "'b" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "dom \<theta> \<union> load_tmps is = {i. i < t}"
is_path_def: fixes G :: "graph"   and p :: "nat list" shows "is_path G p = (p \<noteq> [] \<and> set p \<subseteq> verts G)"
backflows_un: fixes A :: "('b \<times> 'b) set"   and B :: "('b \<times> 'b) set" shows "backflows (A \<union> B) = backflows A \<union> backflows B"
assms_1: shows "\<forall>n. \<forall>asset\<in>support_set p. p asset (Suc n) \<in> borel_measurable (F n)"
ns'_1: shows "g \<turnstile> Entry g-ns'\<rightarrow>n"
dec_sub_from_counter: fixes ck :: "bool"   and mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and d :: "dec"   and res :: "'b state \<times> (v sem_env, v) result"   and extra :: "nat" assumes "evaluate_dec ck mn env s d res" shows "\<forall>count count' s' r. clock s = count + extra \<and> clock s' = count' + extra \<and> res = (s', r) \<and> ck = True \<longrightarrow> evaluate_dec ck mn env (update_clock (\<lambda>_. count) s) d (update_clock (\<lambda>_. count') s', r)"
step: fixes rel :: "(nat \<times> nat) set"   and n :: "nat"   and x :: "nat"   and y :: "nat" assumes "Steps rel n (x, n)"   and "Steps rel n (n, y)" shows "Steps rel (Suc n) (x, y)"
1: shows "x \<in> F \<and> y \<in> G \<and> x \<sqinter> y \<le> z"
lexordp_irreflexive': fixes xs :: "'a list" shows "\<not> lexordp xs xs"
floor_div_pos_int: fixes n :: "int"   and r :: "'a" assumes "0 < n" shows "\<lfloor>r / of_int n\<rfloor> = \<lfloor>r\<rfloor> div n"
assms_2: shows "Mono\<^bsub>L\<^esub> f"
mask_eq: fixes n :: "nat" shows "mask n = (1 << n) - 1"
1: shows "x \<le> (d x \<squnion> d y) \<sqinter> (x \<squnion> y)"
pos_less_divide_eq: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sgn :: "'e \<Rightarrow> 'e"   and abs :: "'e \<Rightarrow> 'e"   and c :: "'e"   and a :: "'e"   and b :: "'e" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero c" shows "less a (divide b c) = less (times a c) b"
segment_eq_compose: fixes a :: "'b"   and b :: "'b" shows "(\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>R b) = (+) a \<circ> (\<lambda>u. u *\<^sub>R (b - a))"
word_add_def: fixes a :: "'b word"   and b :: "'b word" shows "a + b = word_of_int (uint a + uint b)"
left_total_rel_gpv': fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and C :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'l \<Rightarrow> bool" assumes "left_total A"   and "left_total C"   and "left_unique R"   and "right_total R" shows "left_total (rel_gpv'' A C R)"
image_zmset_empty: fixes f :: "'b \<Rightarrow> 'a" shows "Auxiliary.image_zmset f {#}\<^sub>z = {#}\<^sub>z"
calc_ideal_point2_on_circline: fixes z1 :: "complex"   and z2 :: "complex"   and A :: "complex"   and B :: "complex" assumes "(z1, z2) = calc_ideal_point2_cvec A B"   and "is_real A"   and "(cmod A)\<^sup>2 < (cmod B)\<^sup>2" shows "A * z1 * cnj z1 + B * cnj z1 * z2 + cnj B * z1 * cnj z2 + A * z2 * cnj z2 = 0"
closed: shows "\<forall>s. set_pmf (\<O> s) \<subseteq> \<O>\<^sub>s"
arg_cong2: fixes a :: "'d"   and b :: "'d"   and c :: "'e"   and d :: "'e"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "a = b"   and "c = d" shows "f a c = f b d"
