hcju/isabelleps huggingface dataset:
simps_19: fixes x191 :: "char list"   and x192 :: "char list list"   and x193 :: "bool"   and x194 :: "'a exp"   and y191 :: "char list"   and y192 :: "char list list"   and y193 :: "bool"   and y194 :: "'a exp" shows "(INIT x191 (x192,x193) \<leftarrow> x194 = INIT y191 (y192,y193) \<leftarrow> y194) = (x191 = y191 \<and> x192 = y192 \<and> x193 = y193 \<and> x194 = y194)"
sub_distrib_pconjD: fixes prog :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real" assumes "sub_distrib_pconj prog"   and "unitary P"   and "unitary Q" shows "wlp prog P && wp prog Q \<tturnstile> wp prog (P && Q)"
restrict_to_halfspaces_invar_clw_def: shows "restrict_to_halfspaces_invar_clw = restrict_to_halfspaces"
butterfly_add_right: fixes a :: "'b"   and b :: "'a"   and b' :: "'a" shows "butterfly a (b + b') = butterfly a b + butterfly a b'"
index_mat_addrow_1: fixes i :: "nat"   and A :: "'c mat"   and j :: "nat"   and ad :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mul :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and k :: "nat"   and l :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "mat_addrow_gen ad mul a k l A $$ (i, j) = (if k = i then ad (mul a (A $$ (l, j))) (A $$ (i, j)) else A $$ (i, j))"
fresh_2: fixes a :: "name" shows "a \<sharp> TUnit = True"
rev_done: shows "rev done = map proj_thd (take (length done) auxlist)"
ivl_subset_existence_ivl: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl0 x0" shows "{0..t} \<subseteq> existence_ivl0 x0"
independent_bound_general: fixes S :: "('b, 'c) vec set" assumes "vec.independent S" shows "finite S \<and> card S \<le> vec.dim S"
first_entrance_disjoint: fixes A :: "'a set" shows "disjoint_family (first_entrance_set A)"
1: shows "continuous_map (prod_topology (top_of_set {0..1}) X) (prod_topology euclideanreal X) (\<lambda>y. (1 - fst y, snd y))"
wf_trms_subterms: fixes M :: "('fun, 'a) Term.term set" assumes "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s M" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subterms\<^sub>s\<^sub>e\<^sub>t M)"
nonneg_sqrt_nat_or_irrat: fixes x :: "real"   and a :: "nat" assumes "x\<^sup>2 = real a"   and "0 \<le> x" shows "x \<in> \<nat> \<or> x \<notin> \<rat>"
constant_rules_graph_rule: fixes x :: "('b Standard_Constant, nat) labeled_graph \<times> ('b Standard_Constant, nat) labeled_graph"   and C :: "'b set" assumes "x \<in> constant_rules C" shows "graph_rule x"
less_eq_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.plus_ord plus less_eq less" shows "less_eq x y = (plus x y = y)"
aux_2: fixes a :: "'a"   and b :: "'a" assumes "a \<otimes> b = \<zero>"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "a = \<zero> \<or> b = \<zero>"
NotR: fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "F, \<Gamma> \<Rightarrow> \<Delta>" shows "\<Gamma> \<Rightarrow> \<^bold>\<not> F, \<Delta>"
power_increasing: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and n :: "nat"   and N :: "nat"   and a :: "'c" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "n \<le> N"   and "less_eq one a" shows "less_eq (power.power one times a n) (power.power one times a N)"
getEnv_updEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and x :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort"   and ys :: "'varSort"   and y :: "'var" shows "(rho [x \<leftarrow> X]_xs) ys y = (if ys = xs \<and> y = x then Some X else rho ys y)"