simps_9: fixes f :: "'h \<Rightarrow> 'g"   and x21 :: "'h"   and x22 :: "'h list" shows "map f (x21 # x22) = f x21 # map f x22"
bottomI: fixes z :: "'b"   and P :: "'b set" assumes "z \<in> P"   and "\<And>y. y \<in> P \<Longrightarrow> codomain.greater_eq y z" shows "z = codomain.bottom P"
inner_sum_list_left: fixes xs :: "'b list"   and b :: "'b" shows "sum_list xs \<bullet> b = (\<Sum>x\<leftarrow>xs. x \<bullet> b)"
icd_path_shift0: fixes \<pi> :: "nat \<Rightarrow> 'n"   and i :: "nat"   and k :: "nat" assumes "is_path \<pi>" shows "(i icd\<^bsup>\<pi>\<^esup>\<rightarrow> k) = (i - k icd\<^bsup>\<pi> \<guillemotleft> k\<^esup>\<rightarrow> 0)"
sctnbounds_of_ivl_def: fixes M :: "nat"   and X :: "'b set" shows "sctnbounds_of_ivl M X = ivl_rep X \<bind> (\<lambda>(l, u). let ls = (\<lambda>b. Sctn (- b) (- l \<bullet> b)) ` set (take M Basis_list); us = (\<lambda>b. Sctn b (u \<bullet> b)) ` set (take M Basis_list) in RETURN (ls \<union> us))"
asym: fixes b :: "'b list"   and a :: "'b list" assumes "strict_prefix b a"   and "strict_prefix a b" shows "False"
lm2: fixes ls :: "nat"   and key :: "nat" assumes "c_assoc_have_key (c_tl ls) key = 0" shows "c_assoc_have_key ls key = 0"
F2: shows "\<forall>x\<^sub>2 x\<^sub>1. x\<^sub>1 \<subseteq> x\<^sub>2 \<and> x\<^sub>2 \<subseteq> x\<^sub>1 \<longrightarrow> x\<^sub>1 = x\<^sub>2"
sort_quicksort: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.sort_key less_eq (\<lambda>x. x) = linorder.quicksort less_eq"
HEndPhase1_allBlocks: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase1 s s' p" shows "allBlocks s' \<subseteq> allBlocks s \<union> {dblock s' p}"
dim_span_eq_card_independent: fixes B :: "'c set" assumes "cindependent B" shows "cdim (cspan B) = card B"
map_replicate_True: fixes n :: "nat"   and xs :: "bool list" assumes "n = length xs" shows "map2 (\<and>) xs (replicate n True) = xs"
least_cong: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and x' :: "'a"   and A :: "'a set" assumes "weak_partial_order L"   and "x .=\<^bsub>L\<^esub> x'"   and "x \<in> carrier L"   and "x' \<in> carrier L"   and "is_closed\<^bsub>L\<^esub> A" shows "least L x A = least L x' A"
norm_less_lplus: fixes x :: "nat"   and n :: "nat"   and y :: "nat" assumes "x + n < y" shows "x + (n + 1) \<le> y"
E: shows "E \<in> carrier_mat (dim_row A - 2) (dim_col A)"
Meet_singleton: fixes x :: "'b" shows "\<Sqinter>{x} = x"
memoization_correct_2: fixes P :: "heap \<Rightarrow> bool"   and update :: "'c \<Rightarrow> 'd \<Rightarrow> unit Heap"   and lookup :: "'c \<Rightarrow> 'd option Heap"   and empty :: "heap"   and dp :: "'c \<Rightarrow> 'd"   and dp\<^sub>T :: "'c \<Rightarrow> 'd Heap"   and x :: "'c"   and v :: "'d"   and m :: "heap" assumes "dp_consistency_heap_empty P update lookup empty"   and "dp_heap.consistentDP P P dp (heap_mem_defs.lookup' lookup) dp\<^sub>T"   and "execute (dp\<^sub>T x) empty = Some (v, m)" shows "dp_consistency.cmem (heap_mem_defs.lookup' lookup) dp m"
eval_BinOp_arg2_indepI: fixes G :: "prog"   and s0 :: "st"   and e1 :: "expr"   and v1 :: "val"   and s1 :: "state"   and binop :: "binop"   and e2 :: "expr"   and v2 :: "val" assumes "G\<turnstile>Norm s0 \<midarrow>e1-\<succ>v1\<rightarrow> s1"   and "\<not> need_second_arg binop v1" shows "G\<turnstile>Norm s0 \<midarrow>BinOp binop e1 e2-\<succ>eval_binop binop v1 v2\<rightarrow> s1"