f_Equiv_Exec_Stream_expand_shrink_equiv_state_set: fixes k1 :: "nat"   and k2 :: "nat"   and equiv_states :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and localState1 :: "'f \<Rightarrow> 'd"   and c1 :: "'f"   and localState2 :: "'g \<Rightarrow> 'e"   and c2 :: "'g"   and A :: "'h set"   and input_fun1 :: "'h \<Rightarrow> 'i message_af"   and input_fun2 :: "'h \<Rightarrow> 'j message_af"   and output_fun1 :: "'f \<Rightarrow> 'k message_af"   and output_fun2 :: "'g \<Rightarrow> 'k message_af"   and trans_fun1 :: "'i message_af \<Rightarrow> 'f \<Rightarrow> 'f"   and trans_fun2 :: "'j message_af \<Rightarrow> 'g \<Rightarrow> 'g"   and input :: "'h list"   and i :: "nat" assumes "0 < k1"   and "0 < k2"   and "equiv_states (localState1 c1) (localState2 c2)"   and "\<And>input0 m. \<lbrakk>set input0 \<subseteq> A; m \<in> A\<rbrakk> \<Longrightarrow> Equiv_Exec m equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 (f_Exec_Comp trans_fun1 (map input_fun1 input0 \<odot> k1) c1) (f_Exec_Comp trans_fun2 (map input_fun2 input0 \<odot> k2) c2)"   and "set input \<subseteq> A"   and "i < length input" shows "equiv_states (localState1 (f_Exec_Comp_Stream trans_fun1 (map input_fun1 input \<odot> k1) c1 \<div>\<^sub>l k1 ! i)) (localState2 (f_Exec_Comp_Stream trans_fun2 (map input_fun2 input \<odot> k2) c2 \<div>\<^sub>l k2 ! i))"
ax_NormalD: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and t :: "Term.term"   and Q :: "'b assn" assumes "G,A\<turnstile>{P} t\<succ> {Q}" shows "G,A\<turnstile>{Normal P} t\<succ> {Q}"
ipresAbs_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'e"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'f"   and MOD :: "('g, 'h, 'varSort, 'i, 'j, 'var, 'e, 'f, 'k) model_scheme" shows "ipresAbs h hA MOD \<equiv> \<forall>xs x s X. isInBar (xs, s) \<and> wls s X \<longrightarrow> hA (Abs xs x X) = igAbs MOD xs x (h X)"
real_is_int_real: fixes x :: "int" shows "real_is_int (real_of_int x)"
Nml_Nmlize_3: fixes t :: "'a E.term" assumes "E.Arr t" shows "E.Trg \<^bold>\<lfloor>t\<^bold>\<rfloor> = E.Trg t"
use_secure_refinement_helper: fixes \<R>\<^sub>A :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) set"   and \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and P :: "((('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and cm\<^sub>A :: "'Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)"   and mem\<^sub>A :: "'Var\<^sub>A \<Rightarrow> 'Val"   and cm\<^sub>C :: "'Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)"   and mem\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val"   and cm\<^sub>C' :: "'Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)"   and mem\<^sub>C' :: "'Var\<^sub>C \<Rightarrow> 'Val" assumes "secure_refinement \<R>\<^sub>A \<R> P"   and "((cm\<^sub>A, mem\<^sub>A), cm\<^sub>C, mem\<^sub>C) \<in> \<R>"   and "(cm\<^sub>C, mem\<^sub>C) \<leadsto>\<^sub>C (cm\<^sub>C', mem\<^sub>C')" shows "\<exists>cm\<^sub>A' mem\<^sub>A' n. abs.neval (cm\<^sub>A, mem\<^sub>A) n (cm\<^sub>A', mem\<^sub>A') \<and> ((cm\<^sub>A', mem\<^sub>A'), cm\<^sub>C', mem\<^sub>C') \<in> \<R>"
sifum_modes_def: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set" shows "sifum_modes ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars \<equiv> sifum_lang_no_dma ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars \<and> sifum_security dma \<C>_vars (sifum_lang_no_dma.eval\<^sub>w ev\<^sub>A ev\<^sub>B)"
simps_13: fixes f1 :: "'b"   and f2 :: "BitList \<Rightarrow> 'b \<Rightarrow> 'b"   and f3 :: "BitList \<Rightarrow> 'b \<Rightarrow> 'b"   and x2 :: "BitList" shows "rec_BitList f1 f2 f3 (Bit0 x2) = f2 x2 (rec_BitList f1 f2 f3 x2)"
pseudominimal_inD2: fixes P :: "'a set"   and x :: "'a"   and z :: "'a" assumes "pseudominimal_in P x"   and "z \<in> P"   and "x \<succ> z" shows "z = bottom P"
simps: fixes M :: "('d, 'c) Term.term set" shows "fv\<^sub>s\<^sub>e\<^sub>t M = \<Union> (fv ` M)"
pivot_positions_contains: fixes A :: "'b mat"   and f :: "nat \<Rightarrow> nat" assumes "row_echelon_form A"   and "dim_row A \<le> dim_col A"   and "pivot_fun A f (dim_col A)" shows "\<forall>i<length (pivot_positions A). (i, f i) \<in> set (pivot_positions A)"
vcons_in_vcpowerD_1: fixes xs :: "V"   and x :: "V"   and A :: "V"   and n :: "V" assumes "xs #\<^sub>\<circ> x \<in>\<^sub>\<circ> A ^\<^sub>\<times> n"   and "n \<in>\<^sub>\<circ> \<omega>" shows "xs \<in>\<^sub>\<circ> A ^\<^sub>\<times> vcard xs"
empty_neutral_1: fixes x :: "'a multiset" shows "{#} + x = x"
counit_simps_2: shows "local.dom \<epsilon> = f \<star> g"
pt_name2: fixes pi1 :: "(name \<times> name) list"   and pi2 :: "(name \<times> name) list"   and x :: "'d" shows "(pi1 @ pi2) \<bullet> x = pi1 \<bullet> pi2 \<bullet> x"
chaum_ped_\<Sigma>_axioms: shows "chaum_ped_\<Sigma> \<G>"
subspace_diff: fixes S :: "('c \<Rightarrow>\<^sub>0 'b) set"   and x :: "'c \<Rightarrow>\<^sub>0 'b"   and y :: "'c \<Rightarrow>\<^sub>0 'b" assumes "phull.subspace S"   and "x \<in> S"   and "y \<in> S" shows "x - y \<in> S"
igSwapIPresIGWlsSTR_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) model_scheme" shows "igSwapIPresIGWlsSTR MOD \<equiv> \<forall>zs z1 z2 s X. igWls MOD s (igSwap MOD zs z1 z2 X) = igWls MOD s X"
distinct_length_2_or_more: fixes a :: "'a"   and b :: "'a"   and xs :: "'a list" shows "distinct (a # b # xs) = (a \<noteq> b \<and> distinct (a # xs) \<and> distinct (b # xs))"
smult_minus_1: fixes v :: "'c" assumes "v \<in> carrier M" shows "\<ominus>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub> \<odot>\<^bsub>M\<^esub> v = \<ominus>\<^bsub>M\<^esub> v"
UNIV_enum: fixes enum :: "'e list"   and enum_all :: "('e \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('e \<Rightarrow> bool) \<Rightarrow> bool" assumes "class.enum enum enum_all enum_ex" shows "UNIV = set enum"
P3: shows "Coplanar D E F A \<and> Coplanar D E F B \<and> Coplanar D E F C \<and> Coplanar D E F T"
pgen_lossless_gpvD: fixes fail :: "ennreal"   and \<I> :: "('h, 'i) \<I>"   and gpv :: "('j, 'h, 'i) gpv" assumes "pgen_lossless_gpv fail \<I> gpv" shows "expectation_gpv fail \<I> (\<lambda>_. 1) gpv = 1"
Suc: fixes w1 :: "'a list"   and w2 :: "'a list" shows "length w1 = length w2 \<and> length w1 = k \<longrightarrow> x * y * scalar_product w1 w2 = scalar_product (local.times x w1) (local.times y w2)"
white_valid_ref: fixes r :: "'e"   and s :: "'f process_name \<Rightarrow> ('g, 'f, 'h, 'e) Model.local_state" assumes "white r s" shows "obj_at (\<lambda>s. True) r s"
erf_analytic_on: fixes A :: "complex set" shows "erf analytic_on A"