tight_def: fixes \<mu> :: "nat \<Rightarrow> real measure" shows "tight \<mu> \<equiv> (\<forall>n. real_distribution (\<mu> n)) \<and> (\<forall>\<epsilon>>0. \<exists>a b. a < b \<and> (\<forall>n. 1 - \<epsilon> < Sigma_Algebra.measure (\<mu> n) {a<..b}))"
coll_scaleR_left_eq: fixes r :: "real"   and x :: "real \<times> real"   and y :: "real \<times> real" shows "coll 0 (r *\<^sub>R x) y = (r = 0 \<or> coll 0 x y)"
map_simp_3: shows "local.map J.j0 = f0"
assms: fixes v :: "'t" assumes "v \<in> keys p" shows "d (pp_of_term v) \<le> m"
word_div_def: fixes a :: "'b word"   and b :: "'b word" shows "a div b = word_of_int (uint a div uint b)"
A7': fixes a :: "(real, 'a) vec"   and p :: "(real, 'a) vec"   and c :: "(real, 'a) vec"   and b :: "(real, 'a) vec"   and q :: "(real, 'a) vec" assumes "B\<^sub>\<real> a p c"   and "B\<^sub>\<real> b q c" shows "\<exists>x. B\<^sub>\<real> p x b \<and> B\<^sub>\<real> q x a"
scale_scale: fixes a :: "'a"   and b :: "'a"   and x :: "'a" shows "a * (b * x) = a * b * x"
connexE: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "connex A less_eq"   and "x \<in> A"   and "y \<in> A" obtains "less_eq x y"   | "less_eq y x"
distinct_29: fixes x31 :: "'b ptrm"   and x32 :: "'b ptrm"   and x7 :: "'b ptrm" shows "PApp x31 x32 \<noteq> PSnd x7"
closed_compact_differences: fixes S :: "'b set"   and T :: "'b set" assumes "closed S"   and "compact T" shows "closed (\<Union>x\<in>S. \<Union>y\<in>T. {x - y})"
empt: fixes x :: "'a" assumes "is_node x" shows "\<not> memb x empt"
continuous_on_cases: fixes s :: "'b set"   and t :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool" assumes "closed s"   and "closed t"   and "continuous_on s f"   and "continuous_on t g"   and "\<forall>x. x \<in> s \<and> \<not> P x \<or> x \<in> t \<and> P x \<longrightarrow> f x = g x" shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
simE_3: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and \<alpha> :: "freeRes"   and Q' :: "pi" assumes "P \<leadsto>\<^sup>^<Rel> Q"   and "Q \<longmapsto> \<alpha> \<prec> Q'" shows "\<exists>P'. P \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel"
suc_def: fixes A :: "'a set" shows "suc A \<equiv> minim (local.AboveS A)"
axioms_1: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" assumes "Flow c s t f" shows "Preflow c s t f"
principal_dchar_altdef: shows "principal_dchar n = c2dc (principal_char G)"
lookupEvalHeap: fixes v :: "'b"   and h :: "('b \<times> 'c) list"   and f :: "'c \<Rightarrow> 'd" assumes "v \<in> domA h" shows "evalHeap h f v = f (the (map_of h v))"
comm_inv_char: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G"   and "x \<otimes> y = \<one>" shows "inv x = y"
relH_trans: fixes as :: "nat set"   and h1 :: "heap"   and h2 :: "heap"   and h3 :: "heap" assumes "relH as h1 h2"   and "relH as h2 h3" shows "relH as h1 h3"
hom_mult: fixes x :: "'a poly"   and y :: "'a poly" shows "map_poly hom (x * y) = map_poly hom x * map_poly hom y"
functions_of_eq: shows "functions_of P A B = range_permutation A B `` {f}"
mbs': fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "\<not> almost_full_on P (lists A)" shows "\<exists>m\<in>BAD P. \<forall>g. (m, g) \<in> gseq \<longrightarrow> good P g"