cases: fixes a1 :: "'b"   and a2 :: "['b]" assumes "listmem a1 a2" obtains x :: "'b" and xs :: "['b]" where "a1 = xa__" and "a2 = xa__ : xsb__"   | x :: "'b" and xs :: "['b]" and y :: "'b" where "a1 = x" and "a2 = y : xs" and "listmem x xs"
lunit_in_hom_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "prebicategory_with_homs_and_units V H \<a> \<i> src trg"   and "partial_magma.ide V f" shows "partial_magma.in_hom V (prebicategory_with_homs_and_units.lunit V H \<a> \<i> trg f) (H (trg f) f) f"
irreducible_smult_field: fixes c :: "'b"   and p :: "'b poly" shows "irreducible (smult c p) = (c \<noteq> (0::'b) \<and> irreducible p)"
M_4: fixes vs :: "nat \<Rightarrow> nat" shows "\<exists>vs'. ((Suc 0, vs), 0, vs') \<in> (step M)\<^sup>* \<and> P vs vs'"
assms_2: shows "mlf (fromMaybe (p_dst pi) (next_hop (routing_table_semantics rt (p_dst pi)))) \<noteq> None"
SC_siso_imp_SC_WbisT: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "('b, 'c) com" assumes "PL_Indis indis"   and "PL_Indis.SC_siso tval aval indis c" shows "PL_Indis.SC_WbisT tval aval indis c"
freshOutputTransition: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and c :: "name" assumes "P \<Longrightarrow>\<^sub>l a[b] \<prec> P'"   and "c \<sharp> P" shows "c \<sharp> P'"
sep_conj_true_P: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_conj plus sep_disj (\<lambda>s. True) (sep_algebra.sep_conj plus sep_disj (\<lambda>s. True) P) = sep_algebra.sep_conj plus sep_disj (\<lambda>s. True) P"
coeffs_map_poly: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b poly" assumes "(f (lead_coeff p) = (0::'a)) = (p = 0)" shows "coeffs (map_poly f p) = map f (coeffs p)"
conseq_old: fixes P' :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and e' :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and c :: "com"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and Q' :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<exists>k>0. \<forall>l s. P' l s \<longrightarrow> P l s \<and> e' s \<le> k * e s"   and "\<turnstile>\<^sub>1 {P} c { e' \<Down> Q}"   and "\<forall>l s. Q l s \<longrightarrow> Q' l s" shows "\<turnstile>\<^sub>1 {P'} c { e \<Down> Q'}"
assms: fixes s :: "'a list" assumes "s \<in> A" shows "pderivs s r \<subseteq> C"
multiple_block_in_original: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and b :: "'b set"   and n :: "nat" assumes "incidence_system point_set block_collection"   and "b \<in># repeat_mset n block_collection" shows "b \<in># block_collection"
extend_set_inv: fixes h :: "'c \<Rightarrow> 'd" assumes "bij h" shows "extend_set (\<lambda>(x, u). inv h x) = project_set (\<lambda>(x, u). h x)"
induced_def: shows "induced = (\<lambda>A\<in>contained.Partition. THE b. \<exists>a\<in>G. a \<cdot>| L = A \<and> b = \<eta> a)"
eq_conga_out: 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 D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B A D E F" shows "Tarski_neutral_dimensionless.Out Bet E D F"
ceiling_mono: fixes y :: "'c"   and x :: "'c" assumes "y \<le> x" shows "\<lceil>y\<rceil> \<le> \<lceil>x\<rceil>"
lunit_agreement: fixes a :: "'a" assumes "ide a" shows "lunit a = \<l>[a]"
push_bit_of_0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "push_bit n zero = zero"
\<omega>_limit_set_in_compact_subset: fixes K :: "'a set"   and x :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "rev.trapped_forward x K" shows "rev.\<omega>_limit_set x \<subseteq> K"
word_ao_absorbs_6: fixes x :: "'b word"   and y :: "'b word" shows "x OR x AND y = x"
\<tau>: shows "natural_isomorphism D C F' F \<tau>"
infnorm_le_norm: fixes x :: "'a" shows "infnorm x \<le> norm x"
rev_sub_closed: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P" shows "rev_compose R q p \<in> carrier P"
del_arc_in: fixes a :: "'b" assumes "a \<notin> arcs G" shows "del_arc a = G"
mor_def: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2"   and B1' :: "'g set"   and B2' :: "'h set"   and s1' :: "'g \<Rightarrow> ('f, 'g, 'h) F1"   and s2' :: "'h \<Rightarrow> ('f, 'g, 'h) F2"   and f :: "'d \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'h" shows "mor B1 B2 s1 s2 B1' B2' s1' s2' f g = (((\<forall>a\<in>B1. f a \<in> B1') \<and> (\<forall>a\<in>B2. g a \<in> B2')) \<and> (\<forall>z\<in>B1. F1map id f g (s1 z) = s1' (f z)) \<and> (\<forall>z\<in>B2. F2map id f g (s2 z) = s2' (g z)))"
derivation_cases: fixes X :: "'a set"   and x :: "'a" assumes "derivation X"   and "x \<in> X" obtains Y :: "'a set" and y :: "'a" where "Y__ = {y \<in> X. y \<sqsubset> x}" and "extreme Y__ (\<sqsubseteq>) y" and "x = f y"   | Y :: "'a set" where "Y = {y \<in> X. y \<sqsubset> x}" and "f ` Y \<subseteq> Y" and "extreme_bound A (\<sqsubseteq>) Y x"
m: fixes x :: "'a"   and y :: "'a" assumes "I x"   and "I y"   and "x < y" shows "m y < m x"
process_mru_is_None: fixes v_f :: "'c \<Rightarrow> 'e \<Rightarrow> 'd option"   and a :: "'e" shows "(process_mru v_f a = None) = (vote_set v_f {a} = {})"
assms_2: fixes k :: "nat" assumes "k < length xs" shows "xs ! k = ys ! k"
\<delta>_wt: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"
leg1_simps_4: shows "trg (tab\<^sub>1 (Maps.REP S\<^sub>1)) = trg (Maps.REP S\<^sub>1)"
some_betw2: fixes Q :: "'a set"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "Q \<in> \<P>"   and "a \<in> Q"   and "b \<in> Q"   and "c \<in> Q" shows "a = b \<or> a = c \<or> b = c \<or> [[a b c]] \<or> [[b c a]] \<or> [[c a b]]"
freeIn_substInfinitesimalQuadratic: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and c :: "real mpoly"   and d :: "real mpoly"   and At :: "atom" assumes "var \<notin> vars a"   and "var \<notin> vars b"   and "var \<notin> vars c"   and "var \<notin> vars d" shows "freeIn var (substInfinitesimalQuadratic var a b c d At)"
dep_vars_Cons_ddep: fixes n' :: "'node"   and V :: "'var"   and as' :: "'edge list"   and as :: "'edge list"   and V' :: "'var"   and a :: "'edge" assumes "dependent_live_varsp n' V as' as"   and "V' \<in> Use (sourcenode a)"   and "n' = last (targetnodes (a # as))"   and "sourcenode a -{V}a # as'\<rightarrow>\<^sub>d\<^sub>d last (targetnodes (a # as'))" shows "dependent_live_varsp n' V' [] (a # as)"
bij_betw_inv_into_left: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and A' :: "'f set"   and a :: "'e" assumes "bij_betw f A A'"   and "a \<in> A" shows "inv_into A f (f a) = a"
Src_Cod: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Arr V src trg t" shows "bicategorical_language.Src src (bicategorical_language.Cod V src trg t) = bicategorical_language.Src src t"