table_def: fixes n :: "nat"   and V :: "nat set"   and X :: "'b option list set" shows "table n V X = (\<forall>x\<in>X. wf_tuple n V x)"
NR\<^sub>N_recurrence: fixes n :: "nat"   and u :: "nat"   and l :: "nat" assumes "0 < n"   and "p \<in> {0<..<1}" shows "NR\<^sub>N n u l = (ennreal p * (of_nat l + NR\<^sub>N (n - 1) u l) + ennreal q * (of_nat u + (\<Sum>k<n - 1. NR\<^sub>N (k + 1) u l * ennreal (pmf (binomial_pmf (n - 1) q) k)))) / ennreal (1 - q ^ n)"
h: shows "hom_induced p (subtopology X S) {} X T id \<in> Group.iso (homology_group p (subtopology X S)) (relative_homology_group p X T)"
Y_vertex: fixes n :: "nat" shows "Y n \<subseteq> \<^bold>V"
Cauchy_higher_derivative_integral_circlepath_1: fixes z :: "complex"   and r :: "real"   and f :: "complex \<Rightarrow> complex"   and w :: "complex"   and k :: "nat" assumes "continuous_on (cball z r) f"   and "f holomorphic_on ball z r"   and "w \<in> ball z r" shows "(\<lambda>u. f u / (u - w) ^ Suc k) contour_integrable_on circlepath z r"
i_set_mult_subset_imp_mod_0: fixes a :: "nat"   and b :: "nat" assumes "a \<noteq> Suc 0"   and "i_set_mult a \<subseteq> i_set_mult b" shows "a mod b = 0"
K2_line_intersect_twice: fixes a :: "proj2"   and r :: "proj2" assumes "a \<in> hyp2"   and "a \<noteq> r" shows "\<exists>s u. s \<noteq> u \<and> s \<in> S \<and> u \<in> S \<and> proj2_Col a r s \<and> proj2_Col a r u"
has_derivative_inner: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and g' :: "'a \<Rightarrow> 'b" assumes "(f has_derivative f') (at x within s)"   and "(g has_derivative g') (at x within s)" shows "((\<lambda>x. inner (f x) (g x)) has_derivative (\<lambda>h. inner (f x) (g' h) + inner (f' h) (g x))) (at x within s)"
finfun_comp_apply: fixes g :: "'e \<Rightarrow> 'd"   and f :: "'c \<Rightarrow>f 'e" shows "($) (g \<circ>$ f) = g \<circ> ($) f"
Suc_diff: fixes i :: "nat" assumes "i \<le> n" shows "Suc n - i = Suc (n - i)"
bz: shows "(p, q, u, v, d) = bezout (A $ a $ j) (A $ b $ j)"
supp_atm: fixes x :: "NominalSequents.var" shows "supp x = {x}"
simps: fixes n :: "nat"   and as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "inc_inv n (as, lm) (s, l, r) ires = (let lm' = abc_lm_s lm n (Suc (abc_lm_v lm n)) in if s = 0 then False else if s = 1 then inv_locate_a (as, lm) (n, l, r) ires else if s = 2 then inv_locate_b (as, lm) (n, l, r) ires else if s = 3 then inv_after_write (as, lm') (s, l, r) ires else if s = Suc 3 then inv_after_move (as, lm') (s, l, r) ires else if s = Suc 4 then inv_after_clear (as, lm') (s, l, r) ires else if s = Suc (Suc 4) then inv_on_right_moving (as, lm') (s, l, r) ires else if s = Suc (Suc 5) then inv_on_left_moving (as, lm') (s, l, r) ires else if s = Suc (Suc (Suc 5)) then inv_check_left_moving (as, lm') (s, l, r) ires else if s = Suc (Suc (Suc (Suc 5))) then inv_after_left_moving (as, lm') (s, l, r) ires else if s = Suc (Suc (Suc (Suc (Suc 5)))) then inv_stop (as, lm') (s, l, r) ires else False)"
setVar_gState_inv: fixes prog :: "program"   and g :: "'a gState_scheme"   and v :: "varRef"   and val :: "integer"   and p :: "pState" assumes "gState_inv prog g" shows "gState_inv prog (fst (setVar v val g p))"