ideal_prod_el1: fixes I :: "'a set"   and J :: "'a set"   and a :: "'a" assumes "ideal R I"   and "ideal R J"   and "a \<in> I \<diamondsuit>\<^sub>r J" shows "a \<in> I"
homologous_rel_singular_relcycle_1: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and c1 :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and c2 :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "homologous_rel p X S c1 c2"   and "singular_relcycle p X S c1" shows "singular_relcycle p X S c2"
ag_inv_zero: fixes A :: "('d, 'e) aGroup_scheme" assumes "aGroup A" shows "-\<^sub>a\<^bsub>A\<^esub> \<zero>\<^bsub>A\<^esub> = \<zero>\<^bsub>A\<^esub>"
range_pm_of_idx_pm_subset': fixes xs :: "'d list"   and f :: "nat \<Rightarrow>\<^sub>0 'c" shows "Poly_Mapping.range (pm_of_idx_pm xs f) \<subseteq> Poly_Mapping.range f"
fail_def: fixes a :: "'c" shows "fail a \<equiv> \<lambda>s. ((a, s), True)"
Rep_uexpr_inject: fixes x :: "('f, 'e) uexpr"   and y :: "('f, 'e) uexpr" shows "(\<lbrakk>x\<rbrakk>\<^sub>e = \<lbrakk>y\<rbrakk>\<^sub>e) = (x = y)"
inv_image_def: fixes r :: "('e \<times> 'e) set"   and f :: "'d \<Rightarrow> 'e" shows "inv_image r f = {(x, y). (f x, f y) \<in> r}"
tendsto_iff_Liminf_eq_Limsup: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and f0 :: "'b" assumes "F \<noteq> bot" shows "(f \<longlongrightarrow> f0) F = (Liminf F f = f0 \<and> Limsup F f = f0)"
intro: fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" assumes "outstanding_non_volatile_writes_unshared \<S> ts"   and "sharing_consis \<S> ts"   and "read_only_unowned \<S> ts"   and "unowned_shared \<S> ts"   and "no_outstanding_write_to_read_only_memory \<S> ts" shows "valid_sharing \<S> ts"
non_zero: fixes s :: "'a" shows "emeasure (M s) (space (M s)) \<noteq> 0"
bind_sim_select_rule: fixes m :: "'b nres"   and f' :: "'b \<Rightarrow> 'a nres"   and \<Psi> :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c nres"   and \<Phi> :: "'c \<Rightarrow> bool" assumes "m \<bind> f' \<le> SPEC \<Psi>"   and "\<And>x. \<lbrakk>nofail m; inres m x; f' x \<le> SPEC \<Psi>\<rbrakk> \<Longrightarrow> f x \<le> SPEC \<Phi>" shows "m \<bind> f \<le> SPEC \<Phi>"
bound_intersect_2d_upper: fixes prec :: "nat"   and Xs :: "((real \<times> real) \<times> real \<times> real) list"   and x :: "real"   and m :: "real"   and M :: "real" assumes "bound_intersect_2d prec Xs x = Some (m, M)" obtains X :: "(real \<times> real) \<times> real \<times> real" and m' :: "real" where "X \<in> set Xs" and "intersect_segment_xline prec X x = Some (m', M)" and "\<And>X m' M'. \<lbrakk>X \<in> set Xs; intersect_segment_xline prec X x = Some (m', M')\<rbrakk> \<Longrightarrow> M' \<le> M"
approx_mem_monad: fixes x :: "'a star"   and u :: "'a star" assumes "x \<approx> u" shows "u \<in> monad x"
of_nat_take_bit: 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"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and m :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "semiring_1.of_nat one plus zero (take_bit n m) = take_bit n (semiring_1.of_nat one plus zero m)"
SAT_mono: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and \<Gamma> :: "('a, 'b) web_scheme" assumes "\<And>e. f e \<le> g e" shows "SAT \<Gamma> f \<subseteq> SAT \<Gamma> g"
backlinks_subset: fixes X :: "('a \<times> 'a) list"   and Y :: "('a \<times> 'a) list" shows "(set (backlinks X) \<subseteq> set (backlinks Y)) = (set X \<subseteq> set Y)